Collected Papers (on Neutrosophic Theory and Its Applications in Algebra), Volume IX by Florentin Smarandache

ollected

Florentin marandache (author and editor) (on Neutrosophic Theory and Its Applications in Algebra)

nowledge 2022

lobal

Florentin Smarandache (author and editor)

Collected Papers

(on Neutrosophic Theory and Its Applications in Algebra) Volume IX

Peer Reviewers:

Muhammad Saeed

Department of Mathematics, University of Management and Technology, Lahore, PAKISTAN muhammad.saeed@umt.edu.pk

Akbar Rezaei

Department of Mathematics, Payame Noor University, Tehran, IRAN rezaei@pnu.ac.ir

Selçuk Topal

Mathematics Department, Bitlis Eren University, TURKEY s.topal@beu.edu.tr

Madeleine Al Tahan

Department of Mathematics, Lebanese International University, Bekaa, LEBANON madeline.tahan@liu.edu.lb

Florentin Smarandache’s Collected Papers series:

Collected Papers, Vol. I (first edition 1996, second edition 2007) Free download: http://fs.unm.edu/CP1.pdf

Collected Papers, Vol. II (Chişinău, Moldova, 1997) Free download: http://fs.unm.edu/CP2.pdf

Collected Papers, Vol. III (Oradea, Romania, 2000) Free download: http://fs.unm.edu/CP3.pdf

Collected Papers, Vol. IV (100 Collected Papers of Sciences) Multispace & Multistructure. Neutrosophic Transdisciplinarity (Hanko, Finland, 2010) Free download: http://fs.unm.edu/MultispaceMultistructure.pdf

Collected Papers, Vol. V: Papers of Mathematics or Applied mathematics (Brussels, Belgium, 2014) Free download: http://fs.unm.edu/CP5.pdf

Collected Papers, Vol. VI: on Neutrosophic Theory and Applications (Global Knowledge, 2022) Free download: http://fs.unm.edu/CP6.pdf

Collected Papers, Vol. VII: on Neutrosophic Theory and Applications (Global Knowledge, 2022) Free download: http://fs.unm.edu/CP7.pdf

Collected Papers, Vol. VIII: on Neutrosophic Theory and Applications (Global Knowledge, 2022) Free download: http://fs.unm.edu/CP8.pdf

Florentin Smarandache

(author and editor)

Collected Papers

(on Neutrosophic Theory and Its Applications in Algebra)

Volume IX

GLOBAL KNOWLEDGE

Publishing House Miami, United States of America 2022

Publishing:

GLOBAL KNOWLEDGE - Publishing House 848 Brickell Ave Ste 950 Miami Florida 33131, United States https://egk.ccgecon.us info@egk.ccgecon.us

ADSUMUS – Scientific and Cultural Society 13, Dimitrie Cantemir St. 410473 Oradea City, Romania https://adsumus.wordpress.com/ ad.sumus@protonmail.com

NSIA Publishing House

Neutrosphic Science International Association https://www.publishing-nsia.com/ University of New Mexico, Gallup, United States Universidad Regional Autónoma de los Andes (UNIANDES), Babahoyo, Los Rios, Ecuador

Editors:

Publishing: Prof. Dan Florin Lazar lazar.danflorin@yahoo.com Prof. Dr. Maykel Yelandi Leyva Vazquez ub.c.investigacion@uniandes.edu.ec

ISBN 978-1-59973-733 1

Introductory Note

This ninth volume of Collected Papers includes 87 papers comprising 982 pages on Neutrosophic Theory and its applications in Algebra, written between 2014 2022 by the author alone or in collaboration with the following 81 co authors (alphabetically ordered) from 19 countries: E.O. Adeleke, A.A.A. Agboola, Ahmed B. Al Nafee, Ahmed Mostafa Khalil, Akbar Rezaei, S.A. Akinleye, Ali Hassan, Mumtaz Ali, Rajab Ali Borzooei , Assia Bakali, Cenap Özel, Victor Christianto, Chunxin Bo, Rakhal Das, Bijan Davvaz, R. Dhavaseelan, B. Elavarasan, Fahad Alsharari, T. Gharibah, Hina Gulzar, Hashem Bordbar, Le Hoang Son, Emmanuel Ilojide, Tèmítópé Gbóláhàn Jaíyéolá, M. Karthika, Ilanthenral Kandasamy, W.B. Vasantha Kandasamy, Huma Khan, Madad Khan, Mohsin Khan, Hee Sik Kim, Seon Jeong Kim, Valeri Kromov, R. M. Latif, Madeleine Al Tahan, Mehmat Ali Ozturk, Minghao Hu, S.Mirvakili, Mohammad Abobala, Mohammad Hamidi, Mohammed Abdel Sattar, Mohammed A. Al Shumrani, Mohamed Talea, Muhammad Akram, Muhammad Aslam, Muhammad Aslam Malik, Muhammad Gulistan, Muhammad Shabir, G. Muhiuddin, Memudu Olaposi Olatinwo, Osman Anis, Choonkil Park, M. Parimala, Ping Li, K.Porselvi, D. Preethi, S. Rajareega, N. Rajesh, Udhayakumar Ramalingam, Riad K. Al Hamido, Yaser Saber, Arsham Borumand Saeid, Saeid Jafari, Said Broumi, A.A. Salama, Ganeshsree Selvachandran, Songtao Shao, Seok Zun Song, Tahsin Oner, M. Mohseni Takallo, Binod Chandra Tripathy, Tugce Katican, J. Vimala, Xiaohong Zhang, Xiaoyan Mao, Xiaoying Wu, Xingliang Liang, Xin Zhou, Yingcang Ma, Young Bae Jun, Juanjuan Zhang

Florentin Smarandache (author and editor) Collected Papers, IX 5

A.A. Salama, Florentin Smarandache, Valeri Kromov Neutrosophic Closed Set and Neutrosophic Continuous Functions / 25

Mumtaz Ali, Muhammad Shabir, Munazza Naz, Florentin Smarandache Soft neutrosophic semigroups and their generalization / 30

Florentin Smarandache (T, I, F)-Neutrosophic Structures / 49

Akbar Rezaei, Arsham Borumand Saeid, Florentin Smarandache Neutrosophic filters in BE-algebras / 57

Muhammad Aslam Malik, Ali Hassan, Said Broumi, Assia Bakali, Mohamed Talea, Florentin Smarandache Isomorphism of Bipolar Single Valued Neutrosophic Hypergraphs / 72

Muhammad Aslam Malik, Ali Hassan, Said Broumi, F. Smarandache Regular Bipolar Single Valued Neutrosophic Hypergraphs / 96

Mumtaz Ali, Florentin Smarandache Neutrosophic Soluble Groups, Neutrosophic Nilpotent Groups and Their Properties / 102

Florentin Smarandache Operators on Single-Valued Neutrosophic Oversets, Neutrosophic Undersets, and Neutrosophic Offsets / 112

Florentin Smarandache Interval-Valued Neutrosophic Oversets, Neutrosophic Undersets, and Neutrosophic Offsets / 117

Florentin Smarandache Subtraction and Division of Neutrosophic Numbers / 120

S.A. Akinleye, Florentin Smarandache, A.A.A. Agboola On Neutrosophic Quadruple Algebraic Structures / 127

A.A.A. Agboola, B. Davvaz, Florentin Smarandache Neutrosophic Quadruple Algebraic Hyperstructures / 132

Young Bae Jun, Florentin Smarandache, Hashem Bordbar Neutrosophic N-Structures Applied to BCK/BCI-Algebras / 146

Seok Zun Song, Florentin Smarandache, Young Bae Jun Neutrosophic Commutative N-Ideals in BCK-Algebras / 157

R.Dhavaseelan, Saeid Jafari, Florentin Smarandache Compact Open Topology and Evaluation Map via Neutrosophic Sets / 166

Contents

Florentin Smarandache (author and editor) Collected Papers, IX 6

R.Dhavaseelan, M. Parimala, S. Jafari, Florentin Smarandache

On Neutrosophic Semi-Supra Open Set and Neutrosophic Semi-Supra Continuous Functions / 170

Xiaohong Zhang, Yingcang Ma, Florentin Smarandache Neutrosophic Regular Filters and Fuzzy Regular Filters in Pseudo BCI Algebras / 175

Xiaohong Zhang, Florentin Smarandache, Xingliang Liang Neutrosophic Duplet Semi-Group and Cancellable Neutrosophic Triplet Groups / 181

G.Muhiuddin, Hashem Bordbar, Florentin Smarandache, Young Bae Jun Further results on (ε, ε,)-neutrosophic subalgebras and ideals in BCK/BCI-algebras / 196

W.B. Vasantha Kandasamy, Ilanthenral Kandasamy, Florentin Smarandache Algebraic Structure of Neutrosophic Duplets in Neutrosophic Rings <Z U I>, <Q U I> and <R U I> / 204

Xiaohong Zhang, Florentin Smarandache, Mumtaz Ali, Xingliang Liang Commutative Neutrosophic Triplet Group and Neutro-Homomorphism Basic Theorem / 215

Rajab Ali Borzooei, Xiaohong Zhang, Florentin Smarandache, Young Bae Jun Commutative Generalized Neutrosophic Ideals in BCK-Algebras / 237

Young Bae Jun, Seok Zun Song, Florentin Smarandache, Hashem Bordbar Neutrosophic Quadruple BCK/BCI-Algebras / 251

Young Bae Jun, Seon Jeong Kim, Florentin Smarandache Interval Neutrosophic Sets with Applications in BCK/BCI-Algebra / 266

Young Bae Jun, Florentin Smarandache, Seok Zun Song, Hashem Bordbar Neutrosophic Permeable Values and Energetic Subsets with Applications in BCK/BCIAlgebras / 279

Xiaohong Zhang, Xiaoying Wu, Florentin Smarandache, Minghao Hu Left (Right)-Quasi Neutrosophic Triplet Loops (Groups) and Generalized BE-Algebras / 294

Muhammad Akram, Hina Gulzar, Florentin Smarandache, Said Broumi Certain Notions of Neutrosophic Topological K-Algebras / 312

Mumtaz Ali, Florentin Smarandache, Mohsin Khan Study on the Development of Neutrosophic Triplet Ring and Neutrosophic Triplet Field / 327

Rajab Ali Borzooei, M. Mohseni Takallo, Florentin Smarandache, Young Bae Jun Positive implicative BMBJ-neutrosophic ideals in BCK-algebras / 337

Florentin Smarandache (author and editor) Collected Papers, IX 7

Songtao Shao, Xiaohong Zhang, Chunxin Bo, Florentin Smarandache Neutrosophic Hesitant Fuzzy Subalgebras and Filters in Pseudo-BCI Algebras / 353

W.B. Vasantha Kandasamy, Ilanthenral Kandasamy, Florentin Smarandache A Classical Group of Neutrosophic Triplet Groups Using (Z2p, X) / 372

Young Bae Jun, Florentin Smarandache, Mehmat Ali Ozturk

Commutative falling neutrosophic ideals in BCK-algebras / 380

Riad K. Al Hamido, T. Gharibah, S. Jafari, Florentin Smarandache On Neutrosophic Crisp Topology via N-Topology / 390

R.Dhavaseelan, S. Jafari, R. M. Latif, Florentin Smarandache Neutrosophic Rare α-Continuity / 403

R.Dhavaseelan, S. Jafari, N. Rajesh, Florentin Smarandache Neutrosophic Semi-Continuous Multifunctions / 412

Tèmítópé Gbóláhàn Jaíyéolá, Emmanuel Ilojide, Memudu Olaposi Olatinwo, Florentin Smarandache On the Classification of Bol-Moufang Type of Some Varieties of Quasi Neutrosophic Triplet Loop (Fenyves BCI-Algebras) / 422

Mumtaz Ali, Huma Khan, Le Hoang Son, Florentin Smarandache, W. B. Vasantha Kandasamy

New Soft Set Based Class of Linear Algebraic Codes / 438

Florentin Smarandache

Extension of Soft Set to Hypersoft Set, and then to Plithogenic Hypersoft Set / 448

Florentin Smarandache, Xiaohong Zhang, Mumtaz Ali Algebraic Structures of Neutrosophic Triplets, Neutrosophic Duplets, or Neutrosophic Multisets / 451

Florentin Smarandache Neutrosophic Hedge Algebras / 467

G.Muhiuddin, Florentin Smarandache, Young Bae Jun Neutrosophic quadruple ideals in neutrosophic quadruple BCI-algebras / 474

Mohammad Hamidi, Arsham Borumand Saeid, Florentin Smarandache Single-valued neutrosophic filters in EQ-algebras / 487

W.B. Vasantha Kandasamy, Ilanthenral Kandasamy, Florentin Smarandache Semi-Idempotents in Neutrosophic Rings / 501

M.Parimala, M. Karthika, S. Jafari, Florentin Smarandache, R. Udhayakumar Neutrosophic Nano Ideal Topological Structures / 508

Ahmed B. Al Nafee, Riad K. Al Hamido, Florentin Smarandache Separation Axioms in Neutrosophic Crisp Topological Spaces / 515

Muhammad Akram, Hina Gulzar, Florentin Smarandache Neutrosophic Soft Topological K-Algebras / 523

(author and editor) Collected Papers, IX 8

Florentin Smarandache

Smarandache (author and editor)

Mohammed A. Al Shumrani, Florentin Smarandache

Introduction to Non-Standard Neutrosophic Topology / 544

W.B. Vasantha Kandasamy, Ilanthenral Kandasamy, Florentin Smarandache Neutrosophic Quadruple Vector Spaces and Their Properties / 557

Xin Zhou, Ping Li, Florentin Smarandache, Ahmed Mostafa Khalil New Results on Neutrosophic Extended Triplet Groups Equipped with a Partial Order / 565

Xiaohong Zhang, Xiaoying Wu, Xiaoyan Mao, Florentin Smarandache, Choonkil Park On neutrosophic extended triplet groups (loops) and Abel-Grassmann’s groupoids (AG-groupoids) / 578

W.B. Vasantha Kandasamy, Ilanthenral Kandasamy, Florentin Smarandache Neutrosophic Triplets in Neutrosophic Rings / 589

Florentin Smarandache Refined Neutrosophy and Lattices vs. Pair Structures and YinYang Bipolar Fuzzy Set / 598

Xiaohong Zhang, Florentin Smarandache, Yingcang Ma Symmetry in Hyperstructure: Neutrosophic Extended Triplet Semihypergroups and Regular Hypergroups / 614

Yingcang Ma, Xiaohong Zhang, Florentin Smarandache, Juanjuan Zhang The Structure of Idempotents in Neutrosophic Rings and Neutrosophic Quadruple Rings / 631

Florentin Smarandache NeutroAlgebra is a Generalization of Partial Algebra / 646

E.O. Adeleke, A.A.A. Agboola, Florentin Smarandache Refined Neutrosophic Rings I / 656

E.O. Adeleke, A.A.A. Agboola, Florentin Smarandache Refined Neutrosophic Rings II / 661

Florentin Smarandache, Mohammad Abobala n-Refined Neutrosophic Rings / 667

Akbar Rezaei, Florentin Smarandache

On Neutro BE-algebras and Anti-BE-algebras (revisited) / 675

Florentin Smarandache, Akbar Rezaei, Hee Sik Kim A New Trend to Extensions of CI-algebras / 683

Mohammad Hamidi, Florentin Smarandache

Neutro BCK-Algebra / 691

Florentin Smarandache Generalizations and Alternatives of Classical Algebraic Structures to NeutroAlgebraic Structures and AntiAlgebraic Structures / 699

IX 9

Florentin

Collected Papers,

Florentin Smarandache

Introduction to NeutroAlgebraic Structures and AntiAlgebraic Structures (revisited) / 702

Young Bae Jun, Madad Khan, Florentin Smarandache, Seok Zun Song

Length Neutrosophic Subalgebras of BCK/BCI-Algebras / 718

Hashem Bordbar, Rajab Ali Borzooei, Florentin Smarandache, Young Bae Jun A General Model of Neutrosophic Ideals in BCK/BCI-Algebras Based on Neutrosophic Points / 741

Florentin Smarandache

Extension of HyperGraph to n-SuperHyperGraph and to Plithogenic n-SuperHyperGraph, and Extension of HyperAlgebra to n-ary (Classical-/Neutro-/Anti-)HyperAlgebra / 757

M.Parimala, M. Karthika, Florentin Smarandache, Said Broumi

On αω-closed sets and its connectedness in terms of neutrosophic topological spaces / 764

Akbar Rezaei, Florentin Smarandache

The Neutrosophic Triplet of BI-Algebras / 771

W. B. Vasantha Kandasamy, Ilanthenral Kandasamy, Florentin Smarandache NeutroAlgebra of Neutrosophic Triplets using {Zn, x} / 780

K.Porselvi, B. Elavarasan, Florentin Smarandache, Young Bae Jun Neutrosophic N-bi-ideals in semigroups / 795

W.B. Vasantha Kandasamy, Ilanthenral Kandasamy, Florentin Smarandache Neutrosophic Components Semigroups and Multiset Neutrosophic Components Semigroups / 808

Mohammed A. Al Shumrani, Muhammad Gulistan, Florentin Smarandache Further Theory of Neutrosophic Triplet Topology and Applications / 818

Rakhal Das, Florentin Smarandache, Binod Chandra Tripathy Neutrosophic Fuzzy Matrices and Some Algebraic Operations / 830

W.B. Vasantha Kandasamy, Ilanthenral Kandasamy, Florentin Smarandache Neutrosophic Quadruple Algebraic Codes over Z2 and their Properties / 839

V.Christianto, F. Smarandache, Muhammad Aslam

How we can extend the standard deviation notion with neutrosophic interval and quadruple neutrosophic numbers / 853

S.Rajareega, D. Preethi, J. Vimala, Ganeshsree Selvachandran, Florentin Smarandache Some Results on Single Valued Neutrosophic Hypergroup / 858

Rajab Ali Borzooei, Florentin Smarandache, Young Bae Jun Polarity of generalized neutrosophic subalgebras in BCK/BCI-algebras / 864

Florentin Smarandache (author and editor) Collected Papers, IX 10

Tugce Katican, Tahsin Oner, Akbar Rezaei, Florentin Smarandache

Neutrosophic N-Structures Applied to Sheffer Stroke BL-Algebras / 887

Akbar Rezaei, Florentin Smarandache, S. Mirvakili

Applications of (Neutro/Anti)sophications to Semihypergroups / 905

Florentin Smarandache

NeutroGeometry & AntiGeometry are alternatives and generalizations of the NonEuclidean Geometries / 912

Florentin Smarandache, Akbar Rezaei, A.A.A. Agboola, Young Bae Jun, Rajab Ali Borzooei, Bijan Davvaz, Arsham Borumand Saeid, Muhammad Akram, M. Hamidi, S. Mirvakili

On Neutrosophic Quadruple Groups / 933

Florentin Smarandache

Universal NeutroAlgebra and Universal AntiAlgebra / 940

Madeleine Al Tahan, Bijan Davvaz, Florentin Smarandache, Osman Anis

On Some NeutroHyperstructures / 945

M.Hamidi, Florentin Smarandache

Single Valued Neutro Hyper BCK-Subalgebras / 957

M.Parimala, Florentin Smarandache, Madeleine Al Tahan, Cenap Özel

On Complex Neutrosophic Lie Algebras / 968

Florentin Smarandache

Introduction to SuperHyperAlgebra and Neutrosophic SuperHyperAlgebra / 976

Yaser Saber, Fahad Alsharari, Florentin Smarandache, Mohammed Abdel Sattar

On Single Valued Neutrosophic Regularity Spaces / 984

Collected Papers, IX 11

Florentin Smarandache (author and editor)

List of Countries

UNITED STATES OF AMERICA (1)

SAUDI ARABIA (7)

PR CHINA (11)

DENMARK (1) EGYPT (3) INDIA (14) INDONESIA (1) IRAN (8) IRAQ (1) JAPAN (1) SOUTH KOREA (5)

LEBANON (2)

MALAYSIA (1) MOROCCO (3) NIGERIA (6) PAKISTAN (10) SYRIA (3) TURKEY (3)

SR VIETNAM (1)

Florentin Smarandache (author and editor) Collected Papers, IX 12

List of Authors

A

E.O. Adeleke

Department of Mathematics, Federal University of Agriculture, Abeokuta, NIGERIA yemi376@yahoo.com

A.A.A. Agboola

Department of Mathematics, Federal University of Agriculture, Abeokuta, NIGERIA agboolaaaa@funaab.edu.ng

Ahmed B. Al-Nafee

Ministry of Education Open Educational College, Department of Mathematics, Babylon, IRAQ Ahm_math_88@yahoo.com

Ahmed Mostafa Khalil

Department of Mathematics, Faculty of Science, Al Azhar University, Assiut, EGYPT a.khalil@azhar.edu.eg

Akbar Rezaei

Department of Mathematics, Payame Noor University, Tehran, IRAN rezaei@pnu.ac.ir

S.A. Akinleye

Department of Mathematics, Federal University of Agriculture, Abeokuta, NIGERIA akinleye_sa@yahoo.com

Ali Hassan

Department of Mathematics, University of Punjab, Lahore, PAKISTAN alihassan.iiui.math@gmail.com

Mumtaz Ali

Department of Mathematics, Quaid-i-Azam University, Islamabad, PAKISTAN mumtazali770@yahoo.com

Rajab Ali Borzooei

Department of Mathematics, Shahid Beheshti University, Tehran, IRAN borzooei@sbu.ac.ir

Assia Bakali

Ecole Royale Navale, Boulevard Sour Jdid, Casablanca, MOROCCO assiabakali@yahoo.fr

C

Cenap Özel

Department of Mathematics, King Abdulaziz University, Makkah, SAUDI ARABIA

Florentin Smarandache (author and editor) Collected Papers, IX 13

Victor Christianto

Malang Institute of Agriculture (IPM), Malang, INDONESIA victorchristianto@gmail.com

Chunxin Bo

College of Information Engineering, Shanghai Maritime University, Shanghai, PR CHINA

D

Rakhal Das

Department of Mathematics, Tripura University Agartala, Tripura, INDIA rakhal.mathematics@tripurauniv.in

Bijan Davvaz

Department of Mathematics, Yazd University, Yazd, IRAN davvaz@yazd.ac.ir

R.Dhavaseelan

Department of Mathematics, Sona College of Technology, Salem,Tamil Nadu, INDIA dhavaseelan.r@gmail.com

E

B.Elavarasan

Department of Mathematics, Karunya Institute of Technology and Sciences, Coimbatore, Tamilnadu, INDIA elavarasan@karunya.edu

F

Fahad Alsharari

Department of Mathematics, College of Science and Human Studies, Majmaah University, Majmaah, SAUDI ARABIA f.alsharari@mu.edu.sa

G

T.Gharibah

Department of Mathematics, College of Science, Al Baath University, Homs, SYRIA taleb.gharibah@gmail.com

Hina Gulzar

Department of Mathematics, University of the Punjab, New Campus, Lahore, PAKISTAN hinagulzar5@gmail.com

Florentin Smarandache (author and editor) Collected Papers, IX 14

Hashem Bordbar

Department of Mathematics, Shiraz University, Shiraz, IRAN bordbar.amirh@gmail.com

Le Hoang Son

VNU Information Technology Institute, Vietnam National University, Hanoi, SR VIETNAM sonlh@vnu.edu.vn I

Emmanuel Ilojide

Department of Mathematics, College of Physical Sciences, Federal University of Agriculture, Abeokuta, NIGERIA

J

Tèmítópé Gbóláhàn Jaíyéolá

Department of Mathematics, Obafemi Awolowo University, Ile Ife, NIGERIA tjayeola@oauife.edu.ng

K

M.Karthika

Department of Mathematics, Bannari Amman Institute of Technology, Sathyamangalam, Tamil Nadu, INDIA karthikamuthusamy1991@gmail.com

Ilanthenral Kandasamy

School of Computer Science and Engineering, VIT, Vellore, Tamil Nadu, INDIA ilanthenral.k@vit.ac.in

W.B. Vasantha Kandasamy

School of Computer Science and Engineering, VIT, Vellore, Tamil Nadu, INDIA vasantha.wb@vit.ac.in

Huma Khan

Department of Mathematics, University of Management and Technology, Lahore, PAKISTAN humakhan7328@gmail.com

Madad Khan

Department of Mathematics, COMSATS University, Abbottabad Campus, Abbottabad, PAKISTAN madadmath@yahoo.com

Mohsin Khan

Department of Mathematics, Abdul Wali Khan University, Mardan, PAKISTAN mohsinkhan7284@gmail.com

Florentin Smarandache (author and editor) Collected Papers, IX 15

Hee Sik Kim

Department of Mathematics, Hanyang University, Seoul, SOUTH KOREA heekim@hanyang.ac.kr

Seon Jeong Kim

Department of Mathematics, Natural Science of College, Gyeongsang National University, Jinju, SOUTH KOREA skim@gnu.ac.kr

Valeri Kromov

Okayama University of Science, Okayama, JAPAN

L

R.M. Latif

Department of Mathematics & Natural Sciences, Prince Mohammed Bin Fahd University, Al Khobar, SAUDI ARABIA M

Madeleine Al-Tahan

Faculty of Economic Science and Buisness Administration, Lebanese International University, Beirut, LEBANON madeline.tahan@liu.edu.lb

Mehmat Ali Ozturk

Faculty of Arts and Sciences, Adiyaman University, Adiyaman, TURKEY mehaliozturk@gmail.com

Minghao Hu

Department of Mathematics, Shaanxi University of Science & Technology, Xi’an, PR CHINA huminghao@sust.edu.cn

S. Mirvakili

Department of Mathematics, Payame Noor University, Tehran, IRAN

Mohammad Abobala

Tishreen University, Faculty of Science, Department of Mathematics, Lattakia, SYRIA mohammadabobala777@gmail.com

Mohammad Hamidi

Department of Mathematics, Payame Noor University, Tehran, IRAN m.hamidi@pnu.ac.ir

Mohammed Abdel Sattar

Department of Mathematics and Computer Science, Faculty of Science, Beni Suef University, Beni Suef, EGYPT

Mohammed A. Al Shumrani

Department of Mathematics, King Abdulaziz University, Jeddah, SAUDI ARABIA maalshmrani1@kau.edu.sa

Smarandache (author and editor) Collected Papers, IX 16

Florentin

Mohamed Talea

Laboratory of Information processing, Faculty of Science Ben M’Sik, University Hassan II, Casablanca, MOROCCO taleamohamed@yahoo.fr

Muhammad Akram

Department of Mathematics, University of the Punjab, New Campus, Lahore, PAKISTAN m.akram@pucit.edu.pk

Muhammad Aslam

Department of Statistics, Faculty of Science, King Abdulaziz University, Jeddah, SAUDI ARABIA aslam_ravian@hotmail.com

Muhammad Aslam Malik

Department of Mathematics, University of Punjab, Lahore, PAKISTAN aslam@math.pu.edu.pk

Muhammad Gulistan

Department of Mathematics, Hazara University, Mansehra, PAKISTAN gulistanmath@hu.edu.pk

Muhammad Shabir

Department of Mathematics, Quaid-i-Azam University, Islamabad, PAKISTAN mshabirbhatti@yahoo.co.uk

G. Muhiuddin

Department of Mathematics, University of Tabuk, Tabuk, SAUDI ARABIA chishtygm@gmail.com

O

Memudu Olaposi Olatinwo

Department of Mathematics, Faculty of Science, Obafemi Awolowo University, Ile Ife, Osun, NIGERIA

Osman Anis

Department of Mathematics, Lebanese International University, Akkar, LEBANON osman.anis@liu.edu.lb

P

Choonkil Park

Department of Mathematics, Hanyang University, Seoul, SOUTH KOREA

M.Parimala

Department of Mathematics, Bannari Amman Institute of Technology, Sathyamangalam, Tamil Nadu, INDIA rishwanthpari@gmail.com

Ping Li

School of Science, Xi’an Polytechnic University, Xi’an, PR CHINA liping7487@163.com

and editor) Collected Papers, IX 17

Florentin Smarandache (author

K.Porselvi

Department of Mathematics, Karunya Institute of Technology and Sciences, Coimbatore, Tamilnadu, INDIA porselvi94@yahoo.co.in

D.Preethi

Department of Mathematics, Alagappa University, Karaikudi, Tamilnadu, India. preethi06061996@gmail.com

R

S.Rajareega

Department of Mathematics, Alagappa University, Karaikudi, Tamil Nadu, INDIA reega948@gmail.com

N.Rajesh

Department of Mathematics, Rajah Serfoji Govt. College, Thanjavur, Tamil Nadu, INDIA nrajesh_topology@yahoo.co.in

Udhayakumar Ramalingam

Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Tamil Nadu, INDIA udhayaram_v@yahoo.co.in

Riad K.

Al-Hamido

Department of Mathematics, College of Science, Al Baath University, Homs, SYRIA riad hamido1983@hotmail.com

S

Yaser Saber

Department of Mathematics, College of Science and Human Studies, Majmaah University, Majmaah, SAUDI ARABIA y.sber@mu.edu.sa

Arsham Borumand Saeid

Department of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, IRAN arsham@uk.ac.ir

Saeid Jafari

College of Vestsjaelland South, Slagelse, DENMARK jafaripersia@gmail.com

Said Broumi

Laboratory of Information Processing, Faculty of Science Ben M’Sik, University Hassan II, Casablanca, MOROCCO broumisaid78@gmail.com

A. A. Salama

Department of Math. and Computer Science, Faculty of Sciences, Port Said University, EGYPT drsalama44@gmail.com

Florentin Smarandache (author and editor) Collected Papers, IX 18

(author and editor)

Ganeshsree Selvachandran

Department of Actuarial Science and Applied Statistics, Faculty of Business and Information Science, UCSI University, Jalan Menara Gading, Cheras, Kuala Lumpur, MALAYSIA Ganeshsree@ucsiuniversity.edu

Songtao Shao

Department of Mathematics, Shanghai Maritime University, Shanghai, PR CHINA 201740310005@stu.shmtu.edu.cn

Florentin Smarandache

University of New Mexico, 705 Gurley Ave., Gallup, NM 87301, UNITES STATES OF AMERICA smarand@unm.edu

Seok Zun Song

Department of Mathematics, Jeju National University, Jeju, SOUTH KOREA szsong@jejunu.ac.kr T

Tahsin Oner

Department of Mathematics, Ege University, Izmir, TURKEY

M.Mohseni Takallo

Department of Mathematics, Shahid Beheshti University, Tehran, IRAN mohammad.mohseni1122@gmail.com

Binod Chandra Tripathy

Department of Mathematics, Tripura University Agartala, Tripura, INDIA binodtripathy@tripurauniv.in

Tugce Katican

Department of Mathematics, Ege University, Izmir, TURKEY V

J.Vimala

Department of Mathematics, Alagappa University, Karaikudi, Tamilnadu, INDIA vimaljey@alagappauniversity.ac.in

Xiaohong Zhang

Department of Mathematics, Shaanxi University of Science & Technology, Xi’an, PR CHINA zhangxiaohong@sust.edu.cn

Smarandache

Collected Papers, IX 19

Florentin

Xiaoyan Mao

College of Science and Technology, Ningbo University, Ningbo, PR CHINA

Xiaoying Wu

Department of Mathematics, Shaanxi University of Science & Technology, Xi’an, PR CHINA 46018@sust.edu.cn

Xingliang Liang

Department of Mathematics, Shaanxi University of Science &Technology, Xi’an, PR CHINA liangxingl@sust.edu.cn

Xin Zhou

School of Science, Xi’an Polytechnic University, Xi’an, PR CHINA sxxzx1986@163.com Y

Yingcang Ma

School of Science, Xi’an Polytechnic University, Xi’an, PR CHINA mayingcang@126.com

Young Bae Jun

Department of Mathematics Education, Gyeongsang National University, Jinju, SOUTH KOREA skywine@gmail.com Z

Juanjuan Zhang

School of Science, Xi’an Polytechnic University, Xi’an 710048, PR CHINA 20080712@xpu.edu.cn

Florentin Smarandache (author and editor) Collected Papers, IX 20

List of Papers

1. A.A. Salama, Florentin Smarandache, Valeri Kromov (2014). Neutrosophic Closed Set and Neutrosophic Continuous Functions. Neutrosophic Sets and Systems, 4, 4-8

2. Mumtaz Ali, Muhammad Shabir, Munazza Naz, Florentin Smarandache (2014). Soft neutrosophic semigroups and their generalization. Scientia Magna 10(1), 93 111

3. Florentin Smarandache (2015). (T, I, F) Neutrosophic Structures. Proceedings of the Annual Symposium of the Institute of Solid Mechanics and Session of the Commission of Acoustics, SISOM 2015 Bucharest 21 22 May; Acta Electrotechnica 57(1 2); Neutrosophic Sets and Systems, 8, 3 10

4. Akbar Rezaei, Arsham Borumand Saeid, Florentin Smarandache (2015). Neutrosophic filters in BE algebras. Ratio Mathematica, 29, 65 79

5. Muhammad Aslam Malik, Ali Hassan, Said Broumi, Assia Bakali, Mohamed Talea, Florentin Smarandache (2016). Isomorphism of Bipolar Single Valued Neutrosophic Hypergraphs Critical Review XIII, 79 102

6. Muhammad Aslam Malik, Ali Hassan, Said Broumi, F. Smarandache (2016). Regular Bipolar Single Valued Neutrosophic Hypergraphs Neutrosophic Sets and Systems, 13, 84-89

7. Mumtaz Ali, Florentin Smarandache (2016). Neutrosophic Soluble Groups, Neutrosophic Nilpotent Groups and Their Properties. Annual Symposium of the Institute of Solid Mechanics, SISOM 2015, Robotics and Mechatronics. Special Session and Work Shop on VIPRO Platform and RABOR Rescue Robots, Romanian Academy, Bucharest, 21 22 May 2015; Acta Electrotehnica, 57(1/2), 153 159

8. Florentin Smarandache (2016). Operators on Single Valued Neutrosophic Oversets, Neutrosophic Undersets, and Neutrosophic Offsets. Journal of Mathematics and Informatics 5, 63 67

9. Florentin Smarandache (2016). Interval Valued Neutrosophic Oversets, Neutrosophic Undersets, and Neutrosophic Offsets. International Journal of Science and Engineering Investigations, 5, 54, Paper ID: 55416 01, 4 p.

10. Florentin Smarandache (2016). Subtraction and Division of Neutrosophic Numbers. Critical Review, XIII, 103 110

11. S.A. Akinleye, Florentin Smarandache, A.A.A. Agboola (2016). On Neutrosophic Quadruple Algebraic Structures. Neutrosophic Sets and Systems, 12, 122 126

12. A.A.A. Agboola, B. Davvaz, Florentin Smarandache (2017). Neutrosophic Quadruple Algebraic Hyperstructures Annals of Fuzzy Mathematics and Informatics 14(1), 29 42

13. Young Bae Jun, Florentin Smarandache, Hashem Bordbar (2017). Neutrosophic N-Structures Applied to BCK/BCI Algebras. Information, 8, 128; DOI: 10.3390/info8040128

14. Seok Zun Song, Florentin Smarandache, Young Bae Jun (2017). Neutrosophic Commutative N Ideals in BCK Algebras. Information, 8, 130; DOI: 10.3390/info8040130

15. R. Dhavaseelan, Saeid Jafari, Florentin Smarandache (2017). Compact Open Topology and Evaluation Map via Neutrosophic Sets. Neutrosophic Sets and Systems, 16, 35 38

16. R. Dhavaseelan, M. Parimala, S. Jafari, Florentin Smarandache (2017). On Neutrosophic Semi Supra Open Set and Neutrosophic Semi Supra Continuous Functions. Neutrosophic Sets and Systems, 16, 39 43

17. Xiaohong Zhang, Yingcang Ma, Florentin Smarandache (2017). Neutrosophic Regular Filters and Fuzzy Regular Filters in Pseudo BCI Algebras. Neutrosophic Sets and Systems, 17, 10 15

18. Xiaohong Zhang, Florentin Smarandache, Xingliang Liang (2017). Neutrosophic Duplet Semi Group and Cancellable Neutrosophic Triplet Groups. Symmetry, 9, 275; DOI: 10.3390/sym9110275

19. G. Muhiuddin, Hashem Bordbar, Florentin Smarandache, Young Bae Jun (2018). Further results on (ε, ε,) neutrosophic subalgebras and ideals in BCK/BCI algebras. Neutrosophic Sets and Systems, 20, 36 43.

20. W.B. Vasantha Kandasamy, Ilanthenral Kandasamy, Florentin Smarandache (2018). Algebraic Structure of Neutrosophic Duplets in Neutrosophic Rings <Z U I>, <Q U I> and <R U I>. Neutrosophic Sets and Systems 23, 85 95

21. Xiaohong Zhang, Florentin Smarandache, Mumtaz Ali, Xingliang Liang (2018). Commutative Neutrosophic Triplet Group and Neutro Homomorphism Basic Theorem. Italian Journal of Pure and Applied Mathematics, 40, 353 375.

Florentin Smarandache (author and editor) Collected Papers, IX 21

22. Rajab Ali Borzooei, Xiaohong Zhang, Florentin Smarandache, Young Bae Jun (2018). Commutative Generalized Neutrosophic Ideals in BCK Algebras. Symmetry, 10, 350. DOI: 10.3390/sym10080350

23. Young Bae Jun, Seok Zun Song, Florentin Smarandache, Hashem Bordbar (2018). Neutrosophic Quadruple BCK/BCI Algebras. Axioms, 7, 41. DOI: 10.3390/axioms7020041

24. Young Bae Jun, Seon Jeong Kim, Florentin Smarandache (2018). Interval Neutrosophic Sets with Applications in BCK/BCI Algebra. Axioms, 7, 23. DOI: 10.3390/axioms7020023

25. Young Bae Jun, Florentin Smarandache, Seok Zun Song, Hashem Bordbar (2018). Neutrosophic Permeable Values and Energetic Subsets with Applications in BCK/BCI Algebras. Mathematics, 6, 74; DOI: 10.3390/math6050074

26. Xiaohong Zhang, Xiaoying Wu, Florentin Smarandache, Minghao Hu (2018). Left (Right) Quasi Neutrosophic Triplet Loops (Groups) and Generalized BE Algebras. Symmetry, 10, 241; DOI: 10.3390/sym10070241

27. Muhammad Akram, Hina Gulzar, Florentin Smarandache, Said Broumi (2018). Certain Notions of Neutrosophic Topological K Algebras. Mathematics, 6, 234; DOI: 10.3390/math6110

28. Mumtaz Ali, Florentin Smarandache, Mohsin Khan (2018). Study on the Development of Neutrosophic Triplet Ring and Neutrosophic Triplet Field. Mathematics, 6, 46; DOI: 10.3390/math6040046

29. Rajab Ali Borzooei, M. Mohseni Takallo, Florentin Smarandache, Young Bae Jun (2018). Positive implicative BMBJ neutrosophic ideals in BCK algebras. Neutrosophic Sets and Systems, 23, 126 141

30. Songtao Shao, Xiaohong Zhang, Chunxin Bo, Florentin Smarandache (2018). Neutrosophic Hesitant Fuzzy Subalgebras and Filters in Pseudo BCI Algebras. Symmetry, 10, 174; DOI: 10.3390/sym10050174

31. W.B. Vasantha Kandasamy, Ilanthenral Kandasamy, Florentin Smarandache (2018). A Classical Group of Neutrosophic Triplet Groups Using (Z2p, X). Symmetry, 10, 194; DOI: 10.3390/sym10060194

32. Young Bae Jun, Florentin Smarandache, Mehmat Ali Ozturk (2018). Commutative falling neutrosophic ideals in BCK algebras. Neutrosophic Sets and Systems, 20, 44 53

33. Riad K. Al Hamido, T. Gharibah, S. Jafari, Florentin Smarandache (2018). On Neutrosophic Crisp Topology via N Topology. Neutrosophic Sets and Systems, 23, 96 109

34. R. Dhavaseelan, S. Jafari, R. M. Latif, Florentin Smarandache (2018). Neutrosophic Rare α-Continuity. New Trends in Neutrosophic Theory and Applications, II, 336 344

35. R. Dhavaseelan, S. Jafari, N. Rajesh, Florentin Smarandache (2018). Neutrosophic Semi Continuous Multifunctions. New Trends in Neutrosophic Theory and Applications, II, 345 354

36. Tèmítópé Gbóláhàn Jaíyéolá, Emmanuel Ilojide, Memudu Olaposi Olatinwo, Florentin Smarandache (2018). On the Classification of Bol Moufang Type of Some Varieties of Quasi Neutrosophic Triplet Loop (Fenyves BCI Algebras). Symmetry, 10, 427; DOI: 10.3390/sym10100427

37. Mumtaz Ali, Huma Khan, Le Hoang Son, Florentin Smarandache, W. B. Vasantha Kandasamy (2018). New Soft Set Based Class of Linear Algebraic Codes. Symmetry, 10, 510; DOI: 10.3390/sym10100510

38. Florentin Smarandache (2018). Extension of Soft Set to Hypersoft Set, and then to Plithogenic Hypersoft Set Neutrosophic Sets and Systems, 22, 168 170

39. Florentin Smarandache, Xiaohong Zhang, Mumtaz Ali (2019). Algebraic Structures of Neutrosophic Triplets, Neutrosophic Duplets, or Neutrosophic Multisets. Symmetry, 11, 171. DOI: 10.3390/sym11020171

40. Florentin Smarandache (2019). Neutrosophic Hedge Algebras. Broad Research in Artificial Intelligence and Neuroscience, 10(3), 117 123.

41. G. Muhiuddin, Florentin Smarandache, Young Bae Jun (2019). Neutrosophic quadruple ideals in neutrosophic quadruple BCI algebras. Neutrosophic Sets and Systems, 25, 161 173

42. Mohammad Hamidi, Arsham Borumand Saeid, Florentin Smarandache (2019). Single valued neutrosophic filters in EQ algebras. Journal of Intelligent & Fuzzy Systems, 36(1), 805 818

43. W.B. Vasantha Kandasamy, Ilanthenral Kandasamy, Florentin Smarandache (2019). Semi Idempotents in Neutrosophic Rings. Mathematics, 7, 507; DOI: 10.3390/math7060507

44. M. Parimala, M. Karthika, S. Jafari, Florentin Smarandache, R. Udhayakumar (2019). Neutrosophic Nano Ideal Topological Structures. Neutrosophic Sets and Systems, 24, 70 76

45. Ahmed B. Al Nafee, Riad K. Al Hamido, Florentin Smarandache (2019). Separation Axioms in Neutrosophic Crisp Topological Spaces. Neutrosophic Sets and Systems, 25, 25 32

Collected Papers, IX 22

Florentin Smarandache (author and editor)

46. Muhammad Akram, Hina Gulzar, Florentin Smarandache (2019). Neutrosophic Soft Topological K Algebras. Neutrosophic Sets and Systems, 25, 104 124

47. Mohammed A. Al Shumrani, Florentin Smarandache (2019). Introduction to Non Standard Neutrosophic Topology. Symmetry, 11, 0; DOI: 10.3390/sym11050000

48. W.B. Vasantha Kandasamy, Ilanthenral Kandasamy, Florentin Smarandache (2019). Neutrosophic Quadruple Vector Spaces and Their Properties. Mathematics, 7, 758; DOI: 10.3390/math7080758

49. Xin Zhou, Ping Li, Florentin Smarandache, Ahmed Mostafa Khalil (2019). New Results on Neutrosophic Extended Triplet Groups Equipped with a Partial Order. Symmetry, 11, 1514; DOI: 10.3390/sym11121514

50. Xiaohong Zhang, Xiaoying Wu, Xiaoyan Mao, Florentin Smarandache, Choonkil Park (2019). On neutrosophic extended triplet groups (loops) and Abel Grassmann’s groupoids (AG groupoids). Journal of Intelligent & Fuzzy Systems, 37, 5743 5753; DOI:10.3233/JIFS 181742

51. W.B. Vasantha Kandasamy, Ilanthenral Kandasamy, Florentin Smarandache (2019). Neutrosophic Triplets in Neutrosophic Rings. Mathematics, 7, 563; DOI: 10.3390/math7060563

52. Florentin Smarandache (2019). Refined Neutrosophy and Lattices vs. Pair Structures and YinYang Bipolar Fuzzy Set. Mathematics, 7, 353; DOI: 10.3390/math7040353

53. Xiaohong Zhang, Florentin Smarandache, Yingcang Ma (2019). Symmetry in Hyperstructure: Neutrosophic Extended Triplet Semihypergroups and Regular Hypergroups Symmetry, 11, 1217; DOI: 10.3390/sym11101217

54. Yingcang Ma, Xiaohong Zhang, Florentin Smarandache, Juanjuan Zhang (2019). The Structure of Idempotents in Neutrosophic Rings and Neutrosophic Quadruple Rings. Symmetry, 11, 1254; DOI: 10.3390/sym11101254

55. Florentin Smarandache (2020). NeutroAlgebra is a Generalization of Partial Algebra International Journal of Neutrosophic Science 2(1), 8-17

56. E.O. Adeleke, A.A.A. Agboola, Florentin Smarandache (2020). Refined Neutrosophic Rings I. International Journal of Neutrosophic Science 2(2), 77 81 DOI: 10.5281/zenodo.3728222

57. E.O. Adeleke, A.A.A. Agboola, Florentin Smarandache (2020). Refined Neutrosophic Rings II International Journal of Neutrosophic Science 2(2), 89 94. DOI: 10.5281/zenodo.3728235

58. Florentin Smarandache, Mohammad Abobala (2020). n-Refined Neutrosophic Rings International Journal of Neutrosophic Science 3(2), 83 90 DOI:10.5281/zenodo.3828996

59. Akbar Rezaei, Florentin Smarandache (2020). On Neutro BE algebras and Anti BE algebras (revisited). International Journal of Neutrosophic Science 4(1), 8-15. DOI: 10.5281/zenodo.3751862

60. Florentin Smarandache, Akbar Rezaei, Hee Sik Kim (2020). A New Trend to Extensions of CI algebras International Journal of Neutrosophic Science 5(1), 8-15. DOI: 10.5281/zenodo.3788124

61. Mohammad Hamidi, Florentin Smarandache (2020). Neutro BCK Algebra. International Journal of Neutrosophic Science 8(2), 110 117 DOI: 10.5281/zenodo.3902754

62. Florentin Smarandache (2020). Generalizations and Alternatives of Classical Algebraic Structures to NeutroAlgebraic Structures and AntiAlgebraic Structures Journal of Fuzzy Extension & Applications, 1(2), 85 87 DOI: 10.22105/jfea.2020.248816.1008

63. Florentin Smarandache (2020). Introduction to NeutroAlgebraic Structures and AntiAlgebraic Structures (revisited) Neutrosophic Sets and Systems, 31, 2-16

64. Young Bae Jun, Madad Khan, Florentin Smarandache, Seok Zun Song (2020). Length Neutrosophic Subalgebras of BCK/BCI Algebras. Bulletin of the Section of Logic, 49(4), 377 400. DOI: 10.18778/01380680.2020.21

65. Hashem Bordbar, Rajab Ali Borzooei, Florentin Smarandache, Young Bae Jun (2020). A General Model of Neutrosophic Ideals in BCK/BCI Algebras Based on Neutrosophic Points. Bulletin of the Section of Logic, 17. DOI: 10.18778/0138 0680.2020.18

66. Florentin Smarandache (2020). Extension of HyperGraph to n SuperHyperGraph and to Plithogenic n SuperHyperGraph, and Extension of HyperAlgebra to n ary (Classical /Neutro /Anti )HyperAlgebra. Neutrosophic Sets and Systems, 33, 290-296

67. M. Parimala, M. Karthika, Florentin Smarandache, Said Broumi (2020). On αω closed sets and its connectedness in terms of neutrosophic topological spaces International Journal of Neutrosophic Science, 2(2), 82 88 DOI: 10.5281/zenodo.3728230

Smarandache (author and editor) Collected Papers, IX 23

Florentin

68. Akbar Rezaei, Florentin Smarandache (2020). The Neutrosophic Triplet of BI-Algebras Neutrosophic Sets and Systems, 33, 313 321

69. W. B. Vasantha Kandasamy, Ilanthenral Kandasamy, Florentin Smarandache (2020). NeutroAlgebra of Neutrosophic Triplets using {Zn, x}. Neutrosophic Sets and Systems, 38, 510 523

70. K. Porselvi, B. Elavarasan, Florentin Smarandache, Young Bae Jun (2020). Neutrosophic N bi ideals in semigroups. Neutrosophic Sets and Systems, 35, 422 434

71. W.B. Vasantha Kandasamy, Ilanthenral Kandasamy, Florentin Smarandache (2020). Neutrosophic Components Semigroups and Multiset Neutrosophic Components Semigroups Symmetry, 12, 818; DOI: 10.3390/sym12050818

72. Mohammed A. Al Shumrani, Muhammad Gulistan, Florentin Smarandache (2020). Further Theory of Neutrosophic Triplet Topology and Applications Symmetry, 12, 1207; DOI: 10.3390/sym12081207

73. Rakhal Das, Florentin Smarandache, Binod Chandra Tripathy (2020). Neutrosophic Fuzzy Matrices and Some Algebraic Operations Neutrosophic Sets and Systems, 32, 401 409

74. W.B. Vasantha Kandasamy, Ilanthenral Kandasamy, Florentin Smarandache (2020). Neutrosophic Quadruple Algebraic Codes over Z2 and their Properties. Neutrosophic Sets and Systems, 33, 169 182

75. V. Christianto, F. Smarandache, Muhammad Aslam (2020). How we can extend the standard deviation notion with neutrosophic interval and quadruple neutrosophic numbers International Journal of Neutrosophic Science, 2(2), 72 76; DOI: 10.5281/zenodo.3728218

76. S. Rajareega, D. Preethi, J. Vimala, Ganeshsree Selvachandran, Florentin Smarandache (2020). Some Results on Single Valued Neutrosophic Hypergroup Neutrosophic Sets and Systems, 31, 80 85

77. Rajab Ali Borzooei, Florentin Smarandache, Young Bae Jun (2020). Polarity of generalized neutrosophic subalgebras in BCK/BCI algebras Neutrosophic Sets and Systems, 32, 123 145

78. Tugce Katican, Tahsin Oner, Akbar Rezaei, Florentin Smarandache (2021). Neutrosophic N-Structures Applied to Sheffer Stroke BL Algebras. Computer Modeling in Engineering & Sciences, 129(1), 355 372. DOI: 10.32604/cmes.2021.016996

79. Akbar Rezaei, Florentin Smarandache, S. Mirvakili (2021). Applications of (Neutro/Anti)sophications to Semihypergroups. Journal of Mathematics, 6649349, 7. DOI: 10.1155/2021/6649349

80. Florentin Smarandache (2021). NeutroGeometry & AntiGeometry are alternatives and generalizations of the Non Euclidean Geometries Neutrosophic Sets and Systems 46, 457 476

81. Florentin Smarandache, Akbar Rezaei, A.A.A. Agboola, Young Bae Jun, Rajab Ali Borzooei, Bijan Davvaz, Arsham Borumand Saeid, Muhammad Akram, M. Hamidi, S. Mirvakili (2021). On Neutrosophic Quadruple Groups International Journal of Computational Intelligence Systems, 14, 193 DOI: 10.1007/s44196 021 00042-9

82. Florentin Smarandache (2021). Universal NeutroAlgebra and Universal AntiAlgebra. NeutroAlgebra Theory, I, 11 15

83. Madeleine Al Tahan, Bijan Davvaz, Florentin Smarandache, Osman Anis (2021). On Some NeutroHyperstructures Symmetry, 13, 535. DOI: 10.3390/sym13040535

84. M. Hamidi, Florentin Smarandache (2021). Single Valued Neutro Hyper BCK Subalgebras. Journal of Mathematics, Article ID 6656739, 11; DOI: 10.1155/2021/6656739

85. M. Parimala, Florentin Smarandache, Madeleine Al Tahan, Cenap Özel (2022). On Complex Neutrosophic Lie Algebras Palestine Journal of Mathematics, 11(1), 235 242

86. Florentin Smarandache (2022). Introduction to SuperHyperAlgebra and Neutrosophic SuperHyperAlgebra. Journal of Algebraic Hyperstructures and Logical Algebras, 3(2), 17 24 DOI: 10.52547/HATEF.JAHLA.3.2.2

87. Yaser Saber, Fahad Alsharari, Florentin Smarandache, Mohammed Abdel Sattar (2022). On Single Valued Neutrosophic Regularity Spaces. Computer Modeling in Engineering & Sciences, 130(3), 1625 1648; DOI: 10.32604/cmes.2022.017782

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Florentin Smarandache (author and editor)

Neutrosophic Closed Set and Neutrosophic Continuous Functions

A. A. Salama, Florentin Smarandache, Valeri Kromov

A.A. Salama, Florentin Smarandache, Valeri Kromov (2014). Neutrosophic Closed Set and Neutrosophic Continuous Functions. Neutrosophic Sets and Systems, 4, 4-8

Abstract

In this paper, we introduce and study the concept of "neutrosophic closed set "and "neutrosophic continuous function" Possible application to GIS topology rules are touched upon.

Keywords: Neutrosophic Closed Set, Neutrosophic Set; Neutrosophic Topology; Neutrosophic Continuous Function.

1INTRODUCTION

The idea of "neutrosophic set" was first given by Smarandache [11, 12]. Neutrosophic operations have been investigated by Salama at el. [1-10]. Neutrosophy has laid the foundation for a whole family of new mathematical theories, generalizing both their crisp and fuzzy counterparts [9, 13]. Here we shall present the neutrosophic crisp version of these concepts. In this paper, we introduce and study the concept of "neutrosophic closed set "and "neutrosophic continuous function".

2 TERMINOLOGIES

We recollect some relevant basic preliminaries, and in particular the work of Smarandache in [11, 12], and Salama at el. [1 10].

2.1 Definition [5]

A neutrosophic topology (NT for short) an a non empty set X is a family  of neutrosophic subsets in X satisfying the following axioms  1NT ,1 NNO  ,  2NT 12GG  for any 12 , GG  ,  3NT  : ii GGiJ

In this case the pair  , X  is called a neutrosophic topological space ( NTS for short) and any neutrosophic set in  is known as neuterosophic open set ( NOS for short) in X The elements of  are called open neutrosophic sets, A neutrosophic set F is closed if and only if it C (F) is neutrosophic open.

2.1 Definition [5]

The complement of (C (A) for short) of is called a neutrosophic closed set ( for short) in A . NOSA NCS X.

3 Neutrosophic Closed Set .

3.1 Definition

Let  ,X  be a neutrosophic topological space. A neutrosophic set A in  ,X  is said to be neutrosophic closed (in shortly N-closed).

If Ncl (A)  G whenever A  G and G is neutrosophic open; the complement of neutrosophic closed set is Neutrosophic open.

3.1 Proposition

If A and B are neutrosophic closed sets then AB is Neutrosophic closed set.

3.1 Remark

The intersection of two neutrosophic closed (N-closed for short) sets need not be neutrosophic closed set.

3.1 Example

Let  = {a, b, c} and



Florentin Smarandache (author and editor) Collected Papers, IX 25

A = <(0.5,0.5,0.5) , (0.4,0.5,0.5) , (0.4,0.5,0.5)>

B = <(0.3,0.4,0.4) , (0.7,0.5,0.5) , (0.3,0.4,0.4)> Then  = { 0N ,1N , A, B} is a neutrosophic topology on  Define the two neutrosophic sets 1A and 2A as follows,

1A = <(0.5,0.5,0.5),(0.6,0.5,0.5),(0.6,0.5,0.5)>

2A = <(0.7,0.6,0.6((0.3,0.5,0.5),(0.7,0.6,0.6)>

1A and 2A are neutrosophic closed set but 1A  2A is not a neutrosophic closed set.

3.2 Proposition

4. Neutrosophic Continuous Functions

4.1

Definition

i) If BBB B ,,  is a NS in Y, then the preimage of B under , f denoted by ), (1 Bf is a NS in X defined by ),.)( ( ), ()( 1 1 1 1    fffBf B B ii) If AAA A ,,  is a NS in X, then the image of A under , f denoted by ), ( Af is the a NS in Y defined by .))( ), ( ), ()( c AAAfffAf 

be a neutrosophic topological space. If B is neutrosophic closed set and B  A  Ncl (B), then A is N-closed.

3.4 Proposition

Let  ,X 

In a neutrosophic topologicalspace (,), = (the family of all neutrosophic closed sets) iff every neutrosophic subset of (,) is a neutrosophic closed set.

Proof. suppose that every neutrosophic set A of (,) is N closed. Let A, since A  A and A is N closed, Ncl (A)  A. But A  Ncl (A). Hence, Ncl (A) =A. thus, A  . Therefore, T  . If B   then 1 B   . and hence B, That is,   . Therefore = conversely, suppose that A be a neutrosophic set in (,). Let B be a neutrosophic open set in (,). such that A  B. By hypothesis, B is neutrosophic N closed. By definition of neutrosophic closure, Ncl (A)  B. Therefore A is N closed.

3.5 Proposition

Let (,) be a neutrosophic topological space. A neutrosophic set A is neutrosophic open iff B  Nnt (A), whenever B is neutrosophic closed and B  A. Proof

Let A a neutrosophic open set and B be a N-closed, such that B  A. Now, B  A 1 A 1 B and 1 A is a neutrosophic closed set  Ncl (1 A)  1 B. That is, B=1 (1 B)  1 Ncl (1 A). But 1 Ncl (1 A) = Nint (A). Thus, B  Nint (A). Conversely, suppose that A be a neutrosophic set, such that B  Nint (A) whenever B is neutrosophic closed and B  A. Let 1 A  B  1 B  A. Hence by assumption 1 B  Nint (A). that is, 1 Nint (A)  B. But 1 Nint (A) =Ncl (1 A). Hence Ncl(1 A)  B. That is 1 A is neutrosophic closed set. Therefore, A is neutrosophic open set

3.6 Proposition

Here we introduce the properties of images and preimages some of which we shall frequently use in the following sections .

4.1

Corollary

Let A, JiA i  : , be NSs in X, and B, KjB j  : NS in Y, and YXf  : a function. Then (a) ), ()( 2 1 21AfAfAA ), ()( 2 1 1 1 21BfBfBB   (b) )) (( 1 AffA  and if f is injective, then ))(( 1 AffA  (c) BBff  )) (( 1 and if f is surjective, then ,))(( 1 BBff  (d) ), ())( 1 1 i i BfBf ), ())( 1 1 i i BfBf (e) ); ()( i i AfAf ); ()( i i AfAf and if f is injective, then ); ()( i i AfAf (f) NNf (!1)1  NN f 0)0( 1  (g) ,0)0(NN f  NN f 1)1(  if f is subjective. Proof Obvious.

4.2 Definition

Let  1 ,  X and  2 ,  Y be two NTSs, and let YXf  : be a function. Then f is said to be continuous iff the preimage of each NCS in 2 is a NS in 1

4.3 Definition

Let  1 ,  X and  2 ,  Y be two NTSs and let YXf  : be a function. Then f is said to be open iff the image of eachNS in 1 is a NSin 2

If Nint (A)  B  A and if A is neutrosophic open set then B is also neutrosophic open set.

4.1 Example

Let  o X  , and  o Y , be two NTSs (a) If YXf  : is continuous in the usual sense, then in this case, f is continuous in the sense of Definition 5.1 too. Here we consider the NTs on X and Y, respectively, as follows :  o c GG G    :,0, 1 and

Florentin Smarandache (author and editor) Collected Papers, IX 26

 :,0, 2 , In this case we have, for each 2 ,0,   c HH , o H  , )( ), ), ( ,0, 11 1 1 c H H c HHfff f     1 1 )(( ), 0(,     c H fff

(b) If YXf  : is neutrosophic open in the usual sense, then in this case, f is neutrosophic open in the sense of Definition 3.2. Now we obtain some characterizations of neutrosophic continuity:

4.1 Proposition

Let ),(),(: 2 1 YXf  . f is neutrosop continuous iff the preimage of each NS (neutrosophic closed set) in 2 is a NS in 2 .

4.2 Proposition

The following are equivalent to each other: (a) ),(),(: 2 1 YXf  is neutrosophic continuous . (b) )) (()(( 1 1 BfNIntBNIntf  for each CNS B in Y. (c) )) (( )) (( 1 1 B NCl fBf NCl  for each NCB in Y.

4.2 Example

Let  2 ,  Y be a NTS and YXf  : be a function. In this case  2 1 1 :)(     HHf is a NT on X. Indeed, it is the coarsest NT on X which makes the function YXf  : continuous. One may call it the initial neutrosophic crisp topology with respect to . f

4.4 Definition

Let (,) and (,S) be two neutrosophic topological space, then (a)A map  : (,)  (,S) is called N continuous (in short N continuous) if the inverse image of every closed set in (,S) is Neutrosophic closedin (,).

(b)A map :(,) (,S) is called neutrosophic gc irresolute if the inverse image of every Neutrosophic closedset in (,S) is Neutrosophic closedin (,). Equivalently if the inverse image of every Neutrosophic open set in (,S) is Neutrosophic open in (,).

(F)A map :(,)(,S) is said to be perfectly N continuous if the inverse image of every Neutrosophic open set in (,S) is both neutrosophic open and neutrosophic closed in (,).

4.3 Proposition

Let (,) and (,S) be any two neutrosophic topological spaces. Let  : (,)  (,S) be generalized neutrosophic continuous. Then for every neutrosophic set A in , (Ncl(A))  Ncl((A)).

4.4 Proposition

Let (,) and (,S) be any two neutrosophic topological spaces. Let  : (,)  (,S) be generalized neutrosophic continuous. Then for every neutrosophic set A in , Ncl( 1(A))   1(Ncl(A)).

4.5 Proposition

Let (,) and (,S) be any two neutrosophic topological spaces. If A is a Neutrosophic closedset in (,) and if  : (,)  (,S) is neutrosophic continuous and neutrosophic closed then (A) is Neutrosophic closedin (,S).

Proof.

Let G be a neutrosophic-open in (,S). If (A)  G, then A   1(G) in (,). Since A is neutrosophic closedand  1(G)is neutrosophic open in (,), Ncl(A)   1(G), (i.e) (Ncl(A)G. Now by assumption, (Ncl(A)) is neutrosophic closed and Ncl((A))  Ncl((Ncl(A))) = (Ncl(A))  G. Hence, (A) is N closed.

4.5 Proposition

Let (,) and (,S) be any two neutrosophic topological spaces, If  : (,)  (,S) is neutrosophic continuous then it is N continuous.

The converse of proposition 4.5 need not be true. See Example 4.3.

4.3 Example

Let  =a,b,c and  =a,b,c. Define neutrosophic sets A and B as follows A = 50.4,0.4,0.,30.2,0.4,0.,.5)(0.4,0.4,0 B = 6.0,5.0,4.030.3,0.2,0.,60.4,0.5,0.,

(c)A map :(,) (,S) is said to be strongly neutrosophic continuous if  1(A) is both neutrosophic open and neutrosophic closed in (,) for each neutrosophic set A in (,S).

(d)A map  : (,)  (,S) is said to be perfectly neutrosophic continuous if  1 (A) is both neutrosophic open and neutrosophic closed in (,) for each neutrosophic open set A in (,S).

(e)A map :(,)(,S) is said to be strongly N continuous if the inverse image of every Neutrosophic open set in (,S) is neutrosophic open in (,).

Then the family  = 0N,1N, A is a neutrosophic topology on  and S = 0N,1N, B is a neutrosophic topology on . Thus (,) and (,S) are neutrosophic topological spaces. Define  : (,)  (,S) as (a) = b , (b) = a, (c) = c. Clearly f is N continuous. Now  is not neutrosophic continuous, since  1(B)   for B  S.

4.4 Example

Let  = a,b,c. Define the neutrosophic sets A and B as follows.

A = 4.0,5.0,4.050.5,0.5,0.,4.0,5.0,4.0,



 o c HH H



Florentin Smarandache (author and editor) Collected Papers, IX 27

B = 50.3,0.4,0.,50.3,0.4,0. and C = 50.4,0.5,0.,50.5,0.5,0.  = 0N,1N, A ,B and S = 0N, 1N, C are neutrosophic topologies on  Thus (,) and (,S) are neutrosophic topological spaces. Define : (,)  (,S) as follows (a) = b, (b) = b, (c) = c. Clearly  is N continuous. Since D = 7.0,6.0,6.0,.3)(0.4,0.4,0,.7)(0.6,0.6,0 is neutrosophic open in (,S),  1(D) is not neutrosophic open in (,).

4.6

Proposition

Let (,) and (,S) be any two neutrosophic topological space. If  : (,)  (,S) is strongly N continuous then  is neutrosophic continuous. The converse of Proposition 3.19 is not true. See Example 3.3

4.5 Example

Let  =a,b,c. Define the neutrosophic sets A and B as follows.

A = ,.1)(0.1,0.1,0,.9)(0.9,0.9,0 B = ,)(0.1,0.1,0,.8)(0.9,0.1,0 and C = ,)(0.1,0,0.1,.9)(0.9,0.9,0

4.8 Proposition

Let (,) and (,S) be any neutrosophic topological spaces. If : (,)  (,S) is strongly neutrosophic continuous then  is strongly N continuous.

The converse of proposition 3.23 is not true. See Example 4.7

4.7 Example

Let  = a,b,c and Define the neutrosophic sets A and B as follows.

A = .9)(0.9,0.9,0,.1)(0.1,0.1,0,.9)(0.9,0.9,0 B = ,0.99),(0.01,0,0),(0.99,0.99,0.99) (0.99,0.99 and C = .9)(0.9,0.9,0,(0.1,0.1,0.05),.9)(0.9,0.9,0  = 0N, 1N, A ,B and S = 0N, 1N, C are neutrosophic topologies on . Thus (,) and (ٍٍ,S) are neutrosophic topological spaces. Also define  : (,)  (,S) as follows: (a) = a, (b) = (c) = b. Clearly  is strongly N continuous. But  is not strongly neutrosophic continuous. Since D = )9.0,9.0,9.0(,)(0.1,0.1,0,.9)(0.9,0.9,0 be a neutrosophic set in (,S),  1(D) is neutrosophic open and not neutrosophic closed in (,).

4.9 Proposition

 = 0N, 1N, A ,B and S = 0N, 1N, C are neutrosophic topologies on . Thus (,) and (,S) are neutrosophic topological spaces. Also define  :(,) (,S) as follows (a) = a, (b) = c, (c) = b. Clearly  is neutrosophic continuous. But  is not strongly N continuous. Since

D =  .0,9.0,9.099,(0.05,0,0.01),(0.9,0.9,0.99) Is an Neutrosophic open set in (,S),  1(D) is not neutrosophic open in (,).

4.7 Proposition

Let (,) and (, S) be any two neutrosophic topological spaces. If : (,)  (,S) is perfectly N continuous then  is strongly N continuous.

The converse of Proposition 4.7 is not true. See Example 4.6

4.6 Example

Let  = a,b,c. Define the neutrosophic sets A and B as follows.

A = ,.1)(0.1,0.1,0,.9)(0.9,0.9,0

B = ,0.99),(0.01,0,0),(0.99,0.99,0.99) (0.99,0.99

And C = .9)(0.9,0.9,0,(0.1,0.1,0.05),.9)(0.9,0.9,0

 = 0N,1N, A ,B and S = 0N ,1N, C are neutrosophic topologies space on . Thus (,) and (ٍٍ,S) are neutrosophic topological spaces. Also define  : (,)  (,S) as follows (a) = a, (b) = (c) = b. Clearly  is strongly N continuous. But  is not perfectly N continuous. Since D = )9.0,9.0,9.0(,)(0.1,0.1,0,.9)(0.9,0.9,0

Is an Neutrosophic open set in (,S),  1(D) is neutrosophic open and not neutrosophic closed in (,).

Let (,),(,S) and (,R) be any three neutrosophic topological spaces. Suppose  : (,)  (,S), g : (,S)  (,R) be maps. Assume  is neutrosophic gc irresolute and g is N continuous then g   is N continuous.

4.10 Proposition

Let (,), (,S) and (,R) be any three neutrosophic topological spaces. Let  : (,)  (,S), g : (,S)  (,R) be map, such that  is strongly N continuous and g is N continuous. Then the composition g   is neutrosophic continuous.

4.5 Definition

A neutrosophic topological space (,) is said to be neutrosophic 1/2 if every Neutrosophic closed set in (,) is neutrosophic closed in (,).

4.11 Proposition

Let (,),(,S) and (,R) be any neutrosophic topological spaces. Let  : (,)  (,S) and g : (,S)  (,R) be mapping and (,S) be neutrosophic 1/2 if  and g are N continuous then the composition g   is N continuous.

The proposition 4.11 is not valid if (,S) is not neutrosophic 1/2

4.8 Example

Let  = a,b,c. Define the neutrosophic sets A,B and C as follows.

A = .6)(0.4,0.4,0,.3)(0.4,0.4,0

B = .6)(0.4,0.5,0,.3)(0.3,0.4,0 and C = .5)(0.4,0.6,0,.4)(0.5,0.3,0

Florentin Smarandache (author and editor) Collected Papers, IX 28

Then the family  = 0N,1N, A, S = 0N,1N, B and R = 0N, 1N, C are neutrosophic topologies on . Thus (,),(,S) and (,R) are neutrosophic topological spaces. Also define  : (,)  (,S) as (a) = b, (b) = a, (c) = c and g : (,S)  (,R) as g(a) = b, g(b) = c, g(c) = b. Clearly  and g are N continuous function. But g   is not N continuous. For 1 C is neutrosophic closed in (,R).  1(g 1(1 C)) is not N closed in (,). g   is not N continuous.

References

[1] S. A. Alblowi, A. A. Salama and Mohmed Eisa, New Concepts of Neutrosophic Sets, International Journal of Mathematics and Computer Applications Research (IJMCAR),Vol. 3, Issue 3, Oct (2013) 95-102.

[2] I. Hanafy, A.A. Salama and K. Mahfouz, Correlation of Neutrosophic Data, International Refereed Journal of Engineering and Science (IRJES), Vol.(1), Issue 2 .(2012) PP.39-33

[3] I.M. Hanafy, A.A. Salama and K.M. Mahfouz,," Neutrosophic Classical Events and Its Probability" International Journal of Mathematics and Computer Applications Research (IJMCAR) Vol.(3),Issue 1, Mar (2013) pp171-178.

[4] A.A. Salama and S.A. Alblowi, "Generalized Neutrosophic Set and Generalized Neutrosophic Topological Spaces,"Journal Computer Sci. Engineering, Vol. (2) No. (7) (2012)pp 129-132 .

[5] A.A. Salama and S.A. Alblowi, Neutrosophic Set and Neutrosophic Topological Spaces, ISORJ. Mathematics, Vol.(3), Issue(3), (2012) pp-31-35.

[6] A. A. Salama, "Neutrosophic Crisp Points & Neutrosophic Crisp Ideals", Neutrosophic Sets and Systems, Vol.1, No. 1, (2013) pp. 50-54.

[7] A. A Salama and F. Smarandache, "Filters via Neutrosophic Crisp Sets", Neutrosophic Sets and Systems, Vol.1, No. 1, (2013) pp. 34-38.

[8] A.A. Salama, and H.Elagamy, "Neutrosophic Filters" International Journal of Computer Science Engineering and Information Technology Reseearch (IJCSEITR), Vol.3, Issue(1),Mar 2013,(2013) pp 307-312.

[9] A.A. Salama and S.A. Alblowi, Intuitionistic Fuzzy Ideals Spaces, Advances in Fuzzy Mathematics , Vol.(7), Number 1, (2012) pp. 51- 60.

[10] A. A. Salama, F.Smarandache and Valeri Kroumov "Neutrosophic Crisp Sets & Neutrosophic Crisp Topological Spaces" Bulletin of the Research Institute of Technology (Okayama University of Science, Japan), in January-February (2014). (Accepted)

[11] Florentin Smarandache, Neutrosophy and Neutrosophic Logic, First International Conference on Neutrosophy , Neutrosophic Logic , Set, Probability, and Statistics University of New Mexico, Gallup, NM 87301, USA(2002).

[12] F. Smarandache. A Unifying Field in Logics: Neutrosophic Logic. Neutrosophy, Neutrosophic Set, Neutrosophic Probability. American Research Press,

Rehoboth, NM, (1999).

[13] L.A. Zadeh, Fuzzy Sets, Inform and Control 8, (1965) 338-353.

Florentin Smarandache (author and editor) Collected Papers, IX 29

Soft neutrosophic semigroups and their generalization

Mumtaz Ali, Muhammad Shabir, Munazza Naz, Florentin Smarandache

Mumtaz Ali, Muhammad Shabir, Munazza Naz, Florentin Smarandache (2014). Soft neutrosophic semigroups and their generalization. Scientia Magna 10(1), 93-111

Abstract Soft set theory is a general mathematical tool for dealing with uncertain, fuzzy, not clearly deﬁned objects. In this paper we introduced soft neutrosophic semigroup,soft neutosophic bisemigroup, soft neutrosophic N -semigroup with the discuissionf of some of their characteristics. We also introduced a new type of soft neutrophic semigroup, the so called soft strong neutrosophic semigoup which is of pure neutrosophic character. This notion also foound in all the other corresponding notions of soft neutrosophic thoery. We also given some of their properties of this newly born soft structure related to the strong part of neutrosophic theory Keywords Neutrosophic semigroup, neutrosophic bisemigroup, neutrosophic N -semigroup, soft set, soft semigroup, soft neutrosophic semigroup, soft neutrosophic bisemigroup, soft neutrosophic N -semigroup.

§1.Introductionandpreliminaries

FlorentineSmarandacheforthefirsttimeintroducedtheconceptofneutrosophyin1995, whichisbasicallyanewbranchofphilosophywhichactuallystudiestheorigin,nature,and scopeofneutralities.Theneutrosophiclogiccameintobeingbyneutrosophy.Inneutrosophiclogiceachpropositionisapproximatedtohavethepercentageoftruthinasubset T , thepercentageofindeterminacyinasubset I,andthepercentageoffalsityinasubset F Neutrosophiclogicisanextensionoffuzzylogic.Infacttheneutrosophicsetisthegeneralizationofclassicalset,fuzzyconventionalset,intuitionisticfuzzyset,andintervalvaluedfuzzy set.Neutrosophiclogicisusedtoovercometheproblemsofimpreciseness,indeterminate,and inconsistenciesofdateetc.Thetheoryofneutrosophyissoapplicabletoeveryfieldofalgebra.W.B.VasanthaKandasamyandFlorentinSmarandacheintroducedneutrosophicfields, neutrosophicrings,neutrosophicvectorspaces,neutrosophicgroups,neutrosophicbigroupsand neutrosophic N -groups,neutrosophicsemigroups,neutrosophicbisemigroups,andneutrosophic

Smarandache (author and editor) Collected Papers, IX 30

Florentin

N -semigroups,neutrosophicloops,nuetrosophicbiloops,andneutrosophic N -loops,andsoon. Mumtazalietal.introducednuetrosophic LA-semigroups.

Molodtsovintroducedthetheoryofsoftset.Thismathematicaltoolisfreefromparameterizationinadequacy,syndromeoffuzzysettheory,roughsettheory,probabilitytheoryandso on.Thistheoryhasbeenappliedsuccessfullyinmanyﬁeldssuchassmoothnessoffunctions, gametheory,operationresearch,Riemannintegration,Perronintegration,andprobability.Recentlysoftsettheoryattainedmuchattentionoftheresearcherssinceitsappearanceandthe workbasedonseveraloperationsofsoftsetintroducedin[2, 9, 10].Somepropertiesandalgebra maybefoundin[1] Fengetal.introducedsoftsemiringsin[5].Bymeansoflevelsoftsets anadjustableapproachtofuzzysoftsetcanbeseenin[6].Someotherconceptstogetherwith fuzzysetandroughsetwereshownin[7, 8].

Thispaperisabouttointroducedsoftnuetrosophicsemigroup,softneutrosophicgroup, andsoftneutrosophic N -semigroupandtherelatedstrongorpurepartofneutrosophywiththe notionsofsoftsettheory.Intheproceedingsection,wedeﬁnesoftneutrosophicsemigroup,soft neutrosophicstrongsemigroup,andsomeoftheirpropertiesarediscussed.Inthenextsection, softneutrosophicbisemigrouparepresentedwiththeirstrongneutrosophicpart.Alsointhis sectionsomeoftheircharacterizationhavebeenmade.Inthelastsectionsoftneutrosophic N -semigroupandtheircorrespondingstrongtheoryhavebeenconstructedwithsomeoftheir properties.

§2.Deﬁnitionandproperties

Deﬁnition2.1. Let S beasemigroup,thesemigroupgeneratedby S and I i.e. S ∪ I denotedby S ∪ I isdeﬁnedtobeaneutrosophicsemigroupwhere I isindeterminacyelement andtermedasneutrosophicelement.

Itisinterestingtonotethatallneutrosophicsemigroupscontainapropersubsetwhichis asemigroup.

Example2.1. Let Z = {thesetofpositiveandnegativeintegerswithzero}, Z isonly asemigroupundermultiplication.Let N (S)= { Z ∪ I } betheneutrosophicsemigroupunder multiplication.Clearly Z ⊂ N (S)isasemigroup.

Deﬁnition2.2. Let N (S)beaneutrosophicsemigroup.Apropersubset P of N (S)is saidtobeaneutrosophicsubsemigroup,if P isaneutrosophicsemigroupundertheoperations of N (S).Aneutrosophicsemigroup N (S)issaidtohaveasubsemigroupif N (S)hasaproper subsetwhichisasemigroupundertheoperationsof N (S).

Theorem2.1. Let N (S)beaneutrosophicsemigroup.Suppose P1 and P2 beanytwo neutrosophicsubsemigroupsof N (S)then P1 ∪P2 (i.e.theunion)theunionoftwoneutrosophic subsemigroupsingeneralneednotbeaneutrosophicsubsemigroup.

Deﬁnition2.3. Aneutrosophicsemigroup N (S)whichhasanelement e in N (S)such that e ∗ s = s ∗ e = s forall s ∈ N (S),iscalledasaneutrosophicmonoid.

Deﬁnition2.4. Let N (S)beaneutrosophicmonoidunderthebinaryoperation ∗ Suppose e istheidentityin N (S),thatis s ∗ e = e ∗ s = s forall s ∈ N (S).Wecallaproper subset P of N (S)tobeaneutrosophicsubmonoidif

Florentin Smarandache (author and editor) Collected Papers, IX 31

1. P isaneutrosophicsemigroupunder ∗

2. e ∈ P ,i.e., P isamonoidunder ∗.

Deﬁnition2.5. Let N (S)beaneutrosophicsemigroupunderabinaryoperation ∗. P beapropersubsetof N (S). P issaidtobeaneutrosophicidealof N (S)ifthefollowing conditionsaresatisﬁed.

1. P isaneutrosophicsemigroup.

2.Forall p ∈ P andforall s ∈ N (S)wehave p ∗ s and s ∗ p arein P

Deﬁnition2.6. Let N (S)beaneutrosophicsemigroup. P beaneutrosophicidealof N (S), P issaidtobeaneutrosophiccyclicidealorneutrosophicprincipalidealif P canbe generatedbyasingleelement.

Deﬁnition2.7. Let(BN (S), ∗,o)beanonemptysetwithtwobinaryoperations ∗ and o.(BN (S), ∗,o)issaidtobeaneutrosophicbisemigroupif BN (S)= P 1 ∪ P 2whereatleast oneof(P 1, ∗)or(P 2,o)isaneutrosophicsemigroupandotherisjustasemigroup. P 1and P 2 arepropersubsetsof BN (S),i.e. P 1 P 2.

Ifboth(P 1, ∗)and(P 2,o)intheabovedeﬁnitionareneutrosophicsemigroupsthenwe call(BN (S), ∗,o)astrongneutrosophicbisemigroup.Allstrongneutrosophicbisemigroupsare triviallyneutrosophicbisemigroups.

Example2.2. Let(BN (S), ∗,o)= {0, 1, 2, 3,I, 2I, 3I,S(3), ∗,o} =(P1, ∗) ∪ (P2,o) where(P1, ∗)= {0, 1, 2, 3,I, 2I, 3I, ∗} and(P2,o)=(S(3),o).Clearly(P1, ∗)isaneutrosophic semigroupundermultiplicationmodulo4.(P2,o)isjustasemigroup.Thus(BN (S), ∗,o)isa neutrosophicbisemigroup.

Deﬁnition2.8. Let(BN (S)= P 1 ∪ P 2; o, ∗)beaneutrosophicbisemigroup.Aproper subset(T,o, ∗)issaidtobeaneutrosophicsubbisemigroupof BN (S)if

1. T = T 1 ∪ T 2where T 1= P 1 ∩ T and T 2= P 2 ∩ T

2.Atleastoneof(T 1,o)or(T 2, ∗)isaneutrosophicsemigroup.

Deﬁnition2.9. Let(BN (S)= P1 ∪ P2,o, ∗)beaneutrosophicstrongbisemigroup.A propersubset T of BN (S)iscalledthestrongneutrosophicsubbisemigroupif T = T1 ∪ T2 with T1 = P1 ∩ T and T2 = P2 ∩ T andifboth(T1, ∗)and(T2,o)areneutrosophicsubsemigroupsof (P1, ∗)and(P2,o)respectively.Wecall T = T1 ∪T2 tobeaneutrosophicstrongsubbisemigroup, ifatleastoneof(T1, ∗)or(T2,o)isasemigroupthen T = T1 ∪ T2 isonlyaneutrosophic subsemigroup.

Deﬁnition2.10. Let(BN (S)= P1 ∪ P2∗,o)beanyneutrosophicbisemigroup.Let J beapropersubsetof B(NS)suchthat J1 = J ∩ P1 and J2 = J ∩ P2 areidealsof P1 and P2 respectively.Then J iscalledtheneutrosophicbi-idealof BN (S).

Deﬁnition2.11. Let(BN (S), ∗,o)beastrongneutrosophicbisemigroupwhere BN (S)= P1 ∪ P2 with(P1, ∗)and(P2,o)beanytwoneutrosophicsemigroups.Let J beapropersubset of BN (S)where I = I1 ∪ I2 with I1 = J ∩ P1 and I2 = J ∩ P2 areneutrosophicidealsof theneutrosophicsemigroups P1 and P2 respectively.Then I iscalledordeﬁnedasthestrong neutrosophicbi-idealof B(N (S)).

Unionofanytwoneutrosophicbi-idealsingeneralisnotaneutrosophicbi-ideal.Thisis trueofneutrosophicstrongbi-ideals.

Deﬁnition2.12. Let {S(N ), ∗1,..., ∗N } beanonemptysetwith N -binaryoperations

Florentin Smarandache (author and editor) Collected Papers, IX 32

deﬁnedonit.Wecall S(N )aneutrosophic N -semigroup(N apositiveinteger)ifthefollowing conditionsaresatisﬁed.

1. S(N )= S1 ∪ ∪ SN whereeach Si isapropersubsetof S(N )i.e. Si Sj or Sj Si if i = j

2.(Si, ∗i)iseitheraneutrosophicsemigrouporasemigroupfor i =1, 2,...,N Ifallthe N -semigroups(Si, ∗i)areneutrosophicsemigroups(i.e.for i =1, 2,...,N )then wecall S(N )tobeaneutrosophicstrong N -semigroup.

Example2.3. Let S(N )= {S1 ∪S2 ∪S3 ∪S4, ∗1, ∗2, ∗3, ∗4} beaneutrosophic4-semigroup where

S1 = {Z12, semigroupundermultiplicationmodulo12}

S2 = {0, 1, 2, 3,I, 2I, 3I,semigroupundermultiplicationmodulo4},aneutrosophicsemigroup.

,neutrosophicsemigroupundermatrixmultiplicationand S4 = Z ∪ I ,neutrosophicsemigroupundermultiplication.

Deﬁnition2.13. Let S(N )= {S1 ∪ S2 ∪ ∪ SN , ∗1,..., ∗N } beaneutrosophic N semigroup.Apropersubset P = {P1 ∪ P2 ∪ ∪ PN , ∗1, ∗2,..., ∗N } of S(N )issaidtobea neutrosophicNsubsemigroupif Pi = P ∩ Si,i =1, 2,...,N aresubsemigroupsof Si inwhich atleastsomeofthesubsemigroupsareneutrosophicsubsemigroups.

Deﬁnition2.14. Let S(N )= {S1 ∪ S2 ∪ ... ∪ SN , ∗1,..., ∗N } beaneutrosophicstrong N -semigroup.Apropersubset T = {T1 ∪ T2 ∪ ... ∪ TN , ∗1,..., ∗N } of S(N )issaidtobea neutrosophicstrongsub N -semigroupifeach(Ti, ∗i)isaneutrosophicsubsemigroupof(Si, ∗i) for i =1, 2,...,N where Ti = T ∩ Si.

Ifonlyafewofthe(Ti, ∗i)in T arejustsubsemigroupsof(Si, ∗i)(i.e.(Ti, ∗i)arenot neutrosophicsubsemigroupsthenwecall T tobeasub N -semigroupof S(N ).

Deﬁnition2.15. Let S(N )= {S1 ∪ S2 ∪ ... ∪ SN , ∗1,..., ∗N } beaneutrosophic N semigroup.Apropersubset P = {P1 ∪ P2 ∪ ∪ PN , ∗1,..., ∗N } of S(N )issaidtobea neutrosophic N -subsemigroup,ifthefollowingconditionsaretrue,

i. P isaneutrosophicsub N -semigroupof S(N ).

ii.Each Pi = P ∩ Si,i =1, 2,...,N isanidealof Si Then P iscalledordeﬁnedastheneutrosophic N -idealoftheneutrosophic N -semigroup S(N ).

Deﬁnition2.16. Let S(N )= {S1 ∪ S2 ∪ ∪ SN , ∗1,..., ∗N } beaneutrosophicstrong N -semigroup.Apropersubset J = {I1 ∪ I2 ∪ ∪ IN } where It = J ∩ St for t =1, 2,...,N is saidtobeaneutrosophicstrong N -idealof S(N )ifthefollowingconditionsaresatisﬁed.

1.Eachisaneutrosophicsubsemigroupof St,t =1, 2,...,N i.e.Itisaneutrosophic strongN-subsemigroupof S(N ).

2.Eachisatwosidedidealof St for t =1, 2,...,N

Similarlyonecandeﬁneneutrosophicstrong N -leftidealorneutrosophicstrongrightideal of S(N ).

Aneutrosophicstrong N -idealisonewhichisbothaneutrosophicstrong N -leftidealand N -rightidealof S(N ).

 



  

S3 =   

ab cd 

; a,b,c,d ∈ R ∪ I

Florentin Smarandache (author and editor) Collected Papers, IX 33

Throughoutthissubsection U referstoaninitialuniverse, E isasetofparameters, P (U ) isthepowersetof U ,and A ⊂ E.Molodtsov [12] deﬁnedthesoftsetinthefollowingmanner: Deﬁnition2.17. Apair(F,A)iscalledasoftsetover U where F isamappinggiven by F : A −→ P (U ).

Inotherwords,asoftsetover U isaparameterizedfamilyofsubsetsoftheuniverse U For e ∈ A, F (e)maybeconsideredasthesetof e-elementsofthesoftset(F,A),orastheset ofe-approximateelementsofthesoftset.

Example2.4. Supposethat U isthesetofshops. E isthesetofparametersandeach parameterisawordorsenctence.Let E={highrent,normalrent,ingoodcondition,inbad condition}.Letusconsiderasoftset(F,A)whichdescribestheattractivenessofshopsthatMr. Z istakingonrent.Supposethatthereareﬁvehousesintheuniverse U = {h1,h2,h3,h4,h5} underconsideration,andthat A = {e1,e2,e3} bethesetofparameterswhere

e1 standsfortheparameterhighrent.

e2 standsfortheparameternormalrent.

e3 standsfortheparameteringoodcondition.

Supposethat

F (e1)= {h1,h4}

F (e2)= {h2,h5}.

F (e3)= {h3,h4,h5}

Thesoftset(F,A)isanapproximatedfamily {F (ei),i =1, 2, 3} ofsubsetsoftheset U whichgivesusacollectionofapproximatedescriptionofanobject.Thus,wehavethesoftset (F,A)asacollectionofapproximationsasbelow:

(F,A)= {highrent= {h1,h4}, normalrent= {h2,h5}, ingoodcondition= {h3,h4,h5}}

Deﬁnition2.18. Fortwosoftsets(F,A)and(H,B)over U ,(F,A)iscalledasoftsubset of(H,B)if

1. A ⊆ B

2. F (e) ⊆ G(e),forall e ∈ A

Thisrelationshipisdenotedby(F,A) ∼ ⊂ (H,B).Similarly(F,A)iscalledasoftsuperset of(H,B)if(H,B)isasoftsubsetof(F,A)whichisdenotedby(F,A) ∼ ⊃ (H,B).

Deﬁnition2.19. Twosoftsets(F,A)and(H,B)over U arecalledsoftequalif(F,A) isasoftsubsetof(H,B)and(H,B)isasoftsubsetof(F,A).

Deﬁnition2.20. (F,A)over U iscalledanabsolutesoftsetif F (e)= U forall e ∈ A andwedenoteitby U.

Deﬁnition2.21. Let(F,A)and(G,B)betwosoftsetsoveracommonuniverse U such that A ∩ B = φ.Thentheirrestrictedintersectionisdenotedby(F,A) ∩R (G,B)=(H,C) where(H,C)isdeﬁnedas H(c)= F (c) ∩ G(c)forall c ∈ C = A ∩ B.

Deﬁnition2.22. Theextendedintersectionoftwosoftsets(F,A)and(G,B)overa commonuniverse U isthesoftset(H,C),where C = A ∪ B,andforall e ∈ C, H(e)isdeﬁned as

Florentin Smarandache (author and editor) Collected Papers, IX 34

H(e)=       

F (e), if e ∈ A B, G(e), if e ∈ B A, F (e) ∩ G(e), if e ∈ A ∩ B

Wewrite(F,A) ∩ε (G,B)=(H,C).

Deﬁnition2.23. Therestictedunionoftwosoftsets(F,A)and(G,B)overacommon universe U isthesoftset(H,C),where C = A ∪ B,andforall e ∈ C, H(e)isdeﬁnedasthe softset(H,C)=(F,A) ∪R (G,B)where C = A ∩ B and H(c)= F (c) ∪ G(c)forall c ∈ C

Deﬁnition2.24. Theextendedunionoftwosoftsets(F,A)and(G,B)overacommon universe U isthesoftset(H,C),where C = A ∪ B,andforall e ∈ C, H(e)isdeﬁnedas H(e)=        F (e), if e ∈ A B, G(e), if e ∈ B A, F (e) ∪ G(e), if e ∈ A ∩ B.

Wewrite(F,A) ∪ε (G,B)=(H,C). Deﬁnition2.25. Asoftset(F,A)over S iscalledasoftsemigroupover S if(F,A) ◦ (F,A) ⊆ (F,A).

Itiseasytoseethatasoftset(F,A)over S isasoftsemigroupifandonlyif φ = F (a)is asubsemigroupofS.

Deﬁnition2.26. Asoftset(F,A)overasemigroup S iscalledasoftleft(right)ideal over S,if(S,E) ⊆ (F,A) , ((F,A) ⊆ (S,E)) .

Asoftsetover S isasoftidealifitisbothasoftleftandasoftrightidealover S. Proposition2.1. Asoftset(F,A)over S isasoftidealover S ifandonlyif φ = F (a) isanidealof S.

Deﬁnition2.27. Let(G,B)beasoftsubsetofasoftsemigroup(F,A)over S,then (G,B)iscalledasoftsubsemigroup(ideal)of(F,A)if G (b)isasubsemigroup(ideal)of F (b) forall b ∈ A

§3.Softneutrosophicsemigroup

Deﬁnition3.1. Let N (S)beaneutrosophicsemigroupand(F,A)beasoftsetover N (S).Then(F,A)iscalledsoftneutrosophicsemigroupifandonlyif F (e)isneutrosophic subsemigroupof N (S),forall e ∈ A

Equivalently(F,A)isasoftneutrosophicsemigroupover N (S)if(F,A) ◦ (F,A) ⊆ (F,A), where N(N (S),A) =(F,A) = ∼ φ.

Example3.1. Let N (S)= Z + ∪{0}+ ∪{I} beaneutrosophicsemigroupunder +.Consider P = 2Z + ∪ I and R = 3Z + ∪ I areneutrosophicsubsemigroupof N (S). Thenclearlyforall e ∈ A,(F,A)isasoftneutrosophicsemigroupover N (S),where F (x1)= { 2Z + ∪ I }, F (x2)= { 3Z + ∪ I }

Theorem3.1. Asoftneutrosophicsemigroupover N (S)alwayscontainasoftsemigroup over S.

Florentin Smarandache (author and editor) Collected Papers, IX 35

Proof. Theproofofthistheoremisstraightforward.

Theorem3.2. Let(F,A)and(H,A)betwosoftneutrosophicsemigroupsover N (S). Thentheirintersection(F,A) ∩ (H,A)isagainsoftneutrosophicsemigroupover N (S). Proof. Theproofisstaightforward.

Theorem3.3. Let(F,A)and(H,B)betwosoftneutrosophicsemigroupsover N (S). If A ∩ B = φ,then(F,A) ∪ (H,B)isasoftneutrosophicsemigroupover N (S).

Remark3.1. Theextendedunionoftwosoftneutrosophicsemigroups(F,A)and(K,B) over N (S)isnotasoftneutrosophicsemigroupover N (S).

Wetakethefollowingexamplefortheproofofaboveremark.

Example3.2. Let N (S)= Z + ∪ I betheneutrosophicsemigroupunder+.Take P1 = { 2Z + ∪ I } and P2 = { 3Z + ∪ I } tobeanytwoneutrosophicsubsemigroupsof N (S). Thenclearlyforall e ∈ A,(F,A)isasoftneutrosophicsemigroupover N (S),where F (x1)= { 2Z + ∪ I } ,F (x2)= { 3Z + ∪ I }

AgainLet R1 = { 5Z + ∪ I } and R2 = { 4Z + ∪ I } beanotherneutrosophicsubsemigroups of N (S)and(K,B)isanothersoftneutrosophicsemigroupover N (S),where K(x1)= { 5Z + ∪ I },K(x3)= { 4Z + ∪ I }.

Let C = A ∪ B.Theextendedunion(F,A) ∪ε (K,B)=(H,C)where x1 ∈ C,we have H(x1)= F (x1) ∪ K(x1)isnotneutrosophicsubsemigroupasunionoftwoneutrosophic subsemigroupisnotneutrosophicsubsemigroup.

Proposition3.1. Theextendedintersectionoftwosoftneutrosophicsemigroupsover N (S)issoftneutrosophicsemigruopover N (S).

Remark3.2. Therestrictedunionoftwosoftneutrosophicsemigroups(F,A)and(K,B) over N (S)isnotasoftneutrosophicsemigroupover N (S).

Wecaneasilycheckitinaboveexample.

Proposition3.2. Therestrictedintersectionoftwosoftneutrosophicsemigroupsover N (S)issoftneutrosophicsemigroupover N (S).

Proposition3.3. The AND operationoftwosoftneutrosophicsemigroupsover N (S) issoftneutrosophicsemigroupover N (S).

Proposition3.4. The OR operationoftwosoftneutosophicsemigroupover N (S)may notbeasoftnuetrosophicsemigroupover N (S).

Deﬁnition3.2. Let N (S)beaneutrosophicmonoidand(F,A)beasoftsetover N (S). Then(F,A)iscalledsoftneutrosophicmonoidifandonlyif F (e)isneutrosophicsubmonoid of N (S),forall x ∈ A

Example3.3. Let N (S)= Z ∪ I beaneutrosophicmonoidunder+.Let P = 2Z ∪ I and Q = 3Z ∪ I areneutrosophicsubmonoidsof N (S) Then(F,A)isasoftneutrosophic monoidover N (S),where F (x1)= { 2Z ∪ I } ,F (x2)= { 3Z ∪ I }

Theorem3.4. Everysoftneutrosophicmonoidover N (S)isasoftneutrosophicsemigroupover N (S)buttheconverseisnottrueingeneral. Proof. Theproofisstraightforward.

Proposition3.5. Let(F,A)and(K,B)betwosoftneutrosophicmonoidsover N (S). Then

Florentin Smarandache (author and editor) Collected Papers, IX 36

1.Theirextendedunion(F,A) ∪ε (K,B)over N (S)isnotsoftneutrosophicmonoidover N (S).

2.Theirextendedintersection(F,A) ∩ε (K,B)over N (S)issoftneutrosophicmonoidover N (S).

3.Theirrestrictedunion(F,A) ∪R (K,B)over N (S)isnotsoftneutrosophicmonoidover N (S).

4.Theirrestrictedintersection(F,A) ∩ε (K,B)over N (S)issoftneutrosophicmonoid over N (S).

Proposition3.6. Let(F,A)and(H,B)betwosoftneutrosophicmonoidover N (S). Then

1.Their AND operation(F,A) ∧ (H,B)issoftneutrosophicmonoidover N (S).

2.Their OR operation(F,A) ∨ (H,B)isnotsoftneutrosophicmonoidover N (S).

Deﬁnition3.3. Let(F,A)beasoftneutrosophicsemigroupover N (S),then(F,A)is calledFull-softneutrosophicsemigroupover N (S)if F (x)= N (S),forall x ∈ A.Wedenoteit by N (S).

Theorem3.5. EveryFull-softneutrosophicsemigroupover N (S)alwayscontainabsolute softsemigroupover S. Proof. Theproofofthistheoremisstraightforward.

Deﬁnition3.4. Let(F,A)and(H,B)betwosoftneutrosophicsemigroupsover N (S). Then(H,B)isasoftneutrosophicsubsemigroupof(F,A),if

1. B ⊂ A

2. H(a)isneutrosophicsubsemigroupof F (a),forall a ∈ B

Example3.4. Let N (S)= Z ∪I beaneutrosophicsemigroupunder+.Then(F,A)isa softneutrosophicsemigroupover N (S),where F (x1)= { 2Z ∪ I } ,F (x2)= { 3Z ∪ I } ,F (x3)= { 5Z ∪ I }

Let B = {x1,x2}⊂ A. Then(H,B)issoftneutrosophicsubsemigroupof(F,A)over N (S), where H(x1)= { 4Z ∪ I } ,H(x2)= { 6Z ∪ I } .

Theorem3.6. Asoftneutrosophicsemigroupover N (S)havesoftneutrosophicsubsemigroupsaswellassoftsubsemigroupsover N (S). Proof. Obvious.

Theorem3.7. Everysoftsemigroupover S isalwayssoftneutrosophicsubsemigroupof softneutrosophicsemigroupover N (S) Proof. Theproofisobvious.

Theorem3.8. Let(F,A)beasoftneutrosophicsemigroupover N (S)and {(Hi,Bi); i ∈ I} isanonemptyfamilyofsoftneutrosophicsubsemigroupsof(F,A)then

1. ∩i∈I (Hi,Bi)isasoftneutrosophicsubsemigroupof(F,A).

2. ∧i∈I (Hi,Bi)isasoftneutrosophicsubsemigroupof ∧i∈I (F,A).

3. ∪i∈I (Hi,Bi)isasoftneutrosophicsubsemigroupof(F,A)if Bi ∩ Bj = φ,forall i = j. Proof. Straightforward. Deﬁnition3.5. Asoftset(F,A)over N (S)iscalledsoftneutrosophicleft(right)ideal over N (S)if N (S) ◦ (F,A) ⊆ (F,A),where N(N (S),A) =(F,A) = ∼ φ and N (S)isFull-soft neutrosophicsemigroupover N (S).

Florentin Smarandache (author and editor) Collected Papers, IX 37

Asoftsetover N (S)isasoftneutrosophicidealifitisbothasoftneutrosophicleftanda softneutrosophicrightidealover N (S).

Example3.5. Let N (S)= Z ∪ I betheneutrosophicsemigroupundermultiplication. Let P = 2Z ∪ I and Q = 4Z ∪ I areneutrosophicidealsof N (S).Thenclearly(F,A)isa softneutrosophicidealover N (S),where F (x1)= { 2Z ∪ I } ,F (x2)= { 4Z ∪ I }

Proposition3.7. (F,A)issoftneutrosophicidealifandonlyif F (x)isaneutrosophic idealof N (S),forall x ∈ A.

Theorem3.9. Everysoftneutrosophicideal(F,A)over N (S)isasoftneutrosophic semigroupbuttheconverseisnottrue.

Proposition3.8. Let(F,A)and(K,B)betwosoftneutrosophicidealsover N (S). Then

1.Theirextendedunion(F,A) ∪ε (K,B)over N (S)issoftneutrosophicidealover N (S).

2.Theirextendedintersection(F,A) ∩ε (K,B)over N (S)issoftneutrosophicidealover N (S).

3.Theirrestrictedunion(F,A) ∪R (K,B)over N (S)issoftneutrosophicidealover N (S).

4.Theirrestrictedintersection(F,A) ∩ε (K,B)over N (S)issoftneutrosophicidealover N (S).

Proposition3.9.

1.Let(F,A)and(H,B)betwosoftneutrosophicidealover N (S).

2.Their AND operation(F,A) ∧ (H,B)issoftneutrosophicidealover N (S).

3.Their OR operation(F,A) ∨ (H,B)issoftneutrosophicidealover N (S).

Theorem3.10. Let(F,A)and(G,B)betwosoftsemigroups(ideals)over S and T respectively.Then(F,A) × (G,B)isalsoasoftsemigroup(ideal)over S × T. Proof. Theproofisstraightforward.

Theorem3.11. Let(F,A)beasoftneutrosophicsemigroupover N (S)and {(Hi,Bi); i ∈ I} isanonemptyfamilyofsoftneutrosophicidealsof(F,A)then

1. ∩i∈I (Hi,Bi)isasoftneutrosophicidealof(F,A).

2. ∧i∈I (Hi,Bi)isasoftneutrosophicidealof ∧i∈I (F,A).

3. ∪i∈I (Hi,Bi)isasoftneutrosophicidealof(F,A).

4. ∨i∈I (Hi,Bi)isasoftneutrosophicidealof ∨i∈I (F,A).

Deﬁnition3.6. Asoftset(F,A)over N (S)iscalledsoftneutrosophicprincipalideal orsoftneutrosophiccyclicidealifandonlyif F (x)isaprincipalorcyclicneutrosophicidealof N (S),forall x ∈ A.

Proposition3.10. Let(F,A)and(K,B)betwosoftneutrosophicprincipalidealsover N (S).Then

1.Theirextendedunion(F,A) ∪ε (K,B)over N (S)isnotsoftneutrosophicprincipalideal over N (S).

2.Theirextendedintersection(F,A) ∩ε (K,B)over N (S)issoftneutrosophicprincipal idealover N (S).

3.Theirrestrictedunion(F,A) ∪R (K,B)over N (S)isnotsoftneutrosophicprincipal idealover N (S).

Smarandache (author and editor) Collected Papers, IX 38

Florentin

4.Theirrestrictedintersection(F,A) ∩ε (K,B)over N (S)issoftneutrosophicprincipal idealover N (S).

Proposition3.11. Let(F,A)and(H,B)betwosoftneutrosophicprincipalidealsover N (S).Then

1.Their AND operation(F,A) ∧ (H,B)issoftneutrosophicprincipalidealover N (S).

2.Their OR operation(F,A) ∨ (H,B)isnotsoftneutrosophicprincipalidealover N (S).

§3.Softneutrosophicbisemigroup

Deﬁnition3.1. Let {BN (S) , ∗1, ∗2} beaneutrosophicbisemigroupandlet(F,A)be asoftsetover BN (S).Then(F,A)issaidtobesoftneutrosophicbisemigroupover BN (G)if andonlyif F (x)isneutrosophicsubbisemigroupof BN (G)forall x ∈ A

Example3.1. Let BN (S)= {0, 1, 2,I, 2I, Z ∪ I , ×, +} beaneutosophicbisemigroup. Let T = {0,I, 2I, 2Z ∪ I , ×, +},P = {0, 1, 2, 5Z ∪ I , ×, +} and L = {0, 1, 2,Z, ×, +} are neutrosophicsubbisemigroupof BN (S).The(F,A)isclearlysoftneutrosophicbisemigroup over BN (S),where F (x1)= {0,I, 2I, 2Z ∪I , ×, +},F (x2)= {0, 1, 2, 5Z ∪I , ×, +},F (x3)= {0, 1, 2,Z, ×, +}.

Theorem3.1. Let(F,A)and(H,A)betwosoftneutrosophicbisemigroupover BN (S). Thentheirintersection(F,A) ∩ (H,A)isagainasoftneutrosophicbisemigroupover BN (S). Proof. Straightforward.

Theorem3.2. Let(F,A)and(H,B)betwosoftneutrosophicbisemigroupsover BN (S) suchthat A ∩ B = φ,thentheirunionissoftneutrosophicbisemigroupover BN (S). Proof. Straightforward.

Proposition3.1. Let(F,A)and(K,B)betwosoftneutrosophicbisemigroupsover BN (S).Then

1.Theirextendedunion(F,A) ∪ε (K,B)over BN (S)isnotsoftneutrosophicbisemigroup over BN (S).

2.Theirextendedintersection(F,A)∩ε(K,B)over BN (S)issoftneutrosophicbisemigroup over BN (S).

3.Theirrestrictedunion(F,A)∪R (K,B)over BN (S)isnotsoftneutrosophicbisemigroup over BN (S).

4.Theirrestrictedintersection(F,A) ∩ε (K,B)over BN (S)issoftneutrosophicbisemigroupover BN (S).

Proposition3.2. Let(F,A)and(K,B)betwosoftneutrosophicbisemigroupsover BN (S).Then

1.Their AND operation(F,A) ∧ (K,B)issoftneutrosophicbisemigroupover BN (S).

2.Their OR operation(F,A) ∨ (K,B)isnotsoftneutrosophicbisemigroupover BN (S).

Deﬁnition3.2. Let(F,A)beasoftneutrosophicbisemigroupover BN (S),then(F,A) iscalledFull-softneutrosophicbisemigroupover BN (S)if F (x)= BN (S),forall x ∈ A.We denoteitby BN (S).

Deﬁnition3.3. Let(F,A)and(H,B)betwosoftneutrosophicbisemigroupsover BN (S).Then(H,B)isasoftneutrosophicsubbisemigroupof(F,A),if

Florentin Smarandache (author and editor) Collected Papers, IX 39

1. B ⊂ A

2. H(x)isneutrosophicsubbisemigroupof F (x),forall x ∈ B

Example3.2. Let BN (S)= {0, 1, 2,I, 2I, Z ∪ I , ×, +} beaneutosophicbisemigroup. Let T = {0,I, 2I, 2Z ∪ I , ×, +},P = {0, 1, 2, 5Z ∪ I , ×, +} and L = {0, 1, 2,Z, ×, +} are neutrosophicsubbisemigroupof BN (S).The(F,A)isclearlysoftneutrosophicbisemigroup over BN (S),where F (x1)= {0,I, 2I, 2Z ∪I , ×, +},F (x2)= {0, 1, 2, 5Z ∪I , ×, +},F (x3)= {0, 1, 2,Z, ×, +}

Then(H,B)isasoftneutrosophicsubbisemigroupof(F,A),where H (x1)= {0,I, 4Z ∪ I , ×, +},H (x3)= {0, 1, 4Z, ×, +}

Theorem3.3. Let(F,A)beasoftneutrosophicbisemigroupover BN (S)and {(Hi,Bi); i ∈ I} beanon-emptyfamilyofsoftneutrosophicsubbisemigroupsof(F,A)then

1. ∩i∈I (Hi,Bi)isasoftneutrosophicsubbisemigroupof(F,A).

2. ∧i∈I (Hi,Bi)isasoftneutrosophicsubbisemigroupof ∧i∈I (F,A).

3. ∪i∈I (Hi,Bi)isasoftneutrosophicsubbisemigroupof(F,A)if Bi ∩ Bj = φ,forall i = j Proof. Straightforward.

Theorem3.4. (F,A)iscalledsoftneutrosophicbiidealover BN (S)if F (x)isneutrosophicbiidealof BN (S),forall x ∈ A.

Example3.3. Let BN (S)=({ Z ∪ I , 0, 1, 2,I, 2I, +, ×}(× undermultiplication modulo3)) Let T = { 2Z ∪ I , 0,I, 1, 2I, +, ×} and J = { 8Z ∪ I , {0, 1,I, 2I}, +×} are idealsof BN (S) Then(F,A)issoftneutrosophicbiidealover BN (S),where F (x1)= { 2Z ∪ I , 0,I, 1, 2I, +, ×},F (x2)= { 8Z ∪ I , {0, 1,I, 2I}, +×}

Theorem3.5. Everysoftneutrosophicbiideal(F,A)over BS (N )isasoftneutrosophic bisemigroupbuttheconverseisnottrue.

Proposition3.3. Let(F,A)and(K,B)betwosoftneutrosophicbiidealsover BN (S). Then

1.Theirextendedunion(F,A) ∪ε (K,B)over BN (S)isnotsoftneutrosophicbiidealover BN (S).

2.Theirextendedintersection(F,A) ∩ε (K,B)over BN (S)issoftneutrosophicbiideal over BN (S).

3.Theirrestrictedunion(F,A) ∪R (K,B)over BN (S)isnotsoftneutrosophicbiidealover BN (S).

4.Theirrestrictedintersection(F,A) ∩ε (K,B)over BN (S)issoftneutrosophicbiideal over BN (S).

Proposition3.4. Let(F,A)and(H,B)betwosoftneutrosophicbiidealover BN (S). Then

1.Their AND operation(F,A) ∧ (H,B)issoftneutrosophicbiidealover BN (S).

2.Their OR operation(F,A) ∨ (H,B)isnotsoftneutrosophicbiidealover BN (S).

Theorem3.6. Let(F,A)beasoftneutrosophicbisemigroupover BN (S)and {(Hi,Bi); i ∈ I} isanon emptyfamilyofsoftneutrosophicbiidealsof(F,A)then

1. ∩i∈I (Hi,Bi)isasoftneutrosophicbiidealof(F,A).

2. ∧i∈I (Hi,Bi)isasoftneutrosophicbiidealof ∧i∈I (F,A).

Florentin Smarandache (author and editor) Collected Papers, IX 40

Deﬁnition4.1. Let(F,A)beasoftsetoveraneutrosophicbisemigroup BN (S).Then (F,A)issaidtobesoftstrongneutrosophicbisemigroupover BN (G)ifandonlyif F (x)is neutrosophicstrongsubbisemigroupof BN (G)forall x ∈ A

Example4.1. Let BN (S)= {0, 1, 2,I, 2I, Z ∪ I , ×, +} beaneutrosophicbisemigroup. Let T = {0,I, 2I, 2Z ∪ I , ×, +} and R = {0, 1,I, 4Z ∪ I , ×, +} areneutrosophicstrong subbisemigroupsof BN (S) Then(F,A)issoftneutrosophicstrongbisemigroupover BN (S), where F (x1)= {0,I, 2I, 2Z ∪ I , ×, +},F (x2)= {0,I, 1, 4Z ∪ I , ×, +}.

Theorem4.1. Everysoftneutrosophicstrongbisemigroupisasoftneutrosophicbisemigroupbuttheconverseisnottrue.

Proposition4.1. Let(F,A)and(K,B)betwosoftneutrosophicstrongbisemigroups over BN (S).Then

1.Theirextendedunion(F,A) ∪ε (K,B)over BN (S)isnotsoftneutrosophicstrong bisemigroupover BN (S).

2.Theirextendedintersection(F,A) ∩ε (K,B)over BN (S)issoftneutrosophicstong bisemigroupover BN (S).

3.Theirrestrictedunion(F,A) ∪R (K,B)over BN (S)isnotsoftneutrosophicstong bisemigroupover BN (S).

4.Theirrestrictedintersection(F,A) ∩ε (K,B)over BN (S)issoftneutrosophicstrong bisemigroupover BN (S).

Proposition4.2. Let(F,A)and(K,B)betwosoftneutrosophicstrongbisemigroups over BN (S).Then

1.Their AND operation(F,A) ∧ (K,B)issoftneutrosophicstrongbisemigroupover BN (S).

2.Their OR operation(F,A) ∨ (K,B)isnotsoftneutrosophicstrongbisemigroupover BN (S).

Deﬁnition4.2. Let(F,A)and(H,B)betwosoftneutrosophicstrongbisemigroups over BN (S).Then(H,B)isasoftneutrosophicstrongsubbisemigroupof(F,A),if

1. B ⊂ A

2. H(x)isneutrosophicstrongsubbisemigroupof F (x),forall x ∈ B.

Example4.2. Let BN (S)= {0, 1, 2,I, 2I, Z ∪ I , ×, +} beaneutrosophicbisemigroup. Let T = {0,I, 2I, 2Z ∪ I , ×, +} and R = {0, 1,I, 4Z ∪ I , ×, +} areneutrosophicstrong subbisemigroupsof BN (S) Then(F,A)issoftneutrosophicstrongbisemigroupover BN (S), where F (x1)= {0,I, 2I, 2Z ∪ I , ×, +},F (x2)= {0,I, 4Z ∪ I , ×, +}

Then(H,B)isasoftneutrosophicstrongsubbisemigroupof(F,A),where H (x1)= {0,I, 4Z ∪ I , ×, +}

Theorem4.2. Let(F,A)beasoftneutrosophicstrongbisemigroupover BN (S)and {(Hi,Bi); i ∈ I} beanonemptyfamilyofsoftneutrosophicstrongsubbisemigroupsof(F,A) then

1. ∩i∈I (Hi,Bi)isasoftneutrosophicstrongsubbisemigroupof(F,A).

2. ∧i∈I (Hi,Bi)isasoftneutrosophicstrongsubbisemigroupof ∧i∈I (F,A).

§4.Softneutrosophicstrongbisemigroup

Florentin Smarandache (author and editor) Collected Papers, IX 41

3. ∪i∈I (Hi,Bi)isasoftneutrosophicstrongsubbisemigroupof(F,A)if Bi ∩ Bj = φ,for all i = j

Proof. Straightforward.

Deﬁnition4.3. (F,A)over BN (S)iscalledsoftneutrosophicstrongbiidealif F (x)is neutosophicstrongbiidealof BN (S),forall x ∈ A.

Example4.3. Let BN (S)=({ Z ∪I , 0, 1, 2,I, 2I}, +, ×(× undermultiplicationmodulo 3)) Let T = { 2Z ∪ I , 0,I, 1, 2I, +, ×} and J = { 8Z ∪ I , {0, 1,I, 2I}, +×} areneutrosophic strongidealsof BN (S) Then(F,A)issoftneutrosophicstrongbiidealover BN (S),where F (x1)= { 2Z ∪ I , 0,I, 1, 2I, +, ×},F (x2)= { 8Z ∪ I , {0, 1,I, 2I}, +×}

Theorem4.3. Everysoftneutrosophicstrongbiideal(F,A)over BS (N )isasoft neutrosophicbisemigroupbuttheconverseisnottrue.

Theorem4.4. Everysoftneutrosophicstrongbiideal(F,A)over BS (N )isasoft neutrosophicstrongbisemigroupbuttheconverseisnottrue.

Proposition4.3. Let(F,A)and(K,B)betwosoftneutrosophicstrongbiidealsover BN (S).Then

1.Theirextendedunion(F,A)∪ε (K,B)over BN (S)isnotsoftneutrosophicstrongbiideal over BN (S).

2.Theirextendedintersection(F,A) ∩ε (K,B)over BN (S)issoftneutrosophicstrong biidealover BN (S).

3.Theirrestrictedunion(F,A) ∪R (K,B)over BN (S)isnotsoftneutrosophicstrong biidealover BN (S).

4.Theirrestrictedintersection(F,A) ∩ε (K,B)over BN (S)issoftneutrosophicstong biidealover BN (S).

Proposition4.4. Let(F,A)and(H,B)betwosoftneutrosophicstrongbiidealover BN (S).Then

1.Their AND operation(F,A) ∧ (H,B)issoftneutrosophicstrongbiidealover BN (S).

2.Their OR operation(F,A) ∨ (H,B)isnotsoftneutrosophicstrongbiidealover BN (S).

Theorem4.5. Let(F,A)beasoftneutrosophicstrongbisemigroupover BN (S)and {(Hi,Bi); i ∈ I} isanonemptyfamilyofsoftneutrosophicstrongbiidealsof(F,A)then

1. ∩i∈I (Hi,Bi)isasoftneutrosophicstrongbiidealof(F,A).

2. ∧i∈I (Hi,Bi)isasoftneutrosophicstrongbiidealof ∧i∈I (F,A).

§5.Softneutrosophic N -semigroup

Deﬁnition5.1. Let {S(N ), ∗1,..., ∗N } beaneutrosophic N -semigroupand(F,A)be asoftsetover {S(N ), ∗1,..., ∗N }. Then(F,A)istermedassoftneutrosophic N -semigroupif andonlyif F (x)isneutrosophicsub N -semigroup,forall x ∈ A.

Example5.1. Let S(N )= {S1 ∪S2 ∪S3 ∪S4, ∗1, ∗2, ∗3, ∗4} beaneutrosophic4-semigroup where S1 = {Z12, semigroupundermultiplicationmodulo12} S2 = {0, 1, 2, 3,I, 2I, 3I,semigroupundermultiplicationmodulo4},aneutrosophicsemigroup.

Florentin Smarandache (author and editor) Collected Papers, IX 42

S3 =      ab cd   ; a,b,c,d ∈ R ∪ I    ,neutrosophicsemigroupundermatrixmultiplication. S4 = Z ∪ I ,neutrosophicsemigroupundermultiplication.Let T = {T1 ∪ T2 ∪ T3 ∪ T4, ∗1, ∗2, ∗3, ∗4} isaneutosophicsub4-semigroupof S (4),where T1 = {0, 2, 4, 6, 8, 10}⊆ Z12, T2 = {0,I, 2I, 3I}⊂ S2, T3 =      ab cd   ; a,b,c,d ∈ Q ∪ I    ⊂ S3,T 4= { 5Z ∪ I }⊂ S4, theneutrosophicsemigroupundermultiplication.Alsolet P = {P1 ∪ P2 ∪ P3 ∪ P4, ∗1, ∗2, ∗3, ∗4} beanotherneutrosophicsub4-semigroupof S (4),where P1 = {0, 6}⊆ Z12, P2 = {0, 1,I}⊂ S2, P3 =      ab cd   ; a,b,c,d ∈ Z ∪ I    ⊂ S3,P4 = { 2Z ∪ I }⊂ S4 Then(F,A)issoft neutrosophic4-semigroupover S (4),where F (x1)= {0, 2, 4, 6, 8, 10}∪{0,I, 2I, 3I}∪

∈ Z ∪ I    ∪{ 2Z ∪ I }

Theorem5.1. Let(F,A)and(H,A)betwosoftneutrosophic N -semigroupover S(N ). Thentheirintersection(F,A) ∩ (H,A)isagainasoftneutrosophic N -semigroupover S(N ). Proof. Straightforward. Theorem5.2. Let(F,A)and(H,B)betwosoftneutrosophic N -semigroupsover S(N ) suchthat A ∩ B = φ,thentheirunionissoftneutrosophic N -semigroupover S(N ). Proof. Straightforward.

Proposition5.1. Let(F,A)and(K,B)betwosoftneutrosophic N -semigroupsover S(N ).Then

1.Theirextendedunion(F,A) ∪ε (K,B)over S(N )isnotsoftneutrosophic N -semigroup over S(N ).

2.Theirextendedintersection(F,A)∩ε (K,B)over S(N )issoftneutrosophic N -semigroup over S(N ).

3.Theirrestrictedunion(F,A) ∪R (K,B)over S(N )isnotsoftneutrosophic N -semigroup over S(N ).

4.Theirrestrictedintersection(F,A)∩ε (K,B)over S(N )issoftneutrosophic N -semigroup over S(N ).

Proposition5.2. Let(F,A)and(K,B)betwosoftneutrosophic N -semigroupsover S(N ).Then

1.Their AND operation(F,A) ∧ (K,B)issoftneutrosophic N -semigroupover S(N ).

2.Their OR operation(F,A) ∨ (K,B)isnotsoftneutrosophic N -semigroupover S(N ).

Deﬁnition5.2. Let(F,A)beasoftneutrosophic N -semigroupover S(N ),then(F,A) iscalledFull-softneutrosophic N -semigroupover S(N )if F (x)= S(N ),forall x ∈ A.We denoteitby S(N ).

     ab cd   ; a,b,c,d ∈ Q ∪ I    ∪{ 5Z ∪ I }, F (x2)= {0, 6}∪{0, 1,I}∪      ab cd   ; a,b,c,d

Florentin Smarandache (author and editor) Collected Papers, IX 43

Deﬁnition5.3. Let(F,A)and(H,B)betwosoftneutrosophic N -semigroupsover S(N ).Then(H,B)isasoftneutrosophicsub N -semigroupof(F,A),if

1. B ⊂ A

2. H(x)isneutrosophicsub N -semigroupof F (x),forall x ∈ B Example5.2. Let S(N )= {S1 ∪S2 ∪S3 ∪S4, ∗1, ∗2, ∗3, ∗4} beaneutrosophic4-semigroup where

S1 = {Z12, semigroupundermultiplicationmodulo12} S2 = {0, 1, 2, 3,I, 2I, 3I,semigroupundermultiplicationmodulo4},aneutrosophicsemigroup. S3 =      ab cd   ; a,b,c,d ∈ R ∪ I    ,neutrosophicsemigroupundermatrixmultiplication. S4 = Z ∪ I ,neutrosophicsemigroupundermultiplication.Let T = {T1 ∪ T2 ∪ T3 ∪ T4, ∗1, ∗2, ∗3, ∗4} isaneutosophicsub4-semigroupof S (4),where T1 = {0, 2, 4, 6, 8, 10}⊆ Z12, T2 = {0,I, 2I, 3I}⊂ S2, T3 =      ab cd   ; a,b,c,d ∈ Q ∪ I    ⊂ S3,T4 = { 5Z ∪ I }⊂ S4, theneutrosophicsemigroupundermultiplication.Alsolet P = {P1 ∪ P2 ∪ P3 ∪ P4, ∗1, ∗2, ∗3, ∗4} beanotherneutrosophicsub4-semigroupof S (4),where P1 = {0, 6}⊆ Z12, P2 = {0, 1,I}⊂ S2, P3 =      ab cd   ; a,b,c,d ∈ Z ∪ I    ⊂ S3,P4 = { 2Z ∪ I }⊂ S4. Also let R = {R1 ∪ R2 ∪ R3 ∪ R4, ∗1, ∗2, ∗3, ∗4} beaneutrosophicsub4-semigroupos S (4)where R1 = {0, 3, 6, 9} ,R2 = {0,I, 2I} ,R3 =      ab cd   ; a,b,c,d ∈ 2Z ∪ I    ,R4 = { 3Z ∪ I }

Then(F,A)issoftneutrosophic4-semigroupover S (4),where

F (x1)= {0, 2, 4, 6, 8, 10}∪{0,I, 2I, 3I}∪      ab cd   ; a,b,c,d ∈ Q ∪ I    ∪{ 5Z ∪ I },

F (x2)= {0, 6}∪{0, 1,I}∪      ab cd   ; a,b,c,d ∈ Z ∪ I    ∪{ 2Z ∪ I } ,

F (x3)= {0, 3, 6, 9}∪{0,I, 2I}∪      ab cd   ; a,b,c,d ∈ 2Z ∪ I    ∪{ 3Z ∪ I } .

Clearly(H,B)isasoftneutrosophicsub N -semigroupof(F,A) , where

H (x1)=

Theorem5.3. Let(F,A)beasoftneutrosophic N -semigroupover S (N )and {(Hi,Bi); i ∈ I} isanonemptyfamilyofsoftneutrosophicsub N -semigroupsof(F,A)then

{0, 4, 8}∪{0,I, 2I}∪      ab cd   ; a,b,c,d ∈ Z ∪ I    ∪{ 10Z ∪ I }, H (x3)= {0, 6}∪{0,I}∪      ab cd   ; a,b,c,d ∈ 4Z ∪ I    ∪{ 6Z ∪ I }

Florentin Smarandache (author and editor) Collected Papers, IX 44

1. ∩i∈I (Hi,Bi)isasoftneutrosophicsub N -semigroupof(F,A).

2. ∧i∈I (Hi,Bi)isasoftneutrosophicsub N -semigroupof ∧i∈I (F,A).

3. ∪i∈I (Hi,Bi)isasoftneutrosophicsub N -semigroupof(F,A)if Bi ∩ Bj = φ,forall i = j.

Proof. Straightforward.

Deﬁnition5.4. (F,A)over S (N )iscalledsoftneutrosophic N -idealif F (x)isneutosophic N -idealof S (N ),forall x ∈ A

Theorem5.4. Everysoftneutrosophic N -ideal(F,A)over S (N )isasoftneutrosophic N -semigroupbuttheconverseisnottrue.

Proposition5.3. Let(F,A)and(K,B)betwosoftneutrosophic N -idealsover S (N ). Then

1.Theirextendedunion(F,A) ∪ε (K,B)over S (N )isnotsoftneutrosophic N -idealover S (N ).

2.Theirextendedintersection(F,A) ∩ε (K,B)over S (N )issoftneutrosophic N -ideal over S (N ).

3.Theirrestrictedunion(F,A) ∪R (K,B)over S (N )isnotsoftneutrosophic N -idealover S (N ).

4.Theirrestrictedintersection(F,A) ∩ε (K,B)over S (N )issoftneutrosophic N -ideal over S (N ).

Proposition5.4. Let(F,A)and(H,B)betwosoftneutrosophic N -idealover S (N ). Then

1.Their AND operation(F,A) ∧ (H,B)issoftneutrosophic N -idealover S (N ).

2.Their OR operation(F,A) ∨ (H,B)isnotsoftneutrosophic N -idealover S (N ).

Theorem5.5. Let(F,A)beasoftneutrosophic N -semigroupover S (N )and {(Hi,Bi); i ∈ I} isanonemptyfamilyofsoftneutrosophic N -idealsof(F,A)then

1. ∩i∈I (Hi,Bi)isasoftneutrosophic N -idealof(F,A).

2. ∧i∈I (Hi,Bi)isasoftneutrosophic N -idealof ∧i∈I (F,A).

§6.Softneutrosophicstrong N -semigroup

Deﬁnition6.1. Let {S(N ), ∗1,..., ∗N } beaneutrosophic N -semigroupand(F,A)bea softsetover {S(N ), ∗1,..., ∗N } Then(F,A)iscalledsoftneutrosophicstrong N -semigroupif andonlyif F (x)isneutrosophicstrongsub N -semigroup,forall x ∈ A

Example6.1. Let S(N )= {S1 ∪S2 ∪S3 ∪S4, ∗1, ∗2, ∗3, ∗4} beaneutrosophic4-semigroup where

S1 = Z6 ∪ I ,aneutrosophicsemigroup.

S2 = {0, 1, 2, 3,I, 2I, 3I,semigroupundermultiplicationmodulo4},aneutrosophicsemigroup.

; a,b,c,d ∈ R ∪ I    ,neutrosophicsemigroupundermatrixmultiplication.

S3 =

   

 

Florentin Smarandache (author and editor) Collected Papers, IX 45



ab cd

S4 = Z ∪ I ,neutrosophicsemigroupundermultiplication.Let T = {T1 ∪ T2 ∪ T3 ∪ T4, ∗1, ∗2, ∗3, ∗4} isaneutosophicstrongsub4-semigroupof S (4),where T1 = {0, 3, 3I}⊆

Z6 ∪ I , T2 = {0,I, 2I, 3I}⊂ S2, T3 =      ab cd   ; a,b,c,d ∈ Q ∪ I    ⊂ S3,T4 = { 5Z ∪ I }⊂ S4, theneutrosophicsemigroupundermultiplication.Alsolet P = {P1 ∪ P2 ∪ P3 ∪ P4, ∗1, ∗2, ∗3, ∗4} beanotherneutrosophicstrongsub4-semigroupof S (4),where P1 = {0, 2I, 4I} ⊆ Z6 ∪ I , P2 = {0, 1,I}⊂ S2, P3 =      ab cd   ; a,b,c,d ∈ Z ∪ I    ⊂ S3,P4 = { 2Z ∪ I } ⊂ S4. Then(F,A)issoftneutrosophicstrong4-semigroupover S (4),whereThen(F,A)issoft neutrosophic4-semigroupover S (4),where

F (x1)= {0, 3, 3I}∪{0,I, 2I, 3I}∪      ab cd   ; a,b,c,d ∈ Q ∪ I    ∪{ 5Z ∪ I },

F (x2)= {0, 2I, 4I}∪{0, 1,I}∪      ab cd   ; a,b,c,d ∈ Z ∪ I    ∪{ 2Z ∪ I }

Theorem6.1. Everysoftneutrosophicstrong N -semigroupistriviallyasoftneutrosophic N -semigroupbuttheconverseisnottrue.

Proposition6.1. Let(F,A)and(K,B)betwosoftneutrosophicstrong N -semigroups over S(N ).Then

1.Theirextendedunion(F,A) ∪ε (K,B)over S(N )isnotsoftneutrosophicstrong N semigroupover S(N ).

2.Theirextendedintersection(F,A) ∩ε (K,B)over S(N )issoftneutrosophicstrong N -semigroupover S(N ).

3.Theirrestrictedunion(F,A) ∪R (K,B)over S(N )isnotsoftneutrosophicstrong N semigroupover S(N ).

4.Theirrestrictedintersection(F,A) ∩ε (K,B)over S(N )issoftneutrosophicstrong N -semigroupover S(N ).

Proposition6.2. Let(F,A)and(K,B)betwosoftneutrosophicstrong N -semigroups over S(N ).Then

1.Their AND operation(F,A) ∧ (K,B)issoftneutrosophicstrong N -semigroupover S(N ).

2.Their OR operation(F,A) ∨ (K,B)isnotsoftneutrosophicstrong N -semigroupover S(N ).

Deﬁnition6.2. Let(F,A)and(H,B)betwosoftneutrosophicstrong N -semigroups over S (N ).Then(H,B)isasoftneutrosophicstrongsub N -semigroupof(F,A),if

1. B ⊂ A.

2. H(x)isneutrosophicstrongsub N -semigroupof F (x),forall x ∈ B.

Theorem6.2.

1.Let(F,A)beasoftneutrosophicstrong N -semigroupover S (N )and {(Hi,Bi); i ∈ I} isanonemptyfamilyofsoftneutrosophicstongsub N -semigroupsof(F,A)then

2. ∩i∈I (Hi,Bi)isasoftneutrosophicstrongsub N -semigroupof(F,A).

Florentin Smarandache (author and editor) Collected Papers, IX 46

3. ∧i∈I (Hi,Bi)isasoftneutrosophicstrongsub N -semigroupof ∧i∈I (F,A).

4. ∪i∈I (Hi,Bi)isasoftneutrosophicstrongsub N -semigroupof(F,A)if Bi ∩ Bj = φ, forall i = j

Proof. Straightforward.

Deﬁnition6.3. (F,A)over S (N )iscalledsoftneutrosophicstrong N -idealif F (x)is neutosophicstrong N -idealof S (N ),forall x ∈ A.

Theorem6.3. Everysoftneutrosophicstrong N -ideal(F,A)over S (N )isasoft neutrosophicstrong N -semigroupbuttheconverseisnottrue.

Theorem6.4. Everysoftneutrosophicstrong N -ideal(F,A)over S (N )isasoft neutrosophic N -semigroupbuttheconverseisnottrue.

Proposition6.3. Let(F,A)and(K,B)betwosoftneutrosophicstrong N -idealsover S (N ).Then

1.Theirextendedunion(F,A) ∪ε (K,B)over S (N )isnotsoftneutrosophicstrong N idealover S (N ).2.Theirextendedintersection(F,A)∩ε (K,B)over S (N )issoftneutrosophic strong N -idealover S (N ).

3.Theirrestrictedunion(F,A) ∪R (K,B)over S (N )isnotsoftneutrosophicstrong N -idealover S (N ).

4.Theirrestrictedintersection(F,A) ∩ε (K,B)over S (N )issoftneutrosophicstrong N -idealover S (N ).

Proposition6.4. Let(F,A)and(H,B)betwosoftneutrosophicstrong N -idealover S (N ).Then

1.Their AND operation(F,A) ∧ (H,B)issoftneutrosophicstrong N -idealover S (N ).

2.Their OR operation(F,A) ∨ (H,B)isnotsoftneutrosophicstrong N -idealover S (N ).

Theorem6.5. Let(F,A)beasoftneutrosophicstrong N -semigroupover S (N )and {(Hi,Bi); i ∈ I} isanonemptyfamilyofsoftneutrosophicstrong N -idealsof(F,A)then

1. ∩i∈I (Hi,Bi)isasoftneutrosophicstrong N -idealof(F,A).

2. ∧i∈I (Hi,Bi)isasoftneutrosophicstrong N -idealof ∧i∈I (F,A).

Conclusion

Thispaperisanextensionofneutrosphicsemigrouptosoftsemigroup.Wealsoextend neutrosophicbisemigroup,neutrosophic N -semigrouptosoftneutrosophicbisemigroup,and softneutrosophic N -semigroup.Theirrelatedpropertiesandresultsareexplainedwithmany illustrativeexamples,thenotionsrelatedwithstrongpartofneutrosophyalsoestablishedwithin softsemigroup.

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Florentin Smarandache (author and editor) Collected Papers, IX 48

(T, I, F)-Neutrosophic Structures

Florentin Smarandache

Florentin Smarandache (2015). (T, I, F)-Neutrosophic Structures. Proceedings of the Annual Symposium of the Institute of Solid Mechanics and Session of the Commission of Acoustics, SISOM 2015 Bucharest 21-22 May; Acta Electrotechnica 57(1-2); Neutrosophic Sets and Systems 8, 3-10

Abstract. In this paper we introduce for the first time a new type of structures, called (T, I, F)-Neutrosophic Structures, presented from a neutrosophic logic perspective, and we show particular cases of such structures in geometry and in algebra. In any field of knowledge, each structure is composed from two parts: a space, and a set of axioms (or laws) acting (governing) on it. If the space, or at least one of its axioms (laws), has some indeterminacy, that structure is a (T, I, F)-Neutrosophic Structure. The (T, I, F)-Neutrosophic Structures [based on the components T=truth, I=indeterminacy, F=falsehood] are different from the Neutrosophic Algebraic Structures [based on neutrosophic numbers of the form a+bI, where

I=indeterminacy and In = I], that we rename as Neutrosophic I-Algebraic Structures (meaning algebraic structures based on indeterminacy “I” only). But we can combine both and obtain the (T, I, F)-Neutrosophic IAlgebraic Structures, i.e. algebraic structures based on neutrosophic numbers of the form a+bI, but also having indeterminacy related to the structure space (elements which only partially belong to the space, or elements we know nothing if they belong to the space or not) or indeterminacy related to at least one axiom (or law) acting on the structure space. Then we extend them to Refined (T, I, F)-Neutrosophic Refined I-Algebraic Structures.

Keywords: Neurosophy, algebraic structures, neutrosophic sets, neutrosophic logics.

1. Neutrosophic Algebraic Structures [or Neutrosophic I-Algebraic Structures]

A previous type of neutrosophic structures was introduced in algebra by W.B. Vasantha Kandasamy and Florentin Smarandache [1-56], since 2003, and it was called Neutrosophic Algebraic Structures. Later on, more researchers joined the neutrosophic research, such as: Mumtaz Ali, A. A. Salama, Muhammad Shabir, K. Ilanthenral, Meena Kandasamy, H. Wang, Y.-Q. Zhang, R. Sunderraman, Andrew Schumann, Salah Osman, D. Rabounski, V. Christianto, Jiang Zhengjie, Tudor Paroiu, Stefan Vladutescu, Mirela Teodorescu, Daniela Gifu, Alina Tenescu, Fu Yuhua, Francisco Gallego Lupiañez, etc.

The neutrosophic algebraic structures are algebraic structures based on sets of neutrosophic numbers of the form N = a + bI, where a, b are real (or complex) numbers, and a is called the determinate part on N and b is called the indeterminate part of N, while I = indeterminacy, with mI + nI = (m + n)I, 0∙I = 0, In = I for integer n ≥ 1, and I / I = undefined.

When a, b are real numbers, then a + bI is called a neutrosophic real number. While if a, b are complex numbers, then a + bI is called a neutrosophic complex number.

We may say "indeterminacy" for "I" from a+bI, and "degree of indeterminacy" for "I" from (T, I, F) in order to distinguish them.

The neutrosophic algebraic structures studied by VasanthaSmarandache in the period 2003-2015 are: neutrosophic groupoid, neutrosophic semigroup, neutrosophic group, neutrosophic ring, neutrosophic field, neutrosophic vector space, neutrosophic linear algebras etc., which later (between 2006-2011) were generalized by the same researchers to neutrosophic bi-algebraic structures, and more general to neutrosophic N-algebraic structures. Afterwards, the neutrosophic structures were further extended to neutrosophic soft algebraic structures by Florentin Smarandache, Mumtaz Ali, Muhammad Shabir, and Munazza Naz in 2013-2014.

In 2015 Smarandache refined the indeterminacy I into different types of indeterminacies (depending on the problem to solve) such as I1, I2, …, Ip with integer p ≥ 1, and obtained the refined neutrosophic numbers of the form Np = a+b1I1+b2I2+…+bpIp where a, b1, b2, …, bp are real or complex numbers, and a is called the determinate part of Np, while for each k��{1, 2, …, p} Ik is called the k-th indeterminate part of Np, and for each k��{1, 2, …, p}, and similarly

Smarandache (author and editor) Collected Papers, IX 49

Florentin

mIk + nIk = (m + n)Ik, 0∙Ik = 0, Ik n = Ik for integer n ≥ 1, and Ik /Ik = undefined.

The relationships and operations between Ij and Ik, for j ≠ k, depend on each particular problem we need to solve. Then consequently Smarandache [2015] extended the neutrosophic algebraic structures to Refined Neutrosophic Algebraic Structures [or Refined Neutrosophic I-Algebraic Structures], which are algebraic structures based on the sets of the refined neutrosophic numbers a+b1I1+b2I2+…+bpIp

2.(T, I, F)-Neutrosophic Structures.

We now introduce for the first time another type of neutrosophic structures. These structures, in any field of knowledge, are considered from a neutrosophic logic point of view, i.e. from the truth-indeterminacy-falsehood (T, I, F)values. In neutrosophic logic every proposition has a degree of truth (T), a degree of indeterminacy (I), and a degree of falsehood (F), where T, I, F are standard or nonstandard subsets of the non-standard unit interval ] 0, 1+[

In technical applications T, I, and F are only standard subsets of the standard unit interval [0, 1] with: 0 ≤ sup(T) + sup(I) + sup(F) ≤ 3+ where sup(Z) means superior of the subset Z.

In general, each structure is composed from: a space, endowed with a set of axioms (or laws) acting (governing) on it. If the space, or at least one of its axioms, has some indeterminacy, we consider it as a (T, I, F)-Neutrosophic Structure.

Indeterminacy with respect to the space is referred to some elements that partially belong [i.e. with a neutrosophic value (T, I. F)] to the space, or their appurtenance to the space is unknown.

An axiom (or law) which deals with indeterminacy is called neutrosophic axiom (or law). We introduce these new structures because in the world we do not always know exactly or completely the space we work in; and because the axioms (or laws) are not always well defined on this space, or may have indeterminacies when applying them.

3. Refined (T, I, F) Neutrosophic Structures

[or (Tj, Ik, Fl)-Neutrosophic

Structures]

In 2013 Smarandache [76] refined the neutrosophic components (T, I, F) into (T1, T2, …, Tm; I1, I2, …, Ip; F1, F2, …, Fr),

where m, p, r are integers ≥ 1. Consequently, we now [2015] extend the (T, I, F)Neutrosophic Structures to (T1, T2, …, Tm; I1, I2, …, Ip; F1, F2, …, Fr)-Neutrosophic Structures, that we called Refined (T, I, F)-Neutrosophic Structures [or (Tj, Ik, Fl)Neutrosophic Structures]. These are structures whose elements have a refined neutrosophic value of the form (T1, T2, …, Tm; I1, I2, …, Ip; F1, F2, …, Fr) or the space has some indeterminacy of this form

4. (T, I, F)-Neutrosophic I-Algebraic Structures.

The (T, I, F)-Neutrosophic Structures [based on the components T=truth, I=indeterminacy, F=falsehood] are different from the Neutrosophic Algebraic Structures [based on neutrosophic numbers of the form a+bI]. We may rename the last ones as Neutrosophic I-Algebraic Structures (meaning: algebraic structures based on indeterminacy “I” only).

But we can combine both of them and obtain a (T, I, F)Neutrosophic I-Algebraic Structures, i.e. algebraic structures based on neutrosophic numbers of the form a+bI, but also have indeterminacy related to the structure space (elements which only partially belong to the space, or elements we know nothing if they belong to the space or not) or indeterminacy related to at least an axiom (or law) acting on the structure space.

Even more, we can generalize them to Refined (T, I, F)Neutrosophic Refined I-Algebraic Structures, or (Tj, Ik, Fl)Neutrosophic Is-Algebraic Structures.

5. Example of Refined I-Neutrosophic Algebraic Structure

Let the indeterminacy I be split into I1 = contradiction (i.e. truth and falsehood simultaneously), I2 = ignorance (i.e. truth or falsehood), and I3 = unknown, and the corresponding 3-refined neutrosophic numbers of the form a+b1I1+b2I2+b3I3

The (G, *) be a groupoid. Then the 3-refined Ineutrosophic groupoid is generated by I1, I2, I3 and G under *and it is denoted by

N3(G) = {(G∪I1∪I2∪I3), *} = { a+b1I1+b2I2+b3I3 / a, b1, b2,b3 ∈ G }.

Florentin Smarandache (author and editor) Collected Papers, IX 50

6. Example of Refined (T, I, F)-Neutrosophic Structure

Let (T, I, F) be split as (T1, T2; I1, I2; F1, F2, F3). Let H = ( {h1, h2, h3}, # ) be a groupoid, where h1, h2, and h3 are real numbers. Since the elements h1, h2, h3 only partially belong to H in a refined way, we define a refined (T, I, F)-neutrosophic groupoid { or refined (2; 2; 3)neutrosophic groupoid, since T was split into 2 parts, I into 2 parts, and F into 3 parts } as H = {h1(0.1, 0.1; 0.3, 0.0; 0.2, 0.4, 0.1), h2(0.0, 0.1; 0.2, 0.1; 0.2, 0.0, 0.1), h3(0.1, 0.0; 0.3, 0.2; 0.1, 0.4, 0.0)}.

7. Examples of (T, I, F)-Neutrosophic IAlgebraic Structures.

1. Indeterminate Space (due to Unknown Element). And Neutrosophic Number included. Let B = {2+5I, -I, -4, b(0, 0.9, 0)} a neutrosophic set, which contain two neutrosophic numbers, 2+5I and -I, and we know about the element b that its appurtenance to the neutrosophic set is 90% indeterminate.

2. Indeterminate Space (due to Partially Known Element). And Neutrosophic Number included.

Let C = {-7, 0, 2+I(0.5, 0.4, 0.1), 11(0.9, 0, 0) }, which contains a neutrosophic number 2+I, and this neutrosophic number is actually only partially in C; also, the element 11 is also partially in C.

3. Indeterminacy Axiom (Law).

Let D = [0+0I, 1+1I] = {c+dI, where c, d �� [0, 1]}. One defines the binary law # in the following way: # : DD  D x # y = (x1 + x2I) # (y1 + y2I) = [(x1 + x2)/y1] + y2I, but this neutrosophic law is undefined (indeterminate) when y1 = 0.

4. Little Known or Completely Unknown Axiom (Law).

Let us reconsider the same neutrosophic set D as above. But, about the binary neutrosophic law  that D is endowed with, we only know that it associates the neutrosophic numbers 1+I and 0.2+0.3I with the neutrosophic number 0.5+0.4I, i.e. (1+I)(0.2+0.3I) = 0.5+0.4I.

There are many cases in our world when we barely know some axioms (laws).

8.Examples of Refined (T, I, F)-Neutrosophic Refined I-Algebraic Structures.

We combine the ideas from Examples 5 and 6 and we construct the following example. Let’s consider, from Example 5, the groupoid (G, *), where G is a subset of positive real numbers, and its extension to a 3-refined Ineutrosophic groupoid, which was generated by I1, I2, I3 and G under the law * that was denoted by N3(G) = { a+b1I1+b2I2+b3I3 / a, b1, b2,b3 ∈ G }.

We then endow each element from N3(G) with some (2; 2; 3)-refined degrees of membership/indeterminacy/ nonmembership, as in Example 6, of the form (T1, T2; I1, I2; F1, F2, F3), and we obtain a N3(G)(2;2;3) = { a+b1I1+b2I2+b3I3(T1, T2; I1, I2; F1, F2, F3) / a, b1, b2,b3 ∈ G }, where 1 123

a T abbb   , 2 123

0.5a T abbb   12 12 123 123 ,; bb II abbbabbb   31 12 123 123

0.1 0.2 ,, bb FF abbbabbb   , 23 3 123

bb F abbb   

Therefore, N3(G)(2;2;3) is a refined (2; 2; 3)-neutrosophic groupoid and a 3-refined I-neutrosophic groupoid.

9.Neutrosophic Geometric Examples

a) Indeterminate Space.

We might not know if a point P belongs or not to a space S [we write P(0, 1, 0), meaning that P’s indeterminacy is 1, or completely unknown, with respect to S].

Or we might know that a point Q only partially belongs to the space S and partially does not belong to the space S [for example Q(.3, 0.4, 0.5), which means that with respect to S, Q’s membership is 0.3, Q’s indeterminacy is 0.4, and Q’s non membership is 0.5]. Such situations occur when the space has vague or unknown frontiers, or the space contains ambiguous (not well defined) regions.

Florentin Smarandache (author and editor) Collected Papers, IX 51

b) Indeterminate Axiom.

Also, an axiom (α) might not be well defined on the space S, i.e. for some elements of the space the axiom (α) may be valid, for other elements of the space the axiom (α) may be indeterminate (meaning neither valid, nor invalid), while for the remaining elements the axiom (α) may be invalid.

As a concrete example, let’s say that the neutrosophic values of the axiom (α) are (0.6, 0.1, 0.2) = (degree of validity, degree of indeterminacy, degree of invalidity).

10.(T, I, F)-Neutrosophic Geometry as a Par ticular Case of (T, I, F)-Neutrosophic Structures.

As a particular case of (T, I, F)-neutrosophic structures in geometry, one considers a (T, I, F)-Neutrosophic Geometry as a geometry which is defined either on a space with some indeterminacy (i.e. a portion of the space is not known, or is vague, confused, unclear, imprecise), or at least one of its axioms has some indeterminacy (i.e. one does not know if the axiom is verified or not in the given space).

This is a generalization of the Smarandache Geometry (SG) [57-75], where an axiom is validated and invalidated in the same space, or only invalidated, but in multiple ways. Yet the SG has no degree of indeterminacy related to the space or related to the axiom.

A simple Example of a SG is the following that unites Euclidean, Lobachevsky-Bolyai-Gauss, and Riemannian geometries altogether, in the same space, considering the Fifth Postulate of Euclid: in one region of the SG space the postulate is validated (only one parallel trough a point to a given line), in a second region of SG the postulate is invalidated (no parallel through a point to a given line elliptical geometry), and in a third region of SG the postulate is invalidated but in a different way (many parallels through a point to a given line hyperbolic geometry). This simple example shows a hybrid geometry which is partially Euclidean, partially Non-Euclidean Elliptic, and partially Non-Euclidean Hyperbolic. Therefore, the fifth postulate (axiom) of Euclid is true for some regions, and false for others, but it is not indeterminate for any region (i.e. not knowing how many parallels can be drawn through a point to a given line). We can extend this hybrid geometry adding a new space region where one does not know if there are or there are

not parallels through some given points to the given lines (i.e. the Indeterminate component) and we form a more complex (T, I, F)-Neutrosophic Geometry.

12.Neutrosophic Algebraic Examples.

1) Indeterminate Space (due to Unknown Element). Let the set (space) be NH = {4, 6, 7, 9, a}, where the set NH has an unknown element "a", therefore the whole space has some degree of indeterminacy. Neutrosophically, we write a(0, 1, 0), which means the element a is 100% unknown.

2) Indeterminate Space (due to Partially Known Element).

Given the set M = {3, 4, 9(0.7, 0.1, 0.3)}, we have two elements 3 and 4 which surely belong to M, and one writes them neutrosophically as 3(1, 0, 0) and 4(1, 0, 0), while the third element 9 belongs only partially (70%) to M, its appurtenance to M is indeterminate (10%), and does not belong to M (in a percentage of 30%).

Suppose M is endowed with a neutrosophic law* defined in the following way: x1(t1, i1, f1)* x2(t2, i2, f2) = max{x1, x2}( min{t1, t2}, max{i1, i2}, max{f1, f2}), which is a neutrosophic commutative semigroup with unit element 3(1, 0 ,0).

Clearly, if x, y �� M, then x*y �� M. Hence the neutrosophic law * is well defined.

Since max and min operators are commutative and associative, then * is also commutative and associative. If x �� M, then x*x = x. Below, examples of applying this neutrosophic law *: 3*9(0.7, 0.1, 0.3) = 3(1, 0, 0)*9(0.7, 0.1, 0.3) = max{3, 9}( min{1, 0.7}, max{0, 0.1}, max{0, 0.3} ) = 9(0.7, 0.1, 0.3).

3*4 = 3(1, 0, 0)*4(1, 0, 0) = max{3, 4}( min{1, 1}, max{0, 0}, max{0, 0} ) = 4(1, 0, 0).

2) Indeterminate Law (Operation).

For example, let the set (space) be NG = ( {0, 1, 2}, / ), where "/" means division.

NG is a (T, I, F)-neutrosophic groupoid, because the operation "/" (division) is partially defined and undefined (indeterminate). Let's see: 2/1 = 1, which belongs to NG; 1/2 = 0.5, which does not belongs to NG; 1/0 = undefined (indeterminate). So the law defined on the set NG has the properties that:

Florentin Smarandache (author and editor) Collected Papers, IX 52

applying this law to some elements, the results are in NG [well defined law];

 applying this law to other elements, the results are not in NG [not well defined law];  applying this law to again other elements, the results are undefined [indeterminate law].

We can construct many such algebraic structures where at least one axiom has such behavior (such indeterminacy in principal).

12.Websites at UNM for Neutrosophic Algebraic Structures and respectively Neutrosophic Geometries:

http://fs.gallup.unm.edu/neutrosophy.htm and http://fs.gallup.unm.edu/geometries.htm respectively.

cal Communication Uncertainties, Sitech Publishing House and Zip Publishing, Craiova, Romania -Columbus, Ohio, USA, 300 p., 2014.

7. Florentin Smarandache, Stefan Vladutescu, Alina Tenescu, Current Communication Difficulties, Sitech Publishing House and Zip Publishing, Craiova, Romania - Columbus, Ohio, USA, 300 p., 2014.

8. W. B. Vasantha Kandasamy, Florentin Smarandache, Ilanthenral K, New Techniques to Analyze the Prediction of Fuzzy Models, EuropaNova, Brussels, Belgium, 242 p., 2014.

9. W. B. Vasantha Kandasamy, Florentin Smarandache, Ilanthenral K, Pseudo Lattice Graphs and their Applications to Fuzzy and Neutrosophic Models, EuropaNova, Brussels, Belgium, 275 p., 2014.

10.Mumtaz Ali, Florentin Smarandache, Muhammad Shabir, Soft Neutrosophic Algebraic Structures and Their Generalization, Vol. II, EuropaNova, Brussels, Belgium, 288 p., 2014.

11.W. B. Vasantha Kandasamy, Florentin Smarandache, Algebraic Structures on Real and Neutrosophic Semi Open Squares, Education Publisher, Columbus, Ohio, USA, 206 p., 2014.

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12.Florentin Smarandache, Mumtaz Ali, Muhammad Shabir, Soft Neutrosophic Algebraic Structures and Their Generalization, Vol. I, Education Publishing, Columbus, Ohio, USA, 264 p., 2014.

13.Florentin Smarandache, Stefan Vladutescu (coordinators), Communication Neutrosophic Routes, Educational Publisher, Columbus, Ohio, USA, 217 p., 2014.

14.W. B. Vasantha Kandasamy, Florentin Smarandache, Algebraic Structures on Fuzzy Unit Square and Neutrosophic Unit Square, Educational Publisher, Columbus, Ohio, USA, 221 p., 2014.

15.F. Smarandache, Introduction to Neutrosophic Statistics, Sitech and Education Publisher, Craiova, Romania - Educational Publisher, Columbus, Ohio, USA, 123 p., 2014.

16.Florentin Smarandache, Stefan Vladutescu, Neutrosophic Emergencies and Incidencies, Verlag LAP LAMBERT, OmniScriptum, GmbH & Co. KG, Saarbrücken, Deutschland / Germany,248 p., 2013; DOI: 10.13140/2.1.3530.2400.

17.Florentin Smarandache, Introduction to Neutrosophic Measure, Neutrosophic Integral, and Neutrosophic Probability, Sitech & Edu-



Florentin Smarandache (author and editor) Collected Papers, IX 53

cational Publisher, Craiova, Romania - Columbus, Ohio, USA, 140 p., 2013.

18.W. B. Vasantha Kandasamy, Florentin Smarandache, Fuzzy Neutrosophic Models for Social Scientists, Educational Publisher, Columbus, Ohio, USA, 167 pp., 2013.

19.W. B. Vasantha Kandasamy, Florentin Smarandache, Neutrosophic Super Matrices and Quasi Super Matrices, Educational Publisher, Columbus, Ohio, USA, 200 p., 2012.

20.Florentin Smarandache, Tudor Paroiu, Neutrosofia ca reflectarea a realităţii neconvenţionale, Sitech, Craiova, Romania, 130 p., 2012.

21.W. B. Vasantha Kandasamy, Florentin Smarandache, A. Praveen Prakash, Mathematical Analysis of the Problems Faced the People with Disabilities (PWDs) / With Specific Reference to Tamil Nadu (India), Zip Publishing, Columbus, Ohio, USA, 165 p., 2012.

22.Florentin Smarandache, Fu Yuhua, Neutrosophic Interpretation of The Analects of Confucius (弗羅仁汀司馬仁達齊，傅昱華論語的中智學解讀和擴充正反及中智論語

), English-Chinese Bilingual, 英汉双语, Zip Publisher, Columbus, Ohio, USA, 268 p., 2011.

23.W. B. Vasantha Kandasamy, Florentin Smarandache, Neutrosophic Interval Bialgebraic Structures, Zip Publishing, Columbus, Ohio, USA, 195 p., 2011.

24.W. B. Vasantha Kandasamy, Florentin Smarandache, Finite Neutrosophic Complex Numbers, Zip Publisher, Columbus, Ohio, USA, 220 p., 2011.

25.Florentin Smarandache & Fu Yuhua, Neutrosophic Interpretation of Tao Te Ching (English-Chinese bilingual), Translation by Fu Yuhua, Chinese Branch Kappa, Beijing, 208 p., 2011.

26.W.B. Vasantha Kandasamy, Florentin Smarandache, Svenska Fysikarkivet, Neutrosophic Bilinear Algebras and Their Generalization, Stockholm, Sweden, 402 p., 2010.

27.Florentin Smarandache (editor), Multispace&Multistructure. Neutrosophic Transdisciplinarity (100 Collected Papers of Sciences), Vol. IV, North-European Scientific Publishers, Hanko, Finland, 800 p., 2010.

28.W.B. Vasantha Kandasamy, F. Smarandache, K, Ilanthenral, New Classes of Neutrosophic Linear Algebras, CuArt, Slatina, Romania, 286 p., 2010.

29.Florentin Smarandache (editor), Neutrosoph-

ic Physics: More Problems, More Solutions (Collected Papers), North-European Scientific Publishers, Hanko, Finland, 94 p., 2010.

30.F. Smarandache, V. Christianto, Neutrosophic Logic, Wave Mechanics, and Other Stories (Selected Works: 2005-2008), Kogaion Editions, Bucharest, Romania, 129 p., 2009.

31.F. Smarandache and Jiang Zhengjie, Chinese Neutrosophy and Taoist Natural Philosophy [Chinese language], Xiquan Chinese Hse., Beijing, China, 150 p., 2008.

32.Florentin Smarandache, Andrew Schumann, Neutrality and Multi-Valued Logics, A. R Press, Rehoboth, USA, 119 p., 2007.

33.Florentin Smarandache, Salah Osman, Neutrosophy in Arabic Philosophy [English version], Renaissance High Press, Ann Arbor, USA, 291 pp., 2007. - Translated into Arabic language by Dr. Osman Salah, Munsha’t al Ma’arif Publ. Hse., Jalal Huzie & Partners, Alexandria, Egypt, 418 p., 2007.

34.Florentin Smarandache, V. Christianto, Multi-Valued Logic, Neutrosophy, and Schrödinger Equation, Hexis, Phoenix, Arizona, USA, 107 p., 2006.

35.W. B. Vasantha Kandasamy, Florentin Smarandache, Some Neutrosophic Algebraic Structures and Neutrosophic N-Algebraic Structures, Hexis, Phoenix, Arizona, USA, 219 p., 2006.

36.W. B. Vasantha Kandasamy, Florentin Smarandache, N-Algebraic Structures and SN-Algebraic Structures, Hexis, Phoenix, Arizona, USA, 209 p., 2006.

37.W. B. Vasantha Kandasamy, Florentin Smarandache, Neutrosophic Rings, Hexis, Phoenix, Arizona, USA, 154 p., 2006.

38.W. B. Vasantha Kandasamy, Florentin Smarandache, Fuzzy Interval Matrices, Neutrosophic Interval Matrices and Their Applications, Hexis, Phoenix, Arizona, USA, 304 p., 2006.

39.W. B. Vasantha Kandasamy, Florentin Smarandache, Vedic Mathematics, ‘Vedic’ or ‘Mathematics’: A Fuzzy & Neutrosophic Analysis, Automaton, Los Angeles, California, USA, 220 p., 2006.

40.Florentin Smarandache, D. Rabounski, L. Borissova, Neutrosophic Methods in General Relativity, Hexis, Phoenix, Arizona, USA, 78 p., 2005. - Russian translation D. Rabounski, Нейтрософские методы в Общей Теории Относительности, Hexis, Phoenix, Arizona, USA, 105 p., 2006.

41.Florentin Smarandache, H. Wang, Y.-Q.

Florentin Smarandache (author and editor) Collected Papers, IX 54

Zhang, R. Sunderraman, Interval Neutrosophic Sets and Logic: Theory and Applications in Computing, Hexis, Phoenix, Arizona, USA, 87 p., 2005.

42.W. B. Vasantha Kandasamy, Florentin Smarandache, Fuzzy and Neutrosophic Analysis of Women with HIV / AIDS (With Specific Reference to Rural Tamil Nadu in India), translation of the Tamil interviews Meena Kandasamy, Hexis, Phoenix, Arizona, USA, 316 p., 2005.

43.Florentin Smarandache, W. B. Vasantha Kandasamy, K. Ilanthenral, Applications of Bimatrices to some Fuzzy and Neutrosophic Models, Hexis, Phoenix, Arizona, USA, 273 pp., 2005.

44.Florentin Smarandache, Feng Liu, Neutrosophic Dialogues, Xiquan, Phoenix, Arizona, USA, 97 p., 2004.

45.W. B. Vasantha Kandasamy, Florentin Smarandache, Fuzzy Relational Equations & Neutrosophic Relational Equations, Hexis, Phoenix, Arizona, USA, 301 pp., 2004.

46.W. B. Vasantha Kandasamy, Florentin Smarandache, Basic Neutrosophic Algebraic Structures and their Applications to Fuzzy and Neutrosophic Models, Hexis, Phoenix, Arizona, USA, 149 p., 2004.

47.W. B. Vasantha Kandasamy, Florentin Smarandache, Fuzzy Cognitive Maps and Neutrosophic Cognitive Maps, Xiquan, Phoenix, Arizona, USA, 211 p., 2003.

48.Florentin Smarandache (editor), Proceedings of the First International Conference on Neutrosophy, Neutrosophic Logic, Neutrosophic Set, Neutrosophic Probability and Statistics, University of New Mexico, Gallup Campus, Xiquan, Phoenix, Arizona, USA, 147 p., 2002.

49.Florentin Smarandache, Neutrosophy. Neutrosophic Probability, Set, and Logic, American Research Press, Rehoboth, USA, 105 p., 1998. - Republished in 2000, 2003, 2005, A Unifying Field in Logics: Neutrosophic Logic. Neutrosophy, Neutrosophic Set, Neutrosophic Probability and Statistics (second, third, and respectively fourth edition), American Research Press, USA, 156 p.; - Chinese translation by F. Liu, Xiquan Chinese Branch,121 p., 2003; Сущность нейтрософии, Russian partial translation by D.Rabounski, Hexis, Phoenix, Arizona, USA, 32 p., 2006.

II.Neutrosophic Algebraic Structures - Edited Books

50.Florentin Smarandache & Mumtaz Ali - editors, Neutrosophic Sets and Systems, book series, Vol. 1, Educational Publisher, Columbus, Ohio, USA, 70 p., 2013.

51.Florentin Smarandache & Mumtaz Ali - editors, Neutrosophic Sets and Systems, book series, Vol. 2, Educational Publisher, Columbus, Ohio, USA, 110 p., 2014.

52.Florentin Smarandache & Mumtaz Ali - editors, Neutrosophic Sets and Systems, book series, Vol. 3, Educational Publisher, Columbus, Ohio, USA, 76 p., 2014.

53.Florentin Smarandache & Mumtaz Ali - editors, Neutrosophic Sets and Systems, book series, Vol. 4, Educational Publisher, Columbus, Ohio, USA, 74 p., 2014.

54.Florentin Smarandache & Mumtaz Ali - editors, Neutrosophic Sets and Systems, book series, Vol. 5, Educational Publisher, Columbus, Ohio, USA, 76 p., 2014.

55.Florentin Smarandache & Mumtaz Ali - editors, Neutrosophic Sets and Systems, book series, Vol. 6, Educational Publisher, Columbus, Ohio, USA, 83 p., 2014.

56.Florentin Smarandache & Mumtaz Ali - editors, Neutrosophic Sets and Systems, book series, Vol. 7, Educational Publisher, Columbus, Ohio, USA, 88 p., 2015.

III.Neutrosophic Geometries

57.S. Bhattacharya, A Model to the Smarandache Geometries, in “Journal of Recreational Mathematics”, Vol. 33, No. 2, p. 66, 2004-2005; - modified version in “Octogon Mathematical Magazine”, Vol. 14, No. 2, pp. 690-692, October 2006.

58.S. Chimienti and M. Bencze, Smarandache Paradoxist Geometry, in “Bulletin of Pure and Applied Sciences”, Delhi, India, Vol. 17E, No.1, 123-1124, 1998; http://www.gallup.unm edu/~smarandache/prd-geo1.txt.

59.L. Kuciuk and M. Antholy, An Introduction to Smarandache Geometries, in “Mathematics Magazine”, Aurora, Canada, Vol. XII, 2003; online: http://www.mathematicsmagazine.com/12004/Sm_Geom_1_2004.htm; also presented at New Zealand Mathematics Colloquium, Massey University, Palmerston North, New Zealand, December 3-6, 2001, http://atlasconferences.com/c/a/h/f/09.htm; also presented at the International Congress of Mathematicians (ICM 2002), Beijing, China, 20-28 August2002, http://www.icm2002.org cn/B/Schedule_Section04.htm and in Abstracts of Short Communications to the Inter-

Florentin Smarandache (author and editor) Collected Papers, IX 55

national Congress of Mathematicians, International Congress of Mathematicians, 20-28 August 2002, Beijing, China, Higher Education Press, 2002; and in “JP Journal of Geometry and Topology”, Allahabad, India, Vol. 5, No. 1, pp. 77-82, 2005.

60. Linfan Mao, An introduction to Smarandache geometries on maps, presented at 2005 International Conference on Graph Theory and Combinatorics, Zhejiang Normal University, Jinhua, Zhejiang, P. R. China, June 25-30, 2005.

61.Linfan Mao, Automorphism Groups of Maps, Surfaces and Smarandache Geometries, partially post-doctoral research for the Chinese Academy of Science, Am. Res. Press, Rehoboth, 2005.

62.Charles Ashbacher, Smarandache Geometries, in “Smarandache Notions Journal”, Vol. VIII, pp. 212-215, No. 1-2-3, 1997.

63.Linfan Mao, Selected Papers on Mathematical Combinatorics, I, World Academic Press, Liverpool, U.K., 2006.

64.H. Iseri, Partially Paradoxist Smarandache Geometries, http://www.gallup.unm.edu/ ~smarandache/Howard-Iseri-paper.htm.

65.H. Iseri, Smarandache Manifolds, Am. Res. Press, 2002, http://www.gallup.unm.edu/ ~smarandache /Iseri-book1.pdf

66. M. Perez, Scientific Sites, in “Journal of Recreational Mathematics”, Amityville, NY, USA, Vol. 31, No. I, p. 86, 2002-20003.

67.F. Smarandache, Paradoxist Mathematics, in Collected Papers, Vol. II, Kishinev University Press, Kishinev, pp. 5-28, 1997.

68.Linfan Mao, Automorphism groups of maps, surfaces and Smarandache geometries, 2005, http://xxx.lanl.gov/pdf/math/0505318v1

69.Linfan Mao, A new view of combinatorial maps Smarandache’s notion, 2005, http://xxx.lanl. gov/pdf/math/0506232v1

70.Linfan Mao, Parallel bundles in planar map geometries, 2005, http://xxx.lanl.gov/ pdf/math/0506386v1

71.Linfan Mao, Combinatorial Speculations and the Combinatorial Conjecture for Mathematics, 2006, http://xxx.lanl.gov/pdf/math/0606702v2

72.Linfan Mao, Pseudo-Manifold Geometries with Applications, 2006, http://xxx.lanl.gov/ pdf/math/0610307v1

73.Linfan Mao, Geometrical Theory on Combinatorial Manifolds, 2006, http://xxx.lanl.gov/pdf/ math/0612760v1

74.Linfan Mao, A generalization of Stokes theorem on combinatorial manifolds, 2007, http://xxx.lanl.gov/pdf/math/0703400v1

75.D. Rabounski, Smarandache Spaces as a New Extension of the Basic Space-Time of General Relativity, in “Progress in Physics”, Vol. II, p. L1, 2010.

IV.Refined Neutrosophics

76.Florentin Smarandache, n-Valued Refined Neutrosophic Logic and Its Applications in Physics, Progress in Physics, USA, 143-146, Vol. 4, 2013.

Florentin Smarandache (author and editor) Collected Papers, IX 56

Neutrosophic filters in BE-algebras

Akbar Rezaei, Arsham Borumand Saeid, Florentin Smarandache

Akbar Rezaei, Arsham Borumand Saeid, Florentin Smarandache (2015).

Neutrosophic filters in BE-algebras. Ratio Mathematica 29, 65-79

Abstract

In this paper, we introduce the notion of (implicative) neutrosophic filters in BE-algebras. The relation between implicative neutrosophic filters and neutrosophic filters is investigated and we show that in self distributive BEalgebras these notions are equivalent.

Keywords: BE-algebra, neutrosophic set, (implicative) neutrosophic filter.

1 Introduction

Neutrosophic set theory was introduced by Smarandache in 1998 ([10]). Neutrosophic sets are a new mathematical tool for dealing with uncertainties which are free from many difficulties that have troubled the usual theoretical approaches. Research works on neutrosophic set theory for many applications such as infor-mation fussion, probability theory, control theory, decision making, measurement theory, etc. Kandasamy and Smarandache introduced the concept of neutrosophic algebraic structures ([3, 4, 5]). Since then many researchers worked in this area and lots of literatures had been produced about the theory of neutrosophic set. In the neutrosophic set one can have elements which have paraconsistent information (sum of components > 1), others incomplete information (sum of components < 1), others consistent information (in the case when the sum of components =1) and others interval-valued components (with no restriction on their superior or inferior sums).

Florentin Smarandache (author and editor) Collected Papers, IX 57

H.S. Kim and Y.H. Kim introduced the notion of a BE-algebra as a generaliza-tion of a dual BCK-algebra ([6]). B.L. Meng give a procedure which generated a filter by a subset in a transitive BE-algebra ([7]). A. Walendziak introduced the no-tion of a normal filter in BE-algebras and showed that there is a bijection between congruence relations and filters in commutative BE-algebras ([11]). A. Borumand Saeid and et al. defined some types of filters in BE algebras and showed the re-lationship between them ([1]). A. Rezaei and et al. discussed on the relationship between BE-algebras and Hilbert algebras ([9]). Recently, A. Rezaei and et al. introduced the notion of hesitant fuzzy (implicative) filters and get some results on BE-algebras ([8]).

In this paper, we introduce the notion of (implicative) neutrosophic filters and study it in details. In fact, we show that in self distributive BE-algebras concepts of implicative neutrosophic filter and neutrosophic filter are equivalent.

2 Preliminaries

In this section, we cite the fundamental definitions t hat w ill b e u sed i n the sequel:

Definition 2 .1. [6] By a BE-algebra we shall mean an algebra X= (X; ∗, 1 ) of type (2, 0) satisfying the Aollowing axioms: (BE1) x ∗ x =1, (BE2) x ∗ 1=1, (BE3) 1 ∗ x = x, (BE4) x ∗ (y ∗ z) = y ∗ (x ∗ z), for all x, y, z ∈ X.

From now on, X is a BE-algebra, unless otherwise is stated. We introduce a relation “≤” on X by x ≤ y if and only if x ∗ y = 1. A BE-algebra X is said to be self distributive if x ∗ (y ∗ z) = (x ∗ y) ∗ (x ∗ z), for all x, y, z ∈ X. A BE-algebra X is said to be commutative if satisﬁes: (x ∗ y) ∗ y = (y ∗ x) ∗ x, for all x, y ∈ X.

Florentin Smarandache (author and editor) Collected Papers, IX 58

Proposition2.1. [11]If X isacommutativeBE-algebra,thenforall x,y ∈ X, x ∗ y =1 and y ∗ x =1 imply x = y.

Wenotethat“≤”isreﬂexiveby(BE1).If X isselfdistributivethenrelation“≤” isatransitiveorderedseton X,becauseif x ≤ y and y ≤ z,then x ∗ z =1 ∗ (x ∗ z)=(x ∗ y) ∗ (x ∗ z)= x ∗ (y ∗ z)= x ∗ 1=1

Hence x ≤ z.If X iscommutativethenbyProposition2.1,relation“≤”isantisymmetric.Henceif X isacommutativeselfdistributiveBE-algebra,thenrelation “≤”isapartialorderedseton X.

Proposition2.2. [6]InaBE-algebra X,thefollowinghold: (i) x ∗ (y ∗ x)=1, (ii) y ∗ ((y ∗ x) ∗ x)=1, forall x,y ∈ X.

Asubset F of X iscalledafilterof X ifitsatisfies:(F1) 1 ∈ F, (F2) x ∈ F and x ∗ y ∈ F imply y ∈ F .Define

A(x,y)= {z ∈ X : x ∗ (y ∗ z)=1}, whichiscalledanuppersetof x and y.Itiseasytoseethat 1,x,y ∈ A(x,y),for any x,y ∈ X. Everyupperset A(x,y) neednotbeaﬁlterof X ingeneral.

Definition2.2. [1]Anon-emptysubset F of X iscalledanimplicativefilterif satisfiesthefollowingconditions: (IF1) 1 ∈ F , (IF2) x ∗ (y ∗ z) ∈ F and x ∗ y ∈ F implythat x ∗ z ∈ F ,forall x,y,z ∈ X.

Ifwereplace x ofthecondition(IF2)bytheelement1,thenitcanbeeasily observedthateveryimplicativefilterisafilter.However,everyfilterisnotan implicativefilterasshowninthefollowingexample.

Example2.1. Let X = {1,a,b} beaBE-algebrawiththefollowingtable: ∗ 1 ab 1 1 ab a 11 a b 1 a 1

Then F = {1,a} isaﬁlterof X,butitisnotanimplicativeﬁlter,since 1 ∗ (a ∗ b)=1 ∗ a = a ∈ F and 1 ∗ a = a ∈ F but 1 ∗ b = b/ ∈ F

Florentin Smarandache (author and editor) Collected Papers, IX 59

Deﬁnition 2.3. [10] Let X be a set. A neutrosophic subset A of X i s a triple (TA, IA, FA) where TA : X → [0, 1] is the membership function, IA : X → [0, 1] is the indeterminacy function and FA : X → [0, 1] is the nonmembership function. Here for each x ∈ X, TA(x), IA(x) and FA(x) are all standard real numbers in [0, 1]

We note that 0 ≤ TA(x) + IA(x) + FA(x) ≤ 3, for all x ∈ X. The set of neutrosophic subset of X is denoted by NS(X).

Deﬁnition 2.4. [10] Let A and B be two neutrosophic sets on X . Deﬁne A ≤ B if and only if TA(x) ≤ TB (x), IA(x) ≥ IB (x), FA(x) ≥ FB (x), for all x ∈ X.

Deﬁnition 2.5. Let X 1 = (X1; ∗, 1) and X 2 = (X2; ◦, 1 ) be two BE-algebras. Then a mapping f : X1 → X2 is called a homomorphism if, for all x1, x2 ∈ X1 f (x1 ∗ x2) = f (x1) ◦ f (x2) It is clear that if f : X1 → X2 is a homomorphism, then f (1) = 1

3 Neutrosophic Filters

Definition 3.1. A neutrosophic set A of X is called a neutrosophic filter if satisfies the following conditions: (NF1) TA(x) ≤ TA(1),IA(x) ≥ IA(1) and FA(x) ≥ FA(1), (NF2) min{TA(x ∗ y),TA(x)}≤ TA(y), min{IA(x ∗ y),IA(x)}≥ IA(y) and min{FA(x ∗ y),FA(x)}≥ FA(y),forall x,y ∈ X .

Thesetofneutrosophicﬁlterof X isdenotedbyNF(X).

Example3.1. InExample2.1,put TA(1)=0 9, TA(a)= TA(b)=0 5, IA(1)=0 2, IA(a)= IA(b)=0 35 and FA(1)=0 1, FA(a)= FA(b)=0. Then A =(TA,IA,FA) isaneutrosophicﬁlter.

Proposition3.1. Let A ∈ NF(X).Then (i)if x ≤ y,then TA(x) ≤ TA(y), IA(x) ≥ IA(y) and FA(x) ≥ FA(y),

Florentin Smarandache (author and editor) Collected Papers, IX 60

(ii) TA(x) ≤ TA(y ∗ x), IA(x) ≥ IA(y ∗ x) and FA(x) ≥ FA(y ∗ x), (iii) min{TA(x),TA(y)}≤ TA(x ∗ y), min{IA(x),IA(y)}≥ IA(x ∗ y) and min{FA(x),FA(y)}≥ FA(x ∗ y), (iv) TA(x) ≤ TA((x∗y)∗y), IA(x) ≥ IA((x∗y)∗y) and FA(x) ≥ FA((x∗y)∗y), (v) min{TA(x),TA(y)}≤ TA((x ∗ (y ∗ z)) ∗ z), min{IA(x),IA(y)}≥ IA((x ∗ (y ∗ z)) ∗ z) and min{FA(x),FA(y)}≥ FA((x ∗ (y ∗ z)) ∗ z), (vi)if min{TA(y),TA((x ∗ y) ∗ z)}≤ TA(z ∗ x),then TA isorderreversingand IA, FA areorder (i.e.if x ≤ y,then TA(y) ≤ TA(x), IA(y) ≥ IA(x) and FA(y) ≥ FA(x)) (vii)if z ∈ A(x,y),then min{TA(x),TA(y)}≤ TA(z), min{IA(x),IA(y)}≥ IA(z) and min{FA(x),FA(y)}≥ FA(z) (viii)if n i=1 ai ∗ x =1,then n i=1 TA(ai) ≤ TA(x), n i=1 IA(ai) ≥ IA(x) and n i=1 FA(ai) ≥ FA(x) where n i=1 ai ∗ x = an ∗ (an 1 ∗ (... (a1 ∗ x) ... )).

Proof. (i).Let x ≤ y.Then x ∗ y =1 andso TA(x)=min{TA(x),TA(1)} =min{TA(x),TA(x ∗ y)}≤ TA(y), IA(x)=min{IA(x),IA(1)} =min{IA(x),IA(x ∗ y)}≥ IA(y), FA(x)=min{FA(x),FA(1)} =min{FA(x),FA(x ∗ y)}≥ FA(y). (ii).Since x ≤ y ∗ x,byusing(i)theproofisclear. (iii).Byusing(ii)wehave min{TA(x),TA(y)}≤ TA(y) ≤ TA(x ∗ y), min{IA(x),IA(y)}≥ IA(y) ≥ IA(x ∗ y), min{FA(x),FA(y)}≥ FA(y) ≥ FA(x ∗ y). (iv).ItfollowsfromDeﬁnition3.1, TA(x)=min{TA(x),TA(1)} =min{TA(x),TA((x ∗ y) ∗ (x ∗ y))} =min{TA(x),TA(x ∗ ((x ∗ y) ∗ y))} ≤ TA((x ∗ y) ∗ y)

Florentin Smarandache (author and editor) Collected Papers, IX 61

Also,wehave

IA(x)=min{IA(x),IA(1)} =min{IA(x),IA((x ∗ y) ∗ (x ∗ y))} =min{IA(x),IA(x ∗ ((x ∗ y) ∗ y))} ≥ IA((x ∗ y) ∗ y) and

FA(x)=min{FA(x),FA(1)} =min{FA(x),FA((x ∗ y) ∗ (x ∗ y))} =min{FA(x),FA(x ∗ ((x ∗ y) ∗ y))} ≥ FA((x ∗ y) ∗ y).

(v).From(iv)wehave

min{TA(x),TA(y)}≤ min{TA(x),TA((y ∗ (x ∗ z)) ∗ (x ∗ z))} =min{TA(x),TA((x ∗ (y ∗ z)) ∗ (x ∗ z))} =min{TA(x),TA(x ∗ (x ∗ (y ∗ z)) ∗ z))} ≤ TA((x ∗ (y ∗ z)) ∗ z)),

min{IA(x),IA(y)}≥ min{IA(x),IA((y ∗ (x ∗ z)) ∗ (x ∗ z))} =min{IA(x),IA((x ∗ (y ∗ z)) ∗ (x ∗ z))} =min{IA(x),IA(x ∗ (x ∗ (y ∗ z)) ∗ z))} ≥ IA((x ∗ (y ∗ z)) ∗ z)) and min{FA(x),FA(y)}≥ min{FA(x),FA((y ∗ (x ∗ z)) ∗ (x ∗ z))} =min{FA(x),FA((x ∗ (y ∗ z)) ∗ (x ∗ z))} =min{FA(x),FA(x ∗ (x ∗ (y ∗ z)) ∗ z))} ≥ FA((x ∗ (y ∗ z)) ∗ z))

(vi).Let x ≤ y,thatis, x ∗ y =1 TA(y)=min{TA(y),TA(1∗1)} =min{TA(y),TA((x∗y)∗1)}≤ TA(1∗x)= TA(x), IA(y)=min{IA(y),IA(1∗1)} =min{IA(y),IA((x∗y)∗1)}≥ IA(1∗x)= IA(x),

Florentin Smarandache (author and editor) Collected Papers, IX 62

FA(y)=min{FA(y),FA(1 ∗ 1)} =min{FA(y),FA((x ∗ y) ∗ 1)}≥ FA(1 ∗ x)= FA(x)

(vii).Let z ∈ A(x,y). Then x ∗ (y ∗ z)=1.Hence min{TA(x),TA(y)} =min{TA(x),TA(y),TA(1)} =min{TA(x),TA(y),TA(x ∗ (y ∗ z))} ≤ min{TA(y),TA(y ∗ z)} ≤ TA(z)

Also,wehave

min{IA(x),IA(y)} =min{IA(x),IA(y),IA(1)} =min{IA(x),IA(y),IA(x ∗ (y ∗ z))} ≥ min{IA(y),IA(y ∗ z)} ≥ IA(z), and min{FA(x),FA(y)} =min{FA(x),FA(y),FA(1)} =min{FA(x),FA(y),FA(x ∗ (y ∗ z))} ≥ min{FA(y),FA(y ∗ z)} ≥ FA(z).

(viii).Theproofisbyinductionon n.By(vii)itistruefor n =1, 2.Assume thatitsatisﬁesfor n = k,thatis, k i=1 ai ∗ x =1 ⇒ k i=1 TA(ai) ≤ TA(x), k i=1 IA(ai) ≥ IA(x)and k i=1 FA(ai) ≥ FA(x) forall a1,...,ak ,x ∈ X. Supposethat k+1 i=1 ai ∗ x =1,forall a1,...,ak ,ak+1,x ∈ X. Then k+1 i=2 TA(ai) ≤ TA(a1 ∗ x), k+1 i=2 IA(ai) ≥ IA(a1 ∗ x), and k+1 i=2 FA(ai) ≥ FA(a1 ∗ x). Since A isaneutrosophicﬁlterof X,wehave k+1 i=1 TA(ai)=min{( k+1 i=2 TA(ai)),TA(a1)}≤ min{TA(a1 ∗ x),TA(a1)}≤ TA(x),

Florentin Smarandache (author and editor) Collected Papers, IX 63

k+1 i=1 IA(ai)=min{( k+1 i=2 IA(ai)),IA(a1)}≥ min{IA(a1 ∗ x),IA(a1)}≥ IA(x) and k+1 i=1 FA(ai)=min{( k+1 i=2 FA(ai)),FA(a1)}≥ min{FA(a1 ∗ x),FA(a1)}≥ FA(x) ✷

Theorem3.1. If {Ai}i∈I isafamilyofneutrosophicﬁltersin X,then i∈I Ai istoo.

Theorem3.2. Let A ∈ NF(X).Thenthesets

(i) XTA = {x ∈ X : TA(x)= TA(1)}, (ii) XIA = {x ∈ X : IA(x)= IA(1)}, (iii) XFA = {x ∈ X : FA(x)= FA(1)}, areﬁltersof X.

Proof. (i).Obviously, 1 ∈ XhA Let x,x ∗ y ∈ XTA .Then TA(x)= TA(x ∗ y)= TA(1).Now,by(NF1)and(NF2),wehave TA(1)=min{TA(x),TA(x ∗ y)}≤ TA(y) ≤ TA(1) Hence TA(y)= TA(1). Therefore, y ∈ XTA . Theproofsof(ii)and(iii)aresimilarto(i).✷

Definition3.2. Aneutrosophicset A of X iscalledanimplicativeneutrosophic filterof X ifsatisfiesthefollowingconditions:

(INF1) TA(1) ≥ TA(x), (INF2) TA(x ∗ z) ≥ min{TA(x ∗ (y ∗ z)),TA(x ∗ y)}, IA(x ∗ z) ≤ min{IA(x ∗ (y ∗ z)),IA(x ∗ y)} and FA(x ∗ z) ≤ min{FA(x ∗ (y ∗ z)),FA(x ∗ y)},forall x,y,z ∈ X.

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The set of implicative neutrosophic filter of X is denoted by INF(X). If we replace x of the condition (INF2) by the element 1, then it can be easily observed that every implicative neutrosophic filter is a neutrosophic filter. However, every neutrosophic filter is not an implicative neutrosophic filter as shown in the following example.

Example 3.2. Let X = {1, a, b, c, d} be a BE-algebra with the following table: ∗ 1 abcd 1 1 abcd a 11 bcb b 1 a 1 ba c 1 a 11 a d 111 b 1

Then X =(X; ∗, 1) isaBE-algebra.Deﬁneaneutrosophicset A on X as follows:

TA(x)= 0.85 if x =1,a 0.12 otherwise and IA(x)= FA(x)=0.5,forall x ∈ X.

Then clearly A = (TA, IA, FA) is a neutrosophic ﬁlter of X , but it is not an implicative neutrosophic ﬁlter of X, since TA(b ∗ c) ≥ min{TA(b ∗ (d ∗ c)), TA(b ∗ d)}

Theorem 3.3. Let X be a self distributive BE-algebra. Then every neutrosophic ﬁlter is an implicative neutrosophic ﬁlter.

Proof. Let A ∈ NF(X) and x ∈ X Obvious that TA(x) ≤ TA(1), IA(x) ≥ IA(1) and FA(x) ≥ FA(1) By self distributivity and (NF2), we have min{TA(x∗(y ∗z)), TA(x∗y)} = min{TA((x∗y)∗(x∗z)), TA(x∗y)} ≤ TA(x∗z), min{IA(x ∗ (y ∗ z)), IA(x ∗ y)} = min{IA((x ∗ y) ∗ (x ∗ z)), IA(x ∗ y)} ≥ IA(x ∗ z) and min{FA(x∗(y∗z)), FA(x∗y)} = min{FA((x∗y)∗(x∗z)), FA(x∗y)} ≥ FA(x∗z).

Therefore A ∈ INF(X).✷

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Let t ∈ [0, 1]. For a neutrosophic ﬁlter A of X, t-level subset which denoted by U (A; t) is deﬁned as follows:

U (A; t) := {x ∈ A : t ≤ TA(x), IA(x) ≤ t and FA(x) ≤ t} and strong t-level subset which denoted by U (A; t)> as U (A; t)> := {x ∈ A : t < TA(x), IA(x) < t and FA(x) < t}

Theorem 3.4. Let A ∈ NS(X). The following are equivalent: (i) A ∈ NF(X), (ii) (∀t ∈ [0, 1]) U (A; t) = ∅ imply U (A; t) isaﬁlterof X.

Proof. (i)⇒(ii).Let x,y ∈ X besuchthat x,x ∗ y ∈ U (A; t),forany t ∈ [0, 1]. Then t ≤ TA(x) and t ≤ TA(x∗y).Hence t ≤ min{TA(x),TA(x∗y)}≤ TA(y).Also, IA(x) ≤ t and IA(x ∗ y) ≤ t andso t ≥ min{IA(x),IA(x ∗ y)}≥ IA(y).Byasimilarargumentwehave t ≥ min{FA(x),FA(x ∗ y)}≥ FA(y) Therefore, y ∈ U (A; t) (ii)⇒(i).Let U (A; t) beaﬁlterof X,forany t ∈ [0, 1] with U (A; t) = ∅.Put TA(x)= IA(x)= FA(x)= t,forany x ∈ X. Then x ∈ U (A; t).Since U (A; t) isaﬁlterof X,wehave 1 ∈ U (A; t) andso TA(x)= t ≤ TA(1) Now,forany x,y ∈ X,let TA(x ∗ y)= IA(x ∗ y)= FA(x ∗ y)= tx∗y and TA(x)= IA(x)= FA(x)= tx.Put t =min{tx∗y ,tx}.Then x,x ∗ y ∈ U (A; t), so y ∈ U (A; t).Hence t ≤ TA(y), t ≥ IA(y), t ≥ FA(y) andso min{TA(x ∗ y),TA(x)} =min{tx∗y ,tx} = t ≤ TA(y), min{IA(x ∗ y),IA(x)} =min{tx∗y ,tx} = t ≥ IA(y), and min{FA(x ∗ y),FA(x)} =min{tx∗y ,tx} = t ≥ FA(y) Therefore, A ∈ NF(X).✷

Theorem3.5. Let A ∈ NF(X).Thenwehave (∀a,b ∈ X)(∀t ∈ [0, 1])(a,b ∈ U (A; t) ⇒ A(a,b) ⊆ U (A; t))

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Proof. Assume that A ∈ NF(X) Let a, b ∈ X be such that a, b ∈ U (A; t) Then t ≤ TA(a) and t ≤ TA(b) Let c ∈ A(a, b) Hence a ∗ (b ∗ c) = 1 Now, by Proposition 3.1(v) and (BE3), we have

t ≤ min{TA(a), TA(b)} ≤ TA((a ∗ (b ∗ c) ∗ c)) = TA(1 ∗ c) = TA(c), t ≥ min{IA(a), IA(b)} ≥ IA((a ∗ (b ∗ c) ∗ c)) = IA(1 ∗ c) = IA(c) and t ≥ min{FA(a),FA(b)}≥ FA((a ∗ (b ∗ c) ∗ c))= FA(1 ∗ c)= FA(c) Then c ∈ U (A; t).Therefore, A(a,b) ⊆ U (A; t)) ✷

Corolary3.1. Let A ∈ NF(X).Then (∀t ∈ [0, 1])(U (A; t) = ∅⇒ U (A; t)= a,b∈U (A;t) A(a,b)). Proof. Itissufﬁcientprovethat U (A; t) ⊆ a,b∈U (A;t) A(a,b) Forthis,assume that x ∈ U (A; t).Since x ∗ (1 ∗ x)=1,wehave x ∈ A(x, 1).Hence U (A; t) ⊆ A(x, 1) ⊆ x∈U (A;t) A(x, 1) ⊆ x,y∈U (A;t) A(x,y) ✷

Theorem3.6. Let X beaselfdistributiveBE-algebraand A ∈ NF(X).Thenthe followingconditionsareequivalent:

(i) A ∈ INF(X), (ii) TA(y ∗ (y ∗ x)) ≤ TA(y ∗ x), IA(y ∗ (y ∗ x)) ≥ IA(y ∗ x) and FA(y ∗ (y ∗ x)) ≥ FA(y ∗ x), (iii) min{TA((z ∗ (y ∗ (y ∗ x))),TA(z)}≤ TA(y ∗ x), min{IA((z ∗ (y ∗ (y ∗ x))),IA(z)}≥ IA(y ∗ x) and min{FA((z ∗ (y ∗ (y ∗ x))),FA(z)}≥ FA(y ∗ x).

Florentin Smarandache (author and editor) Collected Papers, IX 67

Proof. (i)⇒(ii).Let A ∈ NF(X).By(INF1)and(BE1)wehave

TA(y ∗ (y ∗ x))=min{TA(y ∗ (y ∗ x)),TA(1)} =min{TA(y ∗ (y ∗ x)),TA(y ∗ y)} ≤ TA(y ∗ x),

IA(y ∗ (y ∗ x))=min{IA(y ∗ (y ∗ x)),IA(1)} =min{IA(y ∗ (y ∗ x)),IA(y ∗ y)} ≥ IA(y ∗ x) and FA(y ∗ (y ∗ x))=min{FA(y ∗ (y ∗ x)),FA(1)} =min{FA(y ∗ (y ∗ x)),FA(y ∗ y)} ≥ FA(y ∗ x).

(ii)⇒(iii).Let A beaneutrosophicﬁlterof X satisfyingthecondition(ii).By using(NF2)and(ii)wehave

min{TA(z ∗ (y ∗ (y ∗ x))),TA(z)}≤ TA(y ∗ (y ∗ x)) ≤ TA(y ∗ x), min{IA(z ∗ (y ∗ (y ∗ x))),IA(z)}≥ IA(y ∗ (y ∗ x)) ≥ IA(y ∗ x) and min{FA(z ∗ (y ∗ (y ∗ x))),FA(z)}≥ FA(y ∗ (y ∗ x)) ≥ FA(y ∗ x). (iii)⇒(i).Since x ∗ (y ∗ z)= y ∗ (x ∗ z) ≤ (x ∗ y) ∗ (x ∗ (x ∗ z)), wehave TA(x ∗ (y ∗ z)) ≤ TA((x ∗ y) ∗ (x ∗ (x ∗ z))), IA(x ∗ (y ∗ z)) ≥ IA((x ∗ y) ∗ (x ∗ (x ∗ z))) and FA(x ∗ (y ∗ z)) ≥ FA((x ∗ y) ∗ (x ∗ (x ∗ z))), byProposition3.1(i).Thus min{TA(x ∗ (y ∗ z)),TA(x ∗ y)}≤ min{TA((x ∗ y) ∗ (x ∗ (x ∗ z))),TA(x ∗ y)} ≤ TA(x ∗ z)

Florentin Smarandache (author and editor) Collected Papers, IX 68

min{IA(x ∗ (y ∗ z)), IA(x ∗ y)} ≥ min{IA((x ∗ y) ∗ (x ∗ (x ∗ z))), IA(x ∗ y)} ≥ IA(x ∗ z) and min{FA(x ∗ (y ∗ z)), FA(x ∗ y)} ≥ min{FA((x ∗ y) ∗ (x ∗ (x ∗ z))), FA(x ∗ y)} ≥ FA(x ∗ z)

Therefore, A ∈ INF(X) Let f : X → Y be a homomorphism of BE-algebras and A ∈ NS(X) Deﬁne tree maps T Af : X → [0, 1] such that TAf (x) = TA(f (x)), IAf : X → [0, 1] such that IAf (x) = IA(f (x)) and FAf : X → [0, 1] such that FAf (x) = FA(f (x)), for all x ∈ X. Then TAf , IAf and FAf are well-deﬁne and Af = (TAf , IAf , FAf ) ∈ NS(X) ✷

Theorem 3.7. Let f : X → Y be an onto homomorphism of BE-algebras and A ∈ NS(Y). Then A ∈ NF(Y) (resp. A ∈ INF(Y)) if and only if Af ∈ NF(X) (resp. Af ∈ INF(X)).

Proof. Assumethat A ∈ NF(Y).Forany x ∈ X,wehave

TAf (x)= TA(f (x)) ≤ TA(1Y )= TA(f (1X ))= TAf (1X ), IAf (x)= IA(f (x)) ≥ IA(1Y )= IA(f (1X ))= IAf (1X ) and

FAf (x)= FA(f (x)) ≥ FA(1Y )= FA(f (1X ))= FAf (1X ). Hence(NF1)isvalid.Now,let x,y ∈ X.By(NF1)wehave

min{TAf (x ∗ y),TAf (x)} =min{TA(f (x ∗ y)),TA(f (x))} =min{TA(f (x) ∗ f (y)),TA(f (x))} ≤ TA(f (y)) = TAf (y) Also, min{IAf (x ∗ y),IAf (x)} =min{IA(f (x ∗ y)),IA(f (x))} =min{IA(f (x) ∗ f (y)),IA(f (x))} ≥ IA(f (y)) = IAf (y)

Florentin Smarandache (author and editor) Collected Papers, IX 69

By a similar argument we have min{FAf (x ∗ y), FAf (x)} ≥ FAf (y) Therefore, Af ∈ NF(X)

Conversely, Assume that Af ∈ NF(X) Let y ∈ Y Since f is onto, there exists x ∈ X such that f (x) = y. Then

TA(y) = TA(f (x)) = TAf (x) ≤ TAf (1X ) = TA(f (1X )) = TA(1Y ), IA(y) = IA(f (x)) = IAf (x) ≥ IAf (1X ) = IA(f (1X )) = IA(1Y ) and FA(y) = FA(f (x)) = FAf (x) ≥ FAf (1X ) = FA(f (1X )) = FA(1Y ), Now, let x, y ∈ Y . Then there exists a, b ∈ X such that f (a) = x and f (b) = y. Hence we have min{TA(x ∗ y),TA(x)} =min{TA(f (a) ∗ f (b)), TA(f (a))} =min{TA(f (a ∗ b)),TA(f (a))} =min{TAf (a ∗ b),TAf (a)} ≤ TAf (b) = TA(f (b)) = TA(y).

Also,wehave min{IA(x ∗ y),IA(x)} =min{IA(f (a) ∗ f (b)),IA(f (a))} =min{IA(f (a ∗ b)),IA(f (a))} =min{IAf (a ∗ b),IAf (a)} ≥ IAf (b) = IA(f (b)) = IA(y).

By a similar argument we have min{FA(x ∗ y), FA(x)} ≥ FA(y) Therefore, A ∈ NF(Y).✷

4 Conclusion

F Smarandache as an extension of intuitionistic fuzzy logic introduced the concept of neutrosophic logic and then several researchers have studied of some neutrosophic algebraic structures. In this paper, we applied the theory of neutrosophic sets to BE-algebras and introduced the notions of (implicative) neutrosophic filters and many related properties are investigated.

Florentin Smarandache (author and editor) Collected Papers, IX 70

References

[1]A.BorumandSaeid,A.Rezaei,R.A.Borzooei, SometypesofﬁltersinBEalgebras,Math.Comput.Sci.,7(3)(2013),341–352.

[2]R.A.Borzooei,H.Farahani,M.Moniri, Neutrosophicdeductiveﬁlterson BL-algebras,JournalofIntelligent&FuzzySystems,26(2014),2993–3004.

[3]W.B.V.Kandasamy,K.Ilanthenral,F.Smarandache, Introductiontolinear Bialgebra,Hexis,Phoenix,Arizona,2005.

[4]W.B.V.Kandasamy,F.Smarandache, SomeneutrosophicalgebraicstructuresandneutrosophicN-algebraicstructures,Hexis,Phoenix,Arizona, 2006.

[5]W.B.V.Kandasamy,F.Smarandache, Neutrosophicrings,Hexis,Phoenix, Arizona,2006.

[6]H.S.Kim,Y.H.Kim, OnBE-algebras,Sci,Math,Jpn.,66(1)(2007),113–116.

[7]B.L.Meng, OnﬁltersinBE-algebras,Sci.Math.Jpn.,71(2010),201–207.

[8]A.Rezaei,A.BorumandSaeid, HesitantfuzzyﬁltersinBE-algebras,Int.J. Comput.Int.Sys.,9(1)(2016)110–119.

[9]A.Rezaei,A.BorumandSaeid,R.A.Borzooei, RelationbetweenHilbert algebrasandBE-algebras,Applic.Applic.Math,8(2)(2013),573–584.

[10]F.Smarandache, Neutrosophy,NeutrosophicProbability,Set,andLogic, Amer.Res.Press,Rehoboth,USA,105p.,1998.

[11]A.Walendziak, OnnormalﬁltersandcongruencerelationsinBE-algebras, Commentationesmathematicae,52(2)(2012),199–205.

Florentin Smarandache (author and editor) Collected Papers, IX 71

Isomorphism of Bipolar Single Valued Neutrosophic Hypergraphs

Muhammad Aslam Malik, Ali Hassan, Said Broumi, Assia Bakali, Mohamed Talea, Florentin

Smarandache

Muhammad Aslam Malik, Ali Hassan, Said Broumi, Assia Bakali, Mohamed Talea, Florentin Smarandache (2016). Isomorphism of Bipolar Single Valued Neutrosophic Hypergraphs. Critical Review XIII, 79-102

Abstract

Inthispaper,weintroducethehomomorphism,theweakisomorphism,theco-weak isomorphism,andtheisomorphismofthebipolarsinglevaluedneutrosophichypergraphs. The properties of order, size and degree of vertices are discussed. The equivalence relation of the isomorphism of the bipolar single valued neutrosophic hypergraphs and the weak isomorphism of bipolar single valued neutrosophic hypergraphs,togetherwiththeirpartialorderrelation,isalsoverified.

Keywords

homomorphism, weak-isomorphism, co-weak-isomorphism, isomorphism, bipolar singlevaluedneutrosophichypergraphs.

1 Introduction

The neutrosophic set proposed by Smarandache [8] as a generalization of the fuzzy set [14], intuitionistic fuzzy set [12], interval valued fuzzy set [11] and interval valued intuitionistic fuzzy set [13] theories is a mathematical tool created to deal with incomplete, indeterminate and inconsistent informationintherealworld.Thecharacteristicsoftheneutrosophicsetare the truth membership function (t), the indeterminacy membership function (i),andthefalsitymembershipfunction(f),whichtakevalueswithinthereal standardornon standardunitinterval]0,1+[.

Florentin Smarandache (author and editor) Collected Papers, IX 72

Asubclassoftheneutrosophicset,thesingle valuedneutrosophicset(SVNS), was intoduced by Wang et al. [9]. The same authors [10] also introduced a generalization of the single valued neutrosophic set, namely the interval valuedneutrosophicset(IVNS),inwhichthethreemembershipfunctionsare independent, and their values belong to the unit interval [0, 1]. The IVNS is morepreciseandflexiblethanthesinglevaluedneutrosophicset.

Moreworksonsinglevaluedneutrosophicsets,intervalvaluedneutrosophic setsandtheirapplicationscanbefoundon http://fs.gallup.unm.edu/NSS/.

In this paper, we extend the isomorphism of the bipolar single valued neutrosophichypergraphs,andintroducesomeoftheirrelevantproperties.

1 Preliminaries

Definition 2.1

A hypergraph is an ordered pair H = (X, E), where:

(1) X = {��1, ��2, ,��} is a finite set of vertices.

(2) E = {��1, ��2 , …, ��} is a family of subsets of X

(3) �� are non void for j = 1, 2, 3, ..., m, and ⋃ (��) �� =��

The set X is called 'set of vertices', and E is denominated as the 'set of edges' (or 'hyper edges').

Definition 2.2

A fuzzy hypergraph H = (X, E) is a pair, where X is a finite set and E is a finite family of non trivial fuzzy subsets of X, such that �� =∪�� (��), ��= 1,2,3,…,��.

Remark 2.3

The collection �� ={��1,��2,��3,….,��} is a collection of edge set of H.

Definition 2.4

A fuzzy hypergraph with underlying set X is of the form H = (X, E, R), where �� ={��1,��2,��3,…,��} is the collection of fuzzy subsets of X, that is �� ∶�� → [0,1], j = 1, 2, 3, ..., m, and �� ∶�� →[0,1] is the fuzzy relation of the fuzzy subsets ��, such that: ��(��1,��2, ,��) ≤min(��(��1), ..., ��(��)), (1) for all { ��1,��2, ,��} subsets of X.

Florentin Smarandache (author and editor) Collected Papers, IX 73

Definition 2.5

Let X be a space of points (objects) with generic elements in X denoted by x. A single valued neutrosophic set A (SVNS A) is characterized by its truth member ship function ��(x), its indeterminacy membership function ��(x), and its falsity membership function ��(x). For each point, x ∈ X; ��(x), ��(x), ��(x) ∈ [0, 1].

Definition 2.6

Asinglevaluedneutrosophichypergraphisanorderedpair H = (X, E), where:

(1) X = {��1, ��2, ,��} isafinitesetofvertices. (2) E = {��1, ��2 , …, ��} isafamilyofSVNSsof X. (3)�� ≠ O = (0, 0, 0) for j= 1, 2, 3, ..., m, and⋃ ��(��) �� = ��.

Theset X iscalledsetofverticesand E isthesetofSVN edges(orSVN hyper edges).

Proposition 2.7

The single valued neutrosophic hypergraph is the generalization of fuzzy hypergraphsandintuitionisticfuzzyhypergraphs.

NotethatagivenSVNHGH = (X, E, R),withunderlyingsetX,where E = {��1, ��2 , , ��}, isthecollectionofthenon emptyfamilyofSVNsubsetsof X, and R is theSVNrelationoftheSVNsubsets��,suchthat: ��(��1,��2,… ,��)≤min([��(��1)],…,[��(��)]), (2) ��(��1,��2, ,��)≥max([��(��1)], ,[��(��)]), (3) ��(��1,��2,… ,��)≥max([��(��1)],…,[��(��)]), (4) forall{��1,��2, ,��} subsetsof X.

Definition 2.8

Let X beaspaceofpoints(objects)withgenericelementsin X denotedby x. Abipolarsinglevaluedneutrosophicset A (BSVNS A)ischaracterizedbythe positive truth membership function ��(x), the positive indeterminacy membershipfunction��(x), thepositivefalsitymembershipfunction��(x), the negative truth membership function��(x), the negative indeterminacy membership function��(x), and the negative falsity membership function ��(x)

For each point x∈X;��(x),��(x),��(x) ∈[0, 1], and��(x),��(x),��(x) ∈[ 1,0].

Florentin Smarandache (author and editor) Collected Papers, IX 74

Abipolarsinglevaluedneutrosophichypergraphisanorderedpair H = (X, E), where:

(1) X = {��1, ��2,…,��} isafinitesetofvertices.

(2) E = {��1, ��2 , …, ��} isafamilyofBSVNSsof X.

(3)�� ≠ O = ([0, 0], [0, 0], [0, 0]) for j = 1, 2, 3, ..., m, and ⋃ ��(��) �� = X

Theset X iscalledthe'setofvertices'and E iscalledthe'setofBSVN edges' (or 'IVN-hyper-edges'). Note that a given BSVNHGH = (X, E, R), with underlying set X, where E = { ��1, ��2 , …, ��} is the collection of non empty family of BSVN subsets of X, and R is the BSVN relation of BSVN subsets�� suchthat: ��(��1,��2,… ,��)≤min([��(��1)],….,[��(��)]), (5) ��(��1,��2,… ,��)≥max([��(��1)],….,[��(��)]), (6) ��(��1,��2,… ,��)≥max([��(��1)],….,[��(��)]), (7) ��(��1,��2,… ,��)≥max([��(��1)],….,[��(��)]), (8) ��(��1,��2,… ,��)≤min([��(��1)],….,[��(��)]), (9) ��(��1,��2,… ,��)≤min([��(��1)],….,[��(��)]), (10) forall{��1,��2,…,��} subsetsof X.

Proposition 2.10

The bipolar single valued neutrosophic hypergraph is the generalization of the fuzzy hypergraph, intuitionistic fuzzy hypergraph, bipolar fuzzy hyper graphandintuitionisticfuzzyhypergraph

Example 2.11

ConsidertheBSVNHG H = (X, E, R), withunderlyingset X = {a, b, c}, where E = {A, B},and R definedin Tables below:

H A B

a (0.2, 0.3, 0.9, 0.2, 0.2, 0.3) (0.5, 0.2, 0.7, 0.4, 0.2, 0.3) b (0.5, 0.5, 0.5, 0.4, 0.3, 0.3) (0.1, 0.6, 0.4, 0.9, 0.3, 0.4) c (0.8, 0.8, 0.3, 0.9, 0.2, 0.3) (0.5, 0.9, 0.8, 0.1, 0.2, 0.3)

R �� A 0.2 0.8 0.9 0.1 0.4 0.5 B 0.1 0.9 0.8 0.1 0.5 0.6

Definition 2.9

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Byroutinecalculations, H = (X, E, R) isBSVNHG.

3 Isomorphism of BSVNHGs

Definition 3.1

Ahomomorphism f: H →K betweentwoBSVNHGs H = (X, E, R) and K = (Y, F, S) isamapping f: X → Y whichsatisfiestheconditions: min[��(��)] ≤min[��(��(��))], (11) max[��(��)] ≥max[��(��(��))], (12) max[��(��)] ≥max[��(��(��))], (13) max[��(��)] ≥max[��(��(��))], (14) min[��(��)] ≤min[��(��(��))], (15) min[��(��)] ≤min[��(��(��))], (16)

for all x∈��. ��(��1,��2, ,��)≤ ��(��(��1),��(��2), ,��(��)), (17) ��(��1,��2,…,��)≥��(��(��1),��(��2),…,��(��)), (18) ��(��1,��2,…,��)≥ ��(��(��1),��(��2),…,��(��)), (19) ��(��1,��2, ,��)≥ ��(��(��1),��(��2), ,��(��)), (20) ��(��1,��2, ,��)≤��(��(��1),��(��2), ,��(��)), (21) ��(��1,��2,…,��)≤ ��(��(��1),��(��2),…,��(��)), (22) for all {��1,��2,…,��} subsets of X.

Example 3.2 Consider the two BSVNHGs H = (X, E, R) and K = (Y, F, S) with underlying sets X = {a, b, c} and Y = {x, y, z}, where E = {A, B}, F = {C, D}, RandS, which are defined in Tables given below:

H A B

a (0.2, 0.3, 0.9, 0.2, 0.2, 0.3) (0.5, 0.2, 0.7, 0.4, 0.2, 0.3) b (0.5, 0.5, 0.5, 0.4, 0.3, 0.3) (0.1, 0.6, 0.4, 0.9, 0.3, 0.4) c (0.8, 0.8, 0.3, 0.9, 0.2, 0.3) (0.5, 0.9, 0.8, 0.1, 0.2, 0.3)

K C D

x (0.3, 0.2, 0.2, 0.9, 0.2, 0.3) (0.2, 0.1, 0.3, 0.6, 0.1, 0.2) y (0.2, 0.4, 0.2, 0.4, 0.2, 0.3) (0.3, 0.2, 0.1, 0.7, 0.2, 0.1) z (0.5, 0.8, 0.2, 0.2, 0.1, 0.3) (0.9, 0.7, 0.1, 0.2, 0.1, 0.3)

Florentin Smarandache (author and editor) Collected Papers, IX 76

R ��

A 0.2 0.8 0.9 -0.1 -0.4 -0.5 B 0.1 0.9 0.8 0.1 0.5 0.6 S �� C 0.2 0.8 0.3 0.1 0.2 0.3 D 0.1 0.7 0.3 0.1 0.2 0.3

and f : X → Y defined by: f(a)=x , f(b)=y and f(c)=z. Then, by routine calculations, f: H → K is a homomorphism between H and K

Definition 3.3

A weak isomorphism f : H → K between two BSVNHGs H = (X, E, R) and K = (Y, F, S) is a bijective mapping f : X → Y which satisfies f is homomorphism, such that: min[��(��)] ≤min[��(��(��))], (23) max[��(��)] ≥max[��(��(��))], (24) max[��(��)] ≥max[��(��(��))], (25) max[��(��)] ≥max[��(��(��))], (26) min[��(��)] ≤min[��(��(��))], (27) min[��(��)] ≤min[��(��(��))], (28) for all x∈��.

Note

The weak isomorphism between two BSVNHGs preserves the weights of vertices. Example 3.4

Consider the two BSVNHGs H = (X, E, R) and K = (Y, F, S) with underlying sets X = {a, b, c} and Y = {x, y, z}, where E = {A, B}, F = {C, D}, R and S, which are defined by Tables given below, and f: X → Y defined by: f(a)=x, f(b)=y and f(c)=z Then, by routine calculations, f: H → K is a weak isomorphism between H and K

H A B

a (0.2, 0.3, 0.9, 0.2, 0.2, 0.3) (0.5, 0.2, 0.7, 0.4, 0.2, 0.3) b (0.5, 0.5, 0.5, 0.4, 0.3, 0.3) (0.1, 0.6, 0.4, 0.9, 0.3, 0.4) c (0.8, 0.8, 0.3, 0.9, 0.2, 0.3) (0.5, 0.9, 0.8, 0.1, 0.2, 0.3)

Florentin Smarandache (author and editor) Collected Papers, IX 77

x (0.2, 0.3, 0.2, 0.9, 0.2, 0.3) (0.2, 0.1, 0.8, 0.6, 0.1, 0.4) y (0.2, 0.4, 0.2, 0.4, 0.3, 0.3) (0.1, 0.6, 0.5, 0.6, 0.2, 0.3) z (0.5, 0.8, 0.9, 0.2, 0.2, 0.3) (0.9, 0.9, 0.1, 0.1, 0.3, 0.3)

R ��

A 0.2 0.8 0.9 0.1 0.4 0.3 B 0.1 0.9 0.9 0.1 0.3 0.5

S ��

C 0.2 0.8 0.9 0.1 0.3 0.2 D 0.1 0.9 0.8 0.1 0.3 0.4

Definition 3.5

A co weak isomorphism f: H → K between two BSVNHGs H = (X, E, R) and K = (Y, F, S) is a bijective mapping f: X → Y which satisfies f is homomorphism, such that: ��(��1,��2, ,��)= ��(��(��1),��(��2), ,��(��)), (29) ��(��1,��2, ,��) =��(��(��1),��(��2), ,��(��)), (30) ��(��1,��2,…,��) = ��(��(��1),��(��2),…,��(��)), (31) ��(��1,��2,…,��) = ��(��(��1),��(��2),…,��(��)), (32) ��(��1,��2,…,��) =��(��(��1),��(��2),…,��(��)), (33) ��(��1,��2,…,��)=��(��(��1),��(��2),…,��(��)), (34) for all {��1,��2, ,��} subsets of X.

Note

The co weak isomorphism between two BSVNHGs preserves the weights of edges.

Example 3.6

Consider the two BSVNHGs H = (X, E, R) and K = (Y, F, S) with underlying sets X = {a, b, c} and Y = {x, y, z}, where E = {A, B}, F = {C, D}, R and S, which are defined in Tables given below, and f : X → Y defined by: f(a)=x , f(b)=y and f(c)=z. Then, by routine calculations, f: H → K is a co-weak isomorphism between H and K H A B

a (0.2, 0.3, 0.9, 0.4, 0.2, 0.3) (0.5, 0.2, 0.7, 0.1, 0.2, 0.3) b (0.5, 0.5, 0.5, 0.4, 0.2, 0.3) (0.1, 0.6, 0.4, 0.4, 0.2, 0.3) c (0.8, 0.8, 0.3, 0.1, 0.2, 0.3) (0.5, 0.9, 0.8, 0.4, 0.2, 0.3)

Florentin Smarandache (author and editor) Collected Papers, IX 78

x (0.3, 0.2, 0.2, 0 9, 0.2, 0.3) (0.2, 0.1, 0.3, 0.4, 0.2, 0.3) y (0.2, 0.4, 0.2, 0.4, 0.2, 0.3) (0.3, 0.2,0.1, 0.9, 0.2, 0.3) z (0.5, 0.8, 0.2, 0.1, 0.2, 0.3) (0.9, 0.7, 0.1, 0.1, 0.2, 0.3)

R ��

A 0.2 0.8 0.9 0.1 0.2 0.3 B 0.1 0.9 0.8 0.1 0.2 0.3

S �� C 0.2 0.8 0.9 0.1 0.2 0.3 D 0.1 0.9 0.8 0.1 0.2 0.3

Definition 3.7

An isomorphism f : H → K between two BSVNHGs H = (X, E, R) and K = (Y, F, S) is a bijective mapping f : X → Y which satisfies the conditions: min[��(��)] =min[��(��(��))], (35) max[��(��)] =max[��(��(��))], (36) max[��(��)] =max[��(��(��))], (37) max[��(��)] =max[��(��(��))], (38) min[��(��)] =min[��(��(��))], (39) min[��(��)] =min[��(��(��))], (40)

for all x∈�� (��1,��2,…,��)= ��(��(��1),��(��2),…,��(��)), (41) ��(��1,��2,…,��) =��(��(��1),��(��2),…,��(��)), (42) ��(��1,��2,…,��) = ��(��(��1),��(��2),…,��(��)), (43) ��(��1,��2, ,��) = ��(��(��1),��(��2), ,��(��)), (44) ��(��1,��2, ,��) =��(��(��1),��(��2), ,��(��)), (45) ��(��1,��2,…,��)=��(��(��1),��(��2),…,��(��)), (46) for all {��1,��2,…,��} subsets of X.

Note

The isomorphism between two BSVNHGs preserves the both weights of vertices and weights of edges.

Florentin Smarandache (author and editor) Collected Papers, IX 79

Example 3.8

Consider the two BSVNHGs H = (X, E, R) and K = (Y, F, S) with underlying sets X = {a, b, c} and Y = {x, y, z}, where E = {A, B}, F = {C, D}, R and S, which are defined by Tables given below:

H A B

a (0.2, 0.3, 0.7, 0.2, 0.2, 0.3) (0.5, 0.2, 0.7, 0.6, 0.6, 0.6) b (0.5, 0.5, 0.5, 0.4, 0.3, 0.3) (0.1, 0.6, 0.4, 0.1, 0.2, 0.7) c (0.8, 0.8, 0.3, 0.9, 0.2, 0.4) (0.5, 0.9, 0.8, 0.7, 0.2, 0.3)

K C D

x (0.2, 0.3, 0.2, 0.2, 0.2, 0.4) (0.2, 0.1, 0.8, 0.3, 0.2, 0.3) y (0.2, 0.4, 0.2, 0.6, 0.2, 0.3) (0.1, 0.6, 0.5, 0.1, 0.2, 0.7) z (0.5, 0.8, 0.7, 0.4, 0.3, 0.3) (0.9, 0.9, 0.1, 0.9, 0.6, 0.3)

R �� A 0.2 0.8 0.9 0.1 0.3 0.4 B 0.0 0.9 0.8 0.1 0.7 0.8 S �� C 0.2 0.8 0.9 0.1 0.3 0.4 D 0.0 0.9 0.8 0.1 0.7 0.8 and f : X → Y defined by: f(a)=x , f(b)=y and f(c)=z. Then, by routine calculations, f: H → K is an isomorphism between H and K

Theorem 3.10 Let H = (X, E, R) and K = (Y, F, S) be two BSVNHGs such that H is isomorphic to K, then:

��(��) =(∑��(��(��)),∑��(��(��)),∑��(��(��)), ∑��(��(��)),∑��(��(��)),∑��(��(��))) (47) The size

H is

��(��)=(∑��(��),∑��(��),∑��(��),∑��(��), ∑��(��),∑��(��)) (48)

Definition 3.9 Let H = (X, E, R) be a BSVNHG, then the order of H is denoted and defined by as follows:

denoted and defined by:

Florentin Smarandache (author and editor) Collected Papers, IX 80

(1) O(H) = O(K), (2) S(H) = S(K)

Proof

Let f: H → K be an isomorphism between two BSVNHGs H and K with underlying sets X and Y respectively; then, by definition: min[��(��)] =min[��(��(��))], (49) max[��(��)] =max[��(��(��))], (50) max[��(��)] =max[��(��(��))], (51) max[��(��)] =max[��(��(��))], (52) min[��(��)] =min[��(��(��))], (53) min[��(��)] =min[��(��(��))], (54)

for all x∈�� (��1,��2,…,��)= ��(��(��1),��(��2),…,��(��)), (55) ��(��1,��2,…,��) =��(��(��1),��(��2),…,��(��)), (56) ��(��1,��2,…,��) = ��(��(��1),��(��2),…,��(��)), (57) ��(��1,��2,…,��) = ��(��(��1),��(��2),…,��(��)), (58) ��(��1,��2, ,��) =��(��(��1),��(��2), ,��(��)), (59) ��(��1,��2, ,��)=��(��(��1),��(��2), ,��(��)), (60) for all { ��1,��2,…,��} subsets of X.

Consider: ��(��)=∑min��(��)=∑min��(��(��))=��(��) (61) ��(��)=∑max��(��)=∑max��(��(��))=��(��) (62) Similarly, ��(��)=��(��)and��(��)=��(��) , ��(��)=��(��) and ��(��)=��(��), hence O(H) = O(K).

Next: ��(��)=∑��(��1,��2,…,��) =∑��(��(��1),��(��2),…,��(��)) =��(��) (63) Similarly, ��(��)=∑��(��1,��2,…,��) =∑��(��(��1),��(��2),…,��(��)) =��(��). (64) and ��(��)=��(��), ��(��)=��(��), ��(��)=��(��), ��(��)=��(��), hence ��(��)=��(��).

Florentin Smarandache (author and editor) Collected Papers, IX 81

Remark 3.11

The converse of the above theorem need not to be true in general.

Example 3.12

Consider the two BSVNHGs H = (X, E, R) and K = (Y, F, S) with underlying sets X = {a, b, c, d} and Y = {w, x, y, z}, where E = {A, B}, F = {C, D}, R and S are defined in Tables given below:

H A B

a (0.2, 0.5, 0.3, 0.1, 0.2, 0.3) (0.14, 0.5, 0.3, 0.1, 0.2, 0.3) b (0.0,0.0,0.0, 0.0, 0.0, 0.0) (0.2, 0.5, 0.3, 0.4, 0.2, 0.3) c (0.33, 0.5, 0.3, 0.4, 0.2, 0.3) (0.16, 0.5, 0.3, 0.1, 0.2, 0.3) d (0.5, 0.5, 0.3, 0.1, 0.2, 0.3) (0.0,0.0,0.0, 0.0, 0.0, 0.0)

K C D

w (0.14, 0.5, 0.3, 0.1, 0.2, 0.3) (0.2, 0.5, 0.33, 0.4, 0.2, 0.3) x (0.16, 0.5, 0.3, 0.1, 0.2, 0.3) (0.33,0.5, 0.33, 0.1, 0.2, 0.3) y (0.25, 0.5, 0.3, 0.1, 0.2, 0.3) (0.2, 0.5, 0.33, 0.1, 0.2, 0.3) z (0.5, 0.5, 0.3, 0.4, 0.2, 0.3) (0.0, 0.0, 0.0, 0.0, 0.0, 0.0)

R �� A 0.2 0.5 0.3 0.1 0.2 0.3 B 0.14 0.5 0.3 0.1 0.2 0.3 S �� C 0.14 0.5 0.3 0.1 0.2 0.3 D 0.2 0.5 0.3 0.1 0.2 0.3

where f is defined by: f(a)=w, f(b)=x, f(c)=y, f(d)=z

Here, O(H) = (1.0,2.0, 1.2, 0.7, 0.8, 1.2) = O(K) and S(H)=(0.34, 1.0, 0.9, 0.2, 0.4, 0.9)=S(K), but, by routine calculations, H is not an isomorphism to K.

Corollary 3.13

The weak isomorphism between any two BSVNHGs H and K preserves the orders. Remark 3.14

The converse of the above corollary need not to be true in general.

Example 3.15

Consider the two BSVNHGs H = (X, E, R) and K = (Y, F, S) with underlying sets X = {a, b, c, d} and Y = {w, x, y, z}, where E = {A, B}, F = {C, D}, R and S are defined in Tables given below, where f is defined by: f(a)=w, f(b)=x, f(c)=y, f(d)=z:

Florentin Smarandache (author and editor) Collected Papers, IX 82

H A B

a (0.2, 0.5, 0.3, 0.1, 0.2, 0.3) (0.14, 0.5, 0.3, 0.4, 0.2, 0.3) b (0.0,0.0,0.0,0.0,0.0,0.0) (0.2, 0.5, 0.3, 0.1, 0.2, 0.3)

c (0.33, 0.5, 0.3, 0.4, 0.2, 0.3) (0.16, 0.5, 0.3, 0.1, 0.2, 0.3) d (0.5, 0.5, 0.3, 0.1, 0.2, 0.3) (0.0,0.0,0.0,0.0,0.0,0.0)

K C D

w (0.14, 0.5, 0.3, 0.1, 0.2, 0.3) (0.16, 0.5, 0.3, 0.1, 0.2, 0.3) x (0.0, 0.0, 0.0,0.0,0.0,0.0) (0.16, 0.5, 0.3, 0.1, 0.2, 0.3) y (0.25, 0.5, 0.3, 0.1, 0.2, 0.3) (0.2, 0.5, 0.3, 0.4, 0.2, 0.3) z (0.5, 0.5, 0.3, 0.1, 0.2, 0.3) (0.0, 0.0, 0.0,0.0,0.0,0.0)

Here, O(H)= (1.0, 2.0, 1.2, 0.4, 0.8, 1.2) = O(K), but, by routine calculations, H is not a weak isomorphism to K

Corollary 3.16

The co weak isomorphism between any two BSVNHGs H and K preserves sizes.

Remark 3.17

The converse of the above corollary need not to be true in general.

Example 3.18

Consider the two BSVNHGs H = (X, E, R) and K = (Y, F, S) with underlying sets X = {a, b, c, d} and Y = {w, x, y, z}, where E = {A, B}, F = {C, D},R and S are defined in Tables given below,

H A B

a (0.2, 0.5, 0.3, 0.1, 0.2, 0.3) (0.14, 0.5, 0.3, 0.1, 0.2, 0.3) b (0.0,0.0,0.0,0.0,0.0,0.0) (0.16, 0.5, 0.3, 0.1, 0.2, 0.3) c (0.3, 0.5, 0.3, 0.1, 0.2, 0.3) (0.2, 0.5, 0.3, 0.4, 0.2, 0.3) d (0.5, 0.5, 0.3, 0.1, 0.2, 0.3) (0.0,0.0,0.0,0.0,0.0,0.0)

K C D

w (0.0, 0.0, 0.0, 0.0, 0.0, 0.0) (0.2, 0.5, 0.3, 0.1, 0.2, 0.3) x (0.14,0.5,0.3, 0.1, 0.2, 0.3) (0.25, 0.5, 0.3, 0.1, 0.2, 0.3) y (0.5, 0.5, 0.3, 0.1, 0.2, 0.3) (0.2, 0.5, 0.3, 0.4, 0.2, 0.3) z (0.3, 0.5, 0.3, 0.1, 0.2, 0.3) (0.0,0.0,0.0,0.0,0.0,0.0)

R �� A 0.2 0.5 0.3 0.1 0.2 0.3 B 0.14 0.5 0.3 0.1 0.2 0.3

Florentin Smarandache (author and editor) Collected Papers, IX 83

S ��

C 0.14 0.5 0.3 0.1 0.2 0.3 D 0.2 0.5 0.3 0.1 0.2 0.3

where f is defined by: f(a)=w, f(b)=x, f(c)=y, f(d)=z.

Here S(H)= (0.34, 1.0, 0.6, 0.2, 0.4, 0.6) = S(K), but, by routine calculations, H is not a co weak isomorphism to K.

Definition 3.19

Let H = (X, E, R) be a BSVNHG, then the degree of vertex ��, which is denoted and defined by: deg(��)= (��(��),��(��),��(��), ��(��),��(��), ��(��) (65) where: ��(��)=∑��(��1,��2,…,��), (66) ��(��)=∑��(��1,��2,…,��), (67) ��(��)=∑��(��1,��2, ,��), (68) ��(��)=∑��(��1,��2, ,��), (69) ��(��)=∑��(��1,��2, ,��), (70) ��(��)=∑��(��1,��2,…,��), (71) for �� ≠ ��. Theorem 3.20

If H and K be two isomorphic BSVNHGs, then the degree of their vertices are preserved.

Proof

Let f: H → K be an isomorphism between two BSVNHGs H and K with underlying sets X and Y respectively, then, by definition, we have: min[��(��)] =min[��(��(��))], (72) max[��(��)] =max[��(��(��))], (73) max[��(��)] =max[��(��(��))], (74) max[��(��)] =max[��(��(��))], (75) min[��(��)] =min[��(��(��))], (76) min[��(��)] =min[��(��(��))], (77)

Florentin Smarandache (author and editor) Collected Papers, IX 84

for all x∈��

��(��1,��2,…,��)= ��(��(��1),��(��2),…,��(��)), (78) ��(��1,��2,…,��) =��(��(��1),��(��2),…,��(��)), (79) ��(��1,��2, ,��) = ��(��(��1),��(��2), ,��(��)), (80) ��(��1,��2, ,��) = ��(��(��1),��(��2), ,��(��)), (81) ��(��1,��2,…,��) =��(��(��1),��(��2),…,��(��)), (82) ��(��1,��2,…,��)=��(��(��1),��(��2),…,��(��)), (83) for all { ��1,��2,…,��} subsets of X.

Consider: ��(��)=∑��(��1,��2, ,��) =∑��(��(��1),��(��2),…,��(��)) =��(��(��)), (84) and similarly: ��(��)=��(��(��)), (85) ��(��)=��(��(��)), ��(��)=��(��(��)) (86) ��(��)=��(��(��)), ��(��)=��(��(��)) (87)

Hence: ��(��)=��(��(��)) (88)

Remark 3.21

The converse of the above theorem may not be true in general. Example 3.22

Consider the two BSVNHGs H = (X, E, R) and K = (Y, F, S) with underlying sets X = {a, b} and Y = {x, y}, where E = {A, B}, F = {C, D}, R and S are defined by Tables given below:

A

B a (0.5, 0.5, 0.3, 0.1, 0.2, 0.3) (0.3, 0.5, 0.3, 0.1, 0.2, 0.3) b (0.25, 0.5, 0.3, 0.1, 0.2, 0.3) (0.2, 0.5, 0.3, 0.1, 0.2, 0.3)

K C D

x (0.3, 0.5, 0.3, 0.1, 0.2, 0.3) (0.5,0.5,0.3, 0.1, 0.2, 0.3) y (0.2, 0.5, 0.3, 0.1, 0.2, 0.3) (0.25, 0.5, 0.3, 0.1, 0.2, 0.3) S �� C 0.2 0.5 0.3 -0.1 -0.2 -0.3 D 0.25 0.5 0.3 -0.1 -0.2 -0.3

Florentin Smarandache (author and editor) Collected Papers, IX 85

R ��

A 0.25 0.5 0.3 0.1 0.2 0.3 B 0.2 0.5 0.3 0.1 0.2 0.3

where f is defined by: f(a)=x, f(b)=y, here deg(a) = ( 0.8, 1.0, 0.6, 0.2, 0.4, 0.6) = deg(x) and deg(b) = (0.45, 1.0, 0.6, 0.2, 0.4, 0.6) = deg(y).

But H is not isomorphic to K, i.e. H is neither weak isomorphic, nor co weak isomorphic to K. Theorem 3.23 The isomorphism between BSVNHGs is an equivalence relation.

Proof

Let H = (X, E, R), K = (Y, F, S) and M = (Z, G, W) be BSVNHGs with underlying sets X, Y and Z, respectively: Reflexive

Consider the map (identity map) f : X → X defined as follows: f(x) = x for all x ∈ X, since the identity map is always bijective and satisfies the conditions: min[��(��)] =min[��(��(��))], (89) max[��(��)] =max[��(��(��))], (90) max[��(��)] =max[��(��(��))], (91) max[��(��)] =max[��(��(��))], (92) min[��(��)] =min[��(��(��))], (93) min[��(��)] =min[��(��(��))], (94) for all x∈��. ��(��1,��2, ,��) = ��(��(��1),��(��2), ,��(��)), (95) ��(��1,��2, ,��) =��(��(��1),��(��2), ,��(��)), (96) ��(��1,��2,…,��) = ��(��(��1),��(��2),…,��(��)), (97) ��(��1,��2,…,��) = ��(��(��1),��(��2),…,��(��)), (98) ��(��1,��2,…,��) =��(��(��1),��(��2),…,��(��)), (99) ��(��1,��2,…,��) = ��(��(��1),��(��2),…,��(��)), (100) for all { ��1,��2, ,��} subsets of X. Hence f is an isomorphism of BSVNHG H to itself.

Florentin Smarandache (author and editor) Collected Papers, IX 86

Symmetric

Let f: X → Y be an isomorphism of H and K, then f is a bijective mapping defined as f(x) = y for all x ∈ X.

Then, by definition: min[��(��)] =min[��(��(��))], (101) max[��(��)] =max[��(��(��))], (102) max[��(��)] =max[��(��(��))], (103) max[��(��)] =max[��(��(��))], (104) min[��(��)] =min[��(��(��))], (105) min[��(��)] =min[��(��(��))], (106) for all x∈��. ��(��1,��2, ,��)= ��(��(��1),��(��2), ,��(��)), (107) ��(��1,��2, ,��) =��(��(��1),��(��2), ,��(��)), (108) ��(��1,��2,…,��) = ��(��(��1),��(��2),…,��(��)), (109) ��(��1,��2,…,��) = ��(��(��1),��(��2),…,��(��)), (101) ��(��1,��2,…,��) =��(��(��1),��(��2),…,��(��)), (111) ��(��1,��2,…,��)=��(��(��1),��(��2),…,��(��)), (112) for all {��1,��2, ,��} subsets of X. Since f is bijective, then we have: �� 1(��)=�� forall�� ∈�� Thus, we get: min[��(�� 1(��))] =min[��(��)], (113) max[��(�� 1(��))] =max[��(��)], (114) max[��(�� 1(��))] =max[��(��)], (115) max[��(�� 1(��))] =max[��(��)], (116) min[��(�� 1(��))] =min[��(��)], (117) min[��(�� 1(��))] =min[��(��)], (118) for all x∈��. ��(�� 1(��1),�� 1(��2),…,�� 1(��)) =��(��1 ,��2,…,��), (119) ��(�� 1(��1),�� 1(��2), ,�� 1(��)) =��(��1 ,��2, ,��), (120) ��(�� 1(��1),�� 1(��2),…,�� 1(��)) =��(��1 ,��2,…,��), (121)

Florentin Smarandache (author and editor) Collected Papers, IX 87

��(�� 1(��1),�� 1(��2), ,�� 1(��))=��(��1 ,��2, ,��), (122) ��(�� 1(��1),�� 1(��2),…,�� 1(��))=��(��1 ,��2,…,��), (123) ��(�� 1(��1),�� 1(��2),…,�� 1(��))=��(��1 ,��2,…,��), (124) for all {��1,��2, ,��} subsets of Y. Hence, we have a bijective map �� 1 ∶�� →�� which is an isomorphism from K to H

Transitive

Let �� ∶�� →�� and �� ∶�� →�� be two isomorphism of BSVNHGs of H onto K and K onto M, respectively. Then �� is bijective mapping from X to Z, where �� is defined as (��)(��)=��(��(��))forall�� ∈��.

Since f is an isomorphism, then by definition ��(��)=��forall�� ∈��, which satisfies the conditions:

min[��(��)] =min[��(��(��))], (125) max[��(��)] =max[��(��(��))], (126) max[��(��)] =max[��(��(��))], (127) max[��(��)] =max[��(��(��))], (128) min[��(��)] =min[��(��(��))], (129) min[��(��)] =min[��(��(��))], (130) for all x∈��. ��(��1,��2, ,��)= ��(��(��1),��(��2), ,��(��)), (131) ��(��1,��2,…,��) =��(��(��1),��(��2),…,��(��)), (132) ��(��1,��2,…,��) = ��(��(��1),��(��2),…,��(��)), (133) ��(��1,��2, ,��) = ��(��(��1),��(��2), ,��(��)), (134) ��(��1,��2,…,��) =��(��(��1),��(��2),…,��(��)), (135) ��(��1,��2,…,��)=��(��(��1),��(��2),…,��(��)), (136) for all {��1,��2, ,��} subsets of X.

Since �� ∶�� →�� is an isomorphism, then by definition ��(��)=��forall�� ∈ �� satisfying the conditions: min[��(��)] =min[��(��(��))] , (137) max[��(��)] =max[��(��(��))], (138)

Florentin Smarandache (author and editor) Collected Papers, IX 88

max[��(��)] =max[��(��(��))], (139) max[��(��)] =max[��(��(��))], (140) min[��(��)] =min[��(��(��))], (141) min[��(��)] =min[��(��(��))], (142) for all x∈��. ��(��1,��2,…,��) = ��(��(��1),��(��2),…,��(��)), (143) ��(��1,��2, ,��) =��(��(��1),��(��2), ,��(��)), (144) ��(��1,��2,…,��) = ��(��(��1),��(��2),…,��(��)), (145) ��(��1,��2,…,��)=��(��(��1),��(��2),…,��(��)), (146) ��(��1,��2, ,��) =��(��(��1),��(��2), ,��(��)), (147) ��(��1,��2,…,��) =��(��(��1),��(��2),…,��(��)), (148) for all {��1,��2,…,��} subsets of Y.

Thus, from above equations we conclude that: min[��(��)] = min[��(��(��(��)))], (149) max[��(��)] = max[��(��(��(��)))], (150) max[��(��)] = max[��(��(��(��)))], (151) max[��(��)] = max[�� (��(��(��)))], (152) min[��(��)] = min[��(��(��(��)))], (153) min[��(��)] = min[��(��(��(��)))], (154) for all x∈��. ��(��1, ,��)=��(��(��(��1)), ,��(��(��))), (155) ��(��1, ,��)=��(��(��(��1)), ,��(��(��))), (156) ��(��1, ,��)=��(��(��(��1)), ,��(��(��))), (157) ��(��1,…,��)=��(��(��(��1)),…,��(��(��))), (158) ��(��1,…,��)=��(��(��(��1)),…,��(��(��))), (159) ��(��1,…,��)=��(��(��(��1)),…,��(��(��))), (160) for all {��1,��2,…,��} subsets of X.

Therefore �� is an isomorphism between H and M

Florentin Smarandache (author and editor) Collected Papers, IX 89

Hence, the isomorphism between BSVNHGs is an equivalence relation.

Theorem 3.24

The weak isomorphism between BSVNHGs satisfies the partial order relation.

Proof

Let H = (X, E, R), K = (Y, F, S) and M = (Z, G, W) be BSVNHGs with underlying sets X, Y and Z, respectively:

Reflexive

for all x∈�� (��1,��2,…,��) ≤ ��(��(��1),��(��2),…,��(��)), (167) ��(��1,��2,…,��) ≥��(��(��1),��(��2),…,��(��)), (168) ��(��1,��2,…,��) ≥ ��(��(��1),��(��2),…,��(��)), (169) ��(��1,��2, ,��) ≥ ��(��(��1),��(��2), ,��(��)), (170) ��(��1,��2, ,��) ≤��(��(��1),��(��2), ,��(��)), (171) ��(��1,��2, ,��) ≤��(��(��1),��(��2), ,��(��)), (172) for all {��1,��2,…,��} subsets of X. Hence, f is a weak isomorphism of BSVNHG H to itself.

Anti symmetric

Let f be a weak isomorphism between H onto K, and g be a weak isomorphic between K and H, that is ��:�� →�� is a bijective map defined by: ��(��)= ��forall�� ∈�� satisfying the conditions: min[��(��)] =min[��(��(��))], (173) max[��(��)] =max[��(��(��))], (174) max[��(��)] =max[��(��(��))], (175)

Florentin Smarandache (author and editor) Collected Papers, IX 90

max[��(��)] =max[��(��(��))], (176) min[��(��)] =min[��(��(��))], (177) min[��(��)] =min[��(��(��))], (178) for all x∈�� (��1,��2,…,��)= ��(��(��1),��(��2),…,��(��)), (179) ��(��1,��2,…,��) =��(��(��1),��(��2),…,��(��)), (180) ��(��1,��2,…,��) = ��(��(��1),��(��2),…,��(��)), (181) ��(��1,��2, ,��) = ��(��(��1),��(��2), ,��(��)), (182) ��(��1,��2, ,��) =��(��(��1),��(��2), ,��(��)), (183) ��(��1,��2,…,��)=��(��(��1),��(��2),…,��(��)), (184) for all {��1,��2,…,��} subsets of X.

Since g is also bijective map ��(��)=�� forall�� ∈�� satisfying the conditions: min[��(��)] =min[��(��(��))], (185) max[��(��)] =max[��(��(��))], (186) max[��(��)] =max[��(��(��))], (187) max[��(��)] =max[��(��(��))] , (188) min[��(��)] =min[��(��(��))], (189) min[��(��)] =min[��(��(��))], (190) for all y∈��. ��(��,��2,…,��) ≤ ��(��(��1),��(��2),…,��(��)), (191) ��(��1,��2, ,��) ≥��(��(��1),��(��2), ,��(��)), (192) ��(��1,��2, ,��) ≥ ��(��(��1),��(��2), ,��(��)), (193) ��(��,��2,…,��) ≥ ��(��(��1),��(��2),…,��(��)), (194) ��(��1,��2,…,��) ≤��(��(��1),��(��2),…,��(��)), (195) ��(��1,��2,…,��) ≤ ��(��(��1),��(��2),…,��(��)), (196) for all {��1,��2,…,��} subsets of Y. The above inequalities hold for finite sets X and Y only whenever H and K have same number of edges and corresponding edge have same weights, hence H is identical to K.

Transitive

Let ��:�� →�� and ��:�� →�� be two weak isomorphism of BSVNHGs of H onto K and K onto M, respectively. Then �� is bijective mapping from X to Z, where �� is defined as (��)(��)=��(��(��))forall�� ∈��

Florentin Smarandache (author and editor) Collected Papers, IX 91

Since f is a weak isomorphism, then by definition ��(��)=��forall�� ∈�� which satisfies the conditions:

min[��(��)] =min[��(��(��))], (197) max[��(��)] =max[��(��(��))], (198)

max[��(��)] =max[��(��(��))], (199) max[��(��)] =max[��(��(��))], (200)

min[��(��)] =min[��(��(��))], (201) min[��(��)] =min[��(��(��))], (202) for all x∈��. ��(��1,��2, ,��)≤ ��(��(��1),��(��2), ,��(��)), (203) ��(��1,��2, ,��) ≥��(��(��1),��(��2), ,��(��)), (204) ��(��1,��2,…,��) ≥��(��(��1),��(��2),…,��(��)), (205) ��(��1,��2,…,��) ≥ ��(��(��1),��(��2),…,��(��)), (206) ��(��1,��2,…,��) ≤��(��(��1),��(��2),…,��(��)), (207) ��(��1,��2,…,��)≤��(��(��1),��(��2),…,��(��)), (208) for all {��1,��2, ,��} subsets of X. Since �� ∶�� →�� is a weak isomorphism, then by definition ��(��)=��forall�� ∈ ��, satisfying the conditions: min[��(��)] =min[��(��(��))] , (209) max[��(��)] =max[��(��(��))], (210) max[��(��)] =max[��(��(��))], (211) max[��(��)] =max[��(��(��))], (212) min[��(��)] =min[��(��(��))], (213) min[��(��)] =min[��(��(��))], (214) for all x∈�� (��1,��2,…,��) ≤ ��(��(��1),��(��2),…,��(��)), (215) ��(��1,��2,…,��) ≥��(��(��1),��(��2),…,��(��)), (216) ��(��1,��2,…,��) ≥ ��(��(��1),��(��2),…,��(��)), (217) ��(��1,��2,…,��)≥��(��(��1),��(��2),…,��(��)), (218) ��(��1,��2, ,��) ≤��(��(��1),��(��2), ,��(��)), (219) ��(��1,��2, ,��) ≤��(��(��1),��(��2), ,��(��)), (220) for all {��1,��2,…,��} subsets of Y.

Florentin Smarandache (author and editor) Collected Papers, IX 92

Thus, from above equations, we conclude that: min[��(��)] = min[��(��(��(��)))], (221) max[��(��)] = max[��(��(��(��)))], (222) max[��(��)] = max[��(��(��(��)))], (223) max[��(��)] = max[��(��(��(��)))], (224) min[��(��)] = min[��(��(��(��)))], (225) min[��(��)] = min[��(��(��(��)))], (226) for all x∈��. ��(��1,…,��)≤��(��(��(��1)),…,��(��(��))), (227) ��(��1,…,��)≥��(��(��(��1)),…,��(��(��))), (228) ��(��1, ,��)≥��(��(��(��1)), ,��(��(��))), (229) ��(��1,…,��)≥��(��(��(��1)),…,��(��(��))), (230) ��(��1, ,��)≤��(��(��(��1)), ,��(��(��))), (231) ��(��1,…,��)≤��(��(��(��1)),…,��(��(��))), (232)

for all {��1,��2,…,��} subsets of X. Therefore �� is a weak isomorphism between H and M. Hence, the weak isomorphism between BSVNHGs is a partial order relation.

4 Conclusion

Thebipolarsinglevaluedneutrosophichypergraphcanbeappliedinvarious areas of engineering and computer science. In this paper, the isomorphism between BSVNHGs is proved to be an equivalence relation and the weak isomorphism is proved to be a partial order relation. Similarly, it can be provedthatco weakisomorphisminBSVNHGsisapartialorderrelation.

5 References

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[24] M. Akram, M. M. Yousaf, W. A. Dudek. Self centered interval valued fuzzy graphs.In:AfrikaMatematika,vol.26,Issue5,pp887 898,2015.

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[26] R. Parvathi and M. G. Karunambigai. Intuitionistic Fuzzy Graphs. In: Computational Intelligence. In: Theory and applications, International ConferenceinGermany,Sept18 20,2006.

[27] R. A. Borzooei, H. Rashmanlou. More Results on Vague Graphs, U.P.B. Sci. Bull.,SeriesA,Vol.78,Issue1,2016,109 122.

[28] S. Broumi, M. Talea, F. Smarandache and A. Bakali Single Valued Neutrosophic Graphs: Degree, Order and Size. IEEE International Conference onFuzzySystems(FUZZ),2016,pp.2444 2451.

[29] S.Broumi, M. Talea, A. Bakali, F. Smarandache. Single Valued Neutrosophic Graphs.In:JournalofNewTheory,no.10,68 101(2016).

[30] S. Broumi, M. Talea, A. Bakali, F. Smarandache. On Bipolar Single Valued Neutrosophic Graphs.In:JournalofNewTheory,no.11,84 102(2016).

[31] S.Broumi,M.Talea,A.Bakali,F.Smarandache. Interval Valued Neutrosophic Graphs,SISOM&ACOUSTICS2016,Bucharest12 13May,pp.79 91.

[32] S. Broumi, F. Smarandache, M. Talea and A. Bakali An Introduction to Bipolar Single Valued Neutrosophic Graph Theory. In: Applied Mechanics and Materials,vol.841,2016,pp.184 191.

[33] S. Broumi, M. Talea, A. Bakali, F. Smarandache. Operations on Interval Valued Neutrosophic Graphs (2016),submitted.

[34] S.Broumi,F.Smarandache,M.TaleaandA.Bakali Decision Making Method Based on the Interval Valued Neutrosophic Graph, Future Technologie, 2016, IEEE,pp.44 50.

[35] S. N. Mishra andA.Pal. Product of Interval Valued Intuitionistic fuzzy graph In:AnnalsofPureandAppliedMathematics,Vol.5,No.1(2013)37 46.

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[37] W. B. Vasantha Kandasamy, K. Ilantheral, F. Smarandache. Neutrosophic Graphs: A New Dimension to Graph Theory,2015 http://www.gallup.unm.edu/~smarandache/NeutrosophicGraphs.pdf

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Regular Bipolar Single Valued Neutrosophic Hypergraphs

Muhammad Aslam Malik, Ali Hassan, Said Broumi, F. Smarandache

Abstract. In this paper, we define the regular and totally regular bipolar single valued neutrosophic hypergraphs, and discuss the order and size along with properties of

regular and totally regular bipolar single valued neutrosophic hypergraphs We extend work on completeness of bipolar single valued neutrosophic hypergraphs.

Keywords: bipolar single valued neutrosophic hypergraphs, regular bipolar single valued neutrosophic hypergraphs and totally regu lar bipolar single valued neutrosophic hyper graphs.

1 Introduction

The notion of neutrosophic sets (NSs) was proposed by Smarandache [8] as a generalization of the fuzzy sets [14], intuitionistic fuzzy sets [12], interval valued fuzzy set [11] and interval-valued intuitionistic fuzzy sets [13] theories. The neutrosophic set is a powerful mathematical tool for dealing with incomplete, indeterminate and inconsistent information in real world. The neutrosophic sets are characterized by a truth-membership function (t), an indeterminacy-membership function (i) and a falsity membership function (f) independently, which are within the real standard or nonstandard unit interval ] 0 , 1+[. In order to conveniently use NS in real life applications, Wang et al. [9] introduced the concept of the single-valued neutrosophic set (SVNS), a subclass of the neutrosophic sets. The same authors [10] introduced the concept of the interval valued neutrosophic set (IVNS), which is more precise and flexible than the single valued neutrosophic set. The IVNS is a generalization of the single valued neutrosophic set, in which the three membership functions are independent and their value belong to the unit interval [0, 1]. More works on single valued neutrosophic sets, interval valued neutrosophic sets and their applications can be found on http://fs.gallup.unm.edu/NSS/.

Hypergraph is a graph in which an edge can connect more than two vertices, hypergraphs can be applied to analyse architecture structures and to represent system partitions, Mordesen J.N and P.S Nasir gave the definitions for fuzzy hypergraphs. Parvathy. R and M. G. Karunambigai’s paper introduced the concepts of Intuitionistic fuzzy hypergraphs and analyse its components, Nagoor Gani. A and Sajith

Begum. S defined degree, order and size in intuitionistic fuzzy graphs and extend the properties. Nagoor Gani. A and Latha. R introduced irregular fuzzy graphs and discussed some of its properties.

Regular intuitionistic fuzzy hypergraphs and totally regular intuitionistic fuzzy hypergraphs are introduced by Pradeepa. I and Vimala. S in [0]. In this paper we extend regularity and totally regularity on bipolar single valued neutrosophic hypergraphs.

2 Preliminaries

In this section we discuss the basic concept on neutro sophic set and neutrosophic hyper graphs.

Definition 2.1 Let X be the space of points (objects) with generic elements in X denoted by x. A single valued neu trosophic set A (SVNS A) is characterized by truth mem bership function ��(x), indeterminacy membership func tion ��(x) and a falsity membership function ��(x). For each point x ∈X; ��(x), ��(x), ��(x) ∈ [0, 1].

Definition 2.2 Let X be a space of points (objects) with generic elements in X denoted by x. A bipolar single valued neutrosophic set A (BSVNS A) is characterized by positive truth membership function ��(x), positive indeterminacy membership function ��(x) and a positive falsity membership function ��(x) and negative truth membership function ��(x), negative indeterminacy membership function ��(x) and a negative falsity membership function ��(x)

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Muhammad Aslam Malik, Ali Hassan, Said Broumi, F. Smarandache (2016). Regular Bipolar Single Valued Neutrosophic Hypergraphs. Neutrosophic Sets and Systems, 13, 84-89

For each point x ∈ X; ��(x), ��(x), ��(x) ∈ [0, 1] and ��(x), ��(x), ��(x) ∈ [-1, 0].

Definition 2.3 Let A be a BSVNS on X then support of A is denoted and defined by

Supp(A) = {x : x ∈X, ��(x) > 0, ��(x) > 0, ��(x) > 0, ��(x) < 0, ��(x) < 0, ��(x) < 0}.

Definition 2.4 A hyper graph is an ordered pair H = (X, E), where

(1) X = {��1, ��2, … . , ��} be a finite set of vertices.

(2) E = {��1, ��2 , …., ��} be a family of subsets of X (3) �� for j= 1,2,3,...,m and ⋃��(��)= X

The set X is called set of vertices and E is the set of edges (or hyper edges).

Definition 2.5 A bipolar single valued neutrosophic hypergraph is an ordered pair H = (X, E), where

(1) �� = {��1, ��2, … , ��} be a finite set of vertices.

(2) �� = {��1, ��2, … , ��} be a family of BSVNSs of X

(3) ��≠ O = (0, 0, 0) for j= 1,2,3,...,m and ⋃��(��)= X

The set X is called set of vertices and E is the set of BSVN-edges (or BSVN-hyper edges).

Proposition 2.6 The bipolar single valued neutrosophic hyper graph is the generalization of fuzzy hyper graphs, intuitionistic fuzzy hyper graphs, bipolar fuzzy hyper graphs and single valued neutrosophic hypergraphs.

3 Regular and totally regular BSVNHGs

Definition 3.1 The open neighbourhood of a vertex x in bipolar single valued neutrosophic hypergraphs (BSVNHGs) is the set of adjacent vertices of x, excluding that vertex and is denoted by N(x)

Definition 3.2 The closed neighbourhood of a vertex x in bipolar single valued neutrosophic hypergraphs (BSVNHGs) is the set of adjacent vertices of x, including that vertex and is denoted by N[x]

Example 3.3 Consider a bipolar single valued neutrosophic hypergraphs H = (X, E) where, X = {a, b, c, d, e} and E =

{P, Q, R, S}, which is defined by

P = {(a, 0.1, 0.2, 0.3, -0.4, -0.6 -0.8), (b, 0.4, 0.5, 0.6, -0.4, -0.6 -0.8)}

Q = {(c, 0.1, 0.2, 0.3, -0.4, -0.4 -0.9), (d, 0.4, .5, 0.6, -0.3, -0.5 -0.6), (e, 0.7, 0.8, 0.9, -0.7, -0.9, -0.2)}

R = {(b, 0.1, 0.2, 0.3, -0.2, -0.5, -0.8), (c, 0.4, 0.5, 0.6, -0.9, -0.7 -0.4)}

S = {(a, 0.1, 0.2, 0.3, -0.7, -0.6, -0.9), (d, 0.9, 0.7, 0.6, -0.4, -0.7, -0.9)}

Then the open neighbourhood of a vertex a is the b and d, and closed neighbourhood of a vertex b is b, a and c

Definition 3.4 Let H = (X, E) be a BSVNHG, the open neighbourhood degree of a vertex x, which is denoted and defined by

deg(x) = (��(x), ��(x), ��(x), ��(x), ��(x) , ��(x)) where

��(x) = ∑ ��(��) ��∈��(��)

Example 3.5 Consider a bipolar single valued neutrosoph ic hypergraphs H = (X, E) where, X = {a, b, c, d, e} and E = {P, Q, R, S}, which are defined by

P = {(a, .1, .2, .3, -0.1, -0.2, -0.3), (b, .4, .5, .6, -0.1, -0.2, -0.3)}

Q = {(c, .1, .2, .3, -0.1, -0.2, -0.3), (d, .4, .5, .6, -0.1, -0.2, -0.3), (e, .7, .8, .9, -0.1, -0.2, -0.3)}

R = {(b, .1, .2, .3, -0.1, -0.2, -0.3), (c, .4, .5, .6, -0.1, -0.2, -0.3)}

S = {(a, .1, .2, .3, -0.1, -0.2, -0.3), (d, .4, .5, .6, -0.1, -0.2, -0.3)}

Then the open neighbourhood of a vertex a contain b and d and therefore open neighbourhood degree of a vertex a is (.8, 1, 1.2, -0.2, -0.4, -0.6)

Definition 3.6 Let H = (X, E) be a BSVNHG, the closed neighbourhood degree of a vertex x is denoted and defined by

Florentin Smarandache (author and editor) Collected Papers, IX 97

deg[x] = (��[x], ��[x], ��[x], ��[x], ��[x] , ��[x])

which are defined by

��[x] = ��(x) + ��(x)

Example 3.7 Consider a bipolar single valued neutrosophic hypergraphs H = (X, E) where, X = {a, b, c, d, e} and E = {P, Q, R, S}, which is defined by

P = {(a, 0.1, 0.2, 0.3, -0.1, -0.2, -0.3), (b, 0.4, 0.5, 0.6, -0.1, -0.2, -0.3)}

Q = {(c, 0.1, 0.2, 0.3, -0.1, -0.2, -0.3), (d, 0.4, 0.5, 0.6, -0.1, -0.2, -0.3), (e, 0.7, 0.8, 0.9, -0.1, -0.2, -0.3)}

R = {(b, 0.1, 0.2, 0.3, -0.1, -0.2, -0.3), (c, 0.4, 0.5, 0.6, -0.1, -0.2, -0.3)}

S = {(a, 0.1, 0.2, 0.3, -0.1, -0.2, -0.3), (d, 0.4, 0.5, 0.6, -0.1, -0.2, -0.3)}

The closed neighbourhood of a vertex a contain a, b and d, hence the closed neighbourhood degree of a vertex a is (0.9, .1.2, 1.5, -0.3, -0.6, -0.9)

Definition 3.8 Let H = (X, E) be a BSVNHG, then H is said to be an n-regular BSVNHG if all the vertices have the same open neighbourhood degree n = (n1, n2, n3, n4, n5, n6)

Definition 3.9 Let H = (X, E) be a BSVNHG, then H is said to be m-totally regular BSVNHG if all the vertices have the same closed neighbourhood degree m = (m1, m2, m3, m4, m5, m6).

Proposition 3.10 A regular BSVNHG is the generalization of regular fuzzy hypergraphs, regular intuitionistic fuzzy hypergraphs, regular bipolar fuzzy hypergraphs and regu lar single valued neutrosophic hypergraphs.

Proposition 3.11 A totally regular BSVNHG is the generali zation of totally regular fuzzy hypergraphs, totally regular intuitionistic fuzzy hypergraphs, totally regular bipolar fuzzy hypergraphs and totally regular single valued neu trosophic hypergraphs.

Example 3.12 Consider a bipolar single valued neutrosophic hypergraphs H = (X, E) where, X = {a, b, c, d} and

E = {P, Q, R, S} which is defined by

P = {(a, 0.8, 0.2, 0.3, -0.1, -0.2, -0.3), (b, 0.8, 0.2, 0.3, -0.1, -0.2, -0.3)}

Q = {(b, 0.8, 0.2, 0.3, -0.1, -0.2, -0.3), (c, 0.8, 0.2, 0.3, -0.1, -0.2, -0.3)}

R = {(c, 0.8, 0.2, 0.3, -0.1, -0.2, -0.3), (d, 0.8, 0.2, 0.3, -0.1, -0.2, -0.3)}

S = {(d, 0.8, 0.2, 0.3, -0.1, -0.2, -0.3), (a, 0.8, 0.2, 0.3, -0.1, -0.2, -0.3)}

Here the open neighbourhood degree of every vertex is (1.6, 0.4, 0.6, -0.2, -0.4, -0.6) hence H is regular BSVNHG and closed neighbourhood degree of every vertex is (2.4, 0.6, 0.9, -0.3, -0.6, -0.9), Hence H is both regular and total ly regular BSVNHG.

Theorem 3.13 Let H = (X, E) be a BSVNHG which is both regular and totally regular BSVNHG then E is constant.

Proof: Suppose H is an n-regular and m-totally regular BSVNHG. Then deg(x) = n = (n1, n2, n3, n4,n5, n6) and deg[x] = m = (m1, m2, m3, m4, m5, m6) ∀��∈��. Consider deg[x] = m. Hence by definition, deg(x) + ��(x) = m this implies ��(x) = m n for all ��∈��. Hence E is constant.

Remark 3.14 The converse of above theorem need not to be true in general.

Example 3.15 Consider a bipolar single valued neutrosophic hypergraphs H = (X, E) where, X = {a, b, c, d} and E = {P, Q, R, S}, which is defined by

P = {(a, 0.8, 0.2, 0.3, -0.1, -0.2, -0.3), (b, 0.8, 0.2, 0.3, -0.1, -0.2, -0.3)}

Q = {(b, 0.8, 0.2, 0.3, -0.1, -0.2, -0.3), (d, 0.8, 0.2, 0.3, -0.1, -0.2, -0.3)}

R = {(c, 0.8, 0.2, 0.3, -0.1, -0.2, -0.3), (d, 0.8, 0.2, 0.3, -0.1, -0.2, -0.3)}

S = {(d, 0.8, 0.2, 0.3, -0.1, -0.2, -0.3), (a, 0.8, 0.2, 0.3, -0.1, -0.2, -0.3)}

Here E is constant but deg(a) = (1.6, 0.4, 0.6, -0.2, -0.4,0.6) and deg(d) = (2.4, 0.6, 0.9, -0.3, -0.6, -0.9) i.e deg(a) and deg(d) are not equals hence H is not regular BSVNHG. Next deg[a] = (2.4, 0.6, 0.9, -0.3, -0.6, -0.9) and deg[d]= (3.2, 0.8, 1.2, -.4, -0.8, -1.2), hence deg[a] and deg[d] are not equals hence H is not totally regular BSVNHG, Thus that H is neither regular and nor totally regular BSVNHG

Theorem 3.16 Let H = (X, E) be a BSVNHG then E is constant on X if and only if following are equivalent, (1) H is regular BSVNHG.

(2) H is totally regular BSVNHG.

Proof: Suppose H = (X, E) be a BSVNHG and E is constant in H, that is ��(x) = c = (c , c , c , c , c , c ) ∀�� ∈ ��

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Suppose H is n-regular BSVNHG, then deg(x) = n = (n1, n2, n3, n4, n5, n6) ∀�� ∈ ��, consider deg[x] = deg(x) +��(x) = n +c ∀�� ∈ ��, hence H is totally regular BSVNHG.

Next suppose that H is m totally regular BSVNHG, then deg[x] = m = (m1, m2, m3, m4, m5, m6) for all ��∈��, that is deg(x) + ��(x) = m ∀��∈��, this implies that deg(x) = m c ∀��∈��. Thus H is regular BSVNHG, thus (1) and (2) are equivalent.

Conversely: Assume that (1) and (2) are equivalent. That is H is regular BSVNHG if and only if H is totally regular BSVNHG. Suppose contrary E is not constant, that is ��(x) and ��(y) not equals for some x and y in X. Let H = (X, E) be n-regular BSVNHG, then deg(x) = n = (n1, n2, n3, n4,n5, n6) for all x ∈ ��. Consider deg[x] = deg(x) + ��(x) = n + ��(x) deg[y] = deg(y) + ��( (y) = n + ��(y)

Since ��(x) and ��(y) are not equals for some x and y in X Hence deg[x] and deg[y] are not equals, thus H is not totally regular BSVNHG, which contradict to our assumption. Next let H be totally regular BSVNHG, then deg[x] = deg[y], that is deg(x) + ��(x) = deg(y) + ��(y) and deg(x) –deg(y) = ��(y) – ��(x), since RHS of last equation is non zero, hence LHS of above equation is also nonzero, thus deg(x) and deg(y) are not equals, so H is not regular BSVNHG, which is again contradict to our assumption, thus our supposition was wrong, hence E must be con stant, this completes the proof.

Definition 3.17 Let H = (X, E) be a regular BSVNHG, then the order of BSVNHG H is denoted and defined by O(H) = (p, q, r, s, t, u), where ��=∑ ��(��) ��∈�� , ��= ∑ ��(��) ��∈�� ,��=∑ ��(��), ��∈�� =∑ ��(��) ��∈�� ,��=∑ ��(��) ��∈�� , ��=∑ ��(��) ��∈�� For every x ∈ X and size of regular BSVNHG is denoted and defined by S(H) = ∑ (��) �� =1 , where S(Ei) = (a, b, c, d, e, f) which is defined by a = ∑ �� ∈�� (��) b = ∑ �� ∈�� (��)

c = ∑ ��(��) ��∈�� d = ∑ ��(��) ��∈�� e = ∑ �� ∈�� (��) f = ∑ �� ∈�� (��)

Example 3.18 Consider a bipolar single valued neutrosophic hypergraphs H = (X, E) where, X = {a, b, c, d} and

E = {P, Q, R, S}, which is defined by P = {(a, .8, .2, .3, -.1, -.2, -.3), (b, .8, .2, .3, -.1, -.2, -.3)}

Q = {(b, .8, .2, .3, -.1, -.2, -.3), (c, .8, .2, .3, -.1, -.2, -.3)} R = {(c, .8, .2, .3, -.1, -.2, -.3), (d, .8, .2, .3, -.1, -.2, -.3)} S = {(d, .8, .2, .3, -.1, -.2, -.3), (a, .8, .2, .3, -.1, -.2, -.3)}

Here order and size of H are given (3.2, .8, 1.2, -.4, -.8,1.2) and (6.4, 1.6, 2.4, -.8, -1.6, -2.4) respectively.

Proposition 3.19 The size of an n-regular BSVNHG H = (H, E) is nk/2, where |X|= k.

Proposition 3.20 If H = (X, E) be m-totally regular BSVNHG then 2S(H) + O(H) = mk, where |X|= k

Corollary 3.21 Let H = (X, E) be a n-regular and m-totally regular BSVNHG then O(H) = k(m - n), where |X|=k

Proposition 3.22 The dual of n-regular and m-totally regu lar BSVNHG H = (X, E) is again an n-regular and m totally regular BSVNHG.

Definition 3.23 A bipolar single valued neutrosophic hy pergraph (BSVNHG) is said to be complete BSVNHG if for every x in X, N(x) = {x: x in X-{x}}, that is N(x) contains all remaining vertices of X except x

Example 3.24 Consider a bipolar single valued neutrosophic hypergraphs H = (X, E), where X = {a, b, c, d} and E = {P, Q, R}, which is defined by

P = {(a, 0.4, 0.6, 0.3, -0.5, -0.2, -0.3), (c, 0.8, 0.2, 0.3, -0.1, -0.8, -0.3)}

Q = {(a, 0.8, 0.8, 0.3, -0.1, -0.6, -0.3), (b, 0.8, 0.2, 0.1, -0.1, -0.2, -0.3), (d, 0.8, 0.2, 0.1, -0.1, -0.9, -0.3)}

R = {(c, 0.4, 0.9, 0.9, -0.1, -0.2, -0.3), (d, 0.7, 0.2, 0.1, -0.5, -0.9, -0.3), (b,

Florentin Smarandache (author and editor) Collected Papers, IX 99

0.4, 0.2, 0.1, -0.8, -0.4, -0.2)}. Here N(a) = {b, c, d} , N(b) = {a, c, d}, N(c) = {a, b, d}, N(d) = {a, b, c} hence H is complete BSVNHG.

Remark 3.25 In a complete BSVNHG H = (X, E), the cardi nality of N(x) is same for every vertex.

Theorem 3.26 Every complete BSVNHG H = (X, E) is both regular and totally regular if E is constant in H

Proof: Let H = (X, E) be complete BSVNHG, suppose E is constant in H, so that ��(x) = c = (c1, c2, c3, c4, c5, c6) ∀��∈��, since BSVNHG is complete, then by definition for every vertex x in X, N(x) = {x: x in X-{x}}, the open neighbourhood degree of every vertex is same. That is deg(x) = n = (n1, n2, n3, n4, n5, n6) ∀��∈��. Hence complete BSVNHG is regular BSVNHG. Also, deg[x] = deg(x) + ��(x) = n + c ∀��∈��. Hence H is totally regular BSVNHG.

Remark 3.27 Every complete BSVNHG is totally regular even if E is not constant.

Definition 3.28 A BSVNHG is said to be k-uniform if all the hyper edges have same cardinality.

Example 3.29 Consider a bipolar single valued neutrosophic hypergraphs H = (X, E), where X = {a, b, c, d} and E = {P, Q, R}, which is defined by P = {(a, 0.8, 0.4, 0.2,-0.4, -0.6, -0.2), (b, 0.7, 0.5, 0.3, -0.7, -0.1, -0.2)}

Q = {(b, 0.9, 0.4, 0.8, -0.3, -0.2, -0.9), (c, 0.8, 0.4, 0.2, -0.4, -0.3, -0.7)} R = {(c, 0.8, 0.6, 0.4, -0.3, -0.7, -0.2), (d, 0.8, 0.9, 0.5, -0.4, -0.8, -0.9)}

4 Conclusion

Theoretical concepts of graphs and hypergraphs are utilized by computer science applications. Single valued neu trosophic hypergraphs are more flexible than fuzzy hyper graphs and intuitionistic fuzzy hypergraphs. The concepts of single valued neutrosophic hypergraphs can be applied in various areas of engineering and computer science. In this paper, we defined the regular and totally regular bipolar single valued neutrosophic hyper graphs. We plan to extend our research work to irregular and totally irregular on bipolar single valued neutrosophic hyper graphs.

References

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[16]R. Parvathi and M. G. Karunambigai. Intuitionistic Fuzzy Graphs. In: Computational Intelligence. In: Theory and applications, International Conference in Germany, Sept 18 -20, 2006.

[17]R. A. Borzooei, H. Rashmanlou. More Results On Vague Graphs, U.P.B. Sci. Bull., Series A, Vol. 78, Issue 1, 2016, 109-122.

[18]S. Broumi, M. Talea, F. Smarandache, A. Bakali. Single Valued Neutrosophic Graphs: Degree, Order and Size, FUZZ IEEE Conference (2016), 8 page.

[19]S.Broumi, M. Talea, A. Bakali, F. Smarandache. Single Valued Neutrosophic Graphs. In: Journal of New Theory, no. 10, 68-101 (2016).

[20]S. Broumi, M. Talea, A. Bakali, F. Smarandache. On Bipolar Single Valued Neutrosophic Graphs. In: Journal of New Theory, no. 11, 84-102 (2016).

[21]S. Broumi, M. Talea, A.Bakali, F. Smarandache. Interval Valued Neutrosophic Graphs. SISOM Conference (2016), in press.

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Neutrosophic Soluble Groups, Neutrosophic Nilpotent Groups and Their Properties

Mumtaz Ali, Florentin

Smarandache

Mumtaz Ali, Florentin Smarandache (2016). Neutrosophic Soluble Groups, Neutrosophic Nilpotent Groups and Their Properties. Annual Symposium of the Institute of Solid Mechanics, SISOM 2015, Robotics and Mechatronics. Special Session and Work Shop on VIPRO Platform and RABOR Rescue Robots, Romanian Academy, Bucharest, 21-22 May 2015; Acta Electrotehnica 57(1/2), 153-159

Abstract

The theory of soluble groups and nilpotent groups is old and hence a generalized on. In this paper, we introduced neutrosophic soluble groups and neutrosophic nilpotent groups which have some kind of indeterminacy. These notions are generalized to the classic notions of soluble groups and nilpotent groups. We also derive some new type of series which derived some new notions of soluble groups and nilpotent groups. They are mixed neutrosophic soluble groups and mixed neutrosophic nilpotent groups as well as strong neutrosophic soluble groups and strong neutrosophic nilpotent groups.

Key words: Soluble group, nilpotent group, neutrosophic group, neutrosophic soluble group, neutrosophic nilpotent group.

1. Introduction

Smarandache [15] in 1980 introduced neutrosophy which is a branch of philosophy that studies the origin and scope of neutralities and their interaction with ideational spectra. The concept of neutrosophic set and logic came into being due to neutrosophy, where each proposition is approximated to have the percentage of truth in a subset T, the percentage of indeterminacy in a subset I, and the percentage of falsity in a subset F. Neutrosophic sets are the generalization to all other traditional theories of logics. This mathematical framework is used to handle problems with uncertaint, imprecise, indeterminate, incomplete and inconsistent etc. Kandasamy and Smarandache apply the concept of indeterminacy factor in algebraic structures by inserting the indeterminate element I in the algebraic notions with respect to the opeartaion *. This phenomenon generates the corresponding neutrosophic algebraic notion. They called that indeterminacy element I, a neutrosophic element which is unknown in some sense. This approach a relatively large structure which contain the old classic alegebraic structure. In this way, they studied several neutrosophic algebraic structures in [9,10,11,12]. Some of them are neutrosophic fields, neutrosophic vector spaces, neutrosophic groups, neutrosophic bigroups, neutrosophic N-groups, neutrosophic semigroups, neutrosophic bisemigroups, neutrosophic N-semigroup, neutrosophic loops, neutrosophic biloops, neutrosophic N-loop, neutrosophic groupoids, and neutrosophic bigroupoids and so on. Mumtaz et al.[1] introduced neutrosophic left almost semigroup in short neutrosophic LA-semigroup and their generalization [2]. Further, Mumtaz et al. studied neutrosophic LA-semigroup rings and their generalization.

Groups [5,7] are the most rich algebraic structures in the theory of algebra.They shared common features to all the algebraic structures. Soluble groups [13,14] are important notions in the theory of groups as they are studied on the basis of some kind of series structures of the subgroups of the group. A soluble group is constructed by using abelian

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groups through the extension. A nilpotent group [13] is one whose which has finite length of central series. Thus a nilplotent group is also a soluble group. It is a special type of soluble group because every soluble group has a abelian series. A huge amount of literature on soluble groups and nilpotent groups can be found in [6,8,16,17,18].

In this paper, we introduced neutrosophic soluble groups and neutrosophic nilpotent groups and investigate some of their propertied. The organization of this paper is as follows: In section 1, we give a brief introduction of neutrosophic algebraic structures in terms of I and soluble groups and nilpotent groups. In the next section 2, some basic concept have been studied which we have used in the rest of the paper. In section 3, we introduced neutrosophic soluble groups and investigate some of their basic properties. In section 4, the notions of neutrosophic nilpotent groups are introduced and studied their basic properties. Conclusion is placed in section 5.

2. Fundamental Concepts

Definition 2.1: Let (,) G be a group . Then the neutrosophic group is generated by G and I under denoted by (){,}NGGI . I is called the indeterminateelement with the property 2 II . For an integer n , nI and nI are neutrosophic elements and 0.0 I . 1 I , the inverse of I is not defined and hence does not exist.

Definition 2.2: Let ()NG be a neutrosophic group and H be a neutrosophic subgroup of ()NG . Then H is a neutrosophic normal subgroup of ()NG if xHHx for all ()xNG

Definition 2.3: Let ()NG be a neutrosophic group. Then center of ()NG is denoted by (())CNG and defined as (()){(): CNGxNGaxxa for all ()}aNG .

Definition 2.4: Let be a group and 12,,...,Hn HH be the subgroups of . Then 012 1 {e} ... nn HHHHHG  is called subgroup series of G .

Definition 2.5: Let G be a group and e be the identity element. Then 012 1 {e} ... nn HHHHHG  is called subnormal series. That is j H is normal subgroup of 1 j H  for all j

Definition 2.6: Let 012 1 {e} ... nn HHHHHG  be a subnormal series of . If each j H is normal in for all j , then this subnormal series is called normal series.

The identity element is represented by e and {e} represents the trivial subgroup of G. G G G Smarandache (author and editor) Collected Papers, IX 103

Definition 2.7: A normal series

H H  is an abelian group.

is called an abelian series if the factor group 1 j j

Definition 2.8: A group is called a soluble group if has an abelian series.

Definition 2.9: Let be a soluble group. Then length of the shortest abelian series of is called derived length.

Definition 2.10: Let be a group. The series 012 1 {e} nn HHHHHG  is called central series if 1 j jj

H G Z HH      for all j .

Definition 2.11: A group is called a nilpotent group if has a central series.

3.Neutrosophic Soluble Groups

Definition 3.1: Let () NGGI  be a neutrosophic group and let 12,,...,Hn HH be the neutrosophic subgroups of ()NG . Then a neutrosophic subgroup series is a chain of neutrosophic subgroups such that 012 1 {e} () nn HHHHHNG  .

Example 3.2: Let () NGI  be a neutrosophic group of integers. Then the following are the neutrosophic subgroups series of the group ()NG

012 1 {e} ... nn HHHHHG 

1422

42III  

142 I 

{0} {0} {0}

Florentin Smarandache (author and editor) Collected Papers, IX 104

II ,

. Definition 3.3:Let 012 1 {e} ... () nn HHHHHNG  be a neutrosophic subgroup series of the neutrosophic group ()NG . Then this series of subgroups is called a strong neutrosophic subgroup series if each j H is a neutrosophic subgroup of ()NG for all j .

G G G G G G G Here the identity element is 0 and {0} is the trivial subgroup of Z.

Example 3.4: Let () NGI  be a neutrosophic group. Then the following neutrosophic subgroup series of ()NG is a strong neutrosophic subgroup series: 42III  

Theorem 3.5: Every strong neutrosophic subgroup series is trivially a neutrosophic subgroup series but the converse is not true in general.

Definition 3.6: If some ' j Hs are neutrosophic subgroups and some ' k Hs are just subgroups of ()NG . Then that neutrosophic subgroups series is called mixed neutrosophic subgroup series.

Example 3.7: Let () NGI  be a neutrosophic group. Then the following neutrosophic subgroup series of ()NG is a mixed neutrosophic subgroup series: 1422 II

Theorem 3.8: Every mixed neutrosophic subgroup series is trivially a neutrosophic subgroup series but the converse is not true in general.

Definition 3.9: If ' j Hs in 012 1 {e} () nn HHHHHNG  are only subgroups of the neutrosophic group ()NG , then that series is termed as subgroup series of the neutrosophic group ()NG .

Example 3.10: Let () NGI  be a neutrosophic group. Then the following neutrosophic subgroup series of ()NG is just a subgroup series: 142 I 

Theorem 3.11: A neutrosophic group ()NG has all three type of neutrosophic subgroups series.

Theorem 3.12: Every subgroup series of the group is also a subgroup series of the neutrosophic group ()NG .

Proof: Since is always contained in ()NG . This directly followed the proof.

Definition 3.13:Let 012 1 {e} () nn HHHHHNG  be a neutrosophic subgroup series of the neutrosophic group ()NG . If 012 1 () nn HHHHHNG   (1)

{0} {0} {0} G G {e}

That is each j H is normal in 1 j H  . Then (1) is called a neutrosophic subnormal series of the neutrosophic group ()NG Florentin Smarandache (author and editor) Collected Papers, IX 105

Example 3.14: Let 4 () NGAI  be a neutrosophic group, where 4A is the alternating subgroup of the permutation group 4S . Then the following is the neutrosophic subnormal series of the group ()NG 244 4 C VVIAI 

Definition 3.15: A neutrosophic subnormal series is called strong neutrosophic subnormal series if all ' j Hs are neutrosophic normal subgroups in (1) for all j .

Example 3.16: Let () NGI  be a neutrosophic group of integers. Then the following is a strong neutrosophic subnormal series of ()NG

{0} 42III 

Theorem 3.17: Every strong neutrosophic subnormal series is trivially a neutrosophic subnormal series but the converse is not true in general.

Definition 3.18: A neutrosophic subnormal series is called mixed neutrosophic subnormal series if some ' j Hs are neutrosophic normal subgroups in (1) while some ' k Hs are just normal subgroups in (1) for some j and k.

Example 3.19: Let () NGI  be a neutrosophic group of integers. Thenthe following is a mixed neutrosophic subnormal series of ()NG

{0} 422 II .

Theorem 3.20: Every mixed neutrosophic subnormal series is trivially a neutrosophic subnormal series but the converse is not true in general.

Definition 3.21: A neutrosophic subnormal series is called subnormal series if all ' j Hs are only normal subgroups in (1) for all j .

{e} G

Theorem 3.22: Every subnormal series of the group is also a subnormal series of the neutrosophic group ()NG

Definition 3.23: If j H are all normal neutrosophic subgroups in ()NG . Then the neutrosophic subnormal series (1)is called neutrosophic normal series.

Theorem 3.24: Every neutrosophic normal series is a neutrosophic subnormal series but the converse is not true. For the converse, see the following Example.

Example 3.25: Let 4 () NGAI  be a neutrosophic group, where 4A is the alternating subgroup of the permutation group 4S . Then the following are the neutrosophic subnormal series of the group ()NG Florentin Smarandache (author and editor) Collected Papers, IX 106

This series is not neutrosophic normal series as 2C (cyclic group of order 2) is not normal in 4V (Klein four group). Similarly we can define strong neutrosophic normal series, mixed neutrosophic normal series and normal series respectively on the same lines of the neutrosophic group ()NG

Definition 3.26: The neutrosophic normal series 012 1 {e} ... () nn HHHHHNG  (2) is called neutrosophic abelian series if the factor group 1 j j

We explain it as following: Since 3 2 3

H H  are all abelian for all j .

Example 3.27: Let 3 () NGSI  be a neutrosophic group, where 3S is the permutation group. Then the following is the neutrosophic abelian series of the group ()NG 33 3 A AISI 

SI AI   and 2 is cyclic which is abelian. Thus 3 3

SI AI   is an abelian neutrosophic group. Also, 3 2 3

AI A  and this is factor group is also cyclic and every cyclic group is abelian. Hence 3 3

AI A  is also ablian group. Finally, 3 3 A I which is again abelian group. Therefore the series is a neutrosophic abelian series of the group ()NG . Thus on the same lines, we can define strong neutrosophic abelian series, mixed neutrosophic abelian series and abelian series of the neutrosophic group ()NG

Definition 3.28: A neutrosophic group ()NG is called neutrosophic soluble group if ()NG has a neutrosophic abelian series.

{e} {e} {e}

244 4 C VVIAI 

Example 3.29: Let 3 () NGSI  be a neutrosophic group, where 3S is the permutation group. Then the following is the neutrosophic abelian series of the group ()NG , 33 3 A AISI  Florentin Smarandache (author and editor) Collected Papers, IX 107

Then clearly ()NG is a neutrosophic soluble group.

Theorem 3.30: Every abelian series of a group is also an abelian series of the neutrosophic group ()NG

Theorem 3.31: If a group is a soluble group, then the neutrosophic group ()NG is also soluble neutrosophic group.

Theorem 3.32: If the neutrosophic group ()NG is an abelian neutrosophic group, then ()NG is a neutrosophic soluble group.

Theorem 3.33: If ()C(())NGNG  , then ()NG is a neutrosophic soluble group.

Proof: Suppose the ()C(())NGNG  . Then it follows that ()NG is a neutrosophic abelian group. Hence by above Theorem 3.35, ()NG is a neutrosophic soluble group.

Theorem 3.34: If the neutrosophic group ()NG is a cyclic neutrosophic group, then ()NG is a neutrosophic soluble group.

Definition 3.35: A neutrosophic group ()NG is called strong neutrosophic soluble group if ()NG has a strong neutrosophic abelian series.

Theorem 3.36: Every strong neutrosophic soluble group ()NG is trivially a neutrosophic soluble group but the converse is not true.

Definition 3.37: A neutrosophic group ()NG is called mixed neutrosophic soluble group if ()NG has a mixed neutrosophic abelian series.

Theorem 3.38: Every mixed neutrosophic soluble group ()NG is trivially a neutrosophic soluble group but the converse is not true.

Definition 3.39: A neutrosophic group ()NG is called soluble group if ()NG has an abelian series.

Definition 3.40: Let ()NG be a neutrosophic soluble group. Then length of the shortest neutrosophic abelian series of ()NG is called derived length.

Example 3.41: Let () NGI  be a neutrosophic soluble group. The following is a neutrosophic abelian series of the group ()NG {0} 422 II

Then ()NG has derived length 4.

Remark 3.42: Neutrosophic group of derive length zero is trivial neutrosophic group.

Proposition 3.43: Every neutrosophic subgroup of a neutrosophic soluble group is soluble.

Florentin Smarandache (author and editor) Collected Papers, IX 108

Proposition 3.44: Quotient neutrosophic group of a neutrosophic soluble group is soluble.

4.Neutrosophic Nilpotent Groups

Definition 4.1: Let ()NG be a neutrosophic group. The series 012 1 () nn H HNG H H ... H  (3) is called neutrosophic central series if 1 () j jj

{e}

H NG C HH      for all j .

Definition 4.2: A neutrosophic group ()NG is called a neutrosophic nilpotent group if ()NG has a neutrosophic central series.

Theorem 4.3: Every neutrosophic central series is a neutrosophic abelian series.

Theorem 4.4: If ()C(())NGNG  , then ()NG is a neutrosophic nilpotent group.

Theorem 4.5: Every neutrosophic nilpotent group ()NG is a neutrosophic soluble group.

Theorem 4.6: All neutrosophic abelian groups are neutrosophic nilpotent groups.

Theorem 4.7: All neutrosophic cyclic groups are neutrosophic nilpotent groups.

Theorem 4.8: The direct product of two neutrosophic nilpotent groups is nilpotent.

Definition 4.9: Let ()NG be a neutrosophic group. Then the neutrosophic central series (3) is called strong neutrosophic central series if all ' j Hs are neutrosophic normal subgroupsfor all j

Theorem 4.10: Every strong neutrosophic central series is trivially a neutrosophic central series but the converse is not true in general.

Theorem 4.11: Every strong neutrosophic central series is a strong neutrosophic abelian series.

Definition 4.12: A neutrosophic group ()NG is called strong neutrosophic nilpotent group if ()NG has a strong neutrosophic central series.

Theorem 4.13: Every strong neutrosophic nilpotent group is trivially a neutrosophic nilpotent group.

Theorem 4.14: Every strong neutrosophic nilpotent group is also a strong neutrosophic soluble group.

Smarandache (author and editor) Collected Papers, IX 109

Florentin

Definition 4.15: Let ()NG be a neutrosophic group. Then the neutrosophic central series (3) is called mixed neutrosophic central series if some ' j Hs are neutrosophic normal subgroups while some ' k Hs are just normal subgroups for , jk

Theorem 4.16: Every mixed neutrosophic central series is trivially a neutrosophic central series but the converse is not true in general.

Theorem 4.17: Every mixed neutrosophic central series is a mixed neutrosophic abelian series.

Definition 4.18: A neutrosophic group ()NG is called mixed neutrosophic nilpotent group if ()NG has a mixed neutrosophic central series.

Theorem 4.19: Every mixed neutrosophic nilpotent group is trivially a neutrosophic nilpotent group.

Theorem 4.20: Every mixed neutrosophic nilpotent group is also a mixed neutrosophic soluble group.

Definition 4.21: Let ()NG be a neutrosophic group. Then the neutrosophic central series (3) is called central series if all ' j Hs are only normal subgroups forall j .

Theorem 4.22: Every central series is an abelian series.

Definition 4.23: A neutrosophic group ()NG is called nilpotent group if ()NG has a central series.

Theorem 4.24: Every nilpotent group is also a soluble group.

Theorem 4.25: If G is nilpotent group, then ()NG is also a neutrosophic nilpotent group.

5. Conclusion

In this paper, we initiated the study of neutrosophic soluble groups and neutrosophic nilpotent groups which are the generalization of soluble groups and nilpotent groups. We also investigate their properties. Strong neutrosophic soluble and strong neutrosophic nilpotent groups are introduced which are completely new in their nature and properties. We also study the notions of mixed neutrosophic soluble groups and mixed neutrosophic nilpotent groups. These notions are studied on the basis of their serieses. In future, a lot of study can be carried out on neutrosophic nilpotent groups and neutrosophic soluble groups and their related properties.

References

1. M. Ali, M. Shabir, M. Naz and F. Smarandache, Neutrosophic Left Almost Semigroup, Neutrosophic Sets and Systems, 3 (2014), 18-28.

2. M. Ali, F. Smarandache, M. Shabir, and M. Naz, Neutrosophic Bi-LA-semigroup, and Neutrosophic NLA-semigroup, Neutrosophic Sets and Systems, 4 (2014), 19-24.

3. M. Ali, F. Smarandache, M. Shabir and L. Vladareanu. Generalization of Neutrosophic Rings and Neutrosophic Fields. Neutrosophic Sets and Systems, 5, (2014), 9-14.

4. M. Ali, F. Smarandache, and M. Shabir, New Research on Neutrosophic Algebraic Structures, Europa Nova. ASBL 3E clos du Paranasse Brussels, 1000, Belgium.

5. Bechtell, and Homer, The theory of groups. Addison-Wesley, (1971).

6. F. V. Haeseler, Automatic Sequences (De Gruyter Expositions in Mathematics, 36). (2002), Berlin: Walter de Gruyter.

Florentin Smarandache (author and editor) Collected Papers, IX 110

7. Isaacs, I. Martin, Finite group theory, (2008). American Mathematical Society.

8. J. Joseph, Rotman, An introduction to the theory of groups. Graduate texts in mathematics 148, (4 ed.),

9. (1995).

10.W. B. V. Kandasamy, and F. Smarandache, Basic Neutrosophic Algebraic Structures and their Applications to Fuzzy and Neutrosophic Models, Hexis, 149 pp., 2004.

11.W. B. V. Kandasamy, and F. Smarandache, Neutrosophic Rings, Hexis, Phoenix, 2006.

12.W. B. V. Kandasamy, and F. Smarandache, N-Algebraic Structures and S-N-Algebraic Structures, 209 pp., Hexis, Phoenix, 2006.

13.W. B. V. Kandasamy, and F. Smarandache, Some Neutrosophic Algebraic Structures and Neutrosophic NAlgebraic Structures, 219 p., Hexis, 2006.

14. A. I. Malcev, "Generalized nilpotent algebras and their associated groups", Mat. Sbornik N.S. 25 (67), (1949), 347 366.

15. Palmer, W. Theodore, Banach algebras and the general theory of * algebras. (1994), Cambridge, UK: Cambridge University Press.

16.F. Smarandache, A Unifying Field in Logics. Neutrosophy, Neutrosophic Probability, Set and Logic. Rehoboth: American Research Press, (1999).

17. U. Stammbach, Homology in group theory, Lecture Notes in Mathematics, Volume 359, Springer Verlag, New York, 1973.

18. Tabachnikova, Olga, Smith, and Geoff, Topics in Group Theory (Springer Undergraduate Mathematics Series). Berlin: Springer, (2000).

19. Zassenhaus, and Hans, The theory of groups. (1999). New York: Dover Publications.

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Operators on Single-Valued Neutrosophic Oversets, Neutrosophic Undersets, and Neutrosophic Offsets

Florentin Smarandache

Florentin Smarandache (2016). Operators on Single-Valued Neutrosophic Oversets, Neutrosophic Undersets, and Neutrosophic Offsets. Journal of Mathematics and Informatics 5, 63-67

Abstract. We have defined Neutrosophic Over-/Under-/Off-Set and Logic for the first time in 1995 and published in 2007. During 1995-2016 we presented them to various national and international conferences and seminars. These new notions are totally different from other sets/logics/probabilities.

We extended the neutrosophic set respectively to Neutrosophic Overset {when some neutrosophic component is > 1}, to Neutrosophic Underset {when some neutrosophic component is < 0}, and to Neutrosophic Offset {when some neutrosophic components are off the interval [0, 1], i.e. some neutrosophic component > 1 and other neutrosophic component < 0}.

This is no surprise since our real-world has numerous examples and applications of over-/under-/off-neutrosophic components.

Keywords. neutrosophic overset, neutrosophic underset, neutrosophic offset, neutrosophic over logic, neutrosophic under logic, neutrosophic off logic, neutrosophic over probability, neutrosophic under probability, neutrosophic off probability, over membership (membership degree > 1), under membership (membership degree < 0), off membership (membership degree off the interval [0, 1]).

1.Introduction

In the classical set and logic theories, in the fuzzy set and logic, and in intuitionistic fuzzy set and logic, the degree of membership and degree of non-membership have to belong to, or be included in, the interval [0, 1]. Similarly, in the classical probability and in imprecise probability the probability of an event has to belong to, or respectively be included in, the interval [0, 1].

Yet, we have observed and presented to many conferences and seminars around the globe {see [12]-[33]} and published {see [1]-[8]} that in our real world there are many cases when the degree of membership is greater than 1. The set, which has elements whose membership is over 1, we called it Overset.

Florentin Smarandache (author and editor) Collected Papers, IX 112

Even worst, we observed elements whose membership with respect to a set is under 0, and we called it Underset

In general, a set that has elements whose membership is above 1 and elements whose membership is below 0, we called it Offset (i.e. there are elements whose memberships are off (over and under) the interval [0, 1]).

2.Example of over membership and under membership

In a given company a full-time employer works 40 hours per week. Let’s consider the last week period.

Helen worked part-time, only 30 hours, and the other 10 hours she was absent without payment; hence, her membership degree was 30/40 = 0.75 < 1.

John worked full-time, 40 hours, so he had the membership degree 40/40 = 1, with respect to this company.

But George worked overtime 5 hours, so his membership degree was (40+5)/40 = 45/40 = 1.125 > 1. Thus, we need to make distinction between employees who work overtime, and those who work full-time or part-time. That’s why we need to associate a degree of membership strictly greater than 1 to the overtime workers.

Now, another employee, Jane, was absent without pay for the whole week, so her degree of membership was 0/40 = 0.

Yet, Richard, who was also hired as a full-time, not only didn’t come to work last week at all (0 worked hours), but he produced, by accidentally starting a devastating fire, much damage to the company, which was estimated at a value half of his salary (i.e. as he would have gotten for working 20 hours that week). Therefore, his membership degree has to be less that Jane’s (since Jane produced no damage). Whence, Richard’s degree of membership, with respect to this company, was - 20/40 = - 0.50 < 0.

Consequently, we need to make distinction between employees who produce damage, and those who produce profit, or produce neither damage no profit to the company.

Therefore, the membership degrees > 1 and < 0 are real in our world, so we have to take them into consideration.

Then, similarly, the Neutrosophic Logic/Measure/Probability/Statistics etc. were extended to respectively Neutrosophic Over-/Unde-r/Off-Logic, -Measure, -Probability,Statistics etc. [Smarandache, 2007].

3.Definition of single-valued neutrosophic overset

Let U be a universe of discourse and the neutrosophic set A1  U.

Let T(x), I(x), F(x) be the functions that describe the degrees of membership, indeterminate-membership, and nonmembership respectively, of a generic element x ∈U, withrespecttotheneutrosophicsetA1: T(x), I(x), F(x) : U [0,]  where 0 < 1 <  , and  is called overlimit.

A Single-Valued Neutrosophic Overset A1 is defined as: A1 = {(x, <T(x), I(x), F(x)>), x ∈ U},

Florentin Smarandache (author and editor) Collected Papers, IX 113

such that there exists at least one element in A1 that has at least one neutrosophic component that is > 1, and no element has neutrosophic components that are < 0.

For example: A1 = {(x1, <1.3, 0.5, 0.1>), (x2, <0.2, 1.1, 0.2>)}, since T(x1) = 1.3 > 1, I(x2) = 1.1 > 0, and no neutrosophic component is < 0.

Also O2 = {(a, <0.3, -0.1, 1.1>)}, since I(a) = - 0.1 < 0 and F(a) = 1.1 > 1.

4.Definition of single-valued neutrosophic underset

Let U be a universe of discourse and the neutrosophic set A2  U.

Let T(x), I(x), F(x) be the functions that describe the degrees of membership, indeterminate-membership, and nonmembership respectively, of a generic element x ∈U, withrespecttotheneutrosophicsetA2: T(x), I(x), F(x) : U [,1]  where  <0 < 1, and  is called underlimit.

A Single-Valued NeutrosophicUndersetA2is defined as: A2 = {(x, <T(x), I(x), F(x)>), x ∈ U}, such that there exists at least one element in A2 that has at least one neutrosophic component that is < 0, and no element has neutrosophic components that are > 1.

For example: A2 = {(x1, <-0.4, 0.5, 0.3>), (x2, <0.2, 0.5, -0.2>)}, since T(x1) = -0.4 < 0, F(x2) = -0.2 < 0, and no neutrosophic component is > 1.

5.Definition of single-valued neutrosophic offset

Let U be a universe of discourse and the neutrosophic set A3  U.

Let T(x), I(x), F(x) be the functions that describe the degrees of membership, indeterminate-membership, and nonmembership respectively, of a generic element x ∈U, withrespecttothesetA3: T(x), I(x), F(x) : U [,]  where  < 0 < 1 <  , and  is called under limit, while  is called overlimit.

A Single-Valued Neutrosophic Offset A3 is defined as:

A3 = {(x, <T(x), I(x), F(x)>), x ∈ U}, such that there exist some elements in A3 that have at least one neutrosophic component that is > 1, and at least another neutrosophic component that is < 0.

For examples: A3 = {(x1, <1.2, 0.4, 0.1>), (x2, <0.2, 0.3, -0.7>)}, since T(x1) = 1.2 > 1 and F(x2) = -0.7 < 0. Also, B3 = {(a, <0.3, -0.1, 1.1>)}, since I(a) = - 0.1 < 0 and F(a) = 1.1 > 1.

6.Neutrosophic overset / underset / offset operators

Let U be a universe of discourse and A = {(x, <TA(x), IA(x), FA(x)>), x ∈ U} and and B = {(x, <TB(x), IB(x), FB(x)>), x ∈ U} be two single-valued neutrosophic oversets / undersets / offsets.

TA(x), IA(x), FA(x), TB(x), IB(x), FB(x): U [,] 

Florentin Smarandache (author and editor) Collected Papers, IX 114

where ≤ 0 < 1 ≤, and is called underlimit, while is called overlimit. We take the inequality sign ≤ instead of < on both extremes above, in order to comprise all three cases: overset {when = 0, and 1 < }, underset {when < 0, and 1 = }, and offset{when < 0, and 1 < }.

Neutrosophic Overset / Underset / Offset Union

Then A∪B = {(x, <max{TA(x), TB(x)}, min{IA(x), IB(x)},min{FA(x), FB(x)}>), x∈ U}

Neutrosophic Overset / Underset / Offset Intersection.

Then A∩B = {(x, <min{TA(x), TB(x)}, max{IA(x), IB(x)},max{FA(x), FB(x)}>), x∈ U}

Neutrosophic Overset / Underset / Offset Complement

The complement of the neutrosophic set A is (A)= {(x, <FA(x),  +  - IA(x), TA(x)>), x ∈ U}.

7.Conclusion

The membership degrees over 1 (over membership), or below 0 (undermembership) are part of our real world, sotheydeserve more study in the future. The neutrosophic over set / under set / off set together with neutrosophic over logic / under logic / off logic and especiallyneutrosophic overprobability/ underprobability/ andoffprobabilityhavemany applications in technology, social science, economics and so on that the readers may be interested in exploring.

REFERENCES

1.F.Smarandache, A Unifying Field in Logics: Neutrosophic Logic. Neutrosophy, Neutrosophic Set, Neutrosophic Probability and Statistics, ProQuest Info & Learning, Ann Arbor, MI, USA, pp. 92-93, 2007, http://fs.gallup.unm.edu/ebookneutrosophics6.pdf ; first edition reviewed in ZentralblattfürMathematik (Berlin, Germany): https://zbmath.org/?q=an:01273000

2. Neutrosophy at the University of New Mexico’s website: http://fs.gallup.unm.edu/neutrosophy.htm

3. Neutrosophic Sets and Systems, international journal, in UNM website: http://fs.gallup.unm.edu/NSS; and http://fs.gallup.unm.edu/NSS/NSSNeutrosophicArticles.htm

4.F.Smarandache, Neutrosophic Set – A Generalization of the Intuitionistic Fuzzy Set; various versions of this article were published as follows:

a. in International Journal of Pure and Applied Mathematics, Vol. 24, No. 3, 287297, 2005;

b.in Proceedings of 2006 IEEE International Conference on Granular Computing, edited by Yan-Qing Zhang and Tsau Young Lin, Georgia State University, Atlanta, USA, pp. 38-42, 2006;

c. in Journal of Defense Resources Management, Brasov, Romania, No. 1, 107116, 2010.

d. as A Geometric Interpretation of the Neutrosophic Set – A Generalization of the Intuitionistic Fuzzy Set, in Proceedings of the 2011 IEEE International Conference on Granular Computing, edited by Tzung-Pei Hong, Yasuo Kudo,

         

Smarandache (author

Collected Papers, IX 115

Florentin

and editor)

Mineichi Kudo, Tsau-Young Lin, Been-ChianChien, Shyue-Liang Wang, Masahiro Inuiguchi, GuiLong Liu, IEEE Computer Society, National University of Kaohsiung, Taiwan, 602-606, 8-10 November 2011; http://fs.gallup.unm.edu/IFS-generalized.pdf

5.F.Smarandache, Degree of DependenceandIndependence of the (Sub) Components of Fuzzy Set and Neutrosophic Set, Neutrosophic Setsand Systems, 11 (2016) 9597.

6.F.Smarandache, Vietnam Veteran în StiințeNeutrosofice, instantaneousphotovideo diary, Editura Mingir, Suceava, 2016.

7.F.Smarandache, NeutrosophicOversetApplied in Physics, 69th AnnualGaseous Electronics Conference, Bochum, Germany [through American Physical Society (APS)], October 10, 2016 - Friday, October 14, 2016. Abstract submitted on 12 April 2016.

8.D. P. Popescu, Să nu ne sfiim să gândim diferit - de vorbă cu prof. univ. dr. Florentin Smarandache, Revista “Observatorul”, Toronto, Canada, Tuesday, June 21, 2016, http://www.observatorul.com/default.asp?action=articleviewdetail&ID=15698

9.F. Smarandache, SymbolicNeutrosophicTheory, Europa Nova, Bruxelles, 194 p., 2015; http://fs.gallup.unm.edu/SymbolicNeutrosophicTheory.pdf

10. F.Smarandache, Introduction to Neutrosophic Measure, Neutrosophic Integral, and Neutrosophic Probability, Sitech, 2003; http://fs.gallup.unm.edu/NeutrosophicMeasureIntegralProbability.pdf

11. F.Smarandache, Introduction to Neutrosophic Statistics, Sitech Craiova, 123 pages, 2014, http://fs.gallup.unm.edu/NeutrosophicStatistics.pdf

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Interval-Valued Neutrosophic Oversets, Neutrosophic Undersets, and Neutrosophic Offsets

Florentin Smarandache

Florentin Smarandache (2016). Interval-Valued Neutrosophic Oversets, Neutrosophic Undersets, and Neutrosophic Offsets. International Journal of Science and Engineering Investigations 5, 54, Paper ID: 55416-01, 4 p.

Abstract-We have proposed since 1995 the existence of degrees of membership of an element with respect to a neutrosophic set to also be partially or totally above 1 (overmembership), and partially or totally below 0 (undermembership) in order to better describe our world problems [published in 2007].

Keywords-interval neutrosophic overset, interval neutrosophic underset, interval neutrosophic offset, interval neutrosophic overlogic, interval neutrosophic underlogic, interval neutrosophic offlogic, interval neutrosophic overprobability, interval neutrosophic underprobability, interval neutrosophic offprobability, interval overmembership (interval membership degree partially or totally above 1), interval undermembership (interval membership degree partially or totally below 0), interval offmembership (interval membership degree off the interval [0, 1]).

I. INTRODUCTION

“Neutrosophic” means based on three components T (truthmembership), I (indeterminacy), and F (falsehood-non membership). And “over” means above 1, “under” means below 0, while “offset” means behind/beside the set on both sides of the interval [0, 1], over and under, more and less, supra and below, out of, off the set. Similarly, for “offlogic”, “offmeasure”, “offprobability”, “offstatistics” etc.

It is like a pot with boiling liquid, on a gas stove, when the liquid swells up and leaks out of pot. The pot (the interval [0, 1]) can no longer contain all liquid (i.e., all neutrosophic truth/indeterminate/falsehood values), and therefore some of them fall out of the pot (i.e., one gets neutrosophic truth/indeterminate/falsehood values which are > 1), or the pot cracks on the bottom and the liquid pours down (i.e., one gets neutrosophic truth/indeterminate/falsehood values which are < 0).

Mathematically, they mean getting values off the interval [0, 1].

The American aphorism “think outside the box” has a perfect resonance to the neutrosophic offset, where the box is the interval [0, 1], yet values outside of this interval are permitted.

II. EXAMPLEOF MEMBERSHIP ABOVE 1 AND MEMBERSHIP BELOW 0

Let’s consider a spy agency S = {S1, S2, …, S1000} of a country Atara against its enemy country Batara. Each agent Sj, j ∈ {1, 2, …, 1000}, was required last week to accomplish 5 missions, which represent the full-time contribution/membership.

Last week agent S27 has successfully accomplished his 5 missions, so his membership was T(S27) = 5/5 = 1 = 100% (full-time membership).

Agent S32 has accomplished only 3 missions, so his membership is T(S32) = 3/5 = 0.6 = 60% (part-time membership).

Agent S41 was absent, without pay, due to his health problems; thus T(S41) = 0/5 = 0 = 0% (null-membership).

Agent A53 has successfully accomplished his 5 required missions, plus an extra mission of another agent that was absent due to sickness, therefore T(S53) = (5+1)/5 = 6/5 = 1.2 > 1 (therefore, he has membership above 1, called overmembership).

Yet, agent S75 is a double-agent, and he leaks highly confidential information about country Atara to the enemy country Batara, while simultaneously providing misleading information to the country Atara about the enemy country Batara. Therefore A75 is a negative agent with respect to his country Atara, since he produces damage to Atara, he was estimated to having intentionally done wrongly all his 5 missions, in addition of compromising a mission of another agent of country Atara, thus his membership T(S75) = - (5+1)/5 = - 6/5 = -1.2 < 0 (therefore, he has a membership below 0, called under-membership).

III. DEFINITIONOF INTERVAL-VALUED NEUTROSOPHIC OVERSET

Let U be a universe of discourse and the neutrosophic set A1  U. Let T(x), I(x), F(x) be the functions that describe the degrees of membership, indeterminate-membership, and nonmembership respectively, of a generic element x ∈ U, with respect to the neutrosophic set A1:

T(x), I(x), F(x) : U  P( [0,]  ),

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B. Interval-Valued Neutrosophic Overset / Underset / Offset Intersection

Then A∩B = {(x, <[min{inf(TA(x)), inf(TB(x))}, min{sup(TA(x)), sup(TB(x)}], [max{inf(IA(x)), inf(IB(x))}, max{sup(IA(x)), sup(IB(x)}], [max{inf(FA(x)), inf(FB(x))}, max{sup(FA(x)), sup(FB(x)}]>, x ∈ U}.

C. Interval-Valued Neutrosophic Overset / Underset / Offset Complement

The complement of the neutrosophic set A is C(A) = {(x, <FA(x), [  +  - sup{IA(x)},  +  -inf{IA(x)}], TA(x)>), x ∈ U}.

VII. CONCLUSION

After designing the neutrosophic operators for singlevalued neutrosophic overset/underset/offset, we extended them to interval-valued neutrosophic overset/underset/offset operators. We also presented another example of membership above 1 and membership below 0.

Of course, in many real world problems the neutrosophic union, neutrosophic intersection, and neutrosophic complement for interval-valued neutrosophic overset/underset/offset can be used. Future research will be focused on practical applications.

REFERENCES

[1]Florentin Smarandache, A Unifying Field in Logics: Neutrosophic Logic. Neutrosophy, Neutrosophic Set, Neutrosophic Probability and Statistics, ProQuest Info & Learning, Ann Arbor, MI, USA, pp. 92-93, 2007, http://fs.gallup.unm.edu/ebook-neutrosophics6.pdf ; first edition reviewed in Zentralblatt für Mathematik (Berlin, Germany): https://zbmath.org/?q=an:01273000 .

[2]Neutrosophy at the University of New Mexico’s website: http://fs.gallup.unm.edu/neutrosophy.htm

[3]Neutrosophic Sets and Systems, international journal, in UNM website: http://fs.gallup.unm.edu/NSS; andhttp://fs.gallup.unm.edu/NSS/NSSNeutrosophicArticles.htm

[4]Florentin Smarandache, Neutrosophic Set A Generalization of the Intuitionistic Fuzzy Set; various versions of this article were published as follows: in International Journal of Pure and Applied Mathematics, Vol. 24, No. 3, 287-297, 2005; in Proceedings of 2006 IEEE International Conference on Granular Computing, edited by Yan-Qing ZhangandTsauYoungLin,GeorgiaStateUniversity,Atlanta,USA,pp. 38-42, 2006; in Journal of Defense Resources Management, Brasov, Romania, No. 1, 107-116, 2010. as A Geometric Interpretation of the Neutrosophic Set A Generalization of the Intuitionistic Fuzzy Set, in Proceedings of the 2011 IEEE International Conference on Granular Computing, edited by Tzung-Pei Hong, Yasuo Kudo, Mineichi Kudo, Tsau-Young Lin, Been-Chian Chien, Shyue-Liang Wang, Masahiro Inuiguchi, GuiLong Liu, IEEE Computer Society, National University of Kaohsiung, Taiwan, 602-606, 8-10 November 2011; http://fs.gallup.unm.edu/IFS-generalized.pdf

[5]Florentin Smarandache, Degree of Dependence and Independence ofthe (Sub)Components of Fuzzy Set and Neutrosophic Set, Neutrosophic SetsandSystems(NSS),Vol.11,95-97,2016.

[6]Florentin Smarandache, Vietnam Veteran în Stiințe Neutrosofice, instantaneousphoto-videodiary,EdituraMingir,Suceava,2016.

[7]Florentin Smarandache, Neutrosophic Overset Applied in Physics, 69th Annual Gaseous Electronics Conference, Bochum, Germany [through American Physical Society (APS)], October 10, 2016 - Friday, October 14,2016.Abstractsubmittedon12April2016.

[8]Dumitru P.Popescu, Sănunesfiimsăgândimdiferit - devorbăcuprof. univ. dr. Florentin Smarandache, Revista “Observatorul”, Toronto, Canada, Tuesday, June 21, 2016, http://www.observatorul.com/default.asp?action=articleviewdetail&ID= 15698

[9]F. Smarandache, Operators on Single-Valued Neutrosophic Overset, Neutrosophic Underset, and Neutrosophic Offset, Annals of Pure and AppliedMathematics,submitted,2016.

[10]F. Smarandache, History of Neutrosophic Theory and its Applications (Neutrosophic Over-/Under-/Off-Set and -Logic), International Conference on Virtual Learning, Virtual Learning - Virtual Reality Models and Methodologies Technologies and Software Solutions, Founder and Chairman of Project: Ph. D. Marin Vlada, Partners: Ph. D. Grigore Albeanu, Ph. D. Adrian Adăscăliței, Ph. D. Mircea Dorin Popovici, Prof. Radu Jugureanu, Ph. D. Olimpius Istrate, Institutions: University of Bucharest, National Authority for Scientific Research, SIVECO Romania, Intel Corporation, 2016, http://www.c3.icvl.eu/2016/smarandache

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Subtraction and Division of Neutrosophic Numbers

Florentin Smarandache

Florentin Smarandache (2016). Subtraction and Division of Neutrosophic Numbers. Critical Review XIII, 103-110

Abstract

In this paper, we define the subtraction and the division of neutrosophic singlevalued numbers. The restrictions for these operations are presented for neutrosophic single-valued numbers and neutrosophic single-valued overnumbers / undernumbers / offnumbers. Afterwards, several numeral examplesarepresented.

Keywords

neutrosophic calculus, neutrosophic numbers, neutrosophic summation, neutrosophic multiplication, neutrosophic scalar multiplication, neutrosophic power,neutrosophicsubtraction,neutrosophicdivision.

1 Introduction

Let ��=(��1,��1,��1) and �� =(��2,��2,��2) be two single valued neutrosophic numbers, where ��1,��1,��1,��2,��2,��2 ∈[0,1] , and 0≤��1,��1,��1 ≤3 and 0≤ ��2,��2,��2 ≤3.

Thefollowingoperationalrelationshavebeendefinedandmostlyusedinthe neutrosophicscientificliterature:

1.1 Neutrosophic Summation

��⊕�� =(��1 +��2 ��1��2,��1��2,��1��2) (1)

1.2 Neutrosophic Multiplication

A⊗�� =(��1��2,��1 +��2 ��1��2,��1 +��2 ��1��2) (2)

1.3

Neutrosophic Scalar Multiplication

⋋��=(1 (1 ��1)⋋,��1 ⋋,��1 ⋋ ), (3) where⋋∈ℝ,and⋋>0.

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1.4 Neutrosophic Power

��⋋ =(��1 ⋋,1 (1 ��1)⋋,1 (1 ��1)⋋), (4) where⋋∈ℝ,and⋋>0.

2 Remarks

Actually, the neutrosophic scalar multiplication is an extension of neutrosophicsummation;inthelast,onehas⋋=2. Similarly, the neutrosophic power is an extension of neutrosophic multiplication;inthelast,onehas⋋=2. Neutrosophic summation of numbers is equivalent to neutrosophic union of sets,andneutrosophicmultiplicationofnumbersisequivalenttoneutrosophic intersectionofsets.

That's why, both the neutrosophic summation and neutrosophic multiplication (and implicitly their extensions neutrosophic scalar multiplication and neutrosophic power) can be defined in many ways, i.e. equivalently to their neutrosophic union operators and respectively neutrosophicintersectionoperators.

Ingeneral:

��⊕�� =(��1 ∨��2,��1 ∧��2,��1 ∧��2), (5) or ��⊕�� =(��1 ∨��2,��1 ∨��2,��1 ∨��2), (6) andanalogously: ��⊗�� =(��1 ∧��2,��1 ∨��2,��1 ∨��2) (7) or ��⊗�� =(��1 ∧��2,��1 ∧��2,��1 ∨��2), (8) where"∨"isthefuzzyOR(fuzzyunion)operator,defined,for��,�� ∈[0,1],in threedifferentways,as: �� 1 ∨�� =��+�� , (9) or �� 2 ∨�� =��{��,��}, (10) or �� 3 ∨�� =��{��+��,1}, (11) etc.

Florentin Smarandache (author and editor) Collected Papers, IX 121

While "∧" is the fuzzy AND (fuzzy intersection) operator, defined, for��,�� ∈ [0,1],inthreedifferentways,as:

�� ∧ 1�� =��, (12) or �� ∧ 2�� =��{��,��}, (13) or �� ∧ 3�� =��{��+�� 1,0}, (14) etc.

Into the definitions of��⊕��and��⊗��it's better if one associates 1 ∨ with ∧ 1, since 1 ∨ isopposedto ∧ 1,and 2 ∨ with ∧ 2,and 3 ∨ with ∧ 3,forthesamereason.Butother associationscanalsobeconsidered.

Forexamples:

��⊕�� =(��1 +��2 ��1��2,��1 +��2 ��1��2,��1��2), (15) or ��⊕�� =(��{��1,��2},��{��1,��2},��{��1,��2}), (16) or ��⊕�� =(��{��1,��2},��{��1,��2},��{��1,��2}), (17) or ��⊕�� =(��{��1 +��2,1},��{��1 +��2 1,0},��{��1 +��2 1,0}). (18) wherewehaveassociated 1 ∨ with ∧ 1,and 2 ∨ with ∧ 2,and 3 ∨ with ∧ 3

Let'sassociatethemindifferentways:

��⊕�� =(��1 +��2 ��1��2,��{��1,��2},��{��1,��2}), (19) where 1 ∨ wasassociatedwith ∧ 2 and ∧ 3;or:

��⊕�� =(��{��1,��2},��1,��2,��{��1 +��2 1,0}), (20) where 2 ∨ wasassociatedwith ∧ 1 and ∧ 3;andsoon.

Similarexamplescanbeconstructedfor��⊗��.

3 NeutrosophicSubtraction

Wedefinenow,forthefirsttime,thesubtractionofneutrosophicnumber:

��⊖�� =(��1,��1,��1)⊖(��2,��2,��2)=(��1 ��2 1 ��2 , ��1 ��2 , ��1 ��2)=��, (21)

Florentin Smarandache (author and editor) Collected Papers, IX 122

for all��1,��1,��1,��2,��2,��2 ∈[0,1], with the restrictions that:��2 ≠1, ��2 ≠0, and ��2 ≠0

So,theneutrosophicsubtractiononlypartiallyworks,i.e.when��2 ≠1,��2 ≠0, and��2 ≠0

Therestrictionthat

(��1 ��2 1 ��2 , ��1 ��2 , ��1 ��2)∈([0,1],[0,1],[0,1]) (22)

is set when the classical case when the neutrosophic number components ��,��,��areintheinterval[0,1].

But,forthegeneralcase,whendealingwithneutrosophicoverset/underset /offset[1],ortheneutrosophicnumbercomponentsareintheinterval[Ψ,Ω], whereΨiscalled underlimit andΩiscalled overlimit,withΨ≤0<1≤Ω,i.e. one has neutrosophic overnumbers / undernumbers / offnumbers, then the restriction(22)becomes:

(��1 ��2 1 ��2 , ��1 ��2 , ��1 ��2)∈([Ψ,Ω],[Ψ,Ω],[Ψ,Ω]). (23)

3.1 Proof

The formula for the subtraction was obtained from the attempt to be consistentwiththeneutrosophicaddition. Oneconsidersthemostusedneutrosophicaddition:

(��1,��1,��1)⊕(��2,��2,��2)=(��1 +��2 ��1��2,��1��2,��1��2), (24)

(27)

��1 =��+��2 ��2,whence�� = ��1 ��1 1 ��2 ; ��1 =��2,whence��= ��1 ��2 ; ��1 =��2,whence�� = ��1 ��2 . Florentin Smarandache (author and editor) Collected Papers, IX 123

We consider the ⊖ neutrosophic operation the opposite of the ⊕ neutrosophic operation, as in the set of real numbers the classical subtraction is theoppositeoftheclassicaladdition+. Therefore,let'sconsider: (��1,��1,��1)⊖(��2,��2,��2)=(��,��,��), (25) ⊕(��2,��2,��2) ⊕(��2,��2,��2) where��,��,�� ∈ℝ. Weneutrosophicallyadd⊕(��2,��2,��2)onbothsidesoftheequation.Weget: (��1,��1,��1)=(��,��,��)⊕(��2,��2,��2)=(��+��2 ��2,��2,��2). (26) Or, {

3.2 Checking the Subtraction

With��=(��1,��1,��1),�� =(��2,��2,��2),and�� =(��1 ��2 1 ��2 , ��1 ��2 , ��1 ��2), where��1,��1,��1,��2,��2,��2 ∈[0,1],and��2 ≠1,��2 ≠0,and��2 ≠0,wehave: ��⊖�� =��. (28) Then: ��⊕�� =(��2,��2,��2)⊕(��1 ��2 1 ��2 , ��1 ��2 , ��1 ��2)=(��2 + ��1 ��2 1 ��2 ��2 ⋅ ��1 ��2 1 ��2 ,��2, ��1 ��2 ,��2, ��1 ��2)=(��2 ��2 2+��1 ��2 ��1��2+��2 1 ��2 ,��1,��1)= (��1(1 ��2) 1 ��2 ,��1,��1)=(��1,��1,��1). (29) ��⊖�� =(��1,��1,��1)⊖(��1 ��2 1 ��2 , ��1 ��2 , ��1 ��2)=(��1 ��1 ��2 1 ��2 1 ��1 ��2 1 ��2 , ��1 ��1 ��2 , ��1 ��1 ��2)= (��1 ��1��2 ��1+��2 1 ��2 1 ��2 ��1+��2 1 ��2 ,��2,��2)=( ��1��2+��2 1 ��2 ,��2,��2)= (��2( ��1+1) 1 ��2 ,��2,��2)=(��2,��2,��2) (30)

4 DivisionofNeutrosophicNumbers

Wedefineforthefirsttimethedivisionofneutrosophicnumbers: ��⊘�� =(��1,��1,��1)⊘(��2,��2,��2)=(��1 ��2 , ��1 ��2 1 ��2 , ��1 ��2 1 ��2)=��, (31) where ��1,��1,��1,��2,��2,��2 ∈[0,1], with the restriction that ��2 ≠0, ��2 ≠1, and ��2 ≠1. Similarly,thedivisionofneutrosophicnumbersonlypartiallyworks,i.e.when ��2 ≠0,��2 ≠1,and��2 ≠1. Inthesameway,therestrictionthat (��1 ��2 , ��1 ��2 1 ��2 , ��1 ��2 1 ��2)∈([0,1],[0,1],[0,1]) (32) issetwhenthetraditionalcaseoccurs,whentheneutrosophicnumber components t,i,fareintheinterval[0,1]. But,forthecasewhendealingwithneutrosophicoverset/underset/offset [1],whentheneutrosophicnumbercomponentsareintheinterval[Ψ,Ω], whereΨiscalled underlimit andΩiscalled overlimit,withΨ≤0<1≤Ω,i.e. onehas neutrosophic overnumbers / undernumbers / offnumbers,thenthe restriction(31)becomes:

Florentin Smarandache (author and editor) Collected Papers, IX 124

(��1 ��2 , ��1 ��2 1 ��2 , ��1 ��2 1 ��2)∈([Ψ,Ω],[Ψ,Ω],[Ψ,Ω]) (33)

4.1 Proof

In the same way, the formula for division⊘of neutrosophic numbers was obtained from the attempt to be consistent with the neutrosophic multiplication.

We consider the ⊘ neutrosophic operation the opposite of the ⊗ neutrosophicoperation,asinthesetofrealnumberstheclassicaldivision÷ istheoppositeoftheclassicalmultiplication×

Oneconsidersthemostusedneutrosophicmultiplication: (��1,��1,��1)⊗(��2,��2,��2) =(��1��2,��1 +��2 ��1��2,��1 +��2 ��1��2), (34)

Thus,let'sconsider: (��1,��1,��1)⊘(��2,��2,��2)=(��,��,��), (35) ⨂(��2,��2,��2) ⨂(��2,��2,��2) where��,��,�� ∈ℝ

Weneutrosophicallymultiply⨂bothsidesby(��2,��2,��2).Weget (��1,��1,��1)=(��,��,��)⨂(��2,��2,��2) =(��2,��+��2 ��2,��+��2 ��2). (36)

Or, {

��1 =��2,whence�� = ��1 ��2 ;: ��1 =��+��2 ��2,whence��= ��1 ��2 1 ��2 ; ��1 =��+��2 ��2,whence�� = ��1 ��2 1 ��2 .

4 2 Checking the Division

(37)

With��=(��1,��1,��1),�� =(��2,��2,��2),and�� =(��1 ��2 , ��1 ��2 1 ��2 , ��1 ��2 1 ��2), where��1,��1,��1,��2,��2,��2 ∈[0,1],and��2 ≠0,��2 ≠1,and��2 ≠1,onehas: ��∗�� =��. (38) Then: �� =(��2,��2,��2)×(��1 ��2 , ��1 ��2 1 ��2 , ��1 ��2 1 ��2)=(��2 ⋅ ��1 ��2 ,��2 + ��1 ��2 1 ��2 ��2 ⋅ ��1 ��2 1 ��2 ,��2 + ��1 ��2 1 ��2 ��2 ⋅ ��1 ��2 1 ��2)=

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5 Conclusion

Wehaveobtainedtheformulaforthesubtractionofneutrosophicnumbers⊖ goingbackwordsfromtheformulaofadditionofneutrosophicnumbers ⊕ Similarly,wehavedefinedtheformulafordivisionofneutrosophicnumbers ⊘andweobtaineditbackwordsfromtheneutrosophicmultiplication⨂ We also have taken into account the case when one deals with classical neutrosophicnumbers(i.e.theneutrosophiccomponentst,i,fbelongto[0,1]) aswellasthegeneralcasewhen��,��,��belongto[��,��],wheretheunderlimit �� ≤ 0andtheoverlimit�� ≥ 1.

6 References

[1] Florentin Smarandache, Neutrosophic Overset, Neutrosophic Underset, and Neutrosophic Offset. Similarly for Neutrosophic Over-/Under-/Off- Logic, Probability, and Statistics, 168 p., Pons Editions, Bruxelles, Belgique, 2016; https://hal.archives-ouvertes.fr/hal-01340830 ; https://arxiv.org/ftp/arxiv/papers/1607/1607.00234.pdf

[2] Florentin Smarandache, Neutrosophic Precalculus and Neutrosophic Calculus, EuropaNova, Brussels, Belgium, 154 p., 2015; https://arxiv.org/ftp/arxiv/papers/1509/1509.07723.pdf

(Arabic translation) by Huda E. Khalid and Ahmed K. Essa, Pons Editions, Brussels, 112 p., 2016.

[3] Ye Jun, Bidirectional projection method for multiple attribute group decision making with neutrosophic numbers, Neural Computing and Applications, 2015, DOI: 10.1007/s00521-015 2123-5.

[4] Ye Jun, Multiple-attribute group decision-making method under a neutrosophic number environment, Journal of Intelligent Systems, 2016, 25(3): 377-386.

[5] Ye Jun, Faultdiagnosesofsteamturbineusingtheexponentialsimilaritymeasure of neutrosophic numbers, Journal of Intelligent & Fuzzy Systems, 2016, 30: 1927–1934

)= (��

) 1

)=(��2,��2,��2)=��.

(��1, ��2 ��2 2+��1 ��2 ��1��2+��2 2 1 ��2 , ��2 ��2 2+��1 ��2 ��1��2+��2 2 1 ��2 )= (��1, ��1(1 ��2) 1 ��2 , ��1(1 ��2) 1 ��2 )=(��1,��1,��1)=��. (39) Also: �� = (��1,��1,��1) (��1 ��2, ��1 ��2 1 ��2, ��1 ��2 1 ��2 )

��1 ��1 ��2 , ��1 ��1 ��2 1 ��2 1 ��1 ��2 1 ��2 , ��1 ��1 ��2 1 ��2 1 ��1 ��2 1 ��2 )= (��2, ��1 ��1��2 ��1+��2 1 ��2 1 ��2 ��1+��2 1 ��2 , ��1 ��1��2 ��1+��2 1 ��2 1 ��2 ��1+��2 1 ��2 )=(��2, ��2( ��1+1) 1 ��2 1 ��1 1 ��2 , ��2( ��1+1) 1 ��2

��1

��2

2, ��2(1 ��1)

��1 , ��2(1 ��1

��1

(40)

يكفوسورتوينلا لماكتلاو لضافتلاباسح و يكفوسورتوينلا لماكتلاو لضافتلائدابم

Florentin Smarandache (author and editor) Collected Papers, IX 126

On Neutrosophic Quadruple Algebraic Structures

S.A. Akinleye, F. Smarandache, A.A.A. Agboola

S.A. Akinleye, Florentin Smarandache, A.A.A. Agboola (2016). On Neutrosophic Quadruple Algebraic Structures. Neutrosophic Sets and Systems 12, 122-126

Abstract. In this paper we present the concept of neutrosophic quadruple algebraic structures. Specially, we study neutrosophic quadruple rings and we present their elementary properties.

Keywords: Neutrosophy, neutrosophic quadruple number, neutrosophic quadruple semigroup, neutrosophic quadruple group, neutrosophic quadruple ring, neutrosophic quadruple ideal, neutrosophic quadruple homomorphism.

1 Introduction

The concept of neutrosophic quadruple numbers was introduced by Florentin Smarandache [3]. It was shown in [3]how arithmetic operations of addition, subtraction, multiplication and scalar multiplication could be performed on the set of neutrosophic quadruple numbers. In this paper, we studied neutrosophic sets of quadruple numbers together with binary operations of addition and multiplication and the resulting algebraic structures with their elementary properties are presented. Specially, we studied neutrosophic quadruple rings and we presented their basic properties.

Definition 1.1 [3]

A neutrosophic quadruple number is a number of the form (��,��,��,��), where ��,��,�� have their usual neutrosophic logic meanings and ��,��,��,��∈ℝ or ℂ. The set �� defined by �� = {(��,��,��,��)∶ ��,��,��,��∈ℝorℂ} (1) is called a neutrosophic set of quadruple numbers. For a neutrosophic quadruple number (��,��,��,��), representing any entity which may be a number, an idea, an object, etc., �� is called the known part and (��,��,��) is called the unknown part.

Definition 1.2

Let �� = (��1,��2��,��3��,��4��), �� = (��1,��2��,��3��,��4��)∈��. We define the following:

��+��= (2) (��1+��1,(��2+��2)��,(��3+��3)��,(��4 +��4)��) ��−��= (3) (��1−��1,(��2−��2)��,(��3−��3)��,(��4−��4)��)

Definition 1.3

Let ��=(��1,��2��,��3��,��4��)∈�� and let �� be any scalar which may be real or complex, the scalar product ��.�� is defined by ��.�� = ��.(��1,��2��,��3��,��4��) = (��1,��2��,��3��,��4��) (4) If ��=0, then we have 0.��=(0,0,0,0) and for any non-zero scalars m and n and b= (��1,��2T,��3I,��4F), we have: (��+��)��=��+��, ��(��+��)=��+��, ��(��)=��(��), −��=(−��1,−��2��,−��3��,−��4��)

Definition

1.4

[3] [Absorbance Law]

Let �� be a set endowed with a total order �� < ��, named “x prevailed by y” or “x less strong than y” or “x less preferred than y”. ��≤�� is considered as “�� prevailed by or equal to ��” or “�� less strong than or equal to ��” or “�� less preferred than or equal to ��”.

For any elements ��,��∈��, with ��≤��, absorbance law is defined as ��∙�� = ��∙�� = absorb(��,��) = max{��,��} = �� (5) which means that the bigger element absorbs the smaller element (the big fish eats the small fish). It is clear from (5) that ��∙��=��2 =��(��,��)=��{��,��}=�� (6) and ��1 ∙��2 ···�� = ��{��1,��2,···,��} (7)

Florentin Smarandache (author and editor) Collected Papers, IX 127

Analogously, if �� > ��, we say that “�� prevails to ��” or “�� is stronger than ��” or “�� is preferred to ��”. Also, if ��≥��, we say that “�� prevails or is equal to ��” or “�� is stronger than or equal to ��” or “�� is preferred or equal to ��”

Deﬁnition 1.5

Consider the set {��,��,��}. Suppose in an optimistic way we consider the prevalence order ��>��>��. Then we have:

�� =�� =max{��,��}=��, (8) �� =�� =max{��,��}=��, (9)

�� =�� =max{��,��}=��, (10) �� =��2 = ��, (11) �� =��2 =��, (12) �� =��2 =�� (13)

Analogously, suppose in a pessimistic way we consider the prevalence order ��< ��<��. Then we have:

�� = �� = ��{��,��} = ��, (14) �� = �� = ��{��,��} = ��, (15) �� = �� = ��{��,��} = ��, (16) �� = ��2 = ��, (17) �� = ��2 = ��, (18) �� = ��2 = �� (19)

Definition 1.6

Let ��=(��1,��2��,��3��,��4��), ��=(��1,��2��,��3��,��4��)∈�� Then (20) ��.�� =(��1,��2��,��3��,��4��).(��1,��2��,��3��,��4��) = (��1��1,(��1��2 + ��2��1 + ��2��2)��,(��1��3 + ��2��3 + ��3��1 + ��3��2 + ��3��3)��,(��1��4 + ��2��4,��3��4 +��4��1 +��4��2 + ��4��3 +��4��4)��).

2 Main Results

All neutrosophic quadruple numbers to be considered in this section will be real neutrosophic quadruple numbers i.e ��,��,��,��∈ℝ for any neutrosophic quadruple number (��,��,��,��)∈��.

Theorem 2.1 (��,+) is an abelian group.

Proof. Suppose that ��=(��1,��2��,��3��,��4��), ��=(��1,��2��,��3��, ��=(��1,��2��,��3��,��4�� ∈��

are arbitrary. It can easily be shown that ��+��=��+��∙��+(��+��)= (��+��)+��∙��+(0,0,0,0)=(0,0,0,0)=�� and ��+(−��)=−��+��=(0,0,0,0). Thus, 0=(0,0,0,0) is the additive identity element in (��,+) and for any �� ∈��, �� is the additive inverse. Hence, (��,+) is an abelian group.

Theorem 2.2 (��,.) is a commutative monoid.

Proof. Let ��=(��1,��2��,��3��,��4��), ��=(��1,��2��,��3��, ��=(��1,��2��,��3��,��4�� be arbitrary elements in ��. It can easily be shown that �� =��∙��(��)=(��)��∙��∙(1,0,0,0)=��. Thus, ��=(1,0,0,0) is the multiplicative identity element in (��,.). Hence, (��,.) is a commutative monoid. Theorem 2.3 (��,.) is not a group.

Proof. Let ��=(��,��,��,��) be any arbitrary element in �� Since we cannot ﬁnd any element ��=(��,��,��,��)∈ �� such that ��=�� =�� =(1,0,0,0), it follows that �� 1 does not exist in �� for any given ��,��,��,�� ∈ℝ and consequently, (��,.) cannot be a group.

Example 1. Let ��={(��,��,��,��)∶ ��,��,��,�� ∈ℤ��} Then (��,+) is an abelian group.

Example 2. Let (��2×2,.)={ [(��,��,��,��)(��,��,��,ℎ��) (��,��,��,��)(��,��,��,��)]: a,b,c,d,e,f,g,h,i,j,k,l,m,n,p,q ∈ℝ}

Then (��2×2,.) is a non commutative monoid. Theorem 2.4 (��,+,.) is a commutative ring.

Florentin Smarandache (author and editor) Collected Papers, IX 128

Proof.

It is clear that (��,+) is an abelian group and (��,.) is a semigroup. To complete the proof, suppose that ��=(��1,��2��,��3��,��4��), ��=(��1,��2��,��3��, ��=(��1,��2��,��3��,��4�� ∈�� are arbitrary. It can easily be shown that ��(��+��)=��+ ��,(�� + ��)��=��+�� and �� =��. Hence, (��,+,.) is a commutative ring.

From now on, the ring (��,+,.) will be called neutrosophic quadruple ring and it will be denoted by ��. The zero element of �� will be denoted by (0, 0, 0, 0) and the unity of �� will be denoted by (1, 0, 0, 0).

Example 3.

(i)Let �� be as deﬁned in EXAMPLE 1. Then (��,+,.) is a commutative neutrosophic quadruple ring called a neutrosophic quadruple ring of integers modulo �� It should be noted that ��(ℤ��) has 4�� elements and for ��(ℤ2)we have ��(ℤ2)= ={(0,0,0,0),(1,0,0,0),(0,��,0,0),(0,0,��,0),(0,0,0,��), (0,��,��,��),(0,0,��,��),(0,��,��,0),(0,��,0,��),(1,��,0,0), (1,0,��,0),(1,0,0,��),(1,��,0,��),(1,0,��,��),(1,��,��,0), (1,��,��,��)}

(ii)Let ��2×2 be as deﬁned in EXAMPLE 2. Then (��2×2, ) is a non commutative neutrosophic quadruple ring.

Definition 2.5

Let �� be a neutrosophic quadruple ring. (i)An element �� ∈�� is called idempotent if ��2 =�� (ii)An element �� ∈�� is called nilpotent if there �� ∈��+ such that �� =0

Example 4.

(i)In ��(ℤ2), (1,��,��,��) and (1,��,��,0) are idempotent elements.

(ii)In ��(ℤ4), (2,2��,2��,2��) is a nilpotent element.

Definition 2.6

Let �� be a neutrosophic quadruple ring. �� is called a neutrosophic quadruple integral domain if for ��,�� ∈�� , ��=0 implies that �� =0 or ��=0.

Example 5.

��(ℤ) the neutrosophic quadruple ring of integers is a neutrosophic quadruple integral domain.

Definition 2.7

Let �� be a neutrosophic quadruple ring. An element �� ∈�� is called a zero divisor if there

exists a nonzero element ��∈�� such that ��=0. For example in ��(ℤ2), (0,0,��,��) and (0,��,��,0) are zero divisors even though ℤ2 has no zero divisors.

This is one of the distinct features that characterize neutrosophic quadruple rings.

Definition 2.8

Let �� be a neutrosophic quadruple ring and let �� be a nonempty subset of ��. Then �� is called a neutrosophic quadruple subring of �� if (��,+,.) is itself a neutrosophic quadruple ring. For example, ��(��ℤ) is a neutrosophic quadruple subring of ��(ℤ) for ��= 1,2,3,···.

Theorem

2.9

Let �� be a nonempty subset of a neutrosophic quadruple ring ��. Then �� is a neutrosophic quadruple subring if and only if for all ��,��∈��, the following conditions hold:

(i) ��−�� ∈�� and

(ii) ��∈�� Proof. Same as the classical case and so omitted.

Definition 2.10

Let �� be a neutrosophic quadruple ring. Then the set ��(��) = {�� ∈��:��=��∀��∈��} is called the centre of ��

Theorem 2.11

Let �� be a neutrosophic quadruple ring. Then ��(��) is a neutrosophic quadruple subring of �� Proof. Same as the classical case and so omitted.

Theorem 2.12

Let �� be a neutrosophic quadruple ring and let �� be families of neutrosophic quadruple subrings of ��. Then ⋂ �� =1 �� is a neutrosophic quadruple subring of ��

Definition 2.13

Let �� be a neutrosophic quadruple ring. If there exists a positive integer �� such that �� =0 for

Florentin Smarandache (author and editor) Collected Papers, IX 129

each �� ∈��, then the smallest such positive integer is called the characteristic of ��. If no such positive integer exists, then �� is said to have characteristic zero. For example, ��(ℤ) has characteristic zero and ��(ℤ��) has characteristic ��

Definition 2.14

Let �� be a nonempty subset of a neutrosophic quadruple ring �� is called a neutrosophic quadruple ideal of �� if for all x,y∈��, �� ∈��, the following conditions hold:

(i) ��−�� ∈�� (ii) �� ∈�� and �� ∈��.

Example 6.

(i) ��(3ℤ) is a neutrosophic quadruple ideal of ��(ℤ). (ii)Let �� = {(0,0,0,0),(2,0,0,0),(0,2��,2��,2��),(2,2��,2��,2��)} be a subset of ��(ℤ4). Then �� is a neutrosophic quadruple ideal.

Theorem 2.15

Let �� and �� be neutrosophic quadruple ideals of �� and let {��}��=1 �� be a family of neutrosophic quadruple ideals of ��. Then: (i) ��+��=��. (ii) ��+��=�� for all �� ∈��. (iii) ⋂ �� =1 �� is a neutrosophic quadruple ideal of ��. (iv) ��+�� is a neutrosophic quadruple ideal of ��

Definition 2.16

Let �� be a neutrosophic quadruple ideal of ��. The set ��/��={�� + ��∶ �� ∈ ��} is called a neutrosophic quadruple quotient ring. If ��+�� and ��+�� are two arbitrary elements of ��/�� and if ⊕ and ⊙ are two binary operations on ��/�� deﬁned by: (�� + ��)⊕(�� +4��) = (�� + ��) + ��, (�� + ��)⊙(�� + ��) = (��) + ��,

it can be shown that ⊕ and ⊙ are well deﬁned and that (NQR/NQJ, ⊕, ⊙) is a neutrosophic quadruple ring.

Example 7.

Consider the neutrosophic quadruple ring ��(ℤ) and its neutrosophic quadruple ideal ��(2ℤ). Then

��(ℤ) ��(2ℤ)= {��(2ℤ),(1,0,0,0) + ��(2ℤ),(0,��,0,0) +��(2ℤ),(0,0,��,0)+ ��(2ℤ),(0,0,0,��) +��(2ℤ),(0,��,��,��)+ ��(2ℤ),(0,0,��,��) +��(2ℤ),(0,��,��,0)+ ��(2ℤ),(0,��,0,��) +��(2ℤ),(1,��,0,0)+ ��(2ℤ),(1,0,��,0) +��(2ℤ),(1,0,0,��)+ ��(2ℤ),(1,��,0,��) +��(2ℤ),(1,0,��,��)+ ��(2ℤ),(1,��,��,0) + ��(2ℤ),(1,��,��,��) + ��(2ℤ)}.

which is clearly a neutrosophic quadruple ring.

Definition 2.17

Let �� and �� be two neutrosophic quadruple rings and let ��∶ �� → �� be a mapping deﬁned for all ��,�� ∈ �� as follows:

(i) ��(�� + ��) = ��(��) + ��(��).

(ii) ��(��) = ��(��)��(��).

(iii) ��(��) = ��,��(��) = �� and ��(��) = ��. (iv) ��(1,0,0,0) = (1,0,0,0).

Then �� is called a neutrosophic quadruple homomorphism. Neutrosophic quadruple monomorphism, endomorphism, isomorphism, and other morphisms can be deﬁned in the usual way.

Definition 2.18

Let ��∶ �� → �� be a neutrosophic quadruple ring homomorphism.

(i)The image of �� denoted by �� is deﬁned by the set �� = {�� ∈ �� ∶ �� = ��(��) , for some �� ∈ ��}.

(ii)The kernel of �� denoted by �� is deﬁned by the set �� = {�� ∈ �� ∶ ��(��) = (0,0,0,0)}.

Theorem 2.19

Let ��∶ �� → �� be a neutrosophic quadruple ring homomorphism. Then:

(i) �� is a neutrosophic quadruple subring of �� (ii) �� is not a neutrosophic quadruple ideal of ��. Proof.

(i)Clear.

(ii)Since ��,��,�� cannot have image (0,0,0,0) under ��, it follows that the elements (0,��,0,0),(0,0,��,0),(0,0,0,��) cannot be in the ��. Hence, �� cannot be a neutrosophic quadruple ideal of ��

Florentin Smarandache (author and editor) Collected Papers, IX 130

Example 8.

Consider the projection map

��∶ ��(ℤ2)×��(ℤ2) → ��(ℤ2) deﬁned by ��(��,��) = �� for all ��,�� ∈ ��(ℤ2)

It is clear that �� is a neutrosophic quadruple homomorphism and its kernel is given as �� = {{((0,0,0,0),(0,0,0,0)),((0,0,0,0),(1,0,0,0)), ((0,0,0,0),(0,��,0,0)),((0,0,0,0),(0,0,��,0)), ((0,0,0,0),(0,0,0,��)),((0,0,0,0),(0,��,��,��)), ((0,0,0,0),(0,0,��,��)),((0,0,0,0),(0,��,��,0)), ((0,0,0,0),(0,��,0,��)),((0,0,0,0),(1,��,0,0)), ((0,0,0,0),(1,0,��,0)),((0,0,0,0),(1,0,0,��)), ((0,0,0,0),(1,��,0,��)),((0,0,0,0),(1,0,��,��)), ((0,0,0,0),(1,��,��,0)),((0,0,0,0),(1,��,��,��))}.

Theorem 2.20

Let φ: NQR(Z) → NQR(Z)/NQR(nZ) be a mapping defined by φ(x) = x + NQR(nZ) for all x ∈ NQR(Z) and n = 1, 2, 3, … . Then φ is not a neutrosophic quadruple ring homomorphism.

References

[1]A.A.A. Agboola, On Refined Neutrosophic Algebraic StructuresI, Neutrosophic Sets and Systems 10 (2015), 99-101. [2]F. Smarandache, Neutrosophy/Neutrosophic Probability, Set, and Logic, American Research Press, Rehoboth, USA, 1998, http://fs.gallup.unm.edu/eBook-otherformats.htm [3]F. Smarandache, Neutrosophic Quadruple Numbers, Refined Neutrosophic Quadruple Numbers, Absorbance Law, and the Multiplication of Neutrosophic Quadruple Numbers, Neutrosophic Sets and Systems 10 (2015), 96-98. [4]F. Smarandache, (t,i,f) - Neutrosophic Structures and INeutrosophic Structures, Neutrosophic Sets and Systems 8 (2015), 3-10. [5]F. Smarandache, n-Valued Refined Neutrosophic Logic and Its Applications in Physics, Progress in Physics 4 (2013), 143-146.

Florentin Smarandache (author and editor) Collected Papers, IX 131

Neutrosophic quadruple algebraic hyperstructures

A.A.A. Agboola, B. Davvaz, Florentin Smarandache

A.A.A. Agboola, B. Davvaz, Florentin Smarandache (2017). Neutrosophic quadruple algebraic hyperstructures. Annals of Fuzzy Mathematics and Informatics 14(1), 29-42

Abstract. Theobjectiveofthispaperistodevelopneutrosophicquadruplealgebraichyperstructures.Speciﬁcally,wedevelopneutrosophicquadruplesemihypergroups,neutrosophicquadruplecanonical hypergroupsandneutrosophicquadruplehyperringsandwepresentelementarypropertieswhichcharacterizethem.

Keywords: Neutrosophy, Neutrosophic quadruple number, Neutrosophic quadruple semihypergroup, Neutrosophic quadruple canonical hypergroup, Neutrosophic quadruple hyperrring.

1. Introduction

TheconceptofneutrosophicquadruplenumberswasintroducedbyFlorentin Smarandache[18].Itwasshownin[18]howarithmeticoperationsofaddition,subtraction,multiplicationandscalarmultiplicationcouldbeperformedonthesetof neutrosophicquadruplenumbers.In[1],Akinleyeet.al.introducedthenotion ofneutrosophicquadruplealgebraicstructures.Neutrosophicquadrupleringswere studiedandtheirbasicpropertieswerepresented.Inthepresentpaper,twohyperoperations ˆ +and ˆ × aredeﬁnedontheneutrosophicset NQ ofquadruplenumberstodevelopnewalgebraichyperstructureswhichwecallneutrosophicquadruplealgebraichyperstructures.Speciﬁcally,itisshownthat(NQ, ˆ ×)isaneutrosophicquadruplesemihypergroup,(NQ, ˆ +)isaneutrosophicquadruplecanonical hypergroupand(NQ, ˆ +, ˆ ×)isaneutrosophicquadruplehyperrringandtheirbasic propertiesarepresented.

Deﬁnition1.1 ([18]). Aneutrosophicquadruplenumberisanumberofthe form(a,bT,cI,dF )where T,I,F havetheirusualneutrosophiclogicmeaningsand a,b,c,d ∈ R or C.Theset NQ deﬁnedby NQ = {(a,bT,cI,dF ): a,b,c,d ∈ R or C} (1.1)

Smarandache (author and editor) Collected Papers, IX 132

Florentin

iscalledaneutrosophicsetofquadruplenumbers.Foraneutrosophicquadruple number(a,bT,cI,dF )representinganyentitywhichmaybeanumber,anidea,an object,etc, a iscalledtheknownpartand(bT,cI,dF )iscalledtheunknownpart.

Deﬁnition1.2. Let a =(a1,a2T,a3I,a4F ),b =(b1,b2T,b3I,b4F ) ∈ NQ.We deﬁnethefollowing:

a + b =(a1 + b1, (a2 + b2)T, (a3 + b3)I, (a4 + b4)F ), (1.2)

a b =(a1 b1, (a2 b2)T, (a3 b3)I, (a4 b4)F ) (1.3)

Deﬁnition1.3. Let a =(a1,a2T,a3I,a4F ) ∈ NQ andlet α beanyscalarwhich mayberealorcomplex,thescalarproduct α.a isdeﬁnedby

α.a = α.(a1,a2T,a3I,a4F )=(αa1,αa2T,αa3I,αa4F ) (1.4)

If α =0,thenwehave0.a =(0, 0, 0, 0)andforanynon-zeroscalars m and n and b =(b1,b2T,b3I,b4F ),wehave: (m + n)a = ma + na, m(a + b)= ma + mb, mn(a)= m(na), a =( a1, a2T, a3I, a4F )

Deﬁnition1.4 ([18]). [AbsorbanceLaw]Let X beasetendowedwithatotalorder x<y,named” x prevailedby y”or”x lessstrongerthan y”or”x lesspreferred than y”. x ≤ y isconsideredas”x prevailedbyorequalto y”or”xlessstronger thanorequalto y”or”x lesspreferredthanorequalto y”.

Foranyelements x,y ∈ X,with x ≤ y,absorbancelawisdeﬁnedas

x.y = y.x =absorb(x,y)=max{x,y} = y (1.5) whichmeansthatthebiggerelementabsorbsthesmallerelement(thebigﬁsheats thesmallﬁsh).Itisclearfrom(1.5)that x.x = x 2 =absorb(x,x)=max{x,x} = x and (1.6) x1.x2 xn =max{x1,x2, ,xn}. (1.7)

Analogously,if x>y,wesaythat”x prevailsto y”or”x isstrongerthan y”or ”x ispreferredto y”.Also,if x ≥ y,wesaythat”x prevailsorisequalto y”or”x isstrongerthanorequalto y”or”x ispreferredorequalto y”.

Deﬁnition1.5. Considertheset {T,I,F }.Supposeinanoptimisticwayweconsidertheprevalenceorder T>I>F .Thenwehave:

TI = IT =max{T,I} = T, (1.8)

TF = FT =max{T,F } = T, (1.9)

IF = FI =max{I,F } = I, (1.10)

TT = T 2 = T, (1.11)

II = I 2 = I, (1.12)

FF = F 2 = F. (1.13)

Florentin Smarandache (author and editor) Collected Papers, IX 133

Analogously,supposeinapessimisticwayweconsidertheprevalenceorder T< I<F .Thenwehave:

TI = IT =max{T,I} = I, (1.14)

TF = FT =max{T,F } = F, (1.15)

IF = FI =max{I,F } = F, (1.16)

TT = T 2 = T, (1.17)

II = I 2 = I, (1.18)

FF = F 2 = F. (1.19)

Exceptotherwisestated,wewillconsideronlytheprevalenceorder T<I<F inthispaper.

Deﬁnition1.6. Let a =(a1,a2T,a3I,a4F ),b =(b1,b2T,b3I,b4F ) ∈ NQ.Then a.b =(a1,a2T,a3I,a4F ).(b1,b2T,b3I,b4F ) =(a1b1, (a1b2 + a2b1 + a2b2)T, (a1b3 + a2b3 + a3b1 + a3b2 + a3b3)I, (a1b4 + a2b4,a3b4 + a4b1 + a4b2 + a4b3 + a4b4)F ) (1.20)

Theorem1.7 ([1]). (NQ, +) is anabeliangroup.

Theorem1.8 ([1]). (NQ,.) is acommutativemonoid.

Theorem1.9 ([1]). (NQ,.) is notagroup.

Theorem1.10 ([1]). (NQ, +,.) isacommutativering.

Deﬁnition1.11. Let NQR beaneutrosophicquadrupleringandlet NQS bea nonemptysubsetof NQR.Then NQS iscalledaneutrosophicquadruplesubringof NQR,if(NQS, +,.)isitselfaneutrosophicquadruplering.Forexample, NQR(nZ) isaneutrosophicquadruplesubringof NQR(Z)for n =1, 2, 3, ···

Deﬁnition1.12. Let NQJ beanonemptysubsetofaneutrosophicquadruple ring NQR NQJ iscalledaneutrosophicquadrupleidealof NQR,ifforall x,y ∈ NQJ,r ∈ NQR,thefollowingconditionshold:

(i) x y ∈ NQJ , (ii) xr ∈ NQJ and rx ∈ NQJ .

Deﬁnition1.13 ([1]). Let NQR and NQS betwoneutrosophicquadruplerings andlet φ : NQR → NQS beamappingdeﬁnedforall x,y ∈ NQR asfollows:

(i) φ(x + y)= φ(x)+ φ(y), (ii) φ(xy)= φ(x)φ(y), (iii) φ(T )= T , φ(I)= I and φ(F )= F , (iv) φ(1, 0, 0, 0)=(1, 0, 0, 0).

Then φ is called a neutrosophic quadruple homomorphism. Neutrosophic quadruple monomorphism, endomorphism, isomorphism, and other morphisms can be deﬁned in the usual way.

Deﬁnition 1.14. Let φ : N QR → N QS be a neutrosophic quadruple ring homomorphism.

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Florentin

(i)Theimageof φ denotedby Imφ isdeﬁnedbytheset

Imφ = {y ∈ NQS : y = φ(x), forsome x ∈ NQR}

(ii)Thekernelof φ denotedby Kerφ isdeﬁnedbytheset Kerφ = {x ∈ NQR : φ(x)=(0, 0, 0, 0)}.

Theorem1.15 ([1]). Let φ : NQR → NQS beaneutrosophicquadruplering homomorphism.Then:

(1) Imφ isaneutrosophicquadruplesubringof NQS, (2) Kerφ isnotaneutrosophicquadrupleidealof NQR Theorem1.16 ([1]). Let φ : NQR(Z) → NQR(Z)/NQR(nZ) beamappingdeﬁned by φ(x)= x + NQR(nZ) forall x ∈ NQR(Z) and n =1, 2, 3,....Then φ isnota neutrosophicquadrupleringhomomorphism.

Deﬁnition1.17. Let H beanon-emptysetandlet+beahyperoperationon H Thecouple(H, +)iscalledacanonicalhypergroupifthefollowingconditionshold: (i) x + y = y + x,forall x,y ∈ H, (ii) x +(y + z)=(x + y)+ z,forall x,y,z ∈ H, (iii)thereexistsaneutralelement0 ∈ H suchthat x +0= {x} =0+ x,forall x ∈ H, (iv)forevery x ∈ H,thereexistsauniqueelement x ∈ H suchthat0 ∈ x +( x) ∩ ( x)+ x, (v) z ∈ x + y implies y ∈−x + z and x ∈ z y,forall x,y,z ∈ H.

Anonemptysubset A of H iscalledasubcanonicalhypergroup,if A isacanonical hypergroupunderthesamehyperadditionasthatof H thatis,forevery a,b ∈ A, a b ∈ A.Ifinaddition a + A a ⊆ A forall a ∈ H, A issaidtobenormal.

Deﬁnition1.18. Ahyperringisatripple(R, +,.)satisfyingthefollowingaxioms: (i)(R, +)isacanonicalhypergroup, (ii)(R,.)isasemihypergroupsuchthat x.0=0.x =0forall x ∈ R,thatis,0is abilaterallyabsorbingelement, (iii)forall x,y,z ∈ R, x.(y + z)= x.y + x.z and(x + y).z = x.z + y.z. Thatis,thehyperoperation isdistributiveoverthehyperoperation+.

Deﬁnition1.19. Let(R, +,.)beahyperringandlet A beanonemptysubsetof R. A issaidtobeasubhyperringof R if(A, +,.)isitselfahyperring. Deﬁnition1.20. Let A beasubhyperringofahyperring R.Then

(i) A iscalledalefthyperidealof R if r.a ⊆ A forall r ∈ R,a ∈ A, (ii) A iscalledarighthyperidealof R if a.r ⊆ A forall r ∈ R,a ∈ A, (iii) A iscalledahyperidealof R if A isbothleftandrighthyperidealof R

Deﬁnition 1.21. Let A be a hyperideal of a hyperring R A is said to be normal in R, if r + A r ⊆ A, for all r ∈ R

For full details about hypergroups, canonical hypergroups, hyperrings, neutrosophic canonical hypergroups and neutrosophic hyperrings, the reader should see [3, 14]

Florentin Smarandache (author and editor) Collected Papers, IX 135

2. Developmentofneutrosophicquadruplecanonicalhypergroups andneutrosophicquadruplehyperrings

Inthissection,wedeveloptwoneutrosophichyperquadruplealgebraichyperstructuresnamelyneutrosophicquadruplecanonicalhypergroupandneutrosophic quadruplehyperring.Inwhatfollows,allneutrosophicquadruplenumberswillbe realneutrosophicquadruplenumbersi.e a,b,c,d ∈ R foranyneutrosophicquadruplenumber(a,bT,cI,dF ) ∈ NQ.

Deﬁnition2.1. Let+and behyperoperationson R thatis x + y ⊆ R,x.y ⊆ R for all x,y ∈ R.Let ˆ +and ˆ × behyperoperationson NQ.For x =(x1,x2T,x3I,x4F ),y = (y1,y2T,y3I,y4F ) ∈ NQ with xi,yi ∈ R,i =1, 2, 3, 4,deﬁne: x ˆ +y = {(a,bT,cI,dF ): a ∈ x1 + y1,b ∈ x2 + y2, c ∈ x3 + y3,d ∈ x4 + y4}, (2.1)

x ˆ ×y = {(a,bT,cI,dF ): a ∈ x1.y1,b ∈ (x1.y2) ∪ (x2.y1) ∪ (x2.y2),c ∈ (x1.y3) ∪(x2.y3) ∪ (x3.y1) ∪ (x3.y2) ∪ (x3.y3),d ∈ (x1.y4) ∪ (x2.y4) ∪(x3.y4) ∪ (x4.y1) ∪ (x4.y2) ∪ (x4.y3) ∪ (x4.y4)} (2.2)

Theorem2.2. (NQ, ˆ +) isacanonicalhypergroup.

Proof. Let x =(x1,x2T,x3I,x4F ),y =(y1,y2T,y3I,y4F ),z =(z1,z2T,z3I,z4F ) ∈ NQ bearbitrarywith xi,yi,zi ∈ R,i =1, 2, 3, 4. (i)Toshowthat x ˆ +y = y ˆ +x,let x ˆ +y = {a =(a1,a2T,a3I,a4F ): a1 ∈ x1 + y1,a2 ∈ x2 + y2,a3 ∈ x3 + y3, a4 ∈ x4 + y4}, y ˆ +x = {b =(b1,b2T,b3I,b4F ): b1 ∈ y1 + x1,b2 ∈ y2 + x2,b3 ∈ y3 + b3, b4 ∈ y4 + x4} Since ai,bi ∈ R,i =1, 2, 3, 4,itfollowsthat x ˆ +y = y ˆ +x (ii)Toshowthatthat x ˆ +(y ˆ +z)=(x ˆ +y) ˆ +z,let y ˆ +z = {w =(w1,w2T,w3I,w4F ): w1 ∈ y1 + z1,w2 ∈ y2 + z2, w3 ∈ y3 + z3,w4 ∈ y4 + z4}.Now, x ˆ +(y ˆ +z)= x ˆ +w

= {p =(p1,p2T,p3I,p4F ): p1 ∈ x1 + w1,p2 ∈ x2 + w2,p3 ∈ x3 + w3, p4 ∈ x4 + w4}

= {p =(p1,p2T,p3I,p4F ): p1 ∈ x1 +(y1 + z1),p2 ∈ x2 +(y2 + z2), p3 ∈ x3 +(y3 + z3),p4 ∈ x4 +(y4 + z4)}

Also,let x ˆ +y = {u =(u1,u2T,u3I,u4F ): u1 ∈ x1 + y1,u2 ∈ x2 + y2,u3 ∈ x3 + y3,u4 ∈ x4 + y4} sothat (x ˆ +y) ˆ +z = u ˆ +z

= {q =(q1,q2T,q3I,q4F ): q1 ∈ u1 + z1,q2 ∈ u2 + z2,q3 ∈ u3 + z3, q4 ∈ u4 + z4}

= {q =(q1,q2T,q3I,q4F ): q1 ∈ (x1 + y1)+ z1,q2 ∈ (x2 + y2)+ z2, q3 ∈ (x3 + y3)+ z3,q4 ∈ (x4 + y4)+ z4}

Florentin Smarandache (author and editor) Collected Papers, IX 136

Since ui,pi,qi,wi,xi,yi,zi ∈ R,i =1, 2, 3, 4,itfollowsthat x ˆ +(y ˆ +z)=(x ˆ +y) ˆ +z. (iii)Toshowthat0=(0, 0, 0, 0) ∈ NQ isaneutralelement,consider x ˆ +(0, 0, 0, 0)= {a =(a1,a2T,a3I,a4F ): a1 ∈ x1 +0,a2 ∈ x2 +0,a3 ∈ x3 +0, a4 ∈ x4 +0} = {a =(a1,a2T,a3I,a4F ): a1 ∈{x1},a2 ∈{x2},a3 ∈{x3}, a4 ∈{x4}} = {x}

Similarly,itcanbeshownthat(0, 0, 0, 0) ˆ +x = {x}.Hence0=(0, 0, 0, 0) ∈ NQ isa neutralelement. (iv)Toshowthatthatforevery x ∈ NQ,thereexistsauniqueelement x ∈ NQ suchthat0 ∈ x ˆ +( ˆ x) ∩ ( ˆ x) ˆ +x,consider x+( x) ∩ ( x)+x = {a =(a1,a2T,a3I,a4F ): a1 ∈ x1 x1,a2 ∈ x2 x2, a3 ∈ x3 x3,a4 ∈ x4 x4}∩{b =(b1,b2T,b3I,b4F ): b1 ∈−x1 + x1,b2 ∈−x2 + x2,b3 ∈−x3 + x3,b4 ∈−x4 + x4} = {(0, 0, 0, 0)}.

Thisshowsthatforevery x ∈ NQ,thereexistsauniqueelement ˆ x ∈ NQ such that0 ∈ x ˆ +( ˆ x) ∩ ( ˆ x) ˆ +x. (v)Sinceforall x,y,z ∈ NQ with xi,y1,zi ∈ R,i =1, 2, 3, 4,itfollowsthat z ∈ x ˆ +y implies y ∈ ˆ x ˆ +z and x ∈ z ˆ +( ˆ y).Hence,(NQ, ˆ +)isacanonical hypergroup.

Lemma2.3. Let (NQ, ˆ +) beaneutrosophicquadruplecanonicalhypergroup.Then (1) ˆ ( ˆ x)= x forall x ∈ NQ, (2)0=(0, 0, 0, 0) istheuniqueelementsuchthatforevery x ∈ NQ,thereisan element ˆ x ∈ NQ suchthat 0 ∈ x ˆ +( ˆ x), (3) ˆ 0=0, (4) ˆ (x ˆ +y)= ˆ x ˆ y forall x,y ∈ NQ

Example2.4. Let NQ = {0,x,y} beaneutrosophicquadruplesetandlet ˆ +bea hyperoperationon NQ deﬁnedinthetablebelow. ˆ + 0 x y 0 0 x y x x {0,x,y} y y y y {0,y}

Then (N Q, ˆ +) is a neutrosophic quadruple canonical hypergroup. Theorem 2.5. (N Q, ˆ ×) is a semihypergroup.

Proof. Let x = (x1, x2T, x3I, x4F ), y = (y1, y2T, y3I, y4F ), z = (z1, z2T, z3I, z4F ) ∈ N Q be arbitrary with xi, yi, zi ∈ R, i = 1, 2, 3, 4.

Florentin Smarandache (author and editor) Collected Papers, IX 137

(i) x ˆ ×y = {a =(a1,a2T,a3I,a4F ): a1 ∈ x1y1,a2 ∈ x1y2 ∪ x2y1 ∪ x2y2,a3 ∈ x1y3 ∪x2y3 ∪ x3y1 ∪ x3y2 ∪ x3y3,a4 ∈ x1y4 ∪ x2y4 ∪x3y4 ∪ x4y1 ∪ x4y2 ∪ x4y3 ∪ x4y4} ⊆ NQ.

(ii)Toshowthat x×(y×z)=(x×y)×z,let y×z = {w =(w1,w2T,w3I,w4F ): w1 ∈ y1z1,w2 ∈ y1z2 ∪ y2z1 ∪ y2z2, w3 ∈ y1z3 ∪ y2z3 ∪ y3z1 ∪ y3z2 ∪ y3z3,w4 ∈ y1z4) ∪ y2z4 ∪y3z4 ∪ y4z1 ∪ y4z2 ∪ y4z3 ∪ y4z4} (2.3) sothat x ˆ ×(y ˆ ×z)= x ˆ ×w = {p =(p1,p2T,p3I,p4F ): p1 ∈ x1w1,p2 ∈ x1w2 ∪ x2w1 ∪ x2w2, p3 ∈ x1w3 ∪ x2w3 ∪ x3w1 ∪ x3w2 ∪ x3y3,p4 ∈ x1w4 ∪ x2w4 ∪x3w4 ∪ x4w1 ∪ x4w2 ∪ x4w3 ∪ x4w4}. (2.4)

Also,let x ˆ ×y = {u =(u1,u2T,u3I,u4F ): u1 ∈ x1y1,u2 ∈ x1y2 ∪ x2y1 ∪ x2y2,u3 ∈ x1y3 ∪x2y3 ∪ x3y1 ∪ x3y2 ∪ x3y3,u4 ∈ x1y4 ∪ x2y4 ∪x3y4 ∪ x4y1 ∪ x4y2 ∪ x4y3 ∪ x4y4} (2.5) sothat (x×y)×z = u×z = {q =(q1,q2T,q3I,q4F ): q1 ∈ u1z1,q2 ∈ u1z2 ∪ u2z1 ∪ u2z2, q3 ∈ u1z3 ∪ u2z3 ∪ u3z1 ∪ u3z2 ∪ u3z3,q4 ∈ u1z4 ∪ u2z4 ∪u3z4 ∪ u4z1 ∪ u4z2 ∪ u4z3 ∪ u4z4}.

(2.6)

Substituting wi of(2.3)in(2.4)andalsosubstituting ui of(2.5)in(2.6),where i =1, 2, 3, 4andsince pi,qi,ui,wi,xi,zi ∈ R,itfollowsthat x ˆ ×(y ˆ ×z)=(x ˆ ×y) ˆ ×z Consequently,(NQ, ˆ ×)isasemihypergroupwhichwecallneutrosophicquadruple semihypergroup. Remark2.6. (NQ, ˆ ×)isnotahypergroup. ˆ

Deﬁnition 2.7. Let (N Q, ˆ +) be a neutrosophic quadruple canonical hypergroup. For any subset N H of N Q, we deﬁne N H = { ˆ x : x ∈ N H}

A nonempty subset N H of N Q is called a neutrosophic quadruple subcanonical hypergroup, if the following conditions hold:

(i) 0 = (0, 0, 0, 0) ∈ N H, (ii) x y ⊆ N H for all x, y ∈ N H A neutrosophic quadruple subcanonical hypergroup N H of a netrosophic quadruple canonical hypergroup N Q is said to be normal, if x ˆ +N H ˆ x ⊆ N H for all x ∈ N Q

Florentin Smarandache (author and editor) Collected Papers, IX 138

Deﬁnition2.8. Let(NQ, ˆ +)beaneutrosophicquadruplecanonicalhypergroup. For xi ∈ NQ with i =1, 2, 3 ...,n ∈ N,theheartof NQ denotedby NQω isdeﬁned by NQω = n i=1 xi ˆ xi

InExample 2.4, NQω = NQ

Deﬁnition2.9. Let(NQ1, +)and(NQ2, + )betwoneutrosophicquadruplecanonicalhypergroups.Amapping φ : NQ1 → NQ2 iscalledaneutrosophicquadruple stronghomomorphism,ifthefollowingconditionshold:

(i) φ(x ˆ +y)= φ(x) ˆ + φ(y)forall x,y ∈ NQ1, (ii) φ(T )= T , (iii) φ(I)= I, (iv) φ(F )= F , (v) φ(0)=0.

Ifinaddition φ isabijection,then φ iscalledaneutrosophicquadruplestrong isomorphismandwewrite NQ1 ∼ = NQ2

Deﬁnition2.10. Let φ : NQ1 → NQ2 beaneutrosophicquadruplestronghomomorphismofneutrosophicquadruplecanonicalhypergroups.Thentheset {x ∈ NQ1 : φ(x)=0} iscalledthekernelof φ anditisdenotedby Kerφ.Also,theset {φ(x): x ∈ NQ1} iscalledtheimageof φ anditisdenotedby Imφ

Theorem2.11. (NQ, ˆ +, ˆ ×) isahyperring.

Proof. That(NQ, ˆ +)isacanonicalhypergroupfollowsfromTheorem 2.2.Also, that(NQ, ˆ ×)isasemihypergroupfollowsfromTheorem 2.4 Next,let x =(x1,x2T,x3I,x4F ) ∈ NQ bearbitrarywith xi,yi,zi ∈ R,i = 1, 2, 3, 4.Then x×0= {u =(u1,u2T,u3I,u4F ): u1 ∈ x1 0,u2 ∈ x1 0 ∪ x2 0 ∪ x2 0,u3 ∈ x1 0 ∪x2 0 ∪ x3 0 ∪ x3 0 ∪ x3 0,u4 ∈ x1 0 ∪ x2 0 ∪ x3 0 ∪ x4 0 ∪ x4 0 ∪x4 0 ∪ x4 0} = {u =(u1,u2T,u3I,u4F ): u1 ∈{0},u2 ∈{0},u3 ∈{0},u4 ∈{0}} = {0}.

Similarly, it can be shown that 0 ˆ ×x = {0}. Since x is arbitrary, it follows that x ˆ ×0 = 0 ˆ ×x = {0}, for all x ∈ N Q. Hence, 0 = (0, 0, 0, 0) is a bilaterally absorbing element.

To complete the proof, we have to show that x ˆ ×(y ˆ +z) = (x ˆ ×y) ˆ +(x ˆ ×z), for all x, y, z ∈ N Q To this end, let x = (x1, x2T, x3I, x4F ), y = (y1, y2T, y3I, y4F ), z = (z1, z2T, z3I, z4F ) ∈ N Q be arbitrary with xi, yi, zi ∈ R, i = 1, 2, 3, 4. Let y+z = {w = (w1, w2T, w3I, w4F ) : w1 ∈ y1 + z1, w2 ∈ y2 + z2, w3 ∈ y3 + z3, (2.7) w4 ∈ y4 + z4}

Florentin Smarandache (author and editor) Collected Papers, IX 139

sothat x ˆ ×(y ˆ +z)= x ˆ ×w = {p =(p1,p2T,p3I,p4F ): p1 ∈ x1w1,p2 ∈ x1w2 ∪ x2w1 ∪ x2w2, p3 ∈ x1w3 ∪ x2w3 ∪ x3w1 ∪ x3w2 ∪ x3y3,p4 ∈ x1w4 ∪ x2w4 ∪x3w4 ∪ x4w1 ∪ x4w2 ∪ x4w3 ∪ x4w4} (2.8) Substituting wi,i =1, 2, 3, 4of(2.7)in(2.8),weobtainthefollowing: p1 ∈ x1(y1 + z1), (2.9) p2 ∈ x1(y2 + z2) ∪ x2(y1 + z1) ∪ x2(y2 + z2), (2.10) p3 ∈ x1(y3 + z3) ∪ x2(y3 + z3) ∪ x3(y1 + z1) ∪ x3(y2 + z2) ∪ x3(y3 + z3), (2.11) p4 ∈ x1(y4 + z4) ∪ x2(y4 + z4) ∪ x3(y4 + z4) ∪ x4(y1 + z1) ∪ x4(y2 + z2), ∪x4(y3 + z3) ∪ x4(y4 + z4) (2.12)

Also,let

x ˆ ×y = {u =(u1,u2T,u3I,u4F ): u1 ∈ x1y1,u2 ∈ x1y2 ∪ x2y1 ∪ x2y2, u3 ∈ x1y3 ∪ x2y3 ∪ x3y1 ∪ x3y2 ∪ x3y3,u4 ∈ x1y4 ∪ x2y4 ∪x3y4 ∪ x4y1 ∪ x4y2 ∪ x4y3 ∪ x4y4} (2.13) x ˆ ×z = {v =(v1,v2T,v3I,v4F ): v1 ∈ x1z1,v2 ∈ x1z2 ∪ x2z1 ∪ x2z2, v3 ∈ x1z3 ∪ x2z3 ∪ x3z1 ∪ x3z2 ∪ x3z3,v4 ∈ x1z4 ∪ x2z4 ∪x3z4 ∪ x4z1 ∪ x4z2 ∪ x4z3 ∪ x4z4} (2.14) sothat (x ˆ ×y) ˆ +(x ˆ ×z)= u ˆ +v = {q =(q1,q2T,q3I,q4F ): q1 ∈ u1 + v1,q2 ∈ u2 + v2, q3 ∈ u3 + v3,q4 ∈ u4 + v4} (2.15) Substituting ui of(2.13)and vi of(2.14)in(2.15),weobtainthefollowing: q1 ∈ u1 + v1 ⊆ x1y1 + x1z1 ⊆ x1(y1 + z1), (2.16) q2 ∈ u2 + v2 ⊆ (x1y2 ∪ x2y1 ∪ x2y2) +(x1z2 ∪ x2z1 ∪ x2(z2) ⊆ x1(y2 + z2) ∪ x2(y1 + z1) ∪ x2(y2 + z2), (2.17) q3 ∈ u3 + v3 ⊆ (x1y3 ∪ x2y3 ∪ x3y1) ∪ x3y2 ∪ x3y3) +(x1z3 ∪ x2z3 ∪ x3z1) ∪ x3z2 ∪ x3z3) ⊆ x1(y3 + z3) ∪ x2(y3 + z3) ∪ x3(y1 + z1) ∪ x3(y2 + z2) ∪ x3(y3 + z3). (2.18) q4 ∈ u4 + v4 ⊆ (x1y4 ∪ x2y4 ∪ x3y4) ∪ x4y1 ∪ x4y2) ∪ x4y3 ∪ x4y4) +(x1z4 ∪ x2z4 ∪ x3z4) ∪ x4z1 ∪ x4z2) ∪ x4z3 ∪ x4z4) ⊆ x1(y4 + z4) ∪ x2(y4 + z4) ∪ x3(y4 + z4) ∪ x4(y1 + z1) ∪ x4(y2 + z2) ∪x4(y3 + z3) ∪ x4(y4 + z4) (2.19)

Comparing (2.9), (2.10), (2.11) and (2.12) respectively with (2.16), (2.17), (2.18) and (2.19), we obtain pi = qi, i = 1, 2, 3, 4. Hence, x ˆ ×(y ˆ +z) = (x ˆ ×y) ˆ +(x ˆ ×z), for all

Florentin Smarandache (author and editor) Collected Papers, IX 140

x,y,z ∈ NQ.Thus,(NQ, ˆ +, ˆ ×)isahyperringwhichwecallneutrosophicquadruple hyperring.

Theorem2.12. (NQ, ˆ +, ◦) isaKrasnerhyperringwhere ◦ isanordinarymultiplicativebinaryoperationon NQ.

Deﬁnition2.13. Let(NQ, ˆ +, ˆ ×)beaneutrosophicquadruplehyperring.Anonempty subset NJ of NQ iscalledaneutrosophicquadruplesubhyperringof NQ,if(NJ, ˆ +, ˆ ×) isitselfaneutrosophicquadruplehyperring.

NJ iscalledaneutrosophicquadruplehyperidealifthefollowingconditionshold: (i)(NJ, ˆ +)isaneutrosophicquadruplesubcanonicalhypergroup. (ii)Forall x ∈ NJ and r ∈ NQ, x ˆ ×r,r ˆ ×x ⊆ NJ .

Aneutrosophicquadruplehyperideal NJ of NQ issaidtobenormalin NQ,if x ˆ +NJ ˆ x ⊆ NJ ,forall x ∈ NQ.

Deﬁnition2.14. Let(NQ1, ˆ +, ˆ ×)and(NQ2, ˆ + , ˆ × )betwoneutrosophicquadruplehyperrings.Amapping φ : NQ1 → NQ2 iscalledaneutrosophicquadruple stronghomomorphism,ifthefollowingconditionshold:

(i) φ(x ˆ +y)= φ(x) ˆ + φ(y),forall x,y ∈ NQ1, (ii) φ(x ˆ ×y)= φ(x) ˆ × φ(y),forall x,y ∈ NQ1, (iii) φ(T )= T , (iv) φ(I)= I, (v) φ(F )= F , (vi) φ(0)=0.

Ifinaddition φ isabijection,then φ iscalledaneutrosophicquadruplestrong isomorphismandwewrite NQ1 ∼ = NQ2.

Deﬁnition2.15. Let φ : NQ1 → NQ2 beaneutrosophicquadruplestronghomomorphismofneutrosophicquadruplehyperrings.Thentheset {x ∈ NQ1 : φ(x)=0} iscalledthekernelof φ anditisdenotedby Kerφ.Also,theset {φ(x): x ∈ NQ1} iscalledtheimageof φ anditisdenotedby Imφ.

Example2.16. Let(NQ, ˆ +, ˆ ×)beaneutrosophicquadruplehyperringandlet NX bethesetofallstrongendomorphismsof NQ.If ⊕ and arehyperoperations deﬁnedforall φ,ψ ∈ NX andforall x ∈ NQ as φ ⊕ = {ν(x): ν(x) ∈ φ(x) ˆ +ψ(x)}, φ = {ν(x): ν(x) ∈ φ(x) ˆ ×ψ(x)}, then(NX, ⊕, )isaneutrosophicquadruplehyperring.

3. Characterization of neutrosophic quadruple canonical hypergroups and neutrosophic hyperrings

In this section, we present elementary properties which characterize neutrosophic quadruple canonical hypergroups and neutrosophic quadruple hyperrings. Theorem 3.1. Let N G and N H be neutrosophic quadruple subcanonical hypergroups of a neutrosophic quadruple canonical hypergroup (N Q, ˆ +) Then (1) N G ∩ N H is a neutrosophic quadruple subcanonical hypergroup of N Q,

Florentin Smarandache (author and editor) Collected Papers, IX 141

(2) NG × NH isaneutrosophicquadruplesubcanonicalhypergroupof NQ.

Theorem3.2. Let NH beaneutrosophicquadruplesubcanonicalhypergroupofa neutrosophicquadruplecanonicalhypergroup (NQ, ˆ +).Then

(1) NH+NH = NH , (2) x ˆ +NH = NH ,forall x ∈ NH

Theorem3.3. Let (NQ, ˆ +) beaneutrosophicquadruplecanonicalhypergroup. NQω ,theheartof NQ isanormalneutrosophicquadruplesubcanonicalhypergroup of NQ

Theorem3.4. Let NG and NH beneutrosophicquadruplesubcanonicalhypergroupsofaneutrosophicquadruplecanonicalhypergroup (NQ, ˆ +)

(1) If NG ⊆ NH and NG isnormal,then NG isnormal. (2) If NG isnormal,then NG ˆ +NH isnormal.

Deﬁnition3.5. Let NG and NH beneutrosophicquadruplesubcanonicalhypergroupsofaneutrosophicquadruplecanonicalhypergroup(NQ, ˆ +).Theset NG ˆ +NH isdeﬁnedby NG ˆ +NH = {x ˆ +y : x ∈ NG,y ∈ NH} (3.1)

Itisobviousthat NG ˆ +NH isaneutrosophicquadruplesubcanonicalhypergroup of(NQ, ˆ +).

If x ∈ NH,theset x ˆ +NH isdeﬁnedby x ˆ +NH = {x ˆ +y : y ∈ NH}. (3.2)

If x and y areanytwoelementsof NH and τ isarelationon NH deﬁnedby xτy if x ∈ y ˆ +NH,itcanbeshownthat τ isanequivalencerelationon NH andthe equivalenceclassofanyelement x ∈ NH determinedby τ isdenotedby[x].

Lemma3.6. Forany x ∈ NH ,wehave (1)[x]= x ˆ +NH , (2)[ ˆ x]= ˆ [x]. Proof. (1) [x]= {y ∈ NH : xτy} = {y ∈ NH : y ∈ x ˆ +NH} = x ˆ +NH. (2)Obvious.

Deﬁnition3.7. Let NQ/NH bethecollectionofallequivalenceclassesof x ∈ NH determinedby τ .For[x], [y] ∈ NQ/NH,wedeﬁnetheset[x]⊕[y]as [x] ˆ ⊕[y]= {[z]: z ∈ x ˆ +y} (3.3)

Theorem3.8. (NQ/NH, ˆ ⊕) isaneutrosophicquadruplecanonicalhypergroup. Proof. Sameastheclassicalcaseandsoomitted.

Florentin Smarandache (author and editor) Collected Papers, IX 142

Theorem3.9. Let (NQ, ˆ +) beaneutrosophicquadruplecanonicalhypergroupand let NH beanormalneutrosophicquadruplesubcanonicalhypergroupof NQ.Then, forany x,y ∈ NH ,thefollowingareequivalent:

(1) x ∈ y ˆ +NH , (2) y ˆ x ⊆ NH , (3)(y ˆ x) ∩ NH = ∅ Proof. Sameastheclassicalcaseandsoomitted.

Theorem3.10. Let φ : NQ1 → NQ2 beaneutrosophicquadruplestronghomomorphismofneutrosophicquadruplecanonicalhypergroups.Then

(1) Kerφ isnotaneutrosophicquadruplesubcanonicalhypergroupof NQ1, (2) Imφ isaneutrosophicquadruplesubcanonicalhypergroupof NQ2 Proof. (1)Sinceitisnotpossibletohave φ((0,T, 0, 0))= φ((0, 0, 0, 0)), φ((0, 0,I, 0))= φ((0, 0, 0, 0))and φ((0, 0, 0,F ))= φ((0, 0, 0, 0)),itfollowsthat(0,T, 0, 0), (0, 0,I, 0) and(0, 0, 0,F )cannotbeinthekernelof φ.Consequently, Kerφ cannotbeaneutrosophicquadruplesubcanonicalhypergroupof NQ1 (2)Obvious.

Remark3.11. If φ : NQ1 → NQ2 isaneutrosophicquadruplestronghomomorphismofneutrosophicquadruplecanonicalhypergroups,then Kerφ isasubcanonicalhypergroupof NQ1

Theorem3.12. Let φ : NQ1 → NQ2 beaneutrosophicquadruplestronghomomorphismofneutrosophicquadruplecanonicalhypergroups.Then

(1) NQ1/Kerφ isnotaneutrosophicquadruplecanonicalhypergroup, (2) NQ1/Kerφ isacanonicalhypergroup.

Theorem3.13. Let NH beaneutrosophicquadruplesubcanonicalhypergroupof theneutrosophicquadruplecanonicalhypergroup (NQ, ˆ +).Thenthemapping φ : NQ → NQ/NH deﬁnedby φ(x)= x ˆ +NH isnotaneutrosophicquadruplestrong homomorphism.

Remark3.14. Isomorphismtheoremsdonotholdintheclassofneutrosophic quadruplecanonicalhypergroups.

Lemma3.15. Let NJ beaneutrosophicquadruplehyperidealofaneutrosophic quadruplehyperring (NQ, ˆ +, ˆ ×).Then (1) NJ = NJ , (2) x ˆ +NJ = NJ ,forall x ∈ NJ , (3) x ˆ ×NJ = NJ ,forall x ∈ NJ

Theorem3.16. Let NJ and NK beneutrosophicquadruplehyperidealsofaneutrosophicquadruplehyperring (NQ, +, ×).Then (1) NJ ∩ NK isaneutrosophicquadruplehyperidealof NQ, (2) NJ × NK isaneutrosophicquadruplehyperidealof NQ, (3) NJ ˆ +NK isaneutrosophicquadruplehyperidealof NQ

Theorem 3.17. Let N J be a normal neutrosophic quadruple hyperideal of a neutrosophic quadruple hyperring (N Q, ˆ +, ˆ ×) Then

Florentin Smarandache (author and editor) Collected Papers, IX 143

(1)(x ˆ +NJ ) ˆ +(y ˆ +NJ )=(x ˆ +y) ˆ +NJ ,forall x,y ∈ NJ , (2)(x ˆ +NJ ) ˆ ×(y ˆ +NJ )=(x ˆ ×y) ˆ +NJ ,forall x,y ∈ NJ , (3) x ˆ +NJ = y ˆ +NJ ,forall y ∈ x ˆ +NJ .

Theorem3.18. Let NJ and NK beneutrosophicquadruplehyperidealsofaneutrosophicquadruplehyperring (NQ, ˆ +, ˆ ×) suchthat NJ isnormalin NQ.Then (1) NJ ∩ NK isnormalin NJ , (2) NJ ˆ +NK isnormalin NQ, (3) NJ isnormalin NJ ˆ +NK

Let NJ beaneutrosophicquadruplehyperidealofaneutrosophicquadruple hyperring(NQ, ˆ +, ˆ ×).Forall x ∈ NQ,theset NQ/NJ isdefinedas NQ/NJ = {x ˆ +NJ : x ∈ NQ} (3.4) For[x], [y] ∈ NQ/NJ ,wedefinethehyperoperations ˆ ⊕ and ˆ ⊗ on NQ/NJ asfollows: [x] ˆ ⊕[y]= {[z]: z ∈ x ˆ +y}, (3.5) [x] ˆ ⊗[y]= {[z]: z ∈ x ˆ ×y} (3.6) Itcaneasilybeshownthat(NQ/NH, ˆ ⊕, ˆ ⊗)isaneutrosophicquadruplehyperring. Theorem3.19. Let φ : NQ → NR beaneutrosophicquadruplestronghomomorphismofneutrosophicquadruplehyperringsandlet NJ beaneutrosophicquadruple hyperidealof NQ.Then (1) Kerφ isnotaneutrosophicquadruplehyperidealof NQ, (2) Imφ isaneutrosophicquadruplehyperidealof NR, (3) NQ/Kerφ isnotaneutrosophicquadruplehyperring, (4) NQ/Imφ isaneutrosophicquadruplehyperring, (5) Themapping ψ : NQ → NQ/NJ definedby ψ(x)= x ˆ +NJ ,forall x ∈ NQ isnotaneutrosophicquadruplestronghomomorphism. Remark3.20. Theclassicalisomorphismtheoremsofhyperringsdonotholdin neutrosophicquadruplehyperrings.

4. Conclusion

Wehavedevelopedneutrosophicquadruplealgebraichyperstruturesinthispaper.Inparticular,wehavedevelopednewneutrosophicalgebraichyperstructures namelyneutrosophicquadruplesemihypergroups,neutrosophicquadruplecanonical hypergroupsandneutrosophicquadruplehyperrings.Wehavepresentedelementary propertieswhichcharacterizethenewneutrosophicalgebraichyperstructures.

Acknowledgements. Theauthorsthankalltheanonymousreviewersforuseful observationsandcriticalcommentswhichhaveimprovedthequalityofthepaper.

References

[1]S.A.Akinleye,F.SmarandacheandA.A.A.Agboola,OnNeutrosophicQuadrupleAlgebraic Structures,NeutrosophicSetsandSystems12(2016)122–126.

[2]S.A.Akinleye,E.O.AdelekeandA.A.A.Agboola,IntroductiontoNeutrosophicNearrings, Ann.FuzzyMath.Inform.12(3)(2016)7–19.

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[3]A.A.A.Agboola,OnReﬁnedNeutrosophicAlgebraicStructuresI,NeutrosophicSetsand Systems10(2015)99–101.

[4]A.A.A.AgboolaandB.Davvaz,OnNeutrosophicCanonicalHypergroupsandNeutrosophic Hyperrings,NeutrosophicSetsandSystems2(2014)34–41.

[5]A.AsokkumarandM.Velrajan,Characterizationofregularhyperrings,ItalianJournalof PureandAppliedMathematics22(2007)115–124.

[6]A.R.Bargi,Aclassofhyperrings,J.Disc.Math.Sc.andCryp.(6)(2003)227–233.

[7]P.Corsini,ProlegomenaofHypergroupTheory,Secondedition,Aviaineditore1993.

[8]P.CorsiniandV.Leoreanu,ApplicationsofHyperstructureTheory,AdvancesinMathematics,KluwerAcademicPublishers,Dordrecht2003.

[9]B.Davvaz,PolygroupTheoryandRelatedSystems,WorldSci.Publ.2013.

[10]B.Davvaz,Isomorphismtheoremsofhyperrings,IndianJ.PureAppl.Math.35(3)(2004) 321–333.

[11]B.Davvaz,Approximationsinhyperrings,J.Mult.-ValuedLogicSoftComput.15(5-6) (2009)471–488.

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[14]B.DavvazandV.Leoreanu-Fotea,HyperringTheoryandApplications,InternationalAcademicPress,USA2007.

[15]B.Davvaz,Isomorphismtheoremsofhyperrings,IndianJournalofPureandAppliedMathematics23(3)(2004)321–331.

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[17]F.Smarandache,Neutrosophy/NeutrosophicProbability,Set,andLogic,AmericanResearch Press,Rehoboth,USA1998. http://fs.gallup.unm.edu/eBook-otherformats.htm

[18]F.Smarandache,NeutrosophicQuadrupleNumbers,ReﬁnedNeutrosophicQuadrupleNumbers,AbsorbanceLaw,andtheMultiplicationofNeutrosophicQuadrupleNumbers,NeutrosophicSetsandSystems10(2015)96–98.

[19]F.Smarandache,(t,i,f)-NeutrosophicStructuresandI-NeutrosophicStructures,NeutrosophicSetsandSystems8(2015)3–10.

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Neutrosophic N-Structures Applied to BCK/BCI-Algebras

Young Bae Jun, Florentin Smarandache, Hashem Bordbar

Young Bae Jun, Florentin Smarandache, Hashem Bordbar (2017). Neutrosophic N-Structures Applied to BCK/BCI-Algebras. Information 8, 128; DOI: 10.3390/info8040128

Abstract:

Neutrosophic N -structures with applications in BCK/BCI-algebras is discussed.

The notions of a neutrosophic N -subalgebra and a (closed) neutrosophic N -ideal in a BCK/BCI-algebra are introduced, and several related properties are investigated. Characterizations of a neutrosophic N -subalgebra and a neutrosophic N -ideal are considered, and relations between a neutrosophic N -subalgebra and a neutrosophic N -ideal are stated. Conditions for a neutrosophic N -ideal to be a closed neutrosophic N -ideal are provided.

Keywords: neutrosophic N -structure; neutrosophic N -subalgebra; (closed) neutrosophic N -ideal

1. Introduction

BCK-algebras entered into mathematics in 1966 through the work of Imai and Iséki [1], and they have been applied to many branches of mathematics, such as group theory, functional analysis, probability theory and topology. Such algebras generalize Boolean rings as well as Boolean D-posets (MV-algebras). Additionally, Iséki introduced the notion of a BCI-algebra, which is a generalization of a BCK-algebra (see [2]).

A (crisp) set A in a universe X can be defined in the form of its characteristic function µ A : X → {0, 1} yielding the value 1 for elements belonging to the set A and the value 0 for elements excluded from the set A So far, most of the generalizations of the crisp set have been conducted on the unit interval [0, 1], and they are consistent with the asymmetry observation. In other words, the generalization of the crisp set to fuzzy sets relied on spreading positive information that fit the crisp point {1} into the interval [0, 1] Because no negative meaning of information is suggested, we now feel a need to deal with negative information. To do so, we also feel a need to supply a mathematical tool. To attain such an object, Jun et al. [3] introduced a new function, called a negative-valued function, and constructed N -structures. Zadeh [4] introduced the degree of membership/truth (t) in 1965 and defined the fuzzy set. As a generalization of fuzzy sets, Atanassov [5] introduced the degree of nonmembership/falsehood (f) in 1986 and defined the intuitionistic fuzzy set. Smarandache introduced the degree of indeterminacy/neutrality (i) as an independent component in 1995 (published in 1998) and defined the neutrosophic set on three components:

(t, i, f) = (truth, indeterminacy, falsehood)

Florentin Smarandache (author and editor) Collected Papers, IX 146

Formoredetails,refertothefollowingsite:

http://fs.gallup.unm.edu/FlorentinSmarandache.htm

Inthispaper,wediscussaneutrosophic N -structurewithanapplicationto BCK/BCI-algebras. Weintroducethenotionsofaneutrosophic N -subalgebraanda(closed)neutrosophic N -idealina BCK/BCI-algebra,andinvestigaterelatedproperties.Weconsidercharacterizationsofaneutrosophic N -subalgebraandaneutrosophic N -ideal.Wediscussrelationsbetweenaneutrosophic N -subalgebra andaneutrosophic N -ideal.Weprovideconditionsforaneutrosophic N -idealtobeaclosed neutrosophic N -ideal.

2.Preliminaries

Welet K(τ) betheclassofallalgebraswithtype τ =(2,0).A BCI-algebra referstoasystem X :=(X, ∗, θ) ∈ K(τ) inwhichthefollowingaxiomshold: (I) ((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y)= θ, (II) (x ∗ (x ∗ y)) ∗ y = θ, (III) x ∗ x = θ, (IV) x ∗ y = y ∗ x = θ ⇒ x = y forall x, y, z ∈ X.IfaBCI-algebra X satisﬁes θ ∗ x = θ forall x ∈ X,thenwesaythat X isa BCK-algebra Wecandeﬁneapartialordering by (∀x, y ∈ X)(x y ⇒ x ∗ y = θ)

InaBCK/BCI-algebra X,thefollowinghold: (∀x ∈ X)(x ∗ θ = x) (1) (∀x, y, z ∈ X)((x ∗ y) ∗ z =(x ∗ z) ∗ y) (2)

Anon-emptysubset S ofa BCK/BCI-algebra X iscalleda subalgebra of X if x ∗ y ∈ S forall x, y ∈ S.

Asubset I ofa BCK/BCI-algebra X iscalledan ideal of X ifitsatisﬁesthefollowing: (I1)0 ∈ I, (I2) (∀x, y ∈ X)(x ∗ y ∈ I, y ∈ I ⇒ x ∈ I).

Wereferthereadertothebooks[6,7]forfurtherinformationregardingBCK/BCI-algebras. Foranyfamily {ai | i ∈ Λ} ofrealnumbers,wedeﬁne

{ai | i ∈ Λ} :=

max{ai | i ∈ Λ} if Λ isﬁnite sup{ai | i ∈ Λ} otherwise {ai | i ∈ Λ} := min{ai | i ∈ Λ} if Λ isﬁnite inf{ai | i ∈ Λ} otherwise

Wedenoteby F (X, [ 1,0]) thecollectionoffunctionsfromaset X to [ 1,0].Wesaythatan elementof F (X, [ 1,0]) isa negative-valuedfunction from X to [ 1,0] (briefly, N -function on X). An N -structure referstoanorderedpair (X, f ) of X andan N -function f on X (see[3]).Inwhat follows,welet X denotethenonemptyuniverseofdiscourseunlessotherwisespecified. A neutrosophic N -structure over X (see[8])isdefinedtobethestructure: XN := X (TN ,IN ,FN ) = x (TN (x),IN (x),FN (x)) | x ∈ X (3)

Florentin Smarandache (author and editor) Collected Papers, IX 147

where TN , IN and FN are N -functionson X,whicharecalledthe negativetruthmembershipfunction, the negativeindeterminacymembershipfunction andthe negativefalsitymembershipfunction,respectively, on X. Wenotethateveryneutrosophic N -structure XN over X satisﬁesthecondition: (∀x ∈ X) ( 3 ≤ TN (x)+ IN (x)+ FN (x) ≤ 0)

3.Applicationin BCK/BCI-Algebras

Inthissection,wetakea BCK/BCI-algebra X astheuniverseofdiscourseunless otherwisespeciﬁed. Deﬁnition1. Aneutrosophic N -structure XN over X iscalledaneutrosophic N -subalgebraof X ifthe followingconditionisvalid: (∀x, y ∈ X)    TN (x ∗ y) ≤ {TN (x), TN (y)} IN (x ∗ y) ≥ {IN (x), IN (y)} FN (x ∗ y) ≤ {FN (x), FN (y)}    (4) Example1. ConsideraBCK-algebraX = {θ, a, b, c} withthefollowingCayleytable. ∗ θ abc θθθθθ aa θθ a bba θ b cccc θ

Theneutrosophic N -structure XN = θ ( 0.7, 0.2, 0.6) , a ( 0.5, 0.3, 0.4) , b ( 0.5, 0.3, 0.4) , c ( 0.3, 0.8, 0.5) overXisaneutrosophic N -subalgebraofX.

Let XN beaneutrosophic N -structureover X andlet α, β, γ ∈ [ 1,0] besuchthat 3 ≤ α + β + γ ≤ 0.Considerthefollowingsets: Tα N := {x ∈ X | TN (x) ≤ α} I β N := {x ∈ X | IN (x) ≥ β} Fγ N := {x ∈ X | FN (x) ≤ γ}

Theset XN(α, β,

γ) := {x ∈ X | TN (x) ≤ α, IN (x) ≥ β, FN (x) ≤ γ} iscalledthe (α, β, γ)-levelset

XN.Notethat XN(α, β, γ)= Tα N ∩ I β N ∩ Fγ N

β, γ ∈

Florentin Smarandache (author and editor) Collected Papers, IX 148

Theorem1. Let

beaneutrosophic N -structureover

andlet

[ 1,0] besuchthat 3 ≤ α + β + γ ≤ 0.If XN isaneutrosophic N -subalgebraof X,thenthenonempty (α, β, γ)-levelsetof XN isa subalgebraofX.

Proof. Let α, β, γ ∈ [ 1,0] besuchthat 3 ≤ α + β + γ ≤ 0and XN(α, β, γ) = ∅.If x, y ∈ XN(α, β, γ), then TN (x) ≤ α, IN (x) ≥ β, FN (x) ≤ γ, TN (y) ≤ α, IN (y) ≥ β and FN (y) ≤ γ.Itfollowsfrom Equation(4)that

TN (x ∗ y) ≤ {TN (x), TN (y)}≤ α, IN (x ∗ y) ≥ {IN (x), IN (y)}≥ β,and FN (x ∗ y) ≤ {FN (x), FN (y)}≤ γ

Hence, x ∗ y ∈ XN(α, β, γ),andtherefore XN(α, β, γ) isasubalgebraof X

Theorem2. Let XN beaneutrosophic N -structureover X andassumethat Tα N , I β N and Fγ N aresubalgebrasof Xforall α, β, γ ∈ [ 1,0] with 3 ≤ α + β + γ ≤ 0.ThenXN isaneutrosophic N -subalgebraofX.

Proof. Assumethatthereexist a, b ∈ X suchthat TN (a ∗ b) > {TN (a), TN (b)}.Then TN (a ∗ b) > tα ≥ {TN (a), TN (b)} forsome tα ∈ [ 1,0).Hence a, b ∈ Ttα N but a ∗ b / ∈ Ttα N ,whichisacontradiction.Thus

TN (x ∗ y) ≤ {TN (x), TN (y)}

forall x, y ∈ X.If IN (a ∗ b) < {IN (a), IN (b)} forsome a, b ∈ X,then

IN (a ∗ b) < tβ < {IN (a), IN (b)}

where tβ := 1 2 {IN (a ∗ b)+ {IN (a), IN (b)}}.Thus a, b ∈ I tβ N and a ∗ b / ∈ I tβ N ,whichisa contradiction.Therefore

IN (x ∗ y) ≥ {IN (x), IN (y)}

forall x, y ∈ X.Now,supposethatthereexist a, b ∈ X and tγ ∈ [ 1,0) suchthat

FN (a ∗ b) > tγ ≥ {FN (a), FN (b)}

Then a, b ∈ Ftγ N and a ∗ b / ∈ Ftγ N ,whichisacontradiction.Hence

FN (x ∗ y) ≤ {FN (x), FN (y)}

forall x, y ∈ X.Therefore XN isaneutrosophic N -subalgebraof X

Because [ 1,0] isacompletelydistributivelatticewithrespecttotheusualordering,wehavethe followingtheorem.

Theorem3. If {XNi | i ∈ N} isafamilyofneutrosophic N -subalgebrasof X,then {XNi | i ∈ N}, ⊆ forms acompletedistributivelattice.

Proposition1. Ifaneutrosophic N -structure XN over X isaneutrosophic N -subalgebraof X,then TN (θ) ≤ TN (x),IN (θ) ≥ IN (x) andFN (θ) ≤ FN (x) forallx ∈ X.

Proof. Straightforward.

Theorem4. Let XN beaneutrosophic N -subalgebraof X.Ifthereexistsasequence {an } in X suchthat lim n→∞ TN (an )= 1, lim n→∞ IN (an )= 0 and lim n→∞ FN (an )= 1,then TN (θ)= 1, IN (θ)= 0 and FN (θ)= 1

Proof. ByProposition 1,wehave TN (θ) ≤ TN (x), IN (θ) ≥ IN (x) and FN (θ) ≤ FN (x) forall x ∈ X.Hence TN (θ) ≤ TN (an ), IN (an ) ≤ IN (θ) and FN (θ) ≤ FN (an ) foreverypositiveinteger n.It followsthat

Florentin Smarandache (author and editor) Collected Papers, IX 149

1 ≤ TN (θ) ≤ lim n→∞ TN (an )= 1 0 ≥ IN (θ) ≥ lim n→∞ IN (an )= 0 1 ≤ FN (θ) ≤ lim n→∞ FN (an )= 1

Hence TN (θ)= 1, IN (θ)= 0and FN (θ)= 1.

Proposition2. Ifeveryneutrosophic N -subalgebraXN ofXsatisﬁes: TN (x ∗ y) ≤ TN (y), IN (x ∗ y) ≥ IN (y), FN (x ∗ y) ≤ FN (y) (5) forallx, y ∈ X,thenXN isconstant. Proof. UsingEquations(1)and(5),wehave TN (x)= TN (x ∗ θ) ≤ TN (θ), IN (x)= IN (x ∗ θ) ≥ IN (θ) and FN (x)= FN (x ∗ θ) ≤ FN (θ) forall x ∈ X.ItfollowsfromProposition 1 that TN (x)= TN (θ), IN (x)= IN (θ) and FN (x)= FN (θ) forall x ∈ X.Therefore XN isconstant.

Deﬁnition2. Aneutrosophic N -structure XN over X iscalledaneutrosophic N -idealof X ifthefollowing assertionisvalid: (∀x, y ∈ X)    TN (θ) ≤ TN (x) ≤ {TN (x ∗ y), TN (y)} IN (θ) ≥ IN (x) ≥ {IN (x ∗ y), IN (y)} FN (θ) ≤ FN (x) ≤ {FN (x ∗ y), FN (y)}    (6)

Example2. Theneutrosophic N -structureXN overXinExample 1 isaneutrosophic N -idealofX. Example3. Considera BCI-algebra X := Y × Z where (Y, ∗, θ) isa BCI-algebraand (Z, ,0) istheadjoint BCI-algebraoftheadditivegroup (Z, +,0) ofintegers(see[6]).Let XN beaneutrosophic N -structureover X givenby XN = x (α,0,γ) | x ∈ Y × (N ∪{0}) ∪ x (0,β,0) | x / ∈ Y × (N ∪{0}) where α, γ ∈ [ 1,0) and β ∈ ( 1,0].ThenXN isaneutrosophic N -idealofX. Proposition3. Everyneutrosophic N -idealXN ofXsatisﬁesthefollowingassertions:

Thiscompletestheproof. Proposition4. LetXN

(x, y ∈ X) (x y ⇒ TN (x) ≤ TN (y), IN (x) ≥ IN (y), FN (x) ≤ FN (y)) (7) Proof. Let x, y ∈ X besuchthat x y.Then x ∗ y = θ,andso TN (x) ≤ {TN (x ∗ y), TN (y)} = {TN (θ), TN (y)} = TN (y) IN (x) ≥ {IN (x ∗ y), IN (y)} = {IN (θ), IN (y)} = IN (y) FN (x) ≤ {FN (x ∗ y), FN (y)} = {FN (θ), FN (y)} = FN (y)

beaneutrosophic N -idealofX.Then (1) TN (x ∗ y) ≤ TN ((x ∗ y) ∗ y) ⇔ TN ((x ∗ z) ∗ (y ∗ z)) ≤ TN ((x ∗ y) ∗ z) (2) IN (x ∗ y) ≥ IN ((x ∗ y) ∗ y) ⇔ IN ((x ∗ z) ∗ (y ∗ z)) ≥ IN ((x ∗ y) ∗ z) (3) FN (x ∗ y) ≤ FN ((x ∗ y) ∗ y) ⇔ FN ((x ∗ z) ∗ (y ∗ z)) ≤ FN ((x ∗ y) ∗ z) forallx, y, z ∈ X. Florentin Smarandache (author and editor) Collected Papers, IX 150

Proof. Notethat

((x ∗ (y ∗ z)) ∗ z) ∗ z (x ∗ y) ∗ z (8)

forall x, y, z ∈ X.Assumethat TN (x ∗ y) ≤ TN ((x ∗ y) ∗ y), IN (x ∗ y) ≥ IN ((x ∗ y) ∗ y) and FN (x ∗ y) ≤ FN ((x ∗ y) ∗ y) forall x, y ∈ X.ItfollowsfromEquation(2)andProposition 3 that

TN ((x ∗ z) ∗ (y ∗ z))= TN ((x ∗ (y ∗ z)) ∗ z)

≤ TN (((x ∗ (y ∗ z)) ∗ z) ∗ z) ≤ TN ((x ∗ y) ∗ z)

IN ((x ∗ z) ∗ (y ∗ z))= IN ((x ∗ (y ∗ z)) ∗ z)

≥ IN (((x ∗ (y ∗ z)) ∗ z) ∗ z) ≥ IN ((x ∗ y) ∗ z) and

FN ((x ∗ z) ∗ (y ∗ z))= FN ((x ∗ (y ∗ z)) ∗ z) ≤ FN (((x ∗ (y ∗ z)) ∗ z) ∗ z) ≤ FN ((x ∗ y) ∗ z)

forall x, y ∈ X. Conversely,suppose

TN ((x ∗ z) ∗ (y ∗ z)) ≤ TN ((x ∗ y) ∗ z) IN ((x ∗ z) ∗ (y ∗ z)) ≥ IN ((x ∗ y) ∗ z)

FN ((x ∗ z) ∗ (y ∗ z)) ≤ FN ((x ∗ y) ∗ z) (9) forall x, y, z ∈ X.Ifwesubstitute z for y inEquation(9),then

TN (x ∗ z)= TN ((x ∗ z) ∗ θ)= TN ((x ∗ z) ∗ (z ∗ z)) ≤ TN ((x ∗ z) ∗ z)

IN (x ∗ z)= IN ((x ∗ z) ∗ θ)= IN ((x ∗ z) ∗ (z ∗ z)) ≥ IN ((x ∗ z) ∗ z) FN (x ∗ z)= FN ((x ∗ z) ∗ θ)= FN ((x ∗ z) ∗ (z ∗ z)) ≤ FN ((x ∗ z) ∗ z) forall x, z ∈ X byusing(III)andEquation(1).

Theorem5. Let XN beaneutrosophic N -structureover X andlet α, β, γ ∈ [ 1,0] besuchthat 3 ≤ α + β + γ ≤ 0.If XN isaneutrosophic N -idealof X,thenthenonempty (α, β, γ)-levelsetof XN isanidealofX.

Proof. Assumethat XN(α, β, γ) = ∅ for α, β, γ ∈ [ 1,0] with 3 ≤ α + β + γ ≤ 0.Clearly, θ ∈ XN(α, β, γ).Let x, y ∈ X besuchthat x ∗ y ∈ XN(α, β, γ) and y ∈ XN(α, β, γ).Then TN (x ∗ y) ≤ α, IN (x ∗ y) ≥ β, FN (x ∗ y) ≤ γ, TN (y) ≤ α, IN (y) ≥ β and FN (y) ≤ γ.ItfollowsfromEquation(6)that TN (x) ≤ {TN (x ∗ y), TN (y)}≤ α IN (x) ≥ {IN (x ∗ y), IN (y)}≥ β FN (x) ≤ {FN (x ∗ y), FN (y)}≤ γ sothat x ∈ XN(α, β, γ).Therefore XN(α, β, γ) isanidealof X

Florentin Smarandache (author and editor) Collected Papers, IX 151

Theorem6. Let XN beaneutrosophic N -structureover X andassumethat Tα N , I β N and Fγ N areidealsof X for all α, β, γ ∈ [ 1,0] with 3 ≤ α + β + γ ≤ 0.ThenXN isaneutrosophic N -idealofX. Proof. Ifthereexist a, b, c ∈ X suchthat TN (θ) > TN (a), IN (θ) < IN (b) and FN (θ) > FN (c), respectively,then TN (θ) > at ≥ TN (a), IN (θ) < bi ≤ IN (b) and FN (θ) > c f ≥ FN (c) forsome at, c f ∈ [ 1,0) and bi ∈ ( 1,0].Then θ / ∈ Tat N , θ / ∈ Ibi N and θ / ∈ F c f N .Thisisacontradiction. Hence, TN (θ) ≤ TN (x), IN (θ) ≥ IN (x) and FN (θ) ≤ FN (x) forall x ∈ X.Assumethatthereexist at, bt, ai, bi, a f , b f ∈ X suchthat TN (at ) > {TN (at ∗ bt ), TN (bt )}, IN (ai ) < {IN (ai ∗ bi ), IN (bi )} and FN (a f ) > {FN (a f ∗ b f ), FN (b f )}.Thenthereexist st, s f ∈ [ 1,0) and si ∈ ( 1,0] suchthat

TN (at ) > st ≥ {TN (at ∗ bt ), TN (bt )}

IN (ai ) < si ≤ {IN (ai ∗ bi ), IN (bi )}

FN (a f ) > s f ≥ {FN (a f ∗ b f ), FN (b f )}

Itfollowsthat at ∗ bt ∈ Tst N , bt ∈ Tst N , ai ∗ bi ∈ Isi N , bi ∈ Isi N , a f ∗ b f ∈ F s f N and b f ∈ F s f N .However, at / ∈ Tst N , ai / ∈ Isi N and a f / ∈ F s f N .Thisisacontradiction,andso

TN (x) ≤ {TN (x ∗ y), TN (y)} IN (x) ≥ {IN (x ∗ y), IN (y)}

FN (x) ≤ {FN (x ∗ y), FN (y)}

forall x, y ∈ X.Therefore XN isaneutrosophic N -idealof X.

Proposition5. Foranyneutrosophic N -idealXN ofX,wehave (∀x, y, z ∈ X)    x ∗ y z ⇒     

TN (x) ≤ {TN (y), TN (z)} IN (x) ≥ {IN (y), IN (z)} FN (x) ≤ {FN (y), FN (z)}    (10)

Proof. Let x, y, z ∈ X besuchthat x ∗ y z.Then (x ∗ y) ∗ z = θ,andso

TN (x ∗ y) ≤ {TN ((x ∗ y) ∗ z), TN (z)} = {TN (θ), TN (z)} = TN (z) IN (x ∗ y) ≥ {IN ((x ∗ y) ∗ z), IN (z)} = {IN (θ), IN (z)} = IN (z) FN (x ∗ y) ≤ {FN ((x ∗ y) ∗ z), FN (z)} = {FN (θ), FN (z)} = FN (z) Itfollowsthat

TN (x) ≤ {TN (x ∗ y), TN (y)}≤ {TN (y), TN (z)} IN (x) ≥ {IN (x ∗ y), IN (y)}≥ {IN (y), IN (z)} FN (x) ≤ {FN (x ∗ y), FN (y)}≤ {FN (y), FN (z)}

Thiscompletestheproof. Theorem7. InaBCK-algebra,everyneutrosophic N -idealisaneutrosophic N -subalgebra. Proof. Let XN beaneutrosophic N -idealofa BCK-algebra X.Forany x, y ∈ X,wehave

Florentin Smarandache (author and editor) Collected Papers, IX 152

TN (x ∗ y) ≤ {TN ((x ∗ y) ∗ x), TN (x)} = {TN ((x ∗ x) ∗ y), TN (x)} = {TN (θ ∗ y), TN (x)} = {TN (θ), TN (x)} ≤ {TN (x), TN (y)} IN (x ∗ y) ≥ {IN ((x ∗ y) ∗ x), IN (x)} = {IN ((x ∗ x) ∗ y), IN (x)} = {IN (θ ∗ y), IN (x)} = {IN (θ), IN (x)} ≥ {IN (y), IN (x)} and FN (x ∗ y) ≤ {FN ((x ∗ y) ∗ x), FN (x)} = {FN ((x ∗ x) ∗ y), FN (x)} = {FN (θ ∗ y), FN (x)} = {FN (θ), FN (x)} ≤ {FN (x), FN (y)}

Hence XN isaneutrosophic N -subalgebraofa BCK-algebra X TheconverseofTheorem 7 maynotbetrueingeneral,asseeninthefollowingexample. Example4. ConsideraBCK-algebraX = {θ,1,2,3,4} withthefollowingCayleytable. ∗ θ 1234 θθθθθθ 11 θθθθ 221 θ 1 θ 3333 θθ 44443 θ

LetXN beaneutrosophic N -structureoverX,whichisgivenasfollows: XN = θ ( 0.8,0, 1) , 1 ( 0.8, 0.2, 0.9) , 2 ( 0.2, 0.6, 0.5) , 3 ( 0.7, 0.4, 0.7) , 4 ( 0.4, 0.8, 0.3)

Then XN isaneutrosophic N -subalgebraof X,butitisnotaneutrosophic N -idealof X as TN (2)= 0.2 > 0.7 = {TN (2 ∗ 3), TN (3)}, IN (4)= 0.8 < 0.4 = {IN (4 ∗ 3), IN (3)},or FN (4)= 0.3 > 0.7 = {FN (4 ∗ 3), FN (3)}

Theorem 7 isnotvalidina BCI-algebra;thatis,if X isa BCI-algebra,thenthereisaneutrosophic N -idealthatisnotaneutrosophic N -subalgebra,asseeninthefollowingexample.

Example5. Considertheneutrosophic N -ideal XN of X inExample 3.Ifwetake x :=(θ,0) and y :=(θ,1) inY × (N ∪{0}),thenx ∗ y =(θ,0) ∗ (θ,1)=(θ, 1) / ∈ Y × (N ∪{0}).Hence TN (x ∗ y)= 0 > α = {TN (x), TN (y)} IN (x ∗ y)= β < 0 = {IN (x), IN (y)} or FN (x ∗ y)= 0 > γ = {FN (x), FN (y)}

ThereforeXN isnotaneutrosophic N -subalgebraofX.

Florentin Smarandache (author and editor) Collected Papers, IX 153

Foranyelements ωt, ωi, ω f ∈ X,weconsidersets:

Xωt N := {x ∈ X | TN (x) ≤ TN (ωt )} Xωi N := {x ∈ X | IN (x) ≥ IN (ωi )} X ω f N := x ∈ X | FN (x) ≤ FN (ω f ) Clearly, ωt ∈ Xωt N , ωi ∈ Xωi N and ω f ∈ X ω f N Theorem8. Let ωt, ωi and ω f beanyelementsof X.If XN isaneutrosophic N -idealof X,then Xωt N , Xωi N andXω f N areidealsofX. Proof. Clearly, θ ∈ Xωt N , θ ∈ Xωi N and θ ∈ X ω f N .Let x, y ∈ X besuchthat x ∗ y ∈ Xωt N ∩ Xωi N ∩ X ω f N and y ∈ Xωt N ∩ Xωi N ∩ X ω f N .Then TN (x ∗ y) ≤ TN (ωt ), TN (y) ≤ TN (ωt ) IN (x ∗ y) ≥ IN (ωi ), IN (y) ≥ IN (ωi ) FN (x ∗ y) ≤ FN (ω f ), FN (y) ≤ FN (ω f )

ItfollowsfromEquation(6)that TN (x) ≤ {TN (x ∗ y), TN (y)}≤ TN (ωt ) IN (x) ≥ {IN (x ∗ y), IN (y)}≥ IN (ωi ) FN (x) ≤ {FN (x ∗ y), FN (y)}≤ FN (ω f ) Hence x ∈ Xωt N ∩ Xωi N ∩ X ω f N ,andtherefore Xωt N , Xωi N and X ω f N areidealsof X

Theorem9. Let ωt, ωi, ω f ∈ XandletXN beaneutrosophic N -structureoverX.Then

   (11) (2) IfXN satisﬁesEquation (11) and (∀x ∈ X) (TN (θ) ≤ TN (x), IN (θ) ≥ IN (x), FN (θ) ≤ FN (x)) (12) thenXω Florentin Smarandache (author and editor) Collected Papers, IX 154

(1) IfXωt N ,Xωi N andXω f N areidealsofX,thenthefollowingassertionisvalid: (∀x, y, z ∈ X)    TN (x) ≥ {TN (y ∗ z), TN (z)}⇒ TN (x) ≥ TN (y) IN (x) ≤ {IN (y ∗ z), IN (z)}⇒ IN (x) ≤ IN (y) FN (x) ≥ {FN (y ∗ z), FN (z)}⇒ FN (x) ≥ FN (y) t N ,Xωi N andXω f N areidealsofXforall ωt ∈ Im(TN ), ωi ∈ Im(IN ) and ω f ∈ Im(FN ) Proof. (1)Assumethat Xωt N , Xωi N and X ω f N areidealsof X for ωt, ωi, ω f ∈ X.Let x, y, z ∈ X besuch that TN (x) ≥ {TN (y ∗ z), TN (z)}, IN (x) ≤ {IN (y ∗ z), IN (z)} and FN (x) ≥ {FN (y ∗ z), FN (z)} Then y ∗ z ∈ Xωt N ∩ Xωi N ∩ X ω f N and z ∈ Xωt N ∩ Xωi N ∩ X ω f N ,where ωt = ωi = ω f = x.Itfollows from(I2)that y ∈ Xωt N ∩ Xωi N ∩ X ω f N for ωt = ωi = ω f = x.Hence TN (y) ≤ TN (ωt )= TN (x), IN (y) ≥ IN (ωi )= IN (x) and FN (y) ≤ FN (ω f )= FN (x)

(2)Let ωt ∈ Im(TN ), ωi ∈ Im(IN ) and ω f ∈ Im(FN ) andsupposethat XN satisﬁesEquations (11) and (12).Clearly, θ ∈ Xωt N ∩ Xωi N ∩ X ω f N byEquation (12).Let x, y ∈ X besuchthat x ∗ y ∈ Xωt N ∩ Xωi N ∩ X ω f N and y ∈ Xωt N ∩ Xωi N ∩ X ω f N .Then TN (x ∗ y) ≤ TN (ωt ), TN (y) ≤ TN (ωt ) IN (x ∗ y) ≥ IN (ωi ), IN (y) ≥ IN (ωi ) FN (x ∗ y) ≤ FN (ω f ), FN (y) ≤ FN (ω f ) whichimpliesthat {TN (x ∗ y), TN (y)}≤ TN (ωt ), {IN (x ∗ y), IN (y)}≥ IN (ωi ),and {FN (x ∗ y), FN (y)}≤ FN (ω f ).ItfollowsfromEquation (11) that TN (ωt ) ≥ TN (x), IN (ωi ) ≤ IN (x) and FN (ω f ) ≥ FN (x).Thus, x ∈ Xωt N ∩ Xωi N ∩ X ω f N ,andtherefore Xωt N , Xωi N and X ω f N areidealsof X. Deﬁnition3. Aneutrosophic N -idealXN ofXissaidtobeclosedifitisaneutrosophic N -subalgebraofX. Example6. ConsideraBCI-algebraX = {θ,1, a, b, c} withthefollowingCayleytable. ∗ θ 1 abc θθθ abc 11 θ abc aaa θ cb bbbc θ a cccba θ

LetXN beaneutrosophic N -structureoverXwhichisgivenasfollows: XN = θ ( 0.9, 0.3, 0.8) , 1 ( 0.7, 0.4, 0.7) , a ( 0.6, 0.8, 0.3) , b ( 0.2, 0.6, 0.3) , c ( 0.2, 0.8, 0.5)

ThenXN isaclosedneutrosophic N -idealofX. Theorem10. Let X bea BCI-algebra,Forany α1, α2, γ1, γ2 ∈ [ 1,0) and β1, β2 ∈ ( 1,0] with α1 < α2, γ1 < γ2 and β1 > β2,letXN := X (TN ,IN ,FN ) beaneutrosophic N -structureoverXgivenasfollows:

TN : X → [ 1,0], x → α1 if x ∈ X+ α2 otherwise IN : X → [ 1,0], x → β1 if x ∈ X+ β2 otherwise FN : X → [ 1,0], x → γ1 if x ∈ X+ γ2 otherwise whereX+ = {x ∈ X | θ x}.ThenXN isaclosedneutrosophic N -idealofX. Proof. Because θ ∈ X+,wehave TN (θ)= α1 ≤ TN (x), IN (θ)= β1 ≥ IN (x) and FN (θ)= γ1 ≤ FN (x) forall x ∈ X.Let x, y ∈ X.If x ∈ X+,then TN (x)= α1 ≤ {TN (x ∗ y), TN (y)} IN (x)= β1 ≥ {IN (x ∗ y), IN (y)} FN (x)= γ1 ≤ {FN (x ∗ y), FN (y)}

Florentin Smarandache (author and editor) Collected Papers, IX 155

Supposethat x / ∈ X+.If x ∗ y ∈ X+ then y / ∈ X+,andif y ∈ X+ then x ∗ y / ∈ X+.Ineithercase, wehave

TN (x)= α2 = {TN (x ∗ y), TN (y)}

IN (x)= β2 = {IN (x ∗ y), IN (y)}

FN (x)= γ2 = {FN (x ∗ y), FN (y)}

Forany x, y ∈ X,ifanyoneof x and y doesnotbelongto X+,then

TN (x ∗ y) ≤ α2 = {TN (x), TN (y)}

IN (x ∗ y) ≥ β2 = {IN (x), IN (y)}

FN (x ∗ y) ≤ γ2 = {FN (x), FN (y)}

If x, y ∈ X+,then x ∗ y ∈ X+.Hence

TN (x ∗ y)= α1 = {TN (x), TN (y)}

IN (x ∗ y)= β1 = {IN (x), IN (y)}

FN (x ∗ y)= γ1 = {FN (x), FN (y)}

Therefore XN isaclosedneutrosophic N -idealof X Proposition6. Everyclosedneutrosophic N -idealXN ofaBCI-algebraXsatisﬁesthefollowingcondition: (∀x ∈ X) (TN (θ ∗ x) ≤ TN (x), IN (θ ∗ x) ≥ IN (x), FN (θ ∗ x) ≤ FN (x)) (13) Proof. Straightforward. Weprovideconditionsforaneutrosophic N -idealtobeclosed. Theorem11. Let X bea BCI-algebra.If XN isaneutrosophic N -idealof X thatsatisﬁestheconditionof Equation (13),thenXN isaneutrosophic N -subalgebraandhenceisaclosedneutrosophic N -idealofX.

Proof. Notethat (x ∗ y) ∗ x θ ∗ y forall x, y ∈ X.UsingEquations(10)and(13),wehave TN (x ∗ y) ≤ {TN (x), TN (θ ∗ y)}≤ {TN (x), TN (y)} IN (x ∗ y) ≥ {IN (x), IN (θ ∗ y)}≥ {IN (x), IN (y)} FN (x ∗ y) ≤ {FN (x), FN (θ ∗ y)}≤ {FN (x), FN (y)}

Hence XN is a neutrosophic N -subalgebra and is therefore a closed neutrosophic N -ideal of X

References

1.Imai,Y.;Iséki,K.Onaxiomsystemsofpropositionalcalculi. Proc.Jpn.Acad. 1966, 42,19–21.

2.Iséki,K.Analgebrarelatedwithapropositionalcalculus. Proc.Jpn.Acad. 1966, 42,26–29.

3.Jun,Y.B.;Lee,K.J.;Song,S.Z. N -idealsofBCK/BCI-algebras. J.ChungcheongMath.Soc. 2009, 22,417–437.

4.Zadeh,L.A.Fuzzysets. Inf.Control 1965, 8,338–353.

5.Atanassov,K.Intuitionisticfuzzysets. FuzzySetsSyst. 1986, 20,87–96.

6.Huang,Y.S. BCI-Algebra;SciencePress:Beijing,China,2006.

7.Meng,J.;Jun,Y.B. BCK-Algebras;KyungmoonSaCo.:Seoul,Korea,1994.

8. Khan,M.;Amis,S.;Smarandache,F.;Jun,Y.B.Neutrosophic N -structuresandtheirapplicationsin semigroups. Ann.FuzzyMath.Inform. submitted,2017.

Florentin Smarandache (author and editor) Collected Papers, IX 156

Neutrosophic Commutative N-Ideals in BCK-Algebras

Seok-Zun Song, Florentin Smarandache, Young Bae Jun

Seok-Zun Song, Florentin Smarandache, Young Bae Jun (2017). Neutrosophic Commutative N-Ideals in BCK-Algebras. Information 8, 130; DOI: 10.3390/info8040130

Abstract: The notion of a neutrosophic commutative N -ideal in BCK-algebras is introduced, and several properties are investigated. Relations between a neutrosophic N -ideal and a neutrosophic commutative N -ideal are discussed. Characterizations of a neutrosophic commutative N -ideal are considered.

Keywords: neutrosophic N -structure; neutrosophic N -ideal; neutrosophic commutative N -ideal

1.Introduction

Asageneralizationoffuzzysets,Atanassov[1]introducedthedegreeofnonmembership/ falsehood(f)in1986anddeﬁnedtheintuitionisticfuzzyset.

Smarandacheproposedtheterm“neutrosophic”because“neutrosophic”etymologicallycomes from“neutrosophy”[French neutre < Latin neuter,neutral,andGreek sophia,skill/wisdom]which meansknowledgeofneutralthought,andthisthird/neutralrepresentsthemaindistinction between“fuzzy”/“intuitionisticfuzzy”logic/setand“neutrosophic”logic/set,i.e.,the includedmiddle component(Lupasco–Nicolescu’slogicinphilosophy),i.e.,theneutral/indeterminate/unknown part(besidesthe“truth”/“membership”and“falsehood”/“non-membership”componentsthat bothappearinfuzzylogic/set).Smarandacheintroducedthedegreeofindeterminacy/neutrality (i)asanindependentcomponentin1995(publishedin1998)anddeﬁnedtheneutrosophicseton threecomponents

(t,i,f)=(truth,indeterminacy,falsehood).

Formoredetails,refertothesite http://fs.gallup.unm.edu/FlorentinSmarandache.htm

Junetal.[2]introducedanewfunctionwhichiscallednegative-valuedfunction,and constructed N -structures.Khanetal.[3]introducedthenotionofneutrosophic N -structure andappliedittoasemigroup.Junetal.[4]appliedthenotionofneutrosophic N -structureto BCK/BCI-algebras.Theyintroducedthenotionsofaneutrosophic N -subalgebraanda(closed) neutrosophic N -idealina BCK/BCI-algebra,andinvestigatedrelatedproperties.Theyalsoconsidered characterizationsofaneutrosophic N -subalgebraandaneutrosophic N -ideal,anddiscussedrelations betweenaneutrosophic N -subalgebraandaneutrosophic N -ideal.Theyprovidedconditionsfor aneutrosophic N -idealtobeaclosedneutrosophic N -ideal. BCK-algebrasenteredintomathematicsin 1966throughtheworkofImaiandIséki[5],andhavebeenappliedtomanybranchesofmathematics, suchasgrouptheory,functionalanalysis,probabilitytheoryandtopology.Suchalgebrasgeneralize BooleanringsaswellasBoolean D-posets(= MV-algebras).Also,Isékiintroducedthenotionof a BCI-algebrawhichisageneralizationofa BCK-algebra(see[6]).

Smarandache (author and editor) Collected Papers, IX 157

Florentin

Inthispaper,weintroducethenotionofaneutrosophiccommutative N -idealin BCK-algebras, andinvestigateseveralproperties.Weconsiderrelationsbetweenaneutrosophic N -idealand aneutrosophiccommutative N -ideal.Wediscusscharacterizationsofaneutrosophiccommutative N -ideal.

2.Preliminaries

Bya BCI-algebra,wemeanasystem X :=(X, ∗,0) ∈ K(τ) inwhichthefollowingaxiomshold: (I) ((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y)= 0, (II) (x ∗ (x ∗ y)) ∗ y = 0, (III) x ∗ x = 0, (IV) x ∗ y = y ∗ x = 0 ⇒ x = y

forall x, y, z ∈ X.Ifa BCI-algebra X satisﬁes0 ∗ x = 0forall x ∈ X,thenwesaythat X isa BCK-algebra. Wecandeﬁneapartialordering by (∀x, y ∈ X)(x y ⇒ x ∗ y = 0).

Ina BCK/BCI-algebra X,thefollowinghold: (∀x ∈ X)(x ∗ 0 = x),(1) (∀x, y, z ∈ X)((x ∗ y) ∗ z =(x ∗ z) ∗ y).(2)

A BCK-algebra X issaidtobe commutative ifitsatisﬁesthefollowingequality: (∀x, y ∈ X) (x ∗ (x ∗ y)= y ∗ (y ∗ x)) .(3)

Asubset I ofa BCK/BCI-algebra X iscalledan ideal of X ifitsatisﬁes 0 ∈ I,(4) (∀x, y ∈ X) (x ∗ y ∈ I, y ∈ I ⇒ x ∈ I) .(5)

Asubset I ofa BCK-algebra X iscalleda commutativeideal of X ifitsatisﬁes(4)and (∀x, y, z ∈ X) ((x ∗ y) ∗ z ∈ I, z ∈ I ⇒ x ∗ (y ∗ (y ∗ x)) ∈ I) .(6)

Lemma1. AnidealIiscommutativeifandonlyifthefollowingassertionisvalid. (∀x, y ∈ X) (x ∗ y ∈ I ⇒ x ∗ (y ∗ (y ∗ x)) ∈ I) .(7)

Wereferthereadertothebooks[7,8]forfurtherinformationregarding BCK/BCI-algebras. Foranyfamily {ai | i ∈ Λ} ofrealnumbers,wedeﬁne

{ai | i ∈ Λ} := max{ai | i ∈ Λ} if Λ isﬁnite, sup{ai | i ∈ Λ} otherwise. {ai | i ∈ Λ} := min{ai | i ∈ Λ} if Λ isﬁnite, inf{ai | i ∈ Λ} otherwise.

Denoteby F (X, [ 1,0]) thecollectionoffunctionsfromaset X to [ 1,0].Wesaythatanelementof F (X, [ 1,0]) isa negative-valuedfunction from X to [ 1,0] (brieﬂy, N -function on X).Byan N -structure, wemeananorderedpair (X, f ) of X andan N -function f on X (see[2]).A neutrosophic N -structure overanonemptyuniverseofdiscourse X (see[3])isdeﬁnedtobethestructure

Florentin Smarandache (author and editor) Collected Papers, IX 158

XN := X (TN ,IN ,FN ) = x (TN (x),IN (x),FN (x)) | x ∈ X

where TN , IN and FN are N -functionson X whicharecalledthe negativetruthmembershipfunction, the negativeindeterminacymembershipfunction andthe negativefalsitymembershipfunction,respectively, on X

Notethateveryneutrosophic N -structure XN over X satisﬁesthecondition: (∀x ∈ X) ( 3 ≤ TN (x)+ IN (x)+ FN (x) ≤ 0) .

3.NeutrosophicCommutative N -Ideals

Inwhatfollows,let X denotea BCK-algebraunlessotherwisespeciﬁed.

Deﬁnition1 ([4]). Aneutrosophic N -structure XN over X iscalledaneutrosophic N -idealof X ifthe followingassertionisvalid. (∀x, y ∈ X)    TN (0) ≤ TN (x) ≤ {TN (x ∗ y), TN (y)} IN (0) ≥ IN (x) ≥ {IN (x ∗ y), IN (y)} FN (0) ≤ FN (x) ≤ {FN (x ∗ y), FN (y)}    .(9)

Deﬁnition2. Aneutrosophic N -structure XN over X iscalledaneutrosophiccommutative N -idealof X if thefollowingassertionsarevalid.

(∀x ∈ X) (TN (0) ≤ TN (x), IN (0) ≥ IN (x), FN (0) ≤ FN (x)) ,(10) (∀x, y, z ∈ X)   

TN (x ∗ (y ∗ (y ∗ x))) ≤ {TN ((x ∗ y) ∗ z), TN (z)}

IN (x ∗ (y ∗ (y ∗ x))) ≥ {IN ((x ∗ y) ∗ z), IN (z)} FN (x ∗ (y ∗ (y ∗ x))) ≤ {FN ((x ∗ y) ∗ z), FN (z)}    .(11)

Example1. ConsideraBCK-algebraX = {0,1,2,3,4} withtheCayleytablewhichisgiveninTable 1 Table1. Cayleytableforthebinaryoperation“*”. *01234 000000 110111 222022 333303 444440

Theneutrosophic N -structure XN = 0 ( 0.8, 0.2, 0.9) , 1 ( 0.3, 0.9, 0.5) , 2 ( 0.7, 0.7, 0.4) , 3 ( 0.3, 0.6, 0.7) , 4 ( 0.5, 0.3, 0.1) overXisaneutrosophiccommutative N -idealofX. Theorem1. Everyneutrosophiccommutative N -idealisaneutrosophic N -ideal. Proof. Let XN beaneutrosophiccommutative N -idealof X.Forevery x, z ∈ X,wehave

(8)

Florentin Smarandache (author and editor) Collected Papers, IX 159

TN (x)= TN (x ∗ (0 ∗ (0 ∗ x))) ≤ {TN ((x ∗ 0) ∗ z), TN (z)} = {TN (x ∗ z), TN (z)}, IN (x)= IN (x ∗ (0 ∗ (0 ∗ x))) ≥ {IN ((x ∗ 0) ∗ z), IN (z)} = {IN (x ∗ z), IN (z)}, FN (x)= FN (x ∗ (0 ∗ (0 ∗ x))) ≤ {FN ((x ∗ 0) ∗ z), FN (z)} = {FN (x ∗ z), FN (z)} byputting y = 0in (11) andusing (1).Therefore, XN isaneutrosophiccommutative N -idealof X. TheconverseofTheorem 1 isnottrueingeneralasseeninthefollowingexample. Example2. ConsideraBCK-algebraX = {0,1,2,3,4} withtheCayleytablewhichisgiveninTable 2 Table2. Cayleytableforthebinaryoperation“∗” *01234 000000 110100 222000 333300 444430

Theneutrosophic N -structure XN = 0 ( 0.8, 0.1, 0.7) , 1 ( 0.7, 0.6, 0.6) , 2 ( 0.6, 0.2, 0.4) , 3 ( 0.3, 0.8, 0.4) , 4 ( 0.3, 0.8, 0.4) over X isaneutrosophic N -idealof X.Butitisnotaneutrosophiccommutative N -idealof X since FN (2 ∗ (3 ∗ (3 ∗ 2))= FN (2)= 0.4 0.7 = {FN ((2 ∗ 3) ∗ 0), FN (0)}.

Weconsidercharacterizationsofaneutrosophiccommutative N -ideal.

Theorem2. Let XN beaneutrosophic N -idealof X.Then, XN isaneutrosophiccommutative N -idealof X if andonlyifthefollowingassertionisvalid. (∀x, y ∈ X)    TN (x ∗ (y ∗ (y ∗ x))) ≤ TN (x ∗ y), IN (x ∗ (y ∗ (y ∗ x))) ≥ IN (x ∗ y), FN (x ∗ (y ∗ (y ∗ x))) ≤ FN (x ∗ y)

   .(12)

Proof. Assumethat XN isaneutrosophiccommutative N -idealof X.Theassertion (12) isbytaking z = 0in(11)andusing(1)and(10). Conversely,supposethataneutrosophic N -ideal XN of X satisﬁesthecondition(12).Then, (∀x, y ∈ X)    TN (x ∗ y) ≤ {TN ((x ∗ y) ∗ z), TN (z)} IN (x ∗ y) ≥ {IN ((x ∗ y) ∗ z), IN (z)} FN (x ∗ y) ≤ {FN ((x ∗ y)

.(13) Itfollowsthatthecondition (11) isinducedby (12) and (13).Therefore, XN isaneutrosophic commutative N -idealof X. Lemma2 ([4]). Foranyneutrosophic N -idealXN ofX,wehave



(

X)    x ∗

z ⇒      T

 

Florentin Smarandache (author and editor) Collected Papers, IX 160

∗ z), FN (z)} 



∀x, y, z ∈

N (x) ≤ {TN (y), TN (z)}

N (x) ≥ {IN (y), IN (z)}

N (x) ≤ {FN (y), FN (z)}

 .(14)

Theorem3. Inacommutative BCK-algebra,everyneutrosophic N -idealisaneutrosophiccommutative N -ideal.

Proof. Let XN beaneutrosophic N -idealofacommutative BCK-algebra X.Forany x, y, z ∈ X, wehave

((x ∗ (y ∗ (y ∗ x))) ∗ ((x ∗ y) ∗ z)) ∗ z =((x ∗ (y ∗ (y ∗ x))) ∗ z) ∗ ((x ∗ y) ∗ z) (x ∗ (y ∗ (y ∗ x))) ∗ (x ∗ y) =(x ∗ (x ∗ y)) ∗ (y ∗ (y ∗ x))= 0, thatis, (x ∗ (y ∗ (y ∗ x))) ∗ ((x ∗ y) ∗ z) z.ItfollowsfromLemma 2 that

TN (x ∗ (y ∗ (y ∗ x))) ≤ {TN ((x ∗ y) ∗ z), TN (z)}, IN (x ∗ (y ∗ (y ∗ x))) ≥ {IN ((x ∗ y) ∗ z), IN (z)},

FN (x ∗ (y ∗ (y ∗ x))) ≤ {FN ((x ∗ y) ∗ z), FN (z)}.

Therefore, XN isaneutrosophiccommutative N -idealof X Let XN beaneutrosophic N -structureover X andlet α, β, γ ∈ [ 1,0] besuchthat 3 ≤ α + β + γ ≤ 0.Considerthefollowingsets.

Tα N := {x ∈ X | TN (x) ≤ α},

I β N := {x ∈ X | IN (x) ≥ β},

Fγ N := {x ∈ X | FN (x) ≤ γ}

Theset

XN(α, β, γ) := {x ∈ X | TN (x) ≤ α, IN (x) ≥ β, FN (x) ≤ γ} iscalledthe (α, β, γ)-levelset of XN.Itisclearthat

XN(α, β, γ)= Tα N ∩ I β N ∩ Fγ N Theorem4. If XN isaneutrosophic N -idealof X,then Tα N , I β N and Fγ N arecommutativeidealsof X forall α, β, γ ∈ [ 1,0] with 3 ≤ α + β + γ ≤ 0 whenevertheyarenonempty. Wecall Tα N , I β N and Fγ N levelcommutativeideals of XN

Proof. Assumethat Tα N , I β N and Fγ N arenonemptyforall α, β, γ ∈ [ 1,0] with 3 ≤ α + β + γ ≤ 0. Then, x ∈ Tα N , y ∈ I β N and z ∈ Fγ N forsome x, y, z ∈ X.Thus, TN (0) ≤ TN (x) ≤ α, IN (0) ≥ IN (y) ≥ β, and FN (0) ≤ FN (z) ≤ γ,thatis,0 ∈ Tα N ∩ I β N ∩ Fγ N .Let (x ∗ y) ∗ z ∈ T

α N and z ∈ Tα N

N ((x ∗ y) ∗ z) ≤ α and TN (z) ≤ α

TN (x ∗ (y ∗ (y ∗ x))) ≤ {TN ((x ∗ y) ∗ z), TN (z)}≤ α,

x ∗ (y ∗ (y ∗ x)) ∈ Tα N .If (a ∗ b) ∗ c ∈ I β N and c ∈ I β N

IN ((a ∗ b) ∗ c) ≥ β and IN (c) ≥ β

IN (a ∗ (b ∗ (b ∗ c))) ≥ {IN ((a ∗ b) ∗ c), IN (c)}≥ β, Florentin Smarandache (author and editor) Collected Papers, IX 161

.Then, T

,whichimplythat

thatis,

,then

Thus

andso a ∗ (b ∗ (b ∗ c)) ∈ I β N .Finally,supposethat (u ∗ v) ∗ w ∈ Fγ N and w ∈ Fγ N .Then, FN ((u ∗ v) ∗ w) ≤ γ and FN (w) ≤ γ.Thus, FN (u ∗ (v ∗ (v ∗ w))) ≤ {FN ((u ∗ v) ∗ w), FN (w)}≤ γ, thatis, u ∗ (v ∗ (v ∗ w)) ∈ Fγ N .Therefore, Tα N , I β N and Fγ N arecommutativeidealsof X

Corollary1. Let XN beaneutrosophic N -structureover X andlet α, β, γ ∈ [ 1,0] besuchthat 3 ≤ α + β + γ ≤ 0.If XN isaneutrosophiccommutative N -idealof X,thenthenonempty (α, β, γ)-level setofXN isacommutativeidealofX.

Proof. Straightforward. Lemma3 ([4]). Let XN beaneutrosophic N -structureover X andassumethat Tα N , I β N and Fγ N areidealsof X forall α, β, γ ∈ [ 1,0] with 3 ≤ α + β + γ ≤ 0.ThenXN isaneutrosophic N -idealofX.

Theorem5. Let XN beaneutrosophic N -structureover X andassumethat Tα N , I β N and Fγ N arecommutative idealsof X forall α, β, γ ∈ [ 1,0] with 3 ≤ α + β + γ ≤ 0.Then, XN isaneutrosophiccommutative N -idealofX.

Proof. If Tα N , I β N and Fγ N arecommutativeidealsof X,thentheyareidealsof X.Hence, XN isa neutrosophic N -idealof X byLemma 3.Let x, y ∈ X and α, β, γ ∈ [ 1,0] with 3 ≤ α + β + γ ≤ 0 suchthat TN (x ∗ y)= α, IN (x ∗ y)= β and FN (x ∗ y)= γ.Then, x ∗ y ∈ Tα N ∩ I β N ∩ Fγ N .Since Tα N ∩ I β N ∩ Fγ N isacommutativeidealof X,itfollowsfromLemma 1 that x ∗ (y ∗ (y ∗ x)) ∈ Tα N ∩ I β N ∩ Fγ N . Hence

TN (x ∗ (y ∗ (y ∗ x))) ≤ α = TN (x ∗ y), IN (x ∗ (y ∗ (y ∗ x))) ≥ β = IN (x ∗ y), FN (x ∗ (y ∗ (y ∗ x))) ≤ γ = FN (x ∗ y)

Therefore, XN isaneutrosophiccommutative N -idealof X byTheorem 2

Theorem6. Let f : X → X beaninjectivemapping.Givenaneutrosophic N -structure XN over X, thefollowingareequivalent.

   .(15) (2) Tα N ,I β N andFγ N arecommutativeidealsofXN,satisfyingthefollowingcondition. f (Tα N )= Tα N , f (

∩

γ N .Then TN ( f (x))= TN

x) ≤ α, IN

f (x))= IN

x) ≥ β

≤ γ

x) ∈ Tα N ∩ I β N ∩ Fγ N

(1) XN isaneutrosophiccommutative N -idealofX,satisfyingthefollowingcondition. (∀x ∈ X)    TN ( f (x))= TN (x) IN ( f (x))= IN (x) FN ( f (x))= FN (x) Florentin Smarandache (author and editor) Collected Papers, IX 162

N )= I β N , f (Fγ N )= Fγ N .(16) Proof. Let XN beaneutrosophiccommutative N -idealof X,satisfyingthecondition (15).Then, Tα N , I β N and Fγ N arecommutativeidealsof XN byTheorem 4.Let α ∈ Im(TN ), β ∈ Im(IN ), γ ∈ Im(FN ) and x ∈ T

N ∩ I

(

and FN ( f (x))= FN (x)

Thus, f (

,whichshowsthat f (T

N ) ⊆ T

N , f (I

N ) ⊆ I

N and f (F

N ) ⊆ F

N .Let y ∈ X besuchthat f (y)= x.Then, TN (y)= TN ( f (y))= TN (x) ≤

, IN (y)= IN ( f (y))= IN (x) ≥ β

and FN (y)= FN ( f (y))= FN (x) ≤ γ,whichimplythat y ∈ Tα N ∩ I β N ∩ Fγ N .Thus, x = f (y) ∈ f (Tα N ) ∩ f (I β N ) ∩ f (Fγ N ),andso Tα N ⊆ f (Tα N ), I β N ⊆ f (I β N ) and Fγ N ⊆ f (Fγ N ).Therefore(16)isvalid.

Conversely,assumethat Tα N , I β N and Fγ N arecommutativeidealsof XN,satisfyingthecondition (16) Then, XN isaneutrosophiccommutative N -idealof X byTheorem 5.Let x, y, z ∈ X besuchthat TN (x)= α, IN (y)= β and FN (z)= γ.Notethat

TN (x)= α ⇐⇒ x ∈ Tα N and x / ∈ Tα N forall α > α,

IN (y)= β ⇐⇒ y ∈ I β N and y / ∈ I β N forall β < ˜ β,

FN (z)= γ ⇐⇒ z ∈ Fγ N and z / ∈ Fγ N forall γ > γ

Itfollowsfrom (16) that f (x) ∈ Tα N , f (y) ∈ I β N and f (z) ∈ Fγ N .Hence, TN ( f (x)) ≤ α, IN ( f (y)) ≥ β and FN ( f (z)) ≤ γ.Let α = TN ( f (x)), ˜ β = IN ( f (y)) and γ = FN ( f (z)).If α > α,then f (x) ∈ Tα N = f Tα N ,andthus x ∈ Tα N since f isonetoone.Thisisacontradiction.Hence, TN ( f (x))= α = TN (x). If β < β,then f (y) ∈ I β N = f I β N whichimpliesfromtheinjectivityof f that y ∈ I β N ,acontradiction. Hence, IN ( f (x))= β = IN (x).If γ > ˜ γ,then f (z) ∈ Fγ N = f Fγ N .Since f isonetoone,wehave z ∈ Fγ N whichisacontradiction.Thus, FN ( f (x))= γ = FN (x).Thiscompletestheproof. Foranyelements ωt, ωi, ω f ∈ X,weconsidersets: Xωt N := {x ∈ X | TN (x) ≤ TN (ωt )} , Xωi N := {x ∈ X | IN (x) ≥ IN (ωi )} , X ω f N := x ∈ X | FN (x) ≤ FN (ω f ) Obviously, ωt ∈ Xωt N , ωi ∈ Xωi N and ω f ∈ X ω f N .

Lemma4 ([4]). Let ωt, ωi and ω f beanyelementsof X.If XN isaneutrosophic N -idealof X,then Xωt N , Xωi N andXω f N areidealsofX.

Theorem7. Let ωt, ωi and ω f beanyelementsof X.If XN isaneutrosophiccommutative N -idealof X, thenXωt N ,Xωi N andXω f N arecommutativeidealsofX. Proof. If XN isaneutrosophiccommutative N -idealof X,thenitisaneutrosophic N -idealof X and so Xωt N , Xωi N and X ω f N areidealsof X byLemma 4.Let x ∗ y ∈ Xωt N ∩ Xωi N ∩ X ω f N forany x, y ∈ X.Then, TN (x ∗ y) ≤ TN (ωt ), IN (x ∗ y) ≥ TN (ωi ) and FN (x ∗ y) ≤ FN (ω f ).ItfollowsfromTheorem 2 that

(x ∗ (y ∗ (y ∗ x))) ≤ TN

IN (x ∗ (y ∗ (y ∗ x))) ≥ IN

≥

FN (x ∗ (y ∗ (y ∗ x))) ≤ FN

∗ (

∗ x)) ∈ Xωt N ∩ Xωi N ∩ X ω f

Florentin Smarandache (author and editor) Collected Papers, IX 163

(x ∗ y) ≤ TN (ωt ),

(x ∗ y)

IN (ωi ),

(x ∗ y) ≤ FN (ω f ). Hence, x ∗ (y

N ,andtherefore Xωt N , Xωi N and X ω f N arecommutative idealsof X byLemma 1. Theorem8. Anycommutativeidealof X canberealizedaslevelcommutativeidealsofsomeneutrosophic commutative N -idealofX. Proof. Let A beacommutativeidealof X andlet XN beaneutrosophic N -structureover X inwhich

TN : X → [ 1,0], x → α if x ∈ A, 0otherwise,

IN : X → [ 1,0], x → β if x ∈ A, 1otherwise,

FN : X → [ 1,0], x → γ if x ∈ A, 0otherwise where α, γ ∈ [ 1,0) and β ∈ ( 1,0].Divisionintothefollowingcaseswillverifythat XN is aneutrosophiccommutative N -idealof X.

If (x ∗ y) ∗ z ∈ A and z ∈ A,then x ∗ (y ∗ (y ∗ x) ∈ A.Thus,

TN ((x ∗ y) ∗ z)= TN (z)= TN (x ∗ (y ∗ (y ∗ x)))= α, IN ((x ∗ y) ∗ z)= IN (z)= IN (x ∗ (y ∗ (y ∗ x)))= β, FN ((x ∗ y) ∗ z)= FN (z)= FN (x ∗ (y ∗ (y ∗ x)))= γ, andso(11)isclearlyveriﬁed.

If (x ∗ y) ∗ z / ∈ A and z / ∈ A,then TN ((x ∗ y) ∗ z)= TN (z)= 0, IN ((x ∗ y) ∗ z)= IN (z)= 1and FN ((x ∗ y) ∗ z)= FN (z)= 0.Hence

TN (x ∗ (y ∗ (y ∗ x))) ≤ {TN ((x ∗ y) ∗ z), TN (z)}, IN (x ∗ (y ∗ (y ∗ x))) ≥ {IN ((x ∗ y) ∗ z), IN (z)},

FN (x ∗ (y ∗ (y ∗ x))) ≤ {FN ((x ∗ y) ∗ z), FN (z)}.

If (x ∗ y) ∗ z ∈ A and z / ∈ A,then TN ((x ∗ y) ∗ z)= α, TN (z)= 0, IN ((x ∗ y) ∗ z)= β, IN (z)= 1, FN ((x ∗ y) ∗ z)= γ and FN (z)= 0.Therefore,

TN (x ∗ (y ∗ (y ∗ x))) ≤ {TN ((x ∗ y) ∗ z), TN (z)},

IN (x ∗ (y ∗ (y ∗ x))) ≥ {IN ((x ∗ y) ∗ z), IN (z)},

FN (x ∗ (y ∗ (y ∗ x))) ≤ {FN ((x ∗ y) ∗ z), FN (z)} α N β N γ N

Similarly, if (x ∗ y) ∗ z ∈ / A and z ∈ A, then (11) is veriﬁed. Therefore, X N is a neutrosophic commutative N -ideal of X. Obviously, T = A, I = A and F = A. This completes the proof.

4. Conclusions

In order to deal with the negative meaning of information, Jun et al. [2] have introduced a new function which is called negative-valued function, and constructed N -structures. The concept of neutrosophic set (NS) has been developed by Smarandache in [9,10] as a more general platform which extends the concepts of the classic set and fuzzy set, intuitionistic fuzzy set and interval valued intuitionistic fuzzy set. In this article, we have introduced the notion of a neutrosophic commutative N -ideal in BCK-algebras, and investigated several properties. We have considered relations between a neutrosophic N -ideal and a neutrosophic commutative N -ideal. We have discussed characterizations of a neutrosophic commutative N -ideal.

Florentin Smarandache (author and editor) Collected Papers, IX 164

References

1.Atanassov,K.Intuitionisticfuzzysets. FuzzySetsSyst. 1986, 20,87–96.

2.Jun,Y.B.;Lee,K.J.;Song,S.Z. N -idealsofBCK/BCI-algebras. J.ChungcheongMath.Soc. 2009,22,417–437.

3. Khan,M.;Anis,S.;Smarandache,F.;Jun,Y.B.Neutrosophic N -structuresandtheirapplicationsin semigroups. Ann.FuzzyMath.Inform. 2017,inpress.

4. Jun,Y.B.;Smarandache,F.;Bordbar,H.Neutrosophic N -structuresappliedto BCK/BCI-algebras. Information 2017, 8,128.

5.Imai,Y.;Iséki,K.Onaxiomsystemsofpropositionalcalculi. Proc.Jpn.Acad. 1966, 42,19–21.

6.Iséki,K.Analgebrarelatedwithapropositionalcalculus. Proc.Jpn.Acad. 1966, 42,26–29.

7.Huang,Y.S. BCI-Algebra;SciencePress:Beijing,China,2006.

8.Meng,J.;Jun,Y.B. BCK-Algebras;KyungmoonSaCo.:Seoul,Korea,1994.

9. Smarandache,F. AUnifyingFieldinLogics:NeutrosophicLogic.Neutrosophy,NeutrosophicSet,Neutrosophic Probability;AmericanReserchPress:Rehoboth,NM,USA,1999.

10. Smarandache,F.Neutrosophicset-ageneralizationoftheintuitionisticfuzzyset. Int.J.PureAppl.Math. 2005, 24,287–297.

Florentin Smarandache (author and editor) Collected Papers, IX 165

Compact Open Topology and Evaluation Map via Neutrosophic Sets

R. Dhavaseelan, S. Jafari, F. Smarandache

R. Dhavaseelan, Saeid Jafari, Florentin Smarandache (2017). Compact Open Topology and Evaluation Map via Neutrosophic Sets. Neutrosophic Sets and Systems 16, 35-38

Abstract: The concept of neutrosophic locally compact and neutrosophic compact open topology are introduced and some interesting propositions are discussed.

Keywords: neutrosophic locally Compact Hausdorff space; neutrosophic product topology; neutrosophic compact open topology; neutrosophic homeomorphism; neutrosophic evaluation map; Exponential map.

1IntroductionandPreliminaries

In1965,Zadeh[19]introducedtheusefulnotionofafuzzyset andChang[6]threeyearslaterofferedthenotionoffuzzytopologicalspace.Sincethen,severalauthorshavegeneralizednumerousconceptsofgeneraltopologytothefuzzysetting.The conceptofintuitionisticfuzzysetwasintroducedandstudied byAtanassov[1]andsubsequentlysomeimportantresearchpaperspublishedbyhimandhiscolleagues[2,3,4].Theconcept offuzzycompactopentopologywasintroducedbyS.Dangand A.Behera[9].Theconceptsofintuitionisticevaluationmapsby R.Dhavaseelanetal[9].Aftertheintroductionoftheconcepts ofneutrosophyandneutrosophicsetbyF.Smarandache[[11], [12]],theconceptsofneutrosophiccrispsetandneutrosophic crisptopologicalspaceswereintroducedbyA.A.SalamaandS. A.Alblowi[10].

Inthispaperthenotionofneutrosophiccompactopentopologyisintroduced.Someinterestingpropertiesarediscussed. Moreover,neutrosophiclocalcompactnessandneutrosophic producttopologyaredeveloped.WehavealsoutilizedthenotionoffuzzylocallycompactnessduetoWong[17],Christoph [8]andfuzzyproducttopologyduetoWong[18].

Throughoutthispaperneutrosophictopologicalspaces (X,T ),(Y,S) and (Z,R) willbereplacedby X,Y and Z respectively.

Deﬁnition 1.1. LetT,I,Fberealstandardornonstandard subsets of ]0 , 1+[, with supT = tsup,infT = tinf supI = isup,infI = iinf supF = fsup,infF = finf n sup = tsup + isup + fsup n inf = tinf +iinf +finf .T,I,Fareneutrosophiccomponents.

Definition1.2. LetXbeanonemptyfixedset.Aneutrosophicset[brieflyNS]Aisanobjecthavingtheform A = { x,µA (x),σA (x),γA (x) : x ∈ X},where µA (x),σA (x)

and γA (x) whichrepresentthedegree ofmembershipfunction (namely µA (x)), thedegreeofindeterminacy(namely σA (x)) andthedegreeofnonmembership(namely γA (x))respectively ofeachelement x ∈ X to thesetA.

Remark1.1. (1)Aneutrosophicset A = { x,µA (x),σA (x),γA (x) : x ∈ X} canbeidentiﬁedtoanorderedtriple µA ,σA ,γA in ]0 , 1+[ on X.

(2)Forthesake ofsimplicity,weshallusethesymbol A = µA ,σA ,γA fortheneutrosophicset A = { x,µA (x),σA (x),γA (x) : x ∈ X}

Weintroducetheneutrosophicsets 0N and 1N inX asfollows: Deﬁnition1.3. 0N = { x, 0, 0, 1 : x ∈ X} and 1N = { x, 1, 1, 0 : x ∈ X}.

Deﬁnition1.4. [8]Aneutrosophictopology(NT)onanonempty set X consistsofafamily T ofneutrosophicsetsin X which satisﬁesthefollowing:

(i) 0N , 1N ∈ T ,

(ii) G1 ∩ G2 ∈ T forany G1,G2 ∈ T , (iii) ∪Gi ∈ T forarbitraryfamily {Gi | i ∈ Λ}⊆ T

Inthiscasetheorderedpair (X,T ) orsimply X iscalledaneutrosophictopologicalspace(NTS)andeachneutrosophicsetin T iscalledaneutrosophicopenset(NOS).Thecomplement A ofaNOS A in X iscalledaneutrosophicclosedset(NCS)in X Deﬁnition1.5. [8]Let A beaneutrosophicsubsetofaneutrosophictopologicalspace X.Theneutrosophicinteriorandneutrosophicclosureof A aredenotedanddeﬁnedby

Nint(A)= {G | G isaneutrosophicopensetinXand

Collected Papers, IX 166

Florentin Smarandache (author and editor)

2NeutrosophicLocallyCompactand NeutrosophicCompactOpenTopology

Deﬁnition 2.1. Let X be a nonempty set and x ∈ X a ﬁxed element in X. If r, t ∈ I0 = (0, 1] and s ∈ I1 = [0, 1) are fixed real numbers such that 0 < r + t + s < 3, then xr,t,s = x, r, t, s is called a neutrosophic point (in short NP) in X, where r denotes the degree of membership of xr,t,s, t denotes the degree of indeterminacy and s denotes the degree of nonmembership of xr,t,s and x ∈ X the support of xr,t,s Theneutrosophicpoint xr,t,s iscontainedintheneutrosophic A(xr,t,s ∈ A)ifandonlyif r<µA(x),t<σA(x),s>γA(x).

Deﬁnition2.2. Aneutrosophicset A = x,µA ,σA ,γA ina neutrosophictopologicalspace (X,T ) issaidtobeaneutrosophicneighbourhoodofaneotrosophicpoint xr,t,s,x ∈ X,if thereexistsaneutrosophicopenset B = x,µB ,σB ,γB with xr,t,s ⊆ B ⊆ A

Deﬁnition2.3. Let X and Y beneutrosophictopological spaces.Amapping f : X → Y issaidtobeaneutrosophic homeomorphismif f isbijective,neutrosophiccontinuousand neutrosophicopen.

Deﬁnition2.4. Anneutrosophictopologicalspace (X,T ) is calledaneutrosophicHausdorffspaceor T2-spaceifforany pairofdistinctneutrosophicpoints(i.e.,neutrosophicpointswith distinctsupports) xr,t,s and yu,v,w ,thereexistneutrosophicopen sets U and V suchthat xr,t,s ∈ U ,yu,v,w ∈ V and U ∧ V =0N

Deﬁnition2.5. Anneutrosophictopologicalspace (X,T ) issaid tobeneutrosophiclocallycompactifandonlyifforeveryneutrosophicpoint xr,t,s in X,thereexistsaneutrosophicopenset U ∈ T suchthat xr,t,s ∈ U and U isneutrosophiccompact,i.e., eachneutrosophicopencoverof U hasaﬁnitesubcover.

Deﬁnition2.6. Let A = x,µA(x),σA(x),γA(x) and B = y,µB (y),σB (y),γB (y) beneutrosophicsetsof X and Y respectively.Theproductoftwoneutrosophicsets A and B ina neutrosophictopologicalspace X isdeﬁnedas (A×B)(x,y)= (x,y),min(µA(x),µB (y)),min(σA(x),σB (y)), max(γA(x),γB (y)) forall (x,y) ∈ X × Y .

Deﬁnition2.7. Let f1 : X1 → Y1 and f2 : X2 → Y2.The product f1 × f2 : X1 × X2 → Y1 × Y2 isdeﬁnedby: (f1 × f2)(x1,x2)=(f1(x1),f2(x2)) ∀(x1,x2) ∈ X1 × X2

Lemma2.1. Let fi : Xi → Yi (i =1, 2) befunctionsand U , V areneutrosophicsetsof Y1, Y2,respectively,then (f1 × f2) 1(U × V )= f 1 1 (U ) × f 1 2 (V ) ∀ U × V ∈ Y1 × Y2

Deﬁnition2.8. Amapping f : X → Y isneutrosophiccontinuousiffforeachneutrosophicpoint xr,t,s in X andeachneutrosophicneighbourhood B of f (xr,t,s) in Y ,thereisaneutrosophic neighbourhood A of xr,t,s in X suchthat f (A) ⊆ B

Deﬁnition2.9. Amapping f : X → Y issaidtobeneutrosophic homeomorphismif f isbijective,neutrosophiccontinuousand neutrosophicopen.

Deﬁnition2.10. Aneutrosophictopologicalspace X iscalled aneutrosophicHausdorffspaceor T2 spaceifforanydistinct neutrosophicpoints xr,t,s and yu,v,w ,thereexistsneutrosophic opensets G1 and G2,suchthat xr,t,s ∈ G1,yu,v,w ∈ G2 and G1 ∩ G2 =0∼

Deﬁnition2.11. Aneutrosophictopologicalspace X issaidto beaneutrosophiclocallycompactiffforanyneutrosophicpoint xr,t,s in X,thereexistsaneutrosophicopenset U ∈ T suchthat xr,t,s ∈ U and U isneutrosophiccompactthatis,eachneutrosophicopencoverof U hasaﬁnitesubcover.

Proposition2.1. InaneutrosophicHausdorfftopologicalspace X,thefollowingconditionsareequivalent.

(a) X isaneutrosophiclocallycompact (b)foreachneutrosophicpoint xr,t,s in X,thereexistsaneutrosophicopenset G in X suchthat xr,t,s ∈ G and Ncl(G) isneutrosophiccompact

Proof. (a) ⇒ (b) Byhypothesisforeachneutrosophicpoint xr,t,s in X,thereexistsaneutrosophicopenset G whichisneutrosophiccompact.Since X isneutrosophicHausdorff(neutrosophiccompactsubspaceofneutrosophicHausdorffspaceisneutrosophicclosed), G isneutrosophicclosed,thus G = Ncl(G). Hence xr,t,s ∈ G and Ncl(G) isneutrosophiccompact. (b) ⇒ (a) Proofissimple.

Proposition2.2. Let X beaneutrosophicHausdorfftopological space.Then X isneutrosophiclocallycompactataneutrosophic point xr,t,s in X iffforeveryneutrosophicopenset G containing xr,t,s thereexistsaneutrosophicopenset V suchthat xr,t,s ∈ V , Ncl(V ) isneutrosophiccompactand Ncl(V ) ⊆ G

Proof. Supposethat X isneutrosophiclocallycompactata neutrosophicpoint xr,t,s.ByDeﬁnition2.11,thereexists aneutrosophicopenset G suchthat xr,t,s ∈ G and G is neutrosophiccompact.Since X isaneutrosophicHausdorff space,(neutrosophiccompactsubspaceofneutrosophicHausdorffspaceisneutrosophicclosed), G isneutrosophicclosed. Thus G = Ncl(G).Consideraneutrosophicpoint xr,t,s ∈ G. Since X isneutrosophicHausdorffspace,byDeﬁnition2.10, thereexistneutrosophicopensets C and D suchthat xr,t,s ∈ C, yu,v,w ∈ D and C ∩ D =0∼.Let V = C ∩ G.Hence V ⊆ G implies Ncl(V ) ⊆ Ncl(G)= G.Since Ncl(V ) is neutrosophicclosedand G isneutrosophiccompact,(everyneutrosophicclosedsubsetofaneutrosophiccompactspaceisneutrosophiccompact)itfollowsthat Ncl(V ) isneutrosophiccompact.Thus xr,t,s ∈ Ncl(V ) ⊆ G and Ncl(G) isneutrosophic compact.

G ⊆ A}; Ncl(A)= {G | G isaneutrosophicclosedsetinXand G ⊇ A}

Smarandache (author and editor) Collected Papers, IX 167

Florentin

TheconversefollowsfromProposition2.1(b).

Deﬁnition2.12. Let X and Y betwoneutrosophictopological spaces.Thefunction T : X × Y → Y × X deﬁnedby T (x,y)= (y,x) foreach (x,y) ∈ X × Y iscalledaswitchingmap.

Proposition2.3. Theswitchingmap T : X × Y → Y × X deﬁnedasaboveisneutrosophiccontinuous.

Wenowintroducetheconceptofaneutrosophiccompactopen topologyinthesetofallneutrosophiccontinuousfunctionsfrom aneutrosophictopologicalspace X toaneutrosophictopological space Y .

Deﬁnition2.13. Let X and Y betwoneutrosophictopological spacesandlet Y X = {f : X → Y suchthat f isneutrosophic continuous}.Wegivethisclass Y X atopologycalledtheneutrosophiccompactopentopologyasfollows:Let K = {K ∈ I X : K isneutrosophiccompacton X} and V = {V ∈ I Y : V isneutrosophicopenin Y }.Forany K ∈K and V ∈V ,let SK,V = {f ∈ Y X : f (K) ⊆ V }. Thecollectionofallsuch {SK,V : K ∈K,V ∈V} isaneutrosophicsubbasetogenerateaneutrosophictopologyontheclass Y X .Theclass Y X withthistopologyiscalledaneutrosophic compactopentopologicalspace.

3NeutrosophicEvaluationMapandExponentialMap

Wenowconsidertheneutrosophicproducttopologicalspace Y X ×X anddeﬁneaneutrosophiccontinuousmapfrom Y X ×X into Y

Deﬁnition3.1. Themapping e : Y X × X → Y deﬁnedby e(f,xr,t,s)= f (xr,t,s) foreachneutrosophicpoint xr,t,s ∈ X and f ∈ Y X iscalledtheneutrosophicevaluationmap.

Definition3.2. Let X,Y ,Z beneutrosophictopologicalspaces and f : Z × X → Y beanyfunction.Thentheinducedmap f : X → Y Z isdefinedby (f (xr,t,s))(zt,u,v )= f (zt,u,v ,xr,t,s) forneutrosophicpoint xr,t,s ∈ X and zt,u,v ∈ Z. Conversely,givenafunction f : X → Y Z ,acorresponding function f canalsobedefinedbythesamerule.

Proposition3.1. Let X beaneutrosophiclocallycompactHausdorffspace.Thentheneutrosophicevaluationmap e : Y X × X → Y isneutrosophiccontinuous.

Proof. Consider (f,xr,t,s) ∈ Y X × X,where f ∈ Y X and xr,t,s ∈ X.Let V beaneutrosophicopensetcontaining f (xr,t,s)= e(f,xr,t,s) in Y .Since X isneutrosophiclocallycompactand f isneutrosophiccontinuous,byProposition2.2,thereexistsaneutrosophicopenset U in X suchthat xr,t,s ∈ Ncl(U ) isneutrosophiccompactand f (Ncl(U )) ⊆ V Considertheneutrosophicopenset SNcl(U ),V × U in Y X × X. Clearly (f,xr,t,s) ∈ SNcl(U ),V × U .Let (g,xt,u) ∈ SNcl(U ),V × U

bearbitrary.Thus g(Ncl(U )) ⊆ V .Since xt,u ∈ U ,wehave g(xt,u) ∈ V and e(g,xt,u)= g(xt,u) ∈ V .Thus e(SNcl(U ),V × U ) ⊆ V .Hence e isneutrosophiccontinuous.

Proposition3.2. Let X and Y betwoneutrosophictopological spaceswith Y beingneutrosophiccompact.Let xr,t,s beany neutrosophicpointin X and N beaneutrosophicopensetinthe neutrosophicproductspace X ×Y containing {xr,t,s}×Y .Then thereexistssomeneutrosophicneighbourhood W of xr,t,s in X suchthat {xr,t,s}× Y ⊆ W × Y ⊆ N .

Proposition3.3. Let Z beaneutrosophiclocallycompact Hausdorffspaceand X,Y bearbitraryneutrosophictopological spaces.Thenamap f : Z × X → Y isneutrosophiccontinuous iff f : X → Y Z isneutrosophiccontinuous,where f isdeﬁned bytherule (f (xr,t,s))(zt,u,v )= f (zt,u,v ,xr,t,s)

Proposition3.4. Let X and Z beaneutrosophiclocallycompact Hausdorffspaces.Thenforanyneutrosophictopologicalspace Y ,thefunction E : Y Z×X → (Y Z )X deﬁnedby E(f )= f (that is E(f )(xr,t,s)(zt,u,v )= f (zt,u,v ,xr,t,s)=(f (xr,t,s)(zt,u,v ))) forall f : Z × X → Y isaneutrosophichomeomorphism.

Proof. (a)Clearly E isonto. (b)For E tobeinjective,let E(f )= E(g) for f,g : Z × X → Y .Thus f = g,where f and g aretheinducedmapof f and g,respectively.Nowforanyneutrosophicpoint xr,t,s in X andanyneutrosophicpoint zt,u,v in Z, f (zt,u,v ,xr,t,s)= (f (xr,t,s)(zt,u,v ))=(g(xr,t,s)(zt,u,v ))= g(zt,u,v ,xr,t,s) Thus f = g.

(c)Forprovingtheneutrosophiccontinuityof E,considerany neutrosophicsubbasisneighbourhood V of f in (Y Z )X ,i.e V isoftheform SK,W where K isaneutrosophiccompact subsetof X and W isneutrosophicopenin Y Z .Without lossofgenerality,wemayassumethat W = SL,U ,where L isaneutrosophiccompactsubsetof Z and U isaneutrosophicopensetin Y .Then f (K) ⊆ SL,U = W andthis impliesthat f (K)(L) ⊆ U .Thusforanyneutrosophicpoint xr,t,s in K andforeveryneutrosophicpoint zt,u,v in L,we have (f (xr,t,s))(zt,u,v ) ∈ U ,thatis f (zt,u,v ,xr,t,s) ∈ U andtherefore f (L × K) ⊆ U .Nowsince L isaneutrosophiccompactin Z and K isaneutrosophiccompactin X, L × K isalsoaneutrosophiccompactin Z × X[7]and since U isaneutrosophicopensetin Y ,weconcludethat f ∈ SL×K,U ⊆ Y Z×X .Weassertthat E(SL×K,U ) ⊆ SK,W . Let g ∈ SL×K,U bearbitrary.Thus g(L × K) ⊆ U , i.e g(zt,u,v ,xr,t,s)=(g(xr,t,s))(zt,u,v ) ∈ U forallneutrosophicpoints zt,u,v ∈ L ⊆ Z andforeveryneutrosophicpoint xr,t,s ∈ L ⊆ X.So (g(xr,t,s))(L) ⊆ U foreveryneutrosophicpoint xr,t,s ∈ K ⊆ X ,thatis (g(xr,t,s)) ∈ SL,U = W foreveryneutrosophicpoints xr,t,s ∈ K ⊆ X,thatis g(xr,t,s) ∈ SL,U = W foreveryneutrosophicpoint xr,t,s ∈ K ⊆ U .Hencewehave g(K) ⊆ W ,thatis g = E(g) ∈ SK,W forany g ∈ SL×K,U .

Florentin Smarandache (author and editor) Collected Papers, IX 168

Thus E(SL×K,U ) ⊆ SK,W .Thisprovesthat E isaneutrosophiccontinuous. (d)Forprovingtheneutrosophiccontinuityof E 1,weconsiderthefollowingneutrosophicevaluationmaps: e1 : (Y Z )X × X → Y Z deﬁnedby e1(f,xr,t,s)= f (xr,t,s) where f ∈ (Y Z )X and xr,t,s isanyneutrosophicpointin X and e2 : Y Z × Z → Y deﬁnedby e2(g,zt,u,v )= g(zt,u,v ), where g ∈ Y Z and zt,u,v isaneutrosophicpointin Z.Let denotethecompositionofthefollowingneutrosophiccontinuousfunctions ψ :(Z × X ) × (Y Z )X T −→ (Y Z )X × (Z × X ) i×t

(Y Z )X × (X × Z)

((Y Z )X × X ) × Z e1 ×iZ

(Y Z ) × Z e2 −→ Y ,where i,iZ denotetheneutrosophicidentitymapson (Y Z )X and Z,respectivelyand T,t denote theswitchingmaps.Thus :(Z × X ) × (Y Z )X → Y ,thatis ∈ Y (Z×X)×(Y Z ) X .Weconsiderthemap E : Y (Z×X)×(Y Z ) X → (Y (Z×X))(Y Z ) X (asdeﬁnedinthe statementoftheProposition3.4infactitis E).So E(ψ): (Y Z )X → Y (Z×X).Nowforanyneutrosophicpoints zt,u,v ∈ Z,xr,t,s ∈ X and f ∈ Y (Z×X),againwehave that (E(ψ) ◦ E)(f )(zt,u,v ,xr,t,s)= f (zt,u,v ,xr,t,s);hence E(ψ) ◦ E=identity.Similarlyforany g ∈ (Y Z )X andneutrosophicpoints xr,t,s ∈ X,zt,u,v ∈ Z,sowehavethat (E ◦ E(ψ))(g)(xr,t,s,zt,u,v )=(g(xr,t,s))(zt,u,v );hence E ◦ E(ψ)=identity.Thus E isaneutrosophichomeomorphism.

References

[1]K.Atanassov,Intuitionisticfuzzysets,in:V. Sgurev,Ed., VIIITKR’sSession,Soﬁa(June1983CentralSci.and Techn.Library,Bulg.AcademyofSciences.,1984)

[2]K.Atanassov,Intuitionisticfuzzysets, FuzzySetsandSystems., 20(1986),87-96.

[3]K.Atanassov,ReviewandnewresultsonIntuitionistic fuzzysets,PreprintIM-MFAIS-1-88,Soﬁa.,1988.

[4]K.AtanassovandS.Stoeva,Intuitionisticfuzzysets,in: PolishSyrup.onInterval&FuzzyMathematics,Poznan.,(August1983),23-26.

[5]C.L.Chang,Fuzzy topologicalspaces, J.Math.Anal. Appl., 24(1968),182-190.

[6]F.T.Christoph,Quotientfuzzytopologyandlocalcompactness, J.Math.Anal.Appl., 57(1977),497-504.

[7]S.DangandA.Behera,OnFuzzycompactopentopology,FuzzySetsandSystem., 80(1996),377-381.

[8]R.DhavaseelanandS.Jafari,GeneralizedNeutrosophic closedsets,(Submitted)

[9]R.Dhavaseelan,E.RojaandM.K.Uma,OnIntuitionistic FuzzyEvaluationMap,IJGT,(5)(1-2)2012,55-60.

[10]A.A.SalamaandS.A.Alblowi,NeutrosophicSetandNeutrosophicTopologicalSpaces, IOSRJournalofMathematics,Volume3,Issue4(Sep-Oct.2012),PP31-35

Deﬁnition3.3. Themap E inProposition3.4iscalledtheexponentialmap.

AseasyconsequenceofProposition3.4isasfollows.

Proposition3.5. Let X,Y,Z beneutrosophiclocallycompact Hausdorffspaces.Thenthemap N : Y X × Z Y → Z X deﬁned by N (f,g)= g ◦ f isneutrosophiccontinuous.

Proof. Considerthefollowingcompositions:

,respectivelyand

anexponentialmap

)= g(f (xr,t,s))

[11] F. Smarandache, Neutrosophy and Neutrosophic Logic First International Conference on Neutrosophy, Neutrosophic Logic, Set, Probability, and Statistics University of New Mexico, Gallup, NM 87301, USA(2002) , smarand@unm.edu

[12]F.Smarandache.AUnifyingFieldinLogics:Neutrosophic Logic.Neutrosophy,NeutrosophicSet,NeutrosophicProbability.AmericanResearchPress,Rehoboth,NM,1999.

[13]C.K.Wong,Fuzzypointsandlocalpropertiesoffuzzy topologies, J.Math.Anal.Appl., 46(1974),316-328.

[14]C.K.Wong,Fuzzytopology:productandquotienttheorems, J.Math.Anal.Appl., 45(1974),512-521.

[15]L.A.Zadeh.Fuzzysets, Inform.andControl., 8(1965), 338-353.

−−→

= −→

−−−−→

= −→ Z Y ×

Y X × X

X × Y X × Z Y T −→ Y X × Z Y × X t×iX −−−→ Z Y × Y X × X

(

) i×e2 −−−→ Z Y × Y e2 −→ Z,where T,t denotetheswitchingmaps, iX ,i denotetheneutrosophicidentityfunctionson X and Z Y

e2 denotestheneutrosophicevaluationmaps.Let ϕ = e2 ◦ (i × e2) ◦ (t × iX ) ◦ T .Byproposition3.4,wehave

E : Z X×Y X ×ZY → (Z X )Y X ×ZY .Since ϕ ∈ Z X×Y X ×ZY , E(ϕ) ∈ (Z X )Y X ×ZY .Let N = E(ϕ) thatis, N : Y X × Z Y → Z X isneutrosophiccontinuous.For f ∈ Y X ,g ∈ Z Y andforanyneutrosophicpoint xr,t,s ∈ X,it easytoseethat N (f,g)(xr,t,s

Florentin Smarandache (author and editor) Collected Papers, IX 169

On Neutrosophic Semi-Supra Open Set and Neutrosophic Semi-Supra Continuous Functions

R. Dhavaseelan, M. Parimala, S. Jafari, F. Smarandache

R. Dhavaseelan, M. Parimala, S. Jafari, Florentin Smarandache (2017). On Neutrosophic Semi-Supra Open Set and Neutrosophic Semi-Supra Continuous Functions. Neutrosophic Sets and Systems 16, 39-43

Abstract: Inthispaper,weintroduceandinvestigateanewclass ofsetsandfunctionsbetweentopologicalspacecalledneutrosophic

semi-supraopensetandneutrosophicsemi-supraopencontinuous functionsrespectively.

Keywords: Supra topological spaces; neutrosophic supra-topological spaces; neutrosophic semi-supra open set.

1IntroductionandPreliminaries

IntuitionisticfuzzysetisdeﬁnedbyAtanassov[2]asageneralizationoftheconceptoffuzzysetgivenbyZadesh[14].Using thenotationofintuitionisticfuzzysets,Coker[3]introducedthe notionofanintuitionisticfuzzytopologicalspace.Thesupra topologicalspacesandstudied s-continuousfunctionsand s∗ continuousfunctionswereintroducedbyA.S.Mashhour[6]in 1993.In1987,M.E.AbdEl-Monsefetal.[1]introducedthe fuzzysupratopologicalspacesandstudiedfuzzysupracontinuousfunctionsandobtainedsomepropertiesandcharacterizations.In1996,KeunMin[13]introducedfuzzy s-continuous, fuzzy s-openandfuzzy s-closedmapsandestablishedanumberofcharacterizations.In2008,R.Devietal.[4]introduced theconceptofsupra α-openset,andin1983,A.S.Mashhour etal.introducedthenotionofsupra-semiopenset,suprasemicontinuousfunctionsandstudiedsomeofthebasicpropertiesfor thisclassoffunctions.In1999,NeclaTuran[11]introducedthe conceptofintuitionisticfuzzysupratopologicalspace.Theconceptofintuitionisticfuzzysemi-supraopensetwasintroduced byParimalaandIndirani[7].AftertheintroductionoftheconceptsofneutrosophyandaneutrosophicsebyF.Smarandache [[9],[10]],A.A.SalamaandS.A.Alblowi[8]introducedthe conceptsofneutrosophiccrispsetandneutrosophictopological spaces.

Thepurposeofthispaperistointroduceandinvestigateanew classofsetsandfunctionsbetweentopologicalspacecalledneutrosophicsemi-supraopensetandneutrosophicsemi-supraopen continuousfunctions,respectively.

Deﬁnition1.1. Let T , I, F berealstandardornonstandardsubsetsof ]0 , 1+[,with supT = tsup ,infT = tinf supI = isup ,infI = iinf supF = fsup ,infF = finf

n sup = tsup + isup + fsup n inf = tinf + iinf + finf . T , I, F areneutrosophiccomponents.

Definition1.2. Let X be anonemptyfixedset.Aneutrosophicset[brieflyNS] A isanobjecthavingtheform A = { x,µA (x),σA (x),γA (x) : x ∈ X},where µA (x),σA (x) and γA (x) representthedegreeofmembership function(namely µA (x)),thedegreeofindeterminacy(namely σA (x))andthedegree ofnonmembership(namely γA (x))respectivelyofeachelement x ∈ X totheset A.

Remark1.1. (1)Aneutrosophicset A = { x,µA (x),σA (x),γA (x) : x ∈ X} canbeidentiﬁedto anorderedtriple µA ,σA ,γA in ]0 , 1+[ on X

(2)Forthesakeofsimplicity,weshallusethesymbol A = µA ,σA ,γA fortheneutrosophicset A = { x, µA (x),σA (x),γA (x) : x ∈ X}

Deﬁnition1.3. Let X bea nonemptysetand theneutrosophic sets A and B intheform A = { x,µA (x),σA (x),γA (x) : x ∈ X}, B = { x,µB (x), σB (x),γB (x) : x ∈ X}.Then

(a) A ⊆ B iff µA (x) ≤ µB (x), σA (x) ≤ σB (x) and γA (x) ≥ γB (x) for all x ∈ X;

(b) A = B iff A ⊆ B and B ⊆ A;

(d) A ∩ B = { x,µA (x) ∧ µB (x),σA (x) ∧ σB (x),γA (x) ∨ γB (x) : x ∈ X};

Smarandache (author and editor) Collected Papers, IX 170

Florentin

(e) A ∪ B = { x,µA (x) ∨ µB (x),σA (x) ∨ σB (x),γA (x) ∧ γB (x) : x ∈ X};

(f)[]A = { x,µA (x),σA (x), 1 µA (x) : x ∈ X}; (g) A = { x, 1 γA (x),σA (x),γA (x) : x ∈ X}

Deﬁnition1.4. Let {Ai : i ∈ J } beanarbitraryfamilyofneutrosophicsetsin X.Then

(a) Ai = { x, ∧µAi (x), ∧σAi (x), ∨γAi (x) : x ∈ X}; (b) Ai = { x, ∨µAi (x), ∨σAi (x), ∧γAi (x) : x ∈ X}

Sinceourmainpurposeistoconstructthetoolsfordeveloping neutrosophictopologicalspaces,wemustintroducetheneutrosophicsets 0N and 1N inXasfollows:

Deﬁnition1.5. 0N = { x, 0, 0, 1 : x ∈ X} and 1N = { x, 1, 1, 0 : x ∈ X}

Deﬁnition1.6. [5]Aneutrosophictopology(NT)onanonempty set X isafamily T ofneutrosophicsetsin X satisfyingthefollowingaxioms:

(i) 0N , 1N ∈ T ,

(ii) G1 ∩ G2 ∈ T forany G1,G2 ∈ T ,

(iii) ∪Gi ∈ T forarbitraryfamily {Gi | i ∈ Λ}⊆ T .

Deﬁnition1.7. [5]Let A beaneutrosophicsetinaneutrosophic topologicalspace X.Then Nint(A)= {G | G isaneutrosophicopensetinXand G ⊆ A} iscalledtheneutrosophicinteriorof A; Ncl(A)= {G | G isaneutrosophicclosedsetinXand G ⊇ A} iscalledtheneutrosophicclosureof A

Deﬁnition1.8. Let X beanonemptyset.If r,t,s berealstandardornonstandardsubsetsof ]0 , 1+[,thentheneutrosophic set xr,t,s iscalledaneutrosophicpoint(inshortNP)in X given by xr,t,s (xp)= (r,t,s), if x = xp (0, 0, 1), if x = xp for xp ∈ X iscalledthesupportof xr,t,s ,where r denotesthedegreeofmembershipvalue,t denotesthedegreeofindeterminacy and s isthedegreeofnon-membershipvalueof xr,t,s .

Nowweshalldeﬁnetheimageandpreimageofneutrosophic sets.Let X and Y betwononemptysetsand f : X → Y bea function.

Deﬁnition1.9. [5]

(a)If B = { y,µB (y),σB (y),γB (y) : y ∈ Y } isaneutrosophicsetin Y ,thenthepreimageof B under f ,denotedby f 1(B),istheneutrosophicsetin X deﬁnedby f 1(B)= { x,f 1(µB )(x),f 1(σB )(x),f 1(γB )(x) : x ∈ X}

(b)If A = { x,µA (x),σA (x),γA (x) : x ∈ X} isaneutrosophicsetin X,thentheimageof A under f ,denotedby f (A),istheneutrosophicsetin Y deﬁnedby f (A)= { y,f (µA )(y),f (σA )(y), (1 f (1 γA ))(y) : y ∈ Y }.where f (µA )(y)= supx∈f 1 (y) µA (x), if f 1(y) = ∅, 0, otherwise, f (σA )(y)= supx∈f 1 (y) σA (x), if f 1(y) = ∅, 0, otherwise, (1 f (1 γA ))(y)= inf x∈f 1 (y) γA (x), if f 1(y) = ∅, 1, otherwise,

Forthesakeofsimplicity,letususethesymbol f (γA ) for 1 f (1 γA )

Corollary1.1. [5]Let A , Ai(i ∈ J ) beneutrosophicsetsin X, B, Bi(i ∈ K) beneutrosophicsetsin Y and f : X → Y a function.Then

(a) A1 ⊆ A2 ⇒ f (A1) ⊆ f (A2), (b) B1 ⊆ B2 ⇒ f 1(B1) ⊆ f 1(B2), (c) A ⊆ f 1(f (A)) { Iffisinjective,then A = f 1(f (A)) } , (d) f (f 1(B)) ⊆ B { Iffissurjective,then f (f 1(B))= B }, (e) f 1( Bj )= f 1(Bj ), (f) f 1( Bj )= f 1(Bj ), (g) f ( Ai)= f (Ai), (h) f ( Ai) ⊆ f (Ai) { Iffisinjective,then f ( Ai)= f (Ai)}, (i) f 1(1N )=1N , (j) f 1(0N )=0N , (k) f (1N )=1N ,iffissurjective (l) f (0N )=0N , (m) f (A) ⊆ f (A),iffissurjective, (n) f 1(B)= f 1(B).

Florentin Smarandache (author and editor) Collected Papers, IX 171

Deﬁnition2.1. Aneutrosophicset A inaneutrosophictopologicalspace (X,T ) iscalled

1)aneutrosophicsemiopenset(NSOS)if A ⊆ Ncl(Nint(A)).

2)aneutrosophic α openset (NαOS) if A ⊆ Nint(Ncl(Nint(A)))

3)aneutrosophicpreopenset(NPOS)if A ⊆ Nint(Ncl(A)).

4)aneutrosophicregularopenset(NROS)if A = Nint(Ncl(A))

5)aneutrosophicsemipreopenor β openset (NβOS) if A ⊆ Ncl(Nint(Ncl(A))).

Aneutrosophicset A iscalledaneutrosophicsemiclosedset, neutrosophic α closedset,neutrosophicpreclosedset,neutrosophicregularclosedsetandneutrosophic β closedset,respectively(NSCS,NαCS,NPCS,NRCSandNβCS,resp),ifthe complementof A isaneutrosophicsemiopenset,neutrosophic α-openset,neutrosophicpreopenset,neutrosophicregularopen set,andneutrosophic β-openset,respectively.

Deﬁnition2.2. Let (X,T ) baaneutrosophictopologicalspace. Aneutrosophicset A iscalledaneutrosophicsemi-supraopenset (brieﬂyNSSOS)if A ⊆ s Ncl(s Nint(A)).Thecomplementof aneutrosophicsemi-supraopensetiscalledaneutrosophicsemisupraclosedset.

Proposition2.1. Everyneutrosophicsupraopensetisneutrosophicsemi-supraopenset.

Proof. Let A beaneutrosophicsupraopensetin (X,T ).Since A ⊆ s Ncl(A),weget A ⊆ s Ncl(s Nint(A)).Then s Nint(A) ⊆ s Ncl(s Nint(A)).Hence A ⊆ s Ncl(s Nint(A)).

TheconverseofProposition2.1.,neednotbetrueasshown inExample2.1.

Example2.1. Let X = {a,b}.Deﬁnetheneutrosophicsets A, B and C in X asfollows: A = x, ( a 0 2 , b 0 4 ), ( a 0 2 , b 0 4 ), ( a 0 5 , b 0 6 ) , B = x, ( a 0 6 , b 0 2 ), ( a 0 6 , b 0 2 ), ( a 0 3 , b 0 4 ) and C = x, ( a 0 3 , b 0 4 ), ( a 0 3 , b 0 4 ), ( a 0 4 , b 0 4 ) .Thenthefamilies T = {0N , 1N ,A,B,A ∪ B} isneutrosophictopologyon X. Thus, (X,T ) isaneutrosophictopologicalspace.Then C is calledneutrosophicsemi-supraopenbutnotneutrosophicsupra openset.

Proposition2.2. Everyneutrosophic α-supraopenisneutrosophicsemi-supraopen

Proof. Let A beaneutrosophic α-supraopenin (X,T ),then A ⊆ s Nint(s Ncl(s Nint(A))).Itisobviousthat s Nint(s Ncl(s Nint(A))) ⊆ s Ncl(s Nint(A)).Hence A ⊆ s Ncl(s Nint(A))

TheconverseofProposition2.2.,neednotbetrueasshown inExample2.2.

Example2.2. Let X = {a,b}.Deﬁnetheneutrosophicsets A, B and C in X asfollows: A = x, ( a 0 2 , b 0 3 ), ( a 0 2 , b 0 3 ), ( a 0 5 , b 0 3 ) , B = x, ( a 0 1 , b 0 2 ), ( a 0 1 , b 0 2 ), ( a 0 6 , b 0 5 ) and C = x, ( a 0 2 , b 0 3 ), ( a 0 2 , b 0 3 ), ( a 0 2 , b 0 3 ) .Thenthefamilies T = {0N , 1N ,A,B,A ∪ B} isneutrosophictopologyon X.Thus, (X,T ) isaneutrosophictopologicalspace.Then C iscalledneutrosophicsemi-supraopenbutnotneutrosophic α-supraopenset.

Proposition2.3. Everyneutrosophicregularsupraopensetis neutrosophicsemi-supraopenset

Proof. Let A beaneutrosophicregularsupraopensetin (X,T ) Then A ⊆ (s Ncl(A)).Hence A ⊆ s Ncl(s Nint(A))

TheconverseofProposition2.3.,neednotbetrueasshown inExample2.3.

Example2.3. Let X = {a,b}.Deﬁnetheneutrosophicsets A, B and C in X asfollows: A = x, ( a 0 2 , b 0 3 ), ( a 0 2 , b 0 3 ), ( a 0 5 , b 0 3 ) , B = x, ( a 0 1 , b 0 2 ), ( a 0 1 , b 0 2 ), ( a 0 6 , b 0 5 ) and C = x, ( a 0 2 , b 0 3 ), ( a 0 2 , b 0 3 ), ( a 0 2 , b 0 3 ) .Thenthefamilies T = {0N , 1N ,A,B,A ∪ B} isneutrosophictopologyonX. Thus, (X,T ) isaneutrosophictopologicalspace.Then C is neutrosophicsemi-supraopenbutnotneutrosophicregular-supra openset.

Deﬁnition2.3. Theneutrosophicsemi-supraclosureofaset A is denotedby semi s Ncl(A)= { G:Gisaneutrosophicsemisupraopensetin X and G ⊆ A} andtheneutrosophicsemisuprainteriorofaset A isdenotedby semi s Nint(A)= {G :Gisaneutrosophicsemi-supraclosedsetin X and G ⊇ A}

Remark2.1. Itisclearthat semi s Nint(A) isaneutrosophic semi-supraopensetand semi s Ncl(A) isaneutrosophicsemisupraclosedset.

Proposition2.4. i) semi s Nint(A) =semis-Ncl (A) ii) semi s Ncl(A) =semis-int (A) iii)if A ⊆ B then semi s Ncl(A) ⊆ semi s Ncl(B) and semi s Nint(A) ⊆ semi s Nint(B)

Proof. Itisobvious.

Proposition2.5. (i)Theintersectionofaneutrosophicsupra opensetandaneutrosophicsemi-supraopensetisaneutrosophicsemi-supraopenset.

2MainResults

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Florentin

(ii)Theintersectionofaneutrosophicsemi-supraopensetand aneutrosophicpre-supraopensetisaneutrosophicpre-supra openset.

Proof. Itisobvious.

Deﬁnition2.4. Let (X,T ) and (Y,S) betwoneutrosophicsemisupraopensetsand R beaassociatedsupratopologywith T .A map f :(X,T ) → (Y,S) iscalledneutrosophicsemi-supra continuousmapiftheinverseimageofeachneutrosophicopen setin Y isaneutrosophicsemi-supraopenin X

Proposition2.6. Everyneutrosophicsupracontinuousmapis neutrosophicsemi-supracontinuousmap.

Proof. Let f :(X,T ) → (Y,S) beaneutrosophicsupracontinuousmapand A isaneutrosophicopensetin Y .Then f 1(A) isaneutrosophicopensetin X.Since R isassociatedwith T Then T ⊆ R.Therefore f 1(A) isaneutrosophicsupraopen setin X whichisaneutrosophicsupraopensetin X.Hence f is aneutrosophicsemi-supracontinuousmap.

Remark2.2. Everyneutrosophicsemi-supracontinuousmap neednotbeneutrosophicsupracontinuousmap.

Proposition2.7. Let (X,T ) and (Y,S) betwoneutrosophic topologicalspacesand R beaassociatedneutrosophicsupra topologywith T .Let f beamapfrom X into Y .Thenthe followingareequivalent.

i)fisaneutrosophicsemi-supracontinuousmap.

ii)Theinverseimageofaneutrosophicclosedsetsin Y isa neutrosophicsemiclosedsetin X

iii)Semi-s-Ncl(f 1(A)) ⊆ f 1(Ncl(A)) foreveryneutrosophicset A in Y

iv) f (semi s Ncl(A)) ⊆ Ncl(f (A)) foreveryneutrosophic setAinX.

v) f 1(Nint(B)) ⊆ semi s Nint(f 1(B)) foreveryneutrosophicset B in Y .

Proof. (i) ⇒ (ii): Let A beaneutrosophicclosedsetin Y Then A isneutrosophicopenin Y ,Thus f 1(A)= f 1(A) is neutrosophicsemi-openin X.Itfollowsthat f 1(A) isaneutrosophicsemi-sclosedsetof X. (ii) ⇒ (iii): Let A beanysubsetof X.Since Ncl(A) isneutrosophicclosedin Y thenitfollowsthat f 1(Ncl(A)) isneutrosophicsemi-sclosedin X.Therefore, f 1(Ncl(A))= semi s Ncl(f 1(Ncl(A)) ⊇ semi s Ncl(f 1(A)) (iii) ⇒ (iv): Let A beanysubsetof X.By(iii)weobtain f 1(Ncl(f ((A))) ⊇ semi s Ncl(f 1(f (A))) ⊇ semi s Ncl(A) andhence f (semi s Ncl(A)) ⊆ Ncl(f (A)) (iv) ⇒ (v): Let f (semi s Ncl(A)) ⊆ f (Ncl(A) for everyneutrosophicset A in X.Then semi s Ncl(A)) ⊆ f 1(Ncl(f (A)). semi s Ncl(A) ⊇ f 1(Ncl(f (A)))

and semi s Nint(A) ⊇ f 1(Nint(f (A))).Then semi s Nint(f 1(B)) ⊇ f 1(Nint(B)) Therefore f 1(Nint(B)) ⊆ s Nint(f 1(B)) forevery B in Y (v) ⇒ (i): Let A beaneutrosophicopensetin Y Therefore f 1(Nint(A)) ⊆ semi s Nint(f 1(A)),hence f 1(A) ⊆ semi s Nint(f 1(A)).Butweknowthat semi s Nint(f 1(A)) ⊆ f 1(A), then f 1(A)= semi s Nint(f 1(A)).Therefore f 1(A) isaneutrosophicsemi-sopenset.

Proposition2.8. Ifamap f :(X,T ) → (Y,S) isaneutrosophic semi-s-continuousand g :(Y,S) → (Z,R) isneutrosophiccontinuous,Then g ◦ f isneutrosophicsemi-s-continuous.

Proof. Obvious.

Proposition2.9. Letamap f :(X,T ) → (Y,S) beaneutrosophicsemi-supracontinuousmap,thenoneofthefollowing holds

i) f 1(semi s Nint(A)) ⊆ Nint(f 1(A)) foreveryneutrosophicset A in Y

ii) Ncl(f 1(A)) ⊆ f 1(semi s Ncl(A)) foreveryneutrosophicset A in Y

iii) f (Ncl(B)) ⊆ semi s Ncl(f (B)) foreveryneutrosophic set B in X

Proof. Let A beanyneutrosophicopensetof Y ,thencondition (i)issatisfied,then f 1(semi s Nint(A)) ⊆ Nint(f 1(A)) Weget, f 1(A) ⊆ Nint(f 1(A)).Therefore f 1(A) isaneutrosophicsupraopenset.Everyneutrosophicsupraopensetis aneutrosophicsemisupraopenset.Hence f isaneutrosophic semi-s-continuousfunction.Ifcondition(ii)issatisfied,thenwe caneasilyprovethat f isaneutrosophicsemi-scontinuousfunctionifcondition(iii)issatisfied,and A isanyneutrosophicopen setof Y ,then f 1(A) isasetin X and f (Ncl(f 1(A)) ⊆ semi s Ncl(f (f 1(A))).Thisimplies f (Ncl(f 1(A))) ⊆ semi s Ncl(A).Thisisnothingbutcondition(ii).Hence f isaneutrosophicsemi-s-continuousfunction.

References

[1]M.E.AbdEl-monsefandA.E.Ramadan,Onfuzzysupra topologicalspaces, IndianJ.PureandAppl.Math.no.4, 18(1987),322–329

[2]K.T.Atanassov,Intuitionisticfuzzysets, Fuzzysetsand systems, 20(1986),87-96.

[3]D.Coker,AnintroductiontoIntuitionisticfuzzytopological spaces, Fuzzysetsandsystems, 88(1997)81–89

[4]R.Devi,S.SampathkumarandM.Caldas,Onsupra α-open setsandsupra α-continuousfunctions, GeneralMathematics, Vol.16,Nr.2(2008),77-84.

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[5]R.DhavaseelanandS.Jafari,GeneralizedNeutrosophic closedsets(submitted).

[6]A.S.Mashhour,A.A.Allam,F.H.Khedr,Onsupratopologicalspaces, IndianJ.PureandAppl.Math. no.4,14 (1983),502-510

[7]M.ParimalaandC.Indirani,OnIntuitionisticFuzzysemisupraopensetandintuitionisticfuzzysemi-supracontinuousfunctions, ProcediaComputerScience,47(2015) 319-325.

[8]A.A.SalamaandS.A.Alblowi,Neutrosophicsetandneutrosophictopologicalspaces, IOSRJournalofMathematics,Volume3,Issue4(Sep-Oct.2012),31-35

[9]F.Smarandache,NeutrosophyandNeutrosophicLogic, First International Conference on Neutrosophy, Neutrosophic Logic , Set, Probability, and Statistics University of New Mexico, Gallup, NM 87301, USA(2002), smarand@unm.edu

[10]F.Smarandache.AUnifyingFieldinLogics:Neutrosophic Logic.Neutrosophy,NeutrosophicSet,NeutrosophicProbability.AmericanResearchPress,Rehoboth,NM,1999.

[11]N.Turanl,Onintuitionisticfuzzysupratopologicalspaces, Internationalconferenceonmodelingandsimulation, spain, volII,(1999)69-77.

[12]N.Turanl,Anoverviewofintuitionisticfuzzysupratopologicalspaces, HacettepeJournalofmathematicsand statistics,Volume32(2003),17-26.

[13]WonKeunMin,Onfuzzys-continuousfunctions, Kangweon-Kyungki Math.J. no.1,4(1996),77–82.

[14]L.A.Zadeh,Fuzzysets,Informationandcontrol,8(1965), 338–353.

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Neutrosophic Regular Filters and Fuzzy Regular Filters in Pseudo-BCI Algebras

Zhang, Yingcang Ma, Florentin Smarandache

Abstract. Neutrosophic set is a new mathematical tool for handling problems involving imprecise, indeterminacy and inconsistent data. Pseudo-BCI algebra is a kind of non-classical logic algebra in close connection with various non-commutative fuzzy logics. Recently, we applied neutrosophic set theory to pseudo-BCI algebras. In this paper, we study neutrosophic filters in pseudo-BCI algebras. The concepts of neutrosophic regular filter, neutrosophic closed filter and fuzzy regular

filter in pseudo-BCI algebras are introduced, and some basic properties are discussed. Moreover, the relationships among neutrosophic regular filter, fuzzy filters and anti-grouped neutrosophic filters are presented, and the results are proved: a neutrosophic filter (fuzzy filter) is a neutrosophic regular filter (fuzzy regular filter), if and only if it is both a neutrosophic closed filter (fuzzy closed filter) and an anti-grouped neutrosophic filter (fuzzy anti-grouped filter).

Keywords: Neutrosophic set, Pseudo-BCI algebra, Neutrosophic Filter, Neutrosophic Regular Filter, Fuzzy Regular Filter.

1 Introduction

In 1998, Florentin Smarandache introduced the concept of a neutrosophic set from a philosophical point of view (see [16, 17, 18]). The neutrosophic set is a powerful general formal framework that generalizes the concept of fuzzy set and intuitionistic fuzzy set. In this paper we work with special neutrosophic sets, they are called single valued neutrosophic set (see [21]). The neutrosophic set theory is applied to many scientific fields (see [18, 19, 20]), and also applied to algebraic structures (see [1, 2, 15, 19]), it is similar to the applications of fuzzy set (soft set, rough set) theory in algebraic structures (see [11, 14, and 23]).

In 2008, W. A. Dudek and Y. B. Jun [3] introduced the notion of pseudo-BCI algebra as a generalization of BCI algebra, it is also as a generalization of pseudo-BCK algebra (which is close connection with various noncommutative fuzzy logic formal systems, see [4, 24, 26, 27, 28, and 32]). For non-classical logic algebra systems, the theory of filters (ideals) plays an important role (see [9, 12, 13, 25, and 30]). In [7], the notion of pseudo-BCI filter (ideal) of pseudo-BCI algebras is introduced. In 2009, some special pseudo-BCI filters (ideals) are discussed in [10]. Since then, some articles related filters of pseudoBCI algebras are published (see [29, 31, 33, and 34]).

Recently, we applied neutrosophic set theory to pseudo -BCI algebras in [35]. This paper we further study on the applications of neutrosophic sets to pseudo-BCI algebras. We introduce the new concepts of neutrosophic regular fil-

ter, neutrosophic closed filter and fuzzy regular filter in pseudo-BCI algebras, and investigate their basic properties and present relationships among neutrosophic regular filters, anti-grouped neutrosophic filter and fuzzy filters.

Note that, the notion of pseudo-BCI algebra in this paper is a dual of the original definition in [3], so the notion of filter is a dual of (pseudo-BCI) ideal in [7, 10].

2 Some basic concepts and properties

2.1 On neutrosophic sets

Definition 2.1[17, 18, 19] Let X be a space of points (objects), with a generic element in X denoted by x. A neutrosophic set A in X is characterized by a truth-membership function TA(x), an indeterminacy-membership function IA(x), and a falsity-membership function FA(x). The functions TA(x), IA(x), and FA(x) are real standard or non-standard subsets of ] 0, 1+[. That is, TA(x): X→ ] 0, 1+[, IA(x): X→ ] 0, 1+[, and FA(x): X→ ] 0, 1+[. Thus, there is no restriction on the sum of TA(x), IA(x), and FA(x), so 0 ≤ supTA(x) + supIA(x) + supFA(x) ≤ 3+

Definition 2.2[21] Let X be a space of points (objects) with generic elements in X denoted by x. A simple valued neutrosophic set A in X is characterized by truthmembership function TA(x), indeterminacy-membership function IA(x), and falsity-membership function FA(x). Then, a simple valued neutrosophic set A can be denoted by A={〈x , TA(x), IA(x), FA(x) 〉 | x∈X},

Xiaohong

Florentin Smarandache (author and editor) Collected Papers, IX 175

Xiaohong Zhang, Yingcang Ma, Florentin Smarandache (2017). Neutrosophic Regular Filters and Fuzzy Regular Filters in Pseudo-BCI Algebras. Neutrosophic Sets and Systems 17, 10-15

where TA(x), IA(x), FA(x)∈[0, 1] for each point x in X Therefore, the sum of TA(x), IA(x), and FA(x) satisfies the condition 0 ≤ TA(x) + IA(x) + FA(x) ≤ 3.

Definition 2.3[21] The complement of a simple valued neutrosophic set A is denoted by Ac and is defined as (∀x∈X) ()(),()1(),()(). cccAAA AAA TxFxIxIxFxTx ==−=

Then

Ac={〈x , FA(x), 1 IA(x), TA(x)〉 | x∈X}.

Definition 2.4[21] A simple valued neutrosophic set A is contained in the other simple valued neutrosophic set B, denote A⊆B, if and only if TA(x)≤ TB(x), IA(x) ≤ IB(x), FA(x)≥ FB(x) for any x in X

Definition 2.5[21] Two simple valued neutrosophic sets A and B are equal, written as A = B, if and only if A⊆B and B⊆A.

For convenience, “simple valued neutrosophic set” is abbreviated to “neutrosophic set” later.

Definition 2.6[21] The union of two neutrosophic sets A and B is a neutrosophic set C, written as C=A∪B, whose truth-membership, indeterminacy-membership and falsitymembership functions are related to those of A and B by

TC(x)=max(TA(x), TB(x)), IC(x)=max(IA(x), IB(x)), FC(x)=min(FA(x), FB(x)), ∀x∈X

Definition 2.7[21] The intersection of two neutrosophic sets A and B is a neutrosophic set C, written as C=A∩B, whose truth-membership, indeterminacy-membership and falsity-membership functions are related to those of A and B by

TC(x)= min(TA(x), TB(x)), IC(x)=min(IA(x), IB(x)), FC(x)=max(FA(x), FB(x)), ∀x∈X

Definition 2.8[20] Let A be a neutrosophic set in X and α, β , γ∈[0, 1] with 0≤α+β+γ ≤3 and (α, β , γ)-level set of A denoted by A(α, β , γ) is defined as: A(α, β , γ)={ x∈X | TA(x)≥α, IA(x)≥β , FA(x)≤γ}.

2.2 On pseudo-BCI algebras

Definition 2.9[3] A pseudo-BCI algebra is a structure (X; ≤, →, , 1), where “≤” is a binary relation on X, “→” and “ ” are binary operations on X and “1” is an element of X, verifying the axioms: for all x, y, z∈X, (1) y→z≤(z→x) (y→x), y z≤(z x)→(y x); (2) x≤(x→y) y, x≤(x y)→y; (3) x≤x; (4) x≤y, y≤x ⇒ x=y; (5) x≤y ⇔ x→y =1 ⇔ x y =1.

If (X; ≤, →, , 1) is a pseudo-BCI algebra satisfying x→y = x y for all x, y∈X, then (X; →, 1) is a BCI-algebra.

Proposition 2.1[3, 7, 10] Let (X; ≤, →, , 1) be a pseudoBCI algebra, then X satisfy the following properties (∀x, y, z∈X):

(1)1≤x ⇒ x=1; (2) x≤y ⇒ y→z≤x→z, y z≤x z; (3) x≤y, y≤z ⇒ x≤z; (4) x (y→z)=y→(x z);

(5) x≤y→z ⇔ y≤x z; (6) x→y≤(z→x)→(z→y), x y≤(z x) (z y); (7) x≤y ⇒ z→x≤z→y, z x≤z y; (8)1→x=x, 1 x=x; (9)((y→x) x)→x=y→x, ((y x)→x) x=y x; (10) x→y≤(y→x) 1, x y ≤(y x)→1; (11)(x→y)→1=(x→1) (y 1), (x y) 1=(x 1)→(y→1); (12) x→1=x 1.

Definition 2.10[7] A nonempty subset F of pseudo-BCI algebra X is called a pseudo-BCI filter (briefly, filter) of X if it satisfies: (F1) 1∈F; (F2) x∈F, x→y∈F ⇒ y∈F; (F3) x∈F, x y∈F ⇒ y∈F

Definition 2.11[29] A pseudo-BCI algebra X is said to be anti-grouped pseudo-BCI algebra if it satisfies the following identity: (G1) ∀x, y, z∈X, (x→y)→(x→z)= y→z, (G2) ∀x, y, z∈X, (x y) (x z)= y z

Proposition 2.2 [29] A pseudo-BCI algebra X is an antigrouped pseudo-BCI algebra if and only if it satisfies: ∀x∈X, (x→1)→1=x or (x 1) 1=x.

Definition 2.12[29] A filter F of a pseudo-BCI algebra X is called an anti-grouped filter of X if it satisfies (GF) ∀x∈X, (x→1)→1∈F or (x 1) 1∈F⇒x∈F

Definition 2.13[29] A filter F of a pseudo-BCI algebra X is called a closed filter of X if it satisfies (CF) ∀x∈X, x→1∈F

Definition 2.14[34] A filter F of pseudo-BCI algebra X is said to be regular if it satisfies: (RF1) ∀x, y∈X, y∈F and x→y∈F ⇒ x∈F (RF2) ∀x, y∈X, y∈F and x y∈F ⇒ x∈F

Proposition 2.3 [34] Let X be a pseudo-BCI algebra, F a filter of X. Then F is regular if and only if F is anti-grouped and closed.

Florentin Smarandache (author and editor) Collected Papers, IX 176

Definition 2.15[31, 33] A fuzzy set A in pseudo-BCI algebra X is called fuzzy filter of X if it satisfies: (FF1) ∀x∈X, μA(x)≤μA(1); (FF2) ∀x, y∈X, min{μA(x), μA(x→y)}≤μA(y); (FF3) ∀x, y∈X, min{μA(x), μA(x y)}≤μA(y).

Definition 2.16[31] A fuzzy set A: X →[0, 1] is called a fuzzy closed filter of pseudo-BCI algebra X if it is a fuzzy filter of X such that: (FCF) μA(x→1) ≥ μA(x), x∈X

Definition 2.17[31] A fuzzy set A in pseudo-BCI algebra X is called fuzzy anti-grouped filter of X if it satisfies: (1) ∀x∈X, μA(x)≤μA(1); (2) ∀x, y, z∈X, min{μA(y), μA((x→y)→(x→z))}≤μA(z); (3) ∀x, y, z∈X, min{μA(y), μA((x y) (x z))}≤μA(z).

Proposition 2.4[31] Let A be a fuzzy filter of pseudoBCI algebra X. Then A is a fuzzy anti-grouped filter of X if and only if it satisfies: ∀x∈X, μA(x)≥μA((x→1)→1), μA(x)≥μA((x 1) 1).

Definition 2.18[35] A neutrosophic set A in pseudo-BCI algebra X is called a neutrosophic filter in X if it satisfies: ∀x, y∈X, (NSF1) TA(x)≤TA(1), IA(x)≤IA(1) and FA(x)≥FA(1); (NSF2) min{TA(x), TA(x→y)}≤TA(y), min{IA(x), IA(x→y)} ≤IA(y) and max{FA(x), FA(x→y)}≥FA(y); (NSF3) min{TA(x), TA(x y)}≤TA(y), min{IA(x), IA(x y)} ≤IA(y) and max{FA(x), FA(x y)}≥FA(y).

Proposition 2.5[35] Let A be a neutrosophic filter in pseudo-BCI algebra X, then ∀x, y∈X, (NSF4) x≤y ⇒ TA(x)≤TA(y), IA(x)≤IA(y) and FA(x)≥FA(y).

Definition 2.19[35] A neutrosophic set A in pseudo-BCI algebra X is called anti-grouped neutrosophic filter in X if it satisfies: ∀x, y, z∈X, (1) TA(x)≤TA(1), IA(x)≤IA(1) and FA(x)≥FA(1); (2)min{TA(y), TA((x→y)→(x→z))} ≤ TA(z), min{IA(y), IA((x→y)→(x→z))} ≤ IA(z) and max{FA(x), FA((x→y) →(x→z))} ≥ FA(z); (3)min{TA(y), TA((x y) (x z))} ≤ TA(z), min{IA(y), IA((x y) (x z))} ≤ IA(z) and max{FA(x), FA((x y) (x z))} ≥ FA(z).

Proposition 2.6[35] Let A be a neutrosophic set in pseudo-BCI algebra X. Then A is a neutrosophic filter in X if and only if A satisfies: (i) TA is a fuzzy filter of X; (ii) IA is a fuzzy filter of X; (iii)1 FA is a fuzzy filter of X, where (1 FA)(x) = 1 FA(x), ∀x∈X

Proposition 2.7[35] Let A be a neutrosophic set in pseudo-BCI algebra X. Then A is an anti-grouped neutrosophic filter in X if and only if A satisfies: (i) TA is a fuzzy anti-grouped filter of X;

(ii) IA is a fuzzy anti-grouped filter of X; (iii)1 FA is a fuzzy anti-grouped filter of X, where (1 FA)(x)=1 FA(x), ∀x∈X

3 Neutrosophic regular filters and neutrosophic closed filters

Definition 3.1 A neutrosophic set A in pseudo-BCI algebra X is called a neutrosophic regular filter in X if it is a neutrosophic filter in X such that: ∀x, y∈X, (NSRF1) min{TA(y), TA(x→y)}≤TA(x), min{IA(y), IA(x→y)}≤IA(x) and max{FA(y), FA(x→y)}≥FA(x); (NSRF2) min{TA(y), TA(x y)}≤TA(x), min{IA(y), IA(x y)}≤IA(x) and max{FA(y), FA(x y)}≥FA(x).

Definition 3.2 A neutrosophic set A in pseudo-BCI algebra X is called a neutrosophic closed filter in X if it is a neutrosophic filter in X such that: x∈X, (NSCF) TA(x→1)≥TA(x), IA(x→1)≥IA(x), FA(x→1)≤FA(x).

Proposition 3.1 Let A be a neutrosophic regular filter in pseudo-BCI algebra X. Then A is closed.

Proof: Suppose x∈X. By Definition 2.9 (2) and Proposition 2.1 (12) we have x ≤ (x→1) 1= (x→1)→1.

From this and Proposition 2.5 we get TA(x)≤TA((x→1)→1), IA(x)≤IA((x→1)→1), FA(x)≥FA((x→1)→1).

Moreover, by Definition 2.18 (NSF1) and Definition 3.1 (NSRF1)

TA((x→1)→1)=min{TA(1), TA((x→1)→1)}≤TA(x→1), IA((x→1)→1)=min{IA(1), IA((x→1)→1)}≤IA(x→1), FA((x→1)→1)=max{FA(1), FA((x→1)→1)}≥FA(x→1).

Thus,

TA(x)≤TA((x→1)→1)≤TA(x→1), IA(x)≤IA((x→1)→1)≤IA(x→1), FA(x)≥TA((x→1)→1)≥TA(x→1).

By Definition 3.2 we know that A is closed.

By Proposition 2.4 and Proposition 2.7 we can get the following proposition.

Proposition 3.2 Let A be a neutrosophic filter of pseudo-BCI algebra X. Then A is an anti-grouped neutrosophic filter of X if and only if it satisfies: ∀x∈X, TA(x)≥TA((x→1)→1), TA(x)≥TA((x 1) 1); IA(x)≥IA((x→1)→1), IA(x)≥IA((x 1) 1); FA(x)≤FA((x→1)→1), FA(x)≤FA((x 1) 1).

Proposition 3.3 Let A be a neutrosophic regular filter in pseudo-BCI algebra X. Then A is anti-grouped.

Proof: Suppose x∈X. By Definition 2.9 and Proposition 2.1 we have x→((x→1)→1)= x→((x→1) 1)=1.

From this we get TA(x→((x→1)→1))=TA(1), IA(x→((x→1)→1))=IA(1),

Florentin Smarandache (author and editor) Collected Papers, IX 177

FA(x→((x→1)→1))=FA(1).

Thus, applying Definition 3.1 (NSRF1) we get TA(x)≥min{TA((x→1)→1), TA(x→((x→1)→1))} =min{TA((x→1)→1), TA(1)}=TA((x→1)→1), IA(x)≥min{IA((x→1)→1), IA(x→((x→1)→1))} =min{IA((x→1)→1), IA(1)}=IA((x→1)→1), FA(x)≤max{FA((x→1)→1), FA(x→((x→1)→1))} =max{FA((x→1)→1), FA(1)}=FA((x→1)→1).

Similarly, we can prove that TA(x)≥TA((x 1) 1),IA(x)≥IA((x 1) 1), FA(x)≤FA((x 1) 1).

By Proposition 3.2 we know that A is anti-grouped.

Proposition 3.2 Assume that A is both an anti-grouped neutrosophic filter and a neutrosophic closed filter in pseudo-BCI algebra X. Then A satisfies: ∀x∈X, TA(x)=TA(x→1), IA(x)=IA(x→1), FA(x)=FA(x→1).

Proof: For any x∈X, by Definition 3.2 we have TA(x→1)≥TA(x), IA(x→1)≥IA(x), FA(x→1)≤FA(x). Moreover, ∀x∈X, by Definition 2.19 and Definition 3.2, TA(x)≥min{TA((x→1)→(x→x)), TA(1)} =min{TA((x→1)→1), TA(1)} =TA((x→1)→1)≥TA(x→1), IA(x)≥min{IA((x→1)→(x→x)), IA(1)} =min{IA((x→1)→1), IA(1)} =IA((x→1)→1)≥IA(x→1), FA(x)≤max{FA((x→1)→(x→x)), FA(1)} =max{FA((x→1)→1), FA(1)} =FA((x→1)→1)≤FA(x→1).

That is, TA(x)≥TA(x→1), IA(x)≥IA(x→1), FA(x)≤FA(x→1).

Therefore, ∀x∈X, TA(x)=TA(x→1), IA(x)=IA(x→1), FA(x)=FA(x→1).

Theorem 3.1 Let A be a neutrosophic filter in pseudoBCI algebra X. Then the following conditions are equivalent:

(i) A is both an anti-grouped neutrosophic filter and a neutrosophic closed filter in X; (ii) A satisfies: ∀x∈X, TA(x)=TA(x→1), IA(x)=IA(x→1), FA(x)=FA(x→1).

(iii) A is a neutrosophic regular filter in X

Proof: (i) ⇒ (ii) See Proposition 3.2. (iii) ⇒ (i) See Proposition 3.1 and Proposition 3.3. (ii) ⇒ (iii) Suppose that A satisfies: ∀x∈X, TA(x)=TA(x→1), IA(x)=IA(x→1), FA(x)=FA(x→1).

For any x, y∈X, using Proposition 2.1 (6) we have y→1≤(x→y)→(x→1)

From this, applying Propostion 2.5, TA(y→1)≤TA((x→y)→(x→1)), IA(y→1)≤IA((x→y)→(x→1)), FA(y→1)≥FA((x→y)→(x→1)).

From these, by Definition 2.18 we get

min{TA(y→1), TA(x→y)}

≤ min{TA((x→y)→(x→1)), TA(x→y)}=TA(x→1), min{IA(y→1), IA(x→y)}

≤ min{IA((x→y)→(x→1)), IA(x→y)}=IA(x→1), max{FA(y→1), FA(x→y)}

≥max{FA((x→y)→(x→1)), FA(x→y)}=FA(x→1).

Moreover, by condition (ii), TA(y→1)=TA(y), TA(x→1)=TA(x); IA(y→1)=IA(y), IA(x→1)=IA(x); FA(y→1)=FA(y), FA(x→1)=FA(x). Therefore, min{TA(y), TA(x→y)}≤TA(x), min{IA(y), IA(x→y)}≤ IA(x), max{FA(y), FA(x→y)}≥FA(x).

Similarly, we can get min{TA(y), TA(x y)}≤TA(x), min{IA(y), IA(x y)}≤ IA(x), max{FA(y), FA(x y)}≥FA(x).

By Definition 3.1 we know that A is a neutrosophic regular filter in X.

4 Fuzzy regular filters and neutrosophic filters

Definition 4.1 A fuzzy filter A in pseudo-BCI algebra X is called to be regular if it satisfies: (FRF1) ∀x, y∈X, min{μA(y), μA(x→y)}≤μA(x); (FRF2) ∀x, y∈X, min{μA(y), μA(x y)}≤μA(x).

Lemma 4.1[9, 33] Let X be a pseudo-BCI algebra. Then a fuzzy set μ: X→[0, 1] is a fuzzy filter of X if and only if the level set μ t ={ x∈X | μ(x)≥t} is filter of X for all t∈Im(μ).

Theorem 4.1 Let X be a pseudo-BCI algebra. Then a fuzzy set μ: X→[0, 1] is a fuzzy regular filter of X if and only if the level set μ t ={ x∈X | μ(x)≥t} is regular filter of X for all t∈Im(μ).

Proof: Assume that μ is fuzzy regular filter of X. By Lemma 4.1, for any t∈Im(μ), we have μ t ={x∈X | μ(x)≥t} is filter of X If y∈μ t and x→y∈μ t, then μ(y)≥t, μ( x→y)≥t From this and Definition 4.1 (FRF1) we get μA(x)≥min{μA(y), μA(x→y)}≥ t This means that x∈μ t. Similarly, we can prove that y∈μ t and x y∈μ t⇒ x∈μ t By Definition 2.14 we know that μ t is regular filter of X Conversely, assume that the level set μ t ={ x∈X | μ(x)≥t} is regular filter of X for all t∈Im(μ). By Lemma 4.1 we know that μ: X→[0, 1] is a fuzzy filter of X. Let x, y∈X, denote t0=min{μA(y), μA(x→y)}, then t0∈Im(μ) and μ(y)≥t0, μ( x→y)≥t0 This means that y∈ 0tμ and x→y∈ 0t μ . Since 0tμ is regular filter of X, by Definition 2.14 we have x∈ 0tμ , that is

Florentin Smarandache (author and editor) Collected Papers, IX 178

μ(x)≥ t0=min{μA(y), μA(x→y)}. It follows that Definition 4.1 (FRF1) holds. Similarly, we can prove that ∀x, y∈X, min{μA(y), μA(x y)}≤μA(x). Therefore, μ: X→[0, 1] is a fuzzy regular filter of X.

Similar to Theorem 4.1 we can get the following proposition (the proofs are omitted).

Proposition 4.1 Let X be a pseudo-BCI algebra. Then a fuzzy set μ: X→[0, 1] is a fuzzy closed filter of X if and only if the level set μ t ={ x∈X | μ(x)≥t} is closed filter of X for all t∈Im(μ).

By Theorem 6 in [31] we have

Theorem 4.2 Let μ be a fuzzy filter of pseudo-BCI algebra X. Then the following conditions are equivalent:

(i) μ is fuzzy closed anti-grouped filter of X; (ii) ∀x∈X, μA(x→1)=μA(x).

(iii) μ is a fuzzy regular filter of X

Theorem 4.3 Let A be a neutrosophic set in pseudo-BCI algebra X. Then A is a neutrosophic closed filter in X if and only if A satisfies:

(i) TA is a fuzzy closed filter of X; (ii) IA is a fuzzy closed filter of X; (iii)1 FA is a fuzzy closed filter of X, where (1 FA)(x) =1 FA(x), ∀x∈X

Proof: Assume that A is a neutrosophic closed filter in X.By Definition 3.2 we have (∀x∈X) TA(x→1)≥TA(x), IA(x→1)≥IA(x), FA(x→1)≤FA(x). Thus, (1 FA)(x→1)=1 FA(x→1)≥1 FA(x)=(1 FA)( x). Therefore, using Definition 2.16, we get that TA, IA and 1 FA are fuzzy closed filters of X Conversely, assume that TA, IA and 1 FA are fuzzy closed filters of X. Then, by Definition 2.16, TA(x→1)≥TA(x), IA(x→1)≥IA(x), (1 FA)(x→1)≥(1 FA)(x). Thus, FA(x→1)=1 (1 FA)(x→1)≤1 (1 FA)(x)=FA(x). Hence, applying Definition 3.2 we get that A is a neutrosophic closed filter A in X

By Theorem 4.2, Theorem 4.3, Theorem 3.1 and Proposition 2.7 we can get the following results.

Theorem 4.4 Let A be a neutrosophic set in pseudo-BCI algebra X. Then A is a neutrosophic regular filter in X if and only if A satisfies:

(i) TA is a fuzzy regular filter of X;

(ii) IA is a fuzzy regular filter of X;

(iii)1 FA is a fuzzy regular filter of X, where (1 FA)(x) =1 FA(x), ∀x∈X

Theorem 4.5 Let X be a pseudo-BCI algebra, A be a neutrosophic set in X such that TA(x)≥α0, IA(x)≥β0 and FA(x)≤γ0, ∀x∈X, where α0∈Im(TA), β0∈Im(IA) and γ0∈ Im(FA). Then A is a neutrosophic closed filter in X if and only if (α, β , γ)-level set A(α, β , γ) is closed filter of X for all

α∈Im(TA), β∈Im(IA) and γ∈Im(FA).

Proof: Assume that A is neutrosophic closed filter in X. By Theorem 4.3 and Proposition 4.1, for any α∈Im(TA), β∈Im(IA) and γ∈Im(FA), we have (TA)α ={x∈X | TA(x)≥α}, (IA)β ={x∈X | IA(x)≥β} and (1 FA)1 γ ={x∈X | (1 FA)(x)≥ 1 γ }={x∈X | FA(x)≤ γ } are closed filters of X

Thus (TA)α ∩(IA)β ∩(1 FA)1 γ is a closed filters of X. Moreover, by Definition 2.8, it is easy to verify that (α, β , γ)level set A(α, β , γ) =(TA)α ∩(IA)β ∩(1 FA)1 γ . Therefore, A(α, β , γ) is closed filter of X for all α∈Im(TA), β∈Im(IA) and γ∈ Im(FA).

Conversely, assume that A(α, β , γ) is closed filter of X for all α∈Im(TA), β∈Im(IA) and γ∈Im(FA). Since TA(x)≥α0, IA(x)≥β0 and FA(x)≤γ0, ∀x∈X, then (TA)α ={x∈X | TA(x)≥α}=(TA)α ∩X∩X = (TA)α ∩ (IA) 0β ∩ (1 FA) 0 1 γ = 00 (,,)A α βγ ; (IA)β ={x∈X | IA(x)≥β}=X ∩ (IA)β ∩X = (TA) 0α ∩ (IA) β ∩ (1 FA) 0 1 γ = 00 (,,)A α βγ ; (1 FA) 1 γ ={x∈X | (1 FA)(x)≥1 γ } = X∩X∩{x∈X | FA(x)≤γ} = (TA) 0α ∩ (IA) 0β ∩ {x∈X | FA(x)≤γ} = 00 (,,)A α βγ

Thus, (TA)α ={x∈X | TA(x)≥α}, (IA)β ={x∈X | IA(x)≥β} and (1 FA)1 γ ={x∈X | (1 FA)(x)≥1 γ }={x∈X | FA(x)≤γ } are closed filters of X

From this, applying Proposition 4.1, we know that TA, IA and 1 FA are fuzzy closed filters of X. By Theorem 4.3 we get that A is neutrosophic closed filter in X Similarly, we can get Lemma 4.2 Let X be a pseudo-BCI algebra, A be a neutrosophic set in X such that TA(x)≥α0, IA(x)≥β0 and FA(x)≤γ0, ∀x∈X, where α0∈Im(TA), β0∈Im(IA) and γ0∈ Im(FA). Then A is a (anti-grouped) neutrosophic filter in X if and only if (α, β , γ)-level set A(α, β , γ) is (anti-grouped) filter of X for all α∈Im(TA), β∈Im(IA) and γ∈Im(FA).

Combining Theorem 4.5, Lemma 4.2 and Theorem 3.1 we can get the following theorem.

Theorem 4.6 Let X be a pseudo-BCI algebra, A be a neutrosophic set in X such that TA(x)≥α0, IA(x)≥β0 and FA(x)≤γ0, ∀x∈X, where α0∈Im(TA), β0∈Im(IA) and γ0∈ Im(FA). Then A is a neutrosophic regular filter in X if and only if (α, β , γ)-level set A(α, β , γ) is regular filter of X for all α∈Im(TA), β∈Im(IA) and γ∈Im(FA).

Florentin Smarandache (author and editor) Collected Papers, IX 179

Conclusion

The neutrosophic set theory is applied to many scientific fields, and also applied to algebraic structures.

This paper applied neutrosophic set theory to pseudoBCI algebras, and some new notions of neutrosophic regular filter, neutrosophic closed filter and fuzzy regular filter in pseudo-BCI algebras are introduced.

In addition to studying the basic properties of these new concepts, this paper also considered the relationships between them, and obtained some necessary and sufficient conditions.

Acknowledgment

This work was supported by National Natural Science Foundation of China (Grant No. 61573240, 61473239).

formation Sciences First International Conference on Neutrosophy, Neutrosophic Logic, Set, Probability and Statistics University of New Mexico, Gallup, NM 87301, USA, 2002.

[18]F. Smarandache, Neutrosophic set–a generialization of the intuituionistics fuzzy sets, International Journal of Pure and Applied Mathematics, 24 (3) (2005), 287–297.

[19]F. Smarandache, Neutrosophic Perspectives: Triplets, Duplets, Multisets, Hybrid Operators, Modal Logic, Hedge Algebras. And Applications, Pons Publishing House, Brussels, 2017

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Neutrosophic Duplet Semi-Group and Cancellable Neutrosophic Triplet Groups

Xiaohong Zhang, Florentin Smarandache, Xingliang Liang

Xiaohong Zhang, Florentin Smarandache, Xingliang Liang (2017). Neutrosophic Duplet SemiGroup and Cancellable Neutrosophic Triplet Groups. Symmetry 9, 275; DOI: 10.3390/sym9110275

Abstract: The notions of the neutrosophic triplet and neutrosophic duplet were introduced by Florentin Smarandache. From the existing research results, the neutrosophic triplets and neutrosophic duplets are completely different from the classical algebra structures. In this paper, we further study neutrosophic duplet sets, neutrosophic duplet semi-groups, and cancellable neutrosophic triplet groups. First, some new properties of neutrosophic duplet semi-groups are funded, and the following important result is proven: there is no finite neutrosophic duplet semi-group. Second, the new concepts of weak neutrosophic duplet, weak neutrosophic duplet set, and weak neutrosophic duplet semi-group are introduced, some examples are given by using the mathematical software MATLAB (MathWorks, Inc., Natick, MA, USA), and the characterizations of cancellable weak neutrosophic duplet semi-groups are established. Third, the cancellable neutrosophic triplet groups are investigated, and the following important result is proven: the concept of cancellable neutrosophic triplet group and group coincide. Finally, the neutrosophic triplets and weak neutrosophic duplets in BCI-algebras are discussed.

Keywords: neutrosophic duplet; neutrosophic triplet; weak neutrosophic duplet; semi-group; BCI-algebra

1.Introduction

FlorentinSmarandacheintroducedtheconceptofaneutrosophicsetfromaphilosophical pointofview(see[1 3]).Theneutrosophicsettheoryisappliedtomanyscientificfields andalso appliedtoalgebraicstructures(see[4 10]).Recently,FlorentinSmarandacheandMumtazAli in[11],forthefirsttime,introducedthenotionsofaneutrosophictripletandneutrosophictriplet group.Theneutrosophictripletisagroupofthreeelementsthatsatisfycertainpropertieswith somebinaryoperation;itiscompletelydifferentfromtheclassicalgroupinthestructuralproperties. In2017,FlorentinSmarandachewrotethemonograph[12]thatispresentthelatestdevelopmentsin neutrodophictheories,includingtheneutrosophictriplet,neutrosophictripletgroup,neutrosophic duplet,andneutrosophicdupletset.

Inthispaper,wefocusontheneutrosophicduplet,neutrosophicdupletset,andneutrosophic dupletsemi-group.Wediscusssomenewpropertiesoftheneutrosophicdupletsemi-groupand investigatetheidempotentelementintheneutrosophicdupletsemi-group.Moreover,weintroduce somenewconceptstogeneralizethenotionofneutrosophicdupletsetsanddiscussweak neutrosophicdupletsinBCI-algebras(forBCI-algebraandrelatedgeneralizedlogicalalgebrasystems, pleasesee[13 26]).

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2.1.NeutrosophicTripletandNeutrosophicDuplet

Deﬁnition1. ([11,12])LetNbeasettogetherwithabinaryoperation*.Then,Niscalledaneutrosophic tripletsetifforanya ∈ N,thereexistaneutralof“a”calledneut(a),differentfromtheclassicalalgebraicunitary element,andanoppositeof“a”calledanti(a),withneut(a)andanti(a)belongingtoN,suchthat:

a * neut(a)= neut(a)* a = a;

a * anti(a)= anti(a)* a = neut(a).

Theelements a, neut(a),and anti(a)arecollectivelycalledasaneutrosophictriplet,andwedenote itby(a, neut(a), anti(a)).By neut(a),wemeanneutralof a and,apparently, a isjusttheﬁrstcoordinateof aneutrosophictripletandnotaneutrosophictriplet.Forthesameelement“a”in N,theremaybemore neutralstoit neut(a)andmoreoppositesofit anti(a).

Deﬁnition2. ([11,12])Theelementbin(N,*)isthesecondcomponent,denotedasneut( ),ofaneutrosophic triplet,ifthereexistsotherelementsaandcinNsuchthata*b=b*a=aanda*c=c*a=b.Theformed neutrosophictripletis(a,b,c).

Deﬁnition3. ([11,12])Theelementcin(N,*)isthethirdcomponent,denotedasanti(·),ofaneutrosophic triplet,ifthereexistsotherelementsaandbinNsuchthata*b=b*a=aanda*c=c*a=b.Theformed neutrosophictripletis(a,b,c).

Deﬁnition4. ([11,12])Let(N,*)beaneutrosophictripletset.Then,Niscalledaneutrosophictripletgroup, ifthefollowingconditionsaresatisﬁed:

(1) If(N,*)iswell-deﬁned,i.e.,foranya,b ∈ N,onehasa*b ∈ N.

(2) If(N,*)isassociative,i.e.,(a*b)*c=a*(b*c)foralla,b,c ∈ N.

Theneutrosophictripletgroup,ingeneral,isnotagroupintheclassicalalgebraicway.

Deﬁnition5. ([11,12])Let(N,*)beaneutrosophictripletgroup.Then,Niscalledacommutativeneutrosophic tripletgroupifforalla,b ∈ N,wehavea*b=b*a.

Deﬁnition6. ([12])LetUbeauniverseofdiscourse,andasetA ⊆ U,endowedwithawell-deﬁnedlaw*.Wesay that a,neut(a) ,wherea,neut(a) ∈ A,isaneutrosophicdupletinAif:

(1) neut(a)isdifferentfromtheunitelementofAwithrespecttothelaw*(ifany);

(2) a*neut(a)=neut(a)*a=a;

(3) thereisnoanti(a) ∈ Asuchthata*anti(a)=anti(a)*a=neut(a).

Remark1. Intheabovedeﬁnition,wehaveA ⊆ U.WhenA=U,“neutrosophicdupletinA”issimpliﬁedas “neutrosophicduplet”,withoutcausingconfusion.

Deﬁnition7. ([12])Aneutrosophicdupletset,(D,*),isasetD,endowedwithawell-deﬁnedbinarylaw*, suchthat ∀a ∈ D, ∃ aneutrosophicduplet a,neut(a) suchthatneut(a) ∈ D.Ifassociativelawholdsin neutrosophicdupletset(D,*),thencallitneutrosophicdupletsemi-group.

Remark2. Theabovedefinitionisdifferentfromtheoriginaldefinitionofaneutrosophicdupletsetin[12]. Infact,themeaningofTheoremIX.2.1in[12]isnotconsistentwiththeoriginaldefinitionofaneutrosophic dupletset.TheoriginaldefinitionismodifiedtoensurethatTheoremIX.2.1in[12]isstillcorrect.

Remark3. Inordertoincludericherstructure,theoriginalconceptofaneutrosophictripletisgeneralized toneutrosophicextendedtripletbyFlorentinSmarandache.Foraneutrosophicextendedtripletthatisa neutrosophictriplet,theneutralofx(called“extendedneutral”)isallowedtoalsobeequaltotheclassical

2.BasicConcepts

Florentin Smarandache (author and editor) Collected Papers, IX 182

algebraicunitaryelement(ifany).Therefore,therestriction“differentfromtheclassicalalgebraicunitary element,ifany”isreleased.Asaconsequence,the“extendedopposite”ofxisalsoallowedtobeequaltothe classicalinverseelementfromaclassicalgroup.Thus,aneutrosophicextendedtripletisanobjectoftheform (x,neut(x),anti(x)),forx ∈ N,whereneut(x) ∈ Nistheextendedneutralofx,whichcanbeequalordifferent fromtheclassicalalgebraicunitaryelement,ifany,suchthat:x*neut(x)=neut(x)*x=x,andanti(x) ∈ Nis theextendedoppositeofx,suchthat:x*anti(x)=anti(x)*x=neut(x).Inthispaper,“neutrosophictriplet” means“neutrosophicextendedtriplet”,and“neutrosophicduplet”means“neutrosophicextendedduplet”.

2.2.BCI-Algebras

Deﬁnition8. ([15,22])ABCI-algebraisanalgebra(X; →,1)oftype(2,0)inwhichthefollowingaxioms aresatisﬁed:

(i) (x → y) →((y → z) → (x → z))=1, (ii) x → x=1, (iii) 1 → x=x, (iv) ifx → y=y → x=1,thenx=y.

InanyBCI-algebra(X; →,1)onecandeﬁnearelation ≤ byputting x ≤ y ifandonlyif x → y =1, then ≤ isapartialorderon X Deﬁnition9. ([16,20])Let(X; →,1)beaBCI-algebra.Theset{x|x ≤ 1}iscalledthep-radical(orBCK-part) ofX.ABCI-algebraXiscalledp-semisimpleifitsp-radicalisequalto{1}.

Deﬁnition10. ([16,20])ABCI-algebra(X; →,1)iscalledassociativeif (x → y) → z = x→ (y → z), ∀x,y,z ∈ X.

Proposition1. ([16])Let(X; →,1)beaBCI-algebra.Thenthefollowingareequivalent:

(i) Xisassociative; (ii) x → 1=x, ∀x ∈ X; (iii) x → y=y → x, ∀x,y ∈ X.

Proposition2. ([16,24])Let(X;+, ,1)beanAbelgroup.Deﬁne(X; ≤, →,1),where x → y = x + y, x ≤ y ifandonlyif-x + y =1, ∀x,y ∈ X

Then,(X; ≤, →,1)isaBCI-algebra.

3.NewPropertiesofNeutrosophicDupletSemi-Group

Foraneutrosophicdupletset(D,*),if a ∈ D,then neut(a)maynotbeunique.Thus,thesymbolic neut(a)sometimesmeansoneandsometimesmorethanone,whichisambiguous.Tothisend, thispaper introducesthefollowingnotationstodistinguish: neut(a):denoteanycertainoneofneutralof a; {neut(a)}:denotethesetofallneutralof a

Remark4. Inordernottocauseconfusion,wealwaysassumethat:forthesamea,whenmultipleneut(a)are presentinthesameexpression,theyarealwaysareconsistent.Ofcourse,iftheyareneutralofdifferentelements, theyrefertodifferentobjects(forexample,ingeneral,neut(a)isdifferentfromneut(b)).

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Proposition3. Let(D,*)beaneutrosophicdupletsemi-groupwithrespectto*anda ∈ D.Then,forany x,y ∈ {neut(a)},x*y ∈ {neut(a)}.Thatis,

{neut(a)}*{neut(a)} ⊆ {neut(a)}.

Proof. Forany a ∈ D,byDeﬁnition7,wehave

a * neut(a)= a, neut(a)* a = a

Assume x, y ∈ {neut(a)},then

a * x = x * a = a; a * y = y * a = a.

Fromthis,usingassociativelaw,wecanget a *(x * y)=(x * y)* a = a.

Itfollowsthat x * y isaneutralof a.Thatis, x*y ∈ {neut(a)}.Thismeansthat {neut(a)}*{neut(a)}⊆ {neut(a)}.

Remark5. Ifneut(a)isunique,then neut(a)* neut(a)= neut(a).

But,if neut(a)isnotunique,forexample,assume{neut(a)}={s, t} ∈ D,then neut(a)denoteanyone of s, t.Thus neut(a)* neut(a)representsoneof s*s,and t*t;and{neut(a)}*{neut(a)}= {s*s, s*t, t*s, t*t}.

Proposition3meansthat s*s, s*t, t*s, t*t ∈ {neut(a)}={s, t},thatis, s*s = s,or s*s = t; s*t = s,or s*t = t t * s = s,or t * s = t; t * t = s,or t * t = t.

Inthiscase,theequation neut(a)* neut(a)= neut(a)maynothold.

Proposition4. Let(D,*)beaneutrosophicdupletsemi-groupwithrespectto*andleta,b,c ∈ D.Then

(1) neut(a)*b=neut(a)*c ⇒ a*b=a*c.

(2) b*neut(a)=c*neut(a) ⇒ b*a=c*a.

Proof. (1)Assume neut(a)* b = neut(a)* c.Then a *(neut(a)* b)= a *(neut(a)* c).

Byassociativelaw,wehave

(a *neut(a))* b =(a *neut(a))* c

Thus, a * b = a * c.Thatis,(1)holds. Similarly,wecanprovethat(2)holds.

Theorem1. Let(D,*)beacommutativeneutrosophicdupletsemi-groupwithrespectto*anda,b ∈ D.Then neut(a)* neut(b) ∈ {neut(a * b)}.

Florentin Smarandache (author and editor) Collected Papers, IX 184

Proof. Forany a, b ∈ D,wehave

a * neut(a)* neut(b)* b =(a * neut(a))*(neut(b)* b)= a * b

Fromthisandapplyingthecommutativityandassociativityofoperation*weget

(neut(a)* neut(b))*(a * b)=(a * b)*(neut(a)* neut(b))= a * b

Thismeansthatneut(a)* neut(b) ∈ {neut(a * b)}.

Theorem2. Let(D,*)beaneutrosophicdupletsetwithrespectto*.ThenthereisnoidempotentelementinD, thatis, ∀a ∈ D, a * a = a

Proof. Assumethatthereis a ∈ D suchthat a * a = a.Then a ∈ {neut(a)},and a ∈ {anti(a)},Thisisa contractionwithDeﬁnition6(3).

Sincetheclassicalalgebraicunitaryelementisidempotent,wehave

Corollary1. Let(D,*)beaneutrosophicdupletsetwithrespectto*.Thenthereisnoclassicalunitaryelement inD,thatis,thereisnoe ∈ Dsuchthat ∀a ∈ D,a*e=e*a=a.

Theorem3. Let(D,*)beaneutrosophicdupletsemi-groupwithrespectto*.ThenDisinﬁnite.Thatis,there isnoﬁniteneutrosophicdupletsemi-group.

Proof. Assumethat D isaﬁniteneutrosophicdupletsemi-groupwithrespectto*.Then,forany a ∈ D, a, a * a = a2 , a * a * a = a3 ,..., an ,... ∈ D

Since D isﬁnite,sothereexistsnaturalnumber m, k suchthat am = am+k

Case1:if k = m,then am = a2m,thatis, am = am * am , am isanidempotentelementin D,thisisa contractionwithTheorem2.

Case2:if k > m,thenfrom am = am+k wecanget ak = am * ak m = am+k * akm = a2k = ak * ak

Thismeansthat ak isanidempotentelementin D,thisisacontractionwithTheorem2. Case3:if k <m,thenfrom am = am+k wecanget am = am+k = am * ak = am+k * ak = am+2k ; am = am+2k = am * a2k = am+k * a2k = am+3k ; ...... am = am+mk .

Since m and k arenaturalnumbers,then mk ≥ m.Therefore,from am = am+mk,applyingCase1or Case2,weknowthatthereexistsanidempotentelementin D,thisisacontractionwithTheorem2. Theorem4. Let(D,*)beaneutrosophicdupletsemi-groupwithrespectto*anda ∈ D.Then neut(neut(a)) ∈ {neut(a)}.

Florentin Smarandache (author and editor) Collected Papers, IX 185

Proof. Forany a ∈ D,bythedeﬁnitionof neut( ),wehave

neut(a)* neut(neu(a))= neut(a);

neut(neut(a))* neut(a)= neut(a).

Then

wrong, because the asso

a *(neut(a)* neut(neut(a)))= a * neut(a); (neut(neut(a))* neut(a))* a = neut(a)* a. (b * a)*c = a * c = c,but b *(a * c)= b * c = b.

4. WeakNeutrosophic DupletSet(andSemi-Group)

FromTheorems3and5,wecanseethatthestructureoftheneutrosophicdupletsemi-group isveryscarce.Whatarethereasonsforthat?Thekeyreasonisthatundertheoriginaldeﬁnitionof

Infact,wecanverifythat(D,*)isaneutrosophicdupletsemi-groupbyMATLABprogramming, asshowninFigure 1.

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Example2. .Then,(D,*)isanon-commutative neutrosophicdupletsemi-group.

Table2. Weakneutrosophicdupletsemi-group(2).

111

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*123

1 212 3 333 3

Inthisexample,“1”,“2”,and“3”areidempotentelementsin D,and{neut(1)}={1,2},neut(2)=2, {neut(3)}={2,3}.

Example3. LetD={1,2,3,4}.Theoperation*onDisdeﬁnedasTable 3.Then,(D,*)isacommutative neutrosophicdupletsemi-group.

Table3. Weakneutrosophicdupletsemi-group(3). *1234 1 314 4 2 123 4 3 434 4 4 444 4

Inthisexample,“2”and“4”areidempotentelementsin D,and neut(2)=2,{neut(4)}={1,2,3,4}. neut(1)=2,{anti(1)}= ∅; neut(3)=2,{anti(3)}= ∅.

Example4. LetD={1,2,3,4}.Theoperation*onDisdeﬁnedasTable 4.Then,(D,*)isanon-commutative neutrosophicdupletsemi-group.

Table4. Weakneutrosophicdupletsemi-group(4). *1234 1 223 1 2 223 2 3 223 3 4 123 4

Inthisexample,“2”,“3”,and“4”areidempotentelementsin D,and neut(1)=4,{anti(1)}= ∅ Now,weexplainalloftheneutrosophicdupletsemi-groupswiththreeelements.Intotal,wecan obtain50neutrosophicdupletsemi-groupswiththreeelements,someofwhichmaybeisomorphic. TheyarefundedbyMATLABprogramming,asshowninFigure 2 Deﬁnition12. Aweakneutrosophicdupletsemi-group(D,*)iscalledtobecancellable,ifitsatisﬁes

Theweakneutrosophicdupletsemi-groupsinExamples1–4arenotcancellable.Wegivea cancellableexampleasfollows.

∀

∀a, b, c ∈ D, a *b = a * c ⇒ b = c;

a, b, c ∈ D, b * a = c * a ⇒ b = c

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Inthisexample,foranyelement a in D,and neut(a)=0.

Theorem6. Let(D, *) beacancellableweakneutrosophicdupletsemi-groupwithrespectto*.Then

(1) ∀a ∈ D,neut(a)isunique.

(2) ∀a ∈ D,neut(a)*neut(a)=neut(a).

(3) ∀a ∈ D,neut(a)*neut(a)=neut(a*a).

(4) ∀a,b ∈ D,neut(a)=neut(b).

Proof. (1)Forany a ∈ D,wehave

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Figure 2. Find weak neutrosophic duplet semi-group by MATLAB

Case1:if a ∈ {neut(a)},then a * a = a.Thus a * a = a = a * neut(a).

ByDeﬁnition12,wehave a = neut(a).Thismeansthat{neut(a)}={a},thatis, neut(a)isunique. Case2:if a {neut(a)},assume x, y ∈ {neut(a)},then a * x = a = a * y

ByDeﬁnition12,wehavex = y.Thismeansthat|{neut(a)}|=1,thatis, neut(a)isunique.

(2)If a ∈ {neut(a)},then a * a = a,by(1)weget a = neut(a),so neut(a)* neut(a)= neut(a). If a {neut(a)},bythesamewaywithProposition3,wecanprovethat

{neut(a)}*{neut(a)} ⊆ {neut(a)}.

Using(1)wehave neut(a)* neut(a)= neut(a).

(3)Forany a ∈ D,since(byassociativelaw)

(neut(a)* neut(a))*(a * a)= a * a;

(a * a)*(neut(a)* neut(a))= a * a

Thismeansthat neut(a)* neut(a) ∈ {neut(a * a)},butby(1)|{neut(a)}|=1,thus neut(a)* neut(a)=neut(a*a).

(4)Forany a, b ∈ D,since(byassociativelaw)

a * neut(a)* neut(b)*b = a * b.

Fromthis,applyingDeﬁnition12, neut(a)* neut(b)* b = b

neut(a)* neut(b)* b = b = neut(b)* b. ApplyingDeﬁnition12again, neut(a)* neut(b)= neut(b).

Similarly,wecanget neut(a)* neut(b)= neut(a).

Hence, neut(a)= neut(b).

Theorem7. Let(D,)beacancellableweakneutrosophicdupletsemi-groupwithrespectto*.IfDisaﬁniteset, thenDisasinglepointset,thatis,|D|=1.

Proof. ByTheorem6,weknowthat{neut(a)| a ∈ D}isasinglepointset.Denote neut(a)= e (∀a ∈ D). Assumethat D isaﬁniteset,if|D| = 1,thenthereexists x ∈ D suchthat x = e.Denote|D|= n, D ={a1, a2,..., an}.Inthetableofoperation*,considerthelineinwhichthe x islocated: x * a1, x * a2,..., x * an

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Since D iscancellable,then x * a1, x * a2, ... , x * anaredifferentfromeachother.Thus, ∃ai such that x * ai = e.Itfollowsthat x, neut(x)= e isnotaneutrosophicduplet.ApplyingDeﬁnition11, x ∈ {neut(x)}={e}.Thatis, x = e.Thisisacontractionwiththehypothesis x = e.Hence|D|=1. ApplyingTheorems2and6,wecangetthefollowingtheorem.

Theorem8. Let(D,*)beaneutrosophicdupletsemi-groupwithrespectto*.ThenDisnotcancellable.Thatis, thereisnocancellableneutrosophicdupletsemi-group.

5.OnCancellableNeutrosophicTripetGroups

Deﬁnition13. Aneutrosophictripletgroup(D,*)iscalledtobecancellable,ifitsatisﬁes

∀a, b, c ∈ D, a * b = a * c ⇒ b = c; ∀a, b, c ∈ D, b * a = c * a ⇒ b = c

Example7. LetD={1,2,3,4}.Theoperation*onDisdeﬁnedasTable 5.Then,(D,*)isacancellable neutrosophictripletgroup.

Table5. Cancellableneutrosophictripletgroup. *1234 1 123 4 2 214 3 3 341 2 4 432 1

Inthisexample, neut(1)= neut(2)= neut(3)= neut(4)=1,and anti(1)=1, anti(2)=2, anti(3)=3, anti(4)=4.

Theorem9. Let(D,*)beacancellableneutrosophictripletgroupwithrespectto*.Then

(1) ∀a ∈ D,neut(a)isunique. (2) ∀a ∈ D,anti(a)isunique. (3) ∀a,b ∈ D,neut(a)=neut(b).

(4) (D,*)isagroup,theunitisneut(a), ∀a ∈ D.

Proof. (1)Forany a ∈ D,assume x, y ∈ {neut(a)},then A * x = a = a * y

ByDeﬁnition13,wehave x = y.Thismeansthat|{neut(a)}|=1,thatis, neut(a)isunique. (2)Forany a ∈ D,using(1), neut(a)isunique.Assume x, y ∈ {anti(a)},then a * x = neut(a)= a * y

ByDeﬁnition13,wehave x = y.Thismeansthat|{anti(a)}|=1,thatis, anti(a)isunique. (3)Forany a, b ∈ D,since(byassociativelaw) neut(a)* b = neut(a)* neut(b)* b

Fromthis,applyingDeﬁnition13, neut(a)= neut(a)* neut(b).

Florentin Smarandache (author and editor) Collected Papers, IX 191

Ontheotherhand,since(byassociativelaw)

a * neut(b)= a *(neut(a)* neut(b)).

Fromthis,applyingDeﬁnition13again, neut(b)= neut(a)* neut(b).

Thus, neut(a)= neut(b). (4)Itfollowsfrom(1)~(3).

Sinceanygroupisacancellableneutrosophictripletgroup,byTheorem9(3),wehave Theorem10. Theconceptsofneutrosophictripletgroupandgroupcoincide.

Thefollowingexampleshowsthatthereexistsanon-cancellableneutrosophictripletgroup, inwhich(∀a ∈ D) neut(a)isuniqueand anti(a)isunique.

Example8. LetD={1,2,3,4}.Theoperation*onDisdeﬁnedasTable 6.Then,(D,*)isanon-cancellable neutrosophictripletgroup,but (∀a ∈ D)neut(a)isuniqueandanti(a)isunique.

Table6. Non-cancellableneutrosophictripletgroup. *1234 1 123 4 2 123 4 3 123 4 4 123 4

Inthisexample, neut(1)= anti(1)=1, neut(2)= anti(2)=2, neut(3)= anti(3)=3, neut(4)= anti(4)=4. Deﬁnition14. Aneutrosophictripletgroup(D,*)iscalledtobeweakcancellable,ifitsatisﬁes

∀a, b, c ∈ D,(a * b = a * c and b * a = c * a) ⇒ b = c

Obviously,acancellableneutrosophictripletgroupisweakcancellable,butaweakcancellable neutrosophictripletgroupmaynotbecancellable.Infact,the(D,*)inExample8isweakcancellable, butisnotcancellable.

Theorem11. Let(D,*)beaweakcancellableneutrosophictripletgroupwithrespectto*.Then

(1) ∀a ∈ D,neut(a)isunique. (2) ∀a ∈ D,anti(a)isunique.

Proof. (1)Forany a ∈ D,assume x, y ∈ {neut(a)},then a * x = a = a * y. x* a = a = y * a

ByDeﬁnition14,wehave x = y.Thismeansthat|{neut(a)}|=1,thatis, neut(a)isunique. (2)Forany a ∈ D,using(1), neut(a)isunique.Assume x, y ∈ {anti(a)},then a * x = neut(a)= a * y. x * a = neut(a)= y * a

ByDeﬁnition14,wehave x = y.Thismeansthat|{anti(a)}|=1,thatis, anti(a)isunique.

Florentin Smarandache (author

Collected Papers, IX 192

and editor)

Thefollowingexampleshowsthatthereexistsaneutrosophictripletgroupinwhich(∀a ∈ D) neut(a)isuniqueand anti(a)isunique,butitisnotweakcancellable.

Example9. LetD={1,2,3}.Theoperation*onDisdeﬁnedasTable 7.Then,(D,*)isaneutrosophictriplet group,and (∀a ∈ D)neut(a)isuniqueandanti(a)isunique.However,itisnotweakcancellable,since 2*1=2*2,1*2=2*2,1 = 2.

Table7. Notweakcancellableneutrosophictripletgroup. *123 112 3 222 3 333 2

Inthisexample,wehave

neut(1)= anti(1)=1, neut(2)= anti(2)=2, neut(3)= anti(3)=2.

Thefollowingexampleshowsthatthereexistsacommutativeneutrosophictripletgroupwhich (∃a ∈ D) anti(a)isnotunique.

Example10. Consider(Z6,*),where*isclassicalmultiplication.Then,(Z6,*)isacommutativeneutrosophic tripletgroup,thebinaryoperation*isdeﬁnedinTable 8.Foreacha ∈ Z6,wehaveneut(a)inZ6.Thatis, neut([0])=[0], neut([1])=[1], neut([2])=[4], neut([3])=[3], neut([4])=[4], neut([5])=[1]; {anti([0])}={[0],[1],[2],[3],[4],[5]},

{anti([1])}={[1]}, {anti([2])}={[2],[5]}, {anti([3])}={[1],[3],[5]}, {anti([4])}={[1],[4]}, {anti([5])}={[5]}.

Table8. Cayleytableof(Z6,*).

*[0][1][2][3][4][5] [0][0][0][0][0][0][0] [1][0][1][2][3][4][5] [2][0][2][4][0][2][4] [3][0][3][0][3][0][3] [4][0][4][2][0][4][2] [5][0][5][4][3][2][1]

6.NeutrosophicTripletsandWeakNeutrosophicDupletsinBCI-Algebras

Now,wediscussBCI-algebra(X; →,1).

Theorem12. Let(X; →,1)beaBCI-algebra.Then (1) ∀x ∈ X,if{neut(x)} = ∅ andy ∈ {neut(x)},thenx → 1=x,y → 1=1.

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(2) ∀x ∈ X, if {neut (x)} = ∅ and {anti (x)} = ∅, then z → 1 = x for any z ∈ {anti (x)}.

Proof. (1) Assume y ∈ {neut(x)}, then X → y = y → x = x.

UsingthepropertiesofBCI-algebras,wehave

x → 1= x → (y → y)= y →(x → y)= y → x = x y → 1= y →(x → x)= x →(y → x)= x → x =1.

(2)Assume z ∈ {anti(x)},then Z → x = x → z = neut(x).

Using(1)andthepropertiesofBCI-algebras,wehave 1= neut(x) → 1=(z → x) → 1=(z → 1) → (x → 1)=(z → 1) → x 1= neut(x) → 1=(x → z) → 1=(x → 1) → (z → 1)= x→(z → 1).

Hence, z → 1= x.

Example11. LetD={a,b,c,1}.Theoperation → onDisdeﬁnedasTable 9.Then,(D, →)isaBCI-algebra (itisadualformofI4-2-2 in[16]),and c,1,c isaneutrosophictripletin(D, →).

Table9. NeutrosophictripletinBCI-algebra. → abc 1 a 1 cc 1 bc 11 c cba 1 c 1 abc 1

Theorem13. Let(X; →,1)beaBCI-algebra.Then(X, →)isaneutrosophictripletgroupifandonlyif (X; →,1)isanassociativeBCI-algebra.

Proof. Supposethat(X; →)isaneutrosophictripletgroup.Then ∀x ∈ X,{neut(x)} = ∅.ByTheorem12, x → 1= x.UsingProposition1,(X; →,1)isanassociativeBCI-algebra. Conversely,supposethat(X; →,1)isanassociativeBCI-algebra.Then(X; →,1)isagroup. Hence,(X; →)isaneutrosophictripletgroup.

Example12. LetD={a,b,c,1}.Theoperation → onDisdeﬁnedasTable 10.Then,(D; →,1)isaBCI-algebra (itisadualformofI4-1-1 in[16]),and(D, →)isaneutrosophictripletgroup.

Table10. NeutrosophictripletgroupandBCI-algebra.

→ abc 1 a 1 cc 1 bc 11 c cba 1 c 1 abc 1

Theorem14. Let(X; →,1)beaBCI-algebra.Then(X, →)isnotaneutrosophicdupletsemi-group.

Florentin Smarandache (author and editor) Collected Papers, IX 194

Conclusions

This paper is focused on the neutrosophic duplet semi-group. We proved some new properties of the neutrosophic duplet semi-group, and proved that there is no ﬁnite neutrosophic duplet semi-group. We introduced the new concept of weak neutrosophic duplet semi-groups and gave some examples by MATLAB. Moreover, we investigated cancellable neutrosophic triplet groups and proved that the concept of cancellable neutrosophic triplet group and group coincide. Finally, we discussed neutrosophic triplets and weak neutrosophic duplets in BCI-algebras.

Acknowledgments: This work was supported by National Natural Science Foundation of China (Grant Nos. 61573240, 61473239).

References

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2. Smarandache,F.NeutrosophyandNeutrosophicLogic.InProceedingsoftheFirstInternationalConference onNeutrosophy,NeutrosophicLogic,Set,ProbabilityandStatistics,Gallup,NM,USA,1–3December2001.

3. Smarandache,F.Neutrosophicset—Agenerializationoftheintuituionisticsfuzzysets. Int.J.PureAppl.Math. 2005, 3,287–297.

4. Agboola,A.A.A.;Davvaz,B.;Smarandache,F.Neutrosophicquadruplealgebraichyperstructures. Ann.FuzzyMath.Inform. 2017, 1,29–42.

5. Borzooei,R.A.;Farahani,H.;Moniri,M.NeutrosophicdeductiveﬁltersonBL-algebras. J.Intell.FuzzySyst. 2014, 26,2993–3004.

6.Rezaei,A.;Saeid,A.B.;Smarandache,F.NeutrosophicﬁltersinBE-algebras. RatioMath. 2015, 29,65–79.

7. Saeid,A.B.;Jun,Y.B.Neutrosophicsubalgebrasof BCK/BCI-algebrasbasedonneutrosophicpoints. Ann.FuzzyMath.Inform. 2017, 1,87–97.

8. Ye,J.Singlevaluedneutrosophiccross-entropyformulticriteriadecisionmakingproblems. Appl.Math.Model. 2014, 38,1170–1175.[CrossRef]

9. Zhang,X.H.;Ma,Y.C.;Smarandache,F.Neutrosophicregularﬁltersandfuzzyregularﬁltersin pseudo-BCIalgebras. NeutrosophicSetsSyst. 2017, 17,10–15.

10. Zhang,X.H.;Smarandache,F.;Ali,M.;Liang,X.L.Commutativeneutrosophictripletgroupand neutro-homomorphismbasictheorem. Ital.J.PureAppl.Math. 2017,inpress.

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12. Smarandache,F. NeutrosophicPerspectives:Triplets,Duplets,Multisets,HybridOperators,ModalLogic, HedgeAlgebras.AndApplications;PonsPublishingHouse:Brussels,Belgium,2017.

13.Ahn,S.S.;Ko,J.M.RoughfuzzyidealsinBCK/BCI-algebras. J.Comput.Anal.Appl. 2018, 1,75–84.

14.Dudek,W.A.;Jun,Y.B.Pseudo-BCIalgebras. EastAsianMath.J. 2008, 24,187–190.

15.Iséki,K.Analgebrarelatedwithapropositionalcalculus. Proc.Jpn.Acad. 1966, 42,26–29.[CrossRef]

16.Huang,Y BCI-Algebra;SciencePress:Beijing,China,2006.

17.Jun,Y.B.; Kim,H.S.;Neggers,J.Onpseudo-BCIidealsofpseudo-BCIalgebras. Mat.Vesn. 2006, 58,39–46.

18.Kim,H.S.; Kim,Y.H.OnBE-algebras. Sci.Math.Jpn. 2007, 66,113–116.

19.Xin,X.L.;Li,Y.J.;Fu,Y.L.Statesonpseudo-BCIalgebras. Eur.J.PureAppl.Math. 2017, 10,455–472.

20.Zhang,X.H.;Ye,R.F.BZ-algebraandgroup. J.Math.Phys.Sci. 1995, 29,223–233.

21. Zhang,X.H.;Wang,Y.Q.;Dudek,W.A.T-idealsinBZ-algebrasandT-typeBZ-algebras. IndianJ.PureAppl.Math. 2003, 34,1559–1570.

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Zhang,X.H.Fuzzyanti-groupedﬁltersandfuzzynormalﬁltersinpseudo-BCIalgebras. J.Intell.FuzzySyst. 2017, 33,1767–1774.[CrossRef]

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Florentin Smarandache (author and editor) Collected Papers, IX 195

Further results on

neutrosophic subalgebras and ideals in BCK/BCI-algebras

G.Muhiuddin, Hashem Bordbar, Florentin Smarandache, Young Bae Jun

G. Muhiuddin, Hashem Bordbar, Florentin Smarandache, Young Bae Jun (2018). Further results on (ε, ε,)-neutrosophic subalgebras and ideals in BCK/BCI-algebras. Neutrosophic Sets and Systems 20, 36-43.

Abstract: Characterizationsofan (∈, ∈)-neutrosophicidealare considered.Anyidealin a BCK/BCI-algebrawillberealizedas levelneutrosophicidealsofsome (∈, ∈)-neutrosophicideal.Therelationbetween (∈, ∈)-neutrosophicidealand (∈, ∈)-neutrosophic subalgebraina BCK-algebra isdiscussed.Conditionsforan (∈,

Keywords: (∈, ∈)-neutrosophicsubalgebra, (∈, ∈)-neutrosophicideal.

1Introduction

Neutrosophicset(NS)developedbySmarandache[8, 9, 10]introducedneutrosophicset (NS)asamoregeneralplatformwhich extendstheconceptsoftheclassicsetandfuzzyset,intuitionisticfuzzysetandintervalvaluedintuitionisticfuzzyset.Neutrosophicsettheoryisappliedtovariouspartwhichisreferedtothe site

http://fs.gallup.unm.edu/neutrosophy.htm.

Junetal.studiedneutrosophicsubalgebras/idealsin BCK/BCI-algebrasbasedonneutrosophicpoints(see[1],[5] and[7]).

Inthis paper,we characterizean (∈, ∈)-neutrosophicidealina BCK/BCI-algebra.Weshowthatanyidealina BCK/BCI algebracanberealizedaslevelneutrosophicidealsofsome (∈, ∈)-neutrosophicideal.Weinvestigatetherelationbetween (∈, ∈)-neutrosophic idealand (∈, ∈)-neutrosophicsubalgebra ina BCK-algebra.Weprovideconditionsforan (∈, ∈) neutrosophicsubalgebrato bea (∈, ∈)-neutrosophicideal.Using a collectionofidealsin a BCK/BCI-algebra,weestablishan (∈, ∈)-neutrosophicideal.Wediscussequivalencerelationson thefamilyof all (∈, ∈)-neutrosophicideals,andinvestigaterelatedproperties.

2Preliminaries

A BCK/BCI-algebraisanimportantclassoflogicalalgebras introducedbyK.Is´eki(see[2]and[3])andwasextensivelyin-

∈)-neutrosophicsubalgebratobea (∈, ∈)-neutrosophicidealare provided.Usingacollection ofidealsina BCK/BCI-algebra,an (∈, ∈)-neutrosophicidealisestablished.Equivalencerelationson thefamilyofall (∈, ∈)-neutrosophicidealsareintroduced,andrelatedpropertiesareinvestigated.

vestigatedbyseveralresearchers.

Bya BCI-algebra,wemeanaset X withaspecialelement 0 and abinaryoperation ∗ thatsatisﬁesthefollowingconditions:

(I) (∀x,y,z ∈ X )(((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y) =0), (II) (∀x,y ∈ X )((x ∗ (x ∗ y)) ∗ y =0), (III) (∀x ∈ X )(x ∗ x =0), (IV) (∀x,y ∈ X )(x ∗ y =0,y ∗ x =0 ⇒ x = y)

Ifa BCI-algebra X satisﬁesthefollowingidentity: (V) (∀x ∈ X )(0 ∗ x =0), then X iscalleda BCK-algebra. Any BCK/BCI-algebra X satisﬁesthefollowingconditions:

(∀x ∈ X )(x ∗ 0= x) , (2.1) (∀x,y,z ∈ X ) x ≤ y ⇒ x ∗ z ≤ y ∗ z x ≤ y ⇒ z ∗ y ≤ z ∗ x , (2.2) (∀x,y, z ∈ X )((x ∗ y) ∗ z =(x ∗ z) ∗ y) , (2.3) (∀x,y,z ∈ X )((x ∗ z) ∗ (y ∗ z) ≤ x ∗ y) (2.4) where x ≤ y if andonlyif x ∗ y = 0. Anonemptysubset S ofa BCK/BCI-algebra X iscalleda subalgebra of X if x ∗ y ∈ S forall x,y ∈ S. Asubset I ofa BCK/BCI-algebra X iscalled an ideal of X ifitsatisﬁes:

0 ∈ I, (2.5) (∀x ∈ X )(∀y ∈ I )(x ∗ y ∈ I ⇒ x ∈ I ) . (2.6)

(∈, ∈)-

Smarandache (author and editor) Collected Papers, IX 196

Florentin

Wereferthereadertothebooks[4, 6]forfurtherinformation regarding BCK/BCI-algebras.

Foranyfamily {ai | i ∈ Λ} ofrealnumbers,wedeﬁne

{ai | i ∈ Λ} :=sup{ai | i ∈ Λ} and {ai | i ∈ Λ} :=inf{ai | i ∈ Λ}.

If Λ= {1, 2},wewillalsouse a1 ∨ a2 and a1 ∧ a2 insteadof {ai | i ∈ Λ} and {ai | i ∈ Λ},respectively.

Let X beanon-emptyset.A neutrosophicset (NS)in X (see [9])isastructureoftheform:

A∼ := { x; AT (x),AI (x),AF (x) | x ∈ X}

where AT : X → [0, 1] isatruthmembershipfunction, AI : X → [0, 1] isanindeterminatemembershipfunction,and AF : X → [0, 1] isafalsemembershipfunction.Forthesakeof simplicity,weshallusethesymbol A∼ =(AT ,AI ,AF ) forthe neutrosophicset

A∼ := { x; AT (x),AI (x),AF (x) | x ∈ X}

Givenaneutrosophicset A∼ =(AT ,AI ,AF ) inaset X, α,β ∈ (0, 1] and γ ∈ [0, 1),weconsiderthefollowingsets: T∈(A∼; α):= {x ∈ X | AT (x) ≥ α}, I∈(A∼; β):= {x ∈ X | AI (x) ≥ β}, F∈(A∼; γ):= {x ∈ X | AF (x) ≤ γ}. Wesay T∈(A∼; α), I∈(A∼; β) and F∈(A∼; γ) are neutrosophic ∈-subsets.

Aneutrosophicset A∼ =(AT ,AI ,AF ) ina BCK/BCI algebra X iscalledan (∈, ∈)-neutrosophicsubalgebra of X (see [5])ifthefollowingassertionsarevalid.

 (2.9) forall αx,αy ,βx,βy ∈ (0, 1] and γx,γy ∈ [0, 1)

3 (∈, ∈)-neutrosophicsubalgebrasand ideals

Weﬁrstprovidecharacterizationsofan (∈, ∈)-neutrosophic ideal.

Theorem3.1. Givenaneutrosophicset A∼ =(AT ,AI ,AF ) in a BCK/BCI-algebra X,thefollowingassertionsareequivalent.

(1) A∼ =(AT ,AI ,AF ) isan (∈, ∈)-neutrosophicidealof X. (2) A∼ =(AT ,AI ,AF ) satisﬁesthefollowingassertions. (∀x ∈ X )  

AT (0) ≥ AT (x), AI (0) ≥ AI (x), AF (0) ≤ AF (x)   (3.1) and (∀x,y ∈ X )  

AT (x) ≥ AT (x ∗ y) ∧ AT (y) AI (x) ≥ AI (x ∗ y) ∧ AI (y) AF (x) ≤ AF (x ∗ y) ∨ AF (y)   (3.2)

Proof. Assumethat A∼ =(AT ,AI ,AF ) isan (∈, ∈) neutrosophicidealof X.Supposethereexist a,b,c ∈ X be suchthat AT (0) <AT (a), AI (0) <AI (b) and AF (0) > AF (c).Then a ∈ T∈(A∼; AT (a)), b ∈ I∈(A∼; AI (b)) and c ∈ F∈(A∼; AF (c)).But

0 / ∈ T∈(A∼; AT (a)) ∩ I∈(A∼; AI (b)) ∩ F∈(A∼; AF (c))

Thisisacontradiction,andthus AT (0) ≥ AT (x), AI (0) ≥ AI (x) and AF (0) ≤ AF (x) forall x ∈ X.Supposethat AT (x) <AT (x ∗ y) ∧ AT (y), AI (a) <AI (a ∗ b) ∧ AI (b) and AF (c) >AF (c ∗ d) ∨ AF (d) forsome x,y,a,b,c,d ∈ X. Taking α := AT (x∗y)∧AT (y), β := AI (a∗b)∧AI (b) and γ := AF (c∗d)∨AF (d) implythat x∗y ∈ T∈(A∼; α), y ∈ T∈(A∼; α), a ∗ b ∈ I∈(A∼; β), b ∈ I∈(A∼; β), c ∗ d ∈ F∈(A∼; γ) and d ∈ F∈(A∼; γ).But x/ ∈ T∈(A∼; α), a/ ∈ I∈(A∼; β) and c/ ∈ F∈(A∼; γ).Thisisimpossible,andso(3.2)isvalid.

Conversely,suppose A∼ =(AT ,AI ,AF ) satisﬁestwoconditions(3.1)and(3.2).Forany x,y,z ∈ X,let α,β ∈ (0, 1] and γ ∈ [0, 1) besuchthat x ∈ T∈(A∼; α), y ∈ I∈(A∼; β) and

(∀x,y ∈ X )         x ∈ T∈(A∼; αx),y ∈ T∈

∼

αy ) ⇒ x ∗ y ∈ T∈

∼; αx ∧ αy ), x ∈ I∈(A∼; βx),y ∈ I∈

∼

y ) ⇒ x ∗ y ∈ I∈(A∼; βx ∧ βy ), x ∈ F∈(A∼; γx),y ∈ F∈(A∼; γy ) ⇒ x ∗ y ∈ F∈(A∼; γx ∨ γy )        

αx,αy ,βx,βy ∈ (0, 1] and γx,γy ∈

∼

(∀x ∈ X )   x ∈ T∈(A∼; αx) ⇒ 0 ∈ T∈

∼

x ∈ I∈(A∼; βx) ⇒ 0 ∈ I∈(A∼; βx) x ∈ F∈(A∼

γx) ⇒ 0 ∈ F∈

A∼

γx)  

      

⇒

    

;

; β

(2.7) forall

[0, 1) Aneutrosophicset A

=(AT ,AI ,AF ) ina BCK/BCI algebra X iscalledan (∈, ∈)-neutrosophicideal of X (see[7]) ifthefollowingassertionsarevalid.

; α

)

;

(

;

(2.8) and (∀x,y ∈ X ) 

x ∗ y ∈ T∈(A∼; αx),y ∈ T∈(A∼; αy )

x ∈ T∈(A∼; αx ∧ αy )

∗ y ∈ I∈(A∼; βx),y ∈ I∈(A∼; βy )

x ∈ I∈(A∼; βx ∧ βy )

∗ y ∈ F∈(A∼; γx),y ∈ F∈(A∼; γy )

x ∈ F∈(A∼; γx ∨ γy )

 

Florentin Smarandache (author and editor) Collected Papers, IX 197

z ∈ F∈(A∼; γ).Itfollowsfrom(3.1)that AT (0) ≥ AT (x) ≥ α, AI (0) ≥ AI (y) ≥ β and AF (0) ≤ AF (z) ≤ γ andsothat 0 ∈ T∈(A∼; α)∩I∈(A∼; β)∩F∈(A∼; γ).Let a,b,c,d,x,y ∈ X besuchthat a ∗ b ∈ T∈(A∼; αa), b ∈ T∈(A∼; αb), c ∗ d ∈ I∈(A∼; βc), d ∈ I∈(A∼; βd), x ∗ y ∈ F∈(A∼; γx),and y ∈ F∈(A∼; γy ) for αa,αb,βc,βd ∈ (0, 1] and γx,γy ∈ [0, 1).Using(3.2),wehave AT (a) ≥ AT (a ∗ b) ∧ AT (b) ≥ αa ∧ αb AI (c) ≥ AI (c ∗ d) ∧ AI (d) ≥ βc ∧ βd AF (x) ≤ AF (x ∗ y) ∨ AF (y) ≤ γx ∨ γy

Hence a ∈ T∈(A∼; αa ∧ αb), c ∈ I∈(A∼; βc ∧ βd) and x ∈ F∈(A∼; γx ∨ γy ).Therefore A∼ =(AT ,AI ,AF ) isan (∈, ∈) neutrosophicidealof X

Theorem3.2. Let A∼ =(AT ,AI ,AF ) beaneutrosophicset ina BCK/BCI-algebra X.Thenthefollowingassertionsare equivalent.

(1) A∼ =(AT ,AI ,AF ) isan (∈, ∈)-neutrosophicidealof X

(2) Thenonemptyneutrosophic ∈-subsets T∈(A∼; α), I∈(A∼; β) and F∈(A∼; γ) areidealsof X forall α,β ∈ (0, 1] and γ ∈ [0, 1).

Proof. Let A∼ =(AT ,AI ,AF ) bean (∈, ∈)-neutrosophicideal of X andassumethat T∈(A∼; α), I∈(A∼; β) and F∈(A∼; γ) are nonemptyfor α,β ∈ (0, 1] and γ ∈ [0, 1).Thenthereexist x,y,z ∈ X suchthat x ∈ T∈(A∼; α), y ∈ I∈(A∼; β) and z ∈ F∈(A∼; γ).Itfollowsfrom(2.8)that 0 ∈ T∈(A∼; α) ∩ I∈(A∼; β) ∩ F∈(A∼; γ). Let x,y,a,b,u,v ∈ X besuchthat x ∗ y ∈ T∈(A∼; α), y ∈ T∈(A∼; α), a ∗ b ∈ I∈(A∼; β), b ∈ I∈(A∼; β), u ∗ v ∈ F∈(A∼; γ) and v ∈ F∈(A∼; γ).Then AT (x) ≥ AT (x ∗ y) ∧ AT (y) ≥ α ∧ α = α AI (a) ≥ AI (a ∗ b) ∧ AI (b) ≥ β ∧ β = β AF (u) ≤ AF (u ∗ v) ∨ AF (v) ≤ γ ∨ γ = γ by(3.2),andso x ∈ T∈(A∼; α), a ∈ I∈(A∼; β) and u ∈ F∈(A∼; γ).Hencethenonemptyneutrosophic ∈-subsets T∈(A∼; α), I∈(A∼; β) and F∈(A∼; γ) areidealsof X forall α,β ∈ (0, 1] and γ ∈ [0, 1) Conversely,let A∼ =(AT ,AI ,AF ) beaneutrosophic setin X forwhich T∈(A∼; α), I∈(A∼; β) and F∈(A∼; γ) arenonemptyandareidealsof X forall α,β ∈ (0, 1] and γ ∈ [0, 1).Assumethat AT (0) <AT (x), AI (0) <AI (y) and AF (0) >AF (z) forsome x,y,z ∈ X.Then x ∈ T∈(A∼; AT (x)), y ∈ I∈(A∼; AI (y)) and z ∈ F∈(A∼; AF (z)), thatis, T∈(A∼; α), I∈(A∼; β) and F∈(A∼; γ) arenonempty. But 0 / ∈ T∈(A∼; AT (x)) ∩ I∈(A∼; AI (y)) ∩ F∈(A∼; AF (z)), whichisacontradictionsince T∈(A∼; AT (x)), I∈(A∼; AI (y)) and F∈(A∼; AF (z)) areidealsof X.Hence AT (0) ≥ AT (x), AI (0) ≥ AI (x) and AF (0) ≤ AF (x) forall x ∈ X.Suppose

that AT (x) <AT (x ∗ y) ∧ AT (y), AI (a) <AI (a ∗ b) ∧ AI (b), AF (u) >AF (u ∗ v) ∨ AF (v)

forsome x,y,a,b,u,v ∈ X.Taking α := AT (x ∗ y) ∧ AT (y), β := AI (a ∗ b) ∧ AI (b) and γ := AF (u ∗ v) ∨ AF (v) implythat α,β ∈ (0, 1], γ ∈ [0, 1), x ∗ y ∈ T∈(A∼; α), y ∈ T∈(A∼; α), a ∗ b ∈ I∈(A∼; β), b ∈ I∈(A∼; β), u ∗ v ∈ F∈(A∼; γ) and v ∈ F∈(A∼; γ).But x/ ∈ T∈(A∼; α), a/ ∈ I∈(A∼; β) and u/ ∈ F∈(A∼; γ).Thisisacontradictionsince T∈(A∼; α), I∈(A∼; β) and F∈(A∼; γ) areidealsof X.Thus

AT (x) ≥ AT (x ∗ y) ∧ AT (y), AI (x) ≥ AI (x ∗ y) ∧ AI (y), AF (x) ≤ AF (x ∗ y) ∨ AF (y)

forall x,y ∈ X.Therefore A∼ =(AT ,AI ,AF ) isan (∈, ∈)-neutrosophicidealof X byTheorem 3.1

Proposition3.3. Every (∈, ∈)-neutrosophicideal A∼ = (AT ,AI ,AF ) ofa BCK/BCI-algebra X satisﬁesthefollowingassertions.

(∀x,y ∈ X )  x ≤ y ⇒   

AT (x) ≥ AT (y) AI (x) ≥ AI (y) AF (x) ≤ AF (y)   , (3.3) (∀x,y,z ∈ X )  x ∗ y ≤ z ⇒   

AT (x) ≥ AT (y) ∧ AT (z) AI (x) ≥ AI (y) ∧ AI (z) AF (x) ≤ AF (y) ∨ AF (z)   (3.4)

Proof. Let x,y ∈ X besuchthat x ≤ y.Then x ∗ y =0,andso AT (x) ≥ AT (x ∗ y) ∧ AT (y)= AT (0) ∧ AT (y)= AT (y), AI (x) ≥ AI (x ∗ y) ∧ AI (y)= AI (0) ∧ AI (y)= AI (y), AF (x) ≤ AF (x ∗ y) ∨ AF (y)= AF (0) ∨ AF (y)= AF (y) byTheorem 3.1.Hence(3.3)isvalid.Let x,y,z ∈ X besuch that x ∗ y ≤ z.Then (x ∗ y) ∗ z =0,andthus AT (x) ≥ AT (x ∗ y) ∧ AT (y) ≥ (AT ((x ∗ y) ∗ z) ∧ AT (z)) ∧ AT (y) ≥ (AT (0) ∧ AT (z)) ∧ AT (y) ≥ AT (z) ∧ AT (y), AI (x) ≥ AI (x ∗ y) ∧ AI (y) ≥ (AI ((x ∗ y) ∗ z) ∧ AI (z)) ∧ AI (y) ≥ (AI (0) ∧ AI (z)) ∧ AI (y) ≥ AI (z) ∧ AI (y)

Florentin Smarandache (author and editor) Collected Papers, IX 198

AF (x) ≤ AF (x ∗ y) ∨ AF (y)

≤ (AF ((x ∗ y) ∗ z) ∨ AF (z)) ∨ AF (y)

≤ (AF (0) ∨ AF (z)) ∨ AF (y) ≤ AF (z) ∨ AF (y)

byTheorem 3.1

Theorem3.4. Anyidealofa BCK/BCI-algebra X canberealizedaslevelneutrosophicidealsofsome (∈, ∈)-neutrosophic idealof X.

Proof. Let I beanidealofa BCK/BCI-algebra X andlet A∼ =(AT ,AI ,AF ) beaneutrosophicsetin X givenasfollows:

AT : X → [0, 1],x → α if x ∈ I, 0 otherwise, AI : X → [0, 1],x → β if x ∈ I, 0 otherwise, AF : X → [0, 1],x → γ if x ∈ I, 1 otherwise

where (α,β,γ) isaﬁxedorderedtriplein (0, 1] × (0, 1] × [0, 1) Then T∈(A∼; α)= I, I∈(A∼; β)= I and F∈(A∼; γ)= I Obviously, AT (0) ≥ AT (x), AI (0) ≥ AI (x) and AF (0) ≤ AF (x) forall x ∈ X.Let x,y ∈ X.If x ∗ y ∈ I and y ∈ I,then x ∈ I.Hence

AT (x ∗ y)= AT (y)= AT (x)= α, AI (x ∗ y)= AI (y)= AI (x)= β, AF (x ∗ y)= AF (y)= AF (x)= γ, andso

AT (x) ≥ AT (x ∗ y) ∧ AT (y), AI (x) ≥ AI (x ∗ y) ∧ AI (y), AF (x) ≤ AF (x ∗ y) ∨ AF (y)

If x ∗ y/ ∈ I and y/ ∈ I,then

AT (x ∗ y)= AT (y)=0, AI (x ∗ y)= AI (y)=0, AF (x ∗ y)= AF (y)=1

If x ∗ y ∈ I and y/ ∈ I,then

AT (x ∗ y)= α and AT (y)=0, AI (x ∗ y)= β and AI (y)=0, AF (x ∗ y)= γ and AF (y)=1,

Itfollowsthat

AT (x) ≥ 0= AT (x ∗ y) ∧ AT (y), AI (x) ≥ 0= AI (x ∗ y) ∧ AI (y), AF (x) ≤ 1= AF (x ∗ y) ∨ AF (y)

Similarly,if x ∗ y/ ∈ I and y ∈ I,then

AT (x) ≥ AT (x ∗ y) ∧ AT (y), AI (x) ≥ AI (x ∗ y) ∧ AI (y), AF (x) ≤ AF (x ∗ y) ∨ AF (y)

Therefore A∼ =(AT ,AI ,AF ) isan (∈, ∈)-neutrosophicideal of X byTheorem 3.1.Thiscompletestheproof.

Lemma3.5([5]). Aneutrosophicset A∼ =(AT ,AI ,AF ) ina BCK/BCI-algebra X isan (∈, ∈)-neutrosophicsubalgebraof X ifandonlyifitsatisﬁes: (∀x,y ∈ X )   AT (x ∗ y) ≥ AT (x) ∧ AT (y) AI (x ∗ y) ≥ AI (x) ∧ AI (y) AF (x ∗ y) ≤ AF (x) ∨ AF (y)   (3.5)

Theorem3.6. Ina BCK-algebra,every (∈, ∈)-neutrosophic idealisan (∈, ∈)-neutrosophicsubalgebra.

Proof. Let A∼ =(AT ,AI ,AF ) bean (∈, ∈)-neutrosophicideal ofa BCK-algebra X.Since x∗y ≤ x forall x,y ∈ X,itfollows fromProposition 3.3 and(3.2)that

AT (x ∗ y) ≥ AT (x) ≥ AT (x ∗ y) ∧ AT (y) ≥ AT (x) ∧ AT (y), AI (x ∗ y) ≥ AI (x) ≥ AI (x ∗ y) ∧ AI (y) ≥ AI (x) ∧ AI (y), AF (x ∗ y) ≤ AF (x) ≤ AF (x ∗ y) ∨ AF (y) ≤ AF (x) ∨ AF (y)

Therefore A∼ =(AT ,AI ,AF ) isan (∈, ∈)-neutrosophicsubalgebraof X byLemma 3.5

ThefollowingexampleshowsthattheconverseofTheorem 3.6 isnottrueingeneral.

Thus

AT (x) ≥ AT (x ∗ y) ∧ AT (y), AI (x) ≥ AI (x ∗ y) ∧ AI (y), AF (x) ≤ AF (x ∗ y) ∨ AF (y)

Example3.7. Consideraset X = {0, 1, 2, 3} withthebinary operation ∗ whichisgiveninTable 1. Then (X ; ∗, 0) isa BCK-algebra(see[6]).Let A∼ =(AT ,AI , AF ) beaneutrosophicsetin X deﬁnedbyTable 2 Itisroutinetoverifythat A∼ =(AT ,AI ,AF ) isan (∈, ∈) neutrosophicsubalgebraof X.Weknowthat I∈(A∼; β) isan idealof X forall β ∈ (0, 1].If α ∈ (0 3, 0 7],then T∈(A∼; α)= {0, 1, 3} isnotanidealof X.Also,if γ ∈ [0 2, 0 8),then F∈(A∼; γ)= {0, 1, 3} isnotanidealof X.Therefore A∼ = (AT ,AI ,AF ) isnotan (∈, ∈)-neutrosophicidealof X byTheorem 3.2.

and

Florentin Smarandache (author and editor) Collected Papers, IX 199

Table1:Cayleytableforthebinaryoperation“∗”

∗ 0 1 2 3 0 0 0 0 0 1 1 0 0 1 2 2 1 0 2 3 3 3 3 0

ingtwocases:

α = {i ∈ ΛT | i<α} and α = {i ∈ ΛT | i<α} Firstcaseimpliesthat x ∈ T∈(A∼; α) ⇔ x ∈ Di forall i<α ⇔ x ∈∩{Di | i<α} (3.9)

Table2:Tabularrepresentationof A∼ =(AT ,AI ,AF )

XAT (x) AI (x) AF (x)

0 0 7 0 9 0 2 1 0 7 0 6 0 2 2 0 3 0 6 0 8 3 0 7 0 4 0 2

Wegiveaconditionforan (∈, ∈)-neutrosophicsubalgebrato bean (∈, ∈)-neutrosophicideal.

Theorem3.8. Let A∼ =(AT ,AI ,AF ) beaneutrosophicset ina BCK-algebra X.If A∼ =(AT ,AI ,AF ) isan (∈, ∈) neutrosophicsubalgebraof X thatsatisﬁesthecondition (3.4), thenitisan (∈, ∈)-neutrosophicidealof X.

Proof. Taking x = y in(3.5)andusing(III)inducethecondition (3.1).Since x ∗ (x ∗ y) ≤ y forall x,y ∈ X,itfollowsfrom(3.4) that AT (x) ≥ AT (x ∗ y) ∧ AT (y), AI (x) ≥ AI (x ∗ y) ∧ AI (y), AF (x) ≤ AF (x ∗ y) ∨ AF (y)

forall x,y ∈ X.Therefore A∼ =(AT ,AI ,AF ) isan (∈, ∈)-neutrosophicidealof X byTheorem 3.1.

Theorem3.9. Let {Dk | k ∈ ΛT ∪ ΛI ∪ ΛF } beacollectionof idealsofa BCK/BCI-algebra X,where ΛT , ΛI and ΛF are nonemptysubsetsof [0, 1],suchthat X = {Dα | α ∈ ΛT }∪{Dβ | β ∈ ΛI }∪{Dγ | γ ∈ ΛF }, (3.6) (∀i,j ∈ ΛT ∪ ΛI ∪ ΛF )(i>j ⇔ Di ⊂ Dj ) (3.7) Let A∼ =(AT ,AI ,AF ) beaneutrosophicsetin X deﬁnedas follows: AT : X → [0, 1],x → {α ∈ ΛT | x ∈ Dα}, AI : X → [0, 1],x → {β ∈ ΛI | x ∈ Dβ }, AF : X → [0, 1],x → {γ ∈ ΛF | x ∈ Dγ }. (3.8)

Then A∼ =(AT ,AI ,AF ) isan (∈, ∈)-neutrosophicidealof X Proof. Let α,β ∈ (0, 1] and γ ∈ [0, 1) besuchthat T∈(A∼; α) = ∅, I∈(A∼; β) = ∅ and F∈(A∼; γ) = ∅.Weconsiderthefollow-

Hence T∈(A∼; α)= ∩{Di | i<α},whichisanidealof X.For thesecondcase,weclaimthat T∈(A∼; α)= ∪{Di | i ≥ α} If x ∈∪{Di | i ≥ α},then x ∈ Di forsome i ≥ α.Thus AT (x) ≥ i ≥ α,andso x ∈ T∈(A∼; α).If x/∈∪{Di | i ≥ α}, then x/ ∈ Di forall i ≥ α.Since α = {i ∈ ΛT | i<α}, thereexists ε> 0 suchthat (α ε,α) ∩ ΛT = ∅.Hence x/ ∈ Di forall i>α ε,whichmeansthatif x ∈ Di then i ≤ α ε. Thus AT (x) ≤ α ε<α,andso x/ ∈ T∈(A∼; α).Therefore T∈(A∼; α)= ∪{Di | i ≥ α} whichisanidealof X since {Dk } formsachain.Similarly,wecanverifythat I∈(A∼; β) isanideal of X.Finally,weconsiderthefollowingtwocases:

γ = {j ∈ ΛF | γ<j} and γ = {j ∈ ΛF | γ<j} Fortheﬁrstcase,wehave x ∈ F∈(A∼; γ) ⇔ x ∈ Dj forall j>γ ⇔ x ∈∩{Dj | j>γ}, (3.10)

andthus F∈(A∼; γ)= ∩{Dj | j>γ} whichisanidealof X Thesecondcaseimpliesthat F∈(A∼; γ)= ∪{Dj | j ≤ γ}.In fact,if x ∈∪{Dj | j ≤ γ},then x ∈ Dj forsome j ≤ γ.Thus AF (x) ≤ j ≤ γ,thatis, x ∈ F∈(A∼; γ).Hence ∪{Dj | j ≤ γ}⊆ F∈(A∼; γ).Nowif x/∈∪{Dj | j ≤ γ},then x/ ∈ Dj for all j ≤ γ.Since γ = {j ∈ ΛF | γ<j},thereexists ε> 0 suchthat (γ,γ+ε)∩ΛF isempty.Hence x/ ∈ Dj forall j<γ+ε, andsoif x ∈ Dj ,then j ≥ γ + ε.Thus AF (x) ≥ γ + ε>γ,and hence x/ ∈ F∈(A∼; γ).Thus F∈(A∼; γ) ⊆∪{Dj | j ≤ γ},and therefore F∈(A∼; γ)= ∪{Dj | j ≤ γ} whichisanidealof X Consequently, A∼ =(AT ,AI ,AF ) isan (∈, ∈)-neutrosophic idealof X byTheorem 3.2

Amapping f : X → Y of BCK/BCI-algebrasiscalled a homomorphism if f (x ∗ y)= f (x) ∗ f (y) forall x,y ∈ X Notethatif f : X → Y isahomomorphismof BCK/BCI algebras,then f (0)=0.Givenahomomorphism f : X → Y of BCK/BCI-algebrasandaneutrosophicset A∼ =(AT ,AI , AF ) in Y ,wedeﬁneaneutrosophicset Af ∼ =(Af T ,Af I ,Af F ) in X,whichiscalledthe inducedneutrosophicset,asfollows:

Af T : X → [0, 1],x → AT (f (x)),

Af I : X → [0, 1],x → AI (f (x)), Af F : X → [0, 1],x → AF (f (x))

Theorem3.10. Let f : X → Y beahomomorphismof BCK/BCI-algebras.If A∼ =(AT ,AI ,AF ) isan (∈, ∈)-neutrosophicidealof Y ,thentheinducedneutrosophicset

Florentin Smarandache (author and editor) Collected Papers, IX 200

Af ∼ =(Af T ,Af I ,Af F ) in X isan (∈, ∈)-neutrosophicidealof X Proof. Forany x ∈ X,wehave

Af T (x)= AT (f (x)) ≤ AT (0)= AT (f (0))= Af T (0),

Af I (x)= AI (f (x)) ≤ AI (0)= AI (f (0))= Af I (0), Af F (x)= AF (f (x)) ≥ AF (0)= AF (f (0))= Af F (0) Let x,y ∈ X.Then

Af T (x ∗ y) ∧ Af T (y)= AT (f (x ∗ y)) ∧ AT (f (y)) = AT (f (x) ∗ f (y)) ∧ AT (f (y))

≤ AT (f (x))= Af T (x),

Af I (x ∗ y) ∧ Af I (y)= AI (f (x ∗ y)) ∧ AI (f (y)) = AI (f (x) ∗ f (y)) ∧ AI (f (y))

≤ AI (f (x))= Af I (x), and Af F (x ∗ y) ∨ Af F (y)= AF (f (x ∗ y)) ∨ AF (f (y))

= AF (f (x) ∗ f (y)) ∨ AF (f (y))

≥ AF (f (x))= Af F (x)

Therefore Af ∼ =(Af T ,Af I ,Af F ) isan (∈, ∈)-neutrosophicideal of X byTheorem 3.1

Theorem3.11. Let f : X → Y beanontohomomorphismof BCK/BCI-algebrasandlet A∼ =(AT ,AI ,AF ) beaneutrosophicsetin Y .Iftheinducedneutrosophicset Af ∼ =(Af T ,Af I , Af F ) in X isan (∈, ∈)-neutrosophicidealof X,then A∼ =(AT , AI ,AF ) isan (∈, ∈)-neutrosophicidealof Y

Proof. Assumethattheinducedneutrosophicset Af ∼ =(Af T , Af I ,Af F ) in X isan (∈, ∈)-neutrosophicidealof X.Forany x ∈ Y ,thereexists a ∈ X suchthat f (a)= x since f isonto. Using(3.1),wehave

AT (x)= AT (f (a))= Af T (a) ≤ Af T (0)= AT (f (0))= AT (0), AI (x)= AI (f (a))= Af I (a) ≤ Af I (0)= AI (f (0))= AI (0), AF (x)= AF (f (a))= Af F (a) ≥ Af F (0)= AF (f (0))= AF (0)

Let x,y ∈ Y .Then f (a)= x and f (b)= y forsome a,b ∈ X Itfollowsfrom(3.2)that

AT (x)= AT (f (a))= Af T (a)

≥ Af T (a ∗ b) ∧ Af T (b)

= AT (f (a ∗ b)) ∧ AT (f (b))

= AT (f (a) ∗ f (b)) ∧ AT (f (b))

= AT (x ∗ y) ∧ AT (y),

AI (x)= AI (f (a))= Af I (a)

≥ Af I (a ∗ b) ∧ Af I (b)

= AI (f (a ∗ b)) ∧ AI (f (b))

= AI (f (a) ∗ f (b)) ∧ AI (f (b))

= AI (x ∗ y) ∧ AI (y), and

AF (x)= AF (f (a))= Af F (a)

≤ Af F (a ∗ b) ∨ Af F (b)

= AF (f (a ∗ b)) ∨ AF (f (b))

= AF (f (a) ∗ f (b)) ∨ AF (f (b))

= AF (x ∗ y) ∨ AF (y)

Therefore A∼ =(AT ,AI ,AF ) isan (∈, ∈)-neutrosophicideal of Y byTheorem 3.1

Let N(∈,∈) (X ) bethecollectionofall (∈, ∈)-neutrosophic idealsof X andlet α,β ∈ (0, 1] and γ ∈ [0, 1).Deﬁnebinary relations Rα T , Rβ I and Rγ F on N(∈,∈) (X ) asfollows:

AT Rα T BT ⇔ T∈(A∼; α)= T∈(B∼; α) AI Rβ I BI ⇔ I∈(A∼; β)= I∈(B∼; β) AF Rγ F BF ⇔ F∈(A∼; γ)= F∈(B∼; γ) (3.11)

forall A∼ =(AT ,AI ,AF ) and B∼ =(BT ,BI ,BF ) in N(∈,∈) (X )

Clearly Rα T , Rβ I and Rγ F areequivalencerelationson N(∈,∈) (X ).Forany A∼ =(AT ,AI ,AF ) ∈N(∈,∈) (X ), let [A∼]T (resp., [A∼]I and [A∼]F )denotetheequivalence classof A∼ =(AT ,AI ,AF ) in N(∈,∈) (X ) under Rα T (resp., Rβ I and Rγ F ).Denoteby N(∈,∈) (X )/Rα T , N(∈,∈) (X )/Rβ I and N(∈,∈) (X )/Rγ F thecollectionofallequivalenceclassesunder Rα T , Rβ I and Rγ F ,respectively,thatis,

N(∈,∈) (X )/Rα T = {[A∼]T | A∼ =(AT ,AI ,AF ) ∈N(∈,∈) (X ), N(∈,∈) (X )/Rβ I = {[A∼]I | A∼ =(AT ,AI ,AF ) ∈N(∈,∈) (X ), N(∈,∈) (X )/Rγ F = {[A∼]F | A∼ =(AT ,AI ,AF ) ∈N(∈,∈) (X ).

Nowlet I(X ) denotethefamilyofallidealsof X.Define maps fα, gβ and hγ from N(∈,∈) (X ) to I(X ) ∪{∅} by fα(A∼)= T∈(A∼; α), gβ (A∼)= I∈(A∼; β) and hγ (A∼)= F∈(A∼; γ), respectively,forall A∼ =(AT ,AI ,AF ) in N(∈,∈) (X ).Then fα, gβ and hγ areclearlywell-defined. Theorem3.12. Forany α,β ∈ (0, 1] and γ ∈ [0, 1),themaps fα, gβ and hγ aresurjectivefrom N(∈,∈) (X ) to I(X ) ∪{∅} Proof. Let 0∼ :=(0T , 0I , 1F ) beaneutrosophicsetin X where 0T , 0I and 1F arefuzzysetsin X definedby 0T (x)=0, 0I (x)=0 and 1F (x)=1 forall x ∈ X.Obviously, 0∼ :=(0T , 0I , 1F ) isan (∈, ∈)-neutrosophicidealof X. Also, fα(0∼)= T∈(0∼; α)= ∅, gβ (0∼)= I∈(0∼; β)= ∅

Florentin Smarandache (author and editor) Collected Papers, IX 201

and hγ (0∼)= F∈(0∼; γ)= ∅.Foranyideal I of X,let A∼ =(AT ,AI ,AF ) bethe (∈, ∈)-neutrosophicidealof X intheproofofTheorem 3.4.Then fα(A∼)= T∈(A∼; α)= I, gβ (A∼)= I∈(A∼; β)= I and hγ (A∼)= F∈(A∼; γ)= I Therefore fα, gβ and hγ aresurjective.

Theorem3.13. Thequotientsets N(∈,∈) (X )/Rα T , N(∈,∈) (X )/Rβ I and N(∈,∈) (X )/Rγ F areequivalentto I(X ) ∪{∅} forany α,β ∈ (0, 1] and γ ∈ [0, 1) Proof. Let A∼ =(AT ,AI ,AF ) ∈N(∈,∈) (X ).Forany α,β ∈ (0, 1] and γ ∈ [0, 1),deﬁne f ∗ α : N(∈,∈) (X )/Rα T →I(X ) ∪{∅}, [A∼]T → fα(A∼), g∗ β : N(∈,∈) (X )/Rβ I →I(X ) ∪{∅}, [A∼]I → gβ (A∼), h∗ γ : N(∈,∈) (X )/Rγ F →I(X ) ∪{∅}, [A∼]F → hγ (A∼)

Assumethat fα(A∼)= fα(B∼), gβ (A∼)= gβ (B∼) and hγ (A∼)= hγ (B∼) for B∼ =(BT ,BI ,BF ) ∈N(∈,∈) (X ) Then T∈(A∼; α)= T∈(B∼; α), I∈(A∼; β)= I∈(B∼; β) and F∈(A∼; γ)= F∈(B∼; γ) whichimplythat AT Rα T BT , AI Rβ I BI and AF Rγ F BF .Hence [A∼]T =[B∼]T , [A∼]I =[B∼]I and [A∼]F =[B∼]F .Therefore f ∗ α, g∗ β and h∗ γ areinjective.Considerthe (∈, ∈)-neutrosophicideal 0∼ :=(0T , 0I , 1F ) of X whichisgivenintheproofofTheorem 3.12.Then f ∗ α([0∼]T )= fα(0∼)= T∈(0∼; α)= ∅, g∗ β ([0∼]I )= gβ (0∼)= I∈(0∼; β)= ∅,and h∗ γ ([0∼]F )= hγ (0∼)= F∈(0∼; γ)= ∅ Foranyideal I of X,considerthe (∈, ∈)-neutrosophicideal A∼ =(AT ,AI ,AF ) of X intheproofofTheorem 3.4.Then f ∗ α([A∼]T )= fα(A∼)= T∈(A∼; α)= I, g∗ β ([A∼]I )= gβ (A∼)= I∈(A∼; β)= I,and h∗ γ ([A∼]F )= hγ (A∼)= F∈(A∼; γ)= I.Hence f ∗ α, g∗ β and h∗ γ aresurjective,andthe proofisover.

Forany α,β ∈ [0, 1],wedeﬁneanotherrelations Rα and Rβ on N(∈,∈) (X ) asfollows:

(A∼,B∼) ∈Rα ⇔ T∈(A∼; α) ∩ F∈(A∼; α) = T∈(B∼; α) ∩ F∈(B∼; α), (A∼,B∼) ∈Rβ ⇔ I∈(A∼; β) ∩ F∈(A∼; β) = I∈(B∼; β) ∩ F∈(B∼; β) (3.12)

forall A∼ =(AT ,AI ,AF ) and B∼ =(BT ,BI ,BF ) in N(∈,∈) (X ).Thentherelations Rα and Rβ arealsoequivalence relationson N(∈,∈) (X ) Theorem3.14. Given α,β ∈ (0, 1),wedeﬁnetwomaps ϕα : N(∈,∈) (X ) →I(X ) ∪{∅}, A∼ → fα(A∼) ∩ hα(A∼), ϕβ : N(∈,∈) (X ) →I(X ) ∪{∅}, A∼ → gβ (A∼) ∩ hβ (A∼) (3.13) foreach A∼ =(AT ,AI ,AF ) ∈N(∈,∈) (X ).Then ϕα and ϕβ aresurjective.

Proof. Considerthe (∈, ∈)-neutrosophicideal 0∼ :=(0T , 0I , 1F ) of X whichisgivenintheproofofTheorem 3.12.Then

ϕα(0∼)= fα(0∼) ∩ hα(0∼)= T∈(0∼; α) ∩ F∈(0∼; α)= ∅, ϕβ (0∼)= gβ (0∼) ∩ hβ (0∼)= I∈(0∼; β) ∩ F∈(0∼; β)= ∅

Foranyideal I of X,considerthe (∈, ∈)-neutrosophicideal A∼ =(AT ,AI ,AF ) of X intheproofofTheorem 3.4.Then ϕα(A∼)= fα(A∼) ∩ hα(A∼) = T∈(A∼; α) ∩ F∈(A∼; α)= I and ϕβ (A∼)= gβ (A∼) ∩ hβ (A∼) = I∈(A∼; β) ∩ F∈(A∼; β)= I. Therefore ϕα and ϕβ aresurjective.

Theorem3.15. Forany α,β ∈ (0, 1),thequotientsets N(∈,∈) (X )/ϕα and N(∈,∈) (X )/ϕβ areequivalentto I(X ) ∪ {∅}

Proof. Given α,β ∈ (0, 1),deﬁnetwomaps ϕ∗ α and ϕ∗ β asfollows:

ϕ∗ α : N(∈,∈) (X )/ϕα →I(X ) ∪{∅}, [A∼]Rα → ϕα(A∼), ϕ∗ β : N(∈,∈) (X )/ϕβ →I(X ) ∪{∅}, [A∼]Rβ → ϕβ (A∼). If ϕ∗ α ([A∼]Rα )= ϕ∗ α ([B∼]Rα ) and ϕ∗ β [A∼]Rβ = ϕ∗ β [B∼]Rβ forall [A∼]Rα , [B∼]Rα ∈N(∈,∈) (X )/ϕα and [A∼]Rβ , [B∼]Rβ ∈N(∈,∈) (X )/ϕβ ,then fα(A∼) ∩ hα(A∼)= fα(B∼) ∩ hα(B∼) and gβ (A∼) ∩ hβ (A∼)= gβ (B∼) ∩ hβ (B∼), thatis, T∈(A∼; α) ∩ F∈(A∼; α)= T∈(B∼; α) ∩ F∈(B∼; α) and I∈(A∼; β) ∩ F∈(A∼; β)= I∈(B∼; β) ∩ F∈(B∼; β). Hence (A∼,B∼) ∈Rα and (A∼,B∼) ∈Rβ .Itfollowsthat [A∼]Rα =[B∼]Rα and [A∼]Rβ =[B∼]Rβ .Thus ϕ∗ α and ϕ∗ β areinjective.Considerthe (∈, ∈)-neutrosophicideal 0∼ :=(0T , 0I , 1F ) of X whichisgivenintheproofofTheorem 3.12.Then ϕ ∗ α ([0∼]Rα )= ϕα(0∼)= fα(0∼) ∩ hα(0∼) = T∈(0∼; α) ∩ F∈(0∼; α)= ∅

Florentin Smarandache (author and editor) Collected Papers, IX 202

and ϕ ∗ β [0∼]Rβ = ϕβ (0∼)= gβ (0∼) ∩ hβ (0∼) = I∈(0∼; β) ∩ F∈(0∼; β)= ∅

Foranyideal I of X,considerthe (∈, ∈)-neutrosophicideal A∼ =(AT ,AI ,AF ) of X intheproofofTheorem 3.4.Then ϕ ∗ α ([A∼]Rα )= ϕα(A∼)= fα(A∼) ∩ hα(A∼) = T∈(A∼; α) ∩ F∈(A∼; α)= I and ϕ ∗ β [A∼]Rβ = ϕβ (A∼) = gβ (A∼) ∩ hβ (A∼) = I∈(A∼; β) ∩ F∈(A∼; β) = I.

Therefore ϕ∗ α and ϕ∗ β are surjective.Thiscompletestheproof.

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Florentin Smarandache (author and editor) Collected Papers, IX 203

Algebraic Structure of Neutrosophic Duplets in Neutrosophic Rings Z

∪ I , Q ∪ I and

W.B. Vasantha Kandasamy, Ilanthenral Kandasamy, Florentin Smarandache

R

∪ I

W.B. Vasantha Kandasamy, Ilanthenral Kandasamy, Florentin Smarandache (2018). Algebraic Structure of Neutrosophic Duplets in Neutrosophic Rings <Z U I>, <Q U I> and <R U I>. Neutrosophic Sets and Systems 23, 85-95

Abstract: The concept of neutrosophy and indeterminacy I was introduced by Smarandache, to deal with neutralies. Since then the notions of neutrosophic rings, neutrosophic semigroups and other algebraic structures have been developed. Neutrosophic duplets and their properties were introduced by Florentin and other researchers have pursued this study.In this paper authors determine the neutrosophic duplets in neutrosophic rings of characteristic zero. The neutrosophic duplets of Z ∪ I , Q ∪ I and R ∪ I ; the neutrosophic ring of integers, neutrosophic ring of rationals and neutrosophic ring of reals respectively have been analysed. It is proved the collection of neutrosophic duplets happens to be infinite in number in these neutrosophic rings. Further the collection enjoys a nice algebraic structure like a neutrosophic subring, in case of the duplets collection {a aI |a ∈ Z} for which 1 I acts as the neutral. For the other type of neutrosophic duplet pairs {a aI, 1 dI } where a ∈ R+ and d ∈ R, this collection under component wise multiplication forms a neutrosophic semigroup. Several other interesting algebraic properties enjoyed by them are obtained in this paper.

Keywords: Neutrosophic ring; neutrosophic duplet; neutrosophic duplet pairs; neutrosophic semigroup; neutrosophic subring

1Introduction

The concept of indeterminacy in the real world data was introduced by Florentin Smarandache [1, 2] as Neutrosophy. Existing neutralities and indeterminacies are dealt by the neutrosophic theory and are applied to real world and engineering problems [3, 4, 5]. Neutrosophic algebraic structures were introduced and studied by [6]. Since then several researchers have been pursuing their research in this direction [7, 8, 9, 10, 11, 12]. Neutrosophic rings [9] and other neutrosophic algebraic structures are elaborately studied in [6, 7, 8, 10]. Related theories of neutrosophic triplet, neutrosophic duplet, and duplet set was studied by Smarandache [13]. Neutrosophic duplets and triplets have interested many and they have studied [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]. Neutrosophic duplet semigroup [18], the neutrosophic triplet group [12], classical group

IX 204

Florentin Smarandache (author and editor) Collected Papers,

of neutrosophic triplet groups[22] and neutrosophic duplets of {Zpn, ×} and {Zpq, ×} [23] have been recently studied.

Herewemainlyintroducetheconceptofneutrosophicdupletsincaseofofneutrosophicringsofcharacteristiczeroandstudyonlythealgebraicpropertiesenjoyedbyneutrosophicduplets,neutralsandneutrosophic dupletpairs.

Inthispaperweinvestigatetheneutrosophicdupletsoftheneutrosophicrings Z ∪ I , Q ∪ I and R ∪ I .Weprovethedupletsforafixedneutralhappenstobeaninfinitecollectionandenjoysanicealgebraic structure.Infactthecollectionofneutralsforfixedduplethappenstobeinfiniteinnumberandtheytooenjoy anicealgebraicstructure.

Thispaperisorganisedintofivesections,sectiononeisintroductoryinnature.Importantresultsinthis paperaregiveninsectiontwoofthispaper.Neutrosophicdupletsoftheneutrosophicring Z ∪ I , and itspropertiesareanalysedinsectionthreeofthispaper.Intheforthsectionneutrosophicdupletsoftherings Q∪I and R∪I ;aredefined anddevelopedandseveraltheoremsareproved.Inthefinalsectiondiscussions, conclusionsandfutureresearchthatcanbecarriedoutisdescribed.

2Results

The basic definition of neutrosophic duplet is recalled from [12]. We just give the notations and describe the neutrosophic rings and neutrosophic semigroups [9].

Notation: Z ∪ I = {a + bI |a,b ∈ Z,I 2 = I } isthecollectionofneutrosophicintegerswhichisa neutrosophicringofintegers. Q ∪ I = {a + bI |a,b ∈ Q,I 2 = I } isthe collectionofneutrosophicrationals and R ∪ I = {a + bI |a,b ∈ R,I 2 = I } isthecollectionofneutrosophicrealswhichareneutrosophicring ofrationalsandrealsrespectively.

Let S be any ring which is commutative and has a unit element 1. Then S ∪ I = {a + bI|a, b ∈ S, I2 = I, +, ×} be the neutrosophic ring. For more refer [9].

Consider U tobetheuniverseofdiscourse,and D asetin U ,whichhasawell-deﬁnedlaw #.

Deﬁnition2.1. Consider a,neut(a) ,where a,and neut(a) belongto D.Itissaidtobeaneutrosophicduplet ifitsatisﬁesthefollowingconditions:

1. neut(a) isnotsameastheunitaryelementof D inrelationwiththe law # (ifany);

2. a# neut(a) =neut(a) # a =a;

3. anti(a) / ∈ Dforwhicha # anti(a)=anti(a) # a= neut(a).

Theresultsprovedinthispaperare

1.Allelementsoftheform a aI and aI a with 1 I astheneutralformsaneutrosophicduplet, a ∈ Z + \{0}.

2.Infact B = {a aI/a ∈ Z \{0}}∪{0},formsaneutrosophicsubringof S.

3.Let S = { Q∪I , +, ×} be theneutrosophic ring.Forevery nI with n ∈ Q\{0} wehave a+bI ∈ Q∪I with a + b =1; a,b ∈ Q \ {0}. suchthat {nI,a + bI } is aneutrosophicduplet.

4.Theidempotent x =1 I actsastheneutralforinﬁnitecollectionofelements a aI where a ∈ Q.

Florentin Smarandache (author and editor) Collected Papers, IX 205

5.Forevery a aI ∈ S where a ∈ Q, 1 dI actsasneutralsfor d ∈ Q.

6.Theorderedpairofneutrosophicduplets B = {(nI,m (m 1)I); n ∈ R,m ∈ R ∪{0}} formsa neutrosophicsemigroupof S = R ∪ I undercomponentwiseproduct.

7.Theorderedpairofneutrosophicduplets D = {(a aI, 1 dI); a ∈ R+; d ∈ R} formsaneutrosophic semigroupunderproducttakencomponentwise.

3Neutrosophicdupletsof Z ∪ I anditsproperties

Inthissectionweﬁndtheneutrosophicdupletsin Z ∪ I .Infactweprovethereareinﬁnitenumberofneutrals foranyrelevantelementin Z ∪ I .Severalinterestingresultsareproved. Firstweillustratesomeoftheneutrosophicdupletsin Z ∪ I

Example3.1. LetS= Z ∪ I = {a + bI |a,b ∈ I,I 2 = I } betheneutrosophicring.Consideranyelement x =9I ∈ Z ∪ I ;weseetheelement 16 15I ∈ Z ∪ I issuchthat 9I × 16 15I =144I 135I =9I = x. Thus 16 15I actsastheneutralof 9I and {9I, 16 15I } isaneutrosophicduplet. Cconsider 15I = y ∈ Z ∪ I ; 15I × 16 15I =15I = y. Thus {15I, 16 15I } isagainaneutrosophic duplet.Let 9I = s ∈ Z ∪ I ; 9I × 16 15I = 144I +135I = 9I = s,so {−9I, 16 15I } isa neutrosophicduplet.Thus {±9I, 16 15I } happenstobeneutrosophicduplets. Further nI ∈ Z ∪ I issuchthat nI × 16 15I =16nI 15nI = nI. Similarly nI × 16 15I = 16nI +15nI = nI. So {nI, 16 15I } isaneutrosophicdupletforall n ∈ Z \{0} Anothernatural questionwhichcomestoonemindiswill 16I 15 actasaneutralfor nI; n ∈ Z \{0},theanswerisyesfor nI × (16I 15)=16nI 15nI = nI.Hencetheclaim.

Wecall 0I =0 asthetrivialneutrosophicdupletas (0,x) isaneutrosophicdupletforall x ∈ Z ∪ I . Inviewofthisexampleweprovethefollowingtheorem.

Theorem3.2. Let S = Z ∪ I = {a + bI |a,b ∈ Z,I 2 = I } beaneutrosophicring.Every ±nI ∈ S; n ∈ Z \{0} hasinﬁnitenumberofneutralsoftheform

• mI (m 1)= x

• m (m 1)I = y

• (m 1) mI = x

• (m 1)I mI = y where m ∈ Z + \{1, 0}.

Proof. Consider nI ∈ Z ∪ I wesee nI × x = nI[mI (m 1)]= nnI nmI + nI = nI. Thus {nI,mI (m 1)} formaninﬁnitecollectionofneutrosophicdupletsforaﬁxed n andvarying m ∈ Z + \{0, 1}. Proofforotherparts(ii),(iii)and(iv)followsbyasimilarargument.

and editor) Collected Papers, IX 206

Florentin Smarandache (author

Thusinviewoftheabovetheoremwecansayforany nI; n ∈ Z \{0}, n isfixed;wehaveaninfinite collectionofneutralspavingwayforaninfinitecollectionofneutrosophicdupletscontributedbyelements x,y, x and y giveninthetheorem.Ontheotherhandforanyfixed x or y or x or y giveninthetheorem wehaveaninfinitecollectionofelementsoftheform nI; n ∈ Z \{0} suchthat {n,x,or y or x or y} isa neutrosophicduplet.

Nowourproblemistoﬁnddoestheseneutralscollection {x,y, x, y} intheoremsatisfyanynicealgebraicstructurein Z ∪ I

Weﬁrstillustratethisusingsomeexamplesbeforeweproposeandproveanytheorem.

Example3.3. Let S = Z ∪ I = {a + bI |a,b ∈ Z,I 2 = I } bethering. {S, ×} isacommutativesemigroup underproduct[].Considertheelement x =5I 4 ∈ Z ∪ I . 5I 4 actsasneutralforallelements nI ∈ Z ∪ I ,n ∈ Z \{0} Consider x × x =5I 4 × 5I 4=25I 20I 20I +16= 15I +16= x2 Now 15I +16 × nI = 15nI +16nI = nI. Thusif {nI,x} aneutrosophicdupletsois {nI,x2}.Consider x 3 = x 2 × x =( 15I +16) × (5I 4) = 75I +80I +60I 64=65I 64= x 3 nI × x 3 =65nI 64nI = nI So {nI, 65I 64} = {nI,x3} isaneutrosophicdupletforall n ∈ Z \{0} Consider

x 4 = x 3 × x =65I 64 × 5I 4 =325I 320I 260I +256= 255I +256= x 4

Clearly

nI × x 4 = nI × ( 255I +25)= 255nI +256nI = nI.

So {nI,x4} isaneutrosophicduplet.Infactonecanproveforany nI ∈ Z ∪ I ; n ∈ Z \{0} then x = m (m 1)I istheneutralof nI then {nI,x}, {nI,x2}, {nI,x3},..., {nI,xr },..., {nI,xt}; t ∈ Z + \{0} areallneutrosophicdupletsfor nI.Thusforanyﬁxed nI thereisaninﬁnitecollectionofneutrals.Wesee ifxisaneutralthenthecyclicsemigroupgeneratedby x denotedby x = {x,x2,x3 ,...} happenstobea collectionofneutralsfor nI ∈ S.

Nowweproceedontogiveexamplesofotherformsofneutrosophicdupletsusing Z ∪ I Example3.4. Let S = { Z ∪ I = {a + bI |a,b ∈ Z,I 2 = I }, +, ×} beaneutrosophicring.Wesee x =1 I ∈ S suchthat (1 I)2 =1 I × 1 I =1 2I + I 2(∵ I 2 = I) =1 I = x.

Thus x isanidempotentof S.Wesee y =5 5I suchthat y × x =(5 5I) × (5 5I)=5 5I 5I +5I =5 5I = y

Thus {5 5I, 1 I } isaneutrosophicdupletsand 1 I istheneutralof 5 5I. y 2 =5 5I × 5 5I =25 25I 25I +25I =25 25I

Florentin Smarandache (author and editor) Collected Papers, IX 207

Wesee {y2 , 1 I } isagainaneutrosophicduplet.

y 3 = y × y 2 =5 5I × (25 25I)=125 125I 125I +125I =125 125I = y 3

Onceagain {y3 , 1 I } isaneutrosophicduplet.Infactwecansayfortheidempotent 1 I thecyclicsemigroup B = {y,y2,y3 ,...} issuchthatforeveryelementin B, 1 I servesastheneutral.

Inviewofalltheseweprovethefollowingtheorem.

Theorem3.5. Let S = { Z ∪ I , +, ×} betheneutrosophicring.

1. 1 I isanidempotentof S

2.Allelementsoftheform a aI and aI a with 1 I astheneutralformsaneutrosophicduplet, a ∈ Z + \{0}

3.Infact B = {a aI/a ∈ Z \{0}}∪{0},formsaneutrosophicsubringof S.

Proof. 1.Let x =1 I ∈ S toshow x isanidempotentof S,wemustshow x × x = x.Wesee 1 I × 1 I =1 2I + I 2 as I 2 = I,weget 1 I × 1 I =1 I;hencetheclaim.

2.Let a aI ∈ S; a ∈ Z. 1 I istheneutralof a aI as a aI × 1 I = a aI aI + aI = a aI. Thus {a aI, 1 I } isaneutrosophicduplet.Onsimilarlines aI a willalsoyieldaneutrosophic dupletwith 1 I.Hencetheresult(ii).

3.Given B = {a aI |a ∈ Z} Toprove B isagroupunder+.Let x = a aI and y = b bI ∈ B; x + y = a aI + b bI =(a + b) (a + b)I as a + b ∈ Z; a + b (a + b)I ∈ B. So B isclosed undertheoperation+.When a =0 weget 0 0I =∈ B and a aI +0= a aI.0actsastheadditive identityof B.Forevery a aI ∈ B wehave (a aI)=( a) ( a)I = a + aI ∈ B issuchthat a aI +( a)+ aI =0 soevery a aI hasanadditiveinverse.Nowweshow {B, ×} isa semigroupunderproduct ×.

(a aI) × (b bI)= ab abI baI + abI = ab abI ∈ B.

Thus B isasemigroupunderproduct.Clearly 1 I ∈ B.Nowwetestthedistributivelaw.let x = a aI,y = b bI and z = c cI ∈ B.

(a aI) × [b bI + c cI]= a aI × [(b + c) (b + c)I = a(b + c) aI(b + c) (b + c)aI + a(b + c)I = a(b + c) aI(b + c) ∈ B

Thus {B, +, ×} isaneutrosophicsubringof S.Finallyweprove Z ∪ I hasneutrosophicdupletsof theform {a aI, 1+ dI }; d ∈ Z \{0}

Florentin Smarandache (author and editor) Collected Papers, IX 208

Theorem3.6. Let S = { Z ∪ I = {a + bI |a,b ∈ Z,I 2 = I }, +, ×} beaneutrosophicring a + bI ∈ S contributestoaneutrosophicdupletifandonlyif a = b.

Proof. Let a + bI ∈ S(a =0,b =0) beanelementwhichcontributesaneutrosophicdupletwith c + dI ∈ S. If {a + bI,c + dI } isaneutrosophicdupletthen (a + bI) × (c + dI)= a + bI,thisimplies

ac +(bd + ad + bc)I = a + bI.

Thisimplies ac = a and bd + ad + bc = b ac = a implies a(c 1)=0 since a =0 wehave c =1.Nowin bd + ad + bc = b substitute c =1;itbecomes bd + ad + b = b whichimplies bd + ad =0 thatis (b + a)d =0; d =0 forif d =0 then c + dI =1 actsasaneutral,forall a + bI ∈ S whichisatrivialneutrosophicduplet. Thus d =0,whichforces a + b =0 or a = b.hence a + bI = a aI. Nowwehavetoﬁnd d.Wehave (a aI)(1+ dI)= a aI + adI adI = a aI

Thisistrueforany d ∈ Z \{0}. Proofoftheconverseisdirect.

Nextweproceedontostudyneutrosophicdupletsof Q ∪ I and R ∪ I

4NeutrosophicDupletsof Q ∪ I and R ∪ I

Inthissectionwestudytheneutrosophicdupletsoftheneutrosophicrings Q∪I = {a+bI |a,b ∈ Q,I 2 = I }; where Q thefieldofrationalsand R ∪ I = {a + bI |a,b, ∈ R,I 2 = I };where R isthefieldofreals.We obtainseveralinterestingresultsinthisdirection.Itisimportanttonote Z ∪ I ⊂ Q ∪ I ⊂ R ∪ I .Hence allneutrosophicdupletsof Z ∪ I willcontinuetobeneutrosophicdupletsof Q ∪ I and R ∪ I .Our analysispertainstotheexistenceofotherneutrosophicdupletsas Z isonlyaringwhereas Q and R arefields. Weenumeratemanyinterestingpropertiesrelatedtothem.

Example4.1. Let S = { Q ∪ I = {a + bI |a,b ∈ Q,I 2 = I }, +, ×} betheneutrosophicringofrationals. Considerforany nI ∈Swehavetheneutral x = 7I 9 + 16 9 ∈ S, suchthat nI × x = nI 7I 9 + 16 9 = nI. Thusfortheelement nI theneutralis 7I 9 + 16 9 ∈ S. Wemakethefollowingobservation 7 9 + 16 9 =1

Infactallelementsoftheform a + bI in Q ∪ I with a + b =1; a,b ∈ Q \{0} canactasneutralsfor nI. Suppose x = 8I 9 + 1 9 ∈ Q ∪ I

Florentin Smarandache (author and editor) Collected Papers, IX 209

thenfor nI = y wesee

Take x = 9I +10 wesee

x × y = nI × 8I 9 + 1 9 = 8In 9 + nI 9 = nI.

x × y = 9I +0 × nI = 9In +10nI = nI andsoon.

Howeverwehaveprovedinsection3ofthispaperforany nI ∈ Z ∪ I thecollectionofallelements a + bI ∈ Z ∪ I with a + b =1; a,b ∈ Z \{0} willactasneutralsof nI. Inviewofalltheseweputforththefollowingtheorem.

Theorem4.2. Let S = { Q ∪ I , +, ×} betheneutrosophicring.Forevery nI with n ∈ Q \{0} wehave a + bI ∈ Q ∪ I with a + b =1; a,b ∈ Q \{0} suchthat {nI,a + bI } isneutrosophicduplet.

Proof. Given nI ∈ Q ∪ I ; n ∈ Q \{0},wehavetoshow a + bI isaneutralwhere a + b =1,a,b, ∈ Q \{0}. consider

nI × (a + bI)= anI + bnI =(a + b)nI = nI as a + b =1.Henceforanyfixed nI ∈ Q ∪ I wehaveaninfinitecollectionofneutrals.Furtherthenumberof suchneutrosophicdupletsareinfiniteinnumberforvarying n andvarying a,b ∈ Q \{0} with a + b =1. Thus thenumberofneutrosophicdupletsincaseofneutrosophicring Q ∪ I containsalltheneutrosophicduplets of Z ∪ I andthenumberofneutrosophicdupletsin Q ∪ I isabiggerinfinitethanthatoftheneutrosophic dupletsin Z ∪ I .Furtherall a + bI where a,b ∈ Q \ Z with a + b =1 happenstocontributetoneutrosophic dupletswhicharenotin Z ∪ I .

I let

x × y = 5 3 5I 3 × 1 I = 5 3 5I 3 5I 3 + 5I 3 = 5 3 5I 3 = x. Inviewofthisweproposethefollowingtheorem. Theorem4.3. Let S = { Q ∪ I = {a + bI |a,b ∈ Q,I 2 = I }, +, ×} betheneutrosophicringofrationals. 1.Theidempotent x =1 I actsastheneutralforinﬁnitecollectionofelements a aI where a ∈ Q. 2.Forevery a aI ∈ S where a ∈ Q, 1 dI actsasneutralsfor d ∈ Q. Proof.

a aI = x ∈ Q ∪ I ; a ∈ Q

x × y = a aI × 1 I = a aI aI +

= x. Florentin Smarandache (author and editor) Collected Papers, IX 210

Nowweproceedontogiveothertypesofneutrosopohicdupletsin Q ∪ I using 1 I theidempotent whichactsasneutral.Consider x = 5 3 5I 3 ∈ Q ∪

y =1 I,weﬁnd

Considerany

weseefor y =1 I theidempotentin Q ∪ I .

a aI

Thus 1 I actsastheneutralfor a aI;infact {a aI, 1I } isaneutrosophicduplet;forall a ∈ Q. Now consider s = p pI where p ∈ Q and r =1 dI ∈ Q ∪ I ; d ∈ Q.

S × r = p pI × 1 dI = p pI pdI + pdI = p pI = s

Thus {p pI, 1 dI } areneutrosophicdupletsforall p ∈ Q and d ∈ Q.Thecollectionofneutrosophic dupletswhicharein Q ∪ I \{ Z ∪ I } isinfactisofinﬁnitecardinality.

Nextwesearchofothertypesofneutrosophicdupletsin { Q ∪ I }.Suppose a + bI ∈ Q ∪ I andlet c + dI bethepossibleneutralforit,wearrivetheconditionson a,b,c and d

(a + bI) × (c + dI)= a + bI ac + bc + adI + bdI = a + bI ac = a whichispossibleifandonlyif c =1 Hence b + ad + bd = b ad + bd =0 d(a + b)=0 as d =0; a = b.

Thus a + bI = a aI areonlypossibleelementsin Q ∪ I whichcancontributetoneutrosophicdupletsand theneutralsassociatedwiththemisoftheform 1 ± dI and d ∈ Q \{0}. Thuswecansayevenincaseof R the fieldofrealsandfortheassociatedneutrosophicring R ∪ I .Allresultsaretrueincase Q ∪ I and Z ∪ I ; expect R ∪ I \ Q ∪ I hasinfinitedupletsand R ∪ I hasinfinitelymanymoreneutrosophicdupletsthan Q ∪ I .

Thefollowingtheoremonrealneutrsophicringsisbothinnovativeandintersting.

Theorem4.4. Let S = R ∪ I betherealneutrosophicring.Theneutrosophicdupletsarecontributedonlyby elementsoftheform nI and a aI where n ∈ R and a ∈ R+ withneutrals m (m 1)I and 1 dI; m,d ∈ R respectively. Proof. Consider {nI,m(m 1)I } thepair nI × m (m 1)I = nmI nmI + nI = nI forall n,m ∈ R \{1, 0} Thus {nI,m (m 1)I } isaninfinitecollectionofneutrosophicduplets.Wedefine (nI,m (m 1)I) asaneutrosophicdupletpair.Considerthepair {(a aI), (1 dI)}; a ∈ R+,d ∈ R.We see a aI × 1 dI = a aI daI + adI = a aI Thus {(a aI), (1 dI)} formsaninfinitecollectionofneutrosophicduplets.Wecall ((a aI), (1 dI)) as aneutrosophicdupletpair.Hencethetheorem.

Florentin Smarandache (author and editor) Collected Papers, IX 211

Theorem4.5. Let S = R ∪ I betheneutrosophicring

1.Theorderedpairofneutrosophicduplets B = {(nI,m (m 1)I); n ∈ R,m ∈ R ∪{0}} formsa neutrosophicsemigroupof S = R ∪ I undercomponentwiseproduct.

2.Theorderedpairofneutrosophicduplets D = {(a aI, 1 dI); a ∈ R+; d ∈ R} formaneutrosophic semigroupunderproducttakencomponentwise.

Proof. Given B = {(nI,m (m 1)I |n ∈ R,m ∈ (R \{1})}∪ (nI, 0) ⊆ ({ R ∪ I }, { R ∪ I }). Toprove B isaneutrosophicsemigroupof ( R ∪ I , R ∪ I ) .Forany x =(nI, (m (m 1)I) and y =(sI,t 9t 1)I) ∈ B weprove xy = yx ∈ B

x × y = xy =(nI,m (m 1)I × (sI,t (t 1)I) =(nsI, [m (m 1)I] × [t (t 1)I]) (nsI,mt t(m 1)I m(t 1)I +(m 1)(t 1)I) =(nsI,mt (mt 1)I) ∈ B

Itiseasilyveriﬁed xy = yx forall x,y ∈ B. Thus {B, ×} isaneutrosophicsemigroupofneutrosophicduplet pairs.Consider x,y ∈ D;weshow x × y ∈ D.Let x =(a aI, 1 dI) and y =(b bI, 1 cI) ∈ D x × y =(a aI, 1 dI) × (b bI, 1 cI) =(a aI × b bI, ( aI × 1 cI) =(ab abI abI + abI, 1 dI cI + cdI) =(ab abI, 1 (d + c cd)I) ∈ D as x × y isalsointheformof x and y.Hence D theneutrosophicdupletpairsformsaneutrosophicsemigroup undercomponentwiseproduct.

5DiscussionsandConclusions

Inthispaperthenotionofdupletsincaseneutrosophicrings, Z ∪I , Q∪I and R∪I ,havebeenintroduced andanalysed.Itisprovedthatthenumberofneutrosophicdupletsinallthesethreeringshappenstobean infinitecollection.Wefurtherprovethereareinfinitelymanyelementsforwhich 1 I happenstobethe neutral.Hereweestablishthedupletpair {a aI, 1 dI }; a ∈ R+ and d ∈ R happentobeaneutrosophic semigroupundercomponentwiseproduct.Thecollection {a aI } formsaneutrosophicsubring a ∈ Z or Q or R.Forfutureresearchwewanttoanalysewhethertheseneutrosophicringscanhaveneutrosophictriplets andifthatcollectionsenjoysomenicealgebraicproperty.Finallyweleaveitasanopenproblemtofindsome applicationsoftheseneutrosophicdupletswhichformaninfinitecollection.

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[10]A.A.A.Agboola,E.O.Adeleke,andS.A.Akinleye.NeutrosophicringsII.InternationalJournalofMathematicalCombinatorics,2(2012),1–12

[11]F.Smarandache.OperatorsonSingle-ValuedNeutrosophicOversets,NeutrosophicUndersets,andNeutrosophicOffsets.J. Math.Inf.,5(2016),63–67.

[12]F.Smarandache,andM.Ali.Neutrosophictripletgroup.NeuralComput.Appl.,29(2018),595–601,doi:10.1007/s00521016-2535-x.

[13]F Smarandache.NeutrosophicPerspectives:Triplets,Duplets,Multisets,HybridOperators,ModalLogic,HedgeAlgebras andApplications,2nded.;PonsPublishingHouse:Brussels,Belgium,2017;ISBN978-1-59973-531-3.

[14]M.SahinandK.Abdullah.Neutrosophictripletnormedspace.OpenPhys.,15(2017),697–704,doi:10.1515/phys-2017-0082.

[15]F.Smarandache.HybridNeutrosophicTripletRinginPhysicalStructures.Bull.Am.Phys.Soc.,62(2017),17.

[16]F.SmarandacheandM.Ali.NeutrosophicTripletFieldusedinPhysicalApplications.InProceedingsofthe18thAnnual MeetingoftheAPSNorthwestSection,PaciﬁcUniversity,ForestGrove,OR,USA,1–3June2017.

[17]F.Smarandache,andM.Ali.NeutrosophicTripletRinganditsApplications.InProceedingsofthe18thAnnualMeetingof theAPSNorthwestSection,PaciﬁcUniversity,ForestGrove,OR,USA,1–3June2017.

[18]X.H.Zhang,F.Smarandache,andX.L.Liang.NeutrosophicDupletSemi-GroupandCancellableNeutrosophicTriplet Groups.Symmetry,9(2017),275–291,doi:10.3390/sym9110275.

[19]M.Bal,M.M.Shalla,andN.Olgun.NeutrosophicTripletCosetsandQuotientGroups.Symmetry,10(2017),126–139, doi:10.3390/sym10040126.

[20]X.H.Zhang,F.Smarandache,M.Ali,andX.L.Liang.Commutativeneutrosophictripletgroupandneutro-homomorphism basictheorem.Ital.J.PureAppl.Math.(2017),inpress.

[21]W.B.Vasantha,I.Kandasamy,andF.Smarandache,NeutrosophicTripletGroupsandTheirApplicationstoMathematical Modelling;EuropaNova:Brussels,Belgium,2017;ISBN978-1-59973-533-7.

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[22]W.B.Vasantha,I.Kandasamy,andF.Smarandache.AClassicalGroupofNeutrosophicTripletGroupsUsing {Z2p , ×}. Symmetry,10(2018),194,doi:10.3390/sym10060194.

[23]W.B.Vasantha,I.Kandasamy,andF.Smarandache.Neutrosophicdupletsof {Zpn , ×} and {Zpq , ×}.Symmetry10(2018), 345,doi:10.3390/sym10080345.

[24]X.Zhang,Q.Hu,F.Smarandache,X.An,OnNeutrosophicTripletGroups:BasicProperties,NT-Subgroups,andSome Notes.Symmetry10(2018),289,doi:10.3390/sym10070289.

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Florentin

COMMUTATIVE NEUTROSOPHIC TRIPLET GROUP AND NEUTRO-HOMOMORPHISM BASIC THEOREM

Xiaohong Zhang,

Xiaohong Zhang, Florentin Smarandache, Mumtaz Ali, Xingliang Liang (2018). Commutative Neutrosophic Triplet Group and Neutro-Homomorphism Basic Theorem. Italian Journal of Pure and Applied Mathematics 40, 353-375.

Abstract. The neutrosophic triplet is a group of three elements that satisfy certain properties with some binary operations. The neutrosophic triplet group is completely diﬀerent from the classical group in the structural properties. In this paper, we further study neutrosophic triplet group. First, to avoid confusion, some new symbols are introduced, and several basic properties of neutrosophic triplet group are rigorously proved (because the original proof is ﬂawed), and a result about neutrosophic triplet subgroup is revised. Second, some new properties of commutative neutrosophic triplet group are funded, and a new equivalent relation is established. Third, based on the previous results, the following important proposi-tions are proved: from any commutative neutrosophic triplet group, an Abel group can be constructed; from any commutative neutrosophic triplet group, a BCI-algebra can be constructed.

1. Introduction

From a philosophical point of view, Florentin Smarandache introduced the con-cept of a neutrosophic set (see [12, 13, 14]). The neutrosophic set theory is applied to many scientific fields and also applied to algebraic structures (see [1, 3, 7, 10, 11, 15, 17, 19]). Recently, Florentin Smarandache and Mumtaz Ali in [16], for the first time, introduced the notions of neutrosophic triplet and neu-trosophic triplet group. The neutrosophic triplet is a group of three elements that satisfy certain properties with some binary operation. The neutrosophic triplet group is completely different from the classical group in the structural properties. In 2017, Florentin Smarandache has written the monograph [15] which is present the last developments in neutrodophic theories (including neu-trosophic triplet and neutrosophic triplet group).

In this paper, we further study neutrosophic triplet group. We discuss some new properties of commutative neutrosophic triplet group, and investigate the relationships among commutative neutrosophic triplet group, Abel group (that is, commutative group) and BCIalgebra. Moreover, we establish the quotient structure and neutro-homomorphism basic theorem.

As a guide, it is necessary to give a brief overview of the basic aspects of BCI-algebra and related algebraic systems. In 1966, K. Iseki introduced the concept of BCI-algebra as an algebraic counterpart of the BCI-logic (see [5, 24]). The algebraic structures closely related to BCI algebra are BCK-algebra, BCC-algebra, BZ-algebra, BE-algebra, and so on (see [2, 8, 20, 21, 22, 25]). As a generalization of BCI-algebra, W. A. Dudek and Y. B. Jun [4] introduced the notion of pseudo-BCI algebras. Moreover, pseudo-BCI algebra is also as a generalization of pseudo-BCK algebra (which is close connection with various non-commutative fuzzy logic formal systems, see [18, 22, 23, 24]). Recently, some articles related ﬁlter theory of pseudo-BCI algebras are published (see [26, 27, 28, 29]). As non-classical logic algebras, BCI-algebras are closely related to Abel groups (see [9]); similarly, BZ-algebras (pseudo-BCI algebras) are closely related general groups (see [20, 26]), and some results in [9, 20] will be applied in this paper.

Florentin Smarandache, Mumtaz Ali, Xingliang Liang

Smarandache (author and editor) Collected Papers, IX 215

Florentin

2.1On neutrosophictripletgroup

Deﬁnition2.1 ([16]). Let N beasettogetherwithabinaryoperation ∗.Then, N iscalleda neutrosophictripletset ifforany a ∈ N ,thereexistaneutralof “a”called neut(a),diﬀerentfromtheclassicalalgebraicunitaryelement,andan oppositeof“a”called anti(a),with neut(a)and anti(a)belongingto N ,such that:

a ∗ neut(a)= neut(a) ∗ a = a; a ∗ anti(a)= anti(a) ∗ a = neut(a)

Theelements a, neut(a)and anti(a)arecollectivelycalledasneutrosophic triplet,andwedenoteitby(a,neut(a),anti(a)).By neut(a),wemeanneutralofaandapparently, a isjusttheﬁrstcoordinateofaneutrosophictriplet andnotaneutrosophictriplet.Forthesameelement“a”in N ,theremaybe moreneutralstoit neut(a)andmoreoppositesofit anti(a).

Deﬁnition2.2 ([16]). Theelement b in(N, ∗)isthesecondcomponent,denoted as neut( ),ofaneutrosophictriplet,ifthereexistotherelements a and c in N suchthat a ∗ b = b ∗ a = a and a ∗ c = c ∗ a = b.Theformedneutrosophictriplet is(a,b,c).

Deﬁnition2.3 ([16]). Theelement c in(N, ∗)isthethirdcomponent,denoted as anti( ),ofaneutrosophictriplet,ifthereexistotherelements a and b in N suchthat a ∗ b = b ∗ a = a and a ∗ c = c ∗ a = b.Theformedneutrosophictriplet is(a,b,c).

Deﬁnition2.4 ([16]). Let(N, ∗)beaneutrosophictripletset.Then, N iscalled a neutrosophictripletgroup,ifthefollowingconditionsaresatisﬁed:

(1)If(N, ∗)iswell-deﬁned,i.e.forany a,b ∈ N ,onehas a ∗ b ∈ N

(2)If(N, ∗)isassociative,i.e.(a ∗ b) ∗ c = a ∗ (b ∗ c)forall a,b,c ∈ N .

Deﬁnition2.5 ([16]). Let(N, ∗)beaneutrosophictripletgroup.Then, N iscalleda commutativeneutrosophictripletgroup ifforall a,b ∈ N ,wehave a ∗ b = b ∗ a

Deﬁnition2.6 ([16]). Let(N, ∗)beaneutrosophictripletgroupunder ∗,and let H beasubsetof N .Then, H iscalledaneutrosophictripletsubgroupof N if H itselfisa neutrosophictripletgroup withrespectto ∗

Remark2.7. Inordertoincludericherstructure,theoriginalconceptofneutrosophictripletisgeneralizedtoneutrosophicextendedtripletbyFlorentin Smarandache.Aneutrosophicextendedtripletisaneutrosophictriplet,deﬁnedasabove,butwheretheneutralof x (called“extendedneutral”)isallowed

2.Somebasicconcepts

Florentin Smarandache (author and editor) Collected Papers, IX 216

toalsobe equaltotheclassicalalgebraicunitaryelement(ifany).Therefore,therestriction“diﬀerentfromtheclassicalalgebraicunitaryelementif any”isreleased.Asaconsequence,the“extendedopposite”of x,isalsoallowedtobeequaltotheclassicalinverseelementfromaclassicalgroup.Thus, aneutrosophicextendedtripletisanobjectoftheform(x,neut(x),anti(x)), for x ∈ N ,where neut(x) ∈ N istheextendedneutralof x,whichcanbe equalordiﬀerentfromtheclassicalalgebraicunitaryelementifany,suchthat: x ∗ neut(x)= neut(x) ∗ x = x,and anti(x) ∈ N istheextendedoppositeof x suchthat: x ∗ anti(x)= anti(x) ∗ x = neut(x).Inthispaper,“neutrosophic triplet”meansthat“neutrosophicextendedtriplet”.

2.2OnBCI-algebras

Deﬁnition2.8 ([5,23]). ABCI-algebraisanalgebra(X; →, 1)oftype(2, 0)in whichthefollowingaxiomsaresatisﬁed: (i)(x → y) → ((y → z) → (x → z))=1, (ii) x → x =1, (iii)1 → x = x, (iv)if x → y = y → x =1,then x = y

InanyBCI-algebra(X; →, 1)onecandeﬁnearelation ≤ byputting x ≤ y ifandonlyif x → y =1,then ≤ isapartialorderon X

Deﬁnition2.9 ([9,26]). Let(X; →, 1)beaBCI-algebra.Theset {x|x ≤ 1} is calledthe p-radical (orBCK-part)of X.ABCI-algebra X iscalled p-semisimple ifitsp-radicalisequalto {1}.

Proposition2.10 ([9]). Let (X; →, 1) beaBCI-algebra.Thenthefollowingare equivalent: (i) X isp-semisimple, (ii) x → 1=1 ⇒ x =1, (iii)(x → 1) → 1= x, ∀x ∈ X, (iv)(x → 1) → y =(y → 1) → x forall x,y ∈ X.

Proposition2.11 ([26]). Let (X; →, 1) beaBCI-algebra.Thenthefollowing areequivalent: (S1) X isp-semisimple, (S2) x → y =1 ⇒ x = y forall x,y ∈ X, (S3)(x → y) → (z → y)= z → x forall x,y,z ∈ X, (S4)(x → y) → 1= y → x forall x,y ∈ X, (S5)(x → y) → (a → b)=(x → a) → (y → b) forall x,y,a,b ∈ X.

Proposition2.12 ([9,26]). Let (X; →, 1) bep-semisimpleBCI-algebra;deﬁne + and asfollows:forall x,y ∈ X, x + y def=(x → 1) → y, x def = x → 1

Then (X;+, , 1) isanAbelgroup.

Florentin Smarandache (author and editor) Collected Papers, IX 217

Proposition2.13 ([9,26]). Let (X;+, , 1) beanAbelgroup.Deﬁne (X; ≤, →, 1),where

x → y = x + y,x ≤ y ifandonlyif x + y =1, ∀x,y ∈ X. Then, (X; ≤, →, 1) isaBCI-algebra.

3.Somepropertiesofneutrosophictripletgroup

Asmentionedearlier,foraneutrosophictripletgroup(N, ∗),if a ∈ N ,then neut(a)maynotbeunique,and anti(a)maynotbeunique.Thus,thesymbolic neut(a)sometimesmeansoneandsometimesmorethanone,whichisambiguous.Tothisend,thispaperintroducesthefollowingnotationstodistinguish: neut(a):denoteanycertainoneofneutralof a; {neut(a)}:denotethesetofallneutralof a. Similarly, anti(a):denoteanycertainoneofoppositeof a; {anti(a)}:denotethesetofalloppositeof a

Remark3.1. Inordernottocauseconfusion,wealwaysassumethat:(1) forthesame a,whenmultiple neut(a)(or anti(a))arepresentinthesame expression,theyarealwaysareconsistent.Ofcourse,iftheyareneutral(or opposite)ofdifferentelements,theyrefertodifferentobjects(forexample,in general, neut(a)isdifferentfrom neut(b)).(2)if neut(a)and anti(a)arepresent inthesameexpression,thentheyarematcheachother.

Proposition3.2. Let (N, ∗) beaneutrosophictripletgroupwithrespectto ∗ and a ∈ N .Then

neut(a) ∗ neut(a) ∈{neut(a)} Proof. Forany a ∈ N ,byDeﬁnition2.1wehave a ∗ neut(a)= a,neut(a) ∗ a = a.

Fromthis,usingassociativelaw,wecanget a ∗ (neut(a) ∗ neut(a))=(neut(a) ∗ neut(a)) ∗ a = a.

ByDeﬁnition2.1,itfollowsthat(neut(a) ∗ neut(a))isaneutralof a.Thatis, neut(a) ∗ neut(a) ∈{neut(a)}

Remark3.3. Thisproposition isarevisedversionofTheorem3.21(1)in[16]. If neut(a)isunique,thentheyaresame.But,if neut(a)isnotunique,they arediﬀerent.Forexample,assume {neut(a)} = {s,t},then neut(a)denoteany oneof s,t.Thus neut(a) ∗ neut(a)representsoneof s ∗ s,and t ∗ t.Moreover, Proposition3.2meansthat s ∗ s,t ∗ t ∈{neut(a)} = {s,t},thatis, s ∗ s = s, or s ∗ s = t; t ∗ t = s, or t ∗ t = t.

And,inthiscase,theequation neut(a) ∗neut(a)= neut(a)meansthat s ∗ s = s, t ∗ t = t.So,theyarediﬀerent.

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Proposition3.4. Let (N, ∗) beaneutrosophictripletgroupwithrespectto ∗ and a ∈ N .If neut(a) ∗ neut(a)= neut(a) Then neut(a) ∗ anti(a) ∈{anti(a)}; anti(a) ∗ neut(a) ∈{anti(a)}

Proof. Forany a ∈ N ,byDeﬁnition2.1wehave

a ∗ neut(a)= neut(a) ∗ a = a; a ∗ anti(a)= anti(a) ∗ a = neut(a)

Fromthis,usingassociativelaw,wecanget

a ∗ (neut(a) ∗ anti(a))=(a ∗ neut(a)) ∗ anti(a)= a ∗ anti(a)= neut(a) And, (neut(a) ∗ anti(a)) ∗ a = neut(a) ∗ (anti(a) ∗ a)= neut(a) ∗ neut(a)= neut(a)

ByDeﬁnition2.1,itfollowsthat(neut(a) ∗ anti(a))isaoppositeof a.Thatis, neut(a) ∗ anti(a) ∈{anti(a)}.Inthesameway,wecanget anti(a) ∗ neut(a) ∈ {anti(a)}

Proposition3.5. Let (N, ∗) beaneutrosophictripletgroupwithrespectto ∗ andlet a,b,c ∈ N .Then

(1) a ∗ b = a ∗ c ifandonlyif neut(a) ∗ b = neut(a) ∗ c.

(2) b ∗ a = c ∗ a ifandonlyif b ∗ neut(a)= c ∗ neut(a)

Proof. Assume a ∗ b = a ∗ c.Then anti(a) ∗ (a ∗ b)= anti(a) ∗ (a ∗ c).By associativelaw,wehave

(anti(a) ∗ a) ∗ b =(anti(a) ∗ a) ∗ c.

UsingDeﬁnition2.1weget neut(a) ∗ b = neut(a) ∗ c. Conversely,assume neut(a) ∗ b = neut(a) ∗ c.Then a ∗ (neut(a) ∗ b)= a ∗ (neut(a) ∗ c).Byassociativelaw,wehave

(a ∗ neut(a)) ∗ b =(a ∗ neut(a)) ∗ c.

UsingDeﬁnition2.1weget a ∗ b = a ∗ c.Thatis,(1)holds. Similarly,wecanprovethat(2)holds.

Proposition3.6. Let (N, ∗) beaneutrosophictripletgroupwithrespectto ∗ andlet a,b,c ∈ N

(1) If anti(a) ∗ b = anti(a) ∗ c,then neut(a) ∗ b = neut(a) ∗ c

Florentin Smarandache (author and editor) Collected Papers, IX 219

(2) If b ∗ anti(a)= c ∗ anti(a),then b ∗ neut(a)= c ∗ neut(a)

Proof. Assume anti(a)∗b = anti(a)∗c.Then a∗(anti(a) ∗b)= a∗(anti(a)∗c). Byassociativelaw,wehave

(a ∗ anti(a)) ∗ b =(a ∗ anti(a)) ∗ c.

UsingDeﬁnition2.1weget neut(a) ∗ b = neut(a) ∗ c.Itfollowsthat(1)holds. Similarly,wecanprovethat b ∗ anti(a)= c ∗ anti(a) ⇒ b ∗ neut(a)= c ∗ neut(a).

Theorem3.7. Let (N, ∗) beaneutrosophictripletgroupwithrespectto ∗ and a ∈ N .Then neut(neut(a)) ∈{neut(a)}

Proof. Forany a ∈ N ,byDeﬁnition2.1wehave

neut(a) ∗ neut(neut(a))= neut(a); neut(neut(a)) ∗ neut(a)= neut(a).

Then

a ∗ (neut(a) ∗ neut(neut(a)))= a ∗ neut(a); (neut(neut(a)) ∗ neut(a)) ∗ a = neut(a) ∗ a

ByassociativelawandDeﬁnition2.1,wehave

a ∗ neut(neut(a))= a; neut(neut(a)) ∗ a = a

Fromthis,byDeﬁnition2.1, neut(neut(a)) ∈{neut(a)}

Theorem3.8. Let (N, ∗) beaneutrosophictripletgroupwithrespectto ∗ and a ∈ N .Then neut(anti(a)) ∈{neut(a)}

Proof. Forany a ∈ N ,byDeﬁnition2.1wehave

anti(a) ∗ neut(anti(a))= anti(a); neut(anti(a)) ∗ anti(a)= anti(a).

Then

a ∗ (anti(a) ∗ neut(anti(a)))= a ∗ anti(a); (neut(anti(a)) ∗ anti(a)) ∗ a = anti(a) ∗ a

UsingassociativelawandDeﬁnition2.1,

neut(a) ∗ neut(anti(a))= neut(a); neut(anti(a)) ∗ neut(a)= neut(a).

Itfollowsthat a∗neut(anti(a))= a,neut(anti(a))∗a = a. Thatis, neut(anti(a)) ∈ {neut(a)}

Florentin Smarandache (author and editor) Collected Papers, IX 220

Theorem3.9. Let (N, ∗) beaneutrosophictripletgroupwithrespectto ∗ and a ∈ N .Then

neut(a) ∗ anti(anti(a))= a.

where, neut(a) ∈{neut(a)}, anti(a) ∈{anti(a)},and neut(a) matches anti(a), thatis, a ∗ anti(a)= anti(a) ∗ a = neut(a).

Proof. Forany a ∈ N ,byDeﬁnition2.1wehave

anti(a) ∗ anti(anti(a))= neut(anti(a))

Then

a ∗ (anti(a) ∗ anti(anti(a)))= a ∗ neut(anti(a)). (a ∗ anti(a)) ∗ anti(anti(a))= a ∗ neut(anti(a)). neut(a) ∗ anti(anti(a))= a ∗ neut(anti(a))

Ontheotherhand,byTheorem3.8, neut(anti(a)) ∈{neut(a)}.ByDeﬁnition 2.1,itfollowsthat a∗neut(anti(a))=a.Therefore, neut(a)∗anti(anti(a))=a.

Theorem3.10. Let (N, ∗) beaneutrosophictripletgroupwithrespectto ∗ and a ∈ N .Then anti(neut(a)) ∈{neut(a)}

Proof. Forany a ∈ N ,byDeﬁnition2.1wehave

neut(a) ∗ anti(neut(a))= neut(neut(a)); anti(neut(a)) ∗ neut(a)= neut(neut(a)).

Thus

a ∗ (neut(a) ∗ anti(neut(a)))= a ∗ neut(neut(a)); (anti(neut(a)) ∗ neut(a)) ∗ a = neut(neut(a)) ∗ a.

ApplyingassociativelawandDeﬁnition2.1, a ∗ anti(neut(a))= a ∗ neut(neut(a)); anti(neut(a)) ∗ a = neut(neut(a)) ∗ a.

Ontheotherhand,byTheorem3.7, neut(neut(a)) ∈{neut(a)}.Itfollowsthat a ∗ neut(neut(a))= neut(neut(a)) ∗ a = a. Therefore, a ∗ anti(neut(a)))= anti(neut(a)) ∗ a = a. Thismeansthat anti(neut(a)) ∈{neut(a)}

Theorem3.11. Let (N, ∗) beaneutrosophictripletgroupwithrespectto ∗ and a,b ∈ N .Then neut(a ∗ a) ∈{neut(a)}

Florentin Smarandache (author and editor) Collected Papers, IX 221

Proof. Forany a ∈ N ,byDeﬁnition2.1wehave (a ∗ a) ∗ neut(a ∗ a)= a ∗ a.

Fromthisandapplyingtheassociativityofoperation ∗ andDeﬁnition2.1we get (anti(a) ∗ a) ∗ a ∗ neut(a ∗ a)=(anti(a) ∗ a) ∗ a neut(a) ∗ a ∗ neut(a ∗ a)= neut(a) ∗ a a ∗ neut(a ∗ a)= a

Similarly,wecanprove neut(a ∗ a) ∗ a = a.Thismeansthat neut(a ∗ a) ∈ {neut(a)}.

Now,we notethatProposition3.18in[16]isnottrue.

Example3.12. Consider(Z10,♯),where ♯ isdeﬁnedas a♯b =3ab(mod10). Then,(Z10,♯)isaneutrosophictripletgroupunderthebinaryoperation ♯ with Table1.

Table1Cayleytableofneutrosophictripletgroup(Z10,♯) ♯ 0 1 2 3 4 5 6 7 8 9 0 0 0 0 0 0 0 0 0 0 0 1 0 3 6 9 2 5 8 1 4 7 2 0 6 2 8 4 0 6 2 8 4 3 0 9 8 7 6 5 4 3 2 1 4 0 2 4 6 8 0 2 4 6 8 5 0 5 0 5 0 5 0 5 0 5 6 0 8 6 4 2 0 8 6 4 2 7 0 1 2 3 4 5 6 7 8 9 8 0 4 8 2 6 0 4 8 2 6 9 0 7 4 1 8 5 2 9 6 3

Foreach a ∈ Z10, wehave neut(a)in Z10.Thatis, neut(0)=0,neut(1)=7,neut(2)=2,neut(3)=7,neut(4)=2, neut(5)=5,neut(6)=2,neut(7)=7,neut(8)=2,neut(9)=7.

Let H = {0, 2, 5, 7},then(H,♯)isaneutrosophictripletsubgroupof(Z10,♯), but anti(5) ∈{1, 3, 5, 7, 9}̸⊂ H, anti(0) ∈{0, 1, 2, 3, 4, 5, 6, 7, 8, 9}̸⊂ H

Therefore,Proposition3.18in[16]shouldberevisedtothefollowingform.

Proposition3.13. Let (N, ∗) beaneutrosophictripletgroupand H beasubset of N .Then H isaneutrosophictripletsubgroupof N ifandonlyifthefollowing conditionshold:

Florentin Smarandache (author and editor) Collected Papers, IX 222

(1) a ∗ b ∈ H forall a,b ∈ H

(2) thereexists neut(a) ∈ H forall a ∈ H

(3) thereexists anti(a) ∈ H forall a ∈ H

4.Newpropertiesofcommutativeneutrosophictripletgroup

Theorem4.1. Let (N, ) beacommutativeneutrosophictripletgroupwithrespectto ∗ and a,b ∈ N .Then

{neut(a)}∗{neut(b)}⊆{neut(a ∗ b)}

Proof. Forany a,b ∈ N ,byDeﬁnition2.1and2.4wehave a ∗ neut(a) ∗ neut(b) ∗ b =(a ∗ neut(a)) ∗ (neut(b) ∗ b)= a ∗ b.

Fromthisandapplyingthecommutativityandassociativityofoperation ∗ we get

(neut(a) ∗ neut(b)) ∗ (a ∗ b)=(a ∗ b) ∗ (neut(a) ∗ neut(b))= a ∗ b.

Thismeansthat neut(a)∗neut(b) ∈{neut(a∗b)},thatis, {neut(a)}∗{neut(b)}⊆ {neut(a ∗ b)}

Proposition4.2. Let (N, ∗) beacommutativeneutrosophictripletgroupwith respectto ∗ and H = {neut(a) | a ∈ N }.Then H isaneutrosophictriplet subgroupof N suchthat (∀a ∈ N ) neut(a) ∈ H and unit(h) ∈ H forany h ∈ N .

Proof. Forany h1,h2 ∈ N ,bythedeﬁnitionof H,thereexists a,b ∈ N such that h1 = neut(a), h2 = neut(b).Then,byTheorem4.1wehave h1 ∗ h2 = neut(a) ∗ neut(b) ∈{neut(a ∗ b)}⊆ H.

Moreover,applyingTheorem3.7and3.10, neut(h1)= neut(neut(a)) ∈{neut(a)}⊆ H. anti(h1)= anti(neut(a)) ∈{neut(a)}⊆ H.

UsingProposition3.13weknowthat H isaneutrosophictripletsubgroupof N ,anditsatisﬁes

(∀a ∈ N ) neut(a) ∈ H, and unit(h) ∈ H forany h ∈ N.

Theorem4.3. Let (N, ∗) beacommutativeneutrosophictripletgroupwith respectto ∗ and a,b ∈ N .Then

{anti(a)}∗{anti(b)}⊆{anti(a ∗ b)}

Florentin Smarandache (author and editor) Collected Papers, IX 223

Proof. Forany a,b ∈ N ,byDeﬁnition2.1and2.4wehave

a ∗ anti(a) ∗ anti(b) ∗ b =(a ∗ anti(a)) ∗ (anti(b) ∗ b)= neut(a) ∗ neut(b)

Fromthisandapplyingthecommutativityandassociativityofoperation ∗ we get (anti(a) ∗ anti(b))(a ∗ b)=(a ∗ b) ∗ (anti(a) ∗ anti(b))= neut(a) ∗ neut(b)

ApplyingTheorem4.1, neut(a) ∗ neut(b) ∈{neut(a ∗ b)}.Hence,byDeﬁnition 2.1, anti(a) ∗ anti(b) ∈{anti(a ∗ b)},thatis, {anti(a)}∗{anti(b)}⊆{anti(a ∗ b)}

Theorem4.4. Let (N, ∗) beacommutativeneutrosophictripletgroupwith respectto ∗.Deﬁnebinaryrelation ≈neut on N asfollowing: ∀a,b ∈ N , a ≈neut b iﬀthereexists anti(b) ∈{anti(b)},and p,q ∈ N ,and neut(p) ∈{neut(p)} suchthat

a ∗ anti(b) ∗ neut(p) ∈{neut(q)}

Then ≈neut isreﬂexiveandsymmetric.

Proof. (1)Forany a ∈ N ,byProposition3.2, neut(a) ∗ neut(a) ∈{neut(a)} UsingDeﬁnition2.1weget

a ∗ anti(a) ∗ neut(a)= neut(a) ∗ neut(a) ∈{neut(a)}

Then, a ≈neut a (2)Assume a ≈neut b,thenthereexists p,q ∈ N suchthat

(C1)

a ∗ anti(b) ∗ neut(p)= neut(q).

where anti(b) ∈{anti(b)}, neut(p) ∈ neut(p), neut(q) ∈{neut(q)}.Using Theorem3.10, anti(neut(p)) ∈{neut(p)}.So,wedenote anti(neut(p))= x ∈ {neut(p)}.Thus, b ∗ anti(a) ∗ x = b ∗ anti(a) ∗ anti(neut(p)) = anti(a) ∗ b ∗ anti(neut(p)) (byDefinition 2 5) = anti(a) ∗ (neut(b) ∗ anti(anti(b))) ∗ anti(neut(p)) (byTheorem 3 9) =(anti(a) ∗ anti(anti(b)) ∗ anti(neut(p))) ∗ neut(b)(byDefinition 2 4and 2 5)

∈{anti(a ∗ anti(b) ∗ neut(p))}∗ neut(b) (byTheorem 4.3) ⊆{anti(neut(q))}∗ neut(b) (bytheaboveresult (C1)) ⊆{neut(q)}∗ neut(b) (byTheorem 3.10) ⊆{neut(q ∗ b)} (byTheorem 4.1)

Thismeansthat b ≈neut a

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Deﬁnition4.5. Let(N, ∗)beaneutrosophictripletgroup.Then, N iscalled aneutrosophictripletgroupwithcondition(AN)ifforall a,b ∈ N ,wehave (AN) {anti(a ∗ b)}⊆{anti(a)}∗{anti(b)}

Proposition4.6. Let (N, ∗) beacommutativeneutrosophictripletgroupwith condition(AN)and a,b ∈ N .Then

neut(a ∗ b) ∈{neut(a)}∗{neut(b)}.

Proof. Forany a,b ∈ N ,byDeﬁnition4.5,thereexists anti(a) ∈{anti(a)}, anti(b) ∈{anti(b)} suchthat anti(a ∗ b)= anti(a) ∗ anti(b).

Then

neut(a ∗ b)=(a ∗ b) ∗ anti(a ∗ b)=(a ∗ b) ∗ (anti(a) ∗ anti(b)) =(a ∗ anti(a)) ∗ (b ∗ anti(b))= neut(a) ∗ neut(b)

Thismeansthat neut(a ∗ b) ∈{neut(a)}∗{neut(b)}.

Lemma4.7. Let (N, ∗) beacommutativeneutrosophictripletgroupwithcondition(AN)and a,b ∈ N .Ifthereexists anti(b) ∈{anti(b)}, p,q ∈ N , neut(p) ∈{neut(p)} and neut(q) ∈{neut(q)} suchthat a ∗ anti(b) ∗ neut(p)= neut(q)

Thenforany x ∈{anti(b)},thereexists p1,q1 ∈ N , neut(p1) ∈{neut(p1)} and neut(q1) ∈{neut(q1)} suchthat a ∗ x ∗ neut(p1)= neut(q1)

Proof. Forany x ∈{anti(b)},thereexists y ∈{neut(b)} suchthat b ∗ x = x ∗ b = y.Thus,from a ∗ anti(b) ∗ neut(p)= neut(q)weget a ∗ x ∗ (neut(b) ∗ neut(p)) = a ∗ x ∗ (anti(b) ∗ b) ∗ neut(p) =(a ∗ anti(b) ∗ neut(p)) ∗ (x ∗ b) = neut(q) ∗ y ∈ neut(q) ∗{neut(b)} ⊆{neut(q ∗ b)} (byTheorem 4 1)

Therefore,thereexists p1,q1 ∈ N , neut(p1) ∈{neut(p1)} and neut(q1) ∈ {neut(q1)} suchthat a ∗ x ∗ neut(p1)= neut(q1).

Theorem4.8. Let (N, ∗) beacommutativeneutrosophictripletgroupwith condition(AN).Deﬁnebinaryrelation ≈neut on N asfollowing: ∀a,b ∈ N , a ≈neut b iﬀthereexists anti(b) ∈{anti(b)}, p,q ∈ N ,and neut(p) ∈{neut(p)} suchthat a ∗ anti(b) ∗ neut(p) ∈{neut(q)}. Then ≈neut isanequivalentrelationon N

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Proof. ByTheorem4.4, weonlyprovethat ≈neut istransitive.Assumethat a ≈neut b and b ≈neut c,thenthereexists p,q,r,s ∈ N suchthat

a ∗ anti(b) ∗ neut(p)= neut(q) (C1) b ∗ anti(c) ∗ neut(r)= neut(s) (C2)

where anti(b) ∈{anti(b)}, anti(c) ∈{anti(c)}, neut(p) ∈{neut(p)}, neut(q) ∈ {neut(q)}, neut(r) ∈{neut(r)}, neut(s) ∈{neut(s)}.UsingTheorem3.10and Theorem4.1,wehave neut(p)∗neut(c)∗anti(neut(s))∈{neut(p)}∗{neut(c)}∗{neut(s)}⊆{neut(p∗s∗c)}.

Denote y = neut(p) ∗ neut(c) ∗ anti(neut(s)) ∈{neut(p ∗ s ∗ c)},then a ∗ anti(c) ∗ y = a ∗ anti(c) ∗ neut(p) ∗ neut(c) ∗ anti(neut(s))

= a ∗ anti(c) ∗ neut(p) ∗ anti(neut(s)) ∗ neut(c)(byDeﬁnition2.5)

= a ∗ anti(c) ∗ neut(p) ∗ anti(b ∗ anti(c) ∗ neut(r)) ∗ neut(c) (bytheaboveresult(C2))

∈ a ∗ anti(c) ∗ neut(p) ∗{anti(b) ∗ anti(anti(c)) ∗ anti(neut(r))}∗ neut(c) (byDeﬁnition4.5)

⊆ a ∗ anti(c) ∗ neut(p) ∗{anti(b) ∗ c ∗ anti(neut(r))} (byDeﬁnition2.4,2.5andTheorem3.9)

⊆ a ∗ neut(p) ∗{anti(b) ∗ neut(r) ∗ (anti(c) ∗ c)} (byTheorem3.10,Deﬁnition2.4and2.5)

= a ∗ neut(p) ∗{anti(b) ∗ neut(r) ∗ neut(c)} (byDeﬁnition2.1)

⊆{(a ∗ anti(b) ∗ neut(p)) ∗ neut(r) ∗ neut(c)} (byDeﬁnition2.1)

⊆{neut(q1)∗neut(r)∗neut(c)} (bytheaboveresult(C1)andLemma4.7) ⊆{neut(q1 ∗ r ∗ c)} (byTheorem4.1) Thismeansthat a ≈neut c

5.Commutative neutrosophictripletgroupandAbelgroupwith BCI-algebra

Theorem5.1. Let (N, ∗) beacommutativeneutrosophictripletgroupcondition (AN).Deﬁnebinaryrelation ≈neut on N asTheorem4.8.Thenthefollowing statementsarehold:

(1) a,b,c ∈ N , a ≈neut b ⇒ a ∗ c ≈neut b ∗ c. (2) a ≈neut b ⇒ neut(a) ≈neut neut(b) (3) a ≈neut b ⇒ anti(a) ≈neut anti(b) (4) a,b ∈ N , neut(a) ≈neut neut(b) Proof. (1)Assume a ≈neut b,thenthereexists p,q ∈ N suchthat (C1) a ∗ anti(b) ∗ neut(p)= neut(q), where anti(b) ∈{anti(b)}, neut(p) ∈{neut(p)}, neut(q) ∈{neut(q)}.Thus,

Florentin Smarandache (author and editor) Collected Papers, IX 226

(a ∗ c) ∗ anti(b ∗ c) ∗ neut(p) ∈ (a ∗ c) ∗{anti(b)}∗{anti(c)}∗ neut(p)(byDefinition 4 5) ⊆{a ∗ anti(b) ∗ neut(p)}∗{c ∗ anti(c)} (byDefinition 2 4 and 2 5) = {a ∗ anti(b) ∗ neut(p)}∗{neut(c)} (byDefinition 2.1)

⊆{neut(q1)}∗{neut(c)}(bytheaboveresult (C1) andLemma 4.7) ⊆{neut(q1 ∗ c)} (byTheorem 4.1)

Itfollowsthat a ∗ c ≈neut b ∗ c. (2)Assume a ≈neut b,thenthereexists p,q ∈ N suchthat a ∗ anti(b) ∗ neut(p)= neut(q)

where anti(b) ∈{anti(b)}, neut(p) ∈{neut(p)}, neut(q) ∈{neut(q)}.Then, applyingTheorem3.8andTheorem4.1wehave neut(a)∗anti(neut(b))∗neut(p)∈{neut(a)}∗{neut(b)}∗{neut(p)}⊆{neut(a∗b∗p)}

Thismeansthat neut(a) ≈neut neut(b). (3)Assume a ≈neut b,thenthereexists p,q ∈ N suchthat a ∗ anti(b) ∗ neut(p)= neut(q) where anti(b) ∈{anti(b)}, neut(p) ∈{neut(p)}, neut(q) ∈{neut(q)}.Using Theorem3.10, anti(neut(p)) ∈{neut(p)},anti(neut(q)) ∈{neut(q)}.

ApplyingTheorem4.3wehave

anti(a) ∗ anti(anti(b)) ∗ anti(neut(p)) ∈{anti(a ∗ anti(b) ∗ neut(p))} ⊆{anti(neut(q))}⊆{neut(q)}

Itfollowsthat anti(a) ≈neut anti(b). (4) ∀a,b ∈ N ,since neut(a) ∗ anti(neut(b)) ∗ neut(a) ∈ neut(a) ∗{neut(b)}∗ neut(a)(byTheorem 3 10) ⊆{neut(a ∗ b ∗ a)} (byTheorem 4.1)

Thismeansthat neut(a) ≈neut neut(b).

Theorem5.2. Let (N, ∗) beacommutativeneutrosophictripletgroupwith condition(AN).Deﬁnebinaryrelation ≈neut on N asTheorem4.8.Thenthe quotient N/ ≈neut isanAbelgroupwithrespecttothefollowingoperation:

∀ a,b ∈ N, [a]neut • [b]neut =[a ∗ b]neut.

where [a]neut istheequivalentclassof a,theunitelmentof (N/ ≈neut, •) is 1neut =[neut(a)]neut, ∀a ∈ N , neut(a) ∈{neut(a)}

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Proof. ByTheorem5.1 (1) ∼ (3)weknowthattheoperation“•”iswell deﬁnition.Obviously,(N/ ≈neut, •)isacommutativeneutrosophictripletgroup. Moreover,byTheorem5.1(4)weget

∀a,b ∈ N ,[neut(a)]neut =[neut(b)]neut

∀a,b ∈ N , neut([a]neut)= neut([b]neut).

Thismeansthat neut( )isunique.Denote

1neut =[neut(a)]neut, ∀ a ∈ N,neut(a) ∈{neut(a)}.

Then1neut istheunitelementof(N/ ≈neut, •).Moreover,byTheorem5.1(3) wegetthat anti([a]neut)isunique, ∀a ∈ N .Therefore,(N/ ≈neut, •)isanAbel group.

Theorem5.3. Let (N, ∗) beacommutativeneutrosophictripletgroupwith condition(AN).Deﬁnebinaryrelation ≈neut on N asTheorem4.8.Ifdeﬁnea newoperation“→”onthequotient N/ ≈neut asfollowing:

∀a,b ∈ N, [a]neut → [b]neut =[a]neut • anti([b]neut). Then (N/ ≈neut, →, 1neut) isaBCI-algebra,where 1neut=[neut(a)]neut, ∀a∈N Proof. ByTheorem5.2andProposition2.13wecangettheresult.

Example5.4. Let N = {1, 2, 3, 4, 6, 7, 8, 9}.Theoperation ∗ on N isdeﬁned asTables2.Then,(N, ∗)isaneutrosophictripletgroupwithcondition(AN). Foreach a ∈ N ,wehave neut(a)in N .Thatis, neut(1)=7,neut(2)=2,neut(3)=7,neut(4)=2, neut(6)=2,neut(7)=7,neut(8)=2,neut(9)=7. Moreover,foreach a ∈ N , anti(a)in N .Thatis, anti(1)=9,anti(2) ∈{2, 7},anti(3)=3,anti(4) ∈{1, 6}, anti(6) ∈{4, 9},anti(7)=7,anti(8) ∈{3, 8},anti(9)=1. Itiseasytoverifythat N/ ≈neut= {[2]neut, [1]neut, [3]neut, [4]neut} and(N/ ≈neut, •)isisomorphismto(Z4, +),where [2]neut = {2, 7}, [1]neut = {1, 6}, [3]neut = {3, 8}, [4]neut = {4, 9}.

Table2Cayleytableofneutrosophictripletgroup(N, ∗) ∗ 1 2 3 4 6 7 8 9 1 3 6 9 2 8 1 4 7 2 6 2 8 4 6 2 8 4 3 9 8 7 6 4 3 2 1 4 2 4 6 8 2 4 6 8 6 8 6 4 2 8 6 4 2 7 1 2 3 4 6 7 8 9 8 4 8 2 6 4 8 2 6 9 7 4 1 8 2 9 6 3

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Table3CayleytableofAbelgroup((N/ ≈neut, •)

• [2]neut [1]neut [3]neut [4]neut [2]neut [2]neut [1]neut [3]neut [4]neut [1]neut [1]neut [3]neut [4]neut [2]neut [3]neut [3]neut [4]neut [2]neut [1]neut [4]neut [4]neut [2]neut [1]neut [3]neut

Table4CayleytableofAbelgroup(Z4, +) + 0 1 3 4 0 0 1 2 3 1 1 2 3 0 2 2 3 0 1 3 3 0 1 2

Example5.5. Consider(Z10,♯),where ♯ isdeﬁnedas a♯b =3ab(mod10). Then,(Z10,♯)isaneutrosophictripletgroupwithcondition(AN),thebinary operation ♯ isdeﬁnedinTable1.Foreach ∈ Z10,wehave neut(a)in Z10.That is, neut(0)=0,neut(1)=7,neut(2)=2,neut(3)=7,neut(4)=2, neut(5)=5,neut(6)=2,neut(7)=7,neut(8)=2,neut(9)=7. Moreover,foreach a ∈ Z10, anti(a)in Z10.Thatis, anti(0) ∈{0, 1, 2, 3, 4, 5, 6, 7, 8, 9},anti(1)=9,anti(2) ∈{2, 7}, anti(3)=3,anti(4) ∈{1, 6},anti(5) ∈{1, 3, 5, 7, 9}, anti(6) ∈{4, 9},anti(7)=7,anti(8) ∈{3, 8},anti(9)=1. Itiseasytoverifythat N/ ≈neut= {1neut =[0]neut} and(N/ ≈neut, •)isisomorphismto {1},where [0]neut =1neut = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.

6.Quotientstructureandneutro-homomorphismbasictheorem

Deﬁnition6.1 ([16]). Let(N1, ∗1)and(N2, ∗2)betwoneutrosophictriplet groups.Let f : N1 → N2 beamapping.Then, f iscalled neutro-homomorphism ifforall a,b ∈ N1,wehave:

(1) f (a ∗1 b)= f (a) ∗2 f (b);

(2) f (neut(a))= neut(f (a));

(3) f (anti(a))= anti(f (a)).

Theorem6.2. Let (N, ∗) beacommutativeneutrosophictripletgroupwith respectto ∗, H beaneutrosophictripletsubgroupof N suchthat (∀a ∈ N ) neut(a) ∈ H and (∀a ∈ H) anti(a) ∈ H.Deﬁnebinaryrelation ≈H on N as following:

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∀a,b ∈ N , a ≈H b iﬀthere exists anti(b) ∈{anti(b)}, p ∈ N ,and neut(p) ∈ {neut(p)} suchthat

a ∗ anti(b) ∗ neut(p) ∈ H.

Then ≈H isreﬂexiveandsymmetric.

Proof. (1)Forany a ∈ N ,byProposition3.2andthehypothesis(neut(a) ∈ H forany a ∈ N ),wehave

neut(a) ∗ neut(a) ∈{neut(a)}⊆ H.

ByDeﬁnition2.1weget

a ∗ anti(a) ∗ neut(a)= neut(a) ∗ neut(a) ∈ H.

Then, a ≈H a (2)Assume a ≈H b,thenthereexists p ∈ N suchthat (C2) a ∗ anti(b) ∗ neut(p) ∈ H. where anti(b) ∈{anti(b)}, neut(p) ∈{neut(p)}.Moreover,bythehypothesis (anti(a) ∈ H forany a ∈ H),wehave

(C3) anti(a ∗ anti(b) ∗ neut(p)) ∈ H.

UsingTheorem3.10, anti(neut(p)) ∈{neut(p)}.So,wedenote anti(neut(p))= x ∈{neut(p)}.Thus, b ∗ anti(a) ∗ x

= b ∗ anti(a) ∗ anti(neut(p)) = anti(a) ∗ b ∗ anti(neut(p)) (byDeﬁnition2.5) = anti(a) ∗ (neut(b) ∗ anti(anti(b))) ∗ anti(neut(p))(byTheorem3.9) =(anti(a)∗anti(anti(b))∗anti(neut(p)))∗neut(b)(byDeﬁnition2.4and2.5) ∈{anti(a ∗ anti(b) ∗ neut(p))}∗ neut(b) (byTheorem4.3) ⊆ H (by(C3),thehypothesisandProposition3.13(1)) Thismeansthat b ≈H a

Lemma6.3. Let (N, ∗) beacommutativeneutrosophictripletgroupwithcondition(AN), a,b ∈ N ,and H beaneutrosophictripletsubgroupof N such that (∀a ∈ N ) neut(a) ∈ H and (∀a ∈ H) anti(a) ∈ H.Ifthereexists anti(b) ∈{anti(b)}, p ∈ N ,and neut(p) ∈{neut(p)} suchthat a ∗ anti(b) ∗ neut(p) ∈ H.

Thenforany x ∈{anti(b)},thereexists p1 ∈ N ,and neut(p1) ∈{neut(p1)} suchthat a ∗ x ∗ neut(p1) ∈ H.

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Proof. Forany x ∈{anti(b)},thereexists y ∈{neut(b)} suchthat b∗x = x∗b = y.Since(∀a ∈ N ) neut(a) ∈ H,then y ∈ H.Thus,from a∗anti(b)∗neut(p) ∈ H weget a ∗ x ∗ (neut(b) ∗ neut(p)) = a ∗ x ∗ (anti(b) ∗ b) ∗ neut(p) =(a ∗ anti(b) ∗ neut(p)) ∗ (x ∗ b) =(a ∗ anti(b ∗ neut(p)) ∗ y ∈ H (byProposition 3.13)

Theorem6.4. Let (N, ∗) beacommutativeneutrosophictripletgroupwith condition(AN), H beaneutrosophictripletsubgroupof N suchthat (∀a ∈ N ) neut(a) ∈ H and (∀a ∈ H) anti(a) ∈ H.Deﬁnebinaryrelation ≈H on N as following: ∀a,b ∈ N , a ≈H b iﬀthereexists anti(b) ∈{anti(b)}, p ∈ N ,and neut(p) ∈ {neut(p)} suchthat a ∗ anti(b) ∗ neut(p) ∈ H.

Then ≈H isanequivalentrelationon N Proof. ByTheorem6.2,weonlyprovethat ≈H istransitive.Assumethat a ≈H b and b ≈H c,thenthereexists p,r ∈ N and q,s ∈ N suchthat a ∗ anti(b) ∗ neut(p)= q ∈ H. (C3) b ∗ anti(c) ∗ neut(r)= s ∈ H. (C4)

where anti(b) ∈{anti(b)}, anti(c) ∈{anti(c)}, neut(p) ∈{neut(p)}, neut(r) ∈ {neut(r)}.UsingTheorem4.1andthehypothesis(neut(a) ∈ H forany a ∈ N ), wehave neut(p) ∗ neut(s) ∗ neut(c) ∈ neut(p ∗ s ∗ c) ⊆ H. Denote y = neut(p) ∗ neut(s) ∗ neut(c) ∈ neut(p ∗ s ∗ c),then a ∗ anti(c) ∗ y

= a ∗ anti(c) ∗ neut(p) ∗ neut(s) ∗ neut(c)

= a ∗ anti(c) ∗ neut(p) ∗ (s ∗ anti(s)) ∗ neut(c)(byDeﬁnition2.1)

= a ∗ anti(c) ∗ neut(p) ∗ s ∗ anti(b ∗ anti(c) ∗ neut(r)) ∗ neut(c) (bytheaboveresult(C4)) ∈ a∗anti(c)∗neut(p)∗s∗{anti(b)}∗{anti(anti(c))}∗{anti(neut(r))}neut(c) (byDeﬁnition4.5) = a ∗ anti(c) ∗ neut(p) ∗ s ∗{anti(b)}∗ c ∗{anti(neut(r))} (byTheorem3.9) ⊆ a ∗ anti(c) ∗ neut(p) ∗ s ∗{anti(b)}∗ c ∗{neut(r)} (byTheorem3.10) ⊆{a ∗ anti(b) ∗ neut(p)}∗ s ∗ (anti(c) ∗ c) ∗{neut(r)} (byDeﬁnition2.4and 2.5) ⊆ H ∗ s ∗ neut(c) ∗{neut(r)}

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(byDeﬁnition2.1, theaboveresult(C3)andLemma6.3) ⊆ H (by(C4),thehypothesisandProposition3.13(1)) Itfollowsthat a ≈H c

Theorem6.5. Let (N, ∗) beacommutativeneutrosophictripletgroupwith condition(AN), H beaneutrosophictripletsubgroupof N suchthat (∀a ∈ N ) neut(a) ∈ H and (∀a ∈ H) anti(a) ∈ H.Deﬁnebinaryrelation ≈H on N as following:

∀a,b ∈ N , a ≈H b iﬀthereexists anti(b) ∈{anti(b)}, p ∈ N ,and neut(p) ∈ {neut(p)} suchthat a ∗ anti(b) ∗ neut(p) ∈ H.

Thenthefollowingstatementsarehold:

(1) a,b,c ∈ N , a ≈H b ⇒ a ∗ c ≈H b ∗ c (2) a ≈H b ⇒ neut(a) ≈H neut(b) (3) a ≈H b ⇒ anti(a) ≈H anti(b)

Proof. (1)Assume a ≈H b,thenthereexists p ∈ N suchthat (C2) a ∗ anti(b) ∗ neut(p) ∈ H. where anti(b) ∈{anti(b)}, neut(p) ∈{neut(p)}.Wehave (a ∗ c) ∗ anti(b ∗ c) ∗ neut(p) ∈ (a ∗ c) ∗{anti(b)}∗{anti(c)}∗ neut(p) (byDefinition4.5) ⊆{a ∗ anti(b) ∗ neut(p)}∗{c ∗ anti(c)} (byDefinition2.4and2.5) = {a ∗ anti(b) ∗ neut(p)}∗ neut(c) (byDefinition2.1) ∈ H. (by(C2),thehypothesis,Lemma6.3andProposition3.13(1)) Itfollowsthat a ∗ c ≈H b ∗ c (2)Assume a ≈H b,thenthereexists p ∈ N suchthat a ∗ anti(b) ∗ neut(p) ∈ H,where anti(b) ∈{anti(b)}, neut(p) ∈{neut(p)}.ApplyingTheorem3.8and Theorem4.1wehave neut(a) ∗ anti(neut(b)) ∗ neut(p) ∈ neut(a) ∗{neut(b)}∗ neut(p) ⊆{neut(a∗b∗p)}⊆ H.(bythehypothesis, neut(a) ∈ H forany a ∈ N ) Itfollowsthat neut(a) ≈H neut(b). Assume a ≈H b,thenthereexists p ∈ N suchthat a ∗ anti(b) ∗ neut(p) ∈ H where anti(b) ∈{anti(b)}, neut(p) ∈{neut(p)}.Applyingthehypothesis((∀a ∈ N ) neut(a) ∈ H and(∀a ∈ H) anti(a) ∈ H)andTheorem3.10, anti(a ∗ anti(b) ∗ neut(p)) ∈ H. anti(neut(p)) ∈{neut(p)}⊆ H.

Moreover,byTheorem4.3wehave anti(a) ∗ anti(anti(b)) ∗ anti(neut(p)) ∈{anti(a ∗ anti(b) ∗ neut(p))}⊆ H. Hence, anti(a) ≈H anti(b).

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Theorem6.6. Let (N, ∗) beacommutativeneutrosophictripletgroupwith condition(AN), H beaneutrosophictripletsubgroupof N suchthat (∀a ∈ N ) neut(a) ∈ H and (∀a ∈ H) anti(a) ∈ H.Deﬁnebinaryrelation ≈H on N as Theorem6.5.Thenthequotient N/ ≈H isacommutativeneutrosophictriplet groupwithrespecttothefollowingoperation:

∀a,b ∈ N, [a]H • [b]H =[a ∗ b]H

where [a]H istheequivalentclassof a withrespectto ≈H .Moreover, (N, ∗) is neutron-homomorphismto (N/ ≈H , •) withrespecttothefollowingmapping:

f : N → N/ ≈H ; and ∀a ∈ N,f (a)=[a]H

Proof. ByTheorem6.5weknowthattheoperation“•”iswelldeﬁnition. Obviously,(N/ ≈H , •)isacommutativeneutrosophictripletgroup. Bythedeﬁnitionsofoperation“•”andmapping f wehave

∀a,b ∈ N,f (a ∗ b)=[a ∗ b]H =[a]H • [b]H = f (a) • f (b). Moreover,byTheorem6.5(2)and(3)weget

∀a ∈ N,f (neut(a))=[neut(a)]H = neut([a]H )= neut(f (a)). ∀a ∈ N,f (anti(a))=[anti(a)]H = anti([a]H )= anti(f (a)).

Therefore,(N, ∗)isneutron-homomorphismto(N/ ≈H , •)withrespecttothe mapping f

Theorem6.7. Let (N, ∗) beacommutativeneutrosophictripletgroupwith condition(AN), H beaneutrosophictripletsubgroupof N suchthat (∀a ∈ N ) neut(a) ∈ H and (∀a ∈ H) anti(a) ∈ H.Deﬁnebinaryrelation ≈H on N asTheorem6.5.Ifdeﬁneanewoperation“→”onthequotient N/ ≈H as following: ∀a,b ∈ N , [a]H → [b]H =[a]H • anti([b]H ).Then (N/ ≈H , →, 1H ) is aBCI-algebra,where 1H =[neut(a)]H , ∀a ∈ N .

Proof. ByTheorem6.7andProposition2.13wecangettheresult.

Example6.8. Let N = {1, 2, 3, 4, 6, 7, 8, 9}.Theoperation ∗ on N isdeﬁned asTables2.Then,(N, ∗)isaneutrosophictripletgroupwithcondition(AN). Wecangetthefollowingequation

neut(1)=7,neut(2)=2,neut(3)=7,neut(4)=2, neut(6)=2,neut(7)=7,neut(8)=2,neut(9)=7; anti(1)=9,anti(2) ∈{2, 7},anti(3)=3,anti(4) ∈{1, 6}, anti(6) ∈{4, 9},anti(7)=7,anti(8) ∈{3, 8},anti(9)=1.

Denote H = {2, 3, 7, 8},itiseasytoverifythat H isaneutrosophictriplet subgroupof N suchthat(∀a ∈ N ) neut(a) ∈ H and(∀a ∈ H) anti(a) ∈ H. Moreover, N/ ≈H = {H =[2]H , [1]H } and(N/ ≈H , •)isisomorphismto(Z2, +), where [2]H = {2, 3, 7, 8}, [1]H = {1, 4, 6, 9}

Florentin Smarandache (author and editor) Collected Papers, IX 233

Table5CayleytableofAbelgroup(N/ ≈H , •) • [2]H [1]H [2]H [2]H [1]H [1]H [1]H [2]H

Table6CayleytableofAbelgroup(Z2, +) + 0 1 0 0 1 1 1 0

Thefollowingexample showsthatthebasictheoremofneutro-homomorphism (Theorem6.7)isanaturalandsubstantialgeneralizationofthebasictheorem ofgroup-homomorphism.

Example6.9. Let(N, ∗)beacommutativegroup.Then,(N, ∗)isaneutrosophictripletgroupwithcondition(AN).Obviously,if H isasubgroupof N , thenbinaryrelation ≈H on N istherelationinducedbysubgroup H,thatis, ∀a,b ∈ N,a ≈H b ifandonlyif a ∗ b 1 ∈ H.

Thus,(N, ∗)isgroup-homomorphismto(N/ ≈H , •)=(N/H, •).

7.Conclusion

Thispaperisfocusonneutrosophictripletgroup.Weprovedsomenewpropertiesof(commutative)neutrosophictripletgroup,andconstructedanewequivalentrelationonanycommutativeneutrosophictripletgroupwithcondition (AN).Basedontheseresults,fortheﬁrsttime,wehavedescribedtheinner linkbetweencommutativeneutrosophictripletgroupwithcondition(AN)and AbelgroupwithBCI-algebra.Furthermore,weestablishthequotientstructurebyneutrosophictripletsubgroup,andprovethebasictheoremofneutrohomomorphism,whichisanaturalandsubstantialgeneralizationofthebasic theoremofgroup-homomorphism.Obviously,theseresultswillplayanimportantroleinthefurtherstudyofneutrosophictripletgroup.

Acknowledgment

ThisworkwassupportedbyNationalNaturalScienceFoundationofChina (GrantNo.61573240).

References

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Florentin Smarandache (author and editor) Collected Papers, IX 236

Commutative Generalized Neutrosophic Ideals in BCK-Algebras

Rajab Ali Borzooei, Xiaohong Zhang, Florentin Smarandache, Young Bae Jun (2018). Commutative Generalized Neutrosophic Ideals in BCK-Algebras. Symmetry 10, 350. DOI: 10.3390/sym10080350

Abstract: Theconceptofacommutativegeneralizedneutrosophicidealina BCK-algebraisproposed, andrelatedpropertiesareproved.Characterizationsofacommutativegeneralizedneutrosophic idealareconsidered.Also,someequivalencerelationsonthefamilyofallcommutativegeneralized neutrosophicidealsin BCK-algebrasareintroduced,andsomepropertiesareinvestigated.

Keywords: (commutative)ideal;generalizedneutrosophicset;generalizedneutrosophicideal; commutativegeneralizedneutrosophicideal

1.Introduction

In1965,Zadehintroducedtheconceptoffuzzysetinwhichthedegreeofmembershipisexpressed byonefunction(thatis,truthort).Thetheoryoffuzzysetisappliedtomanyﬁelds,includingfuzzy logicalgebrasystems(suchaspseudo-BCI-algebrasbyZhang[1]).In1986,Atanassovintroduced theconceptofintuitionisticfuzzysetinwhichtherearetwofunctions,membershipfunction(t)and nonmembershipfunction(f).In1995,Smarandacheintroducedthenewconceptofneutrosophic setinwhichtherearethreefunctions,membershipfunction(t),nonmembershipfunction(f)and indeterminacy/neutralitymembershipfunction(i),thatis,therearethreecomponents(t,i,f)= (truth,indeterminacy,falsehood)andtheyareindependentcomponents.

Neutrosophicalgebraicstructuresin BCK/BCI-algebrasarediscussedinthepapers[2 10]. Moreover,Zhangetal.studiedtotallydependent-neutrosophicsets,neutrosophicdupletsemi-group andcancellableneutrosophictripletgroups(see[11,12]).Songetal.proposedthenotionofgeneralized neutrosophicsetandapplieditto BCK/BCI-algebras.

Inthispaper,weproposethenotionofacommutativegeneralizedneutrosophicidealina BCK-algebra,andinvestigaterelatedproperties.Weconsidercharacterizationsofacommutative generalizedneutrosophicideal.Usingacollectionofcommutativeidealsin BCK-algebras,weobtain acommutativegeneralizedneutrosophicideal.Wealsoestablishsomeequivalencerelationsonthe familyofallcommutativegeneralizedneutrosophicidealsin BCK-algebras,anddiscussrelatedbasic propertiesoftheseideals.

2.Preliminaries

Aset X withaconstantelement0andabinaryoperation ∗ iscalleda BCI-algebra,ifitsatisﬁes (∀x, y, z ∈ X):

Rajab Ali Borzooei, Xiaohong Zhang, Florentin Smarandache, Young Bae Jun

Smarandache (author and editor) Collected Papers, IX 237

Florentin

(I) ((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y)= 0, (II) (x ∗ (x ∗ y)) ∗ y = 0, (III) x ∗ x = 0, (IV) x ∗ y = 0, y ∗ x = 0 ⇒ x = y

A BCI-algebra X iscalleda BCK-algebra,ifitsatisﬁes (∀x ∈ X): (V) 0 ∗ x = 0,

Forany BCK/BCI-algebra X,thefollowingconditionshold (∀x, y, z ∈ X): x ∗ 0 = x,(1) x ≤ y ⇒ x ∗ z ≤ y ∗ z, z ∗ y ≤ z ∗ x,(2) (x ∗ y) ∗ z =(x ∗ z) ∗ y,(3) (x ∗ z) ∗ (y ∗ z) ≤ x ∗ y (4)

wheretherelation ≤ isdeﬁnedby: x ≤ y ⇐⇒ x ∗ y = 0.Ifthefollowingassertionisvalidfora BCK-algebra X, ∀x, y ∈ X, x ∗ (x ∗ y)= y ∗ (y ∗ x).(5) then X iscalledacommutative BCK-algebra. Assume I isasubsetofa BCK/BCI-algebra X.Ifthefollowingconditionsarevalid,thenwecall I isanidealof X: 0 ∈ I,(6) (∀x ∈ X) (∀y ∈ I)(x ∗ y ∈ I ⇒ x ∈ I) .(7)

Asubset I ofa BCK-algebra X iscalledacommutativeidealof X ifitsatisﬁes(6)and (∀x, y, z ∈ X) ((x ∗ y) ∗ z ∈ I, z ∈ I ⇒ x ∗ (y ∗ (y ∗ x)) ∈ I) .(8)

Recallthatanycommutativeidealisanideal,buttheinverseisnottrueingeneral(see[7]).

Lemma1 ([7]). Let I beanidealofa BCK-algebra X.Then I iscommutativeidealof X ifandonlyifitsatisﬁes thefollowingconditionforallx, yinX: x ∗ y ∈ I ⇒ x ∗ (y ∗ (y ∗ x)) ∈ I.(9)

Forfurtherinformationregarding BCK/BCI-algebras,pleaseseethebooks[7,13].

Let X beanonemptyset.Afuzzysetin X isafunction µ : X → [0,1],andthecomplementof µ,denotedby µc,isdeﬁnedby µc (x)= 1 µ(x), ∀x ∈ X.Afuzzyset µ ina BCK/BCI-algebra X is calledafuzzyidealof X if (∀x ∈ X)(µ(0) ≥ µ(x)),(10) (∀x, y ∈ X)(µ(x) ≥ min{µ(x ∗ y), µ(y))}.(11)

Assumethat X isanon-emptyset.Aneutrosophicset(NS)in X (see[14])isastructureof theform: A := { x; AT (x), AI (x), AF (x) | x ∈ X}

Florentin Smarandache (author and editor) Collected Papers, IX 238

where AT : X → [0,1] , AI : X → [0,1] ,and AF : X → [0,1] .Weshallusethesymbol A =(AT , AI , AF ) fortheneutrosophicset

A := { x; AT (x), AI (x), AF (x) | x ∈ X}

Ageneralizedneutrosophicset(GNS)inanon-emptyset X isastructureoftheform(see[15]):

A := { x; AT (x), AIT (x), AIF (x), AF (x) | x ∈ X, AIT (x)+ AIF (x) ≤ 1}

where AT : X → [0,1], AF : X → [0,1] , AIT : X → [0,1] ,and AIF : X → [0,1] Weshallusethesymbol A =(AT , AIT , AIF, AF ) forthegeneralizedneutrosophicset

A := { x; AT (x), AIT (x), AIF (x), AF (x) | x ∈ X, AIT (x)+ AIF (x) ≤ 1}

Notethat,foreveryGNS A =(AT , AIT , AIF, AF ) in X,wehave(forall x in X) (∀x ∈ X) (0 ≤ AT (x)+ AIT (x)+ AIF (x)+ AF (x) ≤ 3)

If A =(AT , AIT , AIF, AF ) isaGNSin X,then A =(AT , AIT , Ac IT , Ac T ) and ♦A =(Ac F, Ac IF, AIF, AF ) arealsoGNSsin X

GivenaGNS A =(AT , AIT , AIF, AF ) ina BCK/BCI-algebra X and αT , αIT , βF, β IF ∈ [0,1], wedeﬁnefoursetsasfollows:

UA (T, αT ) := {x ∈ X | AT (x) ≥ αT },

UA (IT, αIT ) := {x ∈ X | AIT (x) ≥ αIT },

LA (F, βF ) := {x ∈ X | AF (x) ≤ βF },

LA (IF, β IF ) := {x ∈ X | AIF (x) ≤ β IF }.

AGNS A =(AT , AIT , AIF, AF ) ina BCK/BCI-algebra X iscalledageneralizedneutrosophic idealof X (see[15])if (∀x ∈ X) AT (0) ≥ AT (x), AIT (0) ≥ AIT (x) AIF (0) ≤ AIF (x), AF (0) ≤ AF (x) ,(12)

T (x) ≥ min{AT (x ∗ y), AT (y)} AIT (x) ≥ min{AIT (x ∗ y), AIT (y)} AIF (x) ≤ max{AIF (x ∗ y), AIF (y)} AF (x) ≤ max{AF (x ∗

     

, AF (y)}      

3.CommutativeGeneralizedNeutrosophicIdeals Unlessspecified, X willalwaysrepresenta BCK-algebrainthefollowingdiscussion. Definition1. AGNS A =(AT , AIT , AIF, AF ) in X iscalledacommutativegeneralizedneutrosophicideal ofXifitsatisfiesthecondition (12) and (∀x, y, z ∈ X)       AT (x ∗ (y ∗ (y ∗ x))) ≥ min{AT ((x ∗ y) ∗ z), AT (z)} AIT (x ∗ (y ∗ (y ∗ x))) ≥ min{AIT ((x ∗ y) ∗ z), AIT (z)} AIF (x ∗ (y ∗ (y ∗ x))) ≤ max{AIF ((x ∗ y) ∗ z), AIF (z)} AF (x ∗ (y ∗ (y ∗ x))) ≤ max{AF ((x ∗ y) ∗ z), AF (z)}       .(14) Example1. DenoteX = {0, a, b, c}.Thebinaryoperation ∗ onXisdefinedinTable 1. Florentin Smarandache (author and editor) Collected Papers, IX 239

(∀x, y ∈ X)

)

.(13)

Table1. Theoperation“∗”. ∗ 0 abc 00000 aa 00 a bba 0 b cccc 0

Wecanverifythat (X, ∗,0) isa BCK-algebra(see[7]).DeﬁneaGNS A =(AT , AIT , AIF, AF ) in X by Table 2

Table2. GNS A =(AT , AIT , AIF, AF )

XAT (x) A IT (x) A IF (x) AF (x) 00.70.60.10.3 a 0.50.50.20.4 b 0.30.20.40.6 c 0.30.20.40.6

ThenA =(AT , AIT , AIF, AF ) isacommutativegeneralizedneutrosophicidealofX. Theorem1. Everycommutativegeneralizedneutrosophicidealisageneralizedneutrosophicideal.

Proof. Assumethat A =(AT , AIT , AIF, AF ) isacommutativegeneralizedneutrosophicidealof X. ∀x, z ∈ X,wehave

AT (x)= AT (x ∗ (0 ∗ (0 ∗ x))) ≥ min{AT ((x ∗ 0) ∗ z), AT (z)} = min{AT (x ∗ z), AT (z)},

AIT (x)= AIT (x ∗ (0 ∗ (0 ∗ x))) ≥ min{AIT ((x ∗ 0) ∗ z), AIT (z)} = min{AIT (x ∗ z), AIT (z)}, AIF (x)= AIF (x ∗ (0 ∗ (0 ∗ x))) ≤ max{AIF ((x ∗ 0) ∗ z), AIF (z)} = max{AIF (x ∗ z), AIF (z)}, and AF (x)= AF (x ∗ (0 ∗ (0 ∗ x))) ≤ max{AF ((x ∗ 0) ∗ z), AF (z)} = max{AF (x ∗ z), AF (z)}. Therefore A =(AT , AIT , AIF, AF ) isageneralizedneutrosophicideal. ThefollowingexampleshowsthattheinverseofTheorem 1 isnottrue. Example2. LetX = {0,1,2,3,4} beasetwiththebinaryoperation ∗ whichisdeﬁnedinTable 3 Table3. Theoperation“∗”. ∗ 01234 000000 110100 222000 333300 444430

Wecanverifythat (X, ∗,0) isa BCK-algebra(see[7]).WedeﬁneaGNS A =(AT , AIT , AIF, AF ) in X byTable 4

Florentin Smarandache (author and editor) Collected Papers, IX 240

Table4. GNS A =(AT , AIT , AIF, AF ).

XAT (x) A IT (x) A IF (x) AF (x) 00.70.60.10.3 10.50.40.20.6 20.30.50.40.4 30.30.40.40.6 40.30.40.40.6

Itisroutinetoverifythat A =(AT , AIT , AIF, AF ) isageneralizedneutrosophicidealof X,but A isnot acommutativegeneralizedneutrosophicidealofXsince

AT (2 ∗ (3 ∗ (3 ∗ 2)))= AT (2)= 0.3 min{AT ((2 ∗ 3) ∗ 0), AT (0)} and/or AIF (2 ∗ (3 ∗ (3 ∗ 2)))= AIF (2)= 0.4 max{AIF ((2 ∗ 3) ∗ 0), AIF (0)}.

AT (x ∗ y) ≤ AT (x ∗ (y ∗ (y ∗ x)))

AIT (x ∗ y) ≤ AIT (x ∗ (y ∗ (y ∗ x))) AIF (x ∗ y) ≥ AIF (x ∗ (y ∗ (y ∗ x))) AF (x ∗ y) ≥ AF (x ∗ (y ∗ (y ∗ x)))

      .(15)

Theorem2. Supposethat A =(AT , AIT , AIF, AF ) isageneralizedneutrosophicidealof X.Then A =(AT , AIT , AIF, AF ) iscommutativeifandonlyifitsatisﬁesthefollowingcondition. (∀x, y ∈ X)      

Proof. Assumethat A =(AT , AIT , AIF, AF ) isacommutativegeneralizedneutrosophicidealof X. Taking z = 0in(14)andusing(12)and(1)induces(15).

Conversely,let A =(AT , AIT , AIF, AF ) beageneralizedneutrosophicidealof X satisfyingthe condition(15).Then

AT (x ∗ (y ∗ (y ∗ x))) ≥ AT (x ∗ y) ≥ min{AT ((x ∗ y) ∗ z), AT (z)}, AIT (x ∗ (y ∗ (y ∗ x))) ≥ AIT (x ∗ y) ≥ min{AIT ((x ∗ y) ∗ z), AIT (z)}, AIF (x ∗ (y ∗ (y ∗ x))) ≤ AIF (x ∗ y) ≤ max{AIF ((x ∗ y) ∗ z), AIF (z)}

z ∈ X

) ofXsatisﬁes: (∀x, y, z ∈ X)     x ∗ y ≤ z ⇒        AT (x) ≥ min{AT (y), AT (z)} AIT (x) ≥ min{AIT (y), AIT (z)} AIF (x) ≤ max{AIF (y), AIF (z)} AF (x) ≤ max{AF (y), AF (z)}     .(16) Weprovideaconditionforageneralizedneutrosophicidealtobecommutative. Florentin Smarandache (author and editor) Collected Papers, IX 241

and

F (x ∗ (y ∗ (y ∗ x))) ≤ AF (x ∗ y) ≤ max{AF ((x ∗ y) ∗ z), AF (z)} forall x, y,

.Therefore A =(AT , AIT , AIF, AF ) isacommutativegeneralizedneutrosophicideal of X Lemma2 ([15]). AnygeneralizedneutrosophicidealA =(AT , AIT , AIF, AF

Theorem3. ForanycommutativeBCK-algebra,everygeneralizedneutrosophicidealiscommutative.

Proof. Assumethat A =(AT , AIT , AIF, AF ) isageneralizedneutrosophicidealofacommutative BCK-algebra X.Notethat ((x ∗ (y ∗ (y ∗ x))) ∗ ((x ∗ y) ∗ z)) ∗ z =((x ∗ (y ∗ (y ∗ x))) ∗ z) ∗ ((x ∗ y) ∗ z) ≤ (x ∗ (y ∗ (y ∗ x))) ∗ (x ∗ y) =(x ∗ (x ∗ y)) ∗ (y ∗ (y ∗ x))= 0, thus, (x ∗ (y ∗ (y ∗ x))) ∗ ((x ∗ y) ∗ z) ≤ z, ∀x, y, z ∈ X.ByLemma 2 weget

AT (x ∗ (y ∗ (y ∗ x))) ≥ min{AT ((x ∗ y) ∗ z), AT (z)}, AIT (x ∗ (y ∗ (y ∗ x))) ≥ min{AIT ((x ∗ y) ∗ z), AIT (z)}, AIF (x ∗ (y ∗ (y ∗ x))) ≤ max{AIF ((x ∗ y) ∗ z), AIF (z)}, AF (x ∗ (y ∗ (y ∗ x))) ≤ max{AF ((x ∗ y) ∗ z), AF (z)}

Therefore A =(AT , AIT , AIF, AF ) isacommutativegeneralizedneutrosophicidealof X.

Lemma3 ([15]) IfaGNS A =(AT , AIT , AIF, AF ) in X isageneralizedneutrosophicidealof X,thenthe sets UA (T, αT ), UA (IT, αIT ), LA (F, βF ) and LA (IF, β IF ) areidealsof X forall αT , αIT , βF, β IF ∈ [0,1] whenevertheyarenon-empty.

Theorem4. IfaGNS A =(AT , AIT , AIF, AF ) in X isacommutativegeneralizedneutrosophicidealof X, thenthesets UA (T, αT ), UA (IT, αIT ), LA (F, βF ) and LA (IF, β IF ) arecommutativeidealsof X forall αT , αIT , βF, β IF ∈ [0,1] whenevertheyarenon-empty.

Thecommutativeideals UA (T, αT ), UA (IT, αIT ), LA (F, βF ) and LA (IF, β IF ) arecalled level neutrosophiccommutativeideals of A =(AT , AIT , AIF, AF ) Proof. Assumethat A =(AT , AIT , AIF, AF ) isacommutativegeneralizedneutrosophicideal of X.Then A =(AT , AIT , AIF, AF ) isageneralizedneutrosophicidealof X.Thus UA (T, αT ), UA (IT, αIT ), LA (F, βF ) and LA (IF, β IF ) areidealsof X whenevertheyarenon-emptyapplying Lemma 3.Supposethat x, y ∈ X and x ∗ y ∈ UA (T, αT ) ∩ UA (IT, αIT ).Using(15), AT (x ∗ (y ∗ (y ∗ x))) ≥ AT (x ∗ y) ≥ αT , AIT (x ∗ (y ∗ (y ∗ x))) ≥ AIT (x ∗ y) ≥ αIT , andso x ∗ (y ∗ (y ∗ x)) ∈ UA (T, αT ) and x ∗ (y ∗ (y ∗ x)) ∈ UA (IT, αIT ).Supposethat a, b ∈ X and a ∗ b ∈ LA (IF, β IF ) ∩ LA (F, βF ).Itfollowsfrom (15) that AIF (a ∗ (b ∗ (b ∗ a))) ≤ AIF (a ∗ b) ≤ β IF and AF (a ∗ (b ∗ (b ∗ a))) ≤ AF (a ∗ b) ≤ βF.Hence a ∗ (b ∗ (b ∗ a)) ∈ LA (IF, β IF ) and a*(b*(b*a)) ∈ LA (F, βF ).Therefore UA (T, αT ), UA (IT, αIT ), LA (F, βF ) and LA (IF, β IF ) arecommutativeideals of X

Lemma4 ([15]). Assumethat A =(AT , AIT , AIF, AF ) isaGNSin X and UA (T, αT ), UA (IT, αIT ), LA (F, βF ) and LA (IF, β IF ) areidealsof X, ∀αT , αIT , βF, β IF ∈ [0,1].Then A =(AT , AIT , AIF, AF ) isa generalizedneutrosophicidealofX.

Theorem5. Let A =(AT , AIT , AIF, AF ) beaGNSin X suchthat UA (T, αT ), UA (IT, αIT ), LA (F, βF ) and LA (IF, β IF ) arecommutativeidealsof X forall αT , αIT , βF, β IF ∈ [0,1].Then A =(AT , AIT , AIF, AF ) isa commutativegeneralizedneutrosophicidealofX.

Florentin Smarandache (author and editor) Collected Papers, IX 242

Proof. Let αT , αIT , βF, β IF ∈ [0,1] besuchthatthenon-emptysets UA (T, αT ), UA (IT, αIT ), LA (F, βF ) and LA (IF, β IF ) arecommutativeidealsof X.Then UA (T, αT ), UA (IT, αIT ), LA (F, βF ) and LA (IF, β IF ) areidealsof X.Hence A =(AT , AIT , AIF, AF ) isageneralizedneutrosophicidealof X applying Lemma 4.Forany x, y ∈ X,let AT (x ∗ y)= αT .Then x ∗ y ∈ UA (T, αT ),andso x ∗ (y ∗ (y ∗ x)) ∈ UA (T, αT ) by(9).Hence AT (x ∗ (y ∗ (y ∗ x))) ≥ αT = AT (x ∗ y).Similarly,wecanshowthat (∀x, y ∈ X)(AIT (x ∗ (y ∗ (y ∗ x))) ≥ AIT (x ∗ y)).

Forany x, y, a, b, ∈ X,let AF (x ∗ y)= βF and AIF (a ∗ b)= β IF.Then x ∗ y ∈ LA (F, βF ) and a ∗ b ∈ LA (IF, β IF ).UsingLemma 1 wehave x ∗ (y ∗ (y ∗ x)) ∈ LA (F, βF ) and a ∗ (b ∗ (b ∗ a)) ∈ LA (IF, β IF ) Thus AF (x ∗ y)= βF ≥ AF (x ∗ (y ∗ (y ∗ x))) and AIF (a ∗ b)= β IF ≥ AIF ((a ∗ b) ∗ b).Therefore A =(AT , AIT , AIF, AF ) isacommutativegeneralizedneutrosophicidealof X

Theorem6. Everycommutativegeneralizedneutrosophicidealcanberealizedaslevelneutrosophic commutativeidealsofsomecommutativegeneralizedneutrosophicidealofX.

Proof. Givenacommutativeideal C of X,deﬁneaGNS A =(AT , AIT , AIF, AF ) asfollows

AT (x)= αT if x ∈ C , 0otherwise, AIT (x)= αIT if x ∈ C , 0otherwise,

AIF (x)= β IF if x ∈ C , 1otherwise, AF (x)= βF if x ∈ C , 1otherwise, where αT , αIT ∈ (0,1] and βF, β IF ∈ [0,1).Let x, y, z ∈ X.If (x ∗ y) ∗ z ∈ C and z ∈ C, then x ∗ (y ∗ (y ∗ x)) ∈ C.Thus

AT (x ∗ (y ∗ (y ∗ x)))= αT = min{AT ((x ∗ y) ∗ z), AT (z)}, AIT (x ∗ (y ∗ (y ∗ x)))= αIT = min{AIT ((x ∗ y) ∗ z), AIT (z)}, AIF (x ∗ (y ∗ (y ∗ x)))= β IF = max{AIF ((x ∗ y) ∗ z), AIF (z)}, AF (x ∗ (y ∗ (y ∗ x)))= βF = max{AF ((x ∗ y) ∗ z), AF (z)}.

Assumethat (x ∗ y) ∗ z / ∈ C and z / ∈ C.Then AT ((x ∗ y) ∗ z)= 0, AT (z)= 0, AIT ((x ∗ y) ∗ z)= 0, AIT (z)= 0, AIF ((x ∗ y) ∗ z)= 1, AIF (z)= 1,and AF ((x ∗ y) ∗ z)= 1, AF (z)= 1.Itfollowsthat

AT (x ∗ (y ∗ (y ∗ x))) ≥ min{AT ((x ∗ y) ∗ z), AT (z)}, AIT (x ∗ (y ∗ (y ∗ x))) ≥ min{AIT ((x ∗ y) ∗ z), AIT (z)},

AIF (x ∗ (y ∗ (y ∗ x))) ≤ max{AIF ((x ∗ y) ∗ z), AIF (z)}, AF (x ∗ (y ∗ (y ∗ x))) ≤ max{AF ((x ∗ y) ∗ z), AF (z)}

Ifexactlyoneof (x ∗ y) ∗ z and z belongsto C,thenexactlyoneof AT ((x ∗ y) ∗ z) and AT (z) is equalto0;exactlyoneof AIT ((x ∗ y) ∗ z) and AIT (z) isequalto0;exactlyoneof AF ((x ∗ y) ∗ z) and AF (z) isequalto1andexactlyoneof AIF ((x ∗ y) ∗ z) and AIF (z) isequalto1.Hence

AT (x ∗ (y ∗ (y ∗ x))) ≥ min{AT ((x ∗ y) ∗ z), AT (z)}, AIT (x ∗ (y ∗ (y ∗ x))) ≥ min{AIT ((x ∗ y) ∗ z), AIT (z)},

AIF (x ∗ (y ∗ (y ∗ x))) ≤ max{AIF ((x ∗ y) ∗ z), AIF (z)}, AF (x ∗ (y ∗ (y ∗ x))) ≤ max{AF ((x ∗ y) ∗ z), AF (z)}

Itisclearthat AT (0) ≥ AT (x), AIT (0) ≥ AIT (x), AIF (0) ≤ AIF (x) and AF (0) ≤ AF (x) forall x ∈ X.Therefore A =(AT , AIT , AIF, AF ) isacommutativegeneralizedneutrosophicidealof X

Florentin Smarandache (author and editor) Collected Papers, IX 243

Obviously, UA (T, αT )= C, UA (IT, αIT )= C, LA (F, βF )= C and LA (IF, β IF )= C.Thiscompletes theproof.

Theorem7. Let {Ct | t ∈ Λ} beacollectionofcommutativeidealsofXsuchthat (1) X = t∈Λ Ct, (2) (∀s, t ∈ Λ) (s > t ⇐⇒ Cs ⊂ Ct ) where Λ isanyindexset.LetA =(AT , AIT , AIF, AF ) beaGNSinXgivenby (∀x ∈ X)

AT (x)= sup{t ∈ Λ | x ∈ Ct } = AIT (x) AIF (x)= inf{t ∈ Λ | x ∈ Ct } = AF (x) .(17)

ThenA =(AT , AIT , AIF, AF ) isacommutativegeneralizedneutrosophicidealofX. Proof. AccordingtoTheorem 5,itissufﬁcienttoshowthat U(T, t), U(IT, t), L(F, s) and L(IF, s) are commutativeidealsof X forevery t ∈ [0, AT (0)= AIT (0)] and s ∈ [AIF (0)= AF (0),1].Inorderto prove U(T, t) and U(IT, t) arecommutativeidealsof X,weconsidertwocases:

(i) t = sup{q ∈ Λ | q < t}, (ii) t = sup{q ∈ Λ | q < t}.

Fortheﬁrstcase,wehave

x ∈ U(T, t) ⇐⇒ (∀q < t)(x ∈ Cq ) ⇐⇒ x ∈ q<t Cq,

x ∈ U(IT, t) ⇐⇒ (∀q < t)(x ∈ Cq ) ⇐⇒ x ∈ q<t Cq

Hence U(T, t)= q<t Cq = U(IT, t),andso U(T, t) and U(IT, t) arecommutativeidealsof X.

Forthesecondcase,weclaimthat U(T, t)= q≥t Cq = U(IT, t).If x ∈ q≥t Cq,then x ∈ Cq for some q ≥ t.Itfollowsthat AIT (x)= AT (x) ≥ q ≥ t andsothat x ∈ U(T, t) and x ∈ U(IT, t) Thisshowsthat q≥t Cq ⊆ U(T, t) and q≥t Cq ⊆ U(IT, t).Now,suppose x / ∈ q≥t Cq.Then x / ∈ Cq, ∀q ≥ t Since t = sup{q ∈ Λ | q < t},thereexists ε > 0suchthat (t ε, t) ∩ Λ = ∅.Thus x / ∈ Cq, ∀q > t ε, thismeansthatif x ∈ Cq,then q ≤ t ε.So AIT (x)= AT (x) ≤ t ε < t,andso x / ∈ U(T, t)= U(IT, t).Therefore U(T, t)= U(IT, t) ⊆ q≥t Cq.Consequently, U(T, t)= U(IT, t)= q≥t Cq which isacommutativeidealof X.Nextweshowthat L(F, s) and L(IF, s) arecommutativeidealsof X Weconsidertwocasesasfollows: (iii) s = inf{r ∈ Λ | s < r}, (iv) s = inf{r ∈ Λ | s < r}. Case(iii)impliesthat

Itfollowsthat L(IF, s)= L(F, s)= s<r Cr,whichisacommutativeidealof X.Case(iv)induces (s, s + ε) ∩ Λ = ∅ forsome ε > 0.If x ∈ s≥r Cr,then x ∈ Cr forsome r ≤ s,andso AIF (x)= AF (x) ≤ r ≤ s,thatis, x ∈ L(IF, s) and x ∈ L(F, s).Hence s≥r Cr ⊆ L(IF, s)= L(F, s).If x / ∈ s≥r Cr,then x / ∈ Cr

∈ L(IF, s) ⇐⇒ (∀s < r)(x ∈ Cr ) ⇐⇒ x ∈ s<r Cr,

∈ U(F, s) ⇐⇒ (∀s < r)(x ∈ Cr ) ⇐⇒ x ∈ s<r Cr

Florentin Smarandache (author and editor) Collected Papers, IX 244

forall r ≤ s whichimpliesthat x / ∈ Cr forall r ≤ s + ε,thatis,if x ∈ Cr then r ≥ s + ε.Hence AIF (x)= AF (x) ≥ s + ε > s,andso x / ∈ L(AIF, s)= L(AF, s).Hence L(AIF, s)= L(AF, s)= s≥r Cr whichisa commutativeidealof X.Thiscompletestheproof.

Assumethta f : X → Y isahomomorphismof BCK/BCI-algebras([7]).ForanyGNS A =(AT , AIT , AIF, AF ) in Y,wedeﬁneanewGNS A f =(A f T , A f IT , A f IF, A f F ) in X,whichiscalledthe induced GNS,by (∀x ∈ X)

A f T (x)= AT ( f (x)), A f IT (x)= AIT ( f (x))

A f IF (x)= AIF ( f (x)), A f F (x)= AF ( f (x)) .(18)

Lemma5 ([15]) Let f : X → Y beahomomorphismof BCK/BCI-algebras.IfaGNS A =(AT , AIT , AIF, AF ) in Y isageneralizedneutrosophicidealof Y,thenthenewGNS A f =(A f T , A f IT , A f IF, A f F ) in X isa generalizedneutrosophicidealofX.

Theorem8. Let f : X → Y beahomomorphismof BCK-algebras.IfaGNS A =(AT , AIT , AIF, AF ) in Y isacommutativegeneralizedneutrosophicidealof Y,thenthenewGNS A f =(A f T , A f IT , A f IF, A f F ) in X isa commutativegeneralizedneutrosophicidealofX.

Proof. Supposethat A =(AT , AIT , AIF, AF ) isacommutativegeneralizedneutrosophicidealof Y Then A =(AT , AIT , AIF, AF ) isageneralizedneutrosophicidealof Y byTheorem 1,andso A f =(A f T , A f IT , A f IF, A f F ) isageneralizedneutrosophicidealof Y byLemma 5.Forany x, y ∈ X,wehave

A f T (x ∗ (y ∗ (y ∗ x)))= AT ( f (x ∗ (y ∗ (y ∗ x))))

= AT ( f (x) ∗ ( f (y) ∗ ( f (y) ∗ f (x))))

≥ AT ( f (x) ∗ f (y))

= AT ( f (x ∗ y))= A f T (x ∗ y),

A f IT (x ∗ (y ∗ (y ∗ x)))= AIT ( f (x ∗ (y ∗ (y ∗ x))))

= AIT ( f (x) ∗ ( f (y) ∗ ( f (y) ∗ f (x))))

≥ AIT ( f (x) ∗ f (y))

= AIT ( f (x ∗ y))= A f IT (x ∗ y),

A f IF (x ∗ (y ∗ (y ∗ x)))= AIF ( f (x ∗ (y ∗ (y ∗ x))))

= AIF ( f (x) ∗ ( f (y) ∗ ( f (y) ∗ f (x))))

≤ AIF ( f (x) ∗ f (y))

= AIF ( f (x ∗ y))= A f IF (x ∗ y), and

A f F (x ∗ (y ∗ (y ∗ x)))= AF ( f (x ∗ (y ∗ (y ∗ x))))

= AF ( f (x) ∗ ( f (y) ∗ ( f (y) ∗ f (x))))

≤ AF ( f (x) ∗ f (y))

= AF ( f (x ∗ y))= A f F (x ∗ y)

Therefore A f =(A f T , A f IT , A f IF, A f F ) isacommutativegeneralizedneutrosophicidealof X

Florentin Smarandache (author and editor) Collected Papers, IX 245

Lemma6 ([15]). Let f : X → Y beanontohomomorphismof BCK/BCI-algebrasandlet A =(AT , AIT , AIF, AF ) beaGNSin Y.IftheinducedGNS A f =(A f T , A f IT , A f IF, A f F ) in X isageneralizedneutrosophic idealofX,thenA =(AT , AIT , AIF, AF ) isageneralizedneutrosophicidealofY.

Theorem9. Assumethta f : X → Y isanontohomomorphismof BCK-algebrasand A =(AT , AIT , AIF, AF ) isaGNSin Y.IftheinducedGNS A f =(A f T , A f IT , A f IF, A f F ) in X isacommutativegeneralized neutrosophicidealofX,thenA =(AT , AIT , AIF, AF ) isacommutativegeneralizedneutrosophicidealofY.

Proof. Supposethat A f =(A f T , A f IT , A f IF, A f F ) isacommutativegeneralizedneutrosophicidealof X.Then A f =(A f T , A f IT , A f IF, A f F ) isageneralizedneutrosophicidealof X,andthus A =(AT , AIT , AIF, AF ) isageneralizedneutrosophicidealof Y.Forany a, b, c ∈ Y,thereexist x, y, z ∈ X suchthat f (x)= a, f (y)= b and f (z)= c.Thus, AT (a ∗ (b ∗ (b ∗ a)))= AT ( f (x) ∗ ( f (y) ∗ ( f (y) ∗ f (x))))= AT ( f (x ∗ (y ∗ (y ∗ x))))

= A f T (x ∗ (y ∗ (y ∗ x))) ≥ A f T (x ∗ y)

= AT ( f (x) ∗ f (y))= AT (a ∗ b),

AIT (a ∗ (b ∗ (b ∗ a)))= AIT ( f (x) ∗ ( f (y) ∗ ( f (y) ∗ f (x))))= AIT ( f (x ∗ (y ∗ (y ∗ x))))

= A f IT (x ∗ (y ∗ (y ∗ x))) ≥ A f IT (x ∗ y)

= AIT ( f (x) ∗ f (y))= AIT (a ∗ b),

AIF (a ∗ (b ∗ (b ∗ a)))= AIF ( f (x) ∗ ( f (y) ∗ ( f (y) ∗ f (x))))= AIF ( f (x ∗ (y ∗ (y ∗ x))))

= A f IF (x ∗ (y ∗ (y ∗ x))) ≤ A f IF (x ∗ y)

= AIF ( f (x) ∗ f (y))= AIF (a ∗ b), and

AF (a ∗ (b ∗ (b ∗ a)))= AF ( f (x) ∗ ( f (y) ∗ ( f (y) ∗ f (x))))= AF ( f (x ∗ (y ∗ (y ∗ x))))

= A f F (x ∗ (y ∗ (y ∗ x))) ≤ A f F (x ∗ y)

= AF ( f (x) ∗ f (y))= AF (a ∗ b)

ItfollowsfromTheorem 2 that A =(AT , AIT , AIF, AF ) isacommutativegeneralizedneutrosophic idealof Y.

Let CGNI(X) denotethesetofallcommutativegeneralizedneutrosophicidealsof X and t ∈ [0,1]. Deﬁnebinaryrelations Ut T , Ut IT , Lt F and Lt IF on CGNI(X) asfollows:

(A, B) ∈ Ut T ⇔ UA (T, t)= UB (T, t), (A, B) ∈ Ut IT ⇔ UA (IT, t)= UB (IT, t), (A, B) ∈ Lt F ⇔ LA (F, t)= LB (F, t), (A, B) ∈ Lt IF ⇔ LA (IF, t)= LB (IF, t) (19) for A =(AT , AIT , AIF, AF ) and B =(BT , BIT , BIF, BF ) in CGNI(X).Thenclearly Ut T , Ut IT , Lt F and Lt IF areequivalencerelationson CGNI(X).Forany A =(AT , AIT , AIF, AF ) ∈ CGNI(X), let [A]Ut T (resp., [A]Ut IT , [A]Lt F and [A]Lt IF )denotetheequivalenceclassof A =(AT , AIT , AIF, AF ) modulo Ut T (resp, Ut IT , Lt F and Lt IF).Denoteby CGNI(X)/Ut T (resp., CGNI(X)/Ut IT , CGNI(X)/Lt F and CGNI(X)/Lt IF)thesystemofallequivalenceclassesmodulo Ut T (resp, Ut IT , Lt F and Lt IF);so CGNI(X)/Ut T = {[A]Ut T | A =(AT , AIT , AIF, AF ) ∈ CGNI(X)},(20)

Florentin Smarandache (author and editor) Collected Papers, IX 246

CGNI(X)/Ut IT = {[A]Ut IT | A =(AT , AIT , AIF, AF ) ∈ CGNI(X)}, (21)

CGNI(X)/Lt F = {[A]Lt F | A =(AT , AIT , AIF, AF ) ∈ CGNI(X)}, (22) and

CGNI(X)/Lt IF = {[A]Lt IF | A =(AT , AIT , AIF, AF ) ∈ CGNI(X)}, (23) respectively.Let CI(X) denotethefamilyofallcommutativeidealsof X andlet t ∈ [0,1].Deﬁnemaps

ft : CGNI(X) → CI(X) ∪{∅}, A → UA (T, t),(24)

gt : CGNI(X) → CI(X) ∪{∅}, A → UA (IT, t),(25)

αt : CGNI(X) → CI(X) ∪{∅}, A → LA (F, t),(26) and

βt : CGNI(X) → CI(X) ∪{∅}, A → LA (IF, t).(27)

Thenthedeﬁnitionsof ft, gt, αt and βt arewell.

Theorem10. Suppose t ∈ (0,1),thedeﬁnitionsof ft, gt, αt and βt areasabove.Thenthemaps ft, gt, αt and βt aresurjectivefromCGNI(X) toCI(X) ∪{∅}.

Proof. Assume t ∈ (0,1).Weknowthat 0∼ =(0T , 0IT , 1IF, 1F ) isin CGNI(X) where 0T , 0IT , 1IF and 1F areconstantfunctionson X deﬁnedby 0T (x)= 0, 0IT (x)= 0, 1IF (x)= 1and 1F (x)= 1forall x ∈ X Obviously ft (0∼)= U0∼ (T, t), gt (0∼)= U0∼ (IT, t), αt (0∼)= L0∼ (F, t) and βt (0∼)= L0∼ (IF, t) are empty.Let G(= ∅) ∈ CGNI(X),andconsiderfunctions:

GT : X → [0,1], G →

GIT : X → [0,1], G →

1if x ∈ G , 0otherwise,

0if x ∈ G , 1otherwise, and

GF : X → [0,1], G →

GIF : X → [0,1], G →

0if x ∈ G , 1otherwise.

Then G∼ = (GT , GIT , GIF, GF ) isacommutativegeneralizedneutrosophicidealof X,and ft (G∼)= UG∼ (T, t)= G, gt (G∼)= UG∼ (IT, t)= G, αt (G∼)= LG∼ (F, t)= G and βt (G∼)= LG∼ (IF, t)= G.Therefore ft, gt, αt and βt aresurjective.

Theorem11. Thequotientsets

CGNI(X)/Ut T ,CGNI(X)/Ut IT ,CGNI(X)/Lt F andCGNI(X)/Lt IF

Florentin Smarandache (author and editor) Collected Papers, IX 247

are equipotent to CI(X) ∪ {∅}. Proof. For t ∈ (0, 1), let ft ∗ (resp, gt ∗ , α∗ t and β∗ t )beamapfrom CGNI(X)/Ut T (resp., CGNI(X)/Ut IT , CGNI(X)/Lt F and CGNI(X)/Lt IF)to CI(X) ∪{∅} deﬁnedby f ∗ t [A]Ut T = ft (A) (resp., g∗ t [A]Ut IT = gt (A) , α∗ t [A]Lt F = αt (A) and β∗ t [A]Lt IF = βt (A))forall A =(AT , AIT , AIF, AF ) ∈ CGNI(X).If UA (T, t)= UB (T, t), UA (IT, t)= UB (IT, t), LA (F, t)= LB (F, t) and LA (IF, t)= LB (IF, t) for A =(AT , AIT , AIF, AF ) and B =(BT , BIT , BF, BIF ) in CGNI(X), then (A, B) ∈ Ut T , (A, B) ∈ Ut IT , (A, B) ∈ Lt F and (A, B) ∈ Lt IF.Hence [A]Ut T =[B]Ut T , [A]Ut IT =[B]Ut IT , [A]Lt F =[B]Lt F and [A]Lt IF =[B]Lt IF .Therefore f ∗ t (resp, g∗ t , α∗ t and β∗ t )isinjective.Nowlet G(= ∅) ∈ CGNI(X).For G∼ = (GT , GIT , GIF, GF ) ∈ CGNI(X),wehave

f ∗ t [G∼]Ut T = ft (G∼)= UG∼ (T, t)= G, g∗ t [G∼]Ut IT = gt (G∼)= UG∼ (IT, t)= G, α∗ t [G∼]Lt F = αt (G∼)= LG∼ (F, t)= G and β∗ t [G∼]Lt IF = βt (G∼)= LG∼ (IF, t)= G

Finally,for 0∼ =(0T , 0IT , 1IF, 1F ) ∈ CGNI(X),wehave f ∗ t [0∼]Ut T = ft (0∼)= U0 (T, t)= ∅, g∗ t [0∼]Ut IT = gt (0∼)= U0∼ (IT, t)= ∅, α∗ t [0∼]Lt F = αt (0∼)= L0 (F, t)= ∅ and β∗ t [0∼]Lt IF = βt (0∼)= L0∼ (IF, t)= ∅ Therefore, f ∗ t (resp, g∗ t , α∗ t and β∗ t )issurjective. ∀t ∈ [0,1],deﬁneanotherrelations Rt and Qt on CGNI(X) asfollows: (A, B) ∈ Rt ⇔ UA (T, t) ∩ LA (F, t)= UB (T, t) ∩ LB (F, t) and (A, B) ∈ Qt ⇔ UA (IT, t) ∩ LA (IF, t)= UB (IT, t) ∩ LB (IF, t) forany A =(AT , AIT , AIF, AF ) and B =(BT , BIT , BIF, BF ) in CGNI(X).Then Rt and Qt are equivalencerelationson CGNI(X)

Florentin Smarandache (author and editor) Collected Papers, IX 248

Theorem12. Supposet ∈ (0,1),considerthefollowingmaps

ϕt : CGNI(X) → CI(X) ∪{∅}, A → ft (A) ∩ αt (A), (28) and

ψt : CGNI(X) → CI(X) ∪{∅}, A → gt (A) ∩ βt (A) (29) foreachA =(AT , AIT , AIF, AF ) ∈ CGNI(X).Then ϕt and ψt aresurjective.

Proof. Assume t ∈ (0,1).For 0∼ =(0T , 0IT , 1IF, 1F ) ∈ CGNI(X),

ϕt (0∼)= ft (0∼) ∩ αt (0∼)= U0∼ (T, t) ∩ L0∼ (F, t)= ∅ and ψt (0∼)= gt (0∼) ∩ βt (0∼)= U0∼ (IT, t) ∩ L0∼ (IF, t)= ∅

Forany G ∈ CI(X),thereexists G∼ = (GT , GIT , GIF, GF ) ∈ CGNI(X) suchthat ϕt (G∼)= ft (G∼) ∩ αt (G∼)= UG∼ (T, t) ∩ LG∼ (F, t)= G and ψt (G∼)= gt (G∼) ∩ βt (G∼)= UG∼ (IT, t) ∩ LG∼ (IF, t)= G.

Therefore ϕt and ψt aresurjective. Theorem13. Forany t ∈ (0,1),thequotientsets CGNI(X)/Rt and CGNI(X)/Qt areequipotentto CI(X) ∪{∅}.

Proof. Let t ∈ (0,1) anddeﬁnemaps

ϕ∗ t : CGNI(X)/Rt → CI(X) ∪{∅}, [A]Rt → ϕt (A) and ψ∗ t : CGNI(X)/Qt → CI(X) ∪{∅}, [A]Qt → ψt (A)

If ϕ∗ t ([A]Rt ) = ϕ∗ t ([B]Rt ) and ψ∗ t [A]Qt = ψ∗ t [B]Qt forall [A]Rt , [B]Rt ∈ CGNI(X)/Rt and [A]Qt , [B]Qt ∈ CGNI(X)/Qt,then ft (A) ∩ αt (A)= ft (B) ∩ αt (B) and gt (A) ∩ βt (A)= gt (B) ∩ βt (B), thatis, UA (T, t) ∩ LA (F, t)= UB (T, t) ∩ LB (F, t) and UA (IT, t) ∩ LA (IF, t)= UB (IT, t) ∩ LB (IF, t). Hence (A, B) ∈ Rt , (A, B) ∈ Qt.So [A]Rt =[B]Rt , [A]Qt =[B]Qt ,whichshowsthat ϕ∗ t and ψ∗ t are injective.For 0∼ =(0T , 0IT , 1IF, 1F ) ∈ CGNI(X), ϕ∗ t ([0∼]Rt ) = ϕt (0∼)= ft (0∼) ∩ αt (0∼)= U0∼ (0T , t) ∩ L0∼ (1F, t)= ∅ and ψ∗ t [0∼]Qt = ψt (0∼)= gt (0∼) ∩ βt (0∼)= U0∼ (0IT , t) ∩ L0∼ (1IF, t)= ∅. If G ∈ CI(X),then G∼ = (GT , GIT , GIF, GF ) ∈ CGNI(X),andso ϕ∗ t ([G∼]Rt ) = ϕt (G∼)= ft (G∼) ∩ αt (G∼)= UG∼ (GT , t) ∩ LG∼ (GF, t)= G

Florentin Smarandache (author and editor) Collected Papers, IX 249

Hence ϕ∗ t and ψ∗ t aresurjective,andtheproofiscomplete.

4. Conclusions

Based on the theory of generalized neutrosophic sets, we proposed the new concept of commutative generalized neutrosophic ideal in a BCK-algebra, and obtained some characterizations. Moreover, we investigated some homomorphism properties related to commutative generalized neutrosophic ideals.

The research ideas of this paper can be extended to a wide range of logical algebraic systems such as pseudo-BCI algebras (see [1,16]). At the same time, the concept of generalized neutrosophic set involved in this paper can be further studied according to the thought in [11,17], which will be the direction of our next research work.

Funding: This research was funded by the National Natural Science Foundation of China grant number 61573240.

References

1. Zhang,X.H.Fuzzyanti-groupedfiltersandfuzzynormalfiltersinpseudo-BCIalgebras. J.Intell.FuzzySyst. 2017, 33,1767–1774.[CrossRef]

2. Jun,Y.B.Neutrosophicsubalgebrasofseveraltypesin BCK/BCI-algebras. Ann.FuzzyMath.Inform. 2017, 14, 75–86.

3. Jun,Y.B.;Kim,S.J.;Smarandache,F.Intervalneutrosophicsetswithapplicationsin BCK/BCI-algebra. Axioms 2018, 7,23.[CrossRef]

4. Jun,Y.B.;Smarandache,F.;Bordbar,H.Neutrosophic N -structuresappliedto BCK/BCI-algebras. Information 2017, 8,128.[CrossRef]

5. Jun,Y.B.;Smarandache,F.;Song,S.Z.;Khan,M.Neutrosophicpositiveimplicative N -idealsin BCK/BCI algebras. Axioms 2018, 7,3.[CrossRef]

6. Khan,M.;Anis,S.;Smarandache,F.;Jun,Y.B.Neutrosophic N -structuresandtheirapplicationsinsemigroups. Ann.FuzzyMath.Inform. 2017, 14,583–598.

7.Meng,J.;Jun,Y.B. BCK-Algebras;KyungMoonSaCo.:Seoul,Korea,1994.

8. Öztürk,M.A.;Jun,Y.B.Neutrosophicidealsin BCK/BCI-algebrasbasedonneutrosophicpoints. J.Int.Math. VirtualInst. 2018, 8,1–17.

9. Saeid,A.B.;Jun,Y.B.Neutrosophicsubalgebrasof BCK/BCI-algebrasbasedonneutrosophicpoints. Ann.FuzzyMath.Inform. 2017, 14,87–97.

10. Song,S.Z.;Smarandache,F.;Jun,Y.B.Neutrosophiccommutative N -idealsin BCK-algebras. Information 2017, 8,130.[CrossRef]

11. Zhang,X.H.;Bo,C.X.;Smarandache,F.;Park,C.Newoperationsoftotallydependent-neutrosophicsetsand totallydependent-neutrosophicsoftsets. Symmetry 2018, 10,187.[CrossRef]

12. Zhang,X.H.;Smarandache,F.;Liang,X.L.Neutrosophicdupletsemi-groupandcancellableneutrosophic tripletgroups. Symmetry 2017, 9,275.[CrossRef]

13.Huang,Y.S. BCI-Algebra;SciencePress:Beijing,China,2006.

14. Smarandache,F. AUnifyingFieldinLogics.Neutrosophy:NeutrosophicProbability,SetandLogic; AmericanResearchPress:Rehoboth,NM,USA,1999.

15. Song,S.Z.;Khan,M.;Smarandache,F.;Jun,Y.B.Anovelextensionofneutrosophicsetsanditsapplication in BCK/BCI-algebras.In NewTrendsinNeutrosophicTheoryandApplications(VolumeII);PonsEditions; EU:Brussels,Belgium,2018;pp.308–326.

16. Zhang,X.H.;Park,C.;Wu,S.P.Softsettheoreticalapproachtopseudo-BCIalgebras. J.Intell.FuzzySyst. 2018, 34,559–568.[CrossRef]

17. Zhang,X.H.;Bo,C.X.;Smarandache,F.;Dai,J.H.Newinclusionrelationofneutrosophicsetswith applicationsandrelatedlatticestructure. Int.J.Mach.Learn.Cybern. 2018.[CrossRef]

and ψ∗ t [G∼]Qt = ψt (G∼)= gt (G∼) ∩ βt (G∼)= UG∼ (GIT , t) ∩ LG∼ (GIF, t)= G

Florentin Smarandache (author and editor) Collected Papers, IX 250

Neutrosophic Quadruple BCK/BCI-Algebras

Young Bae Jun, Seok-Zun Song, Florentin Smarandache, Hashem Bordbar

Young Bae Jun, Seok-Zun Song, Florentin Smarandache, Hashem Bordbar (2018). Neutrosophic Quadruple BCK/BCI-Algebras. Axioms 7, 41. DOI: 10.3390/axioms7020041

Abstract: The notion of a neutrosophic quadruple BCK/BCI-number is considered, and a neutrosophic quadruple BCK/BCI-algebra, which consists of neutrosophic quadruple BCK/BCI-numbers, is constructed. Several properties are investigated, and a (positive implicative) ideal in a neutrosophic quadruple BCK-algebra and a closed ideal in a neutrosophic quadruple BCI-algebra are studied. Given subsets A and B of a BCK/BCI-algebra, the set NQ(A, B), which consists of neutrosophic quadruple BCK/BCI-numbers with a condition, is established. Conditions for the set NQ(A, B) to be a (positive implicative) ideal of a neutrosophic quadruple BCK-algebra are provided, and conditions for the set NQ(A, B) to be a (closed) ideal of a neutrosophic quadruple BCI-algebra are given. An example to show that the set {0} is not a positive implicative ideal in a neutrosophic quadruple BCK-algebra is provided, and conditions for the set {0 ˜} to be a positive implicative ideal in a neutrosophic quadruple BCK-algebra are then discussed.

Keywords: neutrosophic quadruple BCK/BCI-number; neutrosophic quadruple BCK/BCI-algebra; neutrosophic quadruple subalgebra; (positive implicative) neutrosophic quadruple ideal

1.Introduction

ThenotionofaneutrosophicsetwasdevelopedbySmarandache[1 3]andisamoregeneralplatform thatextendsthenotionsofclassicsets,(intuitionistic)fuzzysets,andintervalvalued(intuitionistic) fuzzysets.Neutrosophicsettheoryisappliedtoadifferentfield(see[4 8]).Neutrosophicalgebraic structuresin BCK/BCI-algebrasarediscussedin[9 16].Neutrosophicquadruplealgebraicstructures andhyperstructuresarediscussedin[17,18].

Inthispaper,wewilluseneutrosophicquadruplenumbersbasedonasetandconstruct neutrosophicquadruple BCK/BCI-algebras.Weinvestigateseveralpropertiesandconsideridealsand positiveimplicativeidealsinneutrosophicquadruple BCK-algebra,andclosedidealsinneutrosophic quadruple BCI-algebra.Givensubsets A and B ofaneutrosophicquadruple BCK/BCI-algebra, weconsider sets NQ(A, B),whichconsistofneutrosophicquadruple BCK/BCI-numberswitha condition.Weprovideconditionsfortheset NQ(A, B) tobea(positiveimplicative)idealofa neutrosophicquadruple BCK-algebraandfortheset NQ(A, B) tobea(closed)idealofaneutrosophic quadruple BCI-algebra.Wegiveanexampletoshowthattheset {0} isnotapositiveimplicativeideal inaneutrosophicquadruple BCK-algebra,andwethenconsiderconditionsfortheset {˜ 0} tobea positiveimplicativeidealinaneutrosophicquadruple BCK-algebra.

Smarandache (author and editor) Collected Papers, IX 251

Florentin

2.Preliminaries

A BCK/BCI-algebraisanimportantclassoflogicalalgebrasintroducedbyIséki(see[19,20]).

Bya BCI-algebra,wemeanaset X withaspecialelement0andabinaryoperation ∗ thatsatisﬁes thefollowingconditions:

(I) (∀x, y, z ∈ X)(((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y)= 0); (II) (∀x, y ∈ X)((x ∗ (x ∗ y)) ∗ y = 0); (III) (∀x ∈ X)(x ∗ x = 0); (IV) (∀x, y ∈ X)(x ∗ y = 0, y ∗ x = 0 ⇒ x = y)

Ifa BCI-algebra X satisﬁestheidentity

(V) (∀x ∈ X)(0 ∗ x = 0),

then X iscalleda BCK-algebra.Any BCK/BCI-algebra X satisﬁesthefollowingconditions:

(∀x ∈ X) (x ∗ 0 = x) (1)

(∀x, y, z ∈ X) (x ≤ y ⇒ x ∗ z ≤ y ∗ z, z ∗ y ≤ z ∗ x) (2) (∀x, y, z ∈ X) ((x ∗ y) ∗ z =(x ∗ z) ∗ y) (3)

(∀x, y, z ∈ X) ((x ∗ z) ∗ (y ∗ z) ≤ x ∗ y) (4)

where x ≤ y ifandonlyif x ∗ y = 0.Any BCI-algebra X satisﬁesthefollowingconditions(see[21]):

(∀x, y ∈ X)(x ∗ (x ∗ (x ∗ y))= x ∗ y),(5) (∀x, y ∈ X)(0 ∗ (x ∗ y)=(0 ∗ x) ∗ (0 ∗ y)).(6)

A BCK-algebra X issaidtobe positiveimplicative ifthefollowingassertionisvalid. (∀x, y, z ∈ X) ((x ∗ z) ∗ (y ∗ z)=(x ∗ y) ∗ z) .(7)

Anonemptysubset S ofa BCK/BCI-algebra X iscalleda subalgebra of X if x ∗ y ∈ S forall x, y ∈ S.Asubset I ofa BCK/BCI-algebra X iscalledan ideal of X ifitsatisﬁes 0 ∈ I,(8) (∀x ∈ X) (∀y ∈ I)(x ∗ y ∈ I ⇒ x ∈ I) .(9)

Asubset I ofa BCI-algebra X iscalleda closedideal (see[21])of X ifitisanidealof X whichsatisfies (∀x ∈ X)(x ∈ I ⇒ 0 ∗ x ∈ I).(10)

Asubset I ofa BCK-algebra X iscalleda positiveimplicativeideal (see[22])of X ifitsatisfies(8)and (∀x, y, z ∈ X)(((x ∗ y) ∗ z ∈ I, y ∗ z ∈ I ⇒ x ∗ z ∈ I) (11)

Observethateverypositiveimplicativeidealisanideal,buttheconverseisnottrue(see[22]). Notealsothata BCK-algebra X ispositiveimplicativeifandonlyifeveryidealof X ispositive implicative(see[22]).

Wereferthereadertothebooks[21,22]forfurtherinformationregarding BCK/BCI-algebras, andtothesite“http://fs.gallup.unm.edu/neutrosophy.htm”forfurtherinformationregarding neutrosophicsettheory.

3.NeutrosophicQuadruple BCK/BCI-Algebras

Weconsiderneutrosophicquadruplenumbersbasedonasetinsteadofrealorcomplexnumbers.

Florentin Smarandache (author and editor) Collected Papers, IX 252

Deﬁnition1. Let X beaset.Aneutrosophicquadruple X-numberisanorderedquadruple (a, xT, yI, zF) wherea, x, y, z ∈ XandT, I, Fhavetheirusualneutrosophiclogicmeanings.

Thesetofallneutrosophicquadruple X-numbersisdenotedby NQ(X),thatis, NQ(X) := {(a, xT, yI, zF) | a, x, y, z ∈ X}, anditiscalledthe neutrosophicquadrupleset basedon X.If X isa BCK/BCI-algebra,aneutrosophic quadruple X-numberiscalleda neutrosophicquadruple BCK/BCI-number andwesaythat NQ(X) is the neutrosophicquadrupleBCK/BCI-set

Let X bea BCK/BCI-algebra.Wedeﬁneabinaryoperation on NQ(X) by (a, xT, yI, zF) (b, uT, vI, wF)=(a ∗ b, (x ∗ u)T, (y ∗ v)I, (z ∗ w)F) forall (a, xT, yI, zF), (b, uT, vI, wF) ∈ NQ(X).Given a1, a2, a3, a4 ∈ X,theneutrosophicquadruple BCK/BCI-number (a1, a2T, a3 I, a4 F) isdenotedby a,thatis, ˜ a =(a1, a2T, a3 I, a4 F), andthezeroneutrosophicquadruple BCK/BCI-number (0,0T,0I,0F) isdenotedby 0,thatis, 0 =(0,0T,0I,0F)

Wedeﬁneanorderrelation“ ”andtheequality“=”on NQ(X) asfollows:

x y ⇔ xi ≤ yi for i = 1,2,3,4 x = y ⇔ xi = yi for i = 1,2,3,4 forall x, y ∈ NQ(X).Itiseasytoverifythat“ ”isanequivalencerelationon NQ(X) Theorem1. IfXisaBCK/BCI-algebra,then (NQ(X); , ˜ 0) isaBCK/BCI-algebra. Proof. Let X bea BCI-algebra.Forany x, y, z ∈ NQ(X),wehave ( ˜ x ˜ y) ( ˜ x ˜ z)=(x1 ∗ y1, (x2 ∗ y2)T, (x3 ∗ y3)I, (x4 ∗ y4)F) (x1 ∗ z1, (x2 ∗ z2)T, (x3 ∗ z3)I, (x4 ∗ z4)F) =((x1 ∗ y1) ∗ (x1 ∗ z1), ((x2 ∗ y2) ∗ (x2 ∗ z2))T, ((x3 ∗ y3) ∗ (x3 ∗ z3))I, ((x4 ∗ y4) ∗ (x4 ∗ z4))T) (z1 ∗ y1, (z2 ∗ y2)T, (z3 ∗ y3)I, (z4 ∗ y4)F) = ˜ z ˜ y x (x y)=(x1, x2T, x3 I, x4 F) (x1 ∗ y1, (x2 ∗ y2)T, (x3 ∗ y3)I, (x4 ∗ y4)F) =(x1 ∗ (x1 ∗ y1), (x2 ∗ (x2 ∗ y2))T, (x3 ∗ (x3 ∗ y3))I, (x4 ∗ (x4 ∗ y4))F) (y1, y2T, y3 I, y4 F) = ˜ y x x =(x1, x2T, x3 I, x4 F) (x1, x2T, x3 I, x4 F) =(x1 ∗ x1, (x2 ∗ x2)T, (x3 ∗ x3)I, (x4 ∗ x4)F) =(0,0T,0I,0F)= ˜ 0.

Florentin Smarandache (author and editor) Collected Papers, IX 253

Assumethat x y = 0and y x = 0.Then

(x1 ∗ y1, (x2 ∗ y2)T, (x3 ∗ y3)I, (x4 ∗ y4)F)=(0,0T,0I,0F) and

(y1 ∗ x1, (y2 ∗ x2)T, (y3 ∗ x3)I, (y4 ∗ x4)F)=(0,0T,0I,0F)

Itfollowsthat x1 ∗ y1 = 0 = y1 ∗ x1, x2 ∗ y2 = 0 = y2 ∗ x2, x3 ∗ y3 = 0 = y3 ∗ x3 and x4 ∗ y4 = 0 = y4 ∗ x4.Hence, x1 = y1, x2 = y2, x3 = y3,and x4 = y4,whichimpliesthat x =(x1, x2T, x3 I, x4 F)=(y1, y2T, y3 I, y4 F)= y.

Therefore,weknowthat (NQ(X); , 0) isa BCI-algebra.Wecallitthe neutrosophicquadruple BCI-algebra.Moreover,if X isa BCK-algebra,thenwehave 0 x =(0 ∗ x1, (0 ∗ x2)T, (0 ∗ x3)I, (0 ∗ x4)F)=(0,0T,0I,0F)= 0.

Hence, (NQ(X); , 0) isa BCK-algebra.Wecallitthe neutrosophicquadrupleBCK-algebra Example1. IfX = {0, a},thentheneutrosophicquadruplesetNQ(X) isgivenasfollows: NQ(X)= {0, ˜ 1, 2, 3, ˜ 4, 5, 6, 7, 8, 9, ˜ 10, ˜ 11, ˜ 12, ˜ 13, ˜ 14, ˜ 15} where 0 =(0,0T,0I,0F), 1 =(0,0T,0I, aF), 2 =(0,0T, aI,0F), 3 =(0,0T, aI, aF), 4 =(0, aT,0I,0F), ˜ 5 =(0, aT,0I, aF), ˜ 6 =(0, aT, aI,0F), ˜ 7 =(0, aT, aI, aF), ˜ 8 =(a,0T,0I,0F), ˜ 9 =(a,0T,0I, aF), 10 =(a,0T, aI,0F), 11 =(a,0T, aI, aF), 12 =(a, aT,0I,0F), 13 =(a, aT,0I, aF), 14 =(a, aT, aI,0F),and 15 =(a, aT, aI, aF).

ConsideraBCK-algebraX = {0, a} withthebinaryoperation ∗,whichisgiveninTable 1. Table1. Cayleytableforthebinaryoperation“∗”. ∗ 0 a 000 aa 0

Then (NQ(X), , 0) isaBCK-algebrainwhichtheoperation isgivenbyTable 2. Table2. Cayleytableforthebinaryoperation“ ”.

Florentin Smarandache (author and editor) Collected Papers, IX 254

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 2 2 2 0 0 2 2 0 0 2 2 0 0 2 2 0 0 3 3 2 1 0 3 2 1 0 3 2 1 0 3 2 1 0 4 4 4 4 4 0 0 0 0 4 4 4 4 0 0 0 0 ˜ 5 ˜ 5 4 ˜ 5 4 1 ˜ 0 1 ˜ 0 ˜ 5 4 ˜ 5 4 1 ˜ 0 1 ˜ 0 6 6 6 4 4 2 2 0 0 6 6 4 4 2 2 0 0 7 7 6 5 4 3 2 1 0 7 6 5 4 3 2 1 0 8 8 8 8 8 8 8 8 8 0 0 0 0 0 0 0 0 9 9 8 8 8 9 8 9 8 9 0 1 0 1 0 1 0 10 10 10 8 8 10 10 8 8 2 2 0 2 2 2 0 0

Table2. Cont.

˜ 0 1 ˜ 2 ˜ 3 4 ˜ 5 ˜ 6 ˜ 7 ˜ 8 ˜ 9 10 11 12 13 14 15 11 11 10 9 8 11 10 9 8 3 2 1 0 3 2 1 0 12 12 12 12 12 ˜ 8 ˜ 8 ˜ 8 ˜ 8 4 4 4 4 ˜ 0 ˜ 0 ˜ 0 ˜ 0 ˜ 13 ˜ 13 ˜ 12 ˜ 13 ˜ 12 9 8 9 8 5 ˜ 4 5 ˜ 4 ˜ 1 0 ˜ 1 0 14 14 14 12 12 10 10 8 8 6 6 4 4 2 2 0 0 15 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

Theorem2. Theneutrosophicquadrupleset NQ(X) basedonapositiveimplicative BCK-algebra X isa positiveimplicativeBCK-algebra.

Proof. Let X beapositiveimplicative BCK-algebra.Then X isa BCK-algebra,so (NQ(X); , ˜ 0) isa BCK-algebrabyTheorem 1.Let x, y, z ∈ NQ(X).Then (xi ∗ zi ) ∗ (yi ∗ zi )=(xi ∗ yi ) ∗ zi forall i = 1,2,3,4since xi, yi, zi ∈ X and X isapositiveimplicative BCK-algebra.Hence, (x z) (y ∗ z)=(x y) z;therefore, NQ(X) basedonapositiveimplicative BCK-algebra X isapositive implicative BCK-algebra.

Proposition1. Theneutrosophicquadrupleset NQ(X) basedonapositiveimplicative BCK-algebra X satisﬁes thefollowingassertions.

(∀ ˜ x, ˜ y, ˜ z ∈ NQ(X))( ˜ x ˜ y ˜ z ⇒ ˜ x ˜ z ˜ y ˜ z) (12) (∀x, y ∈ NQ(X))(x y y ⇒ x y). (13)

Proof. Let x, y, z ∈ NQ(X).If x y z,then 0 =(x y) z =(x z) (y z), so x z y z.Assumethat x y y.UsingEquation(12)impliesthat ˜ x ˜ y ˜ y ˜ y = 0, so x y = ˜ 0,i.e., x y.

Let X bea BCK/BCI-algebra.Given a, b ∈ X andsubsets A and B of X,considerthesets NQ(a, B) := {(a, aT, yI, zF) ∈ NQ(X) | y, z ∈ B} NQ(A, b) := {(a, xT, bI, bF) ∈ NQ(X) | a, x ∈ A} NQ(A, B) := {(a, xT, yI, zF) ∈ NQ(X) | a, x ∈ A; y, z ∈ B}

NQ(A∗ , B) := a∈A NQ(a, B) NQ(A, B∗) := b∈B NQ(A, b)

Florentin Smarandache (author and editor) Collected Papers, IX 255

and NQ(A ∪ B) := NQ(A,0) ∪ NQ(0, B)

Theset NQ(A, A) isdenotedby NQ(A).

Proposition2. LetXbeaBCK/BCI-algebra.Givena, b ∈ XandsubsetsAandBofX,wehave (1) NQ(A∗ , B) andNQ(A, B∗) aresubsetsofNQ(A, B) (1) If 0 ∈ A ∩ BthenNQ(A ∪ B) isasubsetofNQ(A, B)

Proof. Straightforward. Let X bea BCK/BCI-algebra.Given a, b ∈ X andsubalgebras A and B of X, NQ(a, B) and NQ(A, b) maynotbesubalgebrasof NQ(X) since (a, aT, x3 I, x4 F) (a, aT, u3 I, v4 F)=(0,0T, (x3 ∗ u3)I, (x4 ∗ v4)F) / ∈ NQ(a, B) and (x1, x2T, bI, bF) (u1, u2T, bI, bF)=(x1 ∗ u1, (x2 ∗ u2)T,0I,0F) / ∈ NQ(A, b) for (a, aT, x3 I, x4 F) ∈ NQ(a, B), (a, aT, u3 I, v4 F) ∈ NQ(a, B), (x1, x2T, bI, bF) ∈ NQ(A, b), and (u1, u2T, bI, bF) ∈ NQ(A, b).

Theorem3. If A and B aresubalgebrasofa BCK/BCI-algebra X,thentheset NQ(A, B) isasubalgebraof NQ(X),whichiscalledaneutrosophicquadruplesubalgebra.

Proof. Assumethat A and B aresubalgebrasofa BCK/BCI-algebra X.Let ˜ x =(x1, x2T, x3 I, x4 F) and ˜ y =(y1, y2T, y3 I, y4 F) beelementsof NQ(A, B).Then x1, x2, y1, y2 ∈ A and x3, x4, y3, y4 ∈ B, whichimpliesthat x1 ∗ y1 ∈ A, x2 ∗ y2 ∈ A, x3 ∗ y3 ∈ B,and x4 ∗ y4 ∈ B.Hence, x y =(x1 ∗ y1, (x2 ∗ y2)T, (x3 ∗ y3)I, (x4 ∗ y4)F) ∈ NQ(A, B), so NQ(A, B) isasubalgebraof NQ(X)

Theorem4. If A and B areidealsofa BCK/BCI-algebra X,thentheset NQ(A, B) isanidealof NQ(X), whichiscalledaneutrosophicquadrupleideal.

Proof. Assumethat A and B areidealsofa BCK/BCI-algebra X.Obviously, 0 ∈ NQ(A, B). Let x =(x1, x2T, x3 I, x4 F) and y =(y1, y2T, y3 I, y4 F) beelementsof NQ(X) suchthat x y ∈ NQ(A, B) and y ∈ NQ(A, B).Then ˜ x ˜ y =(x1 ∗ y1, (x2 ∗ y2)T, (x3 ∗ y3)I, (x4 ∗ y4)F) ∈ NQ(A, B), so

Sinceeveryidealisasubalgebraina BCK-algebra,wehavethefollowingcorollary. Corollary1. IfAandBareidealsofaBCK-algebraX,thenthesetNQ(A, B) isasubalgebraofNQ(X)

ThefollowingexampleshowsthatCorollary 1 isnottrueina BCI-algebra.

∗ y1 ∈ A, x2 ∗ y2 ∈ A

∗ y

∈

B and x4 ∗ y4 ∈ B.Since y ∈ NQ(A, B),wehave y1, y2 ∈ A and y3, y4

B.Since A and B areidealsof X,itfollowsthat x1, x2 ∈ A and x3, x4 ∈ B. Hence, x =(x1, x2T, x3 I, x4 F) ∈ NQ(A, B),so NQ(A, B) isanidealof NQ(X).

Florentin Smarandache (author and editor) Collected Papers, IX 256

Example2. Considera BCI-algebra (Z, ,0).Ifwetake A = N and B = Z,then NQ(A, B) isanidealof NQ(Z).However,itisnotasubalgebraofNQ(Z) since (2,3T, 5I,6F) (3,5T,6I, 7F)=( 1, 2T, 11I,13F) / ∈ NQ(A, B) for (2,3T, 5I,6F), (3,5T,6I, 7F) ∈ NQ(A, B). Theorem5. If A and B areclosedidealsofa BCI-algebra X,thentheset NQ(A, B) isaclosedidealof NQ(X). Proof. If A and B areclosedidealsofa BCI-algebra X,thentheset NQ(A, B) isanidealof NQ(X) by Theorem 4.Let ˜ x =(x1, x2T, x3 I, x4 F) ∈ NQ(A, B).Then ˜ 0 x =(0 ∗ x1, (0 ∗ x2)T, (0 ∗ x3)I, (0 ∗ x4)F) ∈ NQ(A, B) since0 ∗ x1,0 ∗ x2 ∈ A and0 ∗ x3,0 ∗ x4 ∈ B.Therefore, NQ(A, B) isaclosedidealof NQ(X)

Sinceeveryclosedidealofa BCI-algebra X isasubalgebraof X,wehavethefollowingcorollary. Corollary2. If A and B areclosedidealsofa BCI-algebra X,thentheset NQ(A, B) isasubalgebraof NQ(X).

Inthefollowingexample,weknowthatthereexistideals A and B ina BCI-algebra X suchthat NQ(A, B) isnotaclosedidealof NQ(X)

Example3. Consider BCI-algebras (Y, ∗,0) and (Z, ,0).Then X = Y × Z isa BCI-algebra(see[21]). Let A = Y × N and B = {0}× N.Then A and B areidealsof X,so NQ(A, B) isanidealof NQ(X) by Theorem 4.Let ((0,0), (0,1)T, (0,2)I, (0,3)F) ∈ NQ(A, B).Then ((0,0), (0,0)T, (0,0)I, (0,0)F) ((0,0), (0,1)T, (0,2)I, (0,3)F) =((0,0), (0, 1)T, (0, 2)I, (0, 3)F) / ∈ NQ(A, B) Hence,NQ(A, B) isnotaclosedidealofNQ(X).

Weprovideconditionswheretheset NQ(A, B) isaclosedidealof NQ(X)

Theorem6. LetAandBbeidealsofaBCI-algebraXandlet Γ := {a ∈ NQ(X) | (∀x ∈ NQ(X))(x a ⇒ x = a)}.

Assumethat,if Γ ⊆ NQ(A, B),then |Γ| < ∞.ThenNQ(A, B) isaclosedidealofNQ(X)

Proof. If A and B areidealsof X,then NQ(A, B) isanidealof NQ(X) byTheorem 4 Let a =(a1, a2T, a3 I, a4 F) ∈ NQ(A, B).Forany n ∈ N,denote n(a) := ˜ 0 (˜ 0 a)n.Then n(a) ∈ Γ and n(a)=(0 ∗ (0 ∗ a1)n , (0 ∗ (0 ∗ a2)n )T, (0 ∗ (0 ∗ a3)n )I, (0 ∗ (0 ∗ a4)n )F) =(0 ∗ (0 ∗ an 1 ), (0 ∗ (0 ∗ an 2 ))T, (0 ∗ (0 ∗ an 3 ))I, (0 ∗ (0 ∗ an 4 ))F) = 0 (0 an ) Hence, n(a) an =(0 (0 an )) an =( ˜ 0 an ) ( ˜ 0 an ) = 0 ∈ NQ(A, B),

Florentin Smarandache (author and editor) Collected Papers, IX 257

so n(a) ∈ NQ(A, B),since a ∈ NQ(A, B),and NQ(A, B) isanidealof NQ(X).Since |Γ| < ∞, itfollowsthat k ∈ N suchthat n(a)=(n + k)(a),thatis, n(a)= n(a) (0 a)k,andthus k( ˜ a)= 0 (0 ˜ a)k =(n( ˜ a) (0 ˜ a)k ) n( ˜ a) = n(a) n(a)= 0, i.e., (k 1)( ˜ a) (0 ˜ a)= 0.Since 0 ˜ a ∈ Γ,itfollowsthat 0 ˜ a =(k 1)( ˜ a) ∈ NQ(A, B) Therefore, NQ(A, B) isaclosedidealof NQ(X)

Theorem7. GiventwoelementsaandbinaBCI-algebraX,let

Aa := {x ∈ X | a ∗ x = a} and Bb := {x ∈ X | b ∗ x = b}.(14)

ThenNQ(Aa, Bb ) isaclosedidealofNQ(X)

Proof. Since a ∗ 0 = a and b ∗ 0 = b,wehave0 ∈ Aa ∩ Bb.Thus, ˜ 0 ∈ NQ(Aa, Bb ).If x ∈ Aa and y ∈ Bb,then 0 ∗ x =(a ∗ x) ∗ a = a ∗ a = 0and0 ∗ y =(b ∗ y) ∗ b = b ∗ b = 0.(15)

Let x, y, c, d ∈ X besuchthat x, y ∗ x ∈ Aa and c, d ∗ c ∈ Bb.Then (a ∗ y) ∗ a = 0 ∗ y =(0 ∗ y) ∗ 0 =(0 ∗ y) ∗ (0 ∗ x)= 0 ∗ (y ∗ x)= 0 and (b ∗ d) ∗ b = 0 ∗ d =(0 ∗ d) ∗ 0 =(0 ∗ d) ∗ (0 ∗ c)= 0 ∗ (d ∗ c)= 0, thatis, a ∗ y ≤ a and b ∗ d ≤ b.Ontheotherhand, a = a ∗ (y ∗ x)=(a ∗ x) ∗ (y ∗ x) ≤ a ∗ y and b = b ∗ (d ∗ c)=(b ∗ c) ∗ (d ∗ c) ≤ b ∗ d

Thus, a ∗ y = a and b ∗ d = b,i.e., y ∈ Aa and d ∈ Bb.Hence, Aa and Bb areidealsof X,and NQ(Aa, Bb ) isthereforeanidealof NQ(X) byTheorem 4.Let x =(x1, x2T, x3 I, x4 F) ∈ NQ(Aa, Bb ). Then x1, x2 ∈ Aa,and x3, x4 ∈ Bb.ItfollowsfromEquation (15) that0 ∗ x1 = 0 ∈ Aa,0 ∗ x2 = 0 ∈ Aa, 0 ∗ x3 = 0 ∈ Bb,and0 ∗ x4 = 0 ∈ Bb.Hence, 0 x =(0 ∗ x1, (0 ∗ x2)T, (0 ∗ x3)I, (0 ∗ x4)F) ∈ NQ(Aa, Bb )

Therefore, NQ(Aa, Bb ) isaclosedidealof NQ(X). Proposition3. LetAandBbeidealsofaBCK-algebraX.Then

NQ(A) ∩ NQ(B)= {˜ 0}⇔ (∀x ∈ NQ(A))(∀y ∈ NQ(B))(x y = x).(16) Proof. Notethat NQ(A) and NQ(B) areidealsof NQ(X).Assumethat NQ(A) ∩ NQ(B)= {0}.Let x =(x1, x2T, x3 I, x4 F) ∈ NQ(A) and y =(y1, y2T, y3 I, y4 F) ∈ NQ(B).

Florentin Smarandache (author and editor) Collected Papers, IX 258

Since x (x y) x and x (x y) y,itfollowsthat x (x y) ∈ NQ(A) ∩ NQ(B)= {0}. Obviously, (x y) x ∈{0}.Hence, x y = x. Conversely,supposethat x y = x forall x ∈ NQ(A) and y ∈ NQ(B).If z ∈ NQ(A) ∩ NQ(B), then z ∈ NQ(A) and z ∈ NQ(B),whichisimpliedfromthehypothesisthat z = z z = 0. Hence NQ(A) ∩ NQ(B)= {0}

Theorem8. LetAandBbesubsetsofaBCK-algebraXsuchthat (∀a, b ∈ A ∩ B)(K(a, b) ⊆ A ∩ B) (17) whereK(a, b) := {x ∈ X | x ∗ a ≤ b}.ThenthesetNQ(A, B) isanidealofNQ(X)

Proof. If x ∈ A ∩ B,then0 ∈ K(x, x) since0 ∗ x ≤ x.Hence,0 ∈ A ∩ B byEquation (17),soitisclear that 0 ∈ NQ(A, B).Let ˜ x =(x1, x2T, x3 I, x4 F) and ˜ y =(y1, y2T, y3 I, y4 F) beelementsof NQ(X) such that x y ∈ NQ(A, B) and y ∈ NQ(A, B).Then x y =(x1 ∗ y1, (x2 ∗ y2)T, (x3 ∗ y3)I, (x4 ∗ y4)F) ∈ NQ(A, B), so x1 ∗ y1 ∈ A, x2 ∗ y2 ∈ A, x3 ∗ y3 ∈ B,and x4 ∗ y4 ∈ B.Using(II),wehave x1 ∈ K(x1 ∗ y1, y1) ⊆ A, x2 ∈ K(x2 ∗ y2, y2) ⊆ A, x3 ∈ K(x3 ∗ y3, y3) ⊆ B,and x4 ∈ K(x4 ∗ y4, y4) ⊆ B.Thisimpliesthat ˜ x =(x1, x2T, x3 I, x4 F) ∈ NQ(A, B).Therefore, NQ(A, B) isanidealof NQ(X)

Corollary3. LetAandBbesubsetsofaBCK-algebraXsuchthat (∀a, x, y ∈ X)(x, y ∈ A ∩ B, (a ∗ x) ∗ y = 0 ⇒ a ∈ A ∩ B).(18)

ThenthesetNQ(A, B) isanidealofNQ(X)

Theorem9. LetAandBbenonemptysubsetsofaBCK-algebraXsuchthat (∀a, x, y ∈ X)(x, y ∈ A (or B), a ∗ x ≤ y ⇒ a ∈ A (or B)).(19) ThenthesetNQ(A, B) isanidealofNQ(X)

Proof. AssumethattheconditionexpressedbyEquation (19) isvalidfornonemptysubsets A and B of X.Since0 ∗ x ≤ x forany x ∈ A (or B),wehave0 ∈ A (or B) byEquation (19).Hence,itisclear that 0 ∈ NQ(A, B).Let x =(x1, x2T, x3 I, x4F) and y =(y1, y2T, y3 I, y4F) beelementsof NQ(X) such that x y ∈ NQ(A, B) and y ∈ NQ(A, B).Then ˜ x ˜ y =(x1 ∗ y1, (x2 ∗ y2)T, (x3 ∗ y3)I, (x4 ∗ y4)F) ∈ NQ(A, B), so x1 ∗ y1 ∈ A, x2 ∗ y2 ∈ A, x3 ∗ y3 ∈ B,and x4 ∗ y4 ∈ B.Notethat xi ∗ (xi ∗ yi ) ≤ yi for i = 1,2,3,4. ItfollowsfromEquation(19)that x1, x2 ∈ A and x3, x4 ∈ B.Hence, ˜ x =(x1, x2T, x3 I, x4 F) ∈ NQ(A, B); therefore, NQ(A, B) isanidealof NQ(X).

Theorem10. If A and B arepositiveimplicativeidealsofa BCK-algebra X,thentheset NQ(A, B) isapositive implicativeidealofNQ(X),whichiscalledapositiveimplicativeneutrosophicquadrupleideal.

Florentin Smarandache (author and editor) Collected Papers, IX 259

Proof. Assumethat A and B arepositiveimplicativeidealsofa BCK-algebra X.Obviously, 0 ∈ NQ(A, B). Let x =(x1, x2T, x3 I, x4F), y =(y1, y2T, y3 I, y4F),and z =(z1, z2T, z3 I, z4 F) beelementsof NQ(X) suchthat (x y) z ∈ NQ(A, B) and y z ∈ NQ(A, B).Then ( ˜ x ˜ y) ˜ z =((x1 ∗ y1) ∗ z1, ((x2 ∗ y2) ∗ z2)T, ((x3 ∗ y3) ∗ z3)I, ((x4 ∗ y4) ∗ z4)F) ∈ NQ(A, B), and y z =(y1 ∗ z1, (y2 ∗ z2)T, (y3 ∗ z3)I, (y4 ∗ z4)F) ∈ NQ(A, B), so (x1 ∗ y1) ∗ z1 ∈ A, (x2 ∗ y2) ∗ z2 ∈ A, (x3 ∗ y3) ∗ z3 ∈ B, (x4 ∗ y4) ∗ z4 ∈ B, y1 ∗ z1 ∈ A, y2 ∗ z2 ∈ A, y3 ∗ z3 ∈ B,and y4 ∗ z4 ∈ B.Since A and B arepositiveimplicativeidealsof X,itfollowsthat x1 ∗ z1, x2 ∗ z2 ∈ A and x3 ∗ z3, x4 ∗ z4 ∈ B.Hence, x z =(x1 ∗ z1, (x2 ∗ z2)T, (x3 ∗ z3)I, (x4 ∗ z4)F) ∈ NQ(A, B), so NQ(A, B) isapositiveimplicativeidealof NQ(X)

Theorem11. LetAandBbeidealsofaBCK-algebraXsuchthat (∀x, y, z ∈ X)((x ∗ y) ∗ z ∈ A (or B) ⇒ (x ∗ z) ∗ (y ∗ z) ∈ A (or B)).(20) ThenNQ(A, B) isapositiveimplicativeidealofNQ(X)

Proof. Since A and B areidealsof X,itfollowsfromTheorem 4 that NQ(A, B) isanidealof NQ(X) Let x =(x1, x2T, x3 I, x4 F), y =(y1, y2T, y3 I, y4 F),and z =(z1, z2T, z3 I, z4 F) beelementsof NQ(X) suchthat (x y) z ∈ NQ(A, B) and y z ∈ NQ(A, B).Then (x y) z =((x1 ∗ y1) ∗ z1, ((x2 ∗ y2) ∗ z2)T, ((x3 ∗ y3) ∗ z3)I, ((x4 ∗ y4) ∗ z4)F) ∈ NQ(A, B), and ˜ y ˜ z =(y1 ∗ z1, (y2 ∗ z2)T, (y3 ∗ z3)I, (y4 ∗ z4)F) ∈ NQ(A, B), so (x1 ∗ y1) ∗ z1 ∈ A, (x2 ∗ y2) ∗ z2 ∈ A, (x3 ∗ y3) ∗ z3 ∈ B, (x4 ∗ y4) ∗ z4 ∈ B, y1 ∗ z1 ∈ A, y2 ∗ z2 ∈ A, y3 ∗ z3 ∈ B,and y4 ∗ z4 ∈ B.ItfollowsfromEquation (20) that (x1 ∗ z1) ∗ (y1 ∗ z1) ∈ A, (x2 ∗ z2) ∗ (y2 ∗ z2) ∈ A, (x3 ∗ z3) ∗ (y3 ∗ z3) ∈ B,and (x4 ∗ z4) ∗ (y4 ∗ z4) ∈ B.Since A and B areidealsof X,weget x1 ∗ z1 ∈ A, x2 ∗ z2 ∈ A, x3 ∗ z3 ∈ B,and x4 ∗ z4 ∈ B.Hence, ˜ x ˜ z =(x1 ∗ z1, (x2 ∗ z2)T, (x3 ∗ z3)I, (x4 ∗ z4)F) ∈ NQ(A, B) Therefore, NQ(A, B) isapositiveimplicativeidealof NQ(X). Corollary4. LetAandBbeidealsofaBCK-algebraXsuchthat (∀x, y ∈ X)((x ∗ y) ∗ y ∈ A (or B) ⇒ x ∗ y ∈ A (or B)).(21) ThenNQ(A, B) isapositiveimplicativeidealofNQ(X).

Proof. IftheconditionexpressedinEquation (21) isvalid,thentheconditionexpressedinEquation (20) istrue.Hence, NQ(A, B) isapositiveimplicativeidealof NQ(X) byTheorem 11

Florentin Smarandache (author and editor) Collected Papers, IX 260

Theorem 12. Let A and B be subsets of a BCK-algebra X such that 0 ∈ A ∩ B and ((x ∗ y) ∗ y) ∗ z ∈ A (or B), z ∈ A (or B) ⇒ x ∗ y ∈ A (or B) for all x, y, z ∈ X. Then NQ(A, B) is a positive implicative ideal of NQ(X).

(22)

Proof. Since 0 ∈ A ∩ B, it is clear that 0 ˜ ∈ NQ(A, B). We ﬁrst show that (∀x, y ∈ X)(x ∗ y ∈ A (or B), y ∈ A (or B) ⇒ x ∈ A (or B)). (23) Let x, y ∈ X besuchthat x ∗ y ∈ A (or B)and y ∈ A (or B).Then ((x ∗ 0) ∗ 0) ∗ y = x ∗ y ∈ A (or B)

byEquation (1),which,basedonEquations (1) and (22),impliesthat x = x ∗ 0 ∈ A (or B). Let x =(x1, x2T, x3 I, x4 F), y =(y1, y2T, y3 I, y4F),and z =(z1, z2T, z3 I, z4F) beelementsof NQ(X) suchthat (x y) z ∈ NQ(A, B) and y z ∈ NQ(A, B).Then

(x y) z =((x1 ∗ y1) ∗ z1, ((x2 ∗ y2) ∗ z2)T, ((x3 ∗ y3) ∗ z3)I, ((x4 ∗ y4) ∗ z4)F) ∈ NQ(A, B), and y z =(y1 ∗ z1, (y2 ∗ z2)T, (y3 ∗ z3)I, (y4 ∗ z4)F) ∈ NQ(A, B), so (x1 ∗ y1) ∗ z1 ∈ A, (x2 ∗ y2) ∗ z2 ∈ A, (x3 ∗ y3) ∗ z3 ∈ B, (x4 ∗ y4) ∗ z4 ∈ B, y1 ∗ z1 ∈ A, y2 ∗ z2 ∈ A, y3 ∗ z3 ∈ B,and y4 ∗ z4 ∈ B.Notethat (((xi ∗ zi ) ∗ zi ) ∗ (yi ∗ zi )) ∗ ((xi ∗ yi ) ∗ zi )= 0 ∈ A (or B) for i = 1,2,3,4.Since (xi ∗ yi ) ∗ zi ∈ A for i = 1,2and (xj ∗ yj ) ∗ zj ∈ B for j = 3,4,itfollowsfrom Equation (23) that ((xi ∗ zi ) ∗ zi ) ∗ (yi ∗ zi ) ∈ A for i = 1,2,and ((xj ∗ zj ) ∗ zj ) ∗ (yj ∗ zj ) ∈ B for j = 3,4. Moreover,since yi ∗ zi ∈ A for i = 1,2,and yj ∗ zj ∈ B for j = 3,4,wehave x1 ∗ z1 ∈ A, x2 ∗ z2 ∈ A, x3 ∗ z3 ∈ B,and x4 ∗ z4 ∈ B byEquation(22).Hence, ˜ x ˜ z =(x1 ∗ z1, (x2 ∗ z2)T, (x3 ∗ z3)I, (x4 ∗ z4)F) ∈ NQ(A, B) Therefore, NQ(A, B) isapositiveimplicativeidealof NQ(X).

Theorem13. Let A and B besubsetsofa BCK-algebra X suchthat NQ(A, B) isapositiveimplicativeidealof NQ(X).Thentheset

Ωa := {x ∈ NQ(X) | x a ∈ NQ(A, B)} (24) isanidealofNQ(X) forany a ∈ NQ(X).

Proof. Obviously, 0 ∈ Ωa.Let ˜ x, ˜ y ∈ NQ(X) besuchthat ˜ x ˜ y ∈ Ωa and ˜ y ∈ Ωa.Then ( ˜ x ˜ y) ˜ a ∈ NQ(A, B) and ˜ y ˜ a ∈ NQ(A, B).Since NQ(A, B) isapositiveimplicativeidealof NQ(X),itfollowsfromEquation (11) that ˜ x ˜ a ∈ NQ(A, B) andthereforethat ˜ x ∈ Ωa.Hence, Ωa is anidealof NQ(X)

CombiningTheorems 12 and 13,wehavethefollowingcorollary.

Florentin Smarandache (author and editor) Collected Papers, IX 261

Corollary5. If A and B aresubsetsofa BCK-algebra X satisfying 0 ∈ A ∩ B andtheconditionexpressedin Equation (22),thentheset Ωa inEquation (24) isanidealofNQ(X) forall a ∈ NQ(X).

Theorem14. Foranysubsets A and B ofa BCK-algebra X,iftheset Ωa inEquation (24) isanidealof NQ(X) forall ˜ a ∈ NQ(X),thenNQ(A, B) isapositiveimplicativeidealofNQ(X)

Proof. Since 0 ∈ Ωa,wehave 0 = 0 a ∈ NQ(A, B).Let x, y, z ∈ NQ(X) besuchthat (x y) z ∈ NQ(A, B) and y z ∈ NQ(A, B).Then x y ∈ Ωz and y ∈ Ωz.Since Ωz isan idealof NQ(X),itfollowsthat x ∈ Ωz.Hence, x z ∈ NQ(A, B).Therefore, NQ(A, B) isapositive implicativeidealof NQ(X)

Theorem15. Foranyideals A and B ofa BCK-algebra X andforany a ∈ NQ(X),iftheset Ωa in Equation (24) isanidealofNQ(X),thenNQ(X) isapositiveimplicativeBCK-algebra.

Proof. Let Ω beanyidealof NQ(X).Forany x, y, z ∈ NQ(X),assumethat (x y) z ∈ Ω and y z ∈ Ω.Then x y ∈ Ωz and y ∈ Ωz.Since Ωz isanidealof NQ(X),itfollowsthat x ∈ Ωz Hence, ˜ x ˜ z ∈ Ω,whichshowsthat Ω isapositiveimplicativeidealof NQ(X).Therefore, NQ(X) is apositiveimplicative BCK-algebra.

Ingeneral,theset {˜ 0} isanidealofanyneutrosophicquadruple BCK-algebra NQ(X),butitis notapositiveimplicativeidealof NQ(X) asseeninthefollowingexample.

Example4. ConsideraBCK-algebraX = {0,1,2} withthebinaryoperation ∗,whichisgiveninTable 3

Table3. Cayleytableforthebinaryoperation“∗”. ∗ 012 0000 1100 2210

Thentheneutrosophicquadruple BCK-algebra NQ(X) has81elements.Ifwetake ˜ a =(2,2T,2I,2F) and ˜ b =(1,1T,1I,1F) inNQ(X),then (a b) b =((2 ∗ 1) ∗ 1, ((2 ∗ 1) ∗ 1)T, ((2 ∗ 1) ∗ 1)I, ((2 ∗ 1) ∗ 1)F) =(1 ∗ 1, (1 ∗ 1)T, (1 ∗ 1)I, (1 ∗ 1)F)=(0,0T,0I,0F)= 0, and b b = ˜ 0.However, a ˜ b =(2 ∗ 1, (2 ∗ 1)T, (2 ∗ 1)I, (2 ∗ 1)F)=(1,1T,1I,1F) = 0. Hence, {˜ 0} isnotapositiveimplicativeidealofNQ(X)

Wenowprovideconditionsfortheset {0} tobeapositiveimplicativeidealintheneutrosophic quadruple BCK-algebra. Theorem16. LetNQ(X) beaneutrosophicquadrupleBCK-algebra.Iftheset Ω(a) := {x ∈ NQ(X) | x a} (25) isanidealofNQ(X) forall a ∈ NQ(X),then {0} isapositiveimplicativeidealofNQ(X).

Florentin Smarandache (author and editor) Collected Papers, IX 262

Proof. Weﬁrstshowthat

(∀ ˜ x, ˜ y ∈ NQ(X))(( ˜ x ˜ y) ˜ y = 0 ⇒ ˜ x ˜ y = 0) (26)

Assumethat (x y) y = ˜ 0 forall x, y ∈ NQ(X).Then x y y,so x y ∈ Ω(y).Since y ∈ Ω(y) and Ω(y) isanidealof NQ(X),wehave x ∈ Ω(y).Thus, x y,thatis, x y = ˜ 0. Let u :=(x y) y.Then (( ˜ x ˜ u) ˜ y) ˜ y =(( ˜ x ˜ y) ˜ y) ˜ u = 0, whichimplies,basedonEquations(3)and(26),that (x y) ((x y) y)=(x y) u =(x u) y = 0, thatis, x y (x y) y.Since (x y) y x y,itfollowsthat (x y) y = x y.(27)

Ifweput ˜ y = ˜ x ( ˜ y ( ˜ y ˜ x)) inEquation(27),then x (x (y (y x)))=(x (x (y (y x)))) (x (y (y x))) ( ˜ y ( ˜ y ˜ x)) ( ˜ x ( ˜ y ( ˜ y ˜ x))) (y (y x)) (x y) =(y (x y)) (y x) =((y (x y)) (y x)) (y x) (x (x y)) (y x).

Ontheotherhand, ((x (x y)) (y x)) (x (x (y (y x)))) =((x (x (x (y (y x))))) (x y)) (y x)) =((x (y (y x))) (x y)) (y x)) ( ˜ y ( ˜ y ( ˜ y ˜ x))) ( ˜ y ˜ x))= 0, so ((x (x y)) (y x)) (x (x (y (y x))))= ˜ 0,thatis, ((x (x y)) (y x)) x (x (y (y x)))

Hence, x (x (y (y x)))=((x (x y)) (y x)).(28)

Ifweuse ˜ y ˜ x insteadof ˜ x inEquation(28),then y x =(y x) 0 =(y x) ((y x) (y (y (y x)))) =((y x) ((y x) y)) (y (y x)) =(y x) (y (y x)),

Florentin Smarandache (author and editor) Collected Papers, IX 263

which,bytaking x = y x,impliesthat

˜ y ( ˜ y ˜ x)=( ˜ y ( ˜ y ˜ x)) ( ˜ y ( ˜ y ( ˜ y ˜ x))) =(y (y x)) (y x).

Itfollowsthat (y (y x)) (x y)=((y (y x)) (y x)) (x y) ( ˜ x ( ˜ y ˜ x)) ( ˜ x ˜ y) =(x (x y)) (y x), so, y x =(y (y (y x))) ˜ 0 =( ˜ y ( ˜ y ( ˜ y ˜ x))) (( ˜ y ˜ x) ˜ y) ((y x) ((y x) y)) (y (y x)) =(y x) (y (y x)) (y x) x

Since (y x) x y x,itfollowsthat (y x) x = y x. (29)

BasedonEquation(29),itfollowsthat

((x z) ∗ (y z)) ((x y) z) =((( ˜ x ˜ z) ˜ z) ( ˜ y ˜ z)) (( ˜ x ˜ y) ˜ z) ((x z) y) ((x y) z) = ˜ 0, thatis, (x z) ∗ (y z) (x y) z.Notethat (( ˜ x ˜ y) ˜ z) ((x ˜ z) ( ˜ y ˜ z)) =((x y) z) ((x (y z)) z) (x y) (x (y z)) ( ˜ y ˜ z) ˜ y = 0, whichshowsthat (x y) z (x z) (y z).Hence, (x y) z =(x z) (y z) Therefore, NQ(X) isapositiveimplicative,so {˜ 0} isapositiveimplicativeidealof NQ(X).

4.Conclusions

Wehaveconsideredaneutrosophicquadruple BCK/BCI-numberonasetandestablished neutrosophicquadruple BCK/BCI-algebras,whichconsistofneutrosophicquadruple BCK/BCI-numbers. Wehaveinvestigatedseveralpropertiesandconsideredidealtheoryinaneutrosophicquadruple BCK-algebraandaclosedidealinaneutrosophicquadruple BCI-algebra.Usingsubsets A and B ofaneutrosophicquadruple BCK/BCI-algebra,wehaveconsideredsets NQ(A, B),whichconsistof neutrosophicquadruple BCK/BCI-numberswithacondition.Wehaveprovidedconditionsforthe set NQ(A, B) tobea(positiveimplicative)idealofaneutrosophicquadruple BCK-algebra,andtheset NQ(A, B) tobea(closed)idealofaneutrosophicquadruple BCI-algebra.Wehaveprovidedanexample

Florentin Smarandache (author and editor) Collected Papers, IX 264

to show that the set {0} is not a positive implicative ideal in a neutrosophic quadruple BCK-algebra, and we have considered conditions for the set {0} to be a positive implicative ideal in a neutrosophic quadruple BCK-algebra.

References

1. Smarandache,F.Neutrosophy,NeutrosophicProbability,Set,andLogic,ProQuestInformation&Learning, AnnArbor,Michigan,USA,p.105,1998.Availableonline: http://fs.gallup.unm.edu/eBook-neutrosophics6.pdf (accessedon1September2007).

2. Smarandache,F. AUnifyingFieldinLogics:NeutrosophicLogic.Neutrosophy,NeutrosophicSet,Neutrosophic Probability;AmericanReserchPress:Rehoboth,NM,USA,1999.

3. Smarandache,F.Neutrosophicset—Ageneralizationoftheintuitionisticfuzzyset. Int.J.PureAppl.Math. 2005, 24,287–297.

4. Garg,H.Linguisticsingle-valuedneutrosophicprioritizedaggregationoperatorsandtheirapplicationsto multiple-attributegroupdecision-making. J.AmbientIntell.Humaniz.Comput. 2018,inpress.[CrossRef]

5. Garg,H.Non-linearprogrammingmethodformulti-criteriadecisionmakingproblemsunderinterval neutrosophicsetenvironment. Appl.Intell. 2017,inpress.[CrossRef]

6. Garg,H.SomeNewBiparametricDistanceMeasuresonSingle-ValuedNeutrosophicSetswithApplications toPatternRecognitionandMedicalDiagnosis. Information 2017, 8,162.[CrossRef]

7. Garg,H.Novelsingle-valuedneutrosophicaggregatedoperatorsunderFranknormoperationandits applicationtodecision-makingprocess. Int.J.Uncertain.Quantif. 2016, 6,361–375.

8. Garg,H.;Garg,N.Onsingle-valuedneutrosophicentropyoforder α NeutrosophicSetsSyst. 2016, 14,21–28.

9. Saeid,A.B.;Jun,Y.B.Neutrosophicsubalgebrasof BCK/BCI-algebrasbasedonneutrosophicpoints. Ann.FuzzyMath.Inform. 2017, 14,87–97.

10. Jun,Y.B.Neutrosophicsubalgebrasofseveraltypesin BCK/BCI-algebras. Ann.FuzzyMath.Inform. 2017, 14, 75–86.

11. Jun,Y.B.;Kim,S.J.;Smarandache,F.Intervalneutrosophicsetswithapplicationsin BCK/BCI-algebra. Axioms 2018, 7,23.[CrossRef]

12. Jun,Y.B.;Smarandache,F.;Bordbar,H.Neutrosophic N -structuresappliedto BCK/BCI-algebras. Information 2017, 8,128.[CrossRef]

13. Jun,Y.B.;Smarandache,F.;Song,S.Z.;Khan,M.Neutrosophicpositiveimplicative N -idealsin BCK/BCI-algebras. Axioms 2018, 7,3. [CrossRef]

14. Khan,M.;Anis,S.;Smarandache,F.;Jun,Y.B.Neutrosophic N -structuresandtheirapplicationsin semigroups. Ann.FuzzyMath.Inform. 2017, 14,583–598.

15. Öztürk,M.A.;Jun,Y.B.Neutrosophicidealsin BCK/BCI-algebrasbasedonneutrosophicpoints. J.Inter. Math.VirtualInst. 2018, 8,1–17.

16. Song,S.Z.;Smarandache,F.;Jun,Y.B.Neutrosophiccommutative N -idealsin BCK-algebras. Information 2017, 8,130.[CrossRef]

17. Agboola,A.A.A.;Davvaz,B.;Smarandache,F.Neutrosophicquadruplealgebraichyperstructures. Ann.FuzzyMath.Inform. 2017, 14,29–42.

18. Akinleye,S.A.;Smarandache,F.;Agboola,A.A.A.Onneutrosophicquadruplealgebraicstructures. NeutrosophicSetsSyst. 2016, 12,122–126.

19.Iséki,K.On BCI-algebras. Math.Semin.Notes 1980, 8,125–130.

20.Iséki,K.;Tanaka,S.Anintroductiontothetheoryof BCK-algebras. Math.Jpn. 1978, 23,1–26.

21.Huang,Y. BCI-Algebra;SciencePress:Beijing,China,2006.

22.Meng,J.;Jun,Y.B. BCK-Algebras;KyungmoonsaCo.:Seoul,Korea,1994.

Florentin Smarandache (author and editor) Collected Papers, IX 265

Interval Neutrosophic Sets with Applications in BCK/BCI-Algebra

Young Bae Jun, Seon Jeong Kim, Florentin SmarandacheD

Young Bae Jun, Seon Jeong Kim, Florentin Smarandache (2018). Interval Neutrosophic Sets with Applications in BCK/BCI-Algebra. Axioms 7, 23. DOI: 10.3390/axioms7020023

Abstract: For i, j, k, l, m, n ∈ {1, 2, 3, 4}, the notion of (T(i, j), I(k, l), F(m, n))-interval neutrosophic subalgebra in BCK/BCI-algebra is introduced, and their properties and relations are investigated. The notion of interval neutrosophic length of an interval neutrosophic set is also introduced, and related properties are investigated.

Keywords: interval neutrosophic set; interval neutrosophic subalgebra; interval neutrosophic length

1.Introduction

Intuitionisticfuzzyset,whichisintroducedbyAtanassov[1],isageneralizationofZadeh’s fuzzysets[2],andconsiderbothtruth-membershipandfalsity-membership.Sincethesumofdegree true,indeterminacyandfalseisoneinintuitionisticfuzzysets,incompleteinformationishandled inintuitionisticfuzzysets.Ontheotherhand,neutrosophicsetscanhandletheindeterminate informationandinconsistentinformationthatexistcommonlyinbeliefsystemsinaneutrosophic setsinceindeterminacyisquantiﬁedexplicitlyandtruth-membership,indeterminacy-membership andfalsity-membershipareindependent,whichismentionedin[3].Asaformalframeworkthat generalizestheconceptoftheclassicset,fuzzyset,intervalvaluedfuzzyset,intuitionisticfuzzyset, intervalvaluedintuitionisticfuzzysetandparaconsistentset,etc.,theneutrosophicsetisdevelopedby Smarandache[4,5],whichisappliedtovariousparts,includingalgebra,topology,controltheory, decision-makingproblems,medicinesandinmanyreal-lifeproblems.Theconceptofinterval neutrosophicsetsispresentedbyWangetal.[6],anditismorepreciseandmoreﬂexiblethan thesingle-valuedneutrosophicset.Theintervalneutrosophicsetcanrepresentuncertain,imprecise, incompleteandinconsistentinformation,whichexistsintherealworld. BCK-algebraisintroducedby ImaiandIséki[7],andithasbeenappliedtoseveralbranchesofmathematics,suchasgrouptheory, functionalanalysis,probabilitytheoryandtopology,etc.Asageneralizationof BCK-algebra,Iséki introducedthenotionof BCI-algebra(see[8]).

Inthisarticle,wediscussintervalneutrosophicsetsin BCK/BCI-algebra.Weintroducethenotion of (T(i, j), I(k, l), F(m, n))-intervalneutrosophicsubalgebrain BCK/BCI-algebrafor i, j, k, l, m, n ∈ {1,2,3,4},andinvestigatetheirpropertiesandrelations.Wealsointroducethenotionofinterval neutrosophiclengthofanintervalneutrosophicset,andinvestigaterelatedproperties.

Smarandache

and editor) Collected Papers, IX 266

Florentin

(author

2.Preliminaries

Bya BCI-algebra,wemeanasystem X :=(X, ∗,0) ∈ K(τ) inwhichthefollowingaxiomshold: (I) ((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y)= 0, (II) (x ∗ (x ∗ y)) ∗ y = 0, (III) x ∗ x = 0, (IV) x ∗ y = y ∗ x = 0 ⇒ x = y forall x, y, z ∈ X.Ifa BCI-algebra X satisﬁes0 ∗ x = 0forall x ∈ X,thenwesaythat X is BCK-algebra. Anon-emptysubset S ofa BCK/BCI-algebra X iscalleda subalgebra of X if x ∗ y ∈ S forall x, y ∈ S. Thecollectionofall BCK-algebraandall BCI-algebraaredenotedby BK (X) and BI (X), respectively.Inaddition, B(X) := BK (X) ∪BI (X). Wereferthereadertothebooks[9,10]forfurtherinformationregarding BCK/BCI-algebra.

Bya fuzzystructure overanonemptyset X,wemeananorderedpair (X, ρ) of X andafuzzyset ρ on X

Deﬁnition1 ([11]). Forany (X, ∗,0) ∈B(X),afuzzystructure (X, µ) over (X, ∗,0) iscalleda

• fuzzysubalgebraof (X, ∗,0) withtype 1 (brieﬂy, 1-fuzzysubalgebraof (X, ∗,0))if (∀x, y ∈ X) (µ(x ∗ y) ≥ min{µ(x), µ(y)}) , (1)

• fuzzysubalgebraof (X, ∗,0) withtype 2 (brieﬂy, 2-fuzzysubalgebraof (X, ∗,0))if

(∀x, y ∈ X) (µ(x ∗ y) ≤ min{µ(x), µ(y)}) , (2)

• fuzzysubalgebraof (X, ∗,0) withtype 3 (brieﬂy, 3-fuzzysubalgebraof (X, ∗,0))if

(∀x, y ∈ X) (µ(x ∗ y) ≥ max{µ(x), µ(y)}) , (3)

• fuzzysubalgebraof (X, ∗,0) withtype 4 (brieﬂy, 4-fuzzysubalgebraof (X, ∗,0))if

(∀x, y ∈ X) (µ(x ∗ y) ≤ max{µ(x), µ(y)}) (4)

Let X beanon-emptyset.Aneutrosophicset(NS)in X (see[4])isastructureoftheform: A := { x; AT (x), AI (x), AF (x) | x ∈ X}, where AT : X → [0,1] isatruth-membershipfunction, AI : X → [0,1] isanindeterminatemembership function,and AF : X → [0,1] isafalsemembershipfunction.

Anintervalneutrosophicset(INS) A in X ischaracterizedbytruth-membershipfunction TA, indeterminacymembershipfunction IA andfalsity-membershipfunction FA.Foreachpoint x in X, TA (x), IA (x), FA (x) ∈ [0,1] (see[3,6]).

3.IntervalNeutrosophicSubalgebra

Inwhatfollows,let (X, ∗,0) ∈B(X) and P ∗([0,1]) bethefamilyofallsubintervalsof [0,1] unless otherwisespeciﬁed. Deﬁnition2 ([3,6]). AnintervalneutrosophicsetinanonemptysetXisastructureoftheform: I := { x, I [T](x), I [I](x), I [F](x) | x ∈ X},

Florentin Smarandache (author and editor) Collected Papers, IX 267

where

I [T] : X →P ∗([0,1]), whichiscalledintervaltruth-membershipfunction,

I [I] : X →P ∗([0,1]), whichiscalledintervalindeterminacy-membershipfunction,and

I [F] : X →P ∗([0,1]), whichiscalledintervalfalsity-membershipfunction.

Forthesakeofsimplicity,wewillusethenotation I :=(I [T], I [I], I [F]) fortheinterval neutrosophicset

I := { x, I [T](x), I [I](x), I [F](x) | x ∈ X}.

Givenanintervalneutrosophicset I :=(I [T], I [I], I [F]) in X,weconsiderthefollowingfunctions:

I [T]inf : X → [0,1], x → inf{I [T](x)},

I [I]inf : X → [0,1], x → inf{I [I](x)},

I [F]inf : X → [0,1], x → inf{I [F](x)}, and

I [T]sup : X → [0,1], x → sup{I [T](x)},

I [I]sup : X → [0,1], x → sup{I [I](x)}, I [F]sup : X → [0,1], x → sup{I [F](x)}.

Deﬁnition3. Forany i, j, k, l, m, n ∈{1,2,3,4},anintervalneutrosophicset I :=(I [T], I [I], I [F]) in X iscalleda (T(i, j), I(k, l), F(m, n))-intervalneutrosophicsubalgebraofXifthefollowingassertionsarevalid.

(1) (X, I [T]inf) isani-fuzzysubalgebraof (X, ∗,0) and (X, I [T]sup) isaj-fuzzysubalgebraof (X, ∗,0), (2) (X, I [I]inf) isak-fuzzysubalgebraof (X, ∗,0) and (X, I [I]sup) isanl-fuzzysubalgebraof (X, ∗,0), (3) (X, I [F]inf) isan m-fuzzysubalgebraof (X, ∗,0) and (X, I [F]sup) isan n-fuzzysubalgebraof (X, ∗,0).

Example1. Considera BCK-algebra X = {0,1,2,3} withthebinaryoperation ∗,whichisgiveninTable 1 (see[10]).

Table1. Cayleytableforthebinaryoperation“∗”. ∗ 0123 00000 11001 22102 33330

(1)Let I :=(I [T], I [I], I [F]) beanintervalneutrosophicsetin (X, ∗,0) forwhich I [T], I [I] and I [F] aregivenasfollows:

I [T] : X →P ∗([0,1]) x → 

 [0.4,0.5) if x = 0, (0.3,0.5] if x = 1, [0.2,0.6) if x = 2, [0.1,0.7] if x = 3,

  

Florentin Smarandache (author and editor) Collected Papers, IX 268



I [I] : X →P ∗([0,1]) x →

[0.5,0.8) if x = 0, (0.2,0.7) if x = 1, [0.5,0.6] if x = 2, [0.4,0.8) if x = 3, and

I [F] : X →P ∗([0,1]) x →

[0.4,0.5) if x = 0, (0.2,0.9) if x = 1, [0.1,0.6] if x = 2, (0.4,0.7] if x = 3.

Itisroutinetoverifythat I :=(I [T], I [I], I [F]) isa (T(1,4), I(1,4), F(1,4))-intervalneutrosophic subalgebraof (X, ∗,0).

(2)Let I :=(I [T], I [I], I [F]) beanintervalneutrosophicsetin (X, ∗,0) forwhich I [T], I [I] and I [F] aregivenasfollows:

I [T] : X →P ∗([0,1]) x →

[0.1,0.4) if x = 0, (0.3,0.5) if x = 1, [0.2,0.7] if x = 2, [0.4,0.6) if x = 3,

(0.2,0.5) if x = 0, [0.5,0.8] if x = 1, (0.4,0.5] if x = 2, [0.2,0.6] if x = 3, and

I [I] : X →P ∗([0,1]) x →

I [F] : X →P ∗([0,1]) x →

[0.3,0.4) if x = 0, (0.4,0.7) if x = 1, (0.6,0.8) if x = 2, [0.4,0.6] if x = 3.

Byroutinecalculations,weknowthat I :=(I [T], I [I], I [F]) isa (T(4,4), I(4,4), F(4,4))-interval neutrosophicsubalgebraof (X, ∗,0).

Example2. Considera BCI-algebra X = {0, a, b, c} withthebinaryoperation ∗,whichisgiveninTable 2 (see[10]).

Table2. Cayleytableforthebinaryoperation“∗”. ∗ 0 abc 00 abc aa 0 cb bbc 0 a ccba 0

Let I :=(I [T], I [I], I [F]) beanintervalneutrosophicsetin (X, ∗,0) forwhich I [T], I [I] and I [F] are givenasfollows:

I [T] : X →P ∗([0,1]) x → 

[0.3,0.9) if x = 0, (0.7,0.9) if x = a, [0.7,0.8) if x = b, (0.5,0.8] if x = c,

  







  







  







  







  







       

Florentin Smarandache (author and editor) Collected Papers, IX 269

I [I] : X →P ∗([0,1]) x →

[0.2,0.65) if x = 0, [0.5,0.55] if x = a, (0.6,0.65) if x = b, [0.5,0.55) if x = c, and I [F] : X →P ∗([0,1]) x →

(0.3,0.6) if x = 0, [0.4,0.6] if x = a, (0.4,0.5] if x = b, [0.3,0.5) if x = c.

Routinecalculationsshowthat I :=(I [T], I [I], I [F]) isa (T(4,1), I(4,1), F(4,1))-interval neutrosophicsubalgebraof (X, ∗,0).However,itisnota (T(2,1), I(2,1), F(2,1))-intervalneutrosophic subalgebraof (X, ∗,0) since

I [T]inf(c ∗ a)= I [T]inf(b)= 0.7 > 0.5 = min{I [T]inf(c), I [T]inf(a)} and/or

I [I]inf(a ∗ c)= I [I]inf(b)= 0.6 > 0.5 = min{I [I]inf(a), I [I]inf(c)}.

Inaddition,itisnota (T(4,3), I(4,3), F(4,3))-intervalneutrosophicsubalgebraof (X, ∗,0) since

I [T]sup(a ∗ b)= I [T]sup(c)= 0.8 < 0.9 = max{I [T]inf(a), I [T]inf(c)} and/or

I [F]sup(a ∗ b)= I [F]sup(c)= 0.5 < 0.6 = max{I [F]inf(a), I [F]inf(c)}.

Let I :=(I [T], I [I], I [F]) beanintervalneutrosophicsetin X.Weconsiderthefollowingsets:

U(I [T]inf; αI ) := {x ∈ X |I [T]inf(x) ≥ αI },

L(I [T]sup; αS ) := {x ∈ X |I [T]sup(x) ≤ αS },

U(I [I]inf; β I ) := {x ∈ X |I [I]inf(x) ≥ β I }, L(I [I]sup; βS ) := {x ∈ X |I [I]sup(x) ≤ βS }, and

U(I [F]inf; γI ) := {x ∈ X |I [F]inf(x) ≥ γI }, L(I [F]sup; γS ) := {x ∈ X |I [F]sup(x) ≤ γS }, where αI , αS, β I , βS, γI and γS arenumbersin [0,1]

Theorem1. Ifanintervalneutrosophicset I :=(I [T], I [I], I [F]) in X isa (T(i,4), I(i,4), F(i,4))-interval neutrosophicsubalgebraof (X, ∗,0) for i ∈{1,3},then U(I [T]inf; αI ), L(I [T]sup; αS ), U(I [I]inf; β I ), L(I [I]sup; βS ), U(I [F]inf; γI ) and L(I [F]sup; γS ) areeitheremptyorsubalgebraof (X, ∗,0) forall αI , αS, β I , βS, γI , γS ∈ [0,1]

Proof. Assumethat I :=(I [T], I [I], I [F]) isa (T(1,4), I(1,4), F(1,4))-intervalneutrosophic subalgebraof (X, ∗,0).Then, (X, I [T]inf), (X, I [I]inf) and (X, I [F]inf) are1-fuzzysubalgebraof X; and (X, I [T]sup), (X, I [I]sup) and (X, I [F]sup) are4-fuzzysubalgebraof X.Let αI , αS ∈ [0,1] besuch that U(I [T]inf; αI ) and L(I [T]sup; αS ) arenonempty.Forany x, y ∈ X,if x, y ∈ U(I [T]inf; αI ),then I [T]inf(x) ≥ αI and I [T]inf(y) ≥ αI ,andso

I [T]inf(x ∗ y) ≥ min{I [T]inf(x), I [T]inf(y)}≥ αI ,

  







  







Florentin Smarandache (author and editor) Collected Papers, IX 270

thatis, x ∗ y ∈ U(I [T]inf; αI ).If x, y ∈ L(I [T]sup; αS ),then I [T]sup(x) ≤ αS and I [T]sup(y) ≤ αS, whichimplythat

I [T]sup(x ∗ y) ≤ max{I [T]sup(x), I [T]sup(y)}≤ αS,

thatis, x ∗ y ∈ L(I [T]sup; αS ).Hence, U(I [T]inf; αI ) and L(I [T]sup; αS ) aresubalgebraof (X, ∗,0) forall αI , αS ∈ [0,1].Similarly,wecanprovethat U(I [I]inf; β I ), L(I [I]sup; βS ), U(I [F]inf; γI ) and L(I [F]sup; γS ) areeitheremptyorsubalgebraof (X, ∗,0) forall β I , βS, γI , γS ∈ [0,1].Suppose that I :=(I [T], I [I], I [F]) isa (T(3,4), I(3,4), F(3,4))-intervalneutrosophicsubalgebraof (X, ∗,0).Then, (X, I [T]inf), (X, I [I]inf) and (X, I [F]inf) are3-fuzzysubalgebraof X;and (X, I [T]sup), (X, I [I]sup) and (X, I [F]sup) are4-fuzzysubalgebraof X.Let β I and βS ∈ [0,1] besuchthat U(I [I]inf; β I ) and L(I [I]sup; βS ) arenonempty.Let x, y ∈ U(I [I]inf; β I ).Then, I [I]inf(x) ≥ β I and I [I]inf(y) ≥ β I .Itfollowsthat

I [I]inf(x ∗ y) ≥ max{I [I]inf(x), I [I]inf(y)}≥ β I

andso x ∗ y ∈ U(I [I]inf; β I ).Thus, U(I [I]inf; β I ) isasubalgebraof (X, ∗,0).If x, y ∈ L(I [I]inf; βS ), then I [I]inf(x) ≤ βS and I [I]inf(y) ≤ βS.Hence,

I [I]inf(x ∗ y) ≤ max{I [I]inf(x), I [I]inf(y)}≤ βS,

andso x ∗ y ∈ L(I [I]inf; βS ).Thus, L(I [I]inf; βS ) isasubalgebraof (X, ∗,0).Similarly,wecanshow that U(I [T]inf; αI ), L(I [T]sup; αS ), U(I [F]inf; γI ) and L(I [F]sup; γS ) areeitheremptyorsubalgebraof (X, ∗,0) forall αI , αS, γI , γS ∈ [0,1]

Sinceevery2-fuzzysubalgebraisa4-fuzzysubalgebra,wehavethefollowingcorollary.

Corollary1. Ifanintervalneutrosophicset I :=(I [T], I [I], I [F]) in X isa (T(i,2), I(i,2), F(i,2))-interval neutrosophicsubalgebraof (X, ∗,0) for i ∈{1,3},then U(I [T]inf; αI ), L(I [T]sup; αS ), U(I [I]inf; β I ), L(I [I]sup; βS ), U(I [F]inf; γI ) and L(I [F]sup; γS ) areeitheremptyorsubalgebraof (X, ∗,0) forall αI , αS, β I , βS, γI , γS ∈ [0,1]

ByasimilarwaytotheproofofTheorem 1,wehavethefollowingtheorems.

Theorem2. Ifanintervalneutrosophicset I :=(I [T], I [I], I [F]) in X isa (T(i,4), I(i,4), F(i,4))-interval neutrosophicsubalgebraof (X, ∗,0) for i ∈{2,4},then L(I [T]inf; αI ), L(I [T]sup; αS ), L(I [I]inf; β I ), L(I [I]sup; βS ), L(I [F]inf; γI ) and L(I [F]sup; γS ) areeitheremptyorsubalgebraof (X, ∗,0) forall αI , αS, β I , βS, γI , γS ∈ [0,1].

Corollary2. Ifanintervalneutrosophicset I :=(I [T], I [I], I [F]) in X isa (T(i,2), I(i,2), F(i,2))-interval neutrosophicsubalgebraof (X, ∗,0) for i ∈{2,4},then L(I [T]inf; αI ), L(I [T]sup; αS ), L(I [I]inf; β I ), L(I [I]sup; βS ), L(I [F]inf; γI ) and L(I [F]sup; γS ) areeitheremptyorsubalgebraof (X, ∗,0) forall αI , αS, β I , βS, γI , γS ∈ [0,1]

Theorem3. Ifanintervalneutrosophicset I :=(I [T], I [I], I [F]) in X isa (T(k,1), I(k,1), F(k,1))-interval neutrosophicsubalgebraof (X, ∗,0) for k ∈{1,3},then U(I [T]inf; αI ), U(I [T]sup; αS ), U(I [I]inf; β I ), U(I [I]sup; βS ), U(I [F]inf; γI ) and U(I [F]sup; γS ) areeitheremptyorsubalgebraof (X, ∗,0) forall αI , αS, β I , βS, γI , γS ∈ [0,1] Corollary3. Ifanintervalneutrosophicset I :=(I [T], I [I], I [F]) in X isa (T(k,3), I(k,3), F(k,3))-intervalneutrosophicsubalgebraof (X, ∗,0) for k ∈{1,3},then U(I [T]inf; αI ), U(I [T]sup; αS ), U(I [I]inf; β I ), U(I [I]sup; βS ), U(I [F]inf; γI ) and U(I [F]sup; γS ) areeitheremptyorsubalgebraof (X, ∗,0) forall αI , αS, β I , βS, γI , γS ∈ [0,1].

Florentin Smarandache (author and editor) Collected Papers, IX 271

Theorem4. Ifanintervalneutrosophicset I :=(I [T], I [I], I [F]) in X isa (T(k,1), I(k,1), F(k,1))-interval neutrosophicsubalgebraof (X, ∗,0) for k ∈{2,4},then L(I [T]inf; αI ), U(I [T]sup; αS ), L(I [I]inf; β I ), U(I [I]sup; βS ), L(I [F]inf; γI ) and U(I [F]sup; γS ) areeitheremptyorsubalgebraof (X, ∗,0) forall αI , αS, β I , βS, γI , γS ∈ [0,1].

Corollary4. Ifanintervalneutrosophicset I :=(I [T], I [I], I [F]) in X isa (T(k,3), I(k,3), F(k,3))-intervalneutrosophicsubalgebraof (X, ∗,0) for k ∈{2,4},then L(I [T]inf; αI ), U(I [T]sup; αS ), L(I [I]inf; β I ), U(I [I]sup; βS ), L(I [F]inf; γI ) and U(I [F]sup; γS ) areeitheremptyorsubalgebraof (X, ∗,0) forall αI , αS, β I , βS, γI , γS ∈ [0,1]

Theorem5. Let I :=(I [T], I [I], I [F]) beanintervalneutrosophicsetin X inwhich U(I [T]inf; αI ), L(I [T]sup; αS ), U(I [I]inf; β I ), L(I [I]sup; βS ), U(I [F]inf; γI ) and L(I [F]sup; γS ) arenonemptysubalgebra of (X, ∗,0) forall αI , αS, β I , βS, γI , γS ∈ [0,1].Then, I :=(I [T], I [I], I [F]) isa (T(1,4), I(1,4), F(1,4))-intervalneutrosophicsubalgebraof (X, ∗,0).

Proof. Supposethat (X, I [T]inf) isnota1-fuzzysubalgebraof (X, ∗,0).Then,thereexists x, y ∈ X suchthat

I [T]inf(x ∗ y) < min{I [T]inf(x), I [T]inf(y)}

Ifwetake αI = min{I [T]inf(x), I [T]inf(y)},then x, y ∈ U(I [T]inf; αI ),but x ∗ y / ∈ U(I [T]inf; αI ) Thisisacontradiction,andso (X, I [T]inf) isa1-fuzzysubalgebraof (X, ∗,0).If (X, I [T]sup) isnota 4-fuzzysubalgebraof (X, ∗,0),then

I [T]sup(a ∗ b) > max{I [T]sup(a), I [T]sup(b)}

forsome a, b ∈ X,andso a, b ∈ L(I [T]sup; αS ) and a ∗ b / ∈ L(I [T]sup; αS ) bytaking

αS := max{I [T]sup(a), I [T]sup(b)}

Thisisacontradiction,andtherefore (X, I [T]sup) isa4-fuzzysubalgebraof (X, ∗,0).Similarly,we canverifythat (X, I [I]inf) isa1-fuzzysubalgebraof (X, ∗,0) and (X, I [I]sup) isa4-fuzzysubalgebra of (X, ∗,0);and (X, I [F]inf) isa1-fuzzysubalgebraof (X, ∗,0) and (X, I [F]sup) isa4-fuzzysubalgebra of (X, ∗,0).Consequently, I :=(I [T], I [I], I [F]) isa (T(1,4), I(1,4), F(1,4))-intervalneutrosophic subalgebraof (X, ∗,0)

UsingthesimilarmethodtotheproofofTheorem 5,wegetthefollowingtheorems.

Theorem6. Let I :=(I [T], I [I], I [F]) beanintervalneutrosophicsetin X inwhich L(I [T]inf; αI ), U(I [T]sup; αS ), L(I [I]inf; β I ), U(I [I]sup; βS ), L(I [F]inf; γI ) and U(I [F]sup; γS ) arenonemptysubalgebra of (X, ∗,0) forall αI , αS, β I , βS, γI , γS ∈ [0,1].Then, I :=(I [T], I [I], I [F]) isa (T(4,1), I(4,1), F(4,1))-intervalneutrosophicsubalgebraof (X, ∗,0)

Theorem7. Let I :=(I [T], I [I], I [F]) beanintervalneutrosophicsetin X inwhich L(I [T]inf; αI ), L(I [T]sup; αS ), L(I [I]inf; β I ), L(I [I]sup; βS ), L(I [F]inf; γI ) and L(I [F]sup; γS ) arenonemptysubalgebra of (X, ∗,0) forall αI , αS, β I , βS, γI , γS ∈ [0,1].Then, I :=(I [T], I [I], I [F]) isa (T(4,4), I(4,4), F(4,4))-intervalneutrosophicsubalgebraof (X, ∗,0)

Theorem8. Let I :=(I [T], I [I], I [F]) beanintervalneutrosophicsetin X inwhich U(I [T]inf; αI ), U(I [T]sup; αS ), U(I [I]inf; β I ), U(I [I]sup; βS ), U(I [F]inf; γI ) and U(I [F]sup; γS ) arenonemptysubalgebra of (X, ∗,0) forall αI , αS, β I , βS, γI , γS ∈ [0,1].Then, I :=(I [T], I [I], I [F]) isa (T(1,1), I(1,1), F(1,1))-intervalneutrosophicsubalgebraof (X, ∗,0).

Florentin Smarandache (author and editor) Collected Papers, IX 272

4.IntervalNeutrosophicLengths

Deﬁnition4. Givenanintervalneutrosophicset I :=(I [T], I [I], I [F]) in X,wedeﬁnetheinterval neutrosophiclengthof I asanorderedtriple I :=(I [T] , I [I] , I [F] ) where

I [T] : X → [0,1], x →I [T]sup(x) −I [T]inf(x),

I [I] : X → [0,1], x →I [I]sup(x) −I [I]inf(x), and

I [F] : X → [0,1], x →I [F]sup(x) −I [F]inf(x), whicharecalledintervalneutrosophic T-length,intervalneutrosophic I-lengthandintervalneutrosophic F-lengthof I,respectively.

Example3. Considertheintervalneutrosophicset I :=(I [T], I [I], I [F]) in X,whichisgiveninExample 2. Then,theintervalneutrosophiclengthof I isgivenbyTable 3.

Table3. Intervalneutrosophiclengthof I

X I [T ] I [ I ] I [F ] 00.60.450.3 a 0.20.050.2 b 0.10.050.1 c 0.30.050.2

Theorem9. Ifanintervalneutrosophicset I :=(I [T], I [I], I [F]) in X isa (T(i,3), I(i,3), F(i,3))-interval neutrosophicsubalgebraof (X, ∗,0) for i ∈{2,4},then (X, I [T] ), (X, I [I] ) and (X, I [F] ) are 3-fuzzy subalgebraof (X, ∗,0) Proof. Assumethat I :=(I [T], I [I], I [F]) isa (T(2,3), I(2,3), F(2,3))-intervalneutrosophic subalgebraof (X, ∗,0).Then, (X, I [T]inf), (X, I [I]inf) and (X, I [F]inf) are2-fuzzysubalgebraof X,and (X, I [T]sup), (X, I [I]sup) and (X, I [F]sup) are3-fuzzysubalgebraof X.Thus,

I [T]inf(x ∗ y) ≤ min{I [T]inf(x), I [T]inf(y)},

I [I]inf(x ∗ y) ≤ min{I [I]inf(x), I [I]inf(y)}, I [F]inf(x ∗ y) ≤ min{I [F]inf(x), I [F]inf(y)}, and

I [T]sup(x ∗ y) ≥ max{I [T]sup(x), I [T]sup(y)}, I [I]sup(x ∗ y) ≥ max{I [I]sup(x), I [I]sup(y)}, I [F]sup(x ∗ y) ≥ max{I [F]sup(x), I [F]sup(y)}, forall x, y ∈ X.Itfollowsthat

I [T] (x ∗ y)= I [T]sup(x ∗ y) −I [T]inf(x ∗ y) ≥I [T]sup(x) −I [T]inf(x)= I [T] (x), I [T] (x ∗ y)= I [T]sup(x ∗ y) −I [T]inf(x ∗ y) ≥I [T]sup(y) −I [T]inf(y)= I [T] (y),

I [I] (x ∗ y)= I [I]sup(x ∗ y) −I [I]inf(x ∗ y) ≥I [I]sup(x) −I [I]inf(x)= I [I] (x), I [I] (x ∗ y)= I [I]sup(x ∗ y) −I [I]inf(x ∗ y) ≥I [I]sup(y) −I [I]inf(y)= I [I] (y),

Florentin Smarandache (author and editor) Collected Papers, IX 273

and

I [F] (x ∗ y)= I [F]sup(x ∗ y) −I [F]inf(x ∗ y) ≥I [F]sup(x) −I [F]inf(x)= I [F] (x), I [F] (x ∗ y)= I [F]sup(x ∗ y) −I [F]inf(x ∗ y) ≥I [F]sup(y) −I [F]inf(y)= I [F] (y).

Hence,

I [T] (x ∗ y) ≥ max{I [T] (x), I [T] (y)}, I [I] (x ∗ y) ≥ max{I [I] (x), I [I] (y)}, and

I [F] (x ∗ y) ≥ max{I [F] (x), I [F] (y)}, forall x, y ∈ X.Therefore, (X, I [T] ), (X, I [I] ) and (X, I [F] ) are3-fuzzysubalgebraof (X, ∗,0). Supposethat I :=(I [T], I [I], I [F]) isa (T(4,3), I(4,3), F(4,3))-intervalneutrosophicsubalgebra of (X, ∗,0).Then, (X, I [T]inf), (X, I [I]inf) and (X, I [F]inf) are4-fuzzysubalgebraof X,and (X, I [T]sup), (X, I [I]sup) and (X, I [F]sup) are3-fuzzysubalgebraof X.Hence, I [T]inf(x ∗ y) ≤ max{I [T]inf(x), I [T]inf(y)},

I [I]inf(x ∗ y) ≤ max{I [I]inf(x), I [I]inf(y)},

I [F]inf(x ∗ y) ≤ max{I [F]inf(x), I [F]inf(y)},

(5) and

I [T]sup(x ∗ y) ≥ max{I [T]sup(x), I [T]sup(y)},

I [I]sup(x ∗ y) ≥ max{I [I]sup(x), I [I]sup(y)},

I [F]sup(x ∗ y) ≥ max{I [F]sup(x), I [F]sup(y)},

forall x, y ∈ X.Label(5)impliesthat

I [T]inf(x ∗ y) ≤I [T]inf(x) or I [T]inf(x ∗ y) ≤I [T]inf(y),

I [I]inf(x ∗ y) ≤I [I]inf(x) or I [I]inf(x ∗ y) ≤I [I]inf(y), I [F]inf(x ∗ y) ≤I [F]inf(x) or I [F]inf(x ∗ y) ≤I [F]inf(y).

If I [T]inf(x ∗ y) ≤I [T]inf(x),then

I [T] (x ∗ y)= I [T]sup(x ∗ y) −I [T]inf(x ∗ y) ≥I [T]sup(x) −I [T]inf(x)= I [T] (x).

If I [T]inf(x ∗ y) ≤I [T]inf(y),then

I [T] (x ∗ y)= I [T]sup(x ∗ y) −I [T]inf(x ∗ y) ≥I [T]sup(y) −I [T]inf(y)= I [T] (y)

Itfollowsthat I [T] (x ∗ y) ≥ max{I [T] (x), I [T] (y)}.Therefore, (X, I [T] ) isa3-fuzzy subalgebraof (X, ∗,0).Similarly,wecanshowthat (X, I [I] ) and (X, I [F] ) are3-fuzzysubalgebraof (X, ∗,0)

Corollary5. Ifanintervalneutrosophicset I :=(I [T], I [I], I [F]) in X isa (T(i,3), I(i,3), F(i,3))-interval neutrosophicsubalgebraof (X, ∗,0) for i ∈{2,4},then (X, I [T] ), (X, I [I] ) and (X, I [F] ) are 1-fuzzy subalgebraof (X, ∗,0)

Florentin Smarandache (author and editor) Collected Papers, IX 274

Theorem10. Ifanintervalneutrosophicset I :=(I [T], I [I], I [F]) in X isa (T(3,4), I(3,4), F(3,4))-intervalneutrosophicsubalgebraof (X, ∗,0),then (X, I [T] ), (X, I [I] ) and (X, I [F] ) are 4-fuzzy subalgebraof (X, ∗,0).

Proof. Let I :=(I [T], I [I], I [F]) bea (T(3,4), I(3,4), F(3,4))-intervalneutrosophicsubalgebraof (X, ∗,0).Then, (X, I [T]inf), (X, I [I]inf) and (X, I [F]inf) are3-fuzzysubalgebraof X,and (X, I [T]sup), (X, I [I]sup) and (X, I [F]sup) are4-fuzzysubalgebraof X.Thus,

I [T]inf(x ∗ y) ≥ max{I [T]inf(x), I [T]inf(y)},

I [I]inf(x ∗ y) ≥ max{I [I]inf(x), I [I]inf(y)}, I [F]inf(x ∗ y) ≥ max{I [F]inf(x), I [F]inf(y)}, and

I [T]sup(x ∗ y) ≤ max{I [T]sup(x), I [T]sup(y)}, I [I]sup(x ∗ y) ≤ max{I [I]sup(x), I [I]sup(y)}, I [F]sup(x ∗ y) ≤ max{I [F]sup(x), I [F]sup(y)},

(6) forall x, y ∈ X.ItfollowsfromLabel(6)that

I [T]sup(x ∗ y) ≤I [T]sup(x) or I [T]sup(x ∗ y) ≤I [T]sup(y), I [I]sup(x ∗ y) ≤I [I]sup(x) or I [I]sup(x ∗ y) ≤I [I]sup(y), I [F]sup(x ∗ y) ≤I [F]sup(x) or I [F]sup(x ∗ y) ≤I [F]sup(y).

Assumethat I [T]sup(x ∗ y) ≤I [T]sup(x).Then,

I [T] (x ∗ y)= I [T]sup(x ∗ y) −I [T]inf(x ∗ y) ≤I [T]sup(x) −I [T]inf(x)= I [T] (x).

If I [T]sup(x ∗ y) ≤I [T]sup(y),then

I [T] (x ∗ y)= I [T]sup(x ∗ y) −I [T]inf(x ∗ y) ≤I [T]sup(y) −I [T]inf(y)= I [T] (y)

Hence, I [T] (x ∗ y) ≤ max{I [T] (x), I [T] (y)} forall x, y ∈ X.Byasimilarway,wecan provethat

I [I] (x ∗ y) ≤ max{I [I] (x), I [I] (y)} and I [F] (x ∗ y) ≤ max{I [F] (x), I [F] (y)} forall x, y ∈ X.Therefore, (X, I [T] ), (X, I [I] ) and (X, I [F] ) are4-fuzzysubalgebraof (X, ∗,0)

Theorem11. Ifanintervalneutrosophicset I :=(I [T], I [I], I [F]) in X isa (T(3,2), I(3,2), F(3,2))-intervalneutrosophicsubalgebraof (X, ∗,0),then (X, I [T] ), (X, I [I] ) and (X, I [F] ) are 2-fuzzy subalgebraof (X, ∗,0)

Proof. Assumethat I :=(I [T], I [I], I [F]) isa (T(3,2), I(3,2), F(3,2))-intervalneutrosophic subalgebraof (X, ∗,0).Then, (X, I [T]inf), (X, I [I]inf) and (X, I [F]inf) are3-fuzzysubalgebraof X,and (X, I [T]sup), (X, I [I]sup) and (X, I [F]sup) are2-fuzzysubalgebraof X.Hence, I [T]inf(x ∗ y) ≥ max{I [T]inf(x), I [T]inf(y)}, I [I]inf(x ∗ y) ≥ max{I [I]inf(x), I [I]inf(y)}, I [F]inf(x ∗ y) ≥ max{I [F]inf(x), I [F]inf(y)},

Florentin Smarandache (author and editor) Collected Papers, IX 275

I [T]sup(x ∗ y) ≤ min{I [T]sup(x), I [T]sup(y)},

I [I]sup(x ∗ y) ≤ min{I [I]sup(x), I [I]sup(y)},

I [F]sup(x ∗ y) ≤ min{I [F]sup(x), I [F]sup(y)}, forall x, y ∈ X,whichimplythat

I [T] (x ∗ y)= I [T]sup(x ∗ y) −I [T]inf(x ∗ y) ≤I [T]sup(x) −I [T]inf(x)= I [T] (x),

I [T] (x ∗ y)= I [T]sup(x ∗ y) −I [T]inf(x ∗ y) ≤I [T]sup(y) −I [T]inf(y)= I [T] (y),

I [I] (x ∗ y)= I [I]sup(x ∗ y) −I [I]inf(x ∗ y) ≤I [I]sup(x) −I [I]inf(x)= I [I] (x),

I [I] (x ∗ y)= I [I]sup(x ∗ y) −I [I]inf(x ∗ y) ≤I [I]sup(y) −I [I]inf(y)= I [I] (y), and

I [F] (x ∗ y)= I [F]sup(x ∗ y) −I [F]inf(x ∗ y) ≤I [F]sup(x) −I [F]inf(x)= I [F] (x), I [F] (x ∗ y)= I [F]sup(x ∗ y) −I [F]inf(x ∗ y) ≤I [F]sup(y) −I [F]inf(y)= I [F] (y)

Itfollowsthat

I [T] (x ∗ y) ≤ min{I [T] (x), I [T] (y)},

I [I] (x ∗ y) ≤ min{I [I] (x), I [I] (y)}, and

I [F] (x ∗ y) ≤ min{I [F] (x), I [F] (y)}, forall x, y ∈ X.Hence, (X, I [T] ), (X, I [I] ) and (X, I [F] ) are2-fuzzysubalgebraof (X, ∗,0)

Corollary6. Ifanintervalneutrosophicset I :=(I [T], I [I], I [F]) in X isa (T(3,2), I(3,2), F(3,2))-intervalneutrosophicsubalgebraof (X, ∗,0),then (X, I [T] ), (X, I [I] ) and (X, I [F] ) are 4-fuzzy subalgebraof (X, ∗,0)

Theorem12. Ifanintervalneutrosophicset I :=(I [T], I [I], I [F]) in X isa (T(i,3), I(3,4), F(3,2))-intervalneutrosophicsubalgebraof (X, ∗,0) fori ∈{2,4},then

(1) (X, I [T] ) isa 3-fuzzysubalgebraof (X, ∗,0) (2) (X, I [I] ) isa 4-fuzzysubalgebraof (X, ∗,0) (3) (X, I [F] ) isa 2-fuzzysubalgebraof (X, ∗,0).

Proof. Assumethat I :=(I [T], I [I], I [F]) isa (T(4,3), I(3,4), F(3,2))-intervalneutrosophic subalgebraof (X, ∗,0).Then, (X, I [T]inf) isa4-fuzzysubalgebraof X, (X, I [T]sup) isa3-fuzzy subalgebraof X, (X, I [I]inf) isa3-fuzzysubalgebraof X, (X, I [I]sup) isa4-fuzzysubalgebraof X, (X, I [F]inf) isa3-fuzzysubalgebraof X,and (X, I [F]sup) isa2-fuzzysubalgebraof X.Hence,

I [T]inf(x ∗ y) ≤ max{I [T]inf(x), I [T]inf(y)},(7)

I [T]sup(x ∗ y) ≥ max{I [T]sup(x), I [T]sup(y)},(8)

I [I]inf(x ∗ y) ≥ max{I [I]inf(x), I [I]inf(y)},(9)

I [I]sup(x ∗ y) ≤ max{I [I]sup(x), I [I]sup(y)},(10)

I [F]inf(x ∗ y) ≥ max{I [F]inf(x), I [F]inf(y)},(11)

and

Florentin Smarandache (author and editor) Collected Papers, IX 276

I [F]sup(x ∗ y) ≤ min{I [F]sup(x), I [F]sup(y)}, (12) and for all x, y ∈ X Then, I [T]inf(x ∗ y) ≤I [T]inf(x) or I [T]inf(x ∗ y) ≤I [T]inf(y) byLabel(7).ItfollowsfromLabel(8)that

I [T] (x ∗ y)= I [T]sup(x ∗ y) −I [T]inf(x ∗ y) ≥ I [T]sup(x) −I [T]inf(x)= I [T] (x) or I [T] (x ∗ y)= I [T]sup(x ∗ y) −I [T]inf(x ∗ y) ≥I [T]sup(y) −I [T]inf(y)= I [T] (y), andsothat I [T] (x ∗ y) ≥ max{I [T] (x), I [T] (y)} forall x, y ∈ X.Thus, (X, I [T] ) isa3-fuzzy subalgebraof (X, ∗,0).Thecondition(10)impliesthat I [I]sup(x ∗ y) ≤I [I]sup(x) or I [I]sup(x ∗ y) ≤I [I]sup(y) (13)

CombiningLabels(9)and(13),wehave

I [I] (x ∗ y)= I [I]sup(x ∗ y) −I [I]inf(x ∗ y) ≤ I [I]sup(x) −I [I]inf(x)= I [I] (x) or I [I] (x ∗ y)= I [I]sup(x ∗ y) −I [I]inf(x ∗ y) ≤I [I]sup(y) −I [I]inf(y)= I [I] (y)

Itfollowsthat I [I] (x ∗ y) ≤ max{I [I] (x), I [I] (y)} forall x, y ∈ X.Thus, (X, I [I] ) isa4-fuzzy subalgebraof (X, ∗,0).UsingLabels(11)and(12),wehave

I [F] (x ∗ y)= I [F]sup(x ∗ y) −I [F]inf(x ∗ y) ≤I [F]sup(x) −I [F]inf(x)= I [F] (x) and I [F] (x ∗ y)= I [F]sup(x ∗ y) −I [F]inf(x ∗ y) ≤I [F]sup(y) −I [F]inf(y)= I [F] (y), andso I [F] (x ∗ y) ≤ min{I [F] (x), I [F] (y)} forall x, y ∈ X.Therefore, (X, I [F] ) isa2-fuzzy subalgebraof (X, ∗,0).Similarly,wecanprovethedesiredresultsfor i = 2.

Corollary7. Ifanintervalneutrosophicset I :=(I [T], I [I], I [F]) in X isa (T(i,3), I(3,4), F(3,2))-interval neutrosophicsubalgebraof (X, ∗,0) fori ∈{2,4},then

(1) (X, I [T] ) isa 1-fuzzysubalgebraof (X, ∗,0). (2) (X, I [I] ) and (X, I [F] ) are 4-fuzzysubalgebraof (X, ∗,0).

ByasimilarwaytotheproofofTheorem 12,wehavethefollowingtheorems.

Theorem13. Ifanintervalneutrosophicset I :=(I [T], I [I], I [F]) in X isa (T(i,3), I(3,2), F(3,2))-intervalneutrosophicsubalgebraof (X, ∗,0) fori ∈{2,4},then

(1) (X, I [T] ) isa 3-fuzzysubalgebraof (X, ∗,0) (2) (X, I [I] ) and (X, I [F] ) are 2-fuzzysubalgebraof (X, ∗,0)

Corollary8. Ifanintervalneutrosophicset I :=(I [T], I [I], I [F]) in X isa (T(i,3), I(3,2), F(3,2))-interval neutrosophicsubalgebraof (X, ∗,0) fori ∈{2,4},then

(1) (X, I [T] ) isa 1-fuzzysubalgebraof (X, ∗,0) (2) (X, I [I] ) and (X, I [F] ) are 4-fuzzysubalgebraof (X, ∗,0)

Florentin Smarandache (author and editor) Collected Papers, IX 277

Theorem14. Ifanintervalneutrosophicset I :=(I [T], I [I], I [F]) in X isa (T(i,3), I(3,2), F(2,3))-intervalneutrosophicsubalgebraof (X, ∗,0) fori ∈{2,4},then

(1) (X, I [T] ) and (X, I [F] ) are 3-fuzzysubalgebraof (X, ∗,0) (2) (X, I [I] ) isa 2-fuzzysubalgebraof (X, ∗,0).

Corollary9. Ifanintervalneutrosophicset I :=(I [T], I [I], I [F]) in X isa (T(i,3), I(3,2), F(2,3))-interval neutrosophicsubalgebraof (X, ∗,0) fori ∈{2,4},then

(1) (X, I [T] ) and (X, I [F] ) are 1-fuzzysubalgebraof (X, ∗,0) (2) (X, I [I] ) isa 4-fuzzysubalgebraof (X, ∗,0).

References

1.Atanassov,K.T.Intuitionisticfuzzysets. FuzzySetsSyst. 1986, 20,87–96.

2.Zadeh,L.A.Fuzzysets. Inf.Control 1965, 8,338–353.

3. Wang,H.;Smarandache,F.;Zhang,Y.Q.;Sunderraman,R. IntervalNeutrosophicSetsandLogic:Theoryand ApplicationsinComputing;NeutrosophicBookSeriesNo.5;Hexis:Phoenix,AZ,USA,2005.

4. Smarandache,F. AUnifyingFieldinLogics:NeutrosophicLogic.Neutrosophy,NeutrosophicSet,Neutrosophic Probability;AmericanReserchPress:Rehoboth,NM,USA,1999.

5. Smarandache,F.Neutrosophicset-ageneralizationoftheintuitionisticfuzzyset. Int.J.PureAppl.Math. 2005, 24,287–297.

6. Wang,H.;Zhang,Y.;Sunderraman,R.Truth-valuebasedintervalneutrosophicsets.InProceedingsofthe 2005IEEEInternationalConferenceonGranularComputing,Beijing,China,25–27July2005;Volume1, pp.274–277.doi:10.1109/GRC.2005.1547284.

7.Imai,Y.;Iséki,K.Onaxiomsystemsofpropositionalcalculi. Proc.Jpn.Acad. 1966, 42,19–21.

8.Iséki,K.Analgebrarelatedwithapropositionalcalculus. Proc.Jpn.Acad. 1966, 42,26–29.

9.Huang,Y.S. BCI-Algebra;SciencePress:Beijing,China,2006.

10.Meng,J.;Jun,Y.B. BCK-Algebra;KyungmoonSaCo.:Seoul,Korea,1994.

11. Jun,Y.B.;Hur,K.;Lee,K.J.Hyperfuzzysubalgebraof BCK/BCI-algebra. Ann.FuzzyMath.Inf. 2018, 15,17–28.

Florentin Smarandache (author and editor) Collected Papers, IX 278

Neutrosophic Permeable Values and Energetic Subsets with Applications in BCK/BCI-Algebras

Young Bae Jun, Florentin Smarandache, Seok-Zun Song, Hashem Bordbar (2018). Neutrosophic Permeable Values and Energetic Subsets with Applications in BCK/BCIAlgebras. Mathematics 6, 74; DOI: 10.3390/math6050074

Abstract: The concept of a (∈, ∈)-neutrosophic ideal is introduced, and its characterizations are established. The notions of neutrosophic permeable values are introduced, and related properties are investigated. Conditions for the neutrosophic level sets to be energetic, right stable, and right vanished are discussed. Relations between neutrosophic permeable S and I-values are considered.

Keywords: (∈, ∈)-neutrosophic subalgebra; (∈, ∈)-neutrosophic ideal; neutrosophic (anti-)permeable S-value; neutrosophic (anti-)permeable I-value; S-energetic set; I-energetic set

1.Introduction

Thenotionofneutrosophicset(NS)theorydevelopedbySmarandache(see[1,2])isamoregeneral platformthatextendstheconceptsofclassicandfuzzysets,intuitionisticfuzzysets,andinterval-valued (intuitionistic)fuzzysetsandthatisappliedtovariousparts:patternrecognition,medicaldiagnosis, decision-makingproblems,andsoon(see[3 6]).Smarandache[2]mentionedthatacloudisaNS becauseitsbordersareambiguousandbecauseeachelement(waterdrop)belongswithaneutrosophic probabilitytotheset(e.g.,therearetypesofseparatedwaterdropsaroundacompactmassofwater drops,suchthatwedonotknowhowtoconsiderthem:inoroutofthecloud).Additionally,weare notsurewherethecloudendsnorwhereitbegins,andneitherwhethersomeelementsareorarenot intheset.Thisiswhythepercentageofindeterminacyisrequiredandtheneutrosophicprobability (usingsubsets—notnumbers—ascomponents)shouldbeusedforbettermodeling:itisamoreorganic, smooth,andparticularlyaccurateestimation.Indeterminacyisthezoneofignoranceofaproposition’s value,betweentruthandfalsehood.

Algebraicstructuresplayanimportantroleinmathematicswithwide-rangingapplicationsin severaldisciplinessuchascodingtheory,informationsciences,computersciences,controlengineering, theoreticalphysics,andsoon.NStheoryisalsoappliedtoseveralalgebraicstructures.Inparticular, Junetal.applieditto BCK/BCI-algebras(see[7 12]).Junetal.[8]introducedthenotionsofenergetic subsets,rightvanishedsubsets,rightstablesubsets,and(anti-)permeablevaluesin BCK/BCI-algebras andinvestigatedrelationsbetweenthesesets.

Inthispaper,weintroducethenotionsofneutrosophicpermeable S-values,neutrosophic permeable I-values, (∈, ∈)-neutrosophicideals,neutrosophicanti-permeable S-values, andneutrosophicanti-permeable I-values,whicharemotivatedbytheideaofsubalgebras

Young Bae Jun, Florentin Smarandache, Seok-Zun Song, Hashem BordbarD

Smarandache (author and editor) Collected Papers, IX 279

Florentin

(i.e., S-values)andideals(i.e., I-values),andinvestigatetheirproperties.Weconsidercharacterizations of (∈, ∈)-neutrosophicideals.Wediscussconditionsforthelower(upper)neutrosophic ∈Φ-subsetsto be S-and I-energetic.Weprovideconditionsforatriple (α, β, γ) ofnumberstobeaneutrosophic (anti-)permeable S-or I-value.Weconsiderconditionsfortheupper(lower)neutrosophic ∈Φ-subsetsto berightstable(rightvanished)subsets.Weestablishrelationsbetweenneutrosophic(anti-)permeable S-and I-values.

2.Preliminaries

Analgebra (X; ∗,0) oftype (2,0) iscalleda BCI-algebra ifitsatisﬁesthefollowingconditions:

Ifa BCI-algebra X satisﬁesthefollowingidentity: (V) (∀x ∈ X)(0 ∗ x = 0), then X iscalleda BCK-algebra. Any BCK/BCI-algebra X satisﬁesthefollowingconditions:

(∀x ∈ X) (x ∗ 0 = x) ,(1) (∀x, y, z ∈ X) (x ≤ y ⇒ x ∗ z ≤ y ∗ z, z ∗ y ≤ z ∗ x) ,(2) (∀x, y, z ∈ X) ((x ∗ y) ∗ z =(x ∗ z) ∗ y) ,(3) (∀x, y, z ∈ X) ((x ∗ z) ∗ (y ∗ z) ≤ x ∗ y) ,(4) where x ≤ y ifandonlyif x ∗ y = 0.Anonemptysubset S ofa BCK/BCI-algebra X iscalleda subalgebra of X if x ∗ y ∈ S forall x, y ∈ S.Asubset I ofa BCK/BCI-algebra X iscalledan ideal of X if itsatisﬁesthefollowing: 0 ∈ I,(5) (∀x, y ∈ X) (x ∗ y ∈ I, y ∈ I → x ∈ I) .(6)

Wereferthereadertothebooks[13]and[14]forfurtherinformationregarding BCK/BCI-algebras. Foranyfamily {ai | i ∈ Λ} ofrealnumbers,wedeﬁne

{ai | i ∈ Λ} = sup{ai | i ∈ Λ} and {ai | i ∈ Λ} = inf{ai | i ∈ Λ}.

If Λ = {1,2},wealsouse a1 ∨ a2 and a1 ∧ a2 insteadof {ai | i ∈{1,2}} and {ai | i ∈{1,2}}, respectively.

Welet X beanonemptyset.ANSin X (see[1])isastructureoftheform

A := { x; AT (x), AI (x), AF (x) | x ∈ X},

where AT : X → [0,1] isatruthmembershipfunction, AI : X → [0,1] isanindeterminatemembership function,and AF : X → [0,1] isafalsemembershipfunction.Forthesakeofsimplicity,weusethe symbol A =(AT , AI , AF ) fortheNS

A := { x; AT (x), AI (x), AF (x) | x ∈ X}

Florentin Smarandache (author and editor) Collected Papers, IX 280

Asubset A ofa BCK/BCI-algebra X issaidtobe S-energetic(see[8])ifitsatisﬁes

(∀x, y ∈ X) (x ∗ y ∈ A ⇒{x, y}∩ A = ∅) (7)

Asubset A ofa BCK/BCI-algebra X issaidtobe I-energetic (see[8])ifitsatisﬁes

(∀x, y ∈ X) (y ∈ A ⇒{x, y ∗ x}∩ A = ∅) (8)

Asubset A ofa BCK/BCI-algebra X issaidtobe rightvanished (see[8])ifitsatisﬁes (∀x, y ∈ X) (x ∗ y ∈ A ⇒ x ∈ A) . (9)

Asubset A ofa BCK/BCI-algebra X issaidtobe rightstable (see[8])if A ∗ X := {a ∗ x | a ∈ A, x ∈ X}⊆ A

3.NeutrosophicPermeableValues

GivenaNS A =(AT , AI , AF ) inaset X, α, β ∈ (0,1] and γ ∈ [0,1),weconsiderthefollowingsets: U∈ T (A; α)= {x ∈ X | AT (x) ≥ α}, U∈ T (A; α)∗ = {x ∈ X | AT (x) > α},

U∈ I (A; β)= {x ∈ X | AI (x) ≥ β}, U∈ I (A; β)∗ = {x ∈ X | AI (x) > β},

U∈ F (A; γ)= {x ∈ X | AF (x) ≤ γ}, U∈ F (A; γ)∗ = {x ∈ X | AF (x) < γ},

L∈ T (A; α)= {x ∈ X | AT (x) ≤ α}, L∈ T (A; α)∗ = {x ∈ X | AT (x) < α},

L∈ I (A; β)= {x ∈ X | AI (x) ≤ β}, L∈ I (A; β)∗ = {x ∈ X | AI (x) < β},

L∈ F (A; γ)= {x ∈ X | AF (x) ≥ γ}, L∈ F (A; γ)∗ = {x ∈ X | AF (x) > γ}

Wesay U∈ T (A; α), U∈ I (A; β),and U∈ F (A; γ) are upperneutrosophic ∈Φ-subsets of X,and L∈ T (A; α), L∈ I (A; β),and L∈ F (A; γ) are lowerneutrosophic ∈Φ-subsets of X,where Φ ∈{T, I, F}.Wesay U∈ T (A; α)∗ , U∈ I (A; β)∗,and U∈ F (A; γ)∗ are strongupperneutrosophic ∈Φ-subsets of X,and L∈ T (A; α)∗ , L∈ I (A; β)∗ , and L∈ F (A; γ)∗ are stronglowerneutrosophic ∈Φ-subsets of X,where Φ ∈{T, I, F}.

Deﬁnition1 ([7]). ANS A =(AT , AI , AF ) ina BCK/BCI-algebra X iscalledan (∈, ∈) neutrosophicsubalgebraofXifthefollowingassertionsarevalid:

(10) forallx, y ∈ X, αx, αy, βx, βy ∈ (0,1] and γx, γy ∈ [0,1). Lemma1 ([7]). ANS A =(AT , AI , AF ) ina BCK/BCI-algebra X isan (∈, ∈)-neutrosophicsubalgebraof XifandonlyifA =(AT    .(11) Proposition1. Every (∈, ∈)-neutrosophicsubalgebra A =(AT , AI , AF ) ofa BCK/BCI-algebra X satisﬁes (∀x ∈ X) (AT (0) ≥ AT (x), AI (0) ≥ AI (x), AF (0) ≤ AF (x)) .(12)

Florentin Smarandache (author and editor) Collected Papers, IX 281

x ∈ U∈ T (A; αx ), y ∈ U∈ T (A; αy ) ⇒ x ∗ y ∈ U∈ T (A; αx ∧ αy ), x ∈ U∈ I (A; βx ), y ∈ U∈ I (A; βy ) ⇒ x ∗ y ∈ U∈ I (A; βx ∧ βy ), x ∈ U∈ F (A; γx ), y ∈ U∈ F (A; γy ) ⇒ x ∗ y ∈ U∈ F (A; γx ∨ γy ), , AI , AF ) satisﬁes (∀x, y ∈ X)    AT (x ∗ y) ≥ AT (x) ∧ AT (y) AI (x ∗ y) ≥ AI (x) ∧ AI (y) AF (x ∗ y) ≤ AF (x) ∨ AF (y)

Proof. Straightforward.

Theorem1. If A =(AT , AI , AF ) isan (∈, ∈)-neutrosophicsubalgebraofa BCK/BCI-algebra X,thenthe lowerneutrosophic ∈Φ-subsetsofXareS-energeticsubsetsofX,where Φ ∈{T, I, F} Proof. Let x, y ∈ X and α ∈ (0,1] besuchthat x ∗ y ∈ L∈ T (A; α).Then

α ≥ AT (x ∗ y) ≥ AT (x) ∧ AT (y), andthus AT (x) ≤ α or AT (y) ≤ α;thatis, x ∈ L∈ T (A; α) or y ∈ L∈ T (A; α).Thus {x, y}∩ L∈ T (A; α) = ∅ Therefore L∈ T (A; α) isan S-energeticsubsetof X.Similarly,wecanverifythat L∈ I (A; β) isan S-energetic subsetof X.Welet x, y ∈ X and γ ∈ [0,1) besuchthat x ∗ y ∈ L∈ F (A; γ).Then

γ ≤ AF (x ∗ y) ≤ AF (x) ∨ AF (y).

Itfollowsthat AF (x) ≥ γ or AF (y) ≥ γ;thatis, x ∈ L∈ F (A; γ) or y ∈ L∈ F (A; γ).Hence {x, y}∩ L∈ F (A; γ) = ∅,andtherefore L∈ F (A; γ) isan S-energeticsubsetof X Corollary1. If A =(AT , AI , AF ) isan (∈, ∈)-neutrosophicsubalgebraofa BCK/BCI-algebra X,thenthe stronglowerneutrosophic ∈Φ-subsetsofXareS-energeticsubsetsofX,where Φ ∈{T, I, F}

Proof. Straightforward. TheconverseofTheorem 1 isnottrue,asseeninthefollowingexample.

Example1. Considera BCK-algebra X = {0,1,2,3,4} withthebinaryoperation ∗ thatisgiveninTable 1 (see[14]).

Table1. Cayleytableforthebinaryoperation“∗”. *01234 000000 110000 221001 332102 441110

LetA =(AT , AI , AF ) beaNSinXthatisgiveninTable 2. Table2. Tabulationrepresentationof A =(AT , AI , AF ) XAT (x) A I (x) AF (x) 00.60.8 0.2 10.4 0.5 0.7 20.4 0.5 0.6 30.4 0.5 0.5 40.7 0.8 0.2 If α ∈ [0.4,0.6), β ∈ [0.5,0.8),and γ ∈ (0.2,0.5],then L∈ T (A; α)= {1,2,3}, L∈ I (A; β)= {1,2,3}, andL∈ F (A; γ)= {1,2,3} areS-energeticsubsetsofX.Because AT (4 ∗ 4)= AT (0)= 0.6 0.7 = AT (4) ∧ AT (4)

Florentin Smarandache (author and editor) Collected Papers, IX 282

   (13)

and/or AF (3 ∗ 2)= AF (1)= 0.7 0.6 = AF (3) ∨ AF (2), itfollowsfromLemma 1 thatA =(AT , AI , AF ) isnotan (∈, ∈)-neutrosophicsubalgebraofX. Deﬁnition2. Let A =(AT , AI , AF ) beaNSina BCK/BCI-algebra X and (α, β, γ) ∈ ΛT × ΛI × ΛF, where ΛT , ΛI ,and ΛF aresubsetsof [0,1].Then (α, β, γ) iscalledaneutrosophicpermeable S-valuefor A =(AT , AI , AF ) ifthefollowingassertionisvalid: (∀x, y ∈ X)    x ∗ y ∈ U∈ T (A; α) ⇒ AT (x) ∨ AT (y) ≥ α, x ∗ y ∈ U∈ I (A; β) ⇒ AI (x) ∨ AI (y) ≥ β, x ∗ y ∈ U∈ F (A; γ) ⇒ AF (x) ∧ AF (y) ≤ γ

Example2. LetX = {0,1,2,3,4} beasetwiththebinaryoperation ∗ thatisgiveninTable 3 Table3. Cayleytableforthebinaryoperation“∗”. *01234 000000 110110 222020 333303 444440

Then (X, ∗,0) isaBCK-algebra(see[14]).LetA =(AT, AI, AF) beaNSinXthatisgiveninTable 4. Table4. Tabulationrepresentationof A =(AT , AI , AF )

XAT (x) A I (x) AF (x) 00.20.3 0.7 10.6 0.4 0.6 20.5 0.3 0.4 30.4 0.8 0.5 40.7 0.6 0.2

Itisroutinetoverifythat (α, β, γ) ∈ (0,2,1] × (0.3,1] × [0,0.7) isaneutrosophicpermeable S-valuefor A =(AT , AI , AF ) Theorem2. Let A =(AT , AI , AF ) beaNSina BCK/BCI-algebra X and (α, β, γ) ∈ ΛT × ΛI × ΛF, where ΛT , ΛI ,and ΛF aresubsetsof [0,1].IfA =(AT , AI , AF ) satisﬁesthefollowingcondition: (∀x, y ∈

X)    AT (x ∗ y) ≤ AT (x) ∨ AT (y) AI (x ∗ y) ≤ AI (x) ∨ AI (y) AF (x ∗ y) ≥ AF (x) ∧ AF (y)    ,(14) then (α, β, γ) isaneutrosophicpermeableS-valueforA =(AT , AI , AF ). Proof. Let x, y ∈ X besuchthat x ∗ y ∈ U∈ T (A; α).Then α ≤ AT (x ∗ y) ≤ AT (x) ∨ AT (y) Florentin Smarandache (author and editor) Collected Papers, IX 283

Similarly,if x ∗ y ∈ U∈ I (A; β) for x, y ∈ X,then AI (x) ∨ AI (y) ≥ β.Now,let a, b ∈ X besuchthat a ∗ b ∈ U∈ F (A; γ).Then

γ ≥ AF (a ∗ b) ≥ AF (a) ∧ AF (b)

Therefore (α, β, γ) isaneutrosophicpermeable S-valuefor A =(AT , AI , AF ).

Theorem3. Let A =(AT , AI , AF ) beaNSina BCK-algebra X and (α, β, γ) ∈ ΛT × ΛI × ΛF,where ΛT , ΛI ,and ΛF aresubsetsof [0,1].IfA =(AT , AI , AF ) satisﬁesthefollowingconditions: (∀x ∈ X) (AT (0) ≤ AT (x), AI (0) ≤ AI (x), AF (0) ≥ AF (x)) (15) and (∀x, y ∈ X)    AT (x) ≤ AT (x ∗ y) ∨ AT (y) AI (x) ≤ AI (x ∗ y) ∨ AI (y) AF (x) ≥ AF (x ∗ y) ∧ AF (y)

   ,(16) then (α, β, γ) isaneutrosophicpermeableS-valueforA =(AT , AI , AF )

Proof. Let x, y, a, b, u, v ∈ X besuchthat x ∗ y ∈ U∈ T (A; α), a ∗ b ∈ U∈ I (A; β),and u ∗ v ∈ U∈ F (A; γ).Then

α ≤ AT (x ∗ y) ≤ AT ((x ∗ y) ∗ x) ∨ AT (x)

= AT ((x ∗ x) ∗ y) ∨ AT (x)= AT (0 ∗ y) ∨ AT (x)

= AT (0) ∨ AT (x)= AT (x),

β ≤ AI (a ∗ b) ≤ AI ((a ∗ b) ∗ a) ∨ AI (a)

= AI ((a ∗ a) ∗ b) ∨ AI (a)= AI (0 ∗ b) ∨ AI (a)

= AI (0) ∨ AI (a)= AI (a), and

γ ≥ AF (u ∗ v) ≥ AF ((u ∗ v) ∗ u) ∧ AF (u)

= AF ((u ∗ u) ∗ v) ∧ AF (u)= AF (0 ∗ v) ∧ AF (v)

= AF (0) ∧ AF (v)= AF (v)

byEquations(3),(V),(15),and(16).Itfollowsthat

AT (x) ∨ AT (y) ≥ AT (x) ≥ α, AI (a) ∨ AI (b) ≥ AI (a) ≥ β, AF (u) ∧ AF (v) ≤ AF (u) ≤ γ

Therefore (α, β, γ) isaneutrosophicpermeable S-valuefor A =(AT , AI , AF ).

Theorem4. Let A =(AT , AI , AF ) beaNSina BCK/BCI-algebra X and (α, β, γ) ∈ ΛT × ΛI × ΛF, where ΛT , ΛI ,and ΛF aresubsetsof [0,1].If (α, β, γ) isaneutrosophicpermeable S-valuefor A =(AT , AI , AF ),thenupperneutrosophic ∈Φ-subsetsofXareS-energeticwhere Φ ∈{T, I, F}

Florentin Smarandache (author and editor) Collected Papers, IX 284

Proof. Let x, y, a, b, u, v ∈ X besuchthat x ∗ y ∈ U∈ T (A; α), a ∗ b ∈ U∈ I (A; β),and u ∗ v ∈ U∈ F (A; γ). UsingEquation (13),wehave AT (x) ∨ AT (y) ≥ α, AI (a) ∨ AI (b) ≥ β,and AF (u) ∧ AF (v) ≤ γ. Itfollowsthat

AT (x) ≥ α or AT (y) ≥ α,thatis, x ∈ U∈ T (A; α) or y ∈ U∈ T (A; α);

AI (a) ≥ β or AI (b) ≥ β,thatis, a ∈ U∈ I (A; β) or b ∈ U∈ I (A; β); and

AF (u) ≤ γ or AF (v) ≤ γ,thatis, u ∈ U∈ F (A; γ) or v ∈ U∈ F (A; γ)

Hence {x, y}∩ U∈ T (A; α) = ∅, {a, b}∩ U∈ I (A; β) = ∅,and {u, v}∩ U∈ F (A; γ) = ∅ Therefore U∈ T (A; α), U∈ I (A; β),and U∈ F (A; γ) are S-energeticsubsetsof X.

Deﬁnition3. Let A =(AT , AI , AF ) beaNSina BCK/BCI-algebra X and (α, β, γ) ∈ ΛT × ΛI × ΛF, where ΛT , ΛI ,and ΛF aresubsetsof [0,1].Then (α, β, γ) iscalledaneutrosophicanti-permeable S-valuefor A =(AT , AI , AF ) ifthefollowingassertionisvalid: (∀x, y ∈ X)    x ∗ y ∈ L∈ T (A; α) ⇒ AT (x) ∧ AT (y) ≤ α, x ∗ y ∈ L∈ I (A; β) ⇒ AI (x) ∧ AI (y) ≤ β, x ∗ y ∈ L∈ F (A; γ) ⇒ AF (x) ∨ AF (y) ≥ γ

   .(17)

Example3. LetX = {0,1,2,3,4} beasetwiththebinaryoperation ∗ thatisgiveninTable 5 Table5. Cayleytableforthebinaryoperation“∗”. *01234 000000 110010 221020 333303 444440 Then (X, ∗,0) isaBCK-algebra(see[14]).LetA =(AT, AI, AF) beaNSinXthatisgiveninTable 6. Table6. Tabulationrepresentationof A =(AT , AI , AF )

XAT (x) A I (x) AF (x) 00.70.6 0.4 10.4 0.5 0.6 20.4 0.5 0.6 30.5 0.2 0.7 40.3 0.3 0.9

Itisroutinetoverifythat (α, β, γ) ∈ (0.3,1] × (0.2,1] × [0,0.9) isaneutrosophicanti-permeable S-value forA =(AT , AI , AF ) Theorem5. Let A =(AT , AI , AF ) beaNSina BCK/BCI-algebra X and (α, β, γ) ∈ ΛT × ΛI × ΛF, where ΛT , ΛI ,and ΛF aresubsetsof [0,1].If A =(AT , AI , AF ) isan (∈, ∈)-neutrosophicsubalgebraof X, then (α, β, γ) isaneutrosophicanti-permeableS-valueforA =(AT , AI , AF )

Florentin Smarandache (author and editor) Collected Papers, IX 285

Proof. Let x, y, a, b, u, v ∈ X besuchthat x ∗ y ∈ L∈ T (A; α), a ∗ b ∈ L∈ I (A; β),and u ∗ v ∈ L∈ F (A; γ). UsingLemma 1,wehave

AT (x) ∧ AT (y) ≤ AT (x ∗ y) ≤ α,

AI (a) ∧ AI (b) ≤ AI (a ∗ b) ≤ β,

AF (u) ∨ AF (v) ≥ AF (u ∗ v) ≥ γ,

andthus (α, β, γ) isaneutrosophicanti-permeable S-valuefor A =(AT , AI , AF ). Theorem6. Let A =(AT , AI , AF ) beaNSina BCK/BCI-algebra X and (α, β, γ) ∈ ΛT × ΛI × ΛF, where ΛT , ΛI ,and ΛF aresubsetsof [0,1].If (α, β, γ) isaneutrosophicanti-permeable S-valuefor A =(AT , AI , AF ),thenlowerneutrosophic ∈Φ-subsetsofXareS-energeticwhere Φ ∈{T, I, F}

Proof. Let x, y, a, b, u, v ∈ X besuchthat x ∗ y ∈ L∈ T (A; α), a ∗ b ∈ L∈ I (A; β),and u ∗ v ∈ L∈ F (A; γ) UsingEquation (17),wehave AT (x) ∧ AT (y) ≤ α, AI (a) ∧ AI (b) ≤ β,and AF (u) ∨ AF (v) ≥ γ, whichimplythat

AT (x) ≤ α or AT (y) ≤ α,thatis, x ∈ L∈ T (A; α) or y ∈ L∈ T (A; α);

AI (a) ≤ β or AI (b) ≤ β,thatis, a ∈ L∈ I (A; β) or b ∈ L∈ I (A; β); and

AF (u) ≥ γ or AF (v) ≥ γ,thatis, u ∈ L∈ F (A; γ) or v ∈ L∈ F (A; γ).

Hence {x, y}∩ L∈ T (A; α) = ∅, {a, b}∩ L∈ I (A; β) = ∅,and {u, v}∩ L∈ F (A; γ) = ∅ Therefore L∈ T (A; α), L∈ I (A; β),and L∈ F (A; γ) are S-energeticsubsetsof X

Deﬁnition4. ANS A =(AT , AI , AF ) ina BCK/BCI-algebra X iscalledan (∈, ∈)-neutrosophicidealof Xifthefollowingassertionsarevalid: (∀x ∈ X)

(A; α

.(20) Proof. AssumethatEquation (20) isvalid,andlet x ∈ U∈ T (A; α), a ∈ U∈ I (A; β),and u ∈ U∈ F (A; γ) forany x, a, u ∈ X, α, β ∈ (0,1] and γ ∈ [0,1).Then AT (0) ≥ AT (x) ≥ α, AI (0) ≥ AI (a) ≥ β, and AF (0) ≤ AF (u) ≤ γ.Hence0 ∈ U∈ T (A; α),0 ∈ U∈ I (A; β),and0 ∈ U∈ F (A; γ),andthus Equation (18) isvalid.Let x, y, a, b, u, v ∈ X besuchthat x ∗ y ∈ U∈ T (A; αx ), y ∈ U∈ T (A; αy ), a ∗ b ∈ U∈ I (A; βa ), b ∈ U∈ I (A; βb ), u ∗ v ∈ U∈ F (A; γu ),and v ∈ U∈ F (A; γv ) forall αx, αy, βa, βb ∈ (0,1]

 

(∀x

y ∈ X)    AT (

≥

AI (

≥

∧

  

Florentin Smarandache (author and editor) Collected Papers, IX 286

   x ∈ U∈ T (A; α) ⇒ 0 ∈ U∈ T

) x ∈ U∈ I (A; β) ⇒ 0 ∈ U∈ I (A; β) x ∈ U∈ F (A; γ) ⇒ 0 ∈ U∈ F (A; γ)    ,(18) (∀x, y ∈ X)    x ∗ y ∈ U∈ T (A; αx ), y ∈ U∈ T (A; αy ) ⇒ x ∈ U∈ T (A; αx ∧ αy )

∗ y ∈ U∈ I (A; βx ), y ∈ U∈ I (A; βy ) ⇒ x ∈ U∈ I (A; βx ∧ βy )

∗ y ∈ U∈ F (A; γx ), y ∈ U∈ F (A; γy ) ⇒ x ∈ U∈ F (A; γx ∨ γy )

 ,(19) forall α, β, αx, αy, βx, βy ∈ (0,1] and γ, γx, γy ∈ [0,1). Theorem7. ANS A =(AT , AI , AF ) ina BCK/BCI-algebra X isan (∈, ∈)-neutrosophicidealof X ifand onlyifA =(AT , AI , AF ) satisﬁes

AT (x)

AT (x ∗ y) ∧ AT (y)

AI (x)

AI (x ∗ y)

AI (y)

F (0) ≤ AF (x) ≤ AF (x ∗ y) ∨ AF (y)

and γu, γv ∈ [0,1).Then AT (x ∗ y) ≥ αx, AT (y) ≥ αy, AI (a ∗ b) ≥ βa, AI (b) ≥ βb, AF (u ∗ v) ≤ γu, and AF (v) ≤ γv.ItfollowsfromEquation(20)that

AT (x) ≥ AT (x ∗ y) ∧ AT (y) ≥ αx ∧ αy,

AI (a) ≥ AI (a ∗ b) ∧ AI (b) ≥ βa ∧ βb,

AF (u) ≤ AF (u ∗ v) ∨ AF (v) ≤ γu ∨ γv

Hence x ∈ U∈ T (A; αx ∧ αy ), a ∈ U∈ I (A; βa ∧ βb ),and u ∈ U∈ F (A; γu ∨ γv ).Therefore A =(AT , AI , AF ) isan (∈, ∈)-neutrosophicidealof X.

Conversely,let A =(AT , AI , AF ) bean (∈, ∈)-neutrosophicidealof X.Ifthereexists x0 ∈ X suchthat AT (0) < AT (x0),then x0 ∈ U∈ T (A; α) and0 / ∈ U∈ T (A; α),where α = AT (x0).Thisisa contradiction,andthus AT (0) ≥ AT (x) forall x ∈ X.Assumethat AT (x0) < AT (x0 ∗ y0) ∧ AT (y0) for some x0, y0 ∈ X.Taking α := AT (x0 ∗ y0) ∧ AT (y0) impliesthat x0 ∗ y0 ∈ U∈ T (A; α) and y0 ∈ U∈ T (A; α); but x0 / ∈ U∈ T (A; α).Thisisacontradiction,andthus AT (x) ≥ AT (x ∗ y) ∧ AT (y) forall x, y ∈ X Similarly,wecanverifythat AI (0) ≥ AI (x) ≥ AI (x ∗ y) ∧ AI (y) forall x, y ∈ X.Now,suppose that AF (0) > AF (a) forsome a ∈ X.Then a ∈ U∈ F (A; γ) and0 / ∈ U∈ F (A; γ) bytaking γ = AF (a) Thisisimpossible,andthus AF (0) ≤ AF (x) forall x ∈ X.Supposethereexist a0, b0 ∈ X such that AF (a0) > AF (a0 ∗ b0) ∨ AF (b0),andtake γ := AF (a0 ∗ b0) ∨ AF (b0).Then a0 ∗ b0 ∈ U∈ F (A; γ), b0 ∈ U∈ F (A; γ),and a0 / ∈ U∈ F (A; γ),whichisacontradiction.Thus AF (x) ≤ AF (x ∗ y) ∨ AF (y) forall x, y ∈ X.Therefore A =(AT , AI , AF ) satisﬁesEquation(20).

Lemma2. Every (∈, ∈)-neutrosophicidealA =(AT , AI , AF ) ofaBCK/BCI-algebraXsatisﬁes

(∀x, y ∈ X) (x ≤ y ⇒ AT (x) ≥ AT (y), AI (x) ≥ AI (y), AF (x) ≤ AF (y)) .(21)

Proof. Let x, y ∈ X besuchthat x ≤ y.Then x ∗ y = 0,andthus

AT (x) ≥ AT (x ∗ y) ∧ AT (y)= AT (0) ∧ AT (y)= AT (y), AI (x) ≥ AI (x ∗ y) ∧ AI (y)= AI (0) ∧ AI (y)= AI (y), AF (x) ≤ AF (x ∗ y) ∨ AF (y)= AF (0) ∨ AF (y)= AF (y), byEquation(20).Thiscompletestheproof.

Theorem8. ANS A =(AT , AI , AF ) ina BCK-algebra X isan (∈, ∈)-neutrosophicidealof X ifandonlyif A =(AT , AI , AF ) satisﬁes (∀x, y, z ∈ X) 

 

∗

≤ z ⇒     

 

Florentin Smarandache (author and editor) Collected Papers, IX 287

AT (x) ≥ AT (y) ∧ AT (z) AI (x) ≥ AI (y) ∧ AI (z) AF (x) ≤ AF (y) ∨ AF (z) 

(22) Proof. Let A =(AT , AI , AF ) bean (∈, ∈)-neutrosophicidealof X,andlet x, y, z ∈ X besuchthat x ∗ y ≤ z.UsingTheorem 7 andLemma 2,wehave AT (x) ≥ AT (x ∗ y) ∧ AT (y) ≥ AT (y) ∧ AT (z), AI (x) ≥ AI (x ∗ y) ∧ AI (y) ≥ AI (y) ∧ AI (z), AF (x) ≤ AF (x ∗ y) ∨ AF (y) ≤ AF (y) ∨ AF (z).

Conversely,assumethat A =(AT , AI , AF ) satisﬁesEquation (22).Because0 ∗ x ≤ x forall x ∈ X, itfollowsfromEquation(22)that

AT (0) ≥ AT (x) ∧ AT (x)= AT (x),

AI (0) ≥ AI (x) ∧ AI (x)= AI (x),

AF (0) ≤ AF (x) ∨ AF (x)= AF (x),

forall x ∈ X.Because x ∗ (x ∗ y) ≤ y forall x, y ∈ X,wehave

AT (x) ≥ AT (x ∗ y) ∧ AT (y),

AI (x) ≥ AI (x ∗ y) ∧ AI (y),

AF (x) ≤ AF (x ∗ y) ∨ AF (y),

forall x, y ∈ X byEquation (22).ItfollowsfromTheorem 7 that A =(AT , AI , AF ) isan (∈, ∈)-neutrosophicidealof X

Theorem9. If A =(AT , AI , AF ) isan (∈, ∈)-neutrosophicidealofa BCK/BCI-algebra X,thenthelower neutrosophic ∈Φ-subsetsofXareI-energeticsubsetsofXwhere Φ ∈{T, I, F}

Proof. Let x, a, u ∈ X, α, β ∈ (0,1],and γ ∈ [0,1) besuchthat x ∈ L∈ T (A; α), a ∈ L∈ I (A; β), and u ∈ L∈ F (A; γ).UsingTheorem 7,wehave

α ≥ AT (x) ≥ AT (x ∗ y) ∧ AT (y),

β ≥ AI (a) ≥ AI (a ∗ b) ∧ AI (b),

γ ≤ AF (u) ≤ AF (u ∗ v) ∨ AF (v), forall y, b, v ∈ X.Itfollowsthat

AT (x ∗ y) ≤ α or AT (y) ≤ α,thatis, x ∗ y ∈ L∈ T (A; α) or y ∈ L∈ T (A; α);

AI (a ∗ b) ≤ β or AI (b) ≤ β,thatis, a ∗ b ∈ L∈ T (A; β) or b ∈ L∈ T (A; β); and

AF (u ∗ v) ≥ γ or AF (v) ≥ γ,thatis, u ∗ v ∈ L∈ T (A; γ) or v ∈ L∈ T (A; γ).

Hence {y, x ∗ y}∩ L∈ T (A; α), {b, a ∗ b}∩ L∈ I (A; β),and {v, u ∗ v}∩ L∈ F (A; γ) arenonempty, andtherefore L∈ T (A; α), L∈ I (A; β) and L∈ F (A; γ) are I-energeticsubsetsof X

Corollary2. If A =(AT , AI , AF ) isan (∈, ∈)-neutrosophicidealofa BCK/BCI-algebra X,thenthestrong lowerneutrosophic ∈Φ-subsetsofXareI-energeticsubsetsofXwhere Φ ∈{T, I, F}

Proof. Straightforward.

Theorem10. Let (α, β, γ) ∈ ΛT × ΛI × ΛF,where ΛT , ΛI ,and ΛF aresubsetsof [0,1].If A =(AT , AI , AF ) isan (∈, ∈)-neutrosophicidealofaBCK-algebraX,then

(1) the(strong)upperneutrosophic ∈Φ-subsetsofXarerightstablewhere Φ ∈{T, I, F}; (2) the(strong)lowerneutrosophic ∈Φ-subsetsofXarerightvanishedwhere Φ ∈{T, I, F}

Proof. (1)Let x ∈ X, a ∈ U∈ T (A; α), b ∈ U∈ I (A; β),and c ∈ U∈ F (A; γ).Then AT (a) ≥ α, AI (b) ≥ β, and AF (c) ≤ γ.Because a ∗ x ≤ a, b ∗ x ≤ b,and c ∗ x ≤ c,itfollowsfromLemma 2 that AT (a ∗ x) ≥ AT (a) ≥ α, AI (b ∗ x) ≥ AI (b) ≥ β,and AF (c ∗ x) ≤ AF (c) ≤ γ;thatis, a ∗ x ∈ U∈ T (A; α),

Florentin Smarandache (author and editor) Collected Papers, IX 288

b ∗ x ∈ U∈ I (A; β),and c ∗ x ∈ U∈ F (A; γ).Hencetheupperneutrosophic ∈Φ-subsetsof X arerightstable where Φ ∈{T, I, F}.Similarly,thestrongupperneutrosophic ∈Φ-subsetsof X arerightstablewhere Φ ∈{T, I, F}.

(2)Assumethat x ∗ y ∈ L∈ T (A; α), a ∗ b ∈ L∈ I (A; β),and c ∗ d ∈ L∈ F (A; γ) forany x, y, a, b, c, d ∈ X. Then AT (x ∗ y) ≤ α, AI (a ∗ b) ≤ β,and AF (c ∗ d) ≥ γ.Because x ∗ y ≤ x, a ∗ b ≤ a, and c ∗ d ≤ c,itfollowsfromLemma 2 that α ≥ AT (x ∗ y) ≥ AT (x), β ≥ AI (a ∗ b) ≥ AI (a), and γ ≤ AF (c ∗ d) ≤ AF (c);thatis, x ∈ L∈ T (A; α), a ∈ L∈ I (A; β),and c ∈ L∈ F (A; γ).Thereforethelower neutrosophic ∈Φ-subsetsof X arerightvanishedwhere Φ ∈{T, I, F}.Inasimilarway,weknowthat thestronglowerneutrosophic ∈Φ-subsetsof X arerightvanishedwhere Φ ∈{T, I, F}

Deﬁnition5. Let A =(AT , AI , AF ) beaNSina BCK/BCI-algebra X and (α, β, γ) ∈ ΛT × ΛI × ΛF, where ΛT , ΛI ,and ΛF aresubsetsof [0,1].Then (α, β, γ) iscalledaneutrosophicpermeable I-valuefor A =(AT , AI , AF ) ifthefollowingassertionisvalid: (∀x, y ∈ X)    x ∈ U∈ T (A;

Example4. (1)InExample 2, (α, β, γ) isaneutrosophicpermeableI-valueforA =(AT , AI , AF ). (2)Considera BCI-algebra X = {0,1, a, b, c} withthebinaryoperation ∗ thatisgiveninTable 7 (see[14]). Table7. Cayleytableforthebinaryoperation“∗”. *01abc 000 abc 110 abc aaa 0 cb bbbc 0 a cccba 0

LetA =(AT , AI , AF ) beaNSinXthatisgiveninTable 8

Table8. Tabulationrepresentationof A =(AT , AI , AF ) XAT (x) A I (x) AF (x) 00.330.380.77 10.440.480.66 a 0.550.680.44 b 0.660.580.44 c 0.660.680.55

Itisroutinetocheckthat (α, β, γ) ∈ (0.33,1] × (0.38,1] × [0,0.77) isaneutrosophicpermeable I-value forA =(AT , AI , AF ) Lemma3. IfaNS A =(AT , AI , AF ) ina BCK/BCI-algebra X satisﬁestheconditionofEquation (14),then (∀x ∈ X) (AT (0) ≤ AT (x), AI (0) ≤ AI (x), AF (0) ≥ AF (x)) .(24)

Proof. Straightforward. Theorem11. IfaNS A =(AT , AI , AF ) ina BCK-algebra X satisﬁestheconditionofEquation (14), theneveryneutrosophicpermeable I-valuefor A =(AT , AI , AF ) isaneutrosophicpermeable S-valuefor A =(AT , AI , AF )

α) ⇒ AT (x ∗ y) ∨ AT (y) ≥ α, x ∈ U∈ I (A; β) ⇒ AI (x ∗ y) ∨ AI (y) ≥ β, x ∈ U∈ F (A; γ) ⇒ AF (x ∗ y) ∧ AF (y) ≤ γ    .(23)

Smarandache (author and editor) Collected Papers, IX 289

Florentin

Proof. Let (α, β, γ) beaneutrosophicpermeable I-valuefor A =(AT , AI , AF ).Let x, y, a, b, u, v ∈ X besuchthat x ∗ y ∈ U∈ T (A; α), a ∗ b ∈ U∈ I (A; β),and u ∗ v ∈ U∈ F (A; γ).ItfollowsfromEquations(23), (3),(III),and(V)andLemma 3 that

α ≤ AT ((x ∗ y) ∗ x) ∨ AT (x)= AT ((x ∗ x) ∗ y) ∨ AT (x)

= AT (0 ∗ y) ∨ AT (x)= AT (0) ∨ AT (x)= AT (x),

β ≤ AI ((a ∗ b) ∗ a) ∨ AI (a)= AI ((a ∗ a) ∗ b) ∨ AI (a)

= AI (0 ∗ b) ∨ AI (a)= AI (0) ∨ AI (a)= AI (a), and

γ ≥ AF ((u ∗ v) ∗ u) ∧ AF (u)= AF ((u ∗ u) ∗ v) ∧ AF (u)

= AF (0 ∗ v) ∧ AF (u)= AF (0) ∧ AF (u)= AF (u).

Hence AT (x) ∨ AT (y) ≥ AT (x) ≥ α, AI (a) ∨ AI (b) ≥ AI (a) ≥ β, and AF (u) ∧ AF (v) ≤ AF (u) ≤ γ Therefore (α, β, γ) isaneutrosophicpermeable S-valuefor A =(AT , AI , AF )

GivenaNS A =(AT , AI , AF ) ina BCK/BCI-algebra X,anyupperneutrosophic ∈Φ-subsetsof X maynotbe I-energeticwhere Φ ∈{T, I, F},asseeninthefollowingexample. Example5. Considera BCK-algebra X = {0,1,2,3,4} withthebinaryoperation ∗ thatisgiveninTable 9 (see[14]).

Table9. Cayleytableforthebinaryoperation“∗”. *01234 000000 110000 221010 331100 442120

LetA =(AT , AI , AF ) beaNSinXthatisgiveninTable 10.

Table10. Tabulationrepresentationof A =(AT , AI , AF ) XAT (x) A I (x) AF (x) 00.750.730.34 10.530.450.58 20.670.860.34 30.530.560.58 40.460.560.66

Then U∈ T (A;0.6)= {0,2}, U∈ I (A;0.7)= {0,2},and U∈ F (A;0.4)= {0,2}.Because 2 ∈{0,2} and {1,2 ∗ 1}∩{0,2} = ∅,weknowthat {0,2} isnotanI-energeticsubsetofX.

Wenowprovideconditionsfortheupperneutrosophic ∈Φ-subsetstobe I-energeticwhere Φ ∈{T, I, F} Theorem12. Let A =(AT , AI , AF ) beaNSina BCK/BCI-algebra X and (α, β, γ) ∈ ΛT × ΛI × ΛF, where ΛT , ΛI ,and ΛF aresubsetsof [0,1].If (α, β, γ) isaneutrosophicpermeable I-valuefor A =(AT , AI , AF ),thentheupperneutrosophic ∈Φ-subsetsofXareI-energeticsubsetsofXwhere Φ ∈{T, I, F}.

Florentin Smarandache (author and editor) Collected Papers, IX 290

Proof. Let x, a, u ∈ X and (α, β, γ) ∈ ΛT × ΛI × ΛF,where ΛT , ΛI ,and ΛF aresubsetsof [0,1] such that x ∈ U∈ T (A; α), a ∈ U∈ I (A; β),and u ∈ U∈ F (A; γ).Because (α, β, γ) isaneutrosophicpermeable I-valuefor A =(AT , AI , AF ),itfollowsfromEquation(23)that

AT (x ∗ y) ∨ AT (y) ≥ α, AI (a ∗ b) ∨ AI (b) ≥ β,and AF (u ∗ v) ∧ AF (v) ≤ γ

forall y, b, v ∈ X.Hence

AT (x ∗ y) ≥ α or AT (y) ≥ α,thatis, x ∗ y ∈ U∈ T (A; α) or y ∈ U∈ T (A; α);

AI (a ∗ b) ≥ β or AI (b) ≥ β,thatis, a ∗ b ∈ U∈ I (A; β) or b ∈ U∈ I (A; β); and

AF (u ∗ v) ≤ γ or AF (v) ≤ γ,thatis, u ∗ v ∈ U∈ F (A; γ) or v ∈ U∈ F (A; γ)

Hence {y, x ∗ y}∩ U∈ T (A; α), {b, a ∗ b}∩ U∈ I (A; β),and {v, u ∗ v}∩ U∈ F (A; γ) arenonempty, andthereforetheupperneutrosophic ∈Φ-subsetsof X are I-energeticsubsetsof X where Φ ∈{T, I, F}

Theorem13. Let A =(AT , AI , AF ) beaNSina BCK/BCI-algebra X and (α, β, γ) ∈ ΛT × ΛI × ΛF, where ΛT , ΛI ,and ΛF aresubsetsof [0,1].IfA =(AT , AI , AF ) satisfiesthefollowingcondition: (∀x, y ∈ X)   

AT (x) ≤ AT (x ∗ y) ∨ AT (y)

AI (x) ≤ AI (x ∗ y) ∨ AI (y)

AF (x) ≥ AF (x ∗ y) ∧ AF (y)

   ,(25) then (α, β, γ) isaneutrosophicpermeableI-valueforA =(AT , AI , AF ) Proof. Let x, a, u ∈ X and (α, β, γ) ∈ ΛT × ΛI × ΛF,where ΛT , ΛI ,and ΛF aresubsetsof [0,1] such that x ∈ U∈ T (A; α), a ∈ U∈ I (A; β),and u ∈ U∈ F (A; γ).UsingEquation(25),weobtain α ≤ AT (x) ≤ AT (x ∗ y) ∨ AT (y), β ≤ AI (a) ≤ AI (a ∗ b) ∨ AI (b), γ ≥ AF (u) ≥ AF (u ∗ v) ∧ AF (v), forall y, b, v ∈ X.Therefore (α, β, γ) isaneutrosophicpermeable I-valuefor A =(AT , AI , AF ).

CombiningTheorems 12 and 13,wehavethefollowingcorollary.

Corollary3. Let A =(AT , AI , AF ) beaNSina BCK/BCI-algebra X and (α, β, γ) ∈ ΛT × ΛI × ΛF, where ΛT , ΛI ,and ΛF aresubsetsof [0,1].If A =(AT , AI , AF ) satisﬁestheconditionofEquation (25), thentheupperneutrosophic ∈Φ-subsetsofXareI-energeticsubsetsofXwhere Φ ∈{T, I, F}

Deﬁnition6. Let A =(AT , AI , AF ) beaNSina BCK/BCI-algebra X and (α, β, γ) ∈ ΛT × ΛI × ΛF, where ΛT , ΛI ,and ΛF aresubsetsof [0,1].Then (α, β, γ) iscalledaneutrosophicanti-permeable I-valuefor A =(AT , AI , AF ) ifthefollowingassertionisvalid:

(

X)    x ∈ L∈ T (

) ⇒ A

x ∈

∈

  

Florentin Smarandache (author and editor) Collected Papers, IX 291

∀x, y ∈

A; α

T (x ∗ y) ∧ AT (y) ≤ α,

I (A; β) ⇒ AI (x ∗ y) ∧ AI (y) ≤ β,

∈ L∈ F (A; γ) ⇒ AF (x ∗ y) ∨ AF (y) ≥ γ

.(26)

Theorem14. Let A =(AT , AI , AF ) beaNSina BCK/BCI-algebra X and (α, β, γ) ∈ ΛT × ΛI × ΛF, where ΛT , ΛI ,and ΛF aresubsetsof [0,1].If A =(AT , AI , AF ) satisﬁestheconditionofEquation (19), then (α, β, γ) isaneutrosophicanti-permeableI-valueforA =(AT , AI , AF ).

Proof. Let x, a, u ∈ X besuchthat x ∈ L∈ T (A; α), a ∈ L∈ I (A; β),and u ∈ L∈ F (A; γ).Then

AT (x ∗ y) ∧ AT (y) ≤ AT (x) ≤ α,

AI (a ∗ b) ∧ AI (b) ≤ AI (a) ≤ β,

AF (u ∗ v) ∨ AF (v) ≥ AF (u) ≥ γ,

forall y, b, v ∈ X byEquation (20).Hence (α, β, γ) isaneutrosophicanti-permeable I-valuefor A =(AT, AI, AF)

Theorem15. Let A =(AT , AI , AF ) beaNSina BCK/BCI-algebra X and (α, β, γ) ∈ ΛT × ΛI × ΛF, where ΛT , ΛI ,and ΛF aresubsetsof [0,1].If (α, β, γ) isaneutrosophicanti-permeable I-valuefor A =(AT , AI , AF ),thenthelowerneutrosophic ∈Φ-subsetsofXareI-energeticwhere Φ ∈{T, I, F}

Proof. Let x ∈ L∈ T (A; α), a ∈ L∈ I (A; β),and u ∈ L∈ F (A; γ).Then AT (x ∗ y) ∧ AT (y) ≤ α, AI (a ∗ b) ∧ AI (b) ≤ β,and AF (u ∗ v) ∨ AF (v) ≥ γ forall y, b, v ∈ X byEquation(26).Itfollowsthat

AT (x ∗ y) ≤ α or AT (y) ≤ α,thatis, x ∗ y ∈ L∈ T (A; α) or y ∈ L∈ T (A; α);

AI (a ∗ b) ≤ β or AI (b) ≤ β,thatis, a ∗ b ∈ L∈ I (A; β) or b ∈ L∈ I (A; β); and AF (u ∗ v) ≥ γ or AF (v) ≥ γ,thatis, u ∗ v ∈ L∈ F (A; γ) or v ∈ L∈ F (A; γ)

Hence {y, x ∗ y}∩ L∈ T (A; α), {b, a ∗ b}∩ L∈ I (A; β) and {v, u ∗ v}∩ L∈ F (A; γ) arenonempty, andthereforethelowerneutrosophic ∈Φ-subsetsof X are I-energeticwhere Φ ∈{T, I, F}

CombiningTheorems 14 and 15,weobtainthefollowingcorollary.

Corollary4. Let A =(AT , AI , AF ) beaNSina BCK/BCI-algebra X and (α, β, γ) ∈ ΛT × ΛI × ΛF, where ΛT , ΛI ,and ΛF aresubsetsof [0,1].If A =(AT , AI , AF ) satisﬁestheconditionofEquation (19), thenthelowerneutrosophic ∈Φ-subsetsofXareI-energeticwhere Φ ∈{T, I, F}

Theorem16. If A =(AT , AI , AF ) isan (∈, ∈)-neutrosophicsubalgebraofa BCK-algebra X,thenevery neutrosophicanti-permeable I-valuefor A =(AT , AI , AF ) isaneutrosophicanti-permeable S-valuefor A =(AT , AI , AF )

Proof. Let (α, β, γ) beaneutrosophicanti-permeable I-valuefor A =(AT , AI , AF ). Let x, y, a, b, u, v ∈ X besuchthat x ∗ y ∈ L∈ T (A; α), a ∗ b ∈ L∈ I (A; β),and u ∗ v ∈ L∈ F (A; γ).Itfollows fromEquations(26),(3),(III),and(V)andProposition 1 that

α ≥ AT ((x ∗ y) ∗ x) ∧ AT (x)= AT ((x ∗ x) ∗ y) ∧ AT (x)

= AT (0 ∗ y) ∧ AT (x)= AT (0) ∧ AT (x)= AT (x),

β ≥ AI ((a ∗ b) ∗ a) ∧ AI (a)= AI ((a ∗ a) ∗ b) ∧ AI (a)

= AI (0 ∗ b) ∧ AI (a)= AI (0) ∧ AI (a)= AI (a), and

γ ≤ AF ((u ∗ v) ∗ u) ∨ AF (u)= AF ((u ∗ u) ∗ v) ∨ AF (u)

= AF (0 ∗ v) ∨ AF (u)= AF (0) ∨ AF (u)= AF (u).

Florentin Smarandache (author and editor) Collected Papers, IX 292

4. Conclusions

Using the notions of subalgebras and ideals in BCK/BCI-algebras, Jun et al. [8] introduced the notions of energetic subsets, right vanished subsets, right stable subsets, and (anti-)permeable values in BCK/BCI-algebras, as well as investigated relations between these sets. As a more general platform that extends the concepts of classic and fuzzy sets, intuitionistic fuzzy sets, and interval-valued (intuitionistic) fuzzy sets, the notion of NS theory has been developed by Smarandache (see [1,2]) and has been applied to various parts: pattern recognition, medical diagnosis, decision-making problems, and so on (see [3 6]). In this article, we have introduced the notions of neutrosophic permeable S-values, neutrosophic permeable I-values, (∈, ∈)-neutrosophic ideals, neutrosophic anti-permeable S-values, and neutrosophic anti-permeable I-values, which are motivated by the idea of subalgebras (s-values) and ideals (I-values), and have investigated their properties. We have considered characterizations of (∈, ∈)-neutrosophic ideals and have discussed conditions for the lower (upper) neutrosophic ∈Φ-subsets to be S and I-energetic. We have provided conditions for a triple (α, β, γ) of numbers to be a neutrosophic (anti-)permeable S or I-value, and have considered conditions for the upper (lower) neutrosophic ∈Φ-subsets to be right stable (right vanished) subsets. We have established relations between neutrosophic (anti-)permeable S and I-values.

References

1. Smarandache,F. AUnifyingFieldinLogics:NeutrosophicLogic.Neutrosophy,NeutrosophicSet, NeutrosophicProbability;AmericanReserchPress:Rehoboth,NM,USA,1999.

2. Smarandache,F.Neutrosophicset-ageneralizationoftheintuitionisticfuzzyset. Int.J.PureAppl.Math. 2005, 24,287–297.

3. Garg,H.;Nancy.Somenewbiparametricdistancemeasuresonsingle-valuedneutrosophicsetswith applicationstopatternrecognitionandmedicaldiagnosis. Information 2017, 8,126.

4. Garg,H.;Nancy.Non-linearprogrammingmethodformulti-criteriadecisionmakingproblemsunder intervalneutrosophicsetenvironment. Appl.Intell. 2017,doi:10.1007/s10489-017-1070-5.

5. Garg,H.;Nancy.Linguisticsingle-valuedneutrosophicprioritizedaggregationoperatorsandtheir applicationstomultiple-attributegroupdecision-making. J.AmbientIntell.Humaniz.Comput. 2018, doi:10.1007/s12652-018-0723-5.

6. Nancy;Garg,H.Novelsingle-valuedneutrosophicaggregatedoperatorsunderFranknormoperationand itsapplicationtodecision-makingprocess. Int.J.Uncertain.Quantif. 2016, 6,361–375.

Jun,Y.B.Neutrosophicsubalgebrasofseveraltypesin BCK/BCI-algebras. Ann.FuzzyMath.Inform. 2017, 14,75–86.

8. Jun,Y.B.;Ahn,S.S.;Roh,E.H.Energeticsubsetsandpermeablevalueswithapplicationsin BCK/BCI-algebras. Appl.Math.Sci. 2013, 7,4425–4438.

Jun,Y.B.;Smarandache,F.;Bordbar,H.Neutrosophic N -structuresappliedto BCK/BCI-algebras. Informations 2017, 8,128.

10. Jun,Y.B.;Smarandache,F.;Song,S.Z.;Khan,M.Neutrosophicpositiveimplicative N -idealsin BCK-algebras. Axioms 2018, 7,3.

11. Öztürk,M.A.;Jun,Y.B.Neutrosophicidealsin BCK/BCI-algebrasbasedonneutrosophicpoints. J.Int.Math. VirtualInst. 2018, 8,1–17.

12. Song,S.Z.;Smarandache,F.;Jun,Y.B.Neutrosophiccommutative N -idealsin BCK-algebras. Information 2017, 8,130.

13.Huang,Y.S. BCI-Algebra;SciencePress:Beijing,China,2006.

14.Meng,J.;Jun,Y.B. BCK-Algebras;KyungmoonSaCo.:Seoul,Korea,1994.

Hence AT (x) ∧ AT (y) ≤ AT (x) ≤ α, AI (a) ∧ AI (b) ≤ AI (a) ≤ β, and AF(u) ∨ AF(v) ≥ AF(u) ≥ γ Therefore (α, β, γ) is a neutrosophic anti-permeable S-value for A = (AT , AI , AF ).

Florentin Smarandache (author and editor) Collected Papers, IX 293

Left (Right)-Quasi

Neutrosophic Triplet Loops

(Groups) and Generalized BE-Algebras

Xiaohong Zhang, Xiaoying Wu, Florentin Smarandache, Minghao Hu

Xiaohong Zhang, Xiaoying Wu, Florentin Smarandache, Minghao Hu (2018). Left (Right)-Quasi Neutrosophic Triplet Loops (Groups) and Generalized BE-Algebras. Symmetry 10, 241; DOI: 10.3390/sym10070241

Abstract: The new notion of a neutrosophic triplet group (NTG) is proposed by Florentin Smarandache; it is a new algebraic structure different from the classical group. The aim of this paper is to further expand this new concept and to study its application in related logic algebra systems. Some new notions of left (right)-quasi neutrosophic triplet loops and left (right)-quasi neutrosophic triplet groups are introduced, and some properties are presented. As a corollary of these properties, the following important result are proved: for any commutative neutrosophic triplet group, its every element has a unique neutral element. Moreover, some left (right)-quasi neutrosophic triplet structures in BE-algebras and generalized BE-algebras (including CI-algebras and pseudo CI-algebras) are established, and the adjoint semigroups of the BE-algebras and generalized BE-algebras are investigated for the first time.

Keywords: neutrosophic triplet; quasi neutrosophic triplet loop; quasi neutrosophic triplet group; BE-algebra; CI-algebra

1.Introduction

Thesymmetryexistsintherealworld,andgrouptheoryisamathematicaltoolfordescribing symmetry.Atthesametime,inordertodescribethegeneralizedsymmetry,theconceptofgroupis popularizedindifferentways,forexample,thenotionofageneralizedgroupisintroduced(see[1 4]). Recently,F.Smarandache[5,6]introducedanothernewalgebraicstructure,namely:neutrosophic tripletgroup,whichcomesfromthetheoryoftheneutrosophicset(see[7 11]).Asanewextension oftheconceptofgroup,theneutrosophictripletgrouphasattractedtheattentionofmanyscholars, andaseriesofrelatedpapershavebeenpublished[12 15].

Ontheotherhand,inthelasttwentyyears,thenon-classicallogics,suchasvariousfuzzylogics, havemadegreatprogress.Atthesametime,theresearchonnon-classicallogicalgebrasthatare relatedtoithavealsomadegreatachievements[16 26].AsageneralizationofBCK-algebra,H.S.Kim andY.H.Kim[27]introducedthenotionofBE-algebra.Sincethen,somescholarshavestudiedideals (ﬁlters),congruencerelationsofBE-algebras,andvariousspecialBE-algebrashavebeenproposed, theseresearchresultsareincludedintheliterature[28 31]andmonograph[32].In2013and2016, thenewnotionsofpseudoBE-algebraandcommutativepseudoBE-algebrawereintroduced,andsome newpropertieswereobtained[33,34].SimilartoBCI-algebraasageneralizationofBCK-algebra, B.L.MengintroducedtheconceptofCI-algebra,whichisasageneralizationofBE-algebra,andstudied thestructuresandclosedﬁltersofCI-algebras[35 37].Afterthat,theCI-algebrasandtheirrelated algebraicstructures(suchasQ-algebras,pseudoQ-algebras,pseudoCI-algebras,andpseudoBCHalgebras)havebeenextensivelystudied[38 46].

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Thispaperwillcombinetheabovetwodirectionstostudygeneralneutrosophictripletstructures andtherelationshipsbetweenthesestructuresandgeneralizedBE-algebras.Ontheonehand, weintroducevariousgeneralneutrosophictripletstructures,suchas(l l)-type,(l r)-type,(r l)-type, (r r)-type,(l lr)-type,(r lr)-type,(lr l)-type,and(lr r)-typequasineutrosophictripletloops(groups), andinvestigatetheirbasicproperties.Moreover,wegetanimportantcorollary,namely:thatforany commutativeneutrosophictripletgroup,itseveryelementhasauniqueneutralelement.Ontheother hand,wefurtherstudythepropertiesof(pseudo)BE-algebrasand(pseudo)CI-algebras,andthe generalneutrosophictripletstructuresthatarecontainedinaBE-algebra(CI-algebra)andpseudo BE-algebra(pseudoCI-algebra).Moreover,fortheﬁrsttime,weintroducetheconceptsofadjoint semigroupsofBE-algebrasandgeneralizedBE-algebras(includingCI-algebras,pseudoBE-algebras, andpseudoCI-algebras)anddiscusssomeinterestingtopics.

2.BasicConcepts

Deﬁnition1. ([5,6])LetNbeasettogetherwithabinaryoperation*.Then,Niscalledaneutrosophictriplet setif,foranya∈N,thereexistsaneutralof‘a’,calledneut(a),andanoppositeof‘a’,calledanti(a),withneut(a) andanti(a),belongingtoN,suchthat:

a * neut(a)= neut(a)* a = a;

a * anti(a)= anti(a)* a = neut(a).

Itshouldbenotedthat neut(a)and anti(a)maynotbeuniquehereforsome a∈N.Wecall(a, neut(a), and anti(a))aneutrosophictripletforthedetermined neut(a)and anti(a).

Remark1. Intheoriginaldeﬁnition,theneutralelementisdifferentfromtheunitelementinthetraditional grouptheory.Theabovedeﬁnitionofthispapertakesawaysuchrestriction,pleaseseetheRemark3inRef.[12].

Deﬁnition2. ([5,6,13])Let(N,*)beaneutrosophictripletset.

(1)If*iswell-deﬁned,thatis,foranya,b ∈ N,onehasa*b ∈ N.Then,Niscalledaneutrosophictripletloop. (2)IfNisaneutrosophictripletloop,and*isassociative,thatis,(a*b)*c=a*(b*c)foralla,b,c ∈ N.Then, Niscalledaneutrosophictripletgroup.

(3) IfNisaneutrosophictripletgroup,and*iscommutative,thatis,a*b=b*aforalla,b ∈ N.Then,Nis calledacommutativeneutrosophictripletgroup.

Deﬁnition3. ([27,35,41,42])ACI-algebra(dualQ-algebra)isanalgebra(X; →,1)oftype(2,0),satisfying thefollowingconditions:

(i)x → x=1, (ii)1 → x=x, (iii)x → (y → z)=y → (x → z),forallx,y,z ∈ X.

ACI-algebra(X; →,1)iscalledaBE-algebra,ifitsatisﬁesthefollowingaxiom: (iv)x → 1=1,forallx ∈ X.

ACI-algebra(X; →,1)iscalledadualBCH-algebra,ifitsatisﬁesthefollowingaxiom: (v)x → y=y→x=1 ⇒ x=y.

Abinaryrelation ≤ onCI-algebra(BE-algebra) X, isdeﬁnedby x ≤ y if,andonlyif, x → y =1.

Florentin Smarandache (author and editor) Collected Papers, IX 295

Deﬁnition4. ([33,43,45])Analgebra(X; →, ,1)oftype(2,2,0)iscalledadualpseudoQ-algebraif,forall x,y,z ∈ X,itsatisﬁesthefollowingaxioms:

(dpsQ1)x → x=x x=1, (dpsQ2)1 → x=1 x=x, (dpsQ3)x → (y z)=y (x → z).

AdualpseudoQ-algebraXiscalledapseudoCI-algebra,ifitsatisﬁesthefollowingcondition: (psCI)x → y=1 ⇔ x y=1.

ApseudoCI-algebraXiscalledapseudoBE-algebra,ifitsatisﬁesthefollowingcondition: (psBE)x → 1=x 1=1,forallx ∈ X.

ApseudoCI-algebraXiscalledapseudoBCH-algebra,ifitsatisﬁesthefollowingcondition: (psBCH)x → y=y x=1 ⇒ x=y.

Inadualpseudo-Qalgebra,onecandeﬁnethefollowingbinaryrelations:

x ≤→ y ⇔ x → y =1. x ≤ y ⇔ x y =1.

Obviously,adualpseudo-Qalgebra X isapseudoCI-algebraif,andonlyif, ≤→ = ≤ 3.VariousQuasiNeutrosophicTripletLoops(Groups)

Deﬁnition5. LetNbeasettogetherwithabinaryoperation*(thatis,(N,*)bealoop)anda ∈ N.

(1) Ifexistb,c ∈ N,suchthata*b=aanda*c=b,thenaiscalledanNT-elementwith(r-r)-property; (2) Ifexistb,c ∈ N,suchthata*b=aandc*a=b,thenaiscalledanNT-elementwith(r-l)-property; (3) Ifexistb,c ∈ N,suchthatb*a=aandc*a=b,thenaiscalledanNT-elementwith(l-l)-property; (4) Ifexistb,c ∈ N,suchthatb*a=aanda*c=b,thenaiscalledanNT-elementwith(l-r)-property; (5)Ifexistb,c ∈ N,suchthata*b=b*a=aandc*a=b,thenaiscalledanNT-elementwith(lr-l)-property; (6)Ifexistb,c ∈ N,suchthata*b=b*a=aanda*c=b,thenaiscalledanNT-elementwith(lr-r)-property; (7)Ifexistb,c ∈ N,suchthatb*a=aanda*c=c*a=b,thenaiscalledanNT-elementwith(l-lr)-property; (8)Ifexistb,c ∈ N,suchthata*b=aanda*c=c*a=b,thenaiscalledanNT-elementwith(r-lr)-property; (9) Ifexistb,c ∈ N,suchthata*b=b*a=aanda*c=c*a=b,thenaiscalledanNT-elementwith (lr-lr)-property.

Itiseasytoverifythat,(i)if a isanNT-elementwith(l lr)-property,then a isanNT-elementwith (l l)-propertyand(l r)-property;if a isanNT-elementwith(lr l)-property,then a isanNT-elementwith (l l)-propertyand(r l)-property;andsoon;(ii)aneutrosophictripletloop(N,*)isaneutrosophictriplet groupif,andonlyif,everyelementin N isanNT-elementwith(lr lr)-property;(iii)if*iscommutative, thentheabovepropertiescoincide.Moreover,thefollowingexampleshowsthat(r l)-propertyand (r r)-propertycannotinferto(r lr)-property,and(r r)-propertyand(l lr)-propertycannotinferto (lr lr)-property.

Example1. LetN={a,b,c,d}.Theoperation*onNisdeﬁnedasTable 1.Then,(N,*)isaloop,andaisan NT-elementwith(lr-lr)-property;bisanNT-elementwith(lr-r)-property;cisanNT-elementwith(r-l)-property and(r-r)-property,butcisnotanNT-elementwith(r-lr)-property;anddisanNT-elementwith(r-r)-property and(l-lr)-property,butdisnotanNT-elementwith(lr-lr)-property.

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Table1. Neutrosophictriplet(NT)-elementsinaloop. * abcd a aaad b cabc c cbda d adba

Deﬁnition6. Let(N,*)bealoop(semi-group).IfforeveryelementainN,aisanNT-elementwith (r-r)-property,then(N,*)iscalled(r-r)-quasineutrosophictripletloop(group).Similarly,ifforeveryelement ainN,aisanNT-elementwith(r-l)-,(l-l)-,(l-r)-,(lr-l)-,(lr-r)-,(l-lr)-,(r-lr)-property,then(N,*)iscalled (r-l)-,(l-l)-,(l-r)-,(lr-l)-,(lr-r)-,(l-lr)-,(r-lr)-quasineutrosophictripletloop(group),respectively.Allofthese generalizedneutrosophictripletloops(groups)arecollectivelyknownasquasineutrosophictripletloops(groups).

Remark2. Forquasineutrosophictripletloops(groups),wewillusethenotationslikeneutrosophictripletloops (groups),forexample,todenotea(r-r)-neutralof‘a’byneut(r-r)(a),denotea(r-r)-oppositeof‘a’byanti(r-r)(a), where‘a’isanNT-elementwith(r-r)-property.Ifneut(r-r)(a)andanti(r-r)(a)arenotunique,thendenotetheset ofall(r-r)-neutralof‘a’by{neut(r-r)(a)},denotethesetofall(r-r)-oppositeof‘a’by{anti(r-r)(a)}.

Fortheloop(N,*)inExample1,wecanverifythat(N,*)isa(r r)-quasineutrosophictripletloop, andwehavethefollowing:

neut(r r)(a)= a, anti(r r)(a)= a; neut(r r)(b)= c,{anti(r r)(b)}={a,d}; neut(r r)(c)= a, anti(r r)(c)= d; neut(r r)(d)= b, anti(r r)(d)= c.

Theorem1. If(N,*)isa(l-lr)-quasineutrosophictripletgroup,then(N,*)isaneutrosophictripletgroup. Moreover,if(N,*)isa(r-lr)-quasineutrosophictripletgroup,then(N,*)isaneutrosophictripletgroup.

Proof. Supposethat(N,*)isa(l lr)-quasineutrosophictripletgroup.Forany a ∈ N,byDeﬁnitions5 and6,wehavethefollowing:

neut(l lr)(a)* a = a, anti(l lr)(a)* a = a * anti(l lr)(a)= neut(l lr)(a).

Here, neut(l lr)(a) ∈ {neut(l lr)(a)}, anti(l lr)(a) ∈ {anti(l lr)(a)}.Applyingassociativelawweget thefollowing:

a * neut(l lr)(a)= a *(anti(l lr)(a)* a)=(a * anti(l lr)(a))* a = neut(l lr)(a)* a = a.

Thismeansthat neut(l lr)(a)isarightneutralof‘a’.Fromthearbitrarinessof a,itisknownthat (N,*)isaneutrosophictripletgroup.

Anotherresultcanbeprovedsimilarly. Theorem2. Let(N,*)bea(r-lr)-quasineutrosophictripletgroupsuchthat: (s*p)*a=a*(s*p), ∀ s ∈ {neut(r-lr)(a)}, ∀ p ∈ {anti(r-lr)(a)}.

Then, (1) foranya ∈ N,s ∈ {neut(r-lr)(a)} ⇒ s*s=s. (2) foranya ∈ N,s,t ∈{neut(r-lr)(a)} ⇒ s*t=t. (3) when*iscommutative,foranya ∈ N,neut(r-lr)(a)isunique.

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Proof. (1)Assume s∈{neut(r lr)(a)},then a * s = a,andexist p ∈ N, suchthat p * a = a *p = s.Thus, (s * p)* a = s *(p * a)= s * s,

a *(s * p)=(a * s)* p = a * p=s.

Accordingtothehypothesis,(s * p)* a = a *(s * p),itfollowsthat s * s=s.

(2)Assume s,t∈{neut(r lr)(a)},then a * s = a, a * t = a,andexist p,q ∈ N, suchthat p * a = a * p = s,q * a = a * q = t.Thus, (s * q)* a = s *(q * a)= s * t,

a *(s * q)=(a * s)* q = a * q=t.

Accordingtothehypothesis,(s * p)* a = a *(s * p),itfollowsthat s * t=t.

(3)Suppose a ∈ N,s,t∈{neut(r lr)(a)}.ApplyingTheorem(2)to s and t wehave s * t=t. Moreover, applyingTherorem(2)to t and s wehave t * s=s. Hence,when*iscommutative, s * t=t * s. Therefore, s = t,thatis, neut(r lr)(a)isunique.

Corollary1. Let(N,*)beacommutativeneutrosophictripletgroup.Thenneut(a)isuniqueforanya ∈ N.

Proof. Sinceallneutrosophictripletgroupsare (r-lr)-quasineutrosophictripletgroups,and*is commutative,thentheassumptionconditionsinTheorem2arevalidfor N,soapplyingTheorem2(3), wegetthat neut(a)isuniqueforany a ∈ N.

Thefollowingexamplesshowthattheneutralelementmaybenotuniqueintheneutrosophic tripletloop.

Example2. LetN={1,2,3}.Deﬁnebinaryoperation*onNasfollowingTable 2.Then,(N,*)isacommutative neutrosophictripletloop,and{neut(1)}={1,2}.Since(1*3)*3 = 1*(3*3),so(N,*)isnotaneutrosophic tripletgroup

Table2. Commutativeneutrosophictripletloop.

Example3. LetN={1,2,3,4}.Deﬁnebinaryoperation*onNasfollowingTable 3.Then,(N,*)isa neutrosophictripletloop,and{neut(4)}={2,3}.Since(4*1)*1 = 4*(1*1),so(N,*)isnotaneutrosophic tripletgroup

Table3. Non-commutativeneutrosophictripletloop.

* 123 1 112 2 123 3 233

Florentin Smarandache (author and editor) Collected Papers, IX 298

* 1234 1 3113 2 4224 3 1334 4 3442

4.QuasiNeutrosophicTripletStructuresinBE-AlgebrasandCI-Algebras

FromthedeﬁnitionofBE-algebraandCI-algebra(seeDeﬁnition3),wecanseethat‘1’isaleft neutralelementofeveryelement,thatis,BE-algebrasandCI-algebrasaredirectlyrelatedtoquasi neutrosophictripletstructures.Thissectionwillrevealthevariousinternalconnectionsamongthem.

4.1.BE-Algebras(CI-Algebras)and(l-l)-QuasiNeutrosophicTripletLoops

Theorem3. Let(X; →,1)beaBE-algebra.Then(X, →)isa(l-l)-quasineutrosophictripletloop.And,when |X|>1,(X, →)isnota(lr-l)-quasineutrosophictripletloopwithneutralelement1

Proof. ByDeﬁnition3,forall x ∈ X,1 → x=x and x → x= 1.AccordingDeﬁnition6,weknowthat (X, →)isa(l l)-quasineutrosophictripletloop,suchthat: 1 ∈ {neut(l l)(x)}, x ∈ {anti(l l)(x)},forany x ∈ X.

If |X| >1,thenexist x ∈ X,suchthat x = 1.UsingDeﬁnition3(iv), x → 1 =1 = x,thismeansthat 1isnotarightneutralelementof x.Hence,(X, →)isnota (lr-l)-quasineutrosophictripletloopwith neutralelement1.

Example4. LetX={a,b,c,1}.Deﬁnebinaryoperation*onNasfollowingTable 4.Then,(X; →,1)isa BE-algebra,and(X, →)isa(l-l)-quasineutrosophictripletloop,suchthat:

{neut(l-l)(a)}={1},{anti(l-l)(a)}={a,c};{neut(l-l)(b)}={1},{anti(l-l)(b)}={b,c}; {neut(l-l)(c)}={1},{anti(l-l)(c)}={c};{neut(l-l)(1)}={1},{anti(l-l)(1)}={1}.

Table4. BE-algebraand(l l)-quasineutrosophictripletloop(1). → abc1 a 1bb1 b a1a1 c 1111 1 abc1

Example5. LetX={a,b,c,1}.Deﬁnebinaryoperation*onNasfollowingTable 5.Then,(X; →,1)isa BE-algebra,and(X, →)isa(l-l)-quasineutrosophictripletloopsuchthat:

{neut(l-l)(a)}={1},{anti(l-l)(a)}={a};{neut(l-l)(b)}={1},{anti(l-l)(b)}={b}; {neut(l-l)(c)}={1},{anti(l-l)(c)}={c};{neut(l-l)(1)}={1},{anti(l-l)(1)}={1}.

Table5. BE-algebraand(l l)-quasineutrosophictripletloop(2).

→ abc1 a 1bc1 b a1c1 c ab11 1 abc1

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Deﬁnition7. ([36])Let(X; →,1)beaCI-algebraanda ∈ X.Ifforanyx ∈X,a → x=1impliesa=x,thenais calledanatominX.DenoteA(X)={a ∈ X|aisanatominX},itiscalledthesingularpartofX.ACI-algebra (X; →,1)issaidtobesingularifeveryelementofXisanatom.

Lemma1. ([35 37])If(X; →,1)isaCI-algebra,thenforallx,y ∈ X:

(1) x → ((x → y) → y)=1, (2) 1 → x=1(orequivalently,1 ≤ x)impliesx=1, (3) (x → y) → 1=(x → 1) → (y → 1).

Lemma2. ([36])Let(X; →,1)beaCI-algebra.Ifa,b ∈ XareatomsinX,thenthefollowingaretrue:

(1) a=(a → 1) → 1, (2) (a → b) → 1=b → a, (3) ((a → b) → 1) → 1=a → b, (4) foranyx ∈ X,(a → x) → (b → x)=b → a, (5) foranyx ∈ X,(a → x) → b=(b → x) → a, (6) foranyx ∈ X,(a → x) → (y → b)=(b → x) → (y → a).

Deﬁnition8. Let(X; →,1)beaCI-algebra.Ifforanyx ∈X,x → 1=x,then(X; →,1)issaidtobeastrong singular.

Proposition1. If(X; →,1)isastrongsingularCI-algebra.Then(X; →,1)isasingularCI-algebra.

Proof. Forany x ∈ X,assumethat a → x= 1,where a ∈ X.ByDeﬁnition8,wehave x → 1 =x, a → 1 = a.Hence,applyingDeﬁnition3, a = a → 1 =a → (x → x)= x → (a → x)= x → 1= x

ByDeﬁnition7, x isanatom.Therefore,(X; →,1)issingularCI-algebra.

Proposition2. Let(X; →,1)beaCI-algebra.Then(X; →,1)isastrongsingularCI-algebraif,andonlyif, (X; →,1)isanassociativeBCI-algebra.

Proof. Obviously,everyassociativeBCI-algebraisastrongsingularCI-algebra(see[36]andProposition 1inRef.[12]).

Assumethat(X; →,1)isastrongsingularCI-algebra.

(1) Forany x,y ∈ X,if x → y=y → x= 1,then,byDeﬁnitions8and3,wehavethefollowing: x=x → 1= x → (y → x)= y → (x → x)= y → 1= y.

(2) Forany x,y,z ∈ X,byProposition1andLemma2(4),wecangetthefollowing: (y → z) → ((z → x) → (y → x))=(y → z) → (y → z)=1.

CombiningProof(1)and(2),weknowthat(X; →,1)isaBCI-algebra.Fromthis,applying Deﬁnition8andProposition1inRef.[12],(X; →,1)isanassociativeBCI-algebra.

Theorem4. Let(X; →,1)beaCI-algebra.Then,(X, →)isa(l-l)-quasineutrosophictripletloop.Moreover, (X, →)isaneutrosophictripletgroupif,andonlyif,(X; →,1)isastrongsingularCI-algebra(associative BCI-algebra).

Florentin Smarandache (author and editor) Collected Papers, IX 300

Proof. ItissimilartotheproofofTheorem3,andweknowthat(X, →)isa(l l)-quasineutrosophic tripletloop.

If(X; →,1)isastrongsingularCI-algebra,usingProposition2,(X; →,1)isanassociative BCI-algebra.Hence, → isassociativeandcommutative,itfollowsthat(X, →)isaneutrosophic tripletgroup.

X, →)isaneutrosophictripletgroup,then → isassociative,thus

ByDeﬁnition8weknowthat(X; →,1)isastrongsingularCI-algebra.

Example6. LetX={a,b,c,d,e,1}.Deﬁneoperation → onX,asfollowingTable 6.Then,(X; →,1)isa CI-algebra,and(X, →)isa(l-l)-quasineutrosophictripletloop,suchthat {neut(l-l)(a)}={1},{anti(l-l)(a)}={a,b};{neut(l-l)(b)}={1},{anti(l-l)(b)}={a,b,c}; {neut(l-l)(c)}={1},{anti(l-l)(c)}={c,d,e};{neut(l-l)(d)}={1},{anti(l-l)(d)}={d,e}; {neut(l-l)(e)}={1},{anti(l-l)(e)}={d,e};{neut(l-l)(1)}={1},{anti(l-l)(1)}={1}.

Table6. CI-algebraand(l l)-quasineutrosophictripletloop. → abcde1 a 11ccc1 b 11ccc1 c d11abc d cc111c e cc111c 1 abcde1

4.2.BE-Algebras(CI-Algebras)andTheirAdjointSemi-Groups

I.Fleischer[16]studiedtherelationshipbetweenBCK-algebrasandsemigroups,andW. Huang[17]studiedthecloseconnectionbetweentheBCI-algebrasandsemigroups.Inthissection, wehavestudiedtheadjointsemigroupsoftheBE-algebrasandCI-algebras,andwillgivesome interestingexamples.

ForanyBE-algebraorCI-algebra(X; →,1),andanyelement a in X,weuse pa todenotethe self-mapof X deﬁnedbythefollowing: pa: X → X; → a → x,forall x ∈ X.

Theorem5. Let(X; →,1)beaBE-algebra(orCI-algebra),andM(X)bethesetofﬁniteproductspa * *pb ofself-mapofXwitha, ,b ∈X,where*representsthecompositionoperationofmappings.Then,(M(X),*)is acommutativesemigroupwithidentityp1

Proof. Sincethecompositionoperationofmappingssatisﬁestheassociativelaw,(M(X),*)isa semigroup.Moreover,since p1: X→X → 1 → x,forall x ∈ X.

ApplyingDeﬁnition3(ii),wegetthat p1(x)=x forany x∈X. Hence, p1 *m=p1 *m=m forany m∈M(X).

Conversely,if(

x → 1= x → (x → x)=(x → x) → x= 1 → x = x.

Florentin Smarandache (author and editor) Collected Papers, IX 301

Forany a,b∈X,usingDeﬁnition3(iii)wehave(∀x∈X)thefollowing: (pa * pb)(x)= pa(b → x)= a → (b → x)= b → (a → x)= pb(a → x)=(pb * pa)(x). Therefore,(M(X),*)isacommutativesemigroupwithidentity p1. Now,wecall(M(X),*)theadjointsemigroupof X. Example7. LetX={a,b,c,1}.Deﬁneoperation → onX,asfollowingTable 7.Then,(X; →,1)isa BE-algebra,and pa: X → X; a → 1, b → 1, c → 1,1 → 1.Itisabbreviatedto pa =(1,1,1,1). pb: X → X; a → c, b → 1, c → a,1 → 1.Itisabbreviatedto pb =(c, 1,a, 1). pc: X → X; a → 1, b → 1, c → 1,1 → 1.Itisabbreviatedto pc =(1,1,1,1). p1: X → X; a → a, b → b, c → c,1 → 1.Itisabbreviatedto p1 =(a,b,c, 1).

Wecanverifythat pa * pa = pa, pa * pb = pa, pa * pc = pa; pb * pb =(a, 1,c, 1), pb * pc = pc = pa; pa * (pb * pb)= pa, pb *(pb * pb)= pb, pc *(pb * pb)= pc = pa.Denote pbb =pb * pb =(a, 1,c, 1),then M(X)={pa, pb, pbb, p1},anditsCayleytableisTable 8.Obviously,(M(X),*)isacommutativeneutrosophictriplet groupand

neut(pa) =pa,anti(pa) =pa;neut(pb) =pbb,anti(pb) =pb;neut(pbb) =pbb,anti(pbb) =pbb;neut(p1) =p1,anti(p1) =p1

Table7. BE-algebra. → abc1 a 1111 b c1a1 c 1111 1 abc1

Table8. AdjointsemigroupoftheaboveBE-algebra. * pa pb pbb p1 pa pa pa pa pa pb pa pbb pb pb pbb pa pb pbb pbb p1 pa pb pbb p1

Example8. LetX={a,b,1}.Deﬁneoperation → onX,asfollowingTable 9.Then,(X; →,1)isaCI-algebra,and pa

b. Itisabbreviatedto pa =(1,a,b). pb: X → X; a → b, b

1,1 → a. Itisabbreviatedto pb =(b, 1,a). p1: X → X; a → a, b → b,1 → 1.Itisabbreviatedto p1 =(a,b, 1). Wecanverifythat pa * pa = pb, pa * pb = p1; pb * pb = pa.Then M(X)={pa, pb, p1}anditsCayleytable isTable 10.Obviously,(M(X),*)isacommutativegroupwithidentity p1 and(pa) 1 = pb,(pb) 1 = pa

Florentin Smarandache (author and editor) Collected Papers, IX 302

: X → X; a → 1, b → a

→

Table9. CI-algebra. → ab1 a 1ab b b1a 1 ab1

Table10. AdjointsemigroupoftheaboveCI-algebra. * pa pb p1 pa pb p1 pa pb p1 pa pb p1 pa pb p1

Theorem6. Let(X; →,1)beasingularCI-algebra,andM(X)betheadjointsemigroup.Then(M(X),*)isa commutativegroupwithidentityp1,whereM(X)={pa |a ∈X}and|M(X)|=|X|.

Proof. (1)First,weprovethatforanysingularCI-algebra, a → (b → x)=((a → 1) → b) → x, ∀ a, b, x ∈ X Infact,byDeﬁnition7andLemma2,wehavethefollowing:

((a → 1) → b) → x = ((a → 1) → b) → ((x → 1) → 1) = (x → 1) → (((a → 1) → b) → 1) = (x → 1) → (((a → 1) → 1) → (b → 1)) = (x →1) → (a → (b → 1)) = a → ((x → 1) → (b → 1)) = a → (b → x).

(2)Second,weprovethatforanysingularCI-algebra, a = b ⇒ pa = pb, ∀ a, b ∈ X. Assume pa = pb,a,b ∈ X.Then,forall x in X, pa(x)= pb(x).Hence, a → b = pa(b)= pb(b)= b → b =1.

Fromthis,applyingLemma2(1)and(6)weget a =(a → 1) → 1=(a → 1) → (a → b)=(b → 1) → (a → a)=(b → 1) → 1= b

(3)UsingLemma2(1),weknowthatforany a,b ∈ X, thereexist c ∈ X, suchthat pa * pb = pc, where c =(a → 1) → b.Thismeansthat M(X) ⊆ {pa|a ∈ X}.Bythedeﬁnitionof M(X),{pa a ∈ X} ⊆ M(X).Hence, M(X) = {pa|a ∈ X}.

(4)UsingLemma2(2)and(3),weknowthat |M(X)|=|X|.

5.QuasiNeutrosophicTripletStructuresinPseudoBE-AlgebrasandPseudoCI-Algebras

LiketheaboveSection 4,wecandiscusstherelationshipsbetweenpseudoBE-algebras(pseudo CI-algebras)andquasineutrosophictripletstructures.Thissectionwillgivesomerelatedresultsand examples,butpartofthesimpleproofswillbeomitted.

Florentin Smarandache (author and editor) Collected Papers, IX 303

5.1.PseudoBE-Algebras(PseudoCI-Algebras)and(l-l)-QuasiNeutrosophicTripletLoops

Theorem7. Let(X; →, ,1)bepseudoBE-algebra.Then(X, →)and(X, )are(l-l)-quasineutrosophic tripletloops.And,when|X|>1,(X, →)and(X, )arenot(lr-l)-quasineutrosophictripletloopswithneutral element1

Example9. LetX={a,b,c,1}.Deﬁneoperations → and onXasfollowing Tables 11 and 12.Then,(X; →, ,1)isapseudoBE-algebra,and(X, →)and(X, )are(l-l)-quasineutrosophictripletloops

Table11. PseudoBE-algebra(1). → abc1 a 11b1 b a1c1 c 1111 1 abc1

Table12. PseudoBE-algebra(2). abc1 a 11a1 b a1a1 c 1111 1 abc1

Deﬁnition9. ([44,46])LetabeanelementofapseudoCI-algebra(X; →, ,1).aissaidtobeanatominXif foranyx ∈ X,a → x=1impliesa=x.

ApplyingtheresultsinRef.[44 46]wehavethefollowingpropositions(theproofsareomitted).

Proposition3. If(X; →, ,1)isapseudoCI-algebra,thenforallx,y ∈X

(1) x ≤ (x → y) y,x ≤ (x y) → y, (2) x ≤ y → z ⇔ y ≤ x z, (3) (x → y) → 1=(x → 1) (y 1),(x y) 1=(x 1) → (y → 1), (4) x → 1=x 1, (5) x ≤ yimpliesx → 1=y → 1.

Proposition4. Let(X; →, ,1)beapseudoCI-algebra.Ifa,b ∈XareatomsinX,thenthefollowingaretrue:

(1) a=(a → 1) → 1, (2) foranyx ∈ X,(a → x) x=a,(a x) → x=a, (3) foranyx ∈ X,(a → x) 1=x → a,(a x) → 1=x a, (4) foranyx ∈ X,x → a=(a → 1) (x → 1),x a=(a 1) → (x 1).

Deﬁnition10. ApseudoCI-algebra(X; →, ,1)issaidtobesingularifeveryelementofXisanatom. ApseudoCI-algebra(X; →, ,1)issaidtobestrongsingularifforanyx ∈X,x → 1=x=x 1.

Proposition5. If(X; →, ,1)isastrongsingularpseudoCI-algebra.Then(X; →, ,1)issingular.

Florentin Smarandache (author and editor) Collected Papers, IX 304

Proof. Forany x ∈ X,assumethat a → x= 1,where a ∈ X.ItfollowsfromDeﬁnition10,

x → 1=x=x 1,a → 1=a=a 1.

Hence,applyingDeﬁnition4andProposition3,

a=a → 1=a → (x x)=x (a → x)=x 1=x. ByDeﬁnition9, x isanatom.Therefore, (X; →, ,1) issingularpseudoCI-algebra.

ApplyingTheorem3.11inRef.[46],wecangetthefollowing:

Lemma3. Let(X; →, ,1)beapseudoCI-algebra.Thenthefollowingstatementsareequivalent:

(1) x → (y → z)=(x → y) → z,forallx,y,zinX; (2) x → 1=x=x 1,foreveryxinX; (3) x → y=x y=y → x,forallx,yinX;

(4) x (y z)=(x y) z,forallx,y,zinX.

Proposition6. Let(X; →, ,1)beapseudoCI-algebra.Then(X; →, ,1)isastrongsingularpseudo CI-algebraif,andonlyif, → = and(X; →,1)isanassociativeBCI-algebra.

Proof. WeknowthateveryassociativeBCI-algebraisastrongsingularpseudoCI-algebra.

Now,supposethat(X; →,1)isastrongsingularpseudoCI-algebra.ByDeﬁnition10andLemma3 (3), x → y=x y, ∀x,y ∈ X.Thatis, → = . Hence,(X; →,1)isastrongsingularCI-algebra.Itfollows that(X; →,1)isanassociativeBCI-algebra(usingProposition2).

Theorem8. Let(X; →, ,1)beapseudoCI-algebra.Then(X, →)and(X, )are(l-l)-quasineutrosophic tripletloops.Moreover,(X, →)and(X, )areneutrosophictripletgroupsif,andonlyif,(X; →, ,1)isa strongsingularpseudoCI-algebra(associativeBCI-algebra).

Proof. ApplyingLemma3,andtheproofisomitted.

5.2.PseudoBE-Algebras(PseudoCI-Algebras)andTheirAdjointSemi-Groups

ForanypseudoBE-algebraorpseudoCI-algebra(X; →, ,1)aswellasanyelement a in X, weuse pa → and pa todenotetheself-mapof X,whichisdeﬁnedbythefollowing: pa →: X → X; → a → x,forall x ∈ X. pa : X → X; → a x,forall x ∈ X.

Theorem9. Let(X; →, ,1)beapseudoBE-algebra(orpseudoCI-algebra),and M→(X)={finiteproductspa → * *pb → ofself-mapofX|a, ,b ∈ X}, M (X)={finiteproductspa * *pb ofself-mapofX|a, ,b ∈ X}, M(X)={finiteproductspa → (orpa )* *pb → (orpb )ofself-mapofX|a, ,b ∈ X}, where*representsthecompositionoperationofmappings.Then(M→(X),*),(M (X),*),and(M(X),*) areallsemigroupswiththeidentity p1 = p1 → = p1

Florentin Smarandache (author and editor) Collected Papers, IX 305

Proof. ItissimilartoTheorem5.

Now,wecall(M→(X),*),(M (X),*), and (M(X),*)theadjointsemigroupsof X Example10. LetX={a,b,c,1}.Deﬁneoperations → and onXasfollowing Tables 13 and 14.Then,(X; →, ,1)isapseudoBE-algebra,and pa → =(1, b, b,1), pb → =(a,1, c,1), pc → =(1,1,1,1), p1 → =(a, b, c,1).

Wecanverifythefollowing: pa → * pa → = pa → , pa → * pb → =(1,1, b,1), pa → * pc → = pc → , pa → * p1 → = pa →; pb → * pa → = pc → , pb → * pb → = pb → , pb → * pc → = pc → , pb → * p1 → = pb →; pc → * pa → = pc → , pc → * pb → =pc → , pc → * pc → =pc → , pc → * p1 → = pc →; p1 → * pa → = pa → , p1 → * pb → = pb → , p1 → * pc → = pc → , p1 → * p1 → = p1 → Denote pab → =pa → * pb → =(1,1, b, 1),then pab → * pa → = pc → , pab → * pb → = pab → , pab → * pab → = p→ , pab → * pc → = pc →.Hence, M→(X)={pa → , pb → , pab → , pc → , p1 →}anditsCayleytableisTable 15 Obviously,(M→(X),*)isanon-commutativesemigroup,butitisnotaneutrosophictripletgroup.

Table13. PseudoBE-algebraandadjointsemigroups(1). → abc1 a 1bb1 b a1c1 c 1111 1 abc1

Table14. PseudoBE-algebraandadjointsemigroups(2). abc1 a 1bc1 b a1a1 c 1111 1 abc1

Table15. PseudoBE-algebraandadjointsemigroups(3). * pa → pb → pab → pc → p1 → pa → pa → pab → pab → pc → pa → pb → pc → pb → pc → pc → pb → pab → pc → pab → pc → pc → pab → pc → pc → pc → pc → pc → pc → p1 → pa → pb → pab → pc → p1 →

Similarly,wecanverifythat

pa =(1, b, c,1), pb =(a,1, a,1), pc =(1,1,1,1), p1 =(a, b, c,1). pa * pa = pa , pa * pb = pa * pc =(1,1,1,1), pa * p1 = pa ; pb *pa =(1,1, a,1), pb *pb =pb ,pb *pc =pc ,pb *p1 =pb ;

Florentin Smarandache (author and editor) Collected Papers, IX 306

pc *pa =pc ,pc *pb =pc ,pc *pc =pc ,pc *p1 =pc .

Denote pba = pb * pa =(1,1, a,1),then pba * pa = pba , pa * pba = pc ; pba * pb = pc , pb * pba = pba ; pba * pba = pc ; pba * pc = pc , pc * pba = pc .Hence, M (X)={pa , pb , pba , pc , p1 }anditsCayleytableisTable 16.Obviously,(M (X),*)isanon-commutative semigroup,butitisnotaneutrosophictripletgroup.

Table16. PseudoBE-algebraandadjointsemigroups(4).

* pa pb pba pc p1 pa pa pc pc pc pa pb pba pb pba pc pb pba pba pc pc pc pba pc pc pc pc pc pc p1 pa pb pba pc p1

Now,weconsider M(X).Since pc → =(1,1,1,1)= pc , p1 → =(a, b, c,1)= p1 ; pa → *pa =pa →,pa *pa → =pa →; pa → * pb =(1,1,1,1)= pc → , pb * pa → =(1,1,1,1)= pc →; pa * pb → = pb → * pa =(1,1, c,1); pa *pab → =pab →,pab → *pa =pab →;pb → *pb =pb ,pb *pb → =pb ; pab → * pb =(1,1,1,1)= pc → , pb * pab → =(1,1,1,1)= pc →; pa → * pba =(1,1,1,1)= pc → , pba * pa → =(1,1,1,1)= pc →; pb → *pba =pba ,pba *pb → =pba ; pab → * pba =(1,1,1,1)= pc → , pba * pab → =(1,1,1,1)= pc →

Denote p =(1,1, c,1),then M(X)={pa → , pa , pb → , pb , pab → , pba , p, pc → , p1 →},andTable 17 is itsCayleytable(itisanon-commutativesemigroup,butitisnotaneutrosophictripletgroup).

Table17. PseudoBE-algebraandadjointsemigroups(5). * pa → pa pb → pb pab → pba ppc → p1 → pa → pa → pa → pab → pc → pab → pc → pab → pc → pa → pa pa → pa ppc → pab → pba ppc → pa pb → pc → ppb → pb pc → pba ppc → pb → pb pc → pba pb pb pc → pba pba pc → pb pab → pc → pab → pab → pc → pc → pc → pab → pc → pab → pba pc → pba pba pc → pc → pc → pba pc → pba p pc → pppc → pc → pc → ppc → p pc → pc → pc → pc → pc → pc → pc → pc → pc → pc → p1 → pa → pa pb → pb pab → pba ppc → p1 →

ThefollowingexampleshowsthattheadjointsemigroupsofapseudoBE-algebramaybea commutativeneutrosophictripletgroup.

Florentin Smarandache (author and editor) Collected Papers, IX 307

Example11. LetX={a,b,c,d,1}.Deﬁneoperations → and onXas Tables 18 and 19.Then,(X; →, ,1) isapseudoBE-algebra,aswellasthefollowing: pa → =(1, c, c,1,1), pb → =(d,1,1, d,1), pc → =(d,1,1, d,1), pd → =(1, c, c,1,1), p1 → =(a, b, c, d,1). Wecanverifythefollowing: pa → * pa → = pa → , pa → * pb → = pa → * pc → =(1,1,1,1,1), pa → * pd → = pa → , pa → * p1 → = pa →; pb → * pa → =(1,1,1,1,1), pb → * pb → = pb → * pc → =pb → , pb → * pd → =(1,1,1,1,1), pb → * p1 → = pb →; pc → * pa → =(1,1,1,1,1), pc → * pb → = pc → * pc → =pc → , pc → * pd → =(1,1,1,1,1), pc → * p1 → = pb →; pd → * pa → = pd → , pd → * pb → = pd → * pc → =(1,1,1,1,1), pd → * pd → = pd → , pd → * p1 → = pd → . Denote pab → =pa → * pb → =(1,1,1,1,1),then pab → * pa → = pab → * pb → = pab → * pc → = pab → * pd → = pab → * pab → = pab → * p1 → = pab →.Hence, M→(X)={pa → , pb → , pab → , p1 →}anditsCayleytableis

Table 20.Obviously,(M→(X),*)isacommutativeneutrosophictripletgroup.

Table18. PseudoBE-algebraandcommutativeneutrosophictripletgroups(1). → abcd1 a 1cc11 b d11d1 c d11d1 d 1cc11 1 abcd1

Table19. PseudoBE-algebraandcommutativeneutrosophictripletgroups(2). abcd1 a 1 b c 1 1 b d11d 1 c d11d 1 d 1 b c 1 1 1 a b c d 1

Table20. PseudoBE-algebraandcommutativeneutrosophictripletgroups(3). * pa → pb → pab → p1 → pa → pa → pab → pab → pa → pb → pab → pb → pab → pb → pab → pab → pab → pab → pab → p1 → pa → pb → pab → p1 →

Similarly,wecanverifythefollowing: pa =(1, b, c,1,1), pb =(d,1,1, d,1), pc =(d,1,1, d,1), pd =(1, b, c,1,1), p1 =(a, b, c, d,1). pa *pa =pa ,pa *pb =pa *pc =(1,1,1,1,1), pa *pd =pa ; pb *pa =(1,1,1,1,1), pb *pb =pb *pc =pb ,pb *pd =(1,1,1,1,1). Denote pab =pa * pb =(1,1,1,1,1),then M (X)={pa , pb , pab , p1 }anditsCayleytable isTable 21.Obviously,(M (X),*)isacommutativeneutrosophictripletgroup.

Florentin Smarandache (author and editor) Collected Papers, IX 308

Table21. PseudoBE-algebraandcommutativeneutrosophictripletgroups(4).

* pa pb pab p1 pa pa pab pab pa pb pab pb pab pb pab pab pab pab pab p1 pa pb pab p1

Now,weconsider M(X).Sincethefollowing: pb → = pc → =(d,1,1, d,1)= pb = pc , pa → = pd → =(1, c, c,1,1), pa = pd =(1, b, c,1,1); pa → * pa = pa → , pa * pa → = pa →; pa → * pb =(1,1,1,1,1)= pab → =pab , pb * pa → =(1,1,1,1,1). Hence, M(X)={pa → , pa , pb → , pab → , p1 →},andTable 22 isitsCayleytable(itisacommutative neutrosophictripletgroup).

Table22. PseudoBE-algebraandcommutativeneutrosophictripletgroups(5). * pa → pa pb → pab → p1 → pa → pa → pa → pab → pab → pa → pa pa → pa pab → pab → pa pb → pab → pab → pb → pab → pb → pab → pab → pab → pab → pab → pab → p1 → pa → pa pb → pab → p1 →

Remark 3. Through the discussions of Examples 10 and 11 above, we get the following important revelations: (1)(M→(X), *), (M (X), *), and (M(X), *) are usually three different semi-groups; (2) (M→(X), *) and (M (X), *) are all sub-semi-groups of (M(X), *), which can also be proved from their deﬁnitions; (3) (M→(X), *), (M (X), *), and (M(X), *) may be neutrosophic triplet groups. Under what circumstances they will become neutrosophic triplet groups, will be examined in the next study.

6. Conclusions

In this paper, the concepts of neutrosophic triplet loops (groups) are further generalized, and some new concepts of generalized neutrosophic triplet structures are proposed, including (l l)-type, (l r)-type, (r l)-type, (r r)-type, (l lr)-type, (r lr)-type, (lr l)-type, and (lr r)-type quasi neutrosophic triplet loops (groups), and their basic properties are discussed. In particular, as a corollary of these new properties, an important result is proved. For any commutative neutrosophic triplet group, its every element has only one neutral element. At the same time, the BE-algebras and its various extensions (including CI-algebras, pseudo BE-algebras, and pseudo CI-algebras) have been studied, and some related generalized neutrosophic triplet structures that are contained in these algebras are presented. Moreover, the concept of adjoint semigroups of (generalized) BE-algebras are proposed for the ﬁrst time, abundant examples are given, and some new results are obtained.

Acknowledgments: This work was supported by the National Natural Science Foundation of China (Grant Nos. 61573240, 61473239).

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28.Ahn,S.S.;So,Y.H.OnidealsanduppersetsinBE-algebras. Sci.Math.Jpn. 2008, 68,279–285.

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31. Walendziak,A.OnnormalﬁltersandcongruencerelationsinBE-algebras. Comment.Math. 2012, 52,199–205.

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33. Borzooei,R.A.;Saeid,A.B.;Rezaei,A.;Radfar,A.;Ameri,R.OnpseudoBE-algebras. Discuss.Math.Gen. AlgebraAppl. 2013, 33,95–108.[CrossRef]

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35.Meng,B.L.CI-algebras. Sci.Math.Jpn. 2010, 71,11–17.

36.Meng,B.L.AtomsinCI-algebrasandsingularCI-algebras. Sci.Math.Jpn. 2010, 72,67–72.

37.Meng,B.L.ClosedﬁltersinCI-algebras. Sci.Math.Jpn. 2010, 71,265–270.

38.Kim,K.H.AnoteonCI-algebras. Int.Math.Forum 2011, 6,1–5.

39. Jun,Y.B.;Lee,K.J.;Roh,E.H.IdealsandﬁltersinCI-algebrasbasedonbipolar-valuedfuzzysets. Ann.Fuzz. Math.Inf. 2012, 4,109–121.

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Certain Notions of Neutrosophic Topological K-Algebras

Akram, Hina Gulzar, Florentin Smarandache, Said Broumi

Muhammad Akram, Hina Gulzar, Florentin Smarandache, Said Broumi (2018). Certain Notions of Neutrosophic Topological K-Algebras. Mathematics 6, 234; DOI: 10.3390/math6110

Abstract: The concept of neutrosophic set from philosophical point of view was first considered by Smarandache. A single-valued neutrosophic set is a subclass of the neutrosophic set from a scientific and engineering point of view and an extension of intuitionistic fuzzy sets. In this research article, we apply the notion of single-valued neutrosophic sets to K-algebras. We introduce the notion of single-valued neutrosophic topological K-algebras and investigate some of their properties. Further, we study certain properties, including C5-connected, super connected, compact and Hausdorff, of single-valued neutrosophic topological K-algebras. We also investigate the image and pre-image of single-valued neutrosophic topological K-algebras under homomorphism.

Keywords: K-algebras; single-valued neutrosophic sets; homomorphism; compactness; C5-connectedness

1.Introduction

Anewkindoflogicalalgebra,knownas K-algebra,wasintroducedbyDarandAkramin[1]. A K-algebraisbuiltonagroup G byadjoiningtheinducedbinaryoperationon G.Thegroup G isparticularlyofthetypeinwhicheachnon-identityelementisnotoforder2.Thisalgebraic structureis,ingeneral,non-commutativeandnon-associativewithrightidentityelement[1 3]. Akrametal.[4]introducedfuzzy K-algebras.Theythendevelopedfuzzy K-algebraswithother researchersworldwide.Theconceptsandresultsof K-algebrashavebeenbroadenedtothefuzzy settingframesbyapplyingZadeh’sfuzzysettheoryanditsgeneralizations,namely,interval-valued fuzzysets,intuitionisticfuzzysets,interval-valuedintuitionisticfuzzysets,bipolarfuzzysetsand vaguesets[5].Inhandlinginformationregardingvariousaspectsofuncertainty,non-classicallogicis consideredtobeamorepowerfultoolthantheclassicallogic.Ithasbecomeastrongmathematical toolincomputerscience,medical,engineering,informationtechnology,etc.In1998,Smarandache[6] introducedneutrosophicsetasageneralizationofintuitionisticfuzzyset[7].Aneutrosophicset isidentiﬁedbythreefunctionscalledtruth-membership (T),indeterminacy-membership (I) and falsity-membership (F) functions.Toapplyneutrosophicsetinreal-lifeproblemsmoreconveniently, Smarandache[6]andWangetal.[8]deﬁnedsingle-valuedneutrosophicsetswhichtakesthevalue fromthesubsetof[0,1].Thus,asingle-valuedneutrosophicsetisaninstanceofneutrosophicset.

Algebraicstructureshaveavitalplacewithvastapplicationsinvariousareasoflife.Algebraic structuresprovideamathematicalmodelingofrelatedstudy.Neutrosophicsettheoryhasalsobeen

Muhammad

IX 312

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Papers,

appliedtomanyalgebraicstructures.AgboolaandDavazzintroducedtheconceptofneutrosophic BCI/BCK-algebrasanddiscusselementarypropertiesin[9].Junetal.introducedthenotionof (φ, ψ) neutrosophicsubalgebraofa BCK/BCI-algebra[10].Junetal.[11]definedintervalneutrosophicsets on BCK/BCI-algebra[11].Junetal.[12]proposedneutrosophicpositiveimplicative N-idealsand studytheirextensionproperty[12]Severalsettheoriesandtheirtopologicalstructureshavebeen introducedbymanyresearcherstodealwithuncertainties.Chang[13]wasthefirsttointroducethe notionoffuzzytopology.Later,Lowan[14],PuandLiu[15],andChattopadhyayandSamanta[16] introducedotherconceptsrelatedtofuzzytopology.Coker[17]introducedthenotionofintuitionistic fuzzytopologyasageneralizationoffuzzytopology.SalamaandAlblowi[18]definedthetopological structureofneutrosophicsettheory.AkramandDar[19]introducedtheconceptoffuzzytopological K-algebras.Theyextendedtheirworkonintuitionisticfuzzytopological K-algebras[20].Inthispaper, weintroducethenotionofsingle-valuedneutrosophictopological K-algebrasandinvestigatesome oftheirproperties.Further,westudycertainproperties,including C5-connected,superconnected, compactandHausdorff,ofsingle-valuedneutrosophictopological K-algebras.Wealsoinvestigatethe imageandpre-imageofsingle-valuedneutrosophictopological K-algebrasunderhomomorphism.

2.Preliminaries

Thenotionof K-algebrawasintroducedbyDarandAkramin[1].

Deﬁnition1. [1]Let (G, , e) beagroupinwhicheachnon-identityelementisnotoforder 2.A K-algebraisa structure K =(G, , , e) overaparticulargroup G,where isaninducedbinaryoperation : G × G → G isdeﬁnedby (s, t)= s t = s t 1 , andsatisfythefollowingconditions:

(i) (s t) (s u)=(s ((e u) (e t))) s; (ii) s (s t)=(s (e t) s; (iii) s s = e; (iv) s e = s; and (v) e s = s 1

forall s, t, u ∈ G.Thehomomorphismbetweentwo K-algebras K1 and K2 isamapping f : K1 →K2 such that,forallu,v ∈K1,f (u v)= f (u) f (v)

In[6],Smarandacheinitiatedtheideaofneutrosophicsettheorywhichisageneralizationof intuitionisticfuzzysettheory.Later,SmarandacheandWangetal.introducedasingle-valued neutrosophicset(SNS)asaninstanceofneutrosophicsetin[8].

Deﬁnition2. [8]Let Z beaspaceofpointswithageneralelement s ∈ Z.ASNS A in Z isequippedwith threemembershipfunctions:truthmembershipfunction(TA),indeterminacymembershipfunction(IA)and falsitymembershipfunction(FA),where ∀ s ∈ Z, TA(s), IA(s), FA(s) ∈ [0,1].Thereisnorestrictiononthe sumofthesethreecomponents.Therefore, 0 ≤TA(s)+ IA(s)+ FA(s) ≤ 3

Deﬁnition3. [8]Asingle-valuedneutrosophicemptyset (∅SN ) andsingle-valuedneutrosophicwholeset (1SN ) onZisdeﬁnedas:

Deﬁnition4. [8]IffisamappingfromasetZ1 intoasetZ2, thenthefollowingstatementshold: (i) Let A beaSNSin Z1 and B beaSNSin Z2,thenthepre-imageof B isaSNSin Z1,denotedby f 1(B), deﬁnedas: f 1(B)= {z1 ∈ Z1 : f 1(TB )(z1)= TB ( f (z1)), f 1(IB )(z1)= IB ( f (z1)), f 1(FB )(z1)= FB ( f (z1))}

• ∅SN (u)= {u ∈ Z : (u,0,0,1)}. • 1SN (u)= {u ∈ Z : (u,1,1,0)}

Florentin Smarandache (author and editor) Collected Papers, IX 313

(ii) Let A = {z1 ∈ Z1 : TA(z1), IA(z1), FA(z1)} beaSNSin Z1 and B = {z2 ∈ Z2 : TB (z2), IB (z2), FB (z2)} beaSNSin Z2.Underthemapping f ,theimageof A isaSNSin Z2, denotedby f (A), deﬁnedas: f (A)= {z2 ∈ Z2 : fsup(TA)(z2), fsup(IA)(z2), finf(FA)(z2)}, where forallz2 ∈ Z2.

fsup(TA)(z2)=    supz1∈ f 1 (z2 ) TA (Z1), iff 1 (z2) = ∅, 0, otherwise, fsup(IA)(z2)=    supz1∈ f 1 (z2 ) IA (Z1), iff 1 (z2) = ∅, 0, otherwise, finf(FA)(z2)=    infz1∈ f 1 (z2 ) FA (Z1), iff 1 (z2) = ∅, 0, otherwise

Weformulatethefollowingproposition.

Proposition1. Let f : Z1 → Z2 and A, (Aj, j ∈ J) beaSNSin Z1 and B beaSNSin Z2.Then, f possesses thefollowingproperties:

(i) Iffisonto,thenf (1SN )= 1SN . (ii) f (∅SN )= ∅SN . (iii) f 1(1SN )= 1SN (iv) f 1(∅SN )= ∅SN (v) Iffisonto,thenf ( f 1(B)= B (vi) f 1( n i=1 Ai )= n i=1 f 1(Ai )

3.NeutrosophicTopological K-algebras

Deﬁnition5. Let Z beanonemptyset.Acollection χ ofsingle-valuedneutrosophicsets(SNSs)in Z iscalled asingle-valuedneutrosophictopology(SNT)onZifthefollowingconditionshold:

(a) ∅SN ,1SN ∈ χ

(b)If A, B∈ χ,then A B∈ χ (c)If Ai ∈ χ, ∀i ∈ I,then i∈I Ai ∈ χ

Thepair (Z, χ) iscalledasingle-valuedneutrosophictopologicalspace(SNTS).Eachmemberof χ is saidtobe χ-openorsingle-valuedneutrosophicopenset(SNOS)andcomplimentofeachopensingle-valued neutrosophicsetisasingle-valuedneutrosophicclosedset(SNCS).Adiscretetopologyisatopologywhich containsallsingle-valuedneutrosophicsubsetsofZandindiscreteifitselementsareonly ∅SN , 1SN Deﬁnition6. Let A =(TA, IA, FA) beasingle-valuedneutrosophicsetin K.Then, A iscalleda single-valuedneutrosophicK-subalgebraof K iffollowingconditionsholdfor A:

(i) TA(e) ≥TA(s), IA(e) ≥IA(s), FA(e) ≤FA(s). (ii) TA(s t) ≥ min{TA(s), TA(t)}, IA(s t) ≥ min{IA(s), IA(t)}, FA(s t) ≤ max{FA(s), FA(t)}∀ s, t ∈K.

Florentin Smarandache (author and editor) Collected Papers, IX 314

Example1. Considera K-algebra K =(G, , , e),where G = {e, x, x2 , x3 , x4 , x5 , x6 , x7 , x8} isthecyclic groupoforder 9 andCaley’stablefor isgivenas:

exx2 x3 x4 x5 x6 x7 x8 e ex8 x7 x6 x5 x4 x3 x2 x x xex8 x7 x6 x5 x4 x3 x2 x2 x2 xex8 x7 x6 x5 x4 x3 x3 x3 x2 xex8 x7 x6 x5 x4 x4 x4 x3 x2 xex8 x7 x6 x5 x5 x5 x4 x3 x2 xex8 x7 x6 x6 x6 x5 x4 x3 x2 xex8 x7 x7 x7 x6 x5 x4 x3 x2 xex8 x8 x8 x7 x6 x5 x4 x3 x2 xe

Ifwedeﬁneasingle-valuedneutrosophicset A, B in K suchthat: A = {(e,0.4,0.5,0.8), (s,0.3,0.4,0.7)}, B = {(e,0.3,0.4,0.8), (s,0.2,0.3,0.6)} ∀ s = e ∈ G.

AccordingtoDeﬁnition 5,thefamily {∅SN ,1SN , A, B} ofSNSsof K-algebraisaSNTon K.Wedeﬁne aSNS A = {TA, IA, FA} in K suchthat TA(e)= 0.7, IA(e)= 0.5, FA(e)= 0.2, TA(s)= 0.2, IA(s)= 0.4, FA(s)= 0.6.Clearly, A =(TA, IA, FA) isaSNK-subalgebraof K

Deﬁnition7. Let K =(G, · , , e) bea K-algebraandlet χK beatopologyon K.Let A beaSNSin K andlet χK beatopologyon K.Then,aninducedsingle-valuedneutrosophictopologyon A isacollectionorfamily ofsingle-valuedneutrosophicsubsetsof A whicharetheintersectionwith A andsingle-valuedneutrosophic opensetsin K deﬁnedas χA = {A∩ F : F ∈ χK }.Then, χA iscalledsingle-valuedneutrosophicinduced topologyon A orrelativetopologyandthepair (A, χA) iscalledaninducedtopologicalspaceorsingle-valued neutrosophicsubspaceof (K, χK )

Deﬁnition8. Let (K1, χ1 ) and (K2, χ2 ) betwoSNTSsandlet f : (K1, χ1 ) → (K2, χ2 ).Then, f iscalled single-valuedneutrosophiccontinuousiffollowingconditionshold:

(i) ForeachSNS A∈ χ2 ,f 1(A) ∈ χ1

(ii) ForeachSNK-subalgebra A∈ χ2 ,f 1(A) isaSNK-subalgebra ∈ χ1

Deﬁnition9. Let (K1, χ1 ) and (K2, χ2 ) betwoSNTSsandlet (A, χA) and (B, χB ) betwosingle-valued neutrosophicsubspacesover (K1, χ1 ) and (K2, χ2 ).Let f beamappingfrom (K1, χ1 ) into (K2, χ2 ),then f is amappingfrom (A, χA) to (B, χB ) iff (A) ⊂B

Deﬁnition10. Let f beamappingfrom (A, χA) to (B, χB ).Then, f isrelativelysingle-valuedneutrosophic continuousifforeverySNOSYB in χB ,f 1(YB ) ∩A∈ χA.

Deﬁnition11. Let f beamappingfrom (A, χA) to (B, χB ).Then, f isrelativelysingle-valuedneutrosophic openifforeverySNOSXA in χA,theimagef (XA) ∈ χB

Proposition2. Let (A, χA) and (B, χB ) besingle-valuedneutrosophicsubspacesof (K1, χ1 ) and (K2, χ2 ), where K1 and K2 are K-algebras.If f isasingle-valuedneutrosophiccontinuousfunctionfrom K1 to K2 and f (A) ⊂B.Then,fisrelativelysingle-valuedneutrosophiccontinuousfunctionfrom A into B

Deﬁnition12. Let (K1, χ1 ) and (K2, χ2 ) betwoSNTSs.Amapping f : (K1, χ1 ) → (K2, χ2 ) iscalleda single-valuedneutrosophichomomorphismiffollowingconditionshold:

Florentin Smarandache (author and editor) Collected Papers, IX 315

(i) fisaone-oneandontofunction.

(ii) fisasingle-valuedneutrosophiccontinuousfunctionfrom K1 to K2. (iii) f 1 isasingle-valuedneutrosophiccontinuousfunctionfrom K2 to K1

Theorem1. Let (K1, χ1 ) beaSNTSand (K2, χ2 ) beanindiscreteSNTSonK-algebras K1 and K2, respectively.Then,eachfunction f deﬁnedas f : (K1, χ1 ) → (K2, χ2 ) isasingle-valuedneutrosophic continuousfunctionfrom K1 to K2.If (K1, χ1 ) and (K2, χ2 ) betwodiscreteSNTSs K1 and K2,respectively, theneachhomomorphism f : (K1, χ1 ) → (K2, χ2 ) isasinglevaluesneutrosophiccontinuousfunctionfrom K1 to K2

Proof. Let f beamappingdeﬁnedas f : K1 →K2.Let χ1 beSNTon K1 and χ2 beSNTon K2,where χ2 = {∅SN ,1SN }.Weshowthat f 1(A) isasingle-valuedneutrosophic K-subalgebraof K1,i.e.,for each A∈ χ2 , f 1(A) ∈ χ1 .Since χ2 = {∅SN ,1SN },thenforany u ∈ χ1 ,consider ∅SN ∈ χ2 suchthat f 1(∅SN )(u)= ∅SN ( f (u))= ∅SN (u). Therefore, ( f 1(∅SN ))= ∅SN ∈ χ1 .Likewise, ( f 1(1SN ))= 1SN ∈ χ1 .Hence, f isaSN continuousfunctionfrom K1 to K2.

Now,forthesecondpartofthetheorem,whereboth χ1 and χ2 areSNTSson K1 and K2, respectively,and f : (K1, χ1 ) → (K2, χ2 ) isahomomorphism.Therefore,forall A∈ χ2 and f 1A∈ χ1 , where f isnotausualinversehomomorphism.Toprovethat f 1(A) isasingle-valuedneutrosophic K-subalgebrainof K1.Letfor u, v ∈K1, f 1(TA)(u v)=TA( f (u v))

= TA( f (u) f (v))

≥ min{TA( f (u)) T( f (v))}

= min{ f 1(TA)(u), f 1(TA)(v)}, f 1(IA)(u v)=IA( f (u v))

= IA( f (u) f (v))

≥ min{IA( f (u)) I( f (v))}

= min{ f 1(IA)(u), f 1(IA)(v)}, f 1(FA)(u v)=FA( f (u v))

= FA( f (u) f (v))

≤ max{FA( f (u)) F( f (v))} = max{ f 1(FA)(u), f 1(FA)(v)}

Hence, f isasingle-valuedneutrosophiccontinuousfunctionfrom K1 to K2. Proposition3. Let χ1 and χ2 betwoSNTSson K.Then,eachhomomorphism f : (K, χ1 ) → (K, χ2 ) isa single-valuedneutrosophiccontinuousfunction.

Proof. Let (K, χ1) and (K, χ2 ) betwoSNTSs,where K isa K-algebra.Toprovetheaboveresult, itisenoughtoshowthatresultisfalseforaparticulartopology.Let A =(TA, IA, FA, ) and B = (TB , IB , FB ) betwoSNSsin K.Take χ1 = {∅SN ,1SN , A} and χ2 = {∅SN ,1SN , B}.If f : (K, χ1) → (K, χ2 ),deﬁnedby f (u)= e u,forall u ∈K,then f isahomomorphism.Now,for u ∈A, v ∈ χ2 , ( f 1(B))(u)= B( f (u))= B(e u)= B(u), ∀ u ∈K,i.e., f 1(B)= B.Therefore, ( f 1(B)) / ∈ χ1 .Hence, f isnotasingle-valuedneutrosophic continuousmapping.

Deﬁnition13. Let K =(G, , , e) bea K-algebraand χ beaSNTon K.Let A beasingle-valuedneutrosophic K-algebra(K-subalgebra)of K and χA beaSNTon A.Then, A issaidtobeasingle-valuedneutrosophic topological K-algebra(K-subalgebra)on K iftheselfmapping ρa : (A, χA) → (A, χA) deﬁnedas ρa (u)= u a, ∀a ∈K,isarelativelysingle-valuedneutrosophiccontinuousmapping.

Florentin Smarandache (author and editor) Collected Papers, IX 316

Theorem2. Let χ1 and χ2 betwoSNTSson K1 and K2,respectively,and f : K1 →K2 beahomomorphism suchthat f 1(χ2)= χ1.If A = {TA, IA, FA} isasingle-valuedneutrosophictopological K-algebraof K2, thenf 1(A) isasingle-valuedneutrosophictopologicalK-algebraof K1.

Proof. Let A = {TA, IA, FA} beasingle-valuedneutrosophictopological K-algebraof K2.Toprove that f 1(A) beasingle-valuedneutrosophictopological K-algebraof K1.Letforany u, v ∈K1,

T f 1(A) (u v)= TA( f (u v))

≥ min{TA( f (u)), TA( f (v))}

= min{T f 1(A) (u), T f 1(A) (v)},

I f 1(A) (u v)= IA( f (u v))

≥ min{IA( f (u)), IA( f (v))}

= min{I f 1(A) (u), I f 1(A) (v)},

F f 1(A) (u v)= FA( f (u v)) ≤ max{FA( f (u)), FA( f (v))} = max{F f 1(A) (u), F f 1(A) (v)}

Hence, f 1(A) isasingle-valuedneutrosophic K-algebraof K1. Now,weprovethat f 1(A) issingle-valuedneutrosophictopological K-algebraof K1.Since f isasingle-valuedneutrosophiccontinuousfunction,thenbyproposition3.1, f isalsoarelatively single-valuedneutrosophiccontinuousfunctionwhichmaps ( f 1(A), χ f 1(A) ) to (A, χA).

Let a ∈K1 and Y beaSNSin χA,andlet X beaSNSin χ f 1(A) suchthat f 1(Y)= X.(1)

Wearetoprovethat ρa : ( f 1(A), χ f 1(A) ) → ( f 1(A), χ f 1(A) ) isrelativelysingle-valued neutrosophiccontinuousmapping,thenforany a ∈K1,wehave

T ρ 1 a (X) (u)= T(X) (ρa (u))= T(X) (u a)

= T f 1(Y) (u a)= T(Y) ( f (u a))

= T(Y) ( f (u) f (a))= T(Y) (ρ f (a) ( f (u)))

= T ρ 1 f (a)Y ( f (u))= T f 1 (ρ 1 f (a) (Y)(u)),

I ρ 1 a (X) (u)= I(X) (ρa (u))= I(X) (u a) = I f 1(Y) (u a)= I(Y) ( f (u a)) = I(Y) ( f (u) f (a))= I(Y) (ρ f (a) ( f (u))) = I ρ 1 f (a)Y ( f (u))= I f 1 (ρ 1 f (a) (Y)(u)),

F ρ 1 a (X) (u)= F(X) (ρa (u))= F(X) (u a) = F f 1(Y) (u a)= F(Y) ( f (u a))

= F(Y) ( f (u) f (a))= F(Y) (ρ f (a) ( f (u)))

= F ρ 1 f (a)Y ( f (u))= F f 1 (ρ 1 f (a) (Y)(u)).

Itconcludesthat ρ 1 a (X)= f 1(ρ 1 f (a) (Y)).Thus, ρ 1 a (X) ∩ f 1(A)= f 1(ρ 1 f (a) (Y)) ∩ f 1(A) is aSNSin f 1(A) andaSNSin χ f 1(A).Hence, f 1(A) andasingle-valuedneutrosophictopological K-algebraof K.Hence,theproof.

Theorem3. Let (K1, χ1 ) and (K2, χ2 ) betwoSNTSson K1 and K2,respectively,andlet f beabijective homomorphismof K1 into K2 suchthat f (χ1)= χ2.If A isasingle-valuedneutrosophictopological K-algebra of K1,thenf (A) isasingle-valuedneutrosophictopologicalK-algebraof K2 Proof. Supposethat A = {TA, IA, FA} isaSNtopological K-algebraof K1.Toprovethat f (A) isa single-valuedneutrosophictopological K-algebraof K2,let,for u, v ∈K2,

Florentin Smarandache (author and editor) Collected Papers, IX 317

f (A)=( fsup(TA)(v), fsup(IA)(v), finf(FA)(v)).

Let ao ∈ f 1(u), bo ∈ f 1(v) suchthat

supx∈ f 1(u) TA(x)= TA(ao ),supx∈ f 1(v) TA(x)= TA(bo ), supx∈ f 1(u) IA(x)= IA(ao ),supx∈ f 1(v) IA(x)= IA(bo ), infx∈ f 1(u) FA(x)= FA(ao ),infx∈ f 1(v) FA(x)= FA(bo )

Now,

T f (A) (u v)= sup x∈ f 1(u v) TA(x)

≥TA(ao, bo ) ≥ min{TA(ao ), TA(bo )} = min{ sup x∈ f 1(u) TA(x),sup x∈ f 1(v) TA(x)}

= min{T f (A) (u), T f (A) (v)},

I f (A) (u v)= sup x∈ f 1(u v) IA(x)

≥IA(ao, bo )

≥ min{IA(ao ), IA(bo )} = min{ sup x∈ f 1(u) IA(x),sup x∈ f 1(v) IA(x)}

= min{I f (A) (u), I f (A) (v)},

F f (A) (u v)= inf x∈ f 1(u v) FA(x)

≤FA(ao, bo ) ≤ max{FA(ao ), FA(bo )}

= max{ inf x∈ f 1(u) FA(x),inf x∈ f 1(v) FA(x)}

= max{F f (A) (u), F f (A) (v)}.

Hence, f (A) isasingle-valuedneutrosophic K-subalgebraof K2.Now,weprovethattheself mapping ρb : ( f (A), χ f (A)) → ( f (A), χ f (A)),deﬁnedby ρb (v)= v b,forall b ∈K2,isarelatively single-valuedneutrosophiccontinuousmapping.Let YA beaSNSin χA,thereexistsaSNS“Y”in χ1 suchthat YA = Y ∩A.WeshowthatforaSNSin χ f (A), ρ 1 b (Yf (A) ) ∩ f (A) ∈ χ f (A)

Since f isaninjectivemapping,then f (YA)= f (Y ∩A)= f (Y) ∩ f (A) isaSNSin χ f (A) which showsthat f isrelativelysingle-valuedneutrosophicopen.Inaddition, f issurjective,thenforall b ∈K2, a = f (b),where a ∈K1.

Florentin Smarandache (author and editor) Collected Papers, IX 318

T f 1(ρ 1 b (Yf (A) )) (u)= T f 1(ρ 1 f (a)(Yf (A) )) (u)

= Tρ 1 f (a)(Yf (A) ) ( f (u))

= T(Yf (A) ) (ρ f (a) ( f (u)))

= T(Yf (A) ) ( f (u) f (a))

= T f 1(Yf (A) ) (u a)

= T f 1(Yf (A) ) (ρa (u))

= Tρ 1 ( a) ( f 1(Yf (A) ))(u),

I f 1(ρ 1 b (Yf (A) )) (u)= I f 1(ρ 1 f (a)(Yf (A) )) (u)

= Iρ 1 f (a)(Yf (A) ) ( f (u))

= I(Yf (A) ) (ρ f (a) ( f (u)))

= I(Yf (A) ) ( f (u) f (a))

= I f 1(Yf (A) ) (u a)

= I f 1(Yf (A) ) (ρa (u))

= Iρ 1 ( a) ( f 1(Yf (A) ))(u),

F f 1(ρ 1 b (Yf (A) )) (u)= F f 1(ρ 1 f (a)(Yf (A) )) (u)

= Fρ 1 f (a)(Yf (A) ) ( f (u))

= F(Yf (A) ) (ρ f (a) ( f (u)))

= F(Yf (A) ) ( f (u) f (a))

= F f 1(Yf (A) ) (u a)

= F f 1(Yf (A) ) (ρa (u))

= Fρ 1 ( a) ( f 1(Yf (A) ))(u).

Thisimpliesthat f 1(ρ 1 (b) ((Yf (A) )))= ρ 1 (a) ( f 1(Y(A) )).Since ρa : (A, χA) → (A, χA) isrelatively single-valuedneutrosophiccontinuousmappingand f isrelativelysingle-valuedneutrosophic continuesmappingfrom (A, χA) into ( f (A), χ f (A) ), f 1(ρ 1 (b) ((Yf (A) ))) ∩A = ρ 1 (a) ( f 1(Y(A) )) ∩A is aSNSin χA.Hence, f ( f 1(ρ(b) ((Yf (A) ))) ∩A)= ρ 1 (b) (Yf (A) ) ∩ f (A) isaSNSin χA,whichcompletes theproof.

Example2. Let K =(G, , , e) bea K-algebra,where G = {e, x, x2 , x3 , x4 , x5 , x6 , x7 , x8} isthecyclicgroup oforder 9 andCaley’stablefor isgiveninExample 1.WedeﬁneaSNSas: A = {(e,0.4,0.5,0.8), (s,0.3,0.4,0.6)}, B = {(e,0.3,0.4,0.8), (s,0.2,0.3,0.6)}, forall s = e ∈ G, where A, B∈ [0,1].Thecollection χK = {∅SN ,1SN , A, B} ofSNSsof K isaSNTon K and (K, χK ) isaSNTS.Let C beaSNSin K,deﬁnedas: C = {(e,0.7,0.5,0.2), (s,0.5,0.4,0.6)}, ∀s = e ∈ G.

Now,

Florentin Smarandache (author and editor) Collected Papers, IX 319

Clearly, C isasingle-valuedneutrosophic K-subalgebraof K.Bydirectcalculationsrelativetopology χC isobtainedas χC = {∅A,1A, A}.Then,thepair (C, χC ) isasingle-valuedneutrosophicsubspace of (K, χK ).Weshowthat C isasingle-valuedneutrosophictopological K-subalgebraof K,i.e.,theself mapping ρa : (C, χC ) → (C, χC ) deﬁnedby ρa (u)= u a, ∀a ∈K isrelativelysingle-valuedneutrosophic continuousmapping,i.e.,foraSNOS A in (C, χC ), ρ 1 a (A) ∩C∈ χC .Since ρa ishomomorphism,then ρ 1 a (A) ∩C = A∈ χC .Therefore, ρa : (C, χC ) → (C, χC ) isrelativelysingle-valuedneutrosophiccontinuous mapping.Hence, C isasingle-valuedneutrosophictopologicalK-algebraof K

Example3. Let K =(G, · , , e) bea K-algebra,where G = {e, x, x2 , x3 , x4 , x5 , x6 , x7 , x8} isthecyclicgroup oforder 9 andCaley’stablefor isgiveninExample3.1.WedeﬁneaSNSas:

A = {(e,0.4,0.5,0.8), (s,0.3,0.4,0.6)}, B = {(e,0.3,0.4,0.8), (s,0.2,0.3,0.6)}, D = {(e,0.2,0.1,0.3), (s,0.1,0.1,0.5)},

forall s = e ∈ G, where A, B∈ [0,1].Thecollection χ1 = {∅SN ,1SN , D} and χ2 = {∅SN ,1SN , A, B} of SNSsof K areSNTson K and (K, χ1), (K, χ2) betwoSNTSs.Let C beaSNSin (K, χ2),deﬁnedas: C = {(e,0.7,0.5,0.2), (s,0.5,0.4,0.6)}, ∀s = e ∈ G.

Now,Let f : (K, χ1) → (K, χ2) beahomomorphismsuchthat f 1(χ2)= χ1 (wehavenotconsider K to bedistinct),then,byProposition 3, f isasingle-valuedneutrosophiccontinuousfunctionand f isalsorelatively single-valuedneutrosophiccontinuesmappingfrom (K, χ1) into (K, χ2).Since C isaSNSin (K, χ2) and withrelativetopology χC = {∅A,1A, A} isalsoasingle-valuedneutrosophictopological K-algebraof (K, χ2) Weprovethat f 1(C ) isasingle-valuedneutrosophictopological K-algebrain (K, χ1).Since f isacontinuous function,then,byDeﬁnition 8, f 1(C ) isasingle-valuedneutrosophic K-subalgebrain (K, χ1).Toprovethat f 1(c) isasingle-valuedneutrosophictopologicalK-algebra,thenforb ∈K1 take

ρb : ( f 1(C ), χ f 1(C ) ) → ( f 1(C ), χ f 1(C ) ), for A∈ χ f 1(C), ρ 1 b (A) ∩ f 1(C ) ∈ χ f 1(C) whichshowsthat f 1(C) isasingle-valuedneutrosophic topological K-algebrain (K, χ1).Similarly,wecanshowthat f (C ) isaasingle-valuedneutrosophictopological K-algebrain (K, χ2) byconsideringabijectivehomomorphism.

Deﬁnition14. Let χ beaSNTon K and (K, χ) beaSNTS.Then, (K, χ) iscalledsingle-valuedneutrosophic C5-disconnectedtopologicalspaceifthereexistaSNOSandSNCS H suchthat H =(TH, IH, FH, ) = 1SN and H =(TH, IH, FH, ) = ∅SN ,otherwise (K, χ) iscalledsingle-valuedneutrosophicC5-connected.

Example4. EveryindiscreteSNTspaceon K isC5-connected.

Proposition4. Let (K1, χ1) and (K2, χ2) betwoSNTSsand f : (K1, χ1) → (K2, χ2) beasurjective single-valuedneutrosophiccontinuousmapping.If (K1, χ1) isasingle-valuedneutrosophic C5-connectedspace, then (K2, χ2) isalsoasingle-valuedneutrosophicC5-connectedspace.

Proof. Supposeoncontrarythat (K2, χ2) isasingle-valuedneutrosophic C5-disconnectedspace. Then,byDeﬁnition 14,thereexistbothSNOSandSNCS H besuchthat H = 1SN and H = ∅SN .Since f isasingle-valuedneutrosophiccontinuousandontofunction,so f 1(H)= 1SN or f 1(H)= ∅SN , where f 1(H) isbothSNOSandSNCS.Therefore,

H = f ( f 1(H))= f (1SN )= 1SN (2) and H = f ( f 1(H))= f (∅SN )= ∅SN ,(3)

Florentin Smarandache (author and editor) Collected Papers, IX 320

acontradiction.Hence, (K2, χ2) isasingle-valuedneutrosophic C5-connectedspace.

Corollary1. Let χ beaSNTon K.Then, (K, χ) iscalledasingle-valuedneutrosophic C5-connectedspaceif andonlyiftheredoesnotexistasingle-valuedneutrosophiccontinuousmap f : (K, χ) → (FT , χT ) suchthat f = 1SN andf = ∅SN

Deﬁnition15. Let A = {TA, IA, FA} beaSNSin K.Let χ beaSNTon K.Theinteriorandclosureof A in K isdeﬁnedas:

AInt:TheunionofSNOSswhichcontainedin A

AClo:TheintersectionofSNCSsforwhich A isasubsetoftheseSNCSs.

Remark1. BeingunionofSNOS AInt isaSNOand AClo beingintersectionofSNCSisSNC.

Theorem4. Let A beaSNSinaSNTS (K, χ).Then, AInt issuchanopensetwhichisthelargestopensetof K containedin A

Corollary2. A =(TA, IA, FA) isaSNOSin K ifandonlyif AInt = A and A =(TA, IA, FA) isaSNCS in K ifandonlyif AClo = A

Proposition5. Let A beaSNSin K.Then,followingresultsholdfor A:

(i)(1SN )Int = 1SN . (ii)(∅SN )Clo = ∅SN . (iii) (A)Int = (A)Clo (iv) (A)Clo = (A)Int

Deﬁnition16. Let K bea K-algebraand χ beaSNTon K.ASNOS A in K issaidtobesingle-valued neutrosophicregularopenif A =(AClo )Int.(4)

Remark2. EverySNOSwhichisregularissingle-valuedneutrosophicopenandeverysingle-valued neutrosophicclosedandopensetisasingle-valuedneutrosophicregularopen.

Deﬁnition17. Asingle-valuedneutrosophicsuperconnected K-algebraissucha K-algebrainwhichtheredoes notexistasingle-valuedneutrosophicregularopenset A =(TA, IA, FA) suchthat A = ∅SN and A = 1SN Ifthereexistssuchasingle-valuedneutrosophicregularopenset A =(TA, IA, FA) suchthat A = ∅SN and A = 1SN ,thenK-algebraissaidtobeasingle-valuedneutrosophicsuperdisconnected.

Example5. Let K =(G, , , e) bea K-algebra,where G = {e, x, x2 , x3 , x4 , x5 , x6 , x7 , x8} isthecyclicgroup oforder 9 andCaley’stablefor isgiveninExample 1 WedeﬁneaSNSas: A = {(e,0.2,0.3,0.8), (s,0.1,0.2,0.6)} Let χK = {∅SN ,1SN , A} beaSNTon K andlet B = {(e,0.3,0.3,0.8), (s,0.2,0.2,0.6)} beaSNSin K.here

SNOSs : ∅SN = {0,0,1},1SN = {1,1,0}, A = {(e,0.2,0.3,0.8), (s,0.1,0.2,0.6)}. SNCSs : (∅SN )c =({0,0,1})c =({1,1,0})= 1SN , (1SN )c =({1,1,0})c =({0,0,1})= ∅SN , (A)c =({(e,0.2,0.3,0.8), (s,0.1,0.2,0.6)})c =({(e,0.8,0.3,0.2), (s,0.6,0.2,0.1)})= A (say)

Florentin Smarandache (author and editor) Collected Papers, IX 321

Then,closureof B istheintersectionofclosedsetswhichcontain B.Therefore,

A = BClo . (5)

Now,interiorof B istheunionofopensetswhichcontainin B.Therefore, ∅SN A = A A = B Int . (6) Notethat (BClo )Clo = BClo.Now,ifweconsideraSNS A = {(e,0.2,0.3,0.8), (s,0.1,0.2,0.6)} ina K-algebra K andif χK = {∅SN ,1SN , A} isaSNTon K.Then, (A)Clo = A and (A)Int = A.Consequently,

A =(AClo )Int,(7) whichshowsthat A isaSNregularopensetin K-algebra K.Since A isaSNregularopensetin K and A = ∅SN , A = 1SN , then,byDeﬁnition 17, K-algebra K isasingle-valuedneutrosophicsupperdisconnected K-algebra.

Proposition6. Let K beaK-algebraandlet A beaSNOS.Then,thefollowingstatementsareequivalent: (i) AK-algebraissingle-valuedneutrosophicsuperconnected. (ii)(A)Clo = 1SN ,foreachSNOS A = ∅SN (iii)(A)Int = ∅SN ,foreachSNCS A = 1SN (iv) TheredonotexistSNOSs A, F suchthat A⊆ F and A = ∅SN = F inK-algebra K

Deﬁnition18. Let (K, χ) beaSNTS,where K isa K-algebra.Let S beacollectionofSNOSsin K denotedby S = {(TAj , IAj , FAj ) : j ∈ J}.Let A beaSNOSin K.Then, S iscalledasingle-valuedneutrosophicopen coveringof A if A⊆ S.

Definition19. Let K bea K-algebraand (K, χ) beaSNTS.Let L beafinitesub-collectionof S.If L isalso asingle-valuedneutrosophicopencoveringof A ,thenitiscalledafinitesub-coveringof S and A iscalled single-valuedneutrosophiccompactifeachsingle-valuedneutrosophicopencovering S of A hasafinitesub-cover. Then, (K, χ) iscalledcompactK-algebra.

Remark3. Ifeither K isafinite K-algebraor χ isafinitetopologyon K,i.e.,consistsoffinitenumber ofsingle-valuedneutrosophicsubsetsof K,thentheSNT (K, χ) isasingle-valuedneutrosophiccompact topologicalspace.

Proposition7. Let (K1, χ1) and (K2, χ2) betwoSNTSsand f beasingle-valuedneutrosophiccontinuous mappingfrom K1 into K2.Let A beaSNSin (K1, χ1) .If A issingle-valuedneutrosophiccompactin (K1, χ1), thenf (A) issingle-valuedneutrosophiccompactin (K2, χ2).

Proof. Let f : (K1, χ1) → (K2, χ2) beasingle-valuedneutrosophiccontinuousfunction.Let ´ S =( f 1(Aj : j ∈ J)) beasingle-valuedneutrosophicopencoveringof A since A beaSNSin (K1, χ1) Let ´ L =(Aj : j ∈ J) beasingle-valuedneutrosophicopencoveringof f (A).Since A iscompact,then thereexistsasingle-valuedneutrosophicﬁnitesub-cover n j=1 f 1(Aj ) suchthat

A⊆ n j=1 f 1(Aj )

Wehavetoprovethattherealsoexistsaﬁnitesub-coverof ´ L for f (A) suchthat

Florentin Smarandache (author and editor) Collected Papers, IX 322

f (A) ⊆ n j=1 (Aj )

Now, A⊆ n j=1 f 1(Aj )

f (A) ⊆ f ( n j=1 f 1(Aj ))

f (A) ⊆ n j=1 ( f ( f 1(Aj )))

f (A) ⊆ n j=1 (Aj )

Hence, f (A) issingle-valuedneutrosophiccompactin (K2, χ2)

Deﬁnition20. Asingle-valuedneutrosophicset A ina K-algebra K iscalledasingle-valuedneutrosophic pointif

TA(v)= α ∈ (0,1], ifv=u 0, otherwise,

IA(v)= β ∈ (0,1], ifv=u 0, otherwise,

FA(v)= γ ∈ [0,1), ifv=u 0, otherwise, withsupport u andvalue (α, β, γ),denotedby u(α, β, γ).Thissingle-valuedneutrosophicpointissaid to“belongto"aSNS A, writtenas u(α, β, γ) ∈A if TA(u) ≥ α, IA(u) ≥ β, FA(u) ≤ γ andsaidtobe “quasi-coincidentwith"aSNS A,writtenas u(α, β, γ) q A if TA(u)+ α > 1, IA(u)+ β > 1, FA(u)+ γ < 1. Deﬁnition21. Let K bea K-algebraandlet (K, χ) beaSNTS.Then, (K, χ) iscalledasingle-valued neutrosophicHausdorffspaceifandonlyif,foranytwodistinctsingle-valuedneutrosophicpointsu1, u2 ∈K, thereexistSNOSs B1 =(TB1 , IB1 , FB1 ), B2 =(TB2 , IB2 , FB2 ) suchthatu1 ∈B1, u2 ∈B2,i.e., TB1 (u1)= 1, IB1 (u1)= 1, FB1 (u1)= 0, TB2 (u2)= 1, IB2 (u2)= 1, FB2 (u2)= 0 andsatisfytheconditionthat B1 ∩B2 = ∅SN . Then, (K, χ) iscalledsingle-valuedneutrosophicHausdorff spaceandK-algebraissaidtobaaHausdorffK-algebra.Infact, (K, χ) isaHausdorffK-algebra.

Example6. Let K =(G, · , , e) beaK-algebraandlet (K, χK ) beaSNTSon K,where G = {e, x, x2 , x3 , x4 , x5 , x6 , x7 , x8} isthecyclicgroupoforder 9 andCaley’stablefor isgiveninExample 1 WedeﬁnetwoSNSsas A = {(e,1,1,0), (s,0,0,1)} B = {(e,0,0,1), (s,1,1,0)} Considerasingle-valued neutrosophicpointfore ∈K suchthat

TA(e)= 0.3, ife=u 0, otherwise, IA(e)= 0.2, ife=u 0, otherwise,

Florentin Smarandache (author and editor) Collected Papers, IX 323

FA(e)= 0.4, ife=u 0, otherwise.

Then, e(0.3,0.2,0.4) isasingle-valuedneutrosophicpointwithsupport e andvalue (0.3,0.2,0.4). Thissingle-valuedneutrosophicpointbelongstoSNS “A” butnotSNS “B” . Now,foralls = e ∈K

TB (s)= 0.5, ifs=u 0, otherwise,

IB (s)= 0.4, ifs=u 0, otherwise,

FB (s)= 0.3, ifs=u 0, otherwise.

Then, s(0.5,0.4,0.3) isasingle-valuedneutrosophicpointwithsupport s andvalue (0.5,0.4,0.3).This single-valuedneutrosophicpointbelongstoSNS “B” butnotSNS “A”.Thus, e(0.3,0.2,0.4) ∈A and e(0.3,0.2,0.4) / ∈B, s(0.5,0.4,0.3) ∈B and s(0.5,0.4,0.3) / ∈A and A B = ∅SN .Thus, K-algebraisa HausdorffK-algebraand (K, χK ) isaHausdorfftopologicalspace.

Theorem5. Let (K1, χ1), (K2, χ2) betwoSNTSs.Let f beasingle-valuedneutrosophichomomorphismfrom (K1, χ1) into (K2, χ2).Then, (K1, χ1) isasingle-valuedneutrosophicHausdorffspaceifandonlyif (K2, χ2) isasingle-valuedneutrosophicHausdorffK-algebra.

Proof. Let (K1, χ1), (K2, χ2) betwoSNTSs.Let K1 beasingle-valuedneutrosophicHausdorffspace, then,accordingtotheDeﬁnition 21,thereexisttwoSNOSs X and Y fortwodistinctsingle-valued neutrosophicpoints u1, u2 ∈ χ2 also a, b ∈K1(a = b) suchthat X Y = ∅SN . Now,for w ∈K1,consider ( f 1(u1))(w)= u1( f 1(w)),where u1( f 1(w))= s ∈ (0,1] if w = f 1(a), otherwise0.Thatis, ( f 1(u1))(w)=(( f 1(u))1(w)).Therefore,wehave f 1(u1)=( f 1(u))1. Similarly, f 1(u2)=( f 1(u))2.Now,since f 1 isasingle-valuedneutrosophiccontinuousmapping from K2 into K1,thereexisttwoSNOSs f (X) and f (Y) of u1 and u2,respectively,suchthat f (X) f (Y)= f (∅SN )= ∅SN .Thisimpliesthat K2 isasingle-valuedneutrosophicHausdorff K-algebra.Theconversepartcanbeprovedsimilarly.

Theorem6. Let f beasingle-valuedneutrosophiccontinuousfunctionwhichisbothone-oneandonto,where f isamappingfromasingle-valuedneutrosophiccompact K-algebra K1 intoasingle-valuedneutrosophic HausdorffK-algebra K2.Then,fisahomomorphism.

Proof. Let f : K1 →K2 beasingle-valuedneutrosophiccontinuousbijectivefunctionfrom single-valuedneutrosophiccompact K-algebra K1 intoasingle-valuedneutrosophicHausdorff K-algebra K2.Since f isasingle-valuedneutrosophiccontinuousmappingfrom K1 into K2, f is ahomomorphism.Since f isbijective,weonlyprovethat f issingle-valuedneutrosophicclosed. Let D =(TD , ID , FD ) beasingle-valuedneutrosophicclosedin K1.If D = ∅SN issingle-valued neutrosophicclosedin K1,then f (D)= ∅SN issingle-valuedneutrosophicclosedin K2.However, if D = ∅SN ,then D willbeasingle-valuedneutrosophiccompact,beingsubsetofasingle-valued neutrosophiccompact K-algebra.Then, f (D),beingsingle-valuedneutrosophiccontinuousimageofa single-valuedneutrosophiccompact K-algebra,isalsosingle-valuedneutrosophiccompact.Therefore, K2 isclosed,whichimpliesthatmapping f isclosed.Thus, f isahomomorphism.

Florentin Smarandache (author and editor) Collected Papers, IX 324

4. Conclusions

Non-classical logic is considered as a powerful tool for inspecting uncertainty and indeterminacy found in real world problems. Being a great extension of classical logic, neutrosophic set theory is considered as a useful mathematical tool to cope up with uncertainties in science, technology, and computer science. We have used this mathematical model with a topological structure to investigate the uncertainty in K-algebras. We have introduced the notion of single-valued neutrosophic topological K-algebras and presented certain concepts, including continuous function between two topological on K-algebras, relatively continuous function and homomorphism. We have investigated the image and pre-image of single-valued neutrosophic topological K-algebras under this homomorphism. We have proposed some conclusive concepts, including single-valued neutrosophic compact K-algebras and single-valued neutrosophic Hausdorff K-algebras. We plan to extend our study to: (i) single-valued neutrosophic soft topological K-algebras; and (ii) bipolar neutrosophic soft topological K-algebras.

For other notations and terminologies, readers are referred to [21 26].

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Study on the Development of Neutrosophic Triplet Ring and Neutrosophic Triplet Field

Mumtaz Ali, Florentin Smarandache, Mohsin Khan

Mumtaz Ali, Florentin Smarandache, Mohsin Khan (2018). Study on the Development of Neutrosophic Triplet Ring and Neutrosophic Triplet Field. Mathematics 6, 46; DOI: 10.3390/math6040046

Abstract: Ringsandfieldsaresignificantalgebraicstructuresinalgebraandbothofthemarebased onthegroupstructure.Inthispaper,weattempttoextendthenotionofaneutrosophictripletgroup toaneutrosophictripletringandaneutrosophictripletfield.Weintroduceaneutrosophictriplet ringandstudysomeofitsbasicproperties.Further,wedefinethezerodivisor,neutrosophictriplet subring,neutrosophictripletideal,nilpotentintegralneutrosophictripletdomain,andneutrosophic tripletringhomomorphism.Finally,weintroduceaneutrosophictripletfield.

Keywords: ring;ﬁeld;neutrosophictriplet;neutrosophictripletgroup;neutrosophictripletring; neutrosophictripletﬁeld

1.Introduction

TheconceptofaringfirstarosefromattemptstoproveFermat’slasttheorem[1],startingwithRichard Dedekindinthe1880s.Aftercontributionsfromotherfields,mainlynumbertheory,thenotionofaring wasgeneralizedandfirmlyestablishedduringthe1920sbyEmmyNoetherandWolfgangKrull[2]Modern ringtheory,averyactivemathematicaldiscipline,studiesringsintheirownright.Toexplorerings, mathematicianshavedevisedvariousnotionstobreakringsintosmaller,moreunderstandablepieces, suchasideals,quotientrings,andsimplerings.Inadditiontotheseabstractproperties,ringtheorists alsomakevariousdistinctionsbetweenthetheoriesofcommutativeringsandnoncommutativerings, theformerbelongingtoalgebraicnumbertheoryandalgebraicgeometry.Aparticularlyrichtheory hasbeendevelopedforacertainspecialclassofcommutativerings,knownasﬁelds,whichlies withintherealmofﬁeldtheory.Likewise,thecorrespondingtheoryfornoncommutativerings, thatofnoncommutativedivisionrings,constitutesanactiveresearchinterestfornoncommutative ringtheorists.Sincethediscoveryofamysteriousconnectionbetweennoncommutativeringtheory andgeometryduringthe1980sbyAlainConnes[3 5],noncommutativegeometryhasbecomea particularlyactivedisciplineinringtheory.

Thefoundationofthesubject(i.e.,themappingfromsubfieldstosubgroupsandviceversa)isset upinthecontextofanabsolutelygeneralpairoffields.Inadditiontotheclarificationthatnormally accompaniessuchageneralization,thereareusefulapplicationstoinfinitealgebraicextensionsand totheGaloisTheoryofdifferentialequations[6].Thereisalsoalogicalsimplicitytotheprocedure: everythinghingesonapairofestimatesoffielddegreesandsubgroupindices.Onemightdescribeit asafurtherstepintheDedekind–Artinlinearization[7].

AnearlycontributortothetheoryofnoncommutativeringswastheScottishmathematician Wedderburnwho,in1905,proved“Wedderburn’sTheorem”,namelythateveryﬁnitedivisionringis

and editor) Collected Papers, IX 327

Florentin Smarandache (author

commutativeandsoisaﬁeld[8].Itwasonlyaroundthe1930sthatthetheoriesofcommutativeand noncommutativeringscametogetherandthattheirideasbegantoinﬂuenceeachother.

Neutrosophyisanewbranchofphilosophywhichstudiesthenature,origin,andscopeofneutralities aswellastheirinteractionwithideationalspectra.Theconceptofneutrosophiclogicandaneutrosophic setwasfirstintroducedbyFlorentinSmarandache[9]in1995,whereeachpropositioninneutrosophiclogic isapproximatedtohavethepercentageoftruthinasubset T,thepercentageofindeterminacyinasubset I,andthepercentageoffalsityinasubset F suchthatthisneutrosophiclogiciscalledanextensionoffuzzy logic,especiallytointuitionisticfuzzylogic[10].Thegeneralizationofclassicalsets[9],fuzzysets[11], andintuitionisticfuzzysets[10],etc.,isinfacttheneutrosophicset.Thismathematicaltoolisusedto handleproblemsconsistingofuncertainty,imprecision,indeterminacy,inconsistency,incompleteness, andfalsity.Byutilizingtheideaofneutrosophictheory,VasanthaKandasamyandFlorentinSmarandache studiedneutrosophicalgebraicstructures[12 14]byinsertingaliteralindeterminateelement“I”, where I2 = I,inthealgebraicstructureandthencombining“I”witheachelementofthestructure withrespecttothecorrespondingbinaryoperation,denoted*.Theycallittheneutrosophic element,andthegeneratedalgebraicstructureisthentermedasaneutrosophicalgebraicstructure. Someotherneutrosophicalgebraicstructurescanbeseenasneutrosophicfields[15],neutrosophicvector spaces[16],neutrosophicgroups[17],neutrosophicbigroups[17],neutrosophicN-groups[15], neutrosophicsemigroups[12],neutrosophicbisemigroups[12],neutrosophicN-semigroups[12], neutrosophicloops[12],neutrosophicbiloops[12],neutrosophicN-loop[12],neutrosophicgroupoids[12] andneutrosophicbigroupoids[12]andsoon.

Inthispaper,weintroducetheneutrosophictripletring.Further,wedefinetheneutrosophic tripletzerodivisor,neutrosophictripletsubring,neutrosophictripletideal,nilpotentneutrosophic triplet,integralneutrosophictripletdomain,andneutrosophictripletringhomomorphism. Finally,weintroduceaneutrosophictripletfield.Therestofthepaperisorganizedasfollows. AftertheliteraturereviewinSection 1 andbasicconceptsinSection 2,weintroducetheneutrosophic tripletringinSection 3.Section 4 isabouttheintroductionoftheintegralneutrosophictripletdomain withsomeofitsinterestingproperties,andisalsowherewedeveloptheneutrosophictripletring homomorphism.InSection 5,westudyneutrosophictripletfields.ConclusionsaregiveninSection 6

2.BasicConcepts

Inthissection,alldeﬁnitionsandexampleshavebeentakenfrom[18]toprovidesomebasic conceptsaboutneutrosophictripletsandneutrosophictripletgroups.

Deﬁnition1. Let N beasettogetherwithabinaryoperation ∗.Then N iscalledaneutrosophictripletsetif forany a ∈ N,thereexistsaneutralof“a”called neut(a),differentfromtheclassicalalgebraicunitaryelement, andanoppositeof“a”calledanti(a),withneut(a) andanti(a) belongingtoN,suchthat

a ∗ neut(a)= neut(a) ∗ a = a and

a ∗ anti(a)= anti(a) ∗ a = neut(a)

Theelement a, neut(a),and anti(a) arecollectivelycalledaneutrosophictripletandwedenoteitby (a, neut(a), anti(a)).By neut(a),wemeantheneutralof a,and a isjusttheﬁrstcoordinateofaneutrosophic tripletandnotaneutrosophictriplet[18].

Forthesameelement“a”inN,theremaybemorethanoneneutralneut(a)andmorethanoneopposite anti(a)

Deﬁnition2. Theelement b in (N, ∗) isthesecondcomponent,denotedby neut( ),ofaneutrosophictriplet, ifthereexistotherelements a and c in N suchthat a ∗ b = b ∗ a = a and a ∗ c = c ∗ a = b.Theformed neutrosophictripletis (a, b, c) [12].

Florentin Smarandache (author and editor) Collected Papers, IX 328

Deﬁnition3. Theelement c in (N, ∗) isthethirdcomponent,denotedby anti( ) ofaneutrosophictriplet, ifthereexistotherelements a and b in N suchthat a ∗ b = b ∗ a = a and a ∗ c = c ∗ a = b.Theformed neutrosophictripletis (a, b, c) [12].

Example1. ConsiderZ6 undermultiplicationmodulo 6,where

Z6 = {0,1,2,3,4,5}

Thentheelement 2 givesrisetoaneutrosophictripletbecause neut(2)= 4 = 1,as 2 × 4 = 4 × 2 = 8 ≡ 2(mod6).Also, anti(2)= 2 because 2 × 2 = 4.Thus (2,4,2) isaneutrosophictriplet.Similarly4givesrise toaneutrosophictripletbecause neut(4)= anti(4)= 4 So (4,4,4) isaneutrosophictriplet.However, 3 does notgiverisetoaneutrosophictripletas neut(3)= 5 but anti(3) doesnotexistin Z6,andlastly, 0 gives risetoatrivialneutrosophictripletas neut(0)= anti(0)= 0.Thetrivialneutrosophictripletisdenotedby (0,0,0) [12].

Deﬁnition4. Let (N, ∗) beaneutrosophictripletset.Then N iscalledaneutrosophictripletgroupifthe followingconditionsaresatisﬁed [12].

1.If (N, ∗) iswelldeﬁned,i.e.,foranya, b ∈ N,onehasa ∗ b ∈ N

2.If (N, ∗) isassociative,i.e., (a ∗ b) ∗ c = a ∗ (b ∗ c) foralla, b, c ∈ N.

Theneutrosophictripletgroup,ingeneral,isnotagroupintheclassicalalgebraicsense. Weconsidertheneutrosophicneutralsasreplacingtheclassicalunitaryelement,andtheneutrosophic oppositesasreplacingtheclassicalinverseelements.

Example2. Consider (Z10,#),where # isdeﬁnedas a#b = 3ab(mod10).Then (Z10,#) isaneutrosophic tripletgroupunderthebinaryoperation #,asshownin Table 1 [18].

Table1. Cayleytableofneutrosophictripletgroup (Z10,#) #0123456789 0 0000000000 1 0369258147 2 0628406284 3 0987654321 4 0246802468 5 0505050505 6 0864208642 7 0123456789 8 0482604826 9 0741852963 Itisalsoassociative,i.e., (a#b)#c = a#(b#c).

NowwetaketheLHStoprovetheRHS. (a#b)#c =(3ab)#c = 3(3ab)c = 9abc = 3a(3bc)= 3a(b#c) = a#(b#c)

Florentin Smarandache (author and editor) Collected Papers, IX 329

Foreacha ∈ Z10,wehaveneut(a) inZ10.

Thatis,neut(0) = 0,neut(1) = 7,neut(2) = 2,neut(3) = 7,neut(4) = 2,andsoon.

Similarly,foreacha ∈ Z10,wehaveanti(a) inZ10. Thatis,anti(0) = 0,anti(1) = 9,anti(2) = 2,anti(3) = 3,anti(4) = 1,andsoon.Thus (Z10,#) isa neutrosophictripletgroupwithrespectto #[12].

3.NeutrosophicTripletRings

Inthissection,weintroduceneutrosophictripletringsandstudysomeoftheirbasicproperties andnotions.

Notations1. Sincetheneutrosophictripletringandtheneutrosophictripletﬁeldarealgebraicstructures endowedwithtwointernallaws*and#,inordertoavoidanyconfusion,weusethefollowingnotation: neut ∗ (x) and anti ∗ (x) fortheneutralsandanti’s,respectively,oftheelement x withrespecttothelaw*and neu#(x) andant#(x) fortheneutralsandanti’s,respectively,oftheelement x withrespecttothelaw#.

Deﬁnition5. Let (NTR, ∗,#) beasettogetherwithtwobinaryoperations ∗ and #.Then NTR iscalleda neutrosophictripletringifthefollowingconditionshold:

1. (NTR, ∗) isacommutativeneutrosophictripletgroupwithrespectto ∗;

2. (NTR,#) iswelldeﬁnedandassociative; 3.a#(b ∗ c)=(a#b) ∗ (a#c) and (b ∗ c)#a =(b#a) ∗ (c#a) foralla, b, c ∈ NTR

Remark1. AnNTRingeneralisnotaclassicalring.

Deﬁnition6. Let (NTR, ∗,#) beaneutrosophictripletringandlet a ∈ NTR Wecallthestructureaunitary neutrosophictripletring(UNTR)ifeachelementahasaneut#(a)

Deﬁnition7. Let (NTR, ∗,#) beaneutrosophictripletring.Wecallthestructureacommutativeunitary neutrosophictripletringifitisaUNTRand # iscommutative.

Deﬁnition8. Let (NTR, ∗,#) beaneutrosophictripletringandlet 0 = a ∈ NTR.Ifthereexistsanonzero element b ∈ NTR suchthat b#a = 0,then b iscalledaleftzerodivisorof a.Similarly,anelement b ∈ NTR is calledarightzerodivisorofaifa#b = 0.

Azerodivisorofanelementisonewhichisbothaleftzerodivisorandarightzerodivisorof thatelement.

Theorem1. Let NTR beacommutativeneutrosophictripletringand a, b ∈ NTR suchthata,b, neut#(a), neut#(b), neut(a#b), and anti#(a#b) arecancellableandthat neut#(a), neut#(b) and anti#(a), anti#(b) do existinNTR.Then

1.neut#(a)#neut#(b) = neut#(a#b);and 2.anti#(a)#anti#(b) = anti#(a#b)

Proof.

(1)Considertheleft-handside,with neut#(a)#neut#(b).Multiplyby a totheleftandby b tothe right;thenwehave

a#neut#(a)#neut#(b)#b =(a#neut#(a))#(neut#(b)#b)= a#b,

Florentin Smarandache (author and editor) Collected Papers, IX 330

since#isassociativeNowweconsidertheright-handside;wehave neut#(a#b).Multiplyingby a to theleftandby b totheright,wehave

a#neut#(a#b)#b=(a#b)#neut#(a#b) = a#b, since#isassociativeandcommutative, Thus,LHS= a#b = a#b =RHS.

(2)Consideringtheleft-handside,wehave anti#(a)#anti#(b). Multiplyingby a totheleftandby b totheright,wehave a#anti#(a(#anti#(b)#b =(a#anti#(a))#(anti#(b)#b)= a#b

Nowconsidertheright-handside,wherewehave anti#(a#b). Multiplyingby a totheleftandby b totheright,wehave a#anti#(a#b)#b = (a#b)#anti#(a#b) = a#b, since#isassociativeandcommutative,

Deﬁnition9. Let (NTR, ∗,#) beaneutrosophictripletringandlet S beasubsetof NTR.Then S iscalleda neutrosophictripletsubringofNTRif (S, ∗,#) isaneutrosphictripletring.

Deﬁnition10. Let (NTR, ∗,#) beaneutrosophictripletringand I beasubsetof NTR.Then I iscalleda neutrosophictripletidealofNTRifthefollowingconditionsaresatisﬁed.

1. (I, ∗) isaneutrosophictripletsubgroupof (NTR, ∗);and 2.Forall x ∈ I and r ∈ NTR, x#r ∈ I and r#x ∈ I

Theorem2. Everyneutrosophictripletidealistriviallyaneutrosophictripletsubring,buttheconverseisnot trueingeneral.

Remark2. Let (NTR, ∗,#) beaneutrosophictripletringandleta ∈ NTR.Thenthefollowingaretrue.

1. neut*(a)and anti*(a)ingeneralarenotuniquein NTR

2. neut#(a)and anti#(a)(iftheyexistforsomeelement a)ingeneralarenotuniquein NTR.

Deﬁnition11. Let NTR beaneutrosophictripletringandlet a ∈ NTR.Then a iscalledanilpotentelementif an = 0,forsomepositiveintegern > 1.

Theorem3. Let NTR beacommutativeneutrosophictripletringandlet a ∈ NTR.If a isanilpotent, thefollowingaretrue.

1. (neut ∗ (a))n = neut ∗ (0);and 2. (anti ∗ (a))n = anti ∗ (0).

Proof.

(1)Supposethat a isanilpotentinaneutrosophictripletring NTR.Then,bydeﬁnition, an = 0 forsomepositiveinteger n > 1.

Weprovebymathematicalinduction.

Wecanshowthat neut ∗ (a) ∗ neut ∗ (a)= neut ∗ (a ∗ b) and anti ∗ (a) ∗ anti ∗ (a)= anti ∗ (a ∗ b) inthesamewayaswedidinTheorem 1 abovebyjustreplacingthelaw*by#.

Nowwemake a = b,soweget neut ∗ (a)2 = neut ∗ (a) ∗ neut ∗ (a)= neut(a2)

Florentin Smarandache (author and editor) Collected Papers, IX 331

Weassume,bymathematicalinduction,thatourequalityistrueforanypositiveintegerupto n 1, andweneedtoproveitfor n.

Nowweconsiderleft-handsideof1:

(neut ∗ (a))n =(neut ∗ (a)) ∗ (neut ∗ (a))n 1 = neut ∗ (a ∗ an 1)= neut ∗ (an)= neut ∗ (0)

Thiscompletestheproof.

Theproofof(2)issimilartothatof(1)

4.IntegralNeutrosophicTripletDomainandNeutrosophicTripletRingHomomorphism

Section 4 isabouttheintroductionoftheintegralneutrosophictripletdomainandsomeofits interestingproperties.Moreover,inthissection,wedevelopaneutrosophictripletringhomomorphism.

Deﬁnition12. Let (NTR, ∗,#) beaneutrosophictripletring.Then NTR iscalledacommutativeneutrosophic tripletringifa#b = b#aforalla, b ∈ NTR.

Deﬁnition13. Acommutativeneutrosophictripletring NTR iscalledanintegralneutrosophictripletdomain ifforalla, b ∈ NTR,a#b = 0 impliesa = 0 orb = 0.

Theorem4. LetNTRbeanintegralneutrosophictripletdomain.Thenthefollowingaretrueforalla,b ∈ NTR.

1.Ifneut#(a) andneut#(b) doexist,thenneut#(a)#neut#(b) =0impliesneut#(a)= 0 orneut#(b)= 0; 2.Ifanti#(a) andanti#(b) doexist,thenanti#(a)#anti#(b)= 0 impliesanti#(a)= 0 oranti#(b)= 0.

Proof.

(1)Obvious,since NTR isanintegralneutrosophictripletdomain,and neut#(a) and neut#(b) belongtoNTR.

(2)Obvious,since NTR isanintegralneutrosophictripletdomain,and anti#(a) and anti#(b) belongtoNTR.

Proposition1. Acommutativeneutrosophictripletring NTR isanintegralneutrosophictripletdomainif, andonlyif,whenevera, b, c ∈ NTRsuchthata#b = a#canda = 0, thenb = c

Proof. SupposethatNTRisanintegralneutrosophictripletdomainandlet a, b, c ∈ NTR.Since a = 0 and a ∈ NTR, a isnotazerodivisor,so a iscancellable,i.e., a#b = a#c ⇒ b = c

Reciprocally,let a ∈ NTR,suchthat a = 0;then,byhypothesis, a iscancellable,so a isnotazero divisor.NTRisanintegralneutrosophictripletdomain.

Deﬁnition14. Let (NTR1, ∗,#) and (NTR2, ⊕, ⊗) betwoneutrosophictripletrings.Let f : NTR1 → NTR2 beamapping.Thenfiscalledaneutrosophictripletringhomomorphismifthefollowingconditionsaretrue.

1.f (a ∗ b) = f (a) ⊕ f (b),foralla, b ∈ NTR1.

2.f (a#b) = f (a) ⊗ f (b),foralla, b ∈ NTR1

3.f (neut ∗ (a))= neut⊕( f (a)), foralla ∈ NTR1

4.f (anti ∗ (a))= anti⊕( f (a)), foralla ∈ NTR1

Florentin Smarandache (author and editor) Collected Papers, IX 332

5.NeutrosophicTripletFields

Inthissection,westudyneutrosophictripletﬁeldsandsomeoftheirinterestingproperties.

Deﬁnition15. Let (NTR, ∗,#) beaneutrosophictripletsettogetherwithtwobinaryoperations ∗ and#. Then (NTR, ∗,#) iscalledaneutrosophictripletﬁeldifthefollowingconditionshold.

1. (NTR, ∗) isacommutativeneutrosophictripletgroupwithrespectto*. 2. (NTR,#) isaneutrosophictripletgroupwithrespectto #. 3.a#(b ∗ c) =(a#b) ∗ (a#c) and (b ∗ c)#a =(b#a) ∗ (c#a) foralla, b, c ∈ NTF

Example3. Let X beasetandlet P(X) bethepowersetof X.Then (P(X), ∪, ∩) isaneutrosophictripletﬁeld sinceneut(A)= Aandanti(A)= AforallA ∈ P(X) withrespecttoboth ∪ and ∩.

Proposition2. Aneutrosophictripletﬁeld NTF alwayshasan anti(a) forevery a ∈ NTF withrespecttoboth laws*and#.

Proof. Theproofisstraightforward. Theorem5. Aneutrosophictripletringisnotingeneralaneutrosophictripletﬁeld

Counterexample:

NTR = ({1,2}, ∗,#) *12 1 21 2 12

Neutrosophictriplets: (1,2,1), (2,2,2), ({1,2}, ∗) isacommutativeNTG. #12 1 11 2 11

({1,2},#) iswelldeﬁned,associative,andcommutative.

Fortheelement2thereisno neut#(2) and,consequently,no anti#(2).

Therefore, NTR =({1,2},#)isaneutrosophictripletcommutativesemigroup,butnotaneutrosophic tripletgroup.

Inconclusion, NTR =([1],*,#)isaneutrosophictripletcommutativering,butitisnota neutrosophictripletﬁeld.

Theorem6. AneutrosophictripletﬁeldNTFisnotingeneralanintegralneutrosophictripletdomainNTD

Proof. ConsidertheNTF N =({0,5}, ∗,#),where0 ∗ 0 = 0,0 ∗ 5 = 5 ∗ 0 = 5,5 ∗ 5 = 5.Theneutrosophic tripletswithrespectto*are (0,0,0) and (5,0,5).Hence,weget5 ∗ 5 = 0.

Also0#0 = 0#5 = 5#0 = 5and5#5 = 0.Theneutrosophictripletswithrespectto#are (0,5,0) and (5,0,5).

Aswecansee,5#5 = 0.

Therefore,thisisaNTFwhichisnotanintegralneutrosophictripletdomain.

Theorem7. Assumethat f : NTR1 → NTR2 isaneutrosophictripletringhomomorphism.Thefollowing thenhold.

Florentin Smarandache (author and editor) Collected Papers, IX 333

1. If S isaneutrosophictripletsubring NTR1(∗,#), then f (S) isaneutrosophictripletsubringof NTR2(⊕, ⊗).

2.IfUisaneutrosophictripletsubringofNTR2, thenf 1(U) isaneutrosophictripletsubringofNTR1

3.IfIisaneutrosophictripletidealofNTR2,thenf 1(I) isaneutrosophictripletidealofNTR1

4.Iffisonto,andJisanidealofNTR1,thenf (j) isanidealofNTR2

Proof.

(1)If S isaneutrosophictripletsubring NTR1(∗,#),then f (S) isaneutrosophictripletsubringof NTR2(⊕, ⊗).

Let a, b ∈ S,then a ∗ b ∈ S, neut ∗ (a) ∈ S, anti ∗ (a) ∈ S

Then f (a), f (b) ∈ f (S) and f (a ∗ b) ∈ f (S),but f (a ∗ b) = f (a) ⊕ f (b),since f isahomomorphism. Thus,wehaveprovedthatif f (a), f (b) ∈ f (S),then f (a) ⊕ f (b) ∈ f (S)

Since neut*(a)and anti*(a) ∈ S, f (neut(a)) and f (anti(a)) ∈ f (S) since f isahomomorphism. But f (neut*(a))= neut⊕f (a),and f (anti*(a))= anti⊕f (a). Therefore,if f (a) ∈ f (S),then neut⊕ f (a) = f (neut ∗ (a)) ∈ f (S) and,similarly, anti⊕ f (a) = f (anti ∗ (a)) ∈ f (S).

Now,if a, b ∈ S,then a#b ∈ S.Since a#b ∈ S, f (a#b) ∈ f (S) But f (a#b) = f (a) ⊗ f (b) Therefore,if f (a), f (b) ∈ S,then f (a) ⊗ f (b) = f (a#b) = f (S) (2)Let c, d ∈ U.Then f 1(c), f 1(d) ∈ f 1(U).Also c ⊕ d ∈ U,hence f 1(c ⊕ d) ∈ f 1(U), f 1(c) ∗ f 1(d) ∈ f 1(U) But f 1(c) ∗ f 1(d) = f 1(c ⊕ d), becauseifweapply f onbothsidesweget

f f 1(c) ∗ f 1(d) = f f 1(c ⊕ d) , or f f 1(c) ⊕ f f 1(d) = c ⊕ d or c ⊕ d = c ⊕ d Similarly, f 1(c)# f 1(d) ∈ f 1(U) But f 1(c)# f 1(d) = f 1(c ⊗ d), becauseifweapply f onbothsides,weget

f f 1(c)# f 1(d) = f ( f 1(c ⊗ d)), or f f 1(c) ⊗ f f 1(d) = c ⊗ d, c ⊗ d = c ⊗ d

Florentin Smarandache (author and editor) Collected Papers, IX 334

Since c ∈ U,wehave neut⊕(c)and anti⊕(c) ∈ U, f 1(neut⊕(c))= neut* f 1(c) and f 1(anti⊕(c))= anti* f 1(c) . Weprovethembyapplying f onbothsidesforeachequality.

f(f 1(neut⊕(c)))=f(neut*(f 1((c))), or neut⊕(c) = neut⊕ f f 1(c) , or neut⊕(c)= neut⊕(c). Similarly, f(f 1(anti⊕(c)))=f(anti*(f 1((c))), or anti⊕(c) = anti⊕ f f 1(c) or anti⊕(c)= anti⊕(c)

(3)Let i ∈ I and r ∈ NTR2.Then, i ⊕ r ∈ I,andtherefore, f 1(i ⊕ r) ∈ f 1(I) f 1(i) ∈ f 1(I) and f 1(r) ∈ NTR1.

Weprovethat f 1(i) ∗ f 1(r) = f 1(i ⊕ r) Applying f tobothsides,weget

f ( f 1(i) ∗ f 1(r) = f f 1(i ⊕ r) ; f ( f 1(i)) ⊕ f f 1(r) = i ⊕ r; i ⊕ r = i + r

Therefore,if i ∈ I, r ∈ NTR2,then i ⊕ r ∈ f 1(I) (4)Let j ∈ f ( J) and r ∈ NTR2.Since f isonto,then ∃h ∈ J ⊂ NTR1 suchthat f (h) = j and ∃ s ∈ NTR1 suchthat f (s) = r.Weneedtoprovethat j ⊕ r ∈ f ( J) Applying f 1 tobothsides,weget f 1 (j ⊕ r) ∈ f 1( f ( J)), or f 1(j) ∗ f 1(r) ∈ J or h ∗ s ∈ J

whichistrue,since h ∈ J,whichisanidealin NTR1,while s ∈ NTR1

6.Conclusions

Inthispaper,wepresentedtheneutrosophictripletring.Further,wepresentedthezerodivisor, neutrosophictripletsubring,neutrosophictripletideal,nilpotent,integralneutrosophictripletdomain, andneutrosophictripletringhomomorphism.Finally,wepresentedtheneutrosophictripletfield.Inthe future,wecandevelopneutrosophictripletvectorspaces,neutrosophicmodules,andneutrosophic tripletnearrings,andsoon.

Florentin Smarandache (author and editor) Collected Papers, IX 335

References

1. Kleiner,I.FromNumberstoRing:TheEarlyHistoryofRingTheory.In ElementederMathematik;Springer: Berlin/Heidelberg,Germany,1998;Volume53,pp.18–35.

2. ConferencesoftheMathematicsandStatisticsDepartmentoftheTechnicalUniversityofCatalonia. EmmyNoetherCourse.Availableonline: https://upcommons.upc.edu/bitstream/handle/2117/81399/ CFME-vol-6.pdf?sequence=1&isAllowed=y (accessedon21March2018).

3. Connes,A.Introductiontonon-commutativedifferentialgeometry.In LecturesNotesinPhysics;Springer: Berlin/Heidelberg,Germany,1984;Volume1111,pp.3–16.

4. Connes,A.Non-commutativedifferentialgeometry.In PublicationsMathematics;Springer:Berlin/Heidelberg, Germany,1985;Volume62,pp.257–360.

5. Connes,A.Theactionfunctionalinnon-commutativegeometry.In CommunicationsinMathematicalPhysics; Springer:Berlin/Heidelberg,Germany,1988;Volume11,pp.673–683.

6.Kaplansky,I. AnIntroductiontotheDifferentialAlgebra;Hermann:Paris,France,1957.

7. Kaplansky,I. FieldsandRings,2nded.;TheUniversityofChicagoPress:Chicago,IL,USA,1972;ISBN 0-226-42450-2.

8. Herstein,I.Wedderburn’stheoremandatheoremofJacobson. Am.Math.Mon. 1961, 68,249–251.[CrossRef]

9. Smarandache,F. Neutrosophy:NeutrosophicProbability,Set,andLogic;ProQuestInformation&Learning: AnnArbor,MI,USA,1998;p.105.Availableonline: http://fs.gallup.unm.edu/eBook-neutrosophics6.pdf (accessedon21March2018).

10.Atanassov,K.T.Intuitionisticfuzzysets. FuzzySetsSyst. 1986, 20,87–96.[CrossRef]

11. Zadeh,L.A.Fuzzysets.In FuzzySets,FuzzyLogicandFuzzySystems:SelectedPapersbyLotﬁA.Zadeh; WorldScientiﬁc:RiverEdge,NJ,USA,1996;pp.394–432.

12. Kandasamy,W.B.V.;Smarandache,F. SomeNeutrosophicAlgebraicStructuresandNeutrosophicN-Algebraic Structures;Hexis:Frontigan,France,2006;p.219.

13. Kandasamy,W.B.V.;Smarandache,F. N-AlgebraicStructuresandS-N-AlgebraicStructures;Hexis:Phoenix,AZ, USA,2006;p.209.

14. Kandasamy,W.B.V.;Smarandache,F. BasicNeutrosophicAlgebraicStructuresandTheirApplicationstoFuzzy andNeutrosophicModels;Hexis:Frontigan,France,2004;p.149.

15. Ali,M.;Smarandache,F.;Shabir,M.;Vladareanu,L.GeneralizationofNeutrosophicRingsandNeutrosophic Fields. NeutrosophicSetsSyst. 2014, 5,9–14.

16.Agboola,A.;Akinleye,S.NeutrosophicVectorSpaces. NeutrosophicSetsSyst. 2014, 4,9–18.

17.Agboola,A.;Akwu,A.;Oyebo,Y.Neutrosophicgroupsandsubgroups. Int.J.Math.Comb. 2012, 3,1.

18.Smarandache,F.;Ali,M.Neutrosophictripletgroup. NeuralComput.Appl. 2018, 29,595–601.[CrossRef]

Florentin Smarandache (author and editor) Collected Papers, IX 336

Positive implicative BMBJ-neutrosophic ideals in BCK-algebras

Rajab Ali Borzooei, M. Mohseni Takallo, Florentin Smarandache, Young Bae Jun

Rajab Ali Borzooei, M. Mohseni Takallo, Florentin Smarandache, Young Bae Jun (2018). Positive implicative BMBJ-neutrosophic ideals in BCK-algebras. Neutrosophic Sets and Systems 23, 126-141

Abstract: TheconceptsofapositiveimplicativeBMBJ-neutrosophicidealisintroduced,andseveralpropertiesare investigated.ConditionsforanMBJ-neutrosophicsettobea(positiveimplicative)BMBJ-neutrosophicidealareprovided.RelationsbetweenBMBJ-neutrosophicidealandpositiveimplicativeBMBJ-neutrosophicidealarediscussed. CharacterizationsofpositiveimplicativeBMBJ-neutrosophicidealaredisplayed.

Keywords: MBJ-neutrosophicset;BMBJ-neutrosophicideal;positiveimplicativeBMBJ-neutrosophicideal.

1Introduction

In 1965, L.A. Zadeh [18] introduced the fuzzy set in order to handle uncertainties in many real applications. In 1983, K. Atanassov introdued the notion of intuitionistic fuzzy set as a generalization of fuzzy set. As a more general platform that extends the notions of classic set, (intuitionistic) fuzzy set and interval valued (intuitionistic) fuzzy set, the notion of neutrosophic set is initiated by Smarandache ([13], [14] and [15]). Neutrosophic set is applied to many branchs of sciences. In the aspect of algebraic structures, neutrosophic algebraic structures in BCK/BCI-algebras are discussed in the papers [1], [3], [4], [5], [6], [11], [12], [16] and [17]. In [9], the notion of MBJ-neutrosophic sets is introduced as another generalization of neutrosophic set, and it is applied to BCK/BCI-algebras. Mohseni et al. [9] introduced the concept of MBJ-neutrosophic subalgebras in BCK/BCI-algebras, and investigated related properties. Jun and Roh [7] applied the notion of MBJ-neutrosophic sets to ideals of BCK/BI-algebras, and introduced the concept of MBJ-neutrosophic ideals in BCK/BCI-algebras.

Inthisarticle,weintroducetheconceptsofapositiveimplicativeBMBJ-neutrosophicideal,andinvestigate severalproperties.WeprovideconditionsforanMBJ-neutrosophicsettobea(positiveimplicative)BMBJneutrosophicideal,anddiscussedrelationsbetweenBMBJ-neutrosophicidealandpositiveimplicativeBMBJneutrosophicideal.WeconsidercharacterizationsofpositiveimplicativeBMBJ-neutrosophicideal.

Collected Papers, IX 337

Florentin Smarandache (author and editor)

2Preliminaries

Bya BCI-algebra,wemeanaset X withabinaryoperation ∗ andaspecialelement 0 thatsatisﬁesthe followingconditions:

(I) ((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y)=0,

(II) (x ∗ (x ∗ y)) ∗ y =0, (III) x ∗ x = 0, (IV) x ∗ y =0, y ∗ x =0 ⇒ x = y forall x,y,z ∈ X.Ifa BCI-algebra X satisﬁesthefollowingidentity: (V) (∀x ∈ X)(0 ∗ x =0), then X iscalled a BCK-algebra.

Every BCK/BCI-algebra X satisﬁesthefollowingconditions:

(∀x ∈ X)(x ∗ 0= x) , (2.1) (∀x,y,z ∈ X)(x ≤ y ⇒ x ∗ z ≤ y ∗ z,z ∗ y ≤ z ∗ x) , (2.2) (∀x,y,z ∈ X)((x ∗ y) ∗ z =(x ∗ z) ∗ y) , (2.3) (∀x, y,z ∈ X)((x ∗ z) ∗ (y ∗ z) ≤ x ∗ y) (2.4) where x ≤ y ifandonly if x ∗ y =0

A nonemptysubset S ofa BCK/BCI-algebra X iscalleda subalgebra of X if x ∗ y ∈ S forall x,y ∈ S. Asubset I ofa BCK/BCI-algebra X iscalledan ideal of X ifitsatisﬁes: 0 ∈ I, (2.5) (∀x ∈ X)(∀y ∈ I)(x ∗ y ∈ I ⇒ x ∈ I) . (2.6)

A subset I of a BCK-algebra X is called a positive implicative ideal of X (see [8]) if it satisfies (2.5) and (∀x,y,z ∈ X)(((x ∗ y) ∗ z ∈ I,y ∗ z ∈ I ⇒ x ∗ z ∈ I) . (2.7)

Note from [8] that a subset I of a BCK-algebra X is a positive implicative ideal of X if and only if it is an ideal of X which satisﬁes the condition (∀x,y ∈ X)((x ∗ y) ∗ y ∈ I ⇒ x ∗ y ∈ I). (2.8)

Byan intervalnumber we meana closedsubinterval a =[a ,a+] of I, where 0 ≤ a ≤ a+ ≤ 1. Denote by [I] thesetofall intervalnumbers.Letusdefinewhatisknownas refinedminimum (briefly, rmin)and refinedmaximum (briefly, rmax) oftwoelementsin [I] Wealsodefinethesymbols“ ”,“ ”,“=”incaseof twoelements in [I]. Considertwointervalnumbers a1 := a1 ,a+ 1 and a2 := a2 ,a+ 2 . Then rmin {a1, a2} = min a1 ,a2 , min a+ 1 ,a+ 2 , rmax {˜ a1, ˜ a2} = max a1 ,a2 , max a+ 1 ,a+ 2 , ˜ a1 ˜ a2 ⇔ a1 ≥ a2 ,a+ 1 ≥ a+ 2 ,

Florentin Smarandache (author and editor) Collected Papers, IX 338

andsimilarlywemayhave a1 a2 and a1 =˜ a2.Tosay a1 a2 (resp. a1 ≺ a2)wemean a1 a2 and a1 =˜ a2 (resp. a1 a2 and a1 =˜ a2).Let ai ∈ [I] where i ∈ Λ. Wedeﬁne rinf i∈Λ ai = inf i∈Λ ai , inf i∈Λ a+ i and rsup i∈Λ ai = sup i∈Λ ai , sup i∈Λ a+ i .

Let X beanonemptyset.Afunction A : X → [I] iscalledan interval-valuedfuzzyset (brieﬂy,an IVFset) in X. Let [I]X standforthesetofallIVFsetsin X. Forevery A ∈ [I]X and x ∈ X,A(x)=[A (x),A+(x)] iscalledthe degree of membershipofanelement x to A, where A : X → I and A+ : X → I arefuzzysets in X whicharecalleda lowerfuzzyset andan upperfuzzyset in X, respectively.Forsimplicity,wedenote A =[A ,A+]

Let X be a non-empty set. A neutrosophic set (NS) in X (see [14]) is a structure of the form: A := { x; AT (x),AI (x),AF (x) | x ∈ X} where AT : X → [0, 1] isatruthmembershipfunction, AI : X → [0, 1] isanindeterminatemembership function, and AF : X → [0, 1] isafalsemembershipfunction.

We refer the reader to the books [2, 8] for further information regarding BCK/BCI-algebras, and to the site “http://fs.gallup.unm.edu/neutrosophy.htm” for further information regarding neutrosophic set theory.

Let X be a non-empty set. By an MBJ-neutrosophic set in X (see [9]), we mean a structure of the form:

A := { x; MA(x), BA(x),JA(x) | x ∈ X}

where MA and JA are fuzzysetsin X,whicharecalledatruthmembershipfunctionandafalsemembership function,respectively,and ˜ BA isanIVFsetin X whichiscalledanindeterminateinterval-valuedmembership function.

Forthesakeofsimplicity,weshallusethesymbol A =(MA, BA,JA) fortheMBJ-neutrosophicset A := { x; MA(x), ˜ BA(x),JA(x) | x ∈ X}

    , (2.10)

Let X be a BCK/BCI-algebra. An MBJ-neutrosophic set A = (MA, B A, JA) in X is called a BMBJneutrosophic ideal of X (see [10]) if it satisfies (∀x ∈ X)(MA(x)+ BA (x) ≤ 1, B+ A (x)+ JA(x) ≤ 1), (2.9) (∀x ∈ X)     MA(0) ≥ MA(x) BA (0) ≤ BA (x) B+ A (0) ≥ B+ A (x) JA(0) ≤ JA(x) Florentin Smarandache (author and editor) Collected Papers, IX 339

MA(x) ≥ min{MA(x ∗ y),MA(y)} BA (x) ≤ max{BA (x ∗ y),BA (y)} B+ A (x) ≥ min{B+ A (x ∗ y),B+ A (y)} JA(x) ≤ max{JA(x ∗ y),JA(y)}

3PositiveimplicativeBMBJ-neutrosophicideals

    (2.11)

Inwhatfollows,let X denotea BCK-algebraunlessotherwisespeciﬁed. Deﬁnition3.1. AnMBJ-neutrosophicset A =(MA, BA,JA) in X iscalleda positiveimplicativeBMBJneutrosophic ideal of X if it satisfies (2.9), (2.10) and (∀x,y,z ∈ X)    

MA(x ∗ z) ≥ min{MA((x ∗ y) ∗ z),MA(y ∗ z)} BA (x ∗ z) ≤ max{BA ((x ∗ y) ∗ z),BA (y ∗ z)} B+ A (x ∗ z) ≥ min{B+ A ((x ∗ y) ∗ z),B+ A (y ∗ z)} JA(x ∗ z) ≤ max{JA((x ∗ y) ∗ z),JA(y ∗ z)}

Considera BCK-algebra X = {0, 1, 2, 3, 4} withthebinaryoperation

    . (3.1)

and (∀x,y ∈ X)    

Florentin Smarandache (author and editor) Collected Papers, IX 340

Example3.2.

∗ whichisgiveninTable 1. Let A = (MA, B ˜ A, JA) be an MBJ-neutrosophic set in X defined by Table 2. It is routine to verify that Table1:Cayleytableforthebinaryoperation“∗” ∗ 01234 000000 110000 222002 333303 444440 Table2:MBJ-neutrosophicset A =(MA, BA,JA) XMA(x) BA(x) JA(x) 00 71[0 04, 0 09]0 22 10 61[0 03, 0 08]0 55 20.51[0.02, 0.06]0.55 30.41[0.01, 0.03]0.77 40 31[0 02, 0 05]0 99 A =(MA, BA,JA) isapositiveimplicativeBMBJ-neutrosophicidealof X.

Theorem 3.3. Every positive implicative BMBJ-neutrosophic ideal is a BMBJ-neutrosophic ideal. Proof. The condition (2.11) is induced by taking z = 0 in (3.1) and using (2.1). Hence every positive implica-tive BMBJ-neutrosophic ideal is a BMBJ-neutrosophic ideal. The converse of Theorem 3.3 is not true as seen in the following example.

Example3.4. Considera BCK-algebra X = {0, 1, 2, 3} withthebinaryoperation ∗ whichisgiveninTable 3

Table3:Cayleytableforthebinaryoperation“∗” ∗ 0123 00000 11001 22102 33330 Let A =(MA, ˜ BA,JA) beanMBJ-neutrosophicsetin X deﬁnedbyTable 4

Table4:MBJ-neutrosophicset A =(MA, ˜ BA,JA) XMA(x) BA(x) JA(x) 00 6[0 04, 0 09]0 3 10.5[0.03, 0.08]0.7 20 5[0 03, 0 08]0 7 30 3[0 01, 0 03]0 5

Itisroutinetoverifythat A =(MA, ˜ BA

,JA) isaBMBJ-neutrosophicidealof X

implicativeMBJ-neutrosophicidealof X since MA(2 ∗ 1)=0.5 < 0.6=min{MA((2 ∗ 1) ∗ 1)

A(1 ∗ 1)},

˜ B

(∀x,y ∈ X) x ≤ y ⇒ MA(x) ≥

A(y)

≤

)

(x) ≥

≤

)

Florentin Smarandache (author and editor) Collected Papers, IX 341

.Butitisnotapositive

Lemma3.5. EveryBMBJ-neutrosophicideal

=(MA,

A,JA) of X satisﬁesthefollowingassertion.

,BA (x)

BA (y

, B

A (y),JA(x)

JA(y

(3.2) Proof. Assumethat x ≤ y forall x,y ∈ X.Then x ∗ y =0,andso MA(x) ≥ min{MA(x ∗ y),MA(y)} =min{MA(0),MA(y)} = MA(y),

BA (x) ≤ max{BA (x ∗ y),BA (y)} =max{BA (0),BA (y)} = BA (y),

B+ A (x) ≥ min{B+ A (x ∗ y),B+ A (y)} =min{B+ A (0),B+ A (y)} = B+ A (y), and JA(x) ≤ max{JA(x ∗ y),JA(y)} =max{JA(0),JA(y)} = JA(y).

Thiscompletestheproof. WeprovideconditionsforaBMBJ-neutrosophicidealtobeapositiveimplicativeBMBJ-neutrosophic ideal.

Theorem3.6. AnMBJ-neutrosophicset A =(MA, ˜ BA,JA) in X isapositiveimplicativeBMBJ-neutrosophic idealof X ifandonlyifitisaBMBJ-neutrosophicidealof X andsatisﬁesthefollowingcondition. (∀x,y ∈ X)     MA(x ∗ y) ≥ MA((x ∗ y) ∗ y) BA (x ∗ y) ≤ BA ((x ∗ y) ∗ y) B+ A (x ∗ y) ≥ B+ A ((x ∗ y) ∗ y) JA(x ∗ y) ≤ JA((x ∗ y) ∗ y)

    (3.3)

Proof. Assume that A = (MA, B A, JA) is a positive implicative MBJ-neutrosophic ideal of X. If z is replaced by y in (3.1), then MA(x ∗ y) ≥ min{MA((x ∗ y) ∗ y),MA(y ∗ y)} =min{MA((x ∗ y) ∗ y), MA(0)} = MA((x ∗ y) ∗ y), BA (x ∗ y) ≤ max{BA ((x ∗ y) ∗ y),BA (y ∗ y)} =max{BA ((x ∗ y) ∗ y),BA (0)} = BA ((x ∗ y) ∗ y),

B+ A (x ∗ y) ≥ min{B+ A ((x ∗ y) ∗ y),B+ A (y ∗ y)} =min{B+ A ((x ∗ y) ∗ y), B+ A (0)} = B+ A ((x ∗ y) ∗ y), and JA(x ∗ y) ≤ max{JA((x ∗ y) ∗ y),JA(y ∗ y)} =max{JA((x ∗ y) ∗ y),JA(0)} = JA((x ∗ y) ∗ y) forall x,y ∈ X

Conversely, let A = (MA, B A, JA) be an MBJ-neutrosophic ideal of X satisfying the condition (3.3). Since ((x ∗ z) ∗ z) ∗ (y ∗ z) ≤ (x ∗ z) ∗ y =(x ∗ y) ∗ z

Florentin Smarandache (author and editor) Collected Papers, IX 342

for all x, y, z ∈ X, it follows from Lemma 3.5 that

MA((x ∗ y) ∗ z) ≤ MA(((x ∗ z) ∗ z) ∗ (y ∗ z)), BA ((x ∗ y) ∗ z) ≥ BA (((x ∗ z) ∗ z) ∗ (y ∗ z)), B+ A ((x ∗ y) ∗ z) ≤ B+ A (((x ∗ z) ∗ z) ∗ (y ∗ z)), JA((x ∗ y) ∗ z) ≥ JA(((x ∗ z) ∗ z) ∗ (y ∗ z))

for all x, y, z ∈ X. Using (3.3), (2.11) and (3.4), we have

MA(x ∗ z) ≥ MA((x ∗ z) ∗ z) ≥ min{MA(((x ∗ z) ∗ z) ∗ (y ∗ z)),MA(y ∗ z)}

≥ min{MA((x ∗ y) ∗ z),MA(y ∗ z)},

BA (x ∗ z) ≤ BA ((x ∗ z) ∗ z) ≤ max{BA (((x ∗ z) ∗ z) ∗ (y ∗ z)),BA (y ∗ z)}

≤ max{BA ((x ∗ y) ∗ z),BA (y ∗ z)}, B+ A (x ∗ z) ≥ B+ A ((x ∗ z) ∗ z) ≥ min{B+ A (((x ∗ z) ∗ z) ∗ (y ∗ z)),B+ A (y ∗ z)} ≥ min{B+ A ((x ∗ y) ∗ z),B+ A (y ∗ z)}, and JA(x ∗ z) ≤ JA((x ∗ z) ∗ z) ≤ max{JA(((x ∗ z) ∗ z) ∗ (y ∗ z)),JA(y ∗ z)} ≤ max{JA((x ∗ y) ∗ z),JA(y ∗ z)}

forall x,y,z ∈ X.Therefore A = (MA, ˜ BA,JA) isapositiveimplicativeBMBJ-neutrosophicidealof X GivenanMBJ-neutrosophicset A =(MA, BA,JA) in X,weconsiderthefollowingsets.

U (MA; t):= {x ∈ X | MA(x) ≥ t}, L(BA ; α ):= {x ∈ X | BA (x) ≤ α }, U (B+ A ; α+):= {x ∈ X | B+ A (x) ≥ α+}, L(JA; s):= {x ∈ X | JA(x) ≤ s} where t, s,α ,α+ ∈ [0, 1]

Lemma3.7 ([10]). An MBJ-neutrosophicset A =(MA, ˜ BA,JA) in X is aBMBJ-neutrosophicidealof X ifandonlyifthenon-emptysets U (MA; t), L(BA ; α ), U (B+ A ; α+) and L(JA; s) are idealsof X forall t,s,α .α+ ∈ [0, 1].

Theorem3.8. AnMBJ-neutrosophic set A =(MA, ˜ BA,JA) in X isapositiveimplicativeBMBJ-neutrosophic idealof X ifandonlyifthenon-emptysets U (MA; t), L(BA ; α ), U (B+ A ; α+) and L(JA; s) are positive implicativeidealsof X forall t, s,α .α+ ∈ [0, 1].

Proof. Supposethat A =(MA, ˜ BA,JA) is apositiveimplicativeBMBJ-neutrosophicidealof X.Then A = (MA, B A, JA) is a BMBJ-neutrosophic ideal of X by Theorem 3.3. It follows from Lemma 3.7 that the non-emptysets U (MA; t), L(BA ; α ), U (B+ A ; α+) and L(JA; s) areideals of X forall t,s,α .α+ ∈ [0, 1].Let

(3.4)

Florentin Smarandache (author and editor) Collected Papers, IX 343

x,y,a,b,c,d,u,v ∈ X besuchthat (x ∗ y) ∗ y ∈ U (MA; t), (a ∗ b) ∗ b ∈ L(BA ; α ), (c ∗ d) ∗ d ∈ U (B+ A ; α+) and (u ∗ v) ∗ v ∈ L(JA; s) Using Theorem 3.6, we have

MA(x ∗ y) ≥ MA((x ∗ y) ∗ y) ≥ t, thatis, x ∗ y ∈ U (MA; t), BA (a ∗ b) ≤ BA ((a ∗ b) ∗ b) ≤ α , thatis, a ∗ b ∈ L(BA ; α ), B+ A (c ∗ d) ≥ B+ A ((c ∗ d) ∗ d) ≥ α+ , that is, c ∗ d ∈ U (B+ A ; α+), JA(u ∗ v) ≤ JA((u ∗ v) ∗ v) ≤ s, thatis, u ∗ v ∈ L(JA; s) Therefore U (MA; t), L(BA ; α ), U (B+ A ; α+) and L(JA; s) arepositiveimplicativeidealsof X forall t,s,α .α+ ∈ [0, 1].

Conversely,supposethat thenon-emptysets U (MA; t), L(BA ; α ), U (B+ A ; α+) and L(JA; s) arepositive implicativeidealsof X for all t,s,α .α+ ∈ [0, 1].Then U (MA; t), L(BA ; α ), U (B+ A ; α+) and L(JA; s) are idealsof X forall t, s,α .α+ ∈ [0, 1].ItfollowsfromLemma 3.7 that A =(MA, BA,JA) isaBMBJneutrosophic idealof X. Assumethat MA(x0 ∗ y0) <MA((x0 ∗ y0) ∗ y0)= t0 forsome x0,y0 ∈ X. Then (x0 ∗y0)∗y0 ∈ U (MA; t0) and x0 ∗y0 / ∈ U (MA; t0),whichisacontradiction.Thus MA(x∗y) ≥ MA((x∗y)∗y) forall x,y ∈ X.Similarly,we have B+ A (x ∗ y) ≥ B+ A ((x ∗ y) ∗ y) forall x,y ∈ X.Ifthereexist a0,b0 ∈ X suchthat JA(a0 ∗ b0) >JA((a0 ∗ b0) ∗ b0)= s0,then (a0 ∗ b0) ∗ b0 ∈ L(JA; s0) and a0 ∗ b0 / ∈ L(JA; s0). This isimpossible,andthus JA(a ∗ b) ≤ JA((a ∗ b) ∗ b) forall a,b ∈ X.Bythesimilar way,weknowthat BA (a ∗ b) ≤ BA ((a ∗ b) ∗ b) forall a,b ∈ X.ItfollowsfromTheorem 3.6 that A =(MA, ˜ BA,JA) isapositive implicativeBMBJ-neutrosophicidealof X

Theorem3.9. Let A =(MA, BA,JA) beaBMBJ-neutrosophicidealof X.Then A =(MA, BA,JA) is positiveimplicativeifandonlyifitsatisﬁesthefollowingcondition.

(∀x,y,z ∈ X)     MA((x ∗ z) ∗ (y ∗ z)) ≥ MA((x ∗ y) ∗ z), BA ((x ∗ z) ∗ (y ∗ z)) ≤ BA ((x ∗ y) ∗ z), B+ A ((x ∗ z) ∗ (y ∗ z)) ≥ B+ A ((x ∗ y) ∗ z), JA((x ∗ z) ∗ (y ∗ z)) ≤ JA((x ∗ y) ∗ z).

    (3.5)

Proof. Assumethat A =(MA, ˜ BA, JA) isapositiveimplicativeBMBJ-neutrosophicidealof X.Then A =(MA, BA,JA) isaBMBJ-neutrosophicidealof X byTheorem 3.3,andsatisﬁesthecondition(3.3)by Theorem 3.6.Since ((x ∗ (y ∗ z)) ∗ z) ∗ z =((x ∗ z) ∗ (y ∗ z)) ∗ z ≤ (x ∗ y) ∗ z for all x, y, z ∈ X, it follows from Lemma 3.5 that MA((x ∗ y) ∗ z) ≤ MA(((x ∗ (y ∗ z)) ∗ z) ∗ z), BA ((x ∗ y) ∗ z) ≥ BA (((x ∗ (y ∗ z)) ∗ z) ∗ z), B+ A ((x ∗ y) ∗ z) ≤ B+ A (((x ∗ (y ∗ z)) ∗ z) ∗ z), JA((x ∗ y) ∗ z) ≥ JA(((x ∗ (y ∗ z)) ∗ z) ∗ z) (3.6)

Florentin Smarandache (author and editor) Collected Papers, IX 344

for all x, y, z ∈ X Using (2.3), (3.3) and (3.6), we have

MA((x ∗ z) ∗ (y ∗ z))= MA((x ∗ (y ∗ z)) ∗ z) ≥ MA(((x ∗ (y ∗ z)) ∗ z) ∗ z) ≥ MA((x ∗ y) ∗ z),

BA ((x ∗ z) ∗ (y ∗ z))= BA ((x ∗ (y ∗ z)) ∗ z)

≤ BA (((x ∗ (y ∗ z)) ∗ z) ∗ z)

≤ BA ((x ∗ y) ∗ z),

B+ A ((x ∗ z) ∗ (y ∗ z))= B+ A ((x ∗ (y ∗ z)) ∗ z)

≥ B+ A (((x ∗ (y ∗ z)) ∗ z) ∗ z) ≥ B+ A ((x ∗ y) ∗ z), and

JA((x ∗ z) ∗ (y ∗ z))= JA((x ∗ (y ∗ z)) ∗ z)

≤ JA(((x ∗ (y ∗ z)) ∗ z) ∗ z) ≤ JA((x ∗ y) ∗ z).

Hence (3.5) is valid.

Conversely,let A =(MA, ˜ BA,JA) beaBMBJ-neutrosophicidealof X whichsatisﬁesthecondition(3.5). Ifweput z = y in(3.5)anduse(III)and(2.1),thenweobtainthecondition(3.3).Therefore A =(MA, BA, JA) isapositiveimplicativeBMBJ-neutrosophicidealof X byTheorem 3.6.

Theorem3.10. Let A =(MA, BA,JA) beanMBJ-neutrosophicsetin X.Then A =(MA, BA,JA) isa positiveimplicativeBMBJ-neutrosophicidealof X ifandonlyifitsatisﬁesthecondition (2.9), (2.10) and (∀x,y,z ∈ X)     MA(x ∗ y) ≥ min{MA(((x ∗ y) ∗ y) ∗ z),MA(z)}, BA (x ∗ y) ≤ max{BA (((x ∗ y) ∗ y) ∗ z),BA (z)}, B+ A (x ∗ y) ≥ min{B+ A (((x ∗ y) ∗ y) ∗ z),B+ A (z)}, JA(x ∗ y) ≤ max{JA(((x ∗ y) ∗ y) ∗ z),JA(z)}.

    (3.7) Proof. Assumethat A =(MA, BA,JA) isapositiveimplicativeBMBJ-neutrosophicidealof X.Then A =(MA, BA,JA) isaBMBJ-neutrosophicidealof X (seeTheorem 3.3),andsotheconditions(2.9)and (2.10)arevalid.Using(2.11),(III),(2.1),(2.3)and(3.5),wehave

MA(x ∗ y) ≥ min{MA((x ∗ y) ∗ z),MA(z)}

=min{MA(((x ∗ z) ∗ y) ∗ (y ∗ y)),MA(z)} ≥ min{MA(((x ∗ z) ∗ y) ∗ y),MA(z)} =min{MA(((x ∗ y) ∗ y) ∗ z),MA(z)},

Florentin Smarandache (author and editor) Collected Papers, IX 345

BA (x ∗ y) ≤ max{BA ((x ∗ y) ∗ z),BA (z)}

=max{BA (((x ∗ z) ∗ y) ∗ (y ∗ y)),BA (z)}

≤ max{BA (((x ∗ z) ∗ y) ∗ y),BA (z)}

=max{BA (((x ∗ y) ∗ y) ∗ z),BA (z)},

B+ A (x ∗ y) ≥ min{B+ A ((x ∗ y) ∗ z),B+ A (z)}

=min{B+ A (((x ∗ z) ∗ y) ∗ (y ∗ y)),B+ A (z)}

≥ min{B+ A (((x ∗ z) ∗ y) ∗ y),B+ A (z)} =min{B+ A (((x ∗ y) ∗ y) ∗ z),B+ A (z)}, and

JA(x ∗ y) ≤ max{JA((x ∗ y) ∗ z),JA(z)}

=max{JA(((x ∗ z) ∗ y) ∗ (y ∗ y)),JA(z)}

≤ max{JA(((x ∗ z) ∗ y) ∗ y),JA(z)} =max{JA(((x ∗ y) ∗ y) ∗ z),JA(z)} forall x,y,z ∈ X

Conversely,let A =(MA, BA,JA) beanMBJ-neutrosophicsetin X whichsatisﬁesconditions(2.9), (2.10)and(3.7).Then

MA(x)= MA(x ∗ 0) ≥ min{MA(((x ∗ 0) ∗ 0) ∗ z),MA(z)} =min{MA(x ∗ z),MA(z)}, BA (x)= BA (x ∗ 0) ≤ max{BA (((x ∗ 0) ∗ 0) ∗ z),BA (z)} =max{BA (x ∗ z),BA (z)}, B+ A (x)= B+ A (x ∗ 0) ≥ min{B+ A (((x ∗ 0) ∗ 0) ∗ z),B+ A (z)} =min{B+ A (x ∗ z),B+ A (z)}, and JA(x)= JA(x ∗ 0) ≤ max{JA(((x ∗ 0) ∗ 0) ∗ z),JA(z)} =max{JA(x ∗ z),JA(z)} forall x,z ∈ X.Hence A =(MA, ˜ BA,JA) isaBMBJ-neutrosophicidealof X.Taking z =0 in(3.7)and using(2.1)and(2.10)implythat MA(x ∗ y) ≥ min{MA(((x ∗ y) ∗ y) ∗ 0),MA(0)} =min{MA((x ∗ y) ∗ y),MA(0)} = MA((x ∗ y) ∗ y), BA (x ∗ y) ≤ max{BA (((x ∗ y) ∗ y) ∗ 0),BA (0)} =max{BA ((x ∗ y) ∗ y),BA (0)} = BA ((x ∗ y) ∗ y),

Florentin Smarandache (author and editor) Collected Papers, IX 346

B+ A (x ∗ y) ≥ min{B+ A (((x ∗ y) ∗ y) ∗ 0),B+ A (0)} =min{B+ A ((x ∗ y) ∗ y),B+ A (0)} = B+ A ((x ∗ y) ∗ y), and JA(x ∗ y) ≤ max{JA(((x ∗ y) ∗ y) ∗ 0),JA(0)} =max{JA((x ∗ y) ∗ y),JA(0)} = JA((x ∗ y) ∗ y)

forall x,y ∈ X.ItfollowsfromTheorem 3.6 that A =(MA, ˜ BA,JA) isapositiveimplicativeBMBJneutrosophicidealof X Proposition3.11. EveryBMBJ-neutrosophicideal A =(MA, BA,JA) of X satisﬁesthefollowingassertion. x ∗ y ≤ z ⇒

MA(x) ≥ min{MA(y),MA(z)}, BA (x) ≤ max{BA (y),BA (z)}, B+ A (x) ≥ min{B+ A (y),B+ A (z)}, JA(x) ≤ max{JA(y),JA(z)} (3.8) forall x,y,z ∈ X Proof. Let x,y,z ∈ X besuchthat x ∗ y ≤ z.Then MA(x ∗ y) ≥ min{MA((x ∗ y) ∗ z),MA(z)} =min{MA(0),MA(z)} = MA(z), BA (x ∗ y) ≤ max{BA ((x ∗ y) ∗ z),BA (z)} =max{BA (0),BA (z)} = BA (z), B+ A (x ∗ y) ≥ min{B+ A ((x ∗ y) ∗ z),B+ A (z)} =min{B+ A (0),B+ A (z)} = B+ A (z), and JA(x ∗ y) ≤ max{JA((x ∗ y) ∗ z),JA(z)} =max{JA(0),JA(z)} = JA(z)

Itfollowsthat

MA(x) ≥ min{MA(x ∗ y),MA(y)}≥ min{MA(y),MA(z)}, BA (x) ≤ max{BA (x ∗ y),BA (y)}≤ max{BA (y),BA (z)}, B+ A (x) ≥ min{B+ A (x ∗ y),B+ A (y)}≥ min{B+ A (y),B+ A (z)}, and JA(x) ≤ max{JA(x ∗ y),JA(y)}≤ max{JA(y),JA(z)}

Thiscompletestheproof.

 







Florentin Smarandache (author and editor) Collected Papers, IX 347

WeprovideconditionsforanMBJ-neutrosophicsettobeaBMBJ-neutrosophicidealin BCK/BCI algebras.

Theorem 3.12. Every MBJ-neutrosophic set in X satisfying (2.9), (2.10) and (3.8) is a BMBJ-neutrosophic ideal of X

Proof. Let A =(MA, ˜ BA,JA) beanMBJ-neutrosophicsetin X satisfying(2.9),(2.10)and(3.8).Notethat x ∗ (x ∗ y) ≤ y forall x,y ∈ X.Itfollowsfrom(3.8)that

MA(x) ≥ min{MA(x ∗ y), MA(y)},

BA (x) ≤ max{BA (x ∗ y),BA (y)},

B+ A (x) ≥ min{B+ A (x ∗ y),B+ A (y)}, and JA(x) ≤ max{JA(x ∗ y),JA(y)}

Therefore A =(MA, BA,JA) isaBMBJ-neutrosophic idealof X.

Theorem3.13. AnMBJ-neutrosophicset A =(MA, BA,JA) in X isaBMBJ-neutrosophicidealof X ifand onlyif (MA,BA ) and (B+ A ,JA) areintuitionisticfuzzyidealsof X Proof. Straightforward.

Theorem3.14. Givenanideal I of X,let A =(MA, ˜ BA,JA) beanMBJ-neutrosophicsetin X deﬁnedby

MA(x)= t if x ∈ I, 0 otherwise, BA (x)= α if x ∈ I, 1 otherwise, B+ A (x)= α+ if x ∈ I, 0 otherwise, JA(x)= s if x ∈ I, 1 otherwise, where t,α+ ∈ (0, 1] and s,α ∈ [0, 1) with t + α ≤ 1 and s + α+ ≤ 1.Then A =(MA, BA,JA) isa BMBJ-neutrosophicidealof X suchthat U (MA; t)= L(BA ; α )= U (B+ A ; α+)= L(JA; s)= I.

Proof. Itisclearthat A =(MA, BA,JA) satisﬁesthecondition(2.9)and U (MA; t)= L(BA ; α )= U (B+ A ; α+)= L(JA; s)= I.Let x,y ∈ X.If x ∗ y ∈ I and y ∈ I,then x ∈ I andso MA(x)= t =min{MA(x ∗ y),MA(y)} BA (x)= α =max{BA (x ∗ y),BA (y)}, B+ A (x)= α+ =min{B+ A (x ∗ y),B+ A (y)}, JA(x)= s =max{JA(x ∗ y),JA(y)}

Florentin Smarandache (author and editor) Collected Papers, IX 348

Ifanyoneof x ∗ y and y iscontainedin I,say x ∗ y ∈ I,then MA(x ∗ y)= t, BA (x ∗ y)= α , JA(x ∗ y)= s, MA(y)=0, BA (y)=1, B+ A (y)=0 and JA(y)=1.Hence

MA(x) ≥ 0=min{t, 0} =min{MA(x ∗ y),MA(y)}

BA (x) ≤ 1=max{BA (x ∗ y),BA (y)},

B+ A (x) ≥ 0=min{B+ A (x ∗ y),B+ A (y)},

JA(x) ≤ 1=max{s, 1} =max{JA(x ∗ y),JA(y)}

If x ∗ y/ ∈ I and y/ ∈ I,then MA(x ∗ y)=0= MA(y), BA (x ∗ y)=1= BA (y), B+ A (x ∗ y)=0= B+ A (y) and JA(x ∗ y)=1= JA(y).Itfollowsthat

MA(x) ≥ 0=min{MA(x ∗ y),MA(y)}

BA (x) ≤ 1=max{BA (x ∗ y),BA (y)}, B+ A (x) ≥ 0=min{B+ A (x ∗ y),B+ A (y)}, JA(x) ≤ 1=max{JA(x ∗ y),JA(y)}

Itisobviousthat MA(0) ≥ MA(x), BA (0) ≤ BA (x), B+ A (0) ≥ B+ A (x) and JA(0) ≤ JA(x) forall x ∈ X. Therefore A =(MA, BA,JA) isaBMBJ-neutrosophicidealof X.

Lemma3.15. Foranynon-emptysubset I of X,let A =(MA, BA,JA) beanMBJ-neutrosophicsetin X whichisgiveninTheorem 3.14.If A =(MA, BA,JA) isaBMBJ-neutrosophicidealof X,then I isanideal of X.

Proof. Obviously, 0 ∈ I.Let x,y ∈ X besuchthat x ∗ y ∈ I and y ∈ I.Then MA(x ∗ y)= t = MA(y), BA (x ∗ y)= α = BA (y), B+ A (x ∗ y)= α+ = B+ A (y) and JA(x ∗ y)= s = JA(y).Thus MA(x) ≥ min{MA(x ∗ y),MA(y)} = t, BA (x) ≤ max{BA (x ∗ y),BA (y)} = α , B+ A (x) ≥ min{B+ A (x ∗ y),B+ A (y)} = α+ , JA(x) ≤ max{JA(x ∗ y),JA(y)} = s, andhence x ∈ I.Therefore I isanidealof X.

Theorem3.16. Foranynon-emptysubset I of X,let A =(MA, ˜ BA,JA) beanMBJ-neutrosophicsetin X whichisgiveninTheorem 3.14.If A =(MA, BA,JA) isapositiveimplicativeBMBJ-neutrosophicidealof X,then I isapositiveimplicativeidealof X.

Proof. If A =(MA, ˜ BA,JA) isapositiveimplicativeBMBJ-neutrosophicidealof X,then A =(MA, ˜ BA, JA) isaBMBJ-neutrosophicidealof X andsatisﬁes(3.3)byTheorem 3.6.ItfollowsfromLemma 3.15 that I isanidealof X.Let x,y ∈ X besuchthat (x ∗ y) ∗ y ∈ I.Then MA(x ∗ y) ≥ MA((x ∗ y) ∗ y)= t,BA (x ∗ y) ≤ BA ((x ∗ y) ∗ y)= α , B+ A (x ∗ y) ≥ B+ A ((x ∗ y) ∗ y)= α+,JA(x ∗ y) ≤ JA((x ∗ y) ∗ y)= s, andso x ∗ y ∈ I.Therefore I isapositiveimplicativeidealof X.

Florentin Smarandache (author and editor) Collected Papers, IX 349

Proposition3.17. EverypositiveimplicativeBMBJ-neutrosophicideal A =(MA, BA,JA) of X satisﬁesthe followingcondition. (((x ∗ y) ∗ y) ∗ a) ∗ b =0 ⇒ 

MA(x ∗ y) ≥ min{MA(a),MA(b)}, BA (x ∗ y) ≤ max{BA (a),BA (b)}, B+ A (x ∗ y) ≥ min{B+ A (a),B+ A (b)}, JA(x ∗ y) ≤ max{JA(a),JA(b)} (3.9) forall x,y,a, b ∈ X

Proof. Assumethat A =(MA, BA,JA) isapositiveimplicativeBMBJ-neutrosophicidealof X.Then A =(MA, BA,JA) isaBMBJ-neutrosophicidealof X (seeTheorem 3.3).Let a,b,x,y ∈ X besuchthat (((x ∗ y) ∗ y) ∗ a) ∗ b =0.Then MA(x ∗ y) ≥ MA((x ∗ y) ∗ y) ≥ min{MA(a),MA(b)}, BA (x ∗ y) ≤ ˜ BA((x ∗ y) ∗ y) ≤ max{BA (a),BA (b)}, B+ A (x ∗ y) ≥ B+ A ((x ∗ y) ∗ y) ≥ min{B+ A (a),B+ A (b)}, and JA(x ∗ y) ≤ JA((x ∗ y) ∗ y) ≤ max{JA(a), JA(b)} by Theorem 3.6 and Proposition 3.11. Hence (3.9) is valid.

Theorem3.18. IfanMBJ-neutrosophicset A =(MA, BA,JA) in X satisﬁestheconditions (2.9) and (3.9), then A =(MA, ˜ BA,JA) isapositiveimplicativeBMBJ-neutrosophicidealof X

Proof. Let A =(MA, BA,JA) beanMBJ-neutrosophicsetin X whichsatisﬁestheconditions(2.9)and(3.9). Itisclearthatthecondition(2.10)isinducedbythecondition(3.9).Let x,a,b ∈ X besuchthat x ∗ a ≤ b. Then (((x ∗ 0) ∗ 0) ∗ a) ∗ b =0,andso

MA(x)= MA(x ∗ 0) ≥ min{MA(a),MA(b)},

BA (x)= BA (x ∗ 0) ≤ max{BA (a),BA (b)},

B+ A (x)= B+ A (x ∗ 0) ≥ min{B+ A (a),B+ A (b)}, and JA(x)= JA(x ∗ 0) ≤ max{JA(a),JA(b)} by(2.1)and(3.9).Hence A =(MA, BA,JA) isaBMBJ-neutrosophicidealof X byTheorem 3.12.Since (((x ∗ y) ∗ y) ∗ ((x ∗ y) ∗ y)) ∗ 0=0 forall x,y ∈ X,wehave

MA(x ∗ y) ≥ min{MA((x ∗ y) ∗ y),MA(0)} = MA((x ∗ y) ∗ y),

     

Florentin Smarandache (author and editor) Collected Papers, IX 350

BA (x ∗ y) ≤ max{BA ((x ∗ y) ∗ y),BA (0)} = BA ((x ∗ y) ∗ y), B+ A (x ∗ y) ≥ min{B+ A ((x ∗ y) ∗ y),B+ A (0)} = B+ A ((x ∗ y) ∗ y), and JA(x ∗ y) ≤ max{JA((x ∗ y) ∗ y),JA(0)} = JA((x ∗ y) ∗ y) by(3.9).ItfollowsfromTheorem 3.6 that A =(MA, BA,JA) isapositiveimplicativeMBJ-neutrosophic idealof X.

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Smarandache (author and editor) Collected Papers, IX 352

Florentin

Neutrosophic Hesitant Fuzzy Subalgebras and Filters in Pseudo-BCI Algebras

Songtao Shao, Xiaohong Zhang, Chunxin Bo, Florentin SmarandacheD

Songtao Shao, Xiaohong Zhang, Chunxin Bo, Florentin Smarandache (2018). Neutrosophic Hesitant Fuzzy Subalgebras and Filters in Pseudo-BCI Algebras. Symmetry 10, 174; DOI: 10.3390/sym10050174

Abstract: The notions of the neutrosophic hesitant fuzzy subalgebra and neutrosophic hesitant fuzzy filter in pseudo-BCI algebras are introduced, and some properties and equivalent conditions are investigated. The relationships between neutrosophic hesitant fuzzy subalgebras (filters) and hesitant fuzzy subalgebras (filters) is discussed. Five kinds of special sets are constructed by a neutrosophic hesitant fuzzy set, and the conditions for the two kinds of sets to be filters are given. Moreover, the conditions for two kinds of special neutrosophic hesitant fuzzy sets to be neutrosophic hesitant fuzzy filters are proved.

Keywords: pseudo-BCI algebra; hesitant fuzzy set; neutrosophic set; filter

1. Introduction

G.GeorgescuandA.Iogulescupresentedpseudo-BCKalgebras,whichwasanextensionofthe famousBCKalgebratheory.In[1],thenotionofthepseudo-BCIalgebrawasintroducedbyW.A.Dudek andY.B.Jun.Theyinvestigatedsomepropertiesofpseudo-BCIalgebras.In[2],Y.B.Junetal.presented theconceptofthepseudo-BCIidealinpseudo-BCIalgebrasandresearcheditscharacterizations.Then, someclassesofpseudo-BCIalgebrasandpseudo-ideals(ﬁlters)werestudied;see[3 14].

In1965,Zadehintroducedfuzzysettheory[15].Inthestudyofmodernfuzzylogictheory, algebraicsystemsplayedanimportantrole,suchas[16 22].In2010,Torraintroducedhesitantfuzzy settheory[23].Thehesitantfuzzysetwasausefultooltoexpresspeoples’hesitancyinreallife, anduncertaintyproblemswereresolved.Furthermore,hesitantfuzzysetshavebeenappliedto decisionmakingandalgebraicsystems[24 31].Asageneralizationoffuzzysettheory,Smarandache introducedneutrosophicsettheory[32];theneutrosophicsettheoryisausefultooltodealwith indeterminateandinconsistentdecisioninformation[33,34].Theneutrosophicsetincludesthetruth membership,indeterminacymembershipandfalsitymembership.Then,Wangetal.[35,36]introduced theintervalneutrosophicsetandsingle-valuedneutrosophicset.Ye[37]introducedthesingle-valued neutrosophichesitantfuzzysetasanextensionofthesingle-valuedneutrosophicsetandhesitant fuzzyset.Recently,theneutrosophictripletstructureswereintroducedandresearched[38 40].

Inthispaper,somepreliminaryconceptsinpseudo-BCIalgebras,hesitantfuzzysettheoryand neutrosophicsettheoryarebrieﬂyreviewedinSection 2.InSection 3,thenotionofneutrosophic hesitantfuzzysubalgebrasinpseudo-BCIalgebrasisintroduced.Therelationshipsbetween neutrosophichesitantfuzzysubalgebrasandhesitantfuzzysubalgebrasareinvestigated.Fivekinds

Smarandache (author and editor) Collected Papers, IX 353

Florentin

ofspecialsetsareconstructed.Somepropertiesarestudied.Third,thetwokindsofsetstobefilters aregiven.InSection 4,theconceptofneutrosophichesitantfuzzyfiltersinpseudo-BCIalgebrasis proposed.Theequivalentconditionsoftheneutrosophichesitantfuzzyfiltersintheconstructionof hesitantfuzzyfiltersaregiven.Theconditionsfortwokindsofspecialneutrosophichesitantfuzzy setstobeneutrosophichesitantfuzzyfiltersaregiven.

2.Preliminaries

Letusreviewsomefundamentalnotionsofpseudo-BCIalgebraandinterval-valuedhesitant fuzzyﬁlterinthissection.

Deﬁnition1. ([13])Apseudo-BCIalgebraisastructure(X; →, →,1),where“→”and“ →”arebinary operationsonXand“1”isanelementofX,verifyingtheaxioms: ∀x, y, z ∈ X,

(1) (y → z) → ((z → x) → (y → x))= 1, (y → z) → ((z → x) → (y → x))= 1; (2)x → ((x → y) → y)= 1,x → ((x → y) → y)= 1; (3)x → x = 1; (4)x → y = y → x = 1 =⇒ x = y; (5)x → y = 1 ⇐⇒ x → y = 1

If(X; →, →,1)isapseudo-BCIalgebrasatisfying ∀x, y ∈ X, x → y = x → y,then(X; →,1)isa BCIalgebra.If(X; →, →,1)isapseudo-BCIalgebrasatisfying ∀x ∈ X, x → 1 = 1,then(X; →, →,1) isapseudo-BCKalgebra.

Remark1. ([1])Inanypseudo-BCIalgebra (X; →, →),wecandeﬁneabinaryrelation‘≤’byputting: x ≤ yifandonlyifx → y(orx → y).

Proposition1. ([13])Let (X; →, →) beapseudo-BCIalgebra,then X satisﬁesthefollowingproperties, ∀x, y, z ∈ X,

(1) 1 ≤ x ⇒ x = 1; (2) x ≤ y ⇒ y → z ≤ x → z, y → z ≤ x → z; (3) x ≤ y, y ≤ z ⇒ x ≤ z; (4) x → (y → z)= y → (x → z); (5) x ≤ y → z ⇒ y ≤ x → z; (6) x → y ≤ (z → x) → (z → y), x → y ≤ (z → x) → (z → y); (7) x ≤ y ⇒ z → x ≤ z → y, z → x ≤ z → y; (8) 1 → x = x,1 → x = x; (9) ((y → x) → x) → x = y → x, ((y → x) → x) → x = y → x; (10) x → y ≤ (y → x) → 1, x → y ≤ (y → x) → 1; (11) (x → y) → 1 =(x → 1) → (y → 1), (x → y) → 1 =(x → 1) → (y → 1); (12) x → 1 = x → 1.

Definition2. ([13])AsubsetFofapseudo-BCIalgebraXiscalledafilterofXifitsatisfies: (F1) 1 ∈ F; (F2)x ∈ F, x → y ∈ F ⇒ y ∈ F; (F3)x ∈ F, x → y ∈ F ⇒ y ∈ F.

Deﬁnition3. ([1])Byapseudo-BCIsubalgebraofapseudo-BCI algebra X,wemeanasubset S of X that satisﬁes ∀x, y ∈ S,x → y ∈ S, x → y ∈ S.

Deﬁnition4. ([12])Apseudo-BCKalgebraiscalledatype-2positiveimplicativeifitsatisﬁes: x → (y → z)=(x → y) → (x → z), x → (y → z)=(x → y) → (x → z)

Florentin Smarandache (author and editor) Collected Papers, IX 354

IfXisatype-2positiveimplicativepseudo-BCKalgebra,thenx → y = x → yforallx ∈ X.

Deﬁnition5. ([23])Let X beareferenceset.Ahesitantfuzzyset A on X isdeﬁnedintermsofafunction hA (x) thatreturnsasubsetof [0,1] whenitisappliedtoX,i.e., A = {(x, hA (x))|x ∈ X}.

where hA (x) isasetofsomedifferentvaluesin [0,1],representingthepossiblemembershipdegreesoftheelement x ∈ X.hA (x) iscalledahesitantfuzzyelement,abasisunitofthehesitantfuzzyset.

Example1. Let X = {a, b, c} beareferenceset, hA (a)=[0.1,0.2], hA (b)=[0.3,0.6], hA (c)=[0.7,0.8] Then,Aisconsideredasahesitantfuzzyset, A = {(a, [0.1,0.2]), (b, [0.3,0.6]), (c, [0.7,0.8])}

Definition6. ([13])Afuzzyset µ : X → [0,1] iscalledafuzzypseudo-filter(fuzzyfilter)ofapseudo-BCI algebraXifitsatisfies: (FF1) µ(1) ≥ µ(x), ∀x ∈ X; (FF2) µ(y) ≥ µ(x → y) ∧ µ(x), ∀x, y ∈ X; (FF3) µ(y) ≥ µ(x → y) ∧ µ(x), ∀x, y ∈ X.

Definition7. ([32])LetXbeanon-emptyfixedset,aneutrosophicsetAonXisdefinedas:

A = {(x, TA (x), IA (x), FA (x))|x ∈ X},

where TA (x), IA (x), FA (x) ∈ [0,1],denotingthetruth,indeterminacyandfalsitymembershipdegreeofthe elementx ∈ X,respecting,andsatisfyingthelimit: 0 ≤ TA (x)+ IA (x)+ FA (x) ≤ 3.

Definition8. ([34])LetXbeafixedset;aneutrosophichesitantfuzzysetNonXisdefinedas

N = {(x, tN (x), iN (x), f N (x))|x ∈ X},

inwhich tN (x), ˜ iN (x), f N (x) ∈ P([0,1]),denotingthepossibletruthmembershiphesitantdegrees, indeterminacymembershiphesitantdegreesandfalsitymembershiphesitantdegreesof x ∈ X totheset N,respectively,withtheconditions 0 ≤ δ, γ, η ≤ 1 and 0 ≤ δ+ + γ+ + η+ ≤ 3,where γ ∈ ˜ tN (x), δ ∈ iN (x), η ∈ ˜ f N (x), γ+ ∈ γ∈tN (x) max{γ}, δ+ ∈ δ∈iN (x) max{δ}, η+ ∈ η∈ f N (x) max{η} forx ∈ X.

Example2. Let X = {a, b, c} beareferenceset, hA(a)=([0.4,0.5], [0.1,0.2], [0.2,0.4]), hA(b)= ([0.5,0.6], {0.2,0.3}, [0.3,0.4]), hA(c)=([0.5,0.8], [0.2,0.4], {0.3,0.5}).Then, A isconsideredasaneutrosophic hesitantfuzzyset, A = {(a, [0.4,0.5], [0.1,0.2], [0.2,0.4]), (b, [0.5,0.6], {0.2,0.3}, [0.3,0.4]), (c, [0.5,0.8], [0.2,0.4], {0.3,0.5})}

Conveniently, N(x)= {˜ tN (x), iN (x), ˜ f N (x)} iscalledaneutrosophichesitantfuzzyelement, whichisdenotedbythesimpliﬁedsymbol N(x)= {˜ tN , iN , f N } Deﬁnition9. ([34])Let N1 = {tN1 , iN1 , f N1 } and N2 = {tN2 , iN2 , f N2 } betwoneutrosophichesitantfuzzy sets,then: N1 ∪ N2 = {˜ tN1 ∪ ˜ tN2 , iN1 ∩ iN2 , ˜ f N1 ∩ f N2 }; N1 ∩ N2 = {˜ tN1 ∩ ˜ tN2 , iN1 ∪ iN2 , ˜ f N1 ∪ f N2 }

Florentin Smarandache (author and editor) Collected Papers, IX 355

Inthefollowing,let X beapseudo-BCIalgebra,unlessotherwisespeciﬁed.

Deﬁnition10. Ahesitantfuzzyset A = {(x, hA (x))|x ∈ X} iscalledahesitantfuzzypseudo-subalgebra (hesitantfuzzysubalgebra)ofXifitsatisﬁes: (HFS2)hA (x) ∩ hA (y) ⊆ hA (x → y), ∀x, y ∈ X; (HFS3)hA (x) ∩ hA (y) ⊆ hA (x → y), ∀x, y ∈ X.

Deﬁnition11. Aneutrosophichesitantfuzzyset N = {(x, tN (x), iN (x), f N (x))|x ∈ X} iscalled aneutrosophichesitantfuzzypseudo-subalgebra(neutrosophichesitantfuzzysubalgebra)ofXifitsatisﬁes:

(1) tN (x) ∩ tN (y) ⊆ tN (x → y), tN (x) ∩ tN (y) ⊆ tN (x → y), ∀x, y ∈ X; (2) iN (x) ∪ iN (y) ⊇ iN (x → y), iN (x) ∪ iN (y) ⊇ iN (x → y), ∀x, y ∈ X; (3) f N (x) ∪ f N (y) ⊇ f N (x → y), f N (x) ∪ f N (y) ⊇ f N (x → y), ∀x, y ∈ X.

Example3. LetX = {a, b, c, d,1} withtwobinaryoperationsinTables 1 and 2

Table1. → → a b c d 1 a 1 c 1 1 1 b d 1 1 1 1 c d c 1 1 1 d c c c 1 1 1 a b c d 1

Table2. → → a b c d 1 a 1 d 1 1 1 b d 1 1 1 1 c d d 1 1 1 d c b c 1 1 1 a b c d 1

Then, (X; →, →,1) isapseudo-BCIalgebra.Let: N = {(1, [0,1], {0, 1 16 }, [0, 1 6 ]), (a, [ 1 3 , 1 4 ], [0, 1 2 ], [0, 5 6 ]), (b, [0, 1 2 ], [0, 2 3 ], [0, 2 3 ]), (c, [ 1 3 , 2 3 ], [0, 1 6 ], [0, 1 5 ]), (d, [ 1 3 ,1], [0, 1 3 ], [0, 1 5 ])} then,NisaneutrosophichesitantfuzzysubalgebraofX.

Consideringthreehesitantfuzzysets HtN , HiN , H f N by: HtN = {(x, ˜ tN (x))|x ∈ X}, HiN = {(x,1 ˜ iN (x))|x ∈ X}, H f N = {(x,1 ˜ f N (x))|x ∈ X} Therefore, HtN iscalledageneratedhesitantfuzzysetbyfunction tN (x); HiN iscalledagenerated hesitantfuzzysetbyfunction iN (x); H f N iscalledageneratedhesitantfuzzysetbyfunction f N (x) Theorem1. Let N = {(x, tN (x), ˜ iN (y), f N (x))|x ∈ X} beaneutrosophichesitantfuzzyseton X.Then, N isaneutrosophichesitantfuzzysubalgebraof X ifandonlyifitsatisﬁestheconditions: ∀x ∈ X, HtN and HiN , H f N arehesitantfuzzysubalgebrasofX.

Proof.Necessity: (i)ByDeﬁnition10andDeﬁnition11,wecanobtainthat HtN isahesitantfuzzy subalgebraof X

3.NeutrosophicHesitantFuzzySubalgebrasofPseudo-BCIAlgebras

Florentin Smarandache (author and editor) Collected Papers, IX 356

(ii) ∀x, y ∈ X, (1 iN (x)) ∩ (1 iN (y))= 1 (iN (x) ∪ iN (y)) ⊆ 1 iN (x → y), (1 iN (x)) ∩ (1 ˜ iN (y))= 1 ( ˜ iN (x) ∪ ˜ iN (y)) ⊆ 1 ˜ iN (x → y).

Similarly, (1 fN (x)) ∩ (1 fN (y)) ⊆ 1 fN (x → y), (1 fN (x)) ∩ (1 fN (y)) ⊆ 1 fN (x → y).

Therefore, ∀x ∈ X, HiN = {(x,1 ˜ i(x))|x ∈ X} and H fN = {(x,1 fN (x))|x ∈ X} arehesitantfuzzy subalgebrasof X

Sufﬁciency: (i)Let x, y ∈ HtN .Obviously, tN (x) ∩ tN (y) ⊆ tN (x → y), tN (x) ∩ tN (y) ⊆ ˜ tN (x → y)

(ii)Let x, y ∈ HiN .ByDeﬁnition10,wehave (1 iN (x)) ∩ (1 iN (y)) ⊆ 1 iN (x → y), (1 iN (x)) ∩ (1 iN (y)) ⊆ 1 iN (x → y),thus iN (x) ∪ iN (y) ⊇ iN (x → y), iN (x) ∪ iN (y) ⊇ iN (x → y)

Similarly,Let x, y ∈ H f N ;wehave f N (x) ∪ f N (y) ⊇ f N (x → y), f N (x) ∪ f N (y) ⊇ f (x → y) Thatis, N isaneutrosophichesitantfuzzysubalgebraof X

Theorem2. Let N = {(x, tN (x), iN (x), f N (x))|x ∈ X} beaneutrosophichesitantfuzzyseton X.Then, thefollowingconditionsareequivalent:

(1)N = {(x, tN (x), ˜ iN (x), f N (x))|x ∈ X} isaneutrosophichesitantfuzzysubalgebraofX; (2) ∀λ1, λ2, λ3 ∈ P([0,1]),thenonemptyhesitantfuzzylevelsets HtN (λ1), HiN (λ2), H f N (λ3) are subalgebrasofX,whereP([0,1]) isthepowersetof [0,1],

HtN (λ1)= {x ∈ X|λ1 ⊆ tN (x)}, HiN (λ2)= {x ∈ X|λ2 ⊆ 1 iN (x)}, H f N (λ3)= {x ∈ X|λ3 ⊆ 1 f N (x)}.

Proof. (1)⇒(2)Suppose HtN (λ1), HiN (λ2), H f N (λ3) arenonemptysets.If x, y ∈ HtN (λ1),then λ1 ⊆ tN (x), λ1 ⊆ tN (y).Since N isaneutrosophichesitantfuzzysubalgebraof X,byDeﬁnition 11,wecanobtain:

λ1 ⊆ tN (x) ∩ tN (y) ⊆ tN (x → y), λ1 ⊆ tN (x) ∩ tN (y) ⊆ tN (x → y); then x → y, x → y ∈ HtN (λ1), HtN (λ1) isasubalgebraof X. If x, y ∈ HiN (λ2),then λ2 ⊆ 1 iN (x), λ2 ⊆ 1 iN (y).Since N isaneutrosophichesitantfuzzy subalgebraof X,byDeﬁnition11,wecanobtain: λ2 ⊆ (1 ˜ iN (x)) ∩ (1 ˜ iN (y))= 1 ( ˜ iN (x) ∪ ˜ iN (y)) ⊆ 1 ˜ iN (x → y), λ2 ⊆ (1 ˜ iN (x)) ∩ (1 ˜ iN (y))= 1 ( ˜ iN (x) ∪ ˜ iN (y)) ⊆ 1 ˜ iN (x → y);

Thus, x → y, x → y ∈ HiN (λ2), HiN (λ2) isasubalgebraof X Similarly,wecanobtainthenthat H f N (λ3) isasubalgebraof X (2)⇒(1)Supposethat HtN (λ1), HiN (λ2), H f N (λ3) arenonemptysubalgebrasof X, ∀λ1, λ2, λ3 ∈ P([0,1]).Let x, y ∈ X with tN (x)= µ1, tN (y)= µ2.Let µ1 ∩ µ2 = λ1.Therefore,wehave x, y ∈ H(1) X (λ1).Since H(1) X (λ1) isasubalgebra,wecanobtain x → y, x → y ∈ HtN (λ1).Hence, wecanobtain: tN (x) ∩ tN (y) ⊆ tN (x → y), tN (x) ∩ tN (y) ⊆ tN (x → y); Let x, y ∈ X with i(x)= µ3, i(y)= µ4.Let (1 µ3) ∩ (1 µ4)= λ2.Then,wehave x, y ∈ HiN (λ2).Since HiN (λ2) isasubalgebra,wecanobtain x → y, x → y ∈ H f N (λ2).Hence,wecan obtain (1 ˜ iN (x)) ∩ (1 ˜ iN (y))= 1 ( ˜ iN (x) ∪ ˜ iN (y))= λ2 ⊆ 1 ˜ iN (x → y), (1 ˜ iN (x)) ∩ (1 ˜ iN (y))= 1 ( ˜ iN (x) ∪ ˜ iN (y))= λ2 ⊆ 1 ˜ iN (x → y).Then,wehave ˜ iN (x) ∪ ˜ iN (y) ⊇ ˜ iN (x → y), ˜ iN (x) ∪ ˜ iN (y) ⊇ ˜ iN (x → y)

Similarly,let x, y ∈ X with f N (x)= µ5, f N (y)= µ6;wecanobtain f N (x) ∪ f N (y) ⊇ f N (x → y), ˜ f N (x) ∪ ˜ f N (y) ⊇ ˜ f N (x → y) Thus, N isaneutrosophichesitantfuzzysubalgebraof X

Florentin Smarandache (author and editor) Collected Papers, IX 357

Deﬁnition12. Let N = {(x, tN (x), iN (x), f N (x))|x ∈ X} beaneutrosophichesitantfuzzyseton X. X(1) N (ak , b), X(2) N (ak , b), X(3) N (ak , b), X(4) N (ak , b), X(5) N (a) arecalledgeneratedsubsetsby N: ∀a, b ∈ X, k ∈ N,

X(1) N (ak , b)= {x ∈ X|tN (ak ∗ (b ∗ x))= tN (1), iN (ak ∗ (b ∗ x))= iN (1), f N (ak ∗ (b ∗ x))= f N (1)};

X(2) N (ak , b)= {x ∈ X|tN (ak → (b → x))= tN (1), iN (ak → (b → x))= tN (1), f N (ak → (b → x))= f N (1)}; X(3) N (ak , b)= {x ∈ X|tN (ak → (b → x))= tN (1), iN (ak → (b → x))= tN (1), f N (ak → (b → x))= f N (1)};

X(4) N (ak , b)= {x ∈ X|tN (ak → (b → x))= tN (1), iN (ak → (b → x))= iN (1), f N (ak → (b → x))= f N (1), tN (ak → (b → x))= tN (1), iN (ak → (b → x))= iN (1), f N (ak → (b → x))= f N (1)}; X(5) N (a)= {x ∈ X|tN (a) ⊆ tN (x), iN (a) ⊇ iN (x), f N (a) ⊇ f N (x)} where"a"appears"k"times,"∗"representsanybinaryoperation"→"or" →"onX, ak ∗ (b ∗ x)= a ∗ (a ∗ ( (a ∗ (b ∗ x)) )); ak → (b → x))= a → (a → ( (a → (b → x)) )); ak → (b → x))= a → (a → ( (a → (b → x)) )); ak → (b → x))= a → (a → ( (a → (b → x)) )); ak → (b → x)= a → (a → ( (a → (b → x)) )) Theorem3. Let N = {(x, ˜ tN (x), iN (x), f N (x))|x ∈ X} beaneutrosophichesitantfuzzyseton X.If N satisﬁesthefollowingconditions:

(1) tN (x) ⊆ tN (1), tN (x → y)= tN (x) ∪ tN (y), ∀x, y ∈ X;

(2) iN (x) ⊇ iN (1), iN (x → y)= iN (x) ∩ iN (y), ∀x, y ∈ X;

(3) f N (x) ⊇ f N (1), f N (x → y)= f N (x) ∩ f N (y), ∀x, y ∈ X; thenX(1) N (ak , b)= X,k ∈ N

Proof. ByProposition1,wecanobtain ∀x ∈ X,

tN (ak ∗ (b ∗ x)= tN (1 → (ak ∗ (b ∗ x))) =tN (1) ∪ tN (ak ∗ (b ∗ x)))= tN (1). ˜ iN (ak ∗ (b ∗ x))= ˜ iN (1 → (ak ∗ (b ∗ x))) = ˜ iN (1) ∩ tN (ak ∗ (b ∗ x)))= ˜ iN (1). f N (ak ∗ (b ∗ x))= f N (1 → (ak ∗ (b ∗ x))) = f N (1) ∩ tN (ak ∗ (b ∗ x)))= f N (1).

Thus, x ∈ X(1) N (ak , b), X ⊆ X(1) N (ak , b). Conversely,itiseasytocheckthat X(1) N (ak , b) ⊆ X Finally,wecanobtain X = X(1) N (ak , b)

Corollary1. Let N = {(x, tN (x), iN (x), f N (x))|x ∈ X} beaneutrosophichesitantfuzzyseton X.If N satisﬁesthefollowingconditions:

(1) tN (x) ⊆ tN (1), tN (x → y)= tN (x) ∪ tN (y), ∀x, y ∈ X;

Florentin Smarandache (author and editor) Collected Papers, IX 358

(2) iN (x) ⊇ iN (1), iN (x → y)= iN (x) ∩ iN (y), ∀x, y ∈ X; (3) f N (x) ⊇ f N (1), f N (x → y)= f N (x) ∩ f N (y), ∀x, y ∈ X; thenX(1) N (ak , b)= X,k ∈ N.

Theorem4. Let N = {(x, tN (x), ˜ iN (x), f N (x))|x ∈ X} beaneutrosophichesitantfuzzyseton X N satisﬁes thefollowingconditions:

(1) ˜ tN (1) ⊇ ˜ tN (x), iN (1) ⊆ iN (x), ˜ f N (1) ⊆ ˜ f N (x), ∀x ∈ X; (2)x → y = 1 ⇒ ˜ tN (x) ⊆ ˜ tN (y), iN (x) ⊇ iN (y), ˜ f N (x) ⊇ ˜ f N (y), ∀x, y ∈ X. If ∀a, b, c ∈ X, k ∈ N,b ≤ c,thenX(2) N (ak , c) ⊆ X(2) N (ak , b)

Proof: Let x ∈ X(2) N (ak , c).If b ≤ c,byProposition1,wecanobtain: ˜ tN (1)=˜ tN (ak → (c → x)) = ˜ tN (c → (ak → x)) ⊆ ˜ tN (b → (ak → x)) = ˜ tN (ak → (b → x))

Similarly,wecanobtain: iN (ak → (b → x)) ⊆ iN (ak → (c → x)) ⊆ iN (1); f N (ak → (b → x)) ⊆ f N (ak → (c → x)) ⊆ f N (1) Thatis, x ∈ X(2) N (ak , b), X(2) N (ak , c) ⊆ X(2) N (ak , b). Corollary2. Let N = {(x, ˜ tN (x), iN (x), ˜ f N (x))|x ∈ X} beaneutrosophichesitantfuzzyseton X N satisﬁes thefollowingconditions:

(1) ˜ tN (1) ⊇ ˜ tN (x), iN (1) ⊆ iN (x), f N (1) ⊆ f N (x), ∀x ∈ X; (2)x → y = 1 ⇒ tN (x) ⊆ tN (y), iN (x) ⊇ iN (y), f N (x) ⊇ f N (y), ∀x, y ∈ X. If ∀a, b, c ∈ X, k ∈ N,b ≤ c,thenX(3) N (ak , c) ⊆ X(3) N (ak , b)

Thefollowingexampleshowsthat X(4) N (ak , b) maynotbeaﬁlterof X Example4. LetX = {a, b, c, d,1} withtwobinaryoperationsinTables 3 and 4

Table3. →.

→ a b c d 1 a 1 1 1 1 1 b d 1 1 1 1 c d c 1 1 1 d c c c 1 1 1 a b c d 1

Table4. →. → a b c d 1 a 1 d 1 1 1 b d 1 1 1 1 c d d 1 1 1 d c b c 1 1 1 a b c d 1

Then, (X; →, →,1) isapseudo-BCIalgebra.Let:

Florentin Smarandache (author and editor) Collected Papers, IX 359

N = {(1, [0,1], [ 1 6 , 1 5 ], [0, 1 5 ]), (a, [ 1 3 , 1 4 ], [0, 5 6 ], [0, 3 4 ]), (b, [0, 1 2 ], [ 1 6 , 3 4 ], [0, 1 3 ]), (c, [ 1 3 , 2 3 ], [0, 3 5 ], [0, 1 4 ]), (d, [ 1 3 ,1], [ 1 6 , 1 3 ], [0, 5 6 ])}.

thenX(4) N (c, d)= {a, c, d,1} isnotaﬁlterofX.Sincec → b = c ∈ X(4) N (c, d),butb / ∈ X(4) N (c, d)

Theorem5. Let N = {(x, tN (x), iN (x), f N (x))|x ∈ X} beaneutrosophichesitantfuzzyseton X.Let X beatype-2positiveimplicativepseudo-BCKalgebra.Iffunctions tN (x), iN (x) and f N (x) areinjective,then X(4) N (ak , b) isaﬁlterofXforalla, b ∈ X, k ∈ N

Proof. (1)If X isapseudo-BCKalgebra,thenbyDeﬁnition1andProposition1,wecanobtain 1 ∈ X(4) N (ak , b).

(2)Let x, y ∈ X with x, x → y ∈ X(4) N (ak , b).Thus, ak → (b → x)= 1, ak → (b → (x → y))= 1. Sincefunctions tN , iN and f N areinjective,byDeﬁnition5,wehave:

tN (1)= tN (ak → (b → (x → y)))

= tN (ak → ((b → x) → (b → y)))

= tN ((ak → (b → x)) → (ak → (b → y)))

= tN (1 → (ak → (b → y)))

= tN (ak → ((b → y)).

Similarly,wecanobtain ˜ iN (ak → ((b → y))= ˜ iN (1), f N (ak → ((b → y))= f N (1).Thus,wehave y ∈ X(4) N (ak , b).

(3)Similarly,let x, y ∈ X with x, x → y ∈ X(4) N (ak , b);wehave y ∈ X(4) N (ak , b).

Thismeansthat X(4) N (ak , b) isaﬁlterof X forall a, b ∈ X, k ∈ N. Theorem6. Let N = {(x, ˜ tN (x), iN (x), ˜ f N (x))|x ∈ X)} beaneutrosophichesitantfuzzyseton X.Let X be atype-2positiveimplicativepseudo-BCKalgebra.Iffunctions ˜ tN (x), iN (x) and ˜ f N (x) satisfythefollowing identiﬁes: ∀x, y ∈ X,

(1) ˜ tN (x) ⊆ ˜ tN (1), iN (x) ⊇ iN (1), ˜ f N (x) ⊇ f N (1);

(2) ˜ tN (x → y)= ˜ tN (x) ∩ ˜ tN (y), iN (x → y)= iN (x) ∪ iN (y), ˜ f N (x → y)= ˜ f N (x) ∪ ˜ f N (y); (3) tN (x → y)= tN (x) ∩ tN (y), iN (x → y)= iN (x) ∪ iN (y), f N (x → y)= f N (x) ∪ f N (y); thenX(4) N (ak , b) isaﬁlterofXforalla, b ∈ X, k ∈ N

Proof. (1)If X isapseudo-BCKalgebra,byDeﬁnition1andProposition1,1 ∈ X(4) N (ak , b) (2)Let x, y ∈ X with x, x → y ∈ X(4) N (ak , b).Wehave tN (ak → (b → x))= tN (1), tN (ak → (b → (x → y)))= tN (1).ByDeﬁnition5,wehave:

tN (1)= tN (ak → (b → (x → y)))

= ˜ tN (ak → ((b → x) → (b → y))) = ˜ tN ((ak → (b → x)) → (ak → (b → y))) = ˜ tN (ak → (b → x)) ∩ ˜ t(ak → (b → y)) = ˜ tN (1) ∩ ˜ t(ak → (b → y)) = ˜ tN (ak → (b → y))

Similarly,wecanobtain iN (ak → (b → y))= iN (1), f N (ak → (b → y))= f N (1).Thus,wehave y ∈ X(4) N (ak , b)

(3)Similarly,let x, y ∈ X with x, x → y ∈ X(4) N (ak , b);wehave y ∈ X(4) N (ak , b)

Florentin Smarandache (author and editor) Collected Papers, IX 360

Thismeansthat X(4) N (ak , b) isaﬁlterof X forall a, b ∈ X, k ∈ N

Theorem7. Let N = {(x, tN (x), iN (x), f N (x))|x ∈ X)} beaneutrosophichesitantfuzzyseton X and F be aﬁlterofX.Iffunctions tN (x), iN (x) and f N (x) areinjective,then X(4) N (ak , b)= Fforalla, b ∈ F, k ∈ N

Proof. (1)Let x ∈ X(4) N (ak , b).ByDeﬁnition12,wehave tN (a → (ak 1 → (b → x)))= tN (1), iN (a → (ak 1 → (b → x)))= iN (1), f N (a → (ak 1 → (b → x)))= f N (1).Since F isaﬁlterof X and tN , iN , f N areinjective,thuswecanobtain a → (ak 1 → (b → x))= 1and ak 1 → (b → x) ∈ F. Continuing,wecanobtain b → x ∈ F.Since b ∈ F,thus x ∈ F, X(4) N (ak , b) ⊆ F. (2)Let x ∈ F.When a = 1, b = x,wecanobtain tN (1k → (x → x))= tN (1k → (x → x))= tN (1). Similarly,wehave iN (1k → (x → x))= iN (1k → (x → x))= iN (1), f N (1k → (x → x))= f N (1k → (x → x))= f N (1).Thus,wehave F ⊆ X(4) N (ak , b). Thismeansthat X(4) N (ak , b)= F forall a, b ∈ F, k ∈ N

Theorem8. LetN = {(x, tN (x), iN (x), f N (x))|x ∈ X)} beaneutrosophichesitantfuzzysetonX. (1)IfX(5) N (a) isaﬁlterofX,thenNsatisﬁes: ∀x, y ∈ X, (i) tN (a) ⊆ tN (x → y) ∩ tN (x), iN (a) ⊇ iN (x → y) ∪ iN (x), f N (a) ⊇ f N (x → y) ∪ f N (x) ⇒ tN (a) ⊆ tN (y), iN (a) ⊇ iN (y), f N (a) ⊇ f N (y); (ii) tN (a) ⊆ tN (x → y) ∩ tN (x), iN (a) ⊇ iN (x → y) ∪ iN (x), f N (a) ⊇ f N (x → y) ∪ f N (x) ⇒ tN (a) ⊆ tN (y), iN (a) ⊇ iN (y), f N (a) ⊇ f N (y).

(2)If N satisﬁesConditions(i),(ii)and tN (x) ⊆ tN (1), iN (x) ⊇ iN (1), f N (x) ⊇ f N (1) forall x, y ∈ X, thenX(5) N (a) isaﬁlterofX.

Proof. (1)(i)Let x, y ∈ X with tN (a) ⊆ tN (x → y) ∩ tN (x), ˜ iN (a) ⊇ ˜ iN (x → y) ∪ ˜ iN (x), f N (a) ⊇ f N (x → y) ∪ f N (x);wehave x ∈ X(5) N (a), x → y ∈ X(5) N (a).Since X(5) N (a) isaﬁlter,thuswecanhave y ∈ X(5) N (a), tN (a) ⊆ tN (y), iN (a) ⊇ iN (y), f N (a) ⊇ f N (y).

(ii)Similarly,weknowthat(ii)iscorrect.

(2)Since tN (x) ⊆ tN (1), iN (x) ⊇ iN (1), f N (x) ⊇ f N (1) forall x ∈ X,thus1 ∈ X(5) N (a).Let x, y ∈ X with x, x → y ∈ X(5) N (a);wecanobtain tN (a) ⊆ tN (x), tN (a) ⊆ tN (x → y), iN (a) ⊇ iN (x), iN (a) ⊇ iN (x → y), f N (a) ⊇ f N (x), f N (a) ⊇ f N (x → y).ByCondition(i),wehave tN (a) ⊆ tN (y), iN (a) ⊇ iN (y), f N (a) ⊇ f N (y).Thus,wecanobtain y ∈ X(5) N (a).Similarly,let x, y ∈ X with x, x → y ∈ X(5) N (a), byCondition(1)(ii);wecanobtain y ∈ X(5) N (a) Thismeansthat X(5) N (a) isaﬁlterof X

4.NeutrosophicHesitantFuzzyFiltersofPseudo-BCIAlgebras

Inthefollowing,let X beapseudo-BCIalgebra,unlessotherwisespeciﬁed.

Definition13. ([22])Ahesitantfuzzyset A = {(x, hA (x))|x ∈ X} iscalledahesitantfuzzypseudo-filter (briefly,hesitantfuzzyfilter)ofXifitsatisfies:

(HFF1)hA (x) ⊆ hA (1), ∀x ∈ X; (HFF2)hA (x) ∩ hA (x → y) ⊆ hA (y), ∀x, y ∈ X; (HFF3)hA (x) ∩ hA (x → y) ⊆ hA (y), ∀x, y ∈ X.

Definition14. Aneutrosophichesitantfuzzyset N = {(x, tN (x), iN (x), f N (x))|x ∈ X} iscalleda neutrosophichesitantfuzzypseudo-filter(neutrosophichesitantfuzzyfilter)ofXifitsatisfies:

(NHFF1) tN (x) ⊆ tN (1), iN (x) ⊇ iN (1), f N (x) ⊇ f N (1), ∀x ∈ X; (NHFF2) tN (x → y) ∩ tN (x) ⊆ tN (y), iN (x → y) ∪ iN (x) ⊇ iN (y), f N (x → y) ∪ f N (x) ⊇ f N (y), ∀x, y ∈ X;

Florentin Smarandache (author and editor) Collected Papers, IX 361

(NHFF3) tN (x → y) ∩ tN (x) ⊆ tN (y), iN (x → y) ∪ iN (x) ⊇ iN (y), f N (x → y) ∪ f N (x) ⊇ f N (y), ∀x, y ∈ X.

Aneutrosophichesitantfuzzyset N = {(x, ˜ tN (x), iN (x), ˜ f N (x))|x ∈ X)} iscalledaneutrosophic hesitantfuzzyclosedﬁlterof X ifitisaneutrosophichesitantfuzzyﬁltersuchthat: tN (x → 1) ⊇ tN (x), iN (x → 1) ⊆ iN (x), f N (x → 1) ⊆ f N (x).

Example5. Let X = {a, b, c, d,1} withtwobinaryoperationsinTables 5 and 6.Then, (X; →, →,1) is apseudo-BCIalgebra.Let:

N = {(1, [0,1], [0, 3 7 ], [0, 1 10 ]), (a, [0, 1 4 ], [0, 3 4 ], [0, 1 2 ]), (b, [0, 1 4 ], [0, 3 4 ], [0, 1 2 ]), (c, [0, 1 3 ], [0, 3 5 ], [0, 1 4 ]), (d, [0, 3 4 ]), [0, 3 6 ], [0, 1 5 ])}

Then,NisaneutrosophichesitantfuzzyﬁlterofX.

Table5. → → a b c d 1 a 1 1 1 1 1 b c 1 1 1 1 c a b 1 d 1 d b b c 1 1 1 a b c d 1

Table6. → → a b c d 1 a 1 1 1 1 1 b d 1 1 1 1 c b b 1 d 1 d a b c 1 1 1 a b c d 1

Theorem9. Let N = {(x, ˜ tN (x), iN (y), ˜ f N (x))|x ∈ X} beaneutrosophichesitantfuzzyseton X.Then, N isaneutrosophichesitantfuzzyfilterof X ifandonlyifitsatisfiesthefollowingconditions: ∀x ∈ X, HtN , HiN , H f N arehesitantfuzzyfiltersofX.

Proof.Necessity: If N isaneutrosophichesitantfuzzyfilter: (1)Obviously, HtN isahesitantfuzzyfilterof X. (2)ByDefinition14,wehave (1 iN (x)) ⊆ (1 iN (1)),1 (iN (x) ∪ iN (x → y))=(1 iN (x)) ∩ (1 iN (x → y)) ⊆ (1 iN (y));similarly,byDefinition14,wehave (1 iN (x)) ∩ (1 iN (x → y)) ⊆ (1 iN (y)).Thus, HiN ishesitantfuzzyfilterof X.

(3)Similarly,wehavethat H f N isahesitantfuzzyfilterof X. Sufficiency: If HtN , HiN , H f N arehesitantfuzzyfiltersof X.Itiseasytoprovethat tN (x), ˜ iN (x), ˜ f N (x) satisfiesDefinition14.Therefore, N = {(x, ˜ tN (x), iN (x), ˜ f N (x))|x ∈ X} isaneutrosophic hesitantfuzzyfilterof X

Theorem10. Let N = {(x, tN (x), iN (x), f N (x))|x ∈ X} beaneutrosophichesitantfuzzyseton X.Then, thefollowingareequivalent:

(1)N = {(x, tN (x), iN (x), f N (x))|x ∈ X} isaneutrosophichesitantfuzzyﬁlterofX;

Florentin Smarandache (author and editor) Collected Papers, IX 362

(2) ∀λ1, λ2, λ3 ∈ P([0,1]),thenonemptyhesitantfuzzylevelsets HtN (λ1), HiN (λ2), H f N (λ3) areﬁlters ofX,whereP([0,1]) isthepowersetof [0,1],

HtN (λ1)={x ∈ X|λ1 ⊆ tN (x)};

HiN (λ2)={x ∈ X|λ2 ⊆ 1 ˜ iN (x)};

H f N (λ3)={x ∈ X|λ3 ⊆ 1 f N (x)}.

Proof. (1)⇒(2)(i)Suppose HtN (λ1) = ∅.Let x ∈ HtN (λ1),then λ1 ⊆ ˜ tN (x).Since N isaneutrosophic hesitantfuzzyﬁlterof X,byDeﬁnition14,wehave λ1 ⊆ ˜ tN (x) ⊆ ˜ tN (1).Thus,1 ∈ HtN (λ1)

Let x, y ∈ X with x, x → y ∈ HtN (λ1),then λ1 ⊆ tN (x), λ1 ⊆ tN (x → y).Since N is aneutrosophichesitantfuzzyﬁlterof X,byDeﬁnition14,wehave λ1 ⊆ tN (x → y) ∩ tN (x) ⊆ tN (y) Thus y ∈ HtN (λ1).Similarly,let x, y ∈ X with x, x → y ∈ HtN (λ1).Wehave y ∈ HtN (λ1)

Thus,wecanobtainthat HtN (λ1) isaﬁlterof X

(ii)Suppose HiN (λ2) = ∅.Let x ∈ HiN (λ2),then λ2 ⊆ 1 iN (x).Since N isaneutrosophic hesitantfuzzyﬁlterof X,wehave ˜ iN (1) ⊆ ˜ iN (x).Thus, λ2 ⊆ 1 ˜ iN (x) ⊆ 1 ˜ iN (1),1 ∈ HiN (λ2).

Let x, y ∈ X with x, x → y ∈ HiN (λ2),then λ2 ⊆ 1 iN (x), λ2 ⊆ 1 iN (x → y).Since N is aneutrosophichesitantfuzzyfilterof X,wehave iN (x → y) ∪ iN (x) ⊇ iN (y).Thus,1 (iN (x → y) ∪ iN (x))=(1 iN (x → y)) ∩ (1 iN (x)) ⊆ (1 iN (y)), λ2 ⊆ (1 iN (y)), y ∈ HiN (λ2).Similarly, let x, y ∈ X with x, x → y ∈ HiN (λ2).Wehave y ∈ HiN (λ2). Thus,wecanobtainthat HiN (λ2) isafilterof X. (iii)Wehavethat H f N (λ3) isafilterof X.Theprogressofproofissimilarto(ii). (2)⇒(1)Suppose HtN (λ1) = ∅, HiN (λ2) = ∅, H f N (λ3) = ∅ forall λ1, λ2, λ3 ∈ P([0,1]). (i’)Let x ∈ X with ˜ tN (x)= µ1.Let λ1 = µ1.Since HtN (λ1) isafilterof X,wehave1 ∈ HtN (λ1) Thus, λ1 = µ1 = ˜ tN (x) ⊆ ˜ tN (1) Let x, y ∈ X with ˜ tN (x)= µ1, ˜ tN (x → y)= µ4.Let µ1 ∩ µ4 = λ1.Since HtN (λ1) isafilterof X for all λ1 ∈ P([0,1]),wehave y ∈ HtN (λ1).Thus, λ1 = ˜ tN (x) ∩ ˜ tN (x → y) ⊆ ˜ tN (y)

Similarly,let x, y ∈ X with ˜ tN (x)= µ1, ˜ tN (x → y)= µ4.Wecanobtain ˜ tN (x → y) ∩ ˜ tN (x) ⊆ tN (y)

(ii’)Let x ∈ X with iN (x)= µ2.Let λ2 = 1 µ2.Since HiN (λ2) isafilterof X forall λ2 ∈ P([0,1]), wehave1 ∈ HiN (λ2), λ2 ⊆ 1 iN (1).Thus,1 λ2 = µ2 = iN (x) ⊇ iN (1). Let x, y ∈ X with ˜ iN (x)= µ2, ˜ iN (x → y)= µ5.Let (1 µ2) ∩ (1 µ5)= λ2.Since HiN (λ2) isa filterof X forall λ2 ∈ P([0,1]),wehave y ∈ HiN (λ2), λ2 ⊆ 1 iN (y).Thus, λ2 =(1 µ2) ∩ (1 µ5)= (1 iN (x)) ∩ (1 iN (x → y))= 1 (iN (x) ∪ iN (x → y)) ⊆ (1 iN (y)), iN (x) ∪ iN (x → y) ⊇ iN (y). Similarly,let x, y ∈ X with iN (x)= µ2, iN (x → y)= µ5;wehave iN (x) ∪ iN (x → y) ⊇ iN (y). (iii’)Similarly,wecanobtain f N (x) ⊇ f N (1), f N (x) ∪ f N (x → y) ⊇ f N (y), f N (x) ∪ f N (x → y) ⊇ f N (y). Therefore, N = {(x, tN (x), ˜ iN (x), f N (x))|x ∈ X} isaneutrosophichesitantfuzzyfilterof X Definition15. N = {(x, ˜ tN (x), iN (x), ˜ f N (x))|x ∈ X} isaneutrosophichesitantfuzzyseton X.Definea neutrosophichesitantfuzzysetN∗ = {(x, ˜ t∗ N (x), i∗ N (x), ˜ f ∗ N (x))|x ∈ X} by: t∗ N : X =⇒ P([0,1]), x → ˜ tN (x), x ∈ HtN (λ1) ϕ1, x / ∈ HtN (λ1)

i∗ N : X =⇒ P([0,1]), x → ˜ iN (x), x ∈ HiN (λ2) 1 ϕ2, x / ∈ HiN (λ2)

f ∗ N : X =⇒ P([0,1]), x → ˜ f N (x), x ∈ H f N (λ3) 1 ϕ3, x / ∈ H f N (λ3)

Florentin Smarandache (author and editor) Collected Papers, IX 363

where λ1, λ2, λ3, ϕ1, ϕ2, ϕ3 ∈ P([0,1]), ϕ1 ⊆ λ1, ϕ2 ⊆ λ2, ϕ3 ⊆ λ3.Then, N∗ iscalledagenerated neutrosophichesitantfuzzysetbyhesitantfuzzylevelsetsHtN (λ1), HiN (λ2) andH f N (λ3).

Theorem11. Let N = {(x, tN (x), iN (x), f N (x))|x ∈ X} beaneutrosophichesitantfuzzyﬁlterof X.Then, N∗ isaneutrosophichesitantfuzzyﬁlterofX.

Proof. (1)If N isaneutrosophichesitantfuzzyfilterof X,byTheorem10,weknowthat HtN (λ1), HiN (λ2), H f N (λ3) arefiltersof X.Thus,1 ∈ HtN (λ1),1 ∈ HiN (λ2),1 ∈ H f N (λ3), t∗ N (1)= tN (1) ⊇ t∗ N (x), ˜ i∗ N (1)= ˜ iN (1) ⊆ ˜ i∗ N (x), f ∗ N (1)= f N (1) ⊆ f ∗ N (x), ∀x ∈ X (2)(i)Let x, y ∈ X with x, x → y ∈ HtN (λ1).ByTheorem9,Theorem10andDefinition15,we know λ1 ⊆ ˜ t∗ N (x → y) ∩ ˜ t∗ N (x)= ˜ tN (x → y) ∩ ˜ tN (x) ⊆ ˜ tN (y)= ˜ t∗ N (y) Let x, y ∈ X with x, x → y ∈ HiN (λ2).ByTheorem9,Theorem10andDefinition15,weknow λ2 ⊆ (1 i∗ N (x → y)) ∩ (1 t∗ N (x))=(1 iN (x → y)) ∩ (1 tN (x))= 1 (iN (x → y) ∪ iN (x)) ⊆ 1 iN (y)= 1 t∗ N (y).Thus,wehave1 λ2 ⊇ i∗ N (x → y) ∪ i∗ N (x)= iN (x → y) ∪ iN (x) ⊇ iN (y)= i∗ N (y)

Similarly,let x, y ∈ X with x, x → y ∈ H f N (λ3);wehave1 λ3 ⊇ f ∗ N (x → y) ∪ f ∗ N (x)= f N (x → y) ∪ f N (x) ⊇ f N (y)= f ∗ N (y).

(ii)Let x, y ∈ X with x / ∈ HtN (λ1) or x → y / ∈ HtN (λ1).ByDeﬁnition15,wehave t∗ N (x)= ϕ1 or t∗ N (x → y)= ϕ1.Thus,wecanobtain t∗ N (x) ∩ t∗ N (x → y)= ϕ1 ⊆ t∗ N (y).

Let x, y ∈ X with x / ∈ HiN (λ2) or x → y / ∈ HiN (λ2).ByDeﬁnition15,wehave ˜ i∗ N (x)= 1 ϕ2 or i∗ N (x → y)= 1 ϕ2.Since1 λ2 ⊆ 1 ϕ2;thus,wecanobtain i∗ N (x) ∪ i∗ N (x → y)= 1 ϕ2 ⊇ t∗ N (y)

Similarly,let x, y ∈ X with x / ∈ H f N (λ3) or x → y / ∈ H f N (λ3);wehave f ∗(x) ∪ f ∗(x → y)= 1 ϕ3 ⊇ f ∗(y).

(3)Wecanobtain t∗(x) ∩ t∗(x → y) ⊆ t∗(y), i∗(x) ∪ i∗(x → y) ⊇ i∗(y), f ∗(x) ∪ f ∗(x → y) ⊇ f ∗(y).Theprocessofproofissimilarto(2).

Thus N∗ isaneutrosophichesitantfuzzyfilterof X. Theorem12. Let N = {(x, tN (x), ˜ iN (x), ˜ f N (x))|x ∈ X} beaneutrosophichesitantfuzzyfilterof X.Then, Nsatisfiesthefollowingproperties, ∀x, y, z ∈ X, (1)x ≤ y ⇒ ˜ tN (x) ⊆ ˜ tN (y), iN (x) ⊇ iN (y), ˜ f N (x) ⊇ ˜ f N (y); (2) ˜ tN (x → z) ⊇ ˜ tN (x → (y → z)) ∩ ˜ tN (y), ˜ tN (x → z) ⊇ ˜ tN (x → (y → z)) ∩ ˜ tN (y); iN (x → z) ⊆ iN (x → (y → z)) ∪ iN (y), iN (x → z) ⊆ iN (x → (y → z)) ∪ iN (y); f N (x → z) ⊆ f N (x → (y → z)) ∪ f N (y), f N (x → z) ⊆ f N (x → (y → z)) ∪ f N (y); (3) tN ((x → y) → y) ⊇ tN (x), tN ((x → y) → y) ⊇ tN (x); iN ((x → y) → y) ⊆ iN (x), iN ((x → y) → y) ⊆ iN (x); f N ((x → y) → y) ⊆ f N (x), f N ((x → y) → y) ⊆ f N (x); (4)z ≤ x → y ⇒ tN (x) ∩ tN (z) ⊆ tN (y), iN (x) ∪ iN (z) ⊇ iN (y), f N (x) ∪ f N (z) ⊇ f N (y); z ≤ x → y ⇒ tN (x) ∩ tN (z) ⊆ tN (y), iN (x) ∪ iN (z) ⊇ iN (y), f N (x) ∪ f N (z) ⊇ f N (y)

Proof. (1)Let x, y ∈ X with x ≤ y.ByProposition1,weknow x → y = 1(or x → y = 1).If N is aneutrosophichesitantfuzzyfilterof X,byDefinition14,wehave tN (x)= tN (1) ∩ tN (x)= tN (x → y) ∩ tN (x) ⊆ tN (y) (tN (x)= tN (1) ∩ tN (x)= tN (x → y) ∩ tN (x) ⊆ tN (y)).Thus, tN (x) ⊆ tN (y). Similarly,wehave ˜ iN (x) ⊇ ˜ iN (y), f N (x) ⊇ f N (y). (2)ByProposition1,Definition14,weknow, ∀x, y, z ∈ X, ˜ tN (x → z) ⊇ ˜ tN (y → (x → z)) ∩ ˜ tN (y)= ˜ tN (x → (y → z)) ∩ ˜ tN (y), ˜ tN (x → z) ⊇ ˜ tN (y → (x → z)) ∩ ˜ tN (y)= ˜ tN (x → (y → z)) ∩ ˜ tN (y)

Similarly,wehave, ∀x, y, z ∈ X: iN (x → z) ⊆ iN (x → (y → z)) ∪ iN (y), iN (x → z) ⊆ iN (x → (y → z)) ∪ iN (y); f N (x → z) ⊆ f N (x → (y → z)) ∪ f N (y), f N (x → y) ⊆ f N (x → (y → z)) ∪ f N (y)

Florentin Smarandache (author and editor) Collected Papers, IX 364

(3)ByDeﬁnition1andDeﬁnition14,withregardtothefunction tN (x),wecanobtain, ∀x, y ∈ X,

tN ((x → y) → y) ⊇ tN (x → ((x → y) → y)) ∩ tN (x)

= tN ((x → y) → (x → y)) ∩ tN (x)

= ˜ tN (1) ∩ ˜ tN (x) = tN (x).

Similarly,wehave tN ((x → y) → y) ⊇ tN (x). Withregardtothefunction iN (x),wecanobtain, ∀x, y ∈ X, ˜ iN ((x → y) → y) ⊆ ˜ iN (x → ((x → y) → y)) ∪ ˜ iN (x)

= iN ((x → y) → (x → y)) ∪ iN (x)

= iN (1) ∪ iN (x) = iN (x).

Similarly,wehave iN ((x → y) → y) ⊆ iN (x) Similarly,withregardtothefunction f N (x),wecanobtain f N ((x → y) → y) ⊆ f N (x), f N ((x → y) → y) ⊆ f N (x) (4)Let x, y, z ∈ X with z ≤ x → y.ByRemark1andDeﬁnition14,wecanobtain: tN (x) ∩ tN (z)= tN (x) ∩ (tN (1) ∩ tN (z)) = tN (x) ∩ (tN (z → (x → y)) ∩ tN (z)) ⊆ tN (x) ∩ tN (x → y), ⊆ tN (y)

iN (x) ∪ iN (z)= iN (x) ∪ (iN (1) ∪ iN (z)) = iN (x) ∪ (iN (z → (x → y)) ∪ iN (z)) ⊇ iN (x) ∪ iN (x → y), ⊇ ˜ iN (y).

Similarly,wecanobtain f N (x) ∪ f N (z) ⊇ f N (y).

Let x, y, z ∈ X with z ≤ x → y.Wecanobtain tN (x) ∩ tN (z) ⊆ tN (y), ˜ iN (x) ∪ ˜ iN (z) ⊇ ˜ iN (y), f N (x) ∪ f N (z) ⊇ f N (y).Theprocessoftheproofissimilartotheabove.

Theorem13. Aneutrosophichesitantfuzzyset N = {(x, ˜ tN (x), iN (x), ˜ f N (x))|x ∈ X)} isa neutrosophichesitantfuzzyﬁlterof X ifandonlyifhesitantfuzzysets HtN , HiN , H f N satisfythefollowing conditions,respectively.

(1) tN (x) ⊆ tN (1), tN (x → (y → z)) ∩ tN (y) ⊆ tN (x → z), tN (x → (y → z)) ∩ tN (y) ⊆ tN (x → z), ∀x, y, z ∈ X; (2) iN (x) ⊇ iN (1), iN (x → (y → z)) ∪ iN (y) ⊇ iN (x → z), iN (x → (y → z)) ∪ iN (y) ⊇ iN (x → z), ∀x, y, z ∈ X; (3) f N (x) ⊇ f N (1), f N (x → (y → z)) ∪ f N (y) ⊇ f N (x → z), f N (x → (y → z)) ∪ f N (y) ⊇ f N (x → z), ∀x, y, z ∈ X.

Proof.Necessity: ByTheorem9,Theorem12andDeﬁnition14,(1)∼(3)holds.

Sufﬁciency: (1) ∀x, y, z ∈ X,byProposition1,wecanobtain tN (y)= tN (1 → y) ⊇ tN (1 → (x → y)) ∩ tN (x)= tN (x → y) ∩ tN (x) and tN (y)= tN (1 → y) ⊇ tN (1 → (x → y)) ∩ tN (x)= tN (x → y) ∩ tN (x).Wehave ˜ iN (x) ⊇ ˜ iN (1) forall x ∈ X.Thus, HtN isahesitantfuzzyﬁlterof X (2) ∀x, y, z ∈ X,byProposition1,wecanobtain ˜ iN (y)= ˜ iN (1 → y) ⊆ ˜ iN (1 → (x → y)) ∪ ˜ iN (x)= iN (x → y) ∪ iN (x);thus,wehave (1 iN (x → y)) ∩ (1 iN (x)) ⊆ (1 iN (y))

Florentin Smarandache (author and editor) Collected Papers, IX 365

Similarly,wecanhave (1 iN (x → y)) ∩ (1 iN (x)) ⊆ (1 iN (y)). Itiseasytoobtain (1 ˜ iN (x)) ⊆ (1 tN (1)) forall x ∈ X.Thus, HiN isahesitantfuzzyﬁlter of X.

(3)Wehavethat H f N isahesitantfuzzyfilterof X.Theprocessoftheproofissimilar(2). Therefore, HtN , HiN , H f N arehesitantfuzzyfiltersof X.ByTheorem9,weknowthat N is aneutrosophichesitantfuzzyfilterof X Theorem14. Let N = {(x, ˜ tN (x), iN (x), f N (x))|x ∈ X)} beaneutrosophichesitantfuzzyfilterof X.Then: n ∏ k=1 xk → y = 1 ⇒ tN (y) ⊇ n k=1 tN (xk ), iN (y) ⊆ n i=k iN (xk ), f N (y) ⊆ n k=1 f N (xk ) wheren ∈ N, n ∏ k=1 xk → y = xn → (xn 1 → (··· (x1 → y) ··· ))

Proof. If N isaneutrosophichesitantfuzzyﬁlterof X:

(i)ByTheorem12,weknowthat tN (x1) ⊆ tN (y), iN (x1) ⊇ iN (y), f N (x1) ⊇ f N (y) for n = 1. (ii)ByTheorem12,weknowthat tN (x2) ⊆ tN (x1 → y), iN (x2) ⊇ iN (x1 → y), f N (x2) ⊇ f N (x1 → y) for n = 2.ByDeﬁnition14,wehave tN (x1) ∩ tN (x1 → y) ⊆ tN (y), iN (x1) ∪ iN (x1 → y) ⊇ iN (y), f N (x1) ∪ f N (x1 → y) ⊇ f N (y).Thus, tN (x1) ∩ tN (x2) ⊆ tN (y), iN (x1) ∪ iN (x2) ⊇ iN (y), f N (x1) ∪ f N (x2) ⊇ f N (y).

(iii)Supposethattheaboveformulaistruefor n = j;thus, j ∏ k=1 xk → y = 1, ∀xj, , x1, y ∈ X, andwecanobtain j k=1 tN (xk ) ⊆ tN (y), j k=1 iN (xk ) ⊇ iN (y), j k=1 f N (xk ) ⊇ f N (y).Therefore,supposethat j+1 ∏ k=1 xk → y = 1, ∀xj+1, , x1, y ∈ X,thenwehave j+1 k=2 tN (xk ) ⊆ tN (x1 → y), j+1 k=2 iN (xk ) ⊇ iN (x1 → y), j+1 k=2 f N (xk ) ⊇ f N (x1 → y).ByDeﬁnition14,wecanobtain: tN (y) ⊇ tN (x1) ∩ tN (x1 → y) ⊇ tN (x1) ∩ ( j+1 k=2 tN (xk ))= j+1 k=1 tN (xk ), ˜ iN (y) ⊆ ˜ iN (x1) ∪ ˜ iN (x1 → y) ⊆ ˜ iN (x1) ∪ ( j+1 k=2

˜ iN (xk ))= j+1 k=1

˜ iN (xk ), f N (y) ⊆ f N (x1) ∪ f N (x1 → y) ⊆ f N (x1) ∪ ( j+1 k=2 f N (xk ))= j+1 k=1 f N (xk ), whichcompletetheproof. Corollary3. Let N = {(x, ˜ tN (x), iN (x), ˜ f N (x)))|x ∈ X)} beaneutrosophichesitantfuzzyﬁlterof X.Then: n ∏ k=1 xk ∗ y = 1 ⇒ ˜ tN (y) ⊇ n k=1

˜ tN (xk ), iN (y) ⊆ n k=1 iN (xk ), ˜ f N (y) ⊆ n k=1

˜ f N (xk ) where"∗"representsanybinaryoperation"→"or" →"onX,n ∈ N, n ∏ k=1 xk ∗ y = xn ∗ (xn 1 ∗ ( (x1 ∗ y) )).

Theorem15. Let N = {(x, tN (x), ˜ iN (x), f N (x))|x ∈ X)} beaneutrosophichesitantfuzzyﬁlterof X and X beapseudo-BCKalgebra,thenNisaneutrosophichesitantfuzzysubalgebraofX.

Florentin Smarandache (author and editor) Collected Papers, IX 366

Proof. If N = {(x, tN (x), iN (x), f N (x))|x ∈ X)} isaneutrosophichesitantfuzzyﬁlterof X,thenwe canobtain ∀x, y ∈ X,

tN (x → y) ⊇ tN (y → (x → y)) ∩ tN (y) = ˜ tN (x → (y → y)) ∩ ˜ tN (y)

= tN (x → 1) ∩ tN (y) ⊇ tN (x) ∩ tN (y) ˜ iN (x → y) ⊆ ˜ iN (y → (x → y)) ∪ ˜ iN (y)

= iN (x → (y → y)) ∪ iN (y)

= iN (x → 1) ∪ iN (y) ⊆ iN (x) ∪ iN (y).

f N (x → y) ⊆ f N (y → (x → y)) ∪ f (y) = f N (x → (y → y)) ∪ f N (y) = ˜ f N (x → 1) ∪ ˜ f N (y) ⊆ f N (x) ∪ f N (y).

Similarly,wecanobtain tN (x → y) ⊇ tN (x) ∩ tN (y), iN (x → y) ⊆ iN (x) ∪ iN (y), f N (x → y) ⊆ f N (x) ∪ f N (y).Thus, N isaneutrosophichesitantfuzzysubalgebraof X

Theorem16. Let N = {(x, tN (x), iN (x), f N (x))|x ∈ X)} beaneutrosophichesitantfuzzyclosedﬁlterof X. Then,NisaneutrosophichesitantfuzzysubalgebraofX.

Proof. TheprocessofproofissimilartoTheorem15.

If N = {(x, tN (x), iN (x), f N (x))|x ∈ X)} isaneutrosophichesitantfuzzysubalgebraof X,then N maynotbeaneutrosophichesitantfuzzyﬁlterof X

Example6. Let X = {a, b, c, d,1} withtwobinaryoperationsinTables 1 and 2.Then, (X; →, →,1) is apseudo-BCIalgebra. N isaneutrosophichesitantfuzzysubalgebraof X.However, N isnotaneutrosophic hesitantfuzzyﬁlterofX.Since t(b → a) ∩ t(b)=[ 1 3 , 1 2 ], t(a)=[ 1 3 , 1 4 ],wecannotobtain t(b → a) ∩ t(b) ⊆ t(a).

Deﬁnition16. N = {(x, ˜ tN (x), iN (x), ˜ f N (x))|x ∈ X)} isaneutrosophichesitantfuzzyseton X.Deﬁnea neutrosophichesitantfuzzysetN(a,b) = {(x, ˜ t(a,b) N (x), i(a,b) N (x), ˜ f (a,b) N (x))|x ∈ X} by ∀a, b ∈ X, t(a,b) N : X =⇒ P([0,1]), x → ψ1, a → (b → x)= 1, a → (b → x)= 1; ψ2, otherwise : i(a,b) N : X =⇒ P([0,1]), x → ψ3, a → (b → x)= 1, a → (b → x)= 1; ψ4, otherwise : ˜ f (a,b) N : X =⇒ P([0,1]), x → ψ5, a → (b → x)= 1, a → (b → x)= 1; ψ6, otherwise : where ψ1, ψ2, ψ3, ψ4, ψ5, ψ6 ∈ P([0,1]), ψ1 ⊇ ψ2, ψ3 ⊆ ψ4, ψ5 ⊆ ψ6.Then, N(a,b) iscalledagenerated neutrosophichesitantfuzzyset.

Ageneratedneutrosophichesitantfuzzyset N(a,b) maynotbeaneutrosophichesitantfuzzyﬁlter of X.

Florentin Smarandache (author and editor) Collected Papers, IX 367

Example7. Let X = {a, b, c, d,1} withtwobinaryoperationsinTables 1 and 2.Then, (X; →, →,1) is apseudo-BCIalgebra. N isaneutrosophichesitantfuzzysetof X.However, N(a,b) isnotaneutrosophic hesitantfuzzyﬁlterof X.Since t(1,a) (a → b) ∩ t(1,a) (a)=[0,1], t(1,a) (b)=[ 1 3 , 2 3 ],wecannotobtain t(1,a) (a → b) ∩ t(1,a) (a) ⊆ t(1,a) (b).

Theorem17. Let X beapseudo-BCKalgebra.If X isatype-2positiveimplicativepseudo-BCKalgebra,then N(a,b) isaneutrosophichesitantfuzzyﬁlterofXforalla, b ∈ X.

Proof. If X isapseudo-BCKalgebra,(1)byDeﬁnition1andProposition1,wecanobtain a → (b → 1)= 1 (a → (b → 1)= 1) t(a,b) N (1)= ψ1 ⊇ t(a,b) N (x), i(a,b) N (1)= ψ3 ⊆ i(a,b) N (x), f (a,b) N (1)= ψ5 ⊆ ˜ f (a,b) N (x) forall x ∈ X (2)(i)Let x, y ∈ X with a → (b → x) = 1or a → (b → x) = 1or a → (b → (x → y)) = 1or a → (b → (x → y)) = 1.Thus,wecanobtain: t(a,b) N (x) ∩ t(a,b) N (x → y)= ψ2 ⊆ t(a,b) N (y), t(a,b) N (x) ∩ t(a,b) N (x → y)= ψ2 ⊆ t(a,b) N (y); ˜ i(a,b) N (x) ∪ ˜ i(a,b) N (x → y)= ψ4 ⊇ ˜ i(a,b) N (y), ˜ i(a,b) N (x) ∪ ˜ i(a,b) N (x → y)= ψ4 ⊇ ˜ i(a,b) N (y); f (a,b) N (x) ∪ f (a,b) N (x → y)= ψ6 ⊇ f (a,b) N (y), f (a,b) N (x) ∪ f (a,b) N (x → y)= ψ6 ⊇ f (a,b) N (y).

(ii)Let x, y ∈ X with a → (b → x)= 1, a → (b → x)= 1and a → (b → (x → y))= 1, a → (b → (x → y))= 1.Then,byProposition1andDeﬁnition4,wecanobtain:

t(a,b) N (a → (b → y)) =t(a,b) N (1 → (a → (b → y))) =t(a,b) N ((a → (b → x)) → (a → (b → y))) =t(a,b) N (a → ((b → x) → (b → y))) =t(a,b) N (a → (b → (x → y))) =t(a,b) N (1)

˜ t(a,b) N (a → (b → y)) = ˜ t(a,b) N (1 → (a → (b → y))) = ˜ t(a,b) N (((a → (b → x)) → (a → (b → y))) = ˜ t(a,b) N (a → ((b → x) → (b → y))) = ˜ t(a,b) N (a → (b → (x → y))) = ˜ t(a,b) N (1)

Therefore,wecanobtain, t(a,b) N (y)= ψ1 = t(a,b) N (x) ∩ t(a,b) N (x → y), t(a,b) N (y)= ψ1 = t(a,b) N (x) ∩ t(a,b) N (x → y).

Similarly,wecanobtain, i(a,b) N (y)= ψ3 = i(a,b) N (x) ∪ i(a,b) N (x → y), i(a,b) N (y)= ψ3 = i(a,b) N (x) ∪ i(a,b) N (x → y); f (a,b) N (y)= ψ5 = f (a,b) N (x) ∪ f (a,b) N (x → y), f (a,b) N (y)= ψ5 = f (a,b) N (x) ∪ f (a,b) N (x → y).

Thismeansthat N(a,b) isaneutrosophichesitantfuzzyﬁlterof X

Example8. Let X = {a, b, c, d,1} withtwobinaryoperationsinTables 7 and 8.Then, (X; →, →,1) is atype-2positiveimplicativepseudo-BCIalgebra.Let N beaneutrosophichesitantfuzzyset.Wetake b, c as

Florentin Smarandache (author and editor) Collected Papers, IX 368

anexample;thus,wehave {b, c, d,1} satisfy d → (c → x)= 1, d → (c → x)= 1.Let ψ1 =[0.1,0.4], ψ2 =[0.2,0.3], ψ3 =[0.4,0.5], ψ4 =[0.3,0.6], ψ5 =[0.2,0.8], ψ6 =[0.1,0.9], N(d,c) = {(1, ψ1, ψ3, ψ5), (a, ψ2, ψ4, ψ6), (b, ψ1, ψ3, ψ5), (c, ψ1, ψ3, ψ5), (e, ψ1, ψ3, ψ5)} = {(1, [0.1,0.4], [0.4,0.5], [0.2,0.8]), (a, [0.2,0.3], [0.3,0.6], [0.1,0.9]), (b, [0.1,0.4], [0.4,0.5], [0.2,0.8]), (c, [0.1,0.4], [0.4,0.5], [0.2,0.8]), (d, [0.1,0.4], [0.4,0.5], [0.2,0.8])}

Then,wecanobtainthatN(d,c) isaneutrosophichesitantfuzzyﬁlterofX.

Table7. →

→ a b c d 1 a 1 b c d 1 b a 1 1 1 1 c a d 1 d 1 d a b c 1 1 1 a b c d 1

Table8. → → a b c d 1 a 1 b c d 1 b a 1 1 1 1 c a d 1 d 1 d a b c 1 1 1 a b c d 1

Theorem18. Let N = {(x, ˜ tN (x), i(x), ˜ f (x))|x ∈ X} beaneutrosophichesitantfuzzyﬁlterof X.Then, X(5) N (a)= {x|˜ tN (a) ⊆ ˜ tN (x), iN (a) ⊇ iN (x), ˜ f N (a) ⊇ ˜ f N (x)} isaﬁlterofXforalla ∈ X.

Proof. (1)Let x, y ∈ X with x, x → y ∈ X5 N (a).Then,wehave tN (a) ⊆ tN (x), tN (a) ⊆ tN (x → y) Since N = {(x, tN (x), iN (x), f N (x))|x ∈ X} isaneutrosophichesitantfuzzyﬁlter,thuswehave tN (a) ⊆ tN (x) ∩ tN (x → y) ⊆ tN (y) ⊆ tN (1).Similarly,wecanget iN (a) ⊇ iN (x) ∪ i(x → y) ⊇ iN (y) ⊇ iN (1), f N (a) ⊇ f N (x) ∪ f N (x → y) ⊇ f N (y) ⊇ f N (1)

(2)Similarly,let x, y ∈ X with x, x → y ∈ X(5) N (a);wehave tN (a) ⊆ tN (x) ∩ tN (x → y) ⊆ tN (y) ⊆ tN (1), iN (a) ⊇ iN (x) ∪ iN (x → y) ⊇ iN (y) ⊇ iN (1), f N (a) ⊇ f N (x) ∪ f N (x → y) ⊇ f N (y) ⊇ f N (1)

Thismeansthat X(5) N (a) satisfiestheconditionsofDefinition2(F1),(F2)and(F3); X(5) N (a) isafilter of X

Example9. Let X = {a, b, c, d,1} withtwobinaryoperationsinTables 5 and 6.Then, (X; →, →,1) is apseudo-BCIalgebra.Let:

N = {(1, [0,1], [0, 3 7 ], [0, 1 10 ]), (a, [0, 1 4 ], [0, 3 4 ], [0, 1 2 ]), (b, [0, 1 4 ], [0, 3 4 ], [0, 1 2 ]), (c, [0, 1 3 ], [0, 3 5 ], [0, 1 4 ]), (d, [0, 3 4 ]), [0, 3 6 ], [0, 1 5 ])}

Then, N isaneutrosophichesitantfuzzyﬁlterof X.Let X(5) N (c)= {c, d,1}.Itiseasytogetthat X(5) N (a) is aﬁlter.

5.Conclusions

Inthispaper,theneutrosophichesitantfuzzysettheorywasappliedtopseudo-BCIalgebra, andtheneutrosophichesitantfuzzysubalgebras(filters)inpseudo-BCIalgebrasweredeveloped. Therelationshipsbetweenneutrosophichesitantfuzzysubalgebras(filters)andhesitantfuzzy subalgebras(filters)wasdiscussed,andsomepropertiesweredemonstrated.Infuturework,different typesofneutrosophichesitantfuzzyfilterswillbedefinedanddiscussed.

Florentin Smarandache (author and editor) Collected Papers, IX 369

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24. Faizi,S.;Rashid,T.;Salabun,W.;Zafar,S.DecisionMakingwithUncertaintyUsingHesitantFuzzySets. Int.J.FuzzySyst. 2017, 20,1–11.[CrossRef]

25. Torra,V.;Narukawa,Y.Onhesitantfuzzysetsanddecision.In Proceedingsofthe18thIEEEInternational ConferenceonFuzzySystems,JejuIsland,Korea,20–24August2009;PublishingHouse:JejuIsland,Korea,2009; pp.1378–1382.

26. Wang,F.Q.;Li,X.;Chen,X.H.Hesitantfuzzysoftsetanditsapplicationsinmulticriteriadecisionmaking. J.Appl.Math. 2014, 2014.[CrossRef]

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29. Xu,Z.S.;xia,M.Distanceandsimilaritymeasuresforhesitantfuzzysets. Inf.Sci. 2011, 181,2128–2138. [CrossRef]

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30. AlcantudJCR,TorraV.Decompositiontheoremsandextensionprinciplesforhesitantfuzzysets. Inf.Fusion 2018, 41,48–56.[CrossRef] 31. WangZ.X,LiJ.Correlationcoefficientsofprobabilistichesitantfuzzyelementsandtheirapplicationsto evaluationofthealternatives. Symmetry 2017, 9,259.[CrossRef] 32. Smarandache,F.Aunifyingfieldinlogicsneutrosophy:neutrosophicprobability,setandlogic. Mult.ValuedLog. 1999, 8,489–503. 33. Peng,J.;Wang,J.;Wu,X.Multi-Valuedneutrosophicsetsandpoweraggregationoperatorswiththeir applicationsinmulti-criteriagroupdecision-makingproblems. Int.J.Comput.Int.Syst. 2015, 8,345–363. [CrossRef] 34. Ye,J.Amulticriteriadecision-makingmethodusingaggregationoperatorsforsimplifiedneutrosophicsets. J.Intell.FuzzySyst. 2014, 26,2459–2466. 35. Wang,H.;Smarandache,F.;Zhang,Y.Q.;Sunderraman,R.IntervalNeutrosophicSetsandLogic:Theoryand ApplicationsinComputing. arXiv 2005,arXiv:cs/0505014.[CrossRef] 36. Wang,H.;Smarandache,F.;Sunderraman,R.Single-valuedneutrosophicsets. Rev.AirForceAcad 2013, 17, 10–13.

37. Ye,J.Multiple-attributedecision-makingmethodunderasingle-valuedneutrosophichesitantfuzzy environment. J.Intell.Syst 2014, 24,23–36.[CrossRef]

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Florentin Smarandache (author and editor) Collected Papers, IX 371

A Classical Group of Neutrosophic Triplet Groups Using {Z2p, ×}

Vasantha Kandasamy W.B., Ilanthenral Kandasamy, Florentin Smarandache

W.B. Vasantha Kandasamy, Ilanthenral Kandasamy, Florentin Smarandache (2018). A Classical Group of Neutrosophic Triplet Groups Using (Z2p, X). Symmetry 10, 194; DOI: 10.3390/sym10060194

Abstract: In this paper we study the neutrosophic triplet groups for a ∈ Z2p and prove this collection of triplets (a, neut(a), anti(a)) if trivial forms a semigroup under product, and semi-neutrosophic triplets are included in that collection. Otherwise, they form a group under product, and it is of order (p 1), with (p + 1, p + 1, p + 1) as the multiplicative identity. The new notion of pseudo primitive element is introduced in Z2p analogous to primitive elements in Zp, where p is a prime. Open problems based on the pseudo primitive elements are proposed. Here, we restrict our study to Z2p and take only the usual product modulo 2p

Keywords: neutrosophic triplet groups; semigroup; semi-neutrosophic triplets; classical group of neutrosophic triplets; S-semigroup of neutrosophic triplets; pseudo primitive elements

1.Introduction

FuzzysettheorywasintroducedbyZadehin[1]andwasgeneralizedtotheIntuitionisticFuzzy Set(IFS)byAtanassov[2].Real-world,uncertain,incomplete,indeterminate,andinconsistentdata werepresentedphilosophicallyasaneutrosophicsetbySmarandache[3],whoalsostudiedthenotion ofneutralitiesthatexistinallproblems.Many[4 7]havestudiedneutralitiesinneutrosophicalgebraic structures.Formoreaboutthisliteratureanditsdevelopment,referto[3 10].

Ithasnotbeenfeasibletorelatethisneutrosophicsettoreal-worldproblemsandtheengineering discipline.Toimplementsuchaset,Wangetal.[11]introducedaSingle-ValuedNeutrosophicSet (SVNS),whichwasfurtherdevelopedintoaDoubleValuedNeutrosophicSet(DVNS)[12]andaTriple ReﬁnedIndeterminateNeutrosophicSet(TRINS)[13].Thesesetsarecapableofdealingwiththereal world’sindeterminatedata,andfuzzysetsandIFSsarenot.

Smarandache[14]presentsrecentdevelopmentsinneutrosophictheories,includingtheneutrosophic triplet,therelatedtripletgroup,theneutrosophicduplet,andthedupletset.Thenew,innovative, andinterestingnotionoftheneutrosophictripletgroup,whichisagroupofthreeelements,was introducedbyFlorentinSmarandacheandAli[10].Sincethen,neutrosophictripletshavebeenafieldof interestthatmanyresearchershaveworkedon[15 22].In[21],cancellableneutrosophictripletgroups wereintroduced,anditwasprovedthatitcoincideswiththegroup.Thepaperalsodiscussesweak neutrosophicdupletsinBCIalgebras.Notionssuchastheneutrosophictripletcosetanditsconnection withtheclassicalcoset,neutrosophictripletquotientgroups,andneutrosophictripletnormalsubgroups weredefinedandstudiedby[20].

Usingthenotionofneutrosophictripletgroupsintroducedin[10],whichisdifferentfrom classicalgroups,severalinterestingstructuralpropertiesaredevelopedanddeﬁnedinthispaper. Here,westudytheneutrosophictripletgroupsusingonly {Z2p, ×}, p isaprimeandtheoperation × isproductmodulo2p.Thepropertiesasaneutrosophictripletgroupundertheinheritedoperation ×

Collected Papers, IX 372

Florentin Smarandache (author and editor)

isstudied.Thisleadstothedeﬁnitionofasemi-neutrosophictriplet.However,ithasbeenproved thatsemi-neutrosophictripletsformasemigroupunder ×,buttheneutrosophictripletgroups,which arenontrivialandarenotsemi-neutrosophictriplets,formaclassicalgroupofneutrosophictriplets under ×.

Thispaperisorganizedintofivesections.Section 2 providesbasicconcepts.InSection 3, westudyneutrosophictripletsinthecaseof Z2p,where p isanoddprime.Section 4 definesthe semi-neutrosophictripletandshowsseveralinterestingpropertiesassociatedwiththeclassicalgroup ofneutrosophictriplets.Thefinalsectionprovidestheconclusionsandprobableapplications.

2.BasicConcepts

Werecallherebasicdeﬁnitionsfrom[10]. Deﬁnition1. Consider (S, ×) tobeanonemptysetwithaclosedbinaryoperation. S iscalledaneutrosophic tripletsetifforany x ∈ S therewillexistaneutralof x called neut (x),whichisdifferentfromthealgebraic unitaryelement(classical),andanoppositeof x called anti (x),withboth neut (x) and anti (x) belongingto S suchthat x ∗ neut (x) = neut (x) ∗ x = x and x ∗ anti (x) = anti (x) ∗ x = neut (x)

Theelements x, neut (x),and anti (x) aretogethercalledaneutrosophictripletgroup,denotedby (x, neut (x) , anti (x))

neut (x) denotestheneutralof x. x istheﬁrstcoordinateofaneutrosophictripletgroupandnot aneutrosophictriplet. y isthesecondcomponent,denotedby neut (x),ofaneutrosophictripletif thereareelements x and z ∈ S suchthat x ∗ y = y ∗ x = x and x ∗ z = z ∗ x = y.Thus, (x, y, z) isthe neutrosophictriplet.

Weknowthat (neut (x) , neut (x) , neut (x)) isaneutrosophictripletgroup.Let {S, ∗} bethe neutrosophictripletset.If (S, ∗) iswelldeﬁnedandforall x, y ∈ S, x ∗ y ∈ S,and (x ∗ y) ∗ z = x ∗ (y ∗ z) forall x, y, z ∈ S,then {S, ∗} isdeﬁnedastheneutrosophictripletgroup.Clearly, {S, ∗} isnotagroup intheclassicalsense.

Inthefollowingsection,wedeﬁnethenotionofasemi-neutrosophictriplet,whichisdifferent fromneutrosophicdupletsandtheclassicalgroupofneutrosophictripletsof {Z2p, ×},andderive someofitsinterestingproperties.

3.TheClassicalGroupofNeutrosophicTripletGroupsof {Z2 p , ×} andItsProperties

Herewedeﬁnetheclassicalgroupofneutrosophictripletsusing {Z2p, ×},where p isanodd prime.Thecollectionofallnontrivialneutrosophictripletgroupsformsaclassicalgroupunderthe usualproductmodulo2p,andtheorderofthatgroupis p 1.Wealsoderiveinterestingpropertiesof suchgroups.

Wewillﬁrstillustratethissituationwithsomeexamples.

Example1. Let S = {Z22, ×} bethesemigroupunder × modulo22.Clearly,11and12aretheonly idempotentsorneutralelementsof Z22.Theidempotent 11 ∈ Z22 yieldsonlyatrivialneutrosophictriplet (11,11,11) for 11 × 21 = 11,where 21 isaunitin Z22.Theothernontrivialneutrosophictripletsassociated withtheneutralelement 12 areH = {(2,12,6) , (6,12,2) , (4,12,14) , (14,12,4) , (16,12,20) , (20,12,16) , (12,12,12) , (10,12,10) , (8,12,18) , (18,12,8)}.Itiseasilyveriﬁedthat {H, ×} isaclassicalgroupoforder 10 undercomponent-wisemultiplicationmodulo 22,with (12,12,12) astheidentityelement. (12,12,12) × (12,12,12) = (12,12,12) productmodulo 22.Likewise, (2,12,6) × (2,12,6) = (4,12,14) ,

Florentin Smarandache (author and editor) Collected Papers, IX 373

and (2,12,6) × (4,12,14) = (8,12,18) ; (2,12,6) × (8,12,18) = (16,12,20) , and (2,12,6) × (16,12,20) = (10,12,10) ; (10,12,10) × (2,12,6) = (20,12,16) , and (2,12,6) × (20,12,16) = (18,12,8) ; (2,12,6) × (18,12,8) = (14,12,4) , and (2,12,6) × (14,12,4) = (6,12,2) ; (6,12,2) × (2,12,6) = (12,12,12) , and (2,12,6)10 = (12,12,12)

Thus,Hisacyclicgroupoforder 10. Example2. Let S = {Z14, ×} bethesemigroupunderproductmodulo 14.Theneutralelementsoridempotents ofZ14 are 7 and 8.Theneutrosophictripletsare

H = {(2,8,4) , (4,8,2) , (6,8,6) , (10,8,12) , (12,8,10) , (8,8,8)}, associatedwiththeneutralelement 8.Hisaclassicalgroupoforder 6.Clearly, (10,8,12) × (10,8,12) = (2,8,4), (10,8,12) × (2,8,4) = (6,8,6), (10,8,12) × (6,8,6) = (4,8,2), (10,8,12) × (4,8,2) = (12,8,10),and (10,8,12) × (12,8,10) = (8,8,8)

Thus, H isgeneratedby (10,8,12) as (10,8,12)6 = (8,8,8),and (8,8,8) isthemultiplicativeidentityof theclassicalgroupofneutrosophictriplets.

Example3. Let S = {Z38, ×} bethesemigroupunderproductmodulo 38 19,20 ∈ Z38 aretheidempotents ofZ38 H = {(2,20,10) , (10,20,2) , (4,20,24) , (24,20,4) , (20,20,20) , (8,20,12) , (12,20,8) , (16,20,6) , (6,20,16) , (32,20,22) , (22,20,32) , (18,20,18) , (34,20,14) , (14,20,34) , (26,20,28) , (28,20,26) , (30,2036) , (36,20,30)} istheclassicalgroupofneutrosophictripletswith (20,20,20) astheidentityelementofH. Inviewofalltheseexample,wehavethefollowingresults.

Theorem1. Everysemigroup {Z2p, ×},wherepisanoddprime,hasonlytwoidempotents:pandp + 1 Proof. Clearly, p isaprimeoftheform2n + 1in Z2p p2 = (2n + 1)2 = 4n2 + 4n + 1 = 4n2 + 2n + 2n + 1 = 4n2 + 2n + p = 2n (2n + 1) + p = 2np + p = p

Florentin Smarandache (author and editor) Collected Papers, IX 374

Thus, p isanidempotentin Z2p.Consider p + 1 ∈ Z2p :

(p + 1)2 = p2 + 2p + 1 = p2 + 1 = p + 1as p2 = p.

Thus, p and p + 1aretheonlyidempotentsof Z2p.Infact, Z2p hasnoothernontrivialidempotent. Let x ∈ Z2p beanidempotent.Thisimpliesthat x mustbeevenasalloddelementsotherthan p are units.

Let x = 2n (where n isaninteger),and2 < n < p 1suchthat x2 = 4n2 = x = 2n,whichimplies that2n (2n 1) = 0.

Thisiszeroonlyif2n 1 = p as2n 1isodd.Otherwise,2n = 0,whichisnotpossible,as n isevenand n isnotequalto0, x = 0,so2n 1 = p.Thatis, x = 2n = p + 1istheonlypossibility. Otherwise, x = 0,whichisacontradiction.

Thus, Z2p hasonlytwoidempotents, p and p + 1.

Theorem2. LetG= {Z2p, ×},wherepisanoddprime,bethesemigroupunder ×,productmodulo 2p. 1.Ifa ∈ Z2p hasneut (a) andanti (a),thenaiseven. 2. Theonlynontrivialneutralelementis p + 1 forall a,whichcontributestoneutrosophictripletgroups inG.

Proof. Let a in G besuchthat a × neut (a) = a if a isoddand a = p.Then a 1 existsin Z2p andwe have neut (a) = 1,but neut (a) = 1bydeﬁnition.Hencetheresultistrue. Further,weknow neut (a) × neut (a) = neut (a),thatis neut (a) isanidempotent.Thisispossible ifandonlyif a = p + 1or p Clearly, a = p isruledoutbecause ap = 0foralleven a in Z2p,hencetheclaim. Thus, neut (a) = p + 1istheonlyneutralelementforallrelevant a in Z2p

Deﬁnition2. Let {Z2p, ×} bethesemigroupundermultiplicationmodulo 2p,where p isanoddprime. H = {(a, neut (a) , anti (a)) |a ∈ 2Z2p \{0}} {H, ×} isthecollectionofallneutrosophictripletgroups. H hasthemultiplicativeidentity (p + 1, p + 1, p + 1) underthecomponent-wiseproductmodulo 2p H isdeﬁned astheclassicalgroupofneutrosophictriplets.

Wehavealreadygivenexamplesofthem.Itisimportanttomentionthisdeﬁnitionisvalidonly for Z2p undertheproductmodulo2p where p isanoddprime.

Example4. LetS = {Z46, ×} bethesemigroupunderproductmodulo 46.Let H = {(24,24,24) , (2,24,12) , (12,24,2) , (4,24,6) , (6,24,4) , (8,24,26) , (26,24,8) , (16,24,36) , (36,24,16) , (32,24,18) , (18,24,32) , (22,24,22) , (10,24,30) , (14,24,28) , (28,24,14) , (30,24,10) , (20,24,38) , (38,24,20) , (34,24,44) , (44,24,34) , (40,24,42) , (42,24,40)} betheclassicalgroupofneutrosophictriplets,with (24,24,24) astheidentityunder ×.o (H) = 22

Inviewofallofthis,wehavetodeﬁnethefollowingfor Z2p

Deﬁnition3. Let {Z2p, ×} bethesemigroupunderproductmodulo 2p,where p isanoddprime.Let K = {2,4,...,2p 2} bethesetofallevenelementsof Z2p.For p + 1 ∈ K, x × p + 1 = x, ∀ x ∈ K Therealsoexistsay ∈ Ksuchthatyp 1 = p + 1.WedefinethisyasthepseudoprimitiveelementofK ⊆ Z2p.

Florentin Smarandache (author and editor) Collected Papers, IX 375

Note:Wecandefinepseudoprimitiveelementsonlyfor Z2p where p isanoddprimeandnotfor any Zn,where n isanevenintegerthatisanalogoustoprimitiveelementsin Zp,where p isaprime. Wewillillustratethissituationwithsomeexamples.

Example5. Let {Z6, ×} bethemodulosemigroup.For K = {2,4}, 2 isthepseudoprimitiveelementof K ⊆ Z6

Example6. Let {Z14, ×} bethemodulosemigroupunderproduct ×,modulo 14.Consider K = {2,4,6,8,10,12}⊆ Z14.Then 10 isthepseudoprimitiveelementofK ⊆ Z14

Example7. Let {Z34, ×} bethesemigroupunderproductmodulointeger 34. 10 isthepseudoprimitive elementofK = {2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32}⊆ Z34. Similarly,for {Z38, ×}, 10 isthepseudoprimitiveelementofK = 2Z38 \{0}⊆ Z38. However,inthecaseofZ22, Z58,andZ26, 2 isthepseudoprimitiveelementforthesesemigroups.

Weleaveitasanopenproblemtofindthenumberofsuchpseudoprimitiveelementsof K = {2,4,6,...,2(p 1)} of Z2p Wehavethefollowingtheorem.

Theorem3. LetS = {Z2p, ×} bethesemigroupunderproductmodulo 2p,wherepisanoddprime.

1.K = {2,4, ,2p 2}⊆ Z2p hasapseudoprimitiveelement x ∈ K with x p 1 = p + 1,where p + 1 is themultiplicativeidentityofK.

2.Kisacyclicgroupunder × oforderp 1 generatedbythatx,andp + 1 istheidentityelementofK.

3.SisaSmarandachesemigroup.

Proof. Consider Z2p,where p isanoddprime.Let K = {2,4,6, ,2p 2}⊆ Z2p.Forany x ∈ K, (p + 1)x = px + x = x ispx = 0(mod 2p),where x iseven.Thus, p + 1istheidentityelementof Z2p Thereisa x ∈ K suchthat x p 1 = p + 1usingtheprincipleof2p ≡ 0,where x iseven.This x isthe pseudoprimitiveelementof K

This x ∈ K provespart(2)oftheclaim.

Since K isagroupunder × and K ⊆{Z2p, ×},bythedeﬁnitionofSmarandachesemigroup[4], S isanS-semigroup,so(3)istrue.

Next,weprovethatthefollowingtheoremforourresearchpertainstotheclassicalgroupof neutrosophictripletsandtheirstructure.

Theorem4. LetS = {Z2p, ×} bethesemigroup.Then

H = {(a, neut(a), anti(a)) |a ∈ 2Z2p \{0}},

istheclassicalgroupofneutrosophictriplets,whichiscyclicandoftheorderp 1

Proof. Clearly,fromtheearliertheorem, K = 2Z2p \{0} isacyclicgroupoftheorder p 1,and p + 1 actsastheidentityelementof K

H = {(a, neut(a), anti(a)) |a ∈ K} isaneutrosophictripletgroupscollectionand neut(a)= p + 1 actsastheidentityandistheuniqueelement(neutralelement)forall a ∈ K (neut(a), neut(a), neut(a)) = (p + 1, p + 1, p + 1) actsastheuniqueidentityelementofevery neutrosophictripletgroup h in H

Since K ⊆ Z2p \{0} isacyclicgroupoforder p 1with p + 1astheidentityelementof K, wehave H = {(a, neut (a) , anti (a)) |a ∈ K},tobecyclic.If x ∈ K issuchthat x p 1 = p + 1,thenthat neutrosophictripletgroupelement (x, p + 1, anti(x)) in H willgenerate H asacyclicgroupoforder p 1as a × anti(a)= neut(a).

Florentin Smarandache (author and editor) Collected Papers, IX 376

Hence, H isacyclicgroupoforder p 1.

Next,weproceedtodescribethesemi-neutrosophictripletsinthefollowingsection.

4.Semi-NeutrosophicTripletsandTheirProperties

Inthissection,wedeﬁnethenotionofsemi-neutrosophictripletgroupsandtrivialneutrosophic tripletgroupsandshowsomeinterestingresults.

Example8. Let {Z26, ×} = Sbethesemigroupunderproductmodulo 26 Weseethat 13 ∈ Z26 isanidempotent,but 13 × 25 = 13,where 25 isaunitof Z26.Therefore,forthis 25,wecannotﬁnd anti(13),but 13 × 13 = 13 isanidempotent,and (13,13,13) isaneutrosophictriplet group.Wedonotacceptitasaneutrosophictriplet,asitcannotyieldanyothernontrivialtripletotherthan (13,13,13)

Further,theauthorsof[10]deﬁned (0,0,0) asatrivialneutrosophictripletgroup.

Deﬁnition4. Let S = {Z2p, ×} bethesemigroupunderproductmodulo 2p p ∈ Z2p isanidempotent of Z2p.However, p isnotaneutrosophictripletgroupas p × (2p 1) = 2p p = p.Hence, (p, neut(p), anti(p)) = (p, p, p) isdeﬁnedasasemi-neutrosophictripletgroup.

Proposition1. Let S = {Z2p, ×} bethesemigroupunderproductmodulo 2p (p, p, p) isthe semi-neutrosophictripletgroupofZ2p

Proof. Thisisobviousfromthedeﬁnitionandthefact p2 = p in Z2p underproductmodulo2p.

Example9. Let S = {Z46, ×} bethesemigroupunderproductmodulo 46. T = {(23,23,23) , (0,0,0)} isthe semi-neutrosophictripletgroupandthezeroneutrosophictripletgroup.Clearly, T isasemigroupunder ×,and T isdefinedasthesemigroupofsemi-neutrosophictripletgroupsofordertwoas (23,23,23) × (23,23,23) = (23,23,23) K = {(a, neut (a) , anti (a)) |a ∈ 2Z46 \{0} = {2,4,6,8,10,12,14,16, ,42,44}} isaclassical groupofneutrosophictriplets.

Let P = K ∪ T = K ∪ T.Forevery x ∈ K andforevery y ∈ T, x × y = y × x = (0,0,0) Thus, P isasemigroupunderproduct,and P isdefinedasthesemigroupofneutrosophictriplets. Further,wedeﬁne T astheannihilatingneutrosophictripletsemigroupoftheclassicalgroupof neutrosophictriplets.

Deﬁnition5. Let S = {Z2p, ×},where p isanoddprime,bethesemigroupunderproductmodulo 2p.Let K = {(a, neut (a) , anti (a)) |a ∈ 2Z2p \{0}, ×} betheclassicalgroupofneutrosophictriplets. Let T = {(p, p, p) , (0,0,0)} bethesemigroupofsemi-neutrosophictriplets(asaminomer,wecallthetrivial neutrosophictriplet (0,0,0) asasemi-neutrosophictriplet).Clearly, T ∪ K = T ∪ K = P isdefinedasthe semigroupofneutrosophictripletswitho (P) = o (T) + o (K) = p 1 + 2 = p + 1.

Further,TisdeﬁnedastheannihilatingsemigroupoftheclassicalgroupofneutrosophictripletsK.

Wehaveseenexamplesofclassicalgroupofneutrosophictriplets,andwehavedeﬁnedand studiedthisonlyfor Z2p undertheproductmodulo2p foreveryoddprime p Inthefollowingsection,weidentifyopenproblemsandprobableapplicationsoftheseconcepts.

5.DiscussionsandConclusions

Thispaperstudiestheneutrosophictripletgroupsintroducedby[10]onlyinthecaseof {Z2p, ×}, where p isanoddprime,underproductmodulo2p.Wehaveprovedthetripletsof Z2p arecontributed

Florentin Smarandache (author and editor) Collected Papers, IX 377

onlybyelementsin2Z2p \{0} = {2,4, ... ,2p 2},andthesetripletsunderproductformagroupof order p 1,deﬁnedastheclassicalgroupofneutrosophictriplets.

Further,thenotionofpseudoprimitiveelementisdeﬁnedforelements K1 = 2Z2p \{0} = {2,4,6, ... ,2p 2}⊆ Z2p.This K1 isacyclicgroupoforder p 1with p + 1asitsmultiplicative identity.Basedonthis, K = {(a, neut(a), anti(a)) |a ∈ K1, ×} isprovedtobeacyclicgroupoforder p 1. Wesuggestthefollowingproblems:

1.Howmanypseudoprimitiveelementsaretherein {Z2p, ×},where p isanoddprime?

2. Can {Zn, ×},where n isanycompositenumberdifferentfrom2p,havepseudoprimitive elements?Ifso,whichidempotentservesastheidentity?

Forfutureresearch,onecanapplytheproposedneutrosophictripletgrouptoSVNSanddevelop it for the case of DVNS or TRINS. These neutrosophic triplet groups can be applied to problems where neut(a) and anti(a) are fixed once a is chosen, and vice versa. It can be realized as a special case of Single Valued Neutrosophic Sets (SVNSs) where neutral is always fixed. For every a in K1, the other factor anti(a) is automatically fixed, thereby eliminating the arbitrariness in determining anti(a); however, there is only one case in which a = anti(a) The set 2Z2p \ {0} can be used to model this sort of problem and thereby reduce the arbitrariness in determining anti(a), which is an object of future study.

Abbreviations

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6. Vasantha,W.B.;Smarandache,F. N-AlgebraicStructuresandSN-AlgebraicStructures;Hexis:Phoenix,AZ,USA, 2005;ISBN978-1-931233-05-5.

7. Vasantha,W.B.;Smarandache,F. SomeNeutrosophicAlgebraicStructuresandNeutrosophicN-AlgebraicStructures; Hexis:Phoenix,AZ,USA,2006;ISBN978-1-931233-15-2.

8. Smarandache,F.Neutrosophicset-ageneralizationoftheintuitionisticfuzzyset.InProceedingsofthe2006 IEEEInternationalConferenceonGranularComputing,Atlanta,GA,USA,10–12May2006;pp.38–42.

9. Smarandache,F.OperatorsonSingle-ValuedNeutrosophicOversets,NeutrosophicUndersets,and NeutrosophicOffsets. J.Math.Inf. 2016, 5,63–67.[CrossRef]

The following abbreviations are used in this manuscript: SVNSSingleValuedNeutrosophicSet DVNSDoubleValuedNeutrosophicSet TRINSTripleReﬁnedIndeterminateNeutrosophicSet IFSIntuitionisticFuzzySet

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10.Smarandache,F.;Ali,M.Neutrosophictripletgroup. NeuralComput.Appl. 2018, 29,595–601.[CrossRef]

11. Wang,H.;Smarandache,F.;Zhang,Y.;Sunderraman,R.Singlevaluedneutrosophicsets. Review 2010, 1,10–15.

12. Kandasamy,I.Double-ValuedNeutrosophicSets,theirMinimumSpanningTrees,andClusteringAlgorithm. J.Intell.Syst. 2018, 27,163–182.[CrossRef]

13. Kandasamy,I.;Smarandache,F.TripleReﬁnedIndeterminateNeutrosophicSetsforpersonalityclassiﬁcation. InProceedingsofthe2016IEEESymposiumSeriesonComputationalIntelligence(SSCI),Athens,Greece, 6–9December2016;pp.1–8.

14. Smarandache,F. NeutrosophicPerspectives:Triplets,Duplets,Multisets,HybridOperators,ModalLogic,Hedge AlgebrasandApplications,2nded.;PonsPublishingHouse:Brussels,Belgium,2017;ISBN978-1-59973-531-3.

15.Sahin,M.;Abdullah,K.Neutrosophictripletnormedspace. OpenPhys. 2017, 15,697–704.[CrossRef]

16.Smarandache,F.HybridNeutrosophicTripletRinginPhysicalStructures. Bull.Am.Phys.Soc. 2017, 62,17.

17. Smarandache,F.;Ali,M.NeutrosophicTripletFieldusedinPhysicalApplications.InProceedingsofthe18th AnnualMeetingoftheAPSNorthwestSection,PacificUniversity,ForestGrove,OR,USA,1–3June2017.

18. Smarandache,F.;Ali,M.NeutrosophicTripletRinganditsApplications.InProceedingsofthe18thAnnual MeetingoftheAPSNorthwestSection,PaciﬁcUniversity,ForestGrove,OR,USA,1–3June2017.

19. Zhang,X.H.;Smarandache,F.;Liang,X.L.NeutrosophicDupletSemi-GroupandCancellableNeutrosophic TripletGroups. Symmetry 2017, 9,275–291.[CrossRef]

20. Bal,M.;Shalla,M.M.;Olgun,N.NeutrosophicTripletCosetsandQuotientGroups. Symmetry 2017, 10, 126–139.[CrossRef]

21. Zhang,X.H.;Smarandache,F.;Ali,M.;Liang,X.L.Commutativeneutrosophictripletgroupand neutro-homomorphismbasictheorem. Ital.J.PureAppl.Math. 2017,inpress.

22. Vasantha,W.B.;Kandasamy,I.;Smarandache,F. NeutrosophicTripletGroupsandTheirApplicationsto MathematicalModelling;EuropaNova:Brussels,Belgium,2017;ISBN978-1-59973-533-7.

Florentin Smarandache (author and editor) Collected Papers, IX 379

Commutative falling neutrosophic ideals in BCK-algebras

Young Bae Jun, Florentin Smarandache, Mehmat Ali ¨ Ozt¨urk

Young Bae Jun, Florentin Smarandache, Mehmat Ali Ozturk (2018). Commutative falling neutrosophic ideals in BCK-algebras. Neutrosophic Sets and Systems 20, 44-53

Abstract: Thenotionsofacommutative (∈, ∈)-neutrosophicideal andacommutativefallingneutrosophicidealareintroduced,and severalpropertiesareinvestigated.Characterizationsofacommutative (∈, ∈)-neutrosophicidealareobtained.Relationsbetween commutative (∈, ∈)-neutrosophicidealand (∈, ∈)-neutrosophic idealarediscussed.Conditions foran (∈, ∈)-neutrosophicidealto

beacommutative (∈, ∈)-neutrosophicidealareestablished.Relationsbetweencommutative (∈, ∈)-neutrosophicideal,fallingneutrosophicidealandcommutativefallingneutrosophicidealareconsidered.Conditionsforafallingneutrosophicidealtobecommutativeareprovided.

Keywords: (commutative) (∈, ∈)-neutrosophicideal; neutrosophicrandomset;neutrosophicfallingshadow;(commutative)fallingneutrosophicideal.

1Introduction

Neutrosophicset(NS)developedbySmarandache[11, 12, 13]isamoregeneralplatformwhichextendstheconcepts oftheclassic setandfuzzyset,intuitionisticfuzzysetand intervalvaluedintuitionisticfuzzyset.Neutrosophicset theoryisappliedtovariouspartwhichisreferedtothe sitehttp://fs.gallup.unm.edu/neutrosophy.htm.Jun,Borumand Saeidand Ozturkstudiedneutrosophicsubalgebras/idealsin BCK/BCI-algebrasbasedonneutrosophicpoints(see[1],[6] and[10]). Goodman[2]pointedouttheequivalenceofafuzzy setandaclassofrandomsetsinthestudyofauniﬁedtreatment ofuncertaintymodeledbymeansofcombiningprobabilityand fuzzysettheory.WangandSanchez[16]introducedthetheoryof fallingshadowswhichdirectlyrelatesprobabilityconceptswith themembershipfunctionoffuzzysets.Themathematicalstructureofthetheoryoffallingshadowsisformulatedin[17].Tanet al.[14, 15]establishedatheoreticalapproachtodeﬁneafuzzy inferencerelationandfuzzy setoperationsbasedonthetheoryof fallingshadows.JunandPark[7]consideredafuzzysubalgebra andafuzzyidealasthefallingshadowofthecloudofthesubalgebraandideal.Junetal.[8]introducedthenotionofneutrosophicrandomsetandneutrosophicfallingshadow.Usingthese notions,theyintroducedtheconceptoffallingneutrosophicsubalgebraandfallingneutrosophicidealin BCK/BCI-algebras, andinvestigatedrelatedproperties.Theydiscussedrelationsbetweenfallingneutrosophicsubalgebraandfallingneutrosophic ideal,andestablishedacharacterizationoffallingneutrosophic ideal.

Inthispaper,weintroducetheconceptsofacommutative (∈, ∈)-neutrosophicideal andacommutativefallingneutrosophic ideal,andinvestigateseveralproperties.Weobtaincharacteri-

zationsofacommutative (∈, ∈)-neutrosophicideal, anddiscuss relationsbetweenacommutative (∈, ∈)-neutrosophicidealand an (∈, ∈)-neutrosophicideal. Weprovideconditionsforan (∈, ∈)-neutrosophicideal tobeacommutative (∈, ∈)-neutrosophic ideal,andconsiderrelationsbetweenacommutative (∈, ∈) neutrosophicideal,afallingneutrosophicidealandacommutativefallingneutrosophicideal.Wegiveconditionsforafalling neutrosophicidealtobecommutative.

2Preliminaries

A BCK/BCI-algebraisanimportantclassoflogicalalgebras introducedbyK.Is´eki(see[3]and[4])andwasextensivelyinvestigatedby severalresearchers.

Bya BCI-algebra,wemeanaset X withaspecialelement 0 andabinaryoperation ∗ thatsatisﬁesthefollowingconditions:

Ifa BCI-algebra X satisﬁesthefollowingidentity:

(V) (∀x ∈ X )(0 ∗ x =0),

then X iscalled a BCK-algebra. Any BCK/BCI-algebra X

(I) (∀x,y,z ∈ X )(((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y)=0), (II) (∀x,y ∈ X ) ((x ∗ (x ∗ y)) ∗ y = 0), (III) (∀x ∈ X )(x ∗ x =0), (IV) (∀x,y ∈ X ) (x ∗ y =0,y ∗ x =0 ⇒ x = y

Collected Papers, IX 380

Florentin Smarandache (author and editor)

satisﬁesthefollowingconditions:

(∀x ∈ X )(x ∗ 0= x) , (2.1)

(∀x,y,z ∈ X ) x ≤ y ⇒ x ∗ z ≤ y ∗ z x ≤ y ⇒ z ∗ y ≤ z ∗ x , (2.2)

(∀x,y,z ∈ X )((x ∗ y) ∗ z =(x ∗ z) ∗ y) , (2.3) (∀x,y,z ∈ X )((x ∗ z) ∗ (y ∗ z) ≤ x ∗ y) (2.4)

where x ≤ y ifandonlyif x ∗ y =0. Anonemptysubset S ofa BCK/BCI-algebra X iscalleda subalgebra of X if x ∗ y ∈ S forall x,y ∈ S. Asubset I ofa BCK/BCI-algebra X iscalled an ideal of X ifitsatisﬁes:

0 ∈ I, (2.5) (∀x ∈ X )(∀y ∈ I )(x ∗ y ∈ I ⇒ x ∈ I ) (2.6)

Asubset I ofa BCK-algebra X iscalleda commutativeideal of X ifitsatisﬁes(2.5)and (x ∗ y) ∗ z ∈ I,z ∈ I ⇒ x ∗ (y ∗ (y ∗ x)) ∈ I (2.7) forall x,y,z ∈ X.

Observethateverycommutativeidealisanideal,buttheconverseisnottrue(see[9]).

Wereferthereadertothebooks[5, 9]forfurtherinformation regarding BCK/BCI-algebras. Foranyfamily {ai | i ∈ Λ} ofrealnumbers,wedeﬁne {ai | i ∈ Λ} :=sup{ai | i ∈ Λ} and {ai | i ∈ Λ} :=inf{ai | i ∈ Λ}

If Λ= {1, 2},wewillalsouse a1 ∨ a2 and a1 ∧ a2 insteadof {ai | i ∈ Λ} and {ai | i ∈ Λ},respectively.

Let X beanon-emptyset.A neutrosophicset (NS)in X (see [12])isastructureoftheform:

A := { x; AT (x),AI (x),AF (x) | x ∈ X}

A := { x; AT (x),AI (x),AF (x) | x ∈ X}.

Givenaneutrosophicset A =(AT ,AI ,AF ) inaset X, α,β ∈

(0, 1] and γ ∈ [0, 1),weconsiderthefollowingsets:

T∈(A; α):= {x ∈ X | AT (x) ≥ α}, I∈(A; β):= {x ∈ X | AI (x) ≥ β}, F∈(A; γ):= {x ∈ X | AF (x) ≤ γ}

Wesay T∈(A; α), I∈(A; β) and F∈(A; γ) are neutrosophic ∈ subsets

Aneutrosophicset A =(AT ,AI ,AF ) ina BCK/BCI algebra X iscalledan (∈, ∈)-neutrosophicsubalgebra of X (see [6])ifthefollowingassertionsarevalid. (∀x,y ∈ X )

       

x ∈ T∈(A; αx),y ∈ T∈(A; αy ) ⇒ x ∗ y ∈ T∈(A; αx ∧ αy ), x ∈ I∈(A; βx),y ∈ I∈(A; βy ) ⇒ x ∗ y ∈ I∈(A; βx ∧ βy ), x ∈ F∈(A; γx),y ∈ F∈(A; γy ) ⇒ x ∗ y ∈ F∈(A; γx ∨ γy )

        (2.8) forall αx,αy ,βx,βy ∈ (0, 1] and γx,γy ∈ [0, 1)

Aneutrosophicset A =(AT ,AI ,AF ) ina BCK/BCI algebra X iscalledan (∈, ∈)-neutrosophicideal of X (see[10]) ifthefollowingassertionsarevalid. (∀x ∈ X )   x ∈ T∈(A; αx) ⇒ 0 ∈ T∈(A; αx) x ∈ I∈(A; βx) ⇒ 0 ∈ I∈(A; βx) x ∈ F∈(A; γx) ⇒ 0 ∈ F∈(A; γx)   (2.9) and (∀x,y ∈ X )

       

x ∗ y ∈ T∈(A; αx),y ∈ T∈(A; αy ) ⇒ x ∈ T∈(A; αx ∧ αy ) x ∗ y ∈ I∈(A; βx),y ∈ I∈(A; βy ) ⇒ x ∈ I∈(A; βx ∧ βy ) x ∗ y ∈ F∈(A; γx),y ∈ F∈(A; γy ) ⇒ x ∈ F∈(A; γx ∨ γy )

        (2.10) forall αx,αy ,βx,βy ∈ (0, 1] and γx,γy ∈ [0, 1)

Inwhatfollows,let X and P (X ) denotea BCK/BCI algebraandthepowersetof X,respectively,unlessotherwise speciﬁed.

Foreach x ∈ X and D ∈P (X ),let x := {C ∈P (X ) | x ∈ C}, (2.11) and D := {x | x ∈ D} (2.12)

Anorderedpair (P (X ), B) issaidtobea hyper-measurable structure on X if B isa σ-ﬁeldin P (X ) and X ⊆B

Givenaprobabilityspace (Ω, A,P ) andahyper-measurable structure (P (X ), B) on X,a neutrosophicrandomset on X (see [8])isdeﬁnedtobeatriple ξ :=(ξT ,ξI ,ξF ) inwhich ξT , ξI and ξF aremappingsfrom Ω to P (X ) whichare A B measurables,

Florentin Smarandache (author and editor) Collected Papers, IX 381

thatis, (∀C ∈B)    ξ 1 T (C)= {ωT ∈ Ω | ξT (ωT ) ∈ C}∈A ξ 1 I (C)= {ωI ∈ Ω | ξI (ωI ) ∈ C}∈A ξ 1 F (C)= {ωF ∈ Ω | ξF (ωF ) ∈ C}∈A    . (2.13)

Givenaneutrosophicrandomset

siderfunctions:

,ξ

forall x,y,z ∈ X, αx,αy ,βx,βy ∈ (0, 1] and γx,γy ∈ [0, 1) Example3.2. Consideraset X = {0, 1, 2, 3} withthebinary operation ∗ whichisgiveninTable 1

Table1:Cayleytableforthebinaryoperation“∗” ∗ 0123 00000 11001 22102 33330

Then H :=(HT , HI , HF ) isaneutrosophicseton X,andwe callita neutrosophicfallingshadow (see[8])oftheneutrosophic randomset ξ :=(ξT ,ξI ,ξF ),and ξ :=(ξT ,ξI ,ξF ) iscalleda neutrosophiccloud (see[8])of H :=(HT , HI , HF ).

Forexample,consideraprobabilityspace (Ω, A,P )= ([0, 1], A,m) where A isaBorelﬁeldon [0, 1] and m istheusual Lebesguemeasure.Let H :=(HT , HI , HF ) beaneutrosophic setin X.Thenatriple ξ :=(ξT ,ξI ,ξF ) inwhich

ξT :[0, 1] →P (X ),α → T∈(H ; α), ξI :[0, 1] →P (X ),β → I∈(H ; β), ξF :[0, 1] →P (X ),γ → F∈(H ; γ)

isaneutrosophicrandomsetand ξ :=(ξT ,ξI ,ξF ) isaneutrosophiccloudof H :=(HT , HI , HF ).Wewillcall ξ := (ξT ,ξI ,ξF ) deﬁnedaboveasthe neutrosophiccut-cloud (see[8]) of H :=(HT , HI , HF )

Let (Ω, A,P ) beaprobabilityspaceandlet ξ :=(ξT ,ξI ,ξF ) beaneutrosophicrandomseton X.If ξT (ωT ), ξI (ωI ) and ξF (ωF ) aresubalgebras(resp.,ideals)of X forall ωT ,ωI ,ωF ∈ Ω,thentheneutrosophicfallingshadow H :=(HT , HI , HF ) of ξ :=(ξT ,ξI ,ξF ) iscalleda fallingneutrosophicsubalgebra (resp., fallingneutrosophicideal)of X (see[8]).

3Commutative (∈, ∈)-neutrosophic ideals

Then (X ; ∗, 0) isa BCK-algebra(see[9]).Let A = (AT ,AI ,AF ) beaneutrosophicsetin X deﬁnedbyTable 2

Table2:Tabularrepresentationof A =(AT ,AI ,AF ) XAT (x) AI (x) AF (x) 00.7 0.9 0.2 10.3 0.6 0.8 20 3 0 6 0 8 30 5 0 4 0 7

Itisroutinetoverifythat A =(AT ,AI ,AF ) isacommutative (∈, ∈)-neutrosophicidealof X.

Theorem3.3. Foraneutrosophicset A =(AT ,AI ,AF ) ina BCK-algebra X,thefollowingareequivalent.

(1) Thenon-empty ∈-subsets T∈(A; α), I∈(A; β) and F∈(A; γ) arecommutativeidealsof X forall α,β ∈ (0, 1] and γ ∈ [0, 1)

(2) A =(AT ,AI ,AF ) satisﬁesthefollowingassertions. (∀x ∈ X )   AT (0) ≥ AT (x) AI (0) ≥ AI (x) AF (0) ≤ AF (x)   (3.2)

(3.1)

Deﬁnition3.1. Aneutrosophicset A =(AT ,AI ,AF ) ina BCK-algebra X iscalleda commutative (∈, ∈)-neutrosophic ideal of X ifitsatisﬁesthecondition(2.9)and (x ∗ y) ∗ z ∈ T∈(A; αx),z ∈ T∈(A; αy ) ⇒ x ∗ (y ∗ (y ∗ x)) ∈ T∈(A; αx ∧ αy ) (x ∗ y) ∗ z ∈ I∈(A; βx),z ∈ I∈(A; βy ) ⇒ x ∗ (y ∗ (y ∗ x)) ∈ I∈(A; βx ∧ βy ) (x ∗ y) ∗ z ∈ F∈(A; γx),z ∈ F∈(A; γy ) ⇒ x ∗ (y ∗ (y ∗ x)) ∈ F∈(A; γx ∨ γy )

Proof. Assumethatthenon-empty ∈-subsets T∈(A; α), I∈(A; β) and F∈(A; γ) arecommutativeidealsof X forall α,β ∈ (0, 1] and γ ∈ [0, 1).If AT (0) <AT (a) forsome a ∈ X, then a ∈ T∈(A; AT (a)) and 0 / ∈ T∈(A; AT (a)).Thisisa contradiction,andso AT (0) ≥ AT (x) forall x ∈ X.Similarly,

ξ :=(ξT ,ξI

F ) on X,con-

˜ HT : X → [0, 1],xT → P (ωT | xT ∈ ξT (ωT )), HI : X → [0, 1],xI → P (ωI | xI ∈ ξI (ωI )), HF : X → [0, 1],xF → 1 P (ωF | xF ∈ ξF (ωF ))

andforall x,y,z ∈ X, AT (x ∗ (y ∗ (y ∗ x))) ≥ AT ((x ∗ y) ∗ z) ∧ AT (z) AI (x ∗ (y ∗ (y ∗ x))) ≥ AI ((x ∗ y) ∗ z) ∧ AI (z) AF (x ∗ (y ∗ (y ∗ x))) ≤ AF ((x ∗ y) ∗ z) ∨ AF (z) (3.3)

Florentin Smarandache (author and editor) Collected Papers, IX 382

AI (0) ≥ AI (x) forall x ∈ X.Supposethat AF (0) >AF (a) for some a ∈ X.Then a ∈ F∈(A; AF (a)) and 0 / ∈ F∈(A; AF (a)) Thisisacontradiction,andthus AF (0) ≤ AF (x) forall x ∈ X Therefore(3.2)isvalid.Assumethatthereexist a,b,c ∈ X such that

AT (a ∗ (b ∗ (b ∗ a))) <AT ((a ∗ b) ∗ c) ∧ AT (c)

Taking α := AT ((a ∗ b) ∗ c) ∧ AT (c) impliesthat (a ∗ b) ∗ c ∈ T∈(A; α) and c ∈ T∈(A; α) but a ∗ (b ∗ (b ∗ a)) / ∈ T∈(A; α), whichisacontradiction.Hence

AT (x ∗ (y ∗ (y ∗ x))) ≥ AT ((x ∗ y) ∗ z) ∧ AT (z)

forall x,y,z ∈ X.Bythesimilarway,wecanverifythat

AI (x ∗ (y ∗ (y ∗ x))) ≥ AI ((x ∗ y) ∗ z) ∧ AI (z) forall x,y,z ∈ X.Nowsupposethereare x,y,z ∈ X suchthat

AF (x ∗ (y ∗ (y ∗ x))) >AF ((x ∗ y) ∗ z) ∨ AF (z):= γ. Then (x∗y)∗z ∈ F∈(A; γ) and z ∈ F∈(A; γ) but x∗(y∗(y∗x)) / ∈ F∈(A; γ),acontradiction.Thus

AF (x ∗ (y ∗ (y ∗ x))) ≤ AF ((x ∗ y) ∗ z) ∨ AF (z) forall x,y,z ∈ X.

Conversely,let A =(AT ,AI ,AF ) beaneutrosophicsetin X satisfyingtwoconditions(3.2)and(3.3).Assumethat T∈(A; α), I∈(A; β) and F∈(A; γ) arenonemptyfor α,β ∈ (0, 1] and γ ∈ [0, 1).Let x ∈ T∈(A; α), a ∈ I∈(A; β) and u ∈ F∈(A; γ) for α,β ∈ (0, 1] and γ ∈ [0, 1).Then AT (0) ≥ AT (x) ≥ α, AI (0) ≥ AI (a) ≥ β,and AF (0) ≤ AF (u) ≤ γ by(3.2).It followsthat 0 ∈ T∈(A; α), 0 ∈ I∈(A; β) and 0 ∈ F∈(A; γ).Let a,b,c ∈ X besuchthat (a ∗ b) ∗ c ∈ T∈(A; α) and c ∈ T∈(A; α) for α ∈ (0, 1].Then

AT (a ∗ (b ∗ (b ∗ a))) ≥ AT ((a ∗ b) ∗ c) ∧ AT (c) ≥ α by(3.3),andso a ∗ (b ∗ (b ∗ a)) ∈ T∈(A; α).If (x ∗ y) ∗ z ∈ I∈(A; β) and z ∈ I∈(A; β) forall x,y,z ∈ X and β ∈ (0, 1], then AI ((x ∗ y) ∗ z) ≥ β and AI (z) ≥ β.Hencethecondition (3.3)impliesthat

AI (x ∗ (y ∗ (y ∗ x))) ≥ AI ((x ∗ y) ∗ z) ∧ AI (z) ≥ β, thatis, x ∗ (y ∗ (y ∗ x)) ∈ I∈(A; β).Finally,supposethat (x ∗ y) ∗ z ∈ F∈(A; γ) and z ∈ F∈(A; γ) forall x,y,z ∈ X and γ ∈ (0, 1].Then AF ((x ∗ y) ∗ z) ≤ γ and AF (z) ≤ γ,whichimplyfromthecondition(3.3)that

AF (x ∗ (y ∗ (y ∗ x))) ≤ AF ((x ∗ y) ∗ z) ∨ AF (z) ≤ γ. Hence x ∗ (y ∗ (y ∗ x)) ∈ F∈(A; γ).Thereforethenon-empty ∈

subsets T∈(A; α), I∈(A; β) and F∈(A; γ) arecommutativeideals of X forall α,β ∈ (0, 1] and γ ∈ [0, 1)

Theorem3.4. Let A =(AT ,AI ,AF ) beaneutrosophicsetin a BCK-algebra X.Then A =(AT ,AI ,AF ) isacommutative (∈, ∈)-neutrosophicidealof X ifandonlyifthenon-emptyneutrosophic ∈-subsets T∈(A; α), I∈(A; β) and F∈(A; γ) arecommutativeidealsof X forall α,β ∈ (0, 1] and γ ∈ [0, 1)

Proof. Let A =(AT ,AI ,AF ) beacommutative (∈, ∈) neutrosophicidealof X andassumethat T∈(A; α), I∈(A; β) and F∈(A; γ) arenonemptyfor α,β ∈ (0, 1] and γ ∈ [0, 1).Then thereexist x,y,z ∈ X suchthat x ∈ T∈(A; α), y ∈ I∈(A; β) and z ∈ F∈(A; γ).Itfollowsfrom(2.9)that 0 ∈ T∈(A; α), 0 ∈ I∈(A; β) and 0 ∈ F∈(A; γ).Let x,y,z,a,b,c,u,v,w ∈ X besuchthat

(x ∗ y) ∗ z ∈ T∈(A; α), z ∈ T∈(A; α), (a ∗ b) ∗ c ∈ I∈(A; β), c ∈ I∈(A; β), (u ∗ v) ∗ w ∈ F∈(A; γ), w ∈ F∈(A; γ)

Then

x ∗ (y ∗ (y ∗ x)) ∈ T∈(A; α ∧ α)= T∈(A; α), a ∗ (b ∗ (b ∗ a)) ∈ I∈(A; β ∧ β)= I∈(A; β), u ∗ (v ∗ (v ∗ u)) ∈ F∈(A; γ ∨ γ)= F∈(A; γ)

by(2.10).Hencethenon-emptyneutrosophic ∈-subsets T∈(A; α), I∈(A; β) and F∈(A; γ) arecommutativeidealsof X forall α,β ∈ (0, 1] and γ ∈ [0, 1).

Conversely,let A =(AT ,AI ,AF ) beaneutrosophicsetin X forwhich T∈(A; α), I∈(A; β) and F∈(A; γ) arenonemptyand arecommutativeidealsof X forall α,β ∈ (0, 1] and γ ∈ [0, 1) Obviously,(2.9)isvalid.Let x,y,z ∈ X and αx,αy ∈ (0, 1] besuchthat (x ∗ y) ∗ z ∈ T∈(A; αx) and z ∈ T∈(A; αy ).Then (x ∗ y) ∗ z ∈ T∈(A; α) and z ∈ T∈(A; α) where α = αx ∧ αy . Since T∈(A; α) isacommutativeidealof X,itfollowsthat x ∗ (y ∗ (y ∗ x)) ∈ T∈(A; α)= T∈(A; αx ∧ αy )

Similarly,if (x ∗ y) ∗ z ∈ I∈(A; βx) and z ∈ I∈(A; βy ) forall x,y,z ∈ X and βx,βy ∈ (0, 1],then x ∗ (y ∗ (y ∗ x)) ∈ I∈(A; βx ∧ βy ).

Now,supposethat (x ∗ y) ∗ z ∈ F∈(A; γx) and z ∈ F∈(A; γy ) for all x,y,z ∈ X and γx,γy ∈ [0, 1).Then (x ∗ y) ∗ z ∈ F∈(A; γ) and z ∈ F∈(A; γ) where γ = γx ∨ γy .Hence x ∗ (y ∗ (y ∗ x)) ∈ F∈(A; γ)= F∈(A; γx ∨ γy ) since F∈(A; γ) isacommutativeidealof X.Therefore A = (AT ,AI ,AF ) isacommutative (∈, ∈)-neutrosophicidealof X

Corollary3.5. Let A =(AT ,AI ,AF ) beaneutrosophicsetin a BCK-algebra X.Then A =(AT ,AI ,AF ) isacommuta-

Florentin Smarandache (author and editor) Collected Papers, IX 383

tive (∈, ∈)-neutrosophicidealof X ifandonlyifitsatisﬁestwo conditions (3.2) and (3.3)

Proposition3.6. Everycommutative (∈, ∈)-neutrosophicideal A =(AT ,AI ,AF ) ofa BCK-algebra X satisﬁes: (∀x,y ∈ X )

forall α,β ∈ (0, 1] and γ ∈ [0, 1)

Proof. Itisinducedbytaking z =0 in(3.1).

Theorem3.7. Everycommutative (∈, ∈)-neutrosophicidealof a BCK-algebra X isan (∈, ∈)-neutrosophicidealof X.

Proof. Let A =(AT ,AI ,AF ) beacommutative (∈, ∈) neutrosophicidealofa BCK-algebra X.Assumethat x ∗ y ∈ T∈(A; αx), y ∈ T∈(A; αy ), a ∗ b ∈ I∈(A; βa), b ∈ I∈(A; βb), c ∗ d ∈ F∈(A; γc), d ∈ F∈(A; γd)

forall x,y,a,b,c,d ∈ X.Using(2.1),wehave (x ∗ 0) ∗ y = x ∗ y ∈ T∈(A; αx), (a ∗ 0) ∗ b = a ∗ b ∈ I∈(A; βa), (c ∗ 0) ∗ d = c ∗ d ∈ F∈(A; γc) Itfollowsfrom(3.1),(2.1)and(V)that x = x ∗ 0= x ∗ (0 ∗ (0 ∗ x)) ∈ T∈(A; αx ∧ αy ), a = a ∗ 0= a ∗ (0 ∗ (0 ∗ a)) ∈ I∈(A; βa ∧ βb), c = c ∗ 0= c ∗ (0 ∗ (0 ∗ c)) ∈ F∈(A; γc ∨ γd) Therefore A =(AT ,AI ,AF ) isan (∈, ∈)-neutrosophicidealof X. TheconverseofTheorem 3.7 isnottrueasseeninthefollowingexample.

Example3.8. Consideraset X = {0, 1, 2, 3, 4} withthebinary operation ∗ whichisgiveninTable 3

Then (X ; ∗, 0) isa BCK-algebra(see[9]).Let A = (AT ,AI ,AF ) beaneutrosophicsetin X deﬁnedbyTable 4

Table4:Tabularrepresentationof A =(AT ,AI ,AF ) XAT (x) AI (x) AF (x) 00 660 770 27 10 550 450 37 20 330 660 47 30 330 450 67 40 330 450 67

Routinecalculationsshowthat A =(AT ,AI ,AF ) isan (∈, ∈) neutrosophicidealof X.Butitisnotacommutative (∈, ∈) neutrosophicidealof X since (2 ∗ 3) ∗ 0 ∈ T∈(A;0 6) and 0 ∈ T∈(A;0 5) but 2 ∗ (3 ∗ (3 ∗ 2)) / ∈ T∈(A;0 5 ∧ 0 6), (1 ∗ 3) ∗ 2 ∈ I∈(A;0 55) and 2 ∈ I∈(A;0 63) but 1 ∗ (3 ∗ (3 ∗ 1)) / ∈ I∈(A;0 55 ∧ 0 63),and/or (2 ∗ 3) ∗ 0 ∈ F∈(A;0 43) and 0 ∈ F∈(A;0 39) but 2 ∗ (3 ∗ (3 ∗ 2)) / ∈ F∈(A;0 43 ∨ 0 39)

Weprovideconditionsforan (∈, ∈)-neutrosophicidealtobe acommutative (∈, ∈)-neutrosophicideal.

Theorem3.9. Let A =(AT ,AI ,AF ) bean (∈, ∈)-neutrosophic idealofa BCK-algebra X inwhichthecondition (3.4) isvalid. Then A =(AT ,AI ,AF ) isacommutative (∈, ∈)-neutrosophic idealof X

Proof. Let A =(AT ,AI ,AF ) bean (∈, ∈)-neutrosophicideal of X and x,y,z ∈ X besuchthat (x ∗ y) ∗ z ∈ T∈(A; αx) and z ∈ T∈(A; αy ) for αx,αy ∈ (0, 1].Then x ∗ y ∈ T∈(A; αx ∧ αy ) since A =(AT ,AI ,AF ) isan (∈, ∈)-neutrosophicidealof X Itfollowsfrom(3.4)that x ∗ (y ∗ (y ∗ x)) ∈ T∈(A; αx ∧ αy ) Similarly,if (x ∗ y) ∗ z ∈ I∈(A; βx) and z ∈ I∈(A; βy ),then x ∗ (y ∗ (y ∗ x)) ∈ I∈(A; βx ∧ βy ).Let a,b,c ∈ X and γa,γb ∈ [0, 1) besuchthat (a ∗ b) ∗ c ∈ F∈(A; γa) and c ∈ F∈(A; γa). Then a ∗ b ∈ F∈(A; γa ∨ γb),whichimpliesfrom(3.4)that a ∗ (b ∗ (b ∗ a)) ∈ F∈(A; γa ∨ γb).Therefore A =(AT ,AI ,AF ) isacommutative (∈, ∈)-neutrosophicidealof X

Lemma3.10. Every (∈, ∈)-neutrosophicideal A = (AT ,AI ,AF ) ofa BCK-algebra X satisﬁes:

y,z ∈ T∈(A; α) ⇒ x ∈ T∈(A; α) y,z ∈ I∈(A; β) ⇒ x ∈ I∈(A; β) y,z ∈ F∈(A; γ) ⇒ x ∈ F∈(A; γ) (3.5) forall α,β ∈ [0, 1), γ ∈ (0, 1] and x,y,z ∈ X with x ∗ y ≤ z.

Proof. Forany α,β ∈ [0, 1), γ ∈ (0, 1] and x,y,z ∈ X with x ∗ y ≤ z,let y,z ∈ T∈(A; α), y,z ∈ I∈(A; β) and y,z ∈ F∈(A; γ).Then (x ∗ y) ∗ z =0 ∈ T∈(A; α) ∩ I∈(A; β) ∩ F∈(A; γ)

     

 

x ∗ y ∈ T∈(A; α) ⇒ x ∗ (y ∗ (y ∗ x)) ∈ T∈(A; α) x ∗ y ∈ I∈(A; β) ⇒ x ∗ (y ∗ (y ∗ x)) ∈ I∈(A; β) x ∗ y ∈ F∈(A; γ) ⇒ x ∗ (y ∗ (y ∗ x)) ∈ F∈(A; γ)         (3.4)

Table3:Cayleytableforthebinaryoperation“∗” ∗ 01234 000000 110100 222000 333300 444430

Florentin Smarandache (author and editor) Collected Papers, IX 384

by(2.9).Itfollowsfrom(2.10)that x ∗ y ∈ T∈(A; α) ∩ I∈(A; β) ∩ F∈(A; γ) andsothat x ∈ T∈(A; α) ∩ I∈(A; β) ∩ F∈(A; γ) Thus(3.5)isvalid. Theorem3.11. Inacommutative BCK-algebra,every (∈, ∈) neutrosophicidealisacommutative (∈, ∈)-neutrosophicideal. Proof. Let A =(AT ,AI ,AF ) bean (∈, ∈)-neutrosophicideal ofacommutative BCK-algebra X.Let x,y,z ∈ X besuchthat (x ∗ y) ∗ z ∈ T∈(A; αx) ∩ I∈(A; βx) ∩ F∈(A; γx) and z ∈ T∈(A; αy ) ∩ I∈(A; βy ) ∩ F∈(A; γy ) for αx,αy ,βx,βy ∈ (0, 1] and γx,γy ∈ [0, 1).Notethat ((x ∗ (y ∗ (y ∗ x))) ∗ ((x ∗ y) ∗ z)) ∗ z =((x ∗ (y ∗ (y ∗ x))) ∗ z) ∗ ((x ∗ y) ∗ z) ≤ (x ∗ (y ∗ (y ∗ x))) ∗ (x ∗ y) =(x ∗ (x ∗ y)) ∗ (y ∗ (y ∗ x)) =0 by(2.3),(2.4)and(III),whichimpliesthat (x ∗ (y ∗ (y ∗ x))) ∗ ((x ∗ y) ∗ z) ≤ z. ItfollowsfromLemma 3.10 that x ∗ (y ∗ (y ∗ x)) ∈ T∈(A; αx) ∩ I∈(A; βx) ∩ F∈(A; γx) Therefore A =(AT ,AI ,AF ) isacommutative (∈, ∈) neutrosophicidealof X

4Commutativefallingneutrosophic ideals

Deﬁnition4.1. Let (Ω, A,P ) beaprobabilityspaceandlet ξ := (ξT ,ξI ,ξF ) beaneutrosophicrandomsetona BCK-algebra X.Thentheneutrosophicfallingshadow H :=(HT , HI , HF ) of ξ :=(ξT ,ξI ,ξF ) iscalleda commutativefallingneutrosophic ideal of X if ξT (ωT ), ξI (ωI ) and ξF (ωF ) arecommutativeideals of X forall ωT ,ωI ,ωF ∈ Ω

Example4.2. Consideraset X = {0, 1, 2, 3, 4} withthebinary operation ∗ whichisgiveninTable 5 Then (X ; ∗, 0) isa BCK-algebra(see[9]).Consider (Ω, A,P )=([0, 1], A,m) andlet ξ :=(ξT ,ξI ,ξF ) beaneu-

Table5:Cayleytableforthebinaryoperation“∗” ∗ 01234 000000 110011 221022 333303 444440

Neutrosophic random set on X which is given as follows: ξT :[0, 1] →P (X ),x →    

 {0, 3} if t ∈ [0, 0.25), {0, 4} if t ∈ [0.25, 0.55), {0, 1, 2} if t ∈ [0 55, 0 85), {0, 3, 4} if t ∈ [0 85, 1], ξI :[0, 1] →P (X ),x →    {0, 1, 2} if t ∈ [0, 0 45), {0, 1, 2, 3} if t ∈ [0.45, 0.75), {0, 1, 2, 4} if t ∈ [0.75, 1], and ξF :[0, 1] →P (X ),x →







{0} if t ∈ (0 9, 1], {0, 3} if t ∈ (0 7, 0 9], {0, 4} if t ∈ (0 5, 0 7], {0, 1, 2, 3} if t ∈ (0 3, 0 5], X if t ∈ [0, 0.3].

Then ξT (t),ξI (t) and ξF (t) are commutativeidealsof X for all t ∈ [0, 1].Hencetheneutrosophicfallingshadow H := (HT , HI , HF ) of ξ :=(ξT ,ξI ,ξF ) is acommutativefallingneutrosophicidealof X,anditisgivenasfollows:

HT (x)= 



 1 if x =0, 0.3 if x ∈{1, 2}, 0 4 if x =3, 0 45 if x =4, HI (x)=    1 if x ∈{0, 1, 2}, 0.3 if x = 3, 0.25 if x =4, and HF (x)=    0 if x =0, 0 5 if x ∈{1, 2, 4}, 0 3 if x = 3.

Givenaprobabilityspace (Ω, A,P ),let H :=(HT , HI , HF ) beaneutrosophicfallingshadowofaneutrosophicrandomset

 

   

 

Florentin Smarandache (author and editor) Collected Papers, IX 385

ξ :=(ξT ,ξI ,ξF ).For x ∈ X,let

Ω(x; ξT ):= {ωT ∈ Ω | x ∈ ξT (ωT )}, Ω(x; ξI ):= {ωI ∈ Ω | x ∈ ξI (ωI )}, Ω(x; ξF ):= {ωF ∈ Ω | x ∈ ξF (ωF )} Then Ω(x; ξT ), Ω(x; ξI ), Ω(x; ξF ) ∈A (see[8]).

Proposition4.3. Let H :=(HT , HI , HF ) beaneutrosophic fallingshadowoftheneutrosophicrandomset ξ :=(ξT ,ξI ,ξF ) ona BCK-algebra X.If H :=(HT , HI , HF ) isacommutative fallingneutrosophicidealof X,then

Ω((x ∗ y) ∗ z; ξT ) ∩ Ω(z; ξT ) ⊆ Ω(x ∗ (y ∗ (y ∗ x)); ξT ) Ω((x ∗ y) ∗ z; ξI ) ∩ Ω(z; ξI ) ⊆ Ω(x ∗ (y ∗ (y ∗ x)); ξI ) Ω((x ∗ y) ∗ z; ξF ) ∩ Ω(z; ξF ) ⊆ Ω(x ∗ (y ∗ (y ∗ x)); ξF )

(4.1) and Ω(x ∗ (y ∗ (y ∗ x)); ξT ) ⊆ Ω((x ∗ y) ∗ z; ξT ) Ω(x ∗ (y ∗ (y ∗ x)); ξI ) ⊆ Ω((x ∗ y) ∗ z; ξI ) Ω(x ∗ (y ∗ (y ∗ x)); ξF ) ⊆ Ω((x ∗ y) ∗ z; ξF ) (4.2) forall x,y,z ∈ X

Proof. Let ωT ∈ Ω((x ∗ y) ∗ z; ξT ) ∩ Ω(z; ξT ), ωI ∈ Ω((x ∗ y) ∗ z; ξI ) ∩ Ω(z; ξI ), ωF ∈ Ω((x ∗ y) ∗ z; ξF ) ∩ Ω(z; ξF ) forall x,y,z ∈ X.Then (x ∗ y) ∗ z ∈ ξT (ωT ) and z ∈ ξT (ωT ), (x ∗ y) ∗ z ∈ ξI (ωI ) and z ∈ ξI (ωI ), (x ∗ y) ∗ z ∈ ξF (ωF ) and z ∈ ξF (ωF )

Since ξT (ωT ), ξI (ωI ) and ξF (ωF ) arecommutativeidealsof X, itfollowsfrom(2.7)that x ∗ (y ∗ (y ∗ x)) ∈ ξT (ωT ) ∩ ξI (ωI ) ∩ ξF (ωF )

forall x,y,z ∈ X.Then x ∗ (y ∗ (y ∗ x)) ∈ ξT (ωT ) ∩ ξI (ωI ) ∩ ξF (ωF ).

Notethat ((x ∗ y) ∗ z) ∗ (x ∗ (y ∗ (y ∗ x))) =((x ∗ y) ∗ (x ∗ (y ∗ (y ∗ x)))) ∗ z ≤ ((y ∗ (y ∗ x)) ∗ y) ∗ z =((y ∗ y) ∗ (y ∗ x)) ∗ z =(0 ∗ (y ∗ x)) ∗ z =0 ∗ z =0, whichyields ((x ∗ y) ∗ z) ∗ (x ∗ (y ∗ (y ∗ x))) =0 ∈ ξT (ωT ) ∩ ξI (ωI ) ∩ ξF (ωF ).

Since ξT (ωT ), ξI (ωI ) and ξF (ωF ) arecommutativeidealsand henceidealsof X,itfollowsthat (x ∗ y) ∗ z ∈ ξT (ωT ) ∩ ξI (ωI ) ∩ ξF (ωF ) Hence ωT ∈ Ω((x ∗ y) ∗ z; ξT ), ωI ∈ Ω((x ∗ y) ∗ z; ξI ), ωF ∈ Ω((x ∗ y) ∗ z; ξF ) Therefore(4.2)isvalid.

Givenaprobabilityspace (Ω, A,P ),let F (X ):= {f | f :Ω → X isamapping}. (4.3) Deﬁneabinaryoperation on F (X ) asfollows: (∀ω ∈ Ω)((f g)(ω)= f (ω) ∗ g(ω)) (4.4)

forall f,g ∈F (X ).Then (F (X ); ,θ) isa BCK/BCI algebra(see[7])where θ isgivenasfollows: θ :Ω → X,ω → 0

Foranysubset A of X and gT ,gI ,gF ∈F (X ),considerthe followings:

Ag T := {ωT ∈ Ω | gT (ωT ) ∈ A}, Ag I := {ωI ∈ Ω | gI (ωI ) ∈ A}, Ag F := {ωF ∈ Ω | gF (ωF ) ∈ A} and ξT :Ω →P (F (X )),ωT →{gT ∈F (X ) | gT (ωT ) ∈ A},

ξI :Ω →P (F (X )),ωI →{gI ∈F (X ) | gI (ωI ) ∈ A}, ξF :Ω →P (F (X )),ωF →{gF ∈F (X ) | gF (ωF ) ∈ A} Then Ag T ,Ag I ,Ag F ∈A (see[8]).

ωT ∈

ωI ∈

andsothat ωT ∈ Ω(x ∗ (y ∗ (y ∗ x)); ξT ), ωI ∈ Ω(x ∗ (y ∗ (y ∗ x)); ξI ), ωF ∈ Ω(x ∗ (y ∗ (y ∗ x)); ξF ) Hence(4.1)isvalid.Nowlet

Ω(x ∗ (y ∗ (y ∗ x)); ξT ),

Ω(x ∗ (y ∗ (y ∗ x)); ξI ),

F ∈ Ω(x ∗ (y ∗ (y ∗ x)); ξF )

Florentin Smarandache (author and editor) Collected Papers, IX 386

Theorem4.4. If K isacommutativeidealofa BCK-algebra X,then ξT (ωT )= {gT ∈F (X ) | gT (ωT ) ∈ K}, ξI (ωI )= {gI ∈F (X ) | gI (ωI ) ∈ K}, ξF (ωF )= {gF ∈F (X ) | gF (ωF ) ∈ K} arecommutativeidealsof F (X ) Proof. Assumethat K isacommutativeidealofa BCK-algebra X.Since θ(ωT )=0 ∈ K, θ(ωI )=0 ∈ K and θ(ωF )=0 ∈ K forall ωT ,ωI ,ωF ∈ Ω,wehave θ ∈ ξT (ωT ), θ ∈ ξI (ωI ) and θ ∈ ξF (ωF ).Let fT ,gT ,hT ∈F (X ) besuchthat (fT gT ) hT ∈ ξT (ωT ) and hT ∈ ξT (ωT ) Then (fT (ωT ) ∗ gT (ωT )) ∗ hT (ωT )=((fT gT ) hT )(ωT ) ∈ K and hT (ωT ) ∈ K.Since K isacommutativeidealof X,it followsfrom(2.7)that (fT (gT (gT fT )))(ωT ) = fT (ωT ) ∗ (gT (ωT ) ∗ (gT (ωT ) ∗ fT (ωT ))) ∈ K, thatis, fT (gT (gT fT )) ∈ ξT (ωT ).Hence ξT (ωT ) isa commutativeidealof F (X ).Similarly,wecanverifythat ξI (ωI ) isacommutativeidealof F (X ).Now,let fF ,gF ,hF ∈F (X ) besuchthat (fF gF ) hF ∈ ξF (ωF ) and hF ∈ ξF (ωF ).Then (fF (ωF ) ∗ gF (ωF )) ∗ hF (ωF ) =((fF gF ) hF )(ωF ) ∈ K and hF (ωF ) ∈ K.Then (fF (gF (gF fF )))(ωF ) = fF (ωF ) ∗ (gF (ωF ) ∗ (gF (ωF ) ∗ fF (ωF ))) ∈ K, andso fF (gF (gF fF )) ∈ ξF (ωF ).Hence ξF (ωF ) isa commutativeidealof F (X ).Thiscompletestheproof.

Theorem4.5. Ifweconsideraprobabilityspace (Ω, A,P )= ([0, 1], A,m),theneverycommutative (∈, ∈)-neutrosophicideal ofa BCK-algebraisacommutativefallingneutrosophicideal. Proof. Let H :=(HT , HI , HF ) beacommutative (∈, ∈ )-neutrosophicidealof X.Then T∈(H ; α), I∈(H ; β) and F∈(H ; γ) arecommutativeidealsof X forall α,β ∈ (0, 1] and γ ∈ [0, 1).Henceatriple ξ :=(ξT ,ξI ,ξF ) inwhich ξT :[0, 1] →P (X ),α → T∈(H ; α), ξI :[0, 1] →P (X ),β → I∈(H ; β), ξF :[0, 1] →P (X ),γ → F∈(H ; γ)

isaneutrosophiccut-cloudof H :=(HT , HI , HF ).Therefore H :=(HT , HI , HF ) isacommutativefallingneutrosophicideal

of X

TheconverseofTheorem 4.5 isnottrueasseeninthefollowingexample.

Example4.6. Consideraset X = {0, 1, 2, 3, 4} withthebinary operation ∗ whichisgiveninTable 6

Then (X ; ∗, 0) isa BCK-algebra(see[9]).Consider (Ω, A,P )=([0, 1], A,m) andlet ξ :=(ξT ,ξI ,ξF ) beaneutrosophicrandomseton X whichisgivenasfollows: ξT :[0, 1] →P (X ),x → 

{0, 1} if t ∈ [0, 0.2), {0, 2} if t ∈ [0.2, 0.55), {0, 2, 4} if t ∈ [0 55, 0 75), {0, 1, 2, 3} if t ∈ [0 75, 1], ξI :[0, 1] →P (X ),x → 







Table6:Cayleytableforthebinaryoperation“∗” ∗ 01234 000000 110101 222002 332103 444440

 









   

          

     

{0, 1} if t ∈ [0, 0 34), {0, 4} if t ∈ [0.34, 0.66), {0, 1, 4} if t ∈ [0.66, 0.78), X if t ∈ [0.78, 1], and ξF :[0, 1] →P (X ),x → Florentin Smarandache (author and editor) Collected Papers, IX 387



{0} if t ∈ (0 87, 1], {0, 2} if t ∈ (0 76, 0 87], {0, 4} if t ∈ (0 58, 0 76], {0, 2, 4} if t ∈ (0.33, 0.58], X if t ∈ [0, 0.33]. Then ξT (t),ξI (t) and ξF (t) arecommutativeidealsof X for all t ∈ [0, 1].Hencetheneutrosophicfallingshadow H := (HT , HI , HF ) of ξ :=(ξT ,ξI ,ξF ) isacommutativefallingneutrosophicidealof X,anditisgivenasfollows: HT (x)=

if x =0,

45 if x =1,

8 if x =2, 0 25 if x =3, 0 2 if x =4, HI (x)=

if x =0, 0.68 if x =1, 0 22 if x ∈{2, 3}, 0 66 if x =4,

and HF (x)= 



 0 if x =0, 0.67 if x ∈{1, 3}, 0.31 if x =2, 0.24 if x =4 But H :=(HT , HI , HF ) isnotacommutative (∈, ∈) neutrosophicidealof X since (3 ∗ 4) ∗ 2 ∈ T∈(H ;0.4) and 2 ∈ T∈(H ;0.6), but 3 ∗ (4 ∗ (4 ∗ 3))=3 / ∈ T∈(H ;0 4).

Weproviderelationsbetweenafallingneutrosophicidealand acommutativefallingneutrosophicideal.

Theorem4.7. Let (Ω, A,P ) beaprobabilityspaceandlet ˜ H :=( ˜ HT , ˜ HI , ˜ HF ) beaneutrosophicfallingshadowofaneutrosophicrandomset ξ :=(ξT ,ξI ,ξF ) ona BCK-algebra.If H :=(HT , HI , HF ) isacommutativefallingneutrosophicideal of X,thenitisafallingneutrosophicidealof X

Proof. Let H :=(HT , HI , HF ) beacommutativefallingneutrosophicidealofa BCK-algebra X.Then ξT (ωT ), ξI (ωI ) and ξF (ωF ) arecommutativeidealsof X forall ωT ,ωI ,ωF ∈ Ω. Thus ξT (ωT ), ξI (ωI ) and ξF (ωF ) areidealsof X forall ωT ,ωI , ωF ∈ Ω.Therefore H :=(HT , HI , HF ) isafallingneutrosophicidealof X

ThefollowingexampleshowsthattheconverseofTheorem 4.7 isnottrueingeneral.

Example4.8. Consideraset X = {0, 1, 2, 3, 4} withthebinary operation ∗ whichisgiveninTable 7

and ξF :[0, 1] →P (X ),x →    

  {0} if t ∈ (0.84, 1], {0, 3} if t ∈ (0.76, 0.84], {0, 1, 2, 4} if t ∈ (0.58, 0.76], X if t ∈ [0, 0.58]

Then ξT (t),ξI (t) and ξF (t) areidealsof X forall t ∈ [0, 1]. Hencetheneutrosophicfallingshadow H :=(HT , HI , HF ) of ξ :=(ξT ,ξI ,ξF ) isafallingneutrosophicidealof X.Butit isnotacommutativefallingneutrosophicidealof X becauseif α ∈ [0, 0 27), β ∈ [0, 0 35) and γ ∈ (0 76, 0 84],then ξT (α)= {0, 3}, ξI (β)= {0, 3} and ξF (γ)= {0, 3} arenotcommutative idealsof X respectively.

Sinceeveryidealiscommutativeinacommutative BCK algebra,wehavethefollowingtheorem. Theorem4.9. Let (Ω, A,P ) beaprobabilityspaceandlet H :=(HT , HI , HF ) beaneutrosophicfallingshadowofaneutrosophicrandomset ξ :=(ξT ,ξI ,ξF ) onacommutative BCK algebra.If H :=(HT , HI , HF ) isafallingneutrosophicideal of X,thenitisacommutativefallingneutrosophicidealof X Corollary4.10. Let (Ω, A,P ) beaprobabilityspace.Forany BCK-algebra X whichsatisﬁesoneofthefollowingassertions

(∀x,y ∈ X )(x ≤ y ⇒ x ≤ y ∗ (y ∗ x)), (4.5) (∀x,y ∈ X )(x ≤ y ⇒ x = y ∗ (y ∗ x)), (4.6) (∀x,y ∈ X )(x ∗ (x ∗ y)= y ∗ (y ∗ (x ∗ (x ∗ y)))), (4.7) (∀x,y,z ∈ X )(x,y ≤ z,z ∗ y ≤ z ∗ x ⇒ x ≤ y), (4.8) (∀x,y,z ∈ X )(x ≤ z,z ∗ y ≤ z ∗ x ⇒ x ≤ y), (4.9)

let ˜ H :=( ˜ HT , ˜ HI , ˜ HF ) beaneutrosophicfallingshadowof aneutrosophicrandomset ξ :=(ξT ,ξI ,ξF ) on X.If ˜ H := (HT , HI , HF ) isafallingneutrosophicidealof X,thenitisa commutativefallingneutrosophicidealof X

References

[1]A.BorumandSaeidandY.B.Jun,Neutrosophicsubalgebrasof BCK/BCI-algebrasbasedonneutrosophicpoints, Ann.FuzzyMath.Inform. 14 (2017),no.1,87–97.

Then (X ; ∗, 0) isa BCK-algebra(see[9]).Consider (Ω, A,P )=([0, 1], A,m) andlet ξ :=(ξT ,ξI ,ξF ) beaneutrosophicrandomseton X whichisgivenasfollows: ξT :[0, 1] →P (X ),x →    {0, 3} if t ∈ [0, 0 27), {0, 1, 2, 3} if t ∈ [0 27, 0 66), {0, 1, 2, 4} if t ∈ [0 67, 1], ξI :[0, 1] →P (X ),x → {0, 3} if t ∈ [0, 0 35), {0, 1, 2, 4} if t ∈ [0 35, 1],

[2]I.R.Goodman,Fuzzysetsasequivalenceclassesofrandom sets,in“RecentDevelopmentsinFuzzySetsandPossibilityTheory”(R.Yager,Ed.),Pergamon,NewYork1982,pp. 327–343.

[3]K.Is´eki,On BCI-algebras,Math.SeminarNotes8(1980), 125–130.

[4]K.Is´ekiandS.Tanaka,Anintroductiontothetheoryof BCK-algebras,Math.Japon. 23 (1978),1–26.

[5]Y.Huang, BCI-algebra, SciencePress,Beijing,2006.

 

Table7:Cayleytableforthebinaryoperation“∗” ∗ 01234 000000 110010 221020 333303 444440



Florentin Smarandache (author and editor) Collected Papers, IX 388

[6]Y.B.Jun,Neutrosophicsubalgebrasofseveraltypes in BCK/BCI-algebras,Ann.FuzzyMath.Inform. 14 (2017),no.1,75–86.

[7]Y.B.JunandC.H.Park,Fallingshadowsappliedtosubalgebrasandidealsof BCK/BCI-algebras,HonamMath.J. 34 (2012),no.2,135–144.

[8]Y.B.Jun,F.SmarandacheandH.Bordbar,Neutrosophic fallingshadowsappliedtosubalgebrasandidealsin BCK/BCI-algebras,Ann.FuzzyMath.Inform.(submitted).

[9]J.MengandY.B.Jun, BCK-algebras, KyungmoonsaCo. Seoul,Korea1994.

[10]M.A. ¨ OzturkandY.B.Jun,Neutrosophicidealsin BCK/BCI-algebrasbasedonneutrosophicpoints,J.Inter.Math.VirtualInst. 8 (2018),1–17.

[11]F.Smarandache,Neutrosophy,NeutrosophicProbability,Set,andLogic,ProQuestInformation& Learning,AnnArbor,Michigan,USA,105p.,1998. http://fs.gallup.unm.edu/eBook-neutrosophics6.pdf(last editiononline).

[12]F.Smarandache,AUnifyingFieldinLogics:Neutrosophic Logic.Neutrosophy,NeutrosophicSet,NeutrosophicProbability,AmericanReserchPress,Rehoboth,NM,1999.

[13]F.Smarandache,Neutrosophicset-ageneralizationofthe intuitionisticfuzzyset,Int.J.PureAppl.Math. 24 (2005), no.3,287–297.

[14]S.K.Tan,P.Z.WangandE.S.Lee,Fuzzysetoperations basedonthetheoryoffallingshadows,J.Math.Anal.Appl. 174 (1993),242–255.

[15]S.K.Tan,P.Z.WangandX.Z.Zhang,Fuzzyinferencerelationbasedonthetheoryoffallingshadows,FuzzySetsand Systems 53 (1993),179–188.

[16]P.Z.WangandE.Sanchez,Treatingafuzzysubsetasaprojectablerandomset,in:“FuzzyInformationandDecision” (M.M.Gupta,E.Sanchez,Eds.),Pergamon,NewYork, 1982,pp.212–219.

[17]P.Z.Wang,FuzzySetsandFallingShadowsofRandom Sets,BeijingNormalUniv.Press,People’sRepublicof China,1985.[InChinese]

[18] Abdel-Basset, M., Mohamed, M., Smarandache, F., &

Chang, V. (2018). Neutrosophic Association Rule Mining Algorithm for Big Data Analysis. Symmetry, 10(4), 106.

[19] Abdel-Basset, M., & Mohamed, M. (2018). The Role of Single Valued Neutrosophic Sets and Rough Sets in Smart City: Imperfect and Incomplete Information Systems. Measurement. Volume 124, August 2018, Pages 47-55

[20] Abdel-Basset, M., Gunasekaran, M., Mohamed, M., &

Smarandache, F. A novel method for solving the fully neutrosophic linear programming problems. Neural Computing and Applications, 1-11.

[21] Abdel-Basset, M., Manogaran, G., Gamal, A., &

[22] Abdel-Basset, M., Mohamed, M., & Chang, V. (2018). NMCDA: A framework for evaluating cloudcomputing services. Future Generation Computer Systems, 86, 12-29.

[23] Abdel-Basset, M., Mohamed, M., Zhou, Y., & Hezam, I.

Smarandache, F. (2018). A hybrid approach of neutrosophic setsandDEMATELmethodfordevelopingsupplierselection criteria. Design Automation for Embedded Systems, 1-22. (2017). Multi-criteria group decision making based on neutrosophicanalytichierarchyprocess. Journal of Intelligent & Fuzzy Systems, 33(6), 4055-4066.

[24] Abdel-Basset, M.; Mohamed, M.; Smarandache, F. An Extension of Neutrosophic AHP–SWOT Analysis for StrategicPlanningandDecision-Making. Symmetry 2018, 10, 116.

Collected Papers, IX 389

Florentin

Smarandache (author and editor)

On Neutrosophic Crisp Topology via N-Topology

Riad K. Al-Hamido, T. Gharibah, S. Jafari, F.

Smarandache

Riad K. Al-Hamido, T. Gharibah, S. Jafari, Florentin Smarandache (2018). On Neutrosophic Crisp Topology via N-Topology. Neutrosophic Sets and Systems 23, 96-109

Abstract. In this paper, we extend the neutrosophic crisp topological spaces into N–neutrosophic crisp topological spaces (Nnc-topological space). Moreover, we introduced new types of open and closed sets in N–neutrosophic crisp topological spaces. We also present Nnc semi (open) closed sets, Nnc-preopen (closed) sets and Nnc-α-open (closed) sets and investigate their basic properties.

Keywords: Nnc-topology, N–neutrosophic crisp topological spaces, Nnc-semi (open) closed sets, Nnc-preopen (closed) sets, Nnc α-open (closed) sets, Nncint(A), Nnccl(A).

Introduction

The concept of non-rigid (fuzzy) sets introduced in 1965 by L. A. Zadeh [11] which revolutionized the field of logic and set theory. Since the need for supplementing the classical twovalued logic with respect to notions with rigid extension engendered the concept of fuzzy set. Soon after its advent, this notion has been utilized in different fields of research such as, decision-making problems, modelling of mental processes, that is, establishing a theory of fuzzy algorithms, control theory, fuzzy graphs, fuzzy automatic machine etc., and in general topology. Three years after the presence of the concept of fuzzy set, Chang [3] introduced and developed the theory of fuzzy topological spaces. Many researchers focused on this theory and

Florentin Smarandache (author and editor) Collected Papers, IX 390

they developed it further in different directions. Then another new notion called intuitionistic fuzzy set was established by Atanassov [2] in 1983. Coker [4] introduced the notion of intuitionistic fuzzy topological space F. Smarandache introduced the concepts of neutrosophy and neutrosophic set ([7], [8]). A. A. Salama and S. A. Alblowi [5] introduced the notions of neutrosophic crisp set and neutrosophic crisp topological space. In 2014, A.A. Salama, F. Smarandache and V. Kroumov [6] presented the concept of neutrosophic crisp topological space ( ). W. Al Omeri [1] also investigated neutrosophic crisp sets in the context of neutrosophic crisp topological Spaces. The geometric existence of N topology was given by M. Lellis Thivagar et al. [10], which is a nonempty set equipped with N arbitrary topologies. The notion of Nn-open (closed) sets and N-neutrosophic topological spaces are introduced by M. Lellis Thivagar, S. Jafari,V. Antonysamy and V. Sutha Devi [9]

In this paper, we explore the possibility of expanding the concept of neutrosophic crisp topological spaces into N neutrosophic crisp topological spaces (Nnc topological space). Further, we develop the concept of open (closed) sets, semiopen (semiclosed) sets, preopen (preclosed) sets and α-open (α-closed) sets in the context of N-neutrosophic crisp topological spaces and investigate some of their basic properties.

1.Preliminaries

In this section, we discuss some basic deﬁnitions and properties of N topological spaces and neutrosophic crisp topological spaces which are useful in sequel.

Definition 1.1. [6] Let X be a non empty fixed set. A neutrosophic crisp set (NCS) A is an object having the form 123{, , }, AAAA = where 123 , and AAA are subsets of X satisfying 12 13 23 , and . AAAAAA    = = =

Definition 1.2. [6] Types of NCSs and NN X  in X are as follows:

Florentin Smarandache (author and editor) Collected Papers, IX 391

1.,,or 2.,,or 3.,,or

N N N

X XX X

  

= = = () 4.,,. N  = 2. N X may be defined in many ways as an NCS, asfollows: () () ()

1.,,or 2.,,or 3.,,.

XX XXX XXXX

N N N

  = = = Definition 1.3. [6] Let X be a nonempty set, and the NCSs A and B be in the form Then we may consider two possible definitions for subset AB  which may be defined in two ways: 112233 112233

1.,. 2., ABABABandBA ABABBAandBA   Definition 1.4. [6] Let X be a non empty set and the NCSs A and B in the form 123 123 {, , }, {, , }. AAAABBBB == Then: 1. AB may be defined in two ways as an NCS as follows: 112233 112233

=

)(,,) )(,,). iABABABAB iiABABABAB

)(,,) )(,,). iABABABAB iiABABABAB = Florentin Smarandache (author and editor) Collected Papers, IX 392

= =

1. N may be defined in many ways as an NCS as follows: () () ()

2. AB may be defined in two ways as an NCS, as follows: 112233 112233

Definition 1.5. [6] A neutrosophic crisp topology (NCT) on a non empty set X is a family  ofneutrosophiccrispsubsetsin X satisfying the followingaxioms:

The pair (X, ) is said to be a neutrosophic crisp topological space (NCTS) in X. Moreover, the elements in Γ are said to be neutrosophic crisp open sets (NCOS). A neutrosophic crisp set F is closed (NCCS) if and only if its complement cF is an open neutrosophic crisp set.

Definition 1.6. [6] Let X be a non empty set, and the NCSs A be in the form

{, , } AAAA = . Then cA may be defined in three ways as an NCS:

2.N

Topological Spaces

In this section, we introduce N neutrosophic crisp topological spaces (Nnc topological space) and discuss their basic properties. Moreover, we introduced new types of open and closed sets in the context of Nnc topological spaces. Definition 2.1: Let X be a non empty set. Then ncτ1, ncτ2, ..., ncτN are N arbitrary crisp topologies deﬁned on X and the collection

12 12 1.,. 2., for any and. 3.,{:}. NN jj X AA AA AAjJ     

c cc

= = =

123

123 321 321 ),,or ),,or ),,. cccc

iAAAA iiAAAA iiiAAAA

{:()(),,} N N nc i inciinci i i NGXGABNAB    = = == 

1.,. NNnc XN   1 1 2. {}. inc iinc i GNforallGN   = =  Florentin Smarandache (author and editor) Collected Papers, IX 393

is called Nnc topology on X if the following axioms are satisﬁed:

Then (X, Nncτ) is called Nnc topological space on X. The elements of Nncτ are known as Nnc open (Nnc OS) sets on X and its complement is called Nnc closed (Nnc CS) sets on X. The elements of X are known asNnc sets (Nnc S) on X.

Remark 2.2: Considering N = 2 in Deﬁnition 2.1, we get the required deﬁnition of bi neutrosophic crisp topology on X. The pair (X, 2nc) is called a bi neutrosophic crisp topological space on X.

Remark 2.3: Considering N = 3 in Deﬁnition 2.1, we get the required deﬁnition of tri neutrosophic crisp topology on X. The pair (X, 3nc) is called a tri neutrosophic crisp topological space on X.

Example 2.4: {1,2,3,4}, X = 1 2 3 {,,A},{,,B},{,} nc nc n N c NN NN N X X X     = = = A{3},{2,4},{1},{1},{2},{2,3}, B = =  AB{1,3},{2,4},,,{2},{1,2,3}, AB == Then we get ,,,,, { } 3 nc NNXABABAB  =  which is a tri neutrosophic crisp topology on X The pair (X, 3ncτ) is called a tri neutrosophic crisp topological space on X

Example 2.5: {1,2,3,4}, X = 12{,,A},{,,B} nc NN N nc N XX    == A{3},{2,4},{1},{1},{2},{2,3}, B = =  AB{1,3},{2,4},,,{2},{1,2,3}, AB == Then ,,,,, { } 2 nc NNXABABAB  = 

1 1 3. {}. n n inc iinc i GNforallGN = = 

Florentin Smarandache (author and editor) Collected Papers, IX 394

which is a bi neutrosophic crisp topology on X. The pair (X, 2ncτ) is called a bi neutrosophic crisp topological space on X.

Definition 2.6:Let(X,Nncτ)bea Nnc topologicalspaceonX and A bean Nnc set on X then the Nncint(A) and Nnccl(A) are respectively deﬁned as

(i) Nncint(A) = ∪ {G : G ⊆ A and G is a Nnc open set in X }.

(ii) Nnccl(A) = ∩ {F: A ⊆ F and F is a Nnc closed set in X }.

Proposition 2.7: Let (X, Nncτ) be any Nnc topological space. If A and B are any two Nnc sets in (X, Nncτ), so the Nnc closure operator satisfies the following properties:

(i) A ⊆ Nnccl(A).

(ii) A ⊆ B ⇒ Nnccl(A) ⊆Nnccl(B).

(iii) Nnccl(A ∪ B) = Nnccl(A)∪ Nnccl(B).

Proof

(i) Nnccl(A)=∩ {G : G is a Nnc closed set in X and A ⊆ G }. Thus, A ⊆ Nnccl(A).

(ii) Nnccl(B)=∩ {G : G is a Nnc closed set in X and B ⊆ G }⊇∩{ G : G is a Nnc closed set in X and A ⊆ G } ⊇ Nnccl(A) Thus, Nnccl(A) ⊆Nnccl(B).

(iii) Nnccl(A ∪ B) = ∩{G : G is a Nk closed set in X and A ∪ B ⊆ G} = (∩{G : G isa Nnc closed set in X and A ⊆ G})∪(∩{G : G is a Nnc closedsetin X and B ⊆ G})= Nnccl(A)∪ Nnccl(B). Thus, Nnccl(A ∪ B) = Nnccl(A)∪ Nnccl(B).

Florentin Smarandache (author and editor) Collected Papers, IX 395

Proposition 2.8: Let (X, Nncτ) be any Nnc topological space. If A and B are any two Nnc sets in (X, Nncτ ), then the Nncint(A) operator satisfies the following properties:

(i) Nncint(A) ⊆ A

(ii) A ⊆ B ⇒ Nncint(A) ⊆Nncint(B).

(iii) Nncint(A ∩ B) = Nncint(A)∩ Nncint(B).

(iv) (Nnccl(A))c = Nncint(A)c .

(v) (Nncint(A))c = Nnccl(A)c

Proof

(i) Nncint(A)=∪ {G: G is an Nnc open set in X and G ⊆ A }. Thus, Nncint(A)⊆ A.

(ii) Nncint(B)=∪ {G: G is a Nnc open set in X and G ⊆ B }⊇∪{ G : G is an Nnc open set in X and G ⊆ A } ⊇ Nncint(A). Thus, Nncint(A)⊆ Nncint(B).

(iii) Nncint(A ∩ B) = ∪{G : G is an Nnc open set in X and A ∩ B ⊇ G} = (∪{G : G is a Nnc open set in X and A ⊇ G})∩(∪{G : G is an Nnc-open set in X and B ⊇ G})= Nncint(A) ∩ Nncint(B). Thus, Nncint(A ∩ B) = Nncint(A) ∩ Nncint(B).

(iv) Nnccl(A)=∩ {G: G is an Nnc closed set in X and A ⊆ G}, (Nnccl(A))c =∪{Gc : Gc is an Nnc open set in X and Ac ⊇ Gc} = Nncint(A)c. Thus, (Nnccl(A))c = Nncint(A)c .

(v) Nncint(A)=∪ {G: G is an Nnc open set in X and A ⊇ G}, (Nncint(A))c =∩{Gc : Gc is a Nnc-closed set in X and Ac ⊇ Gc} = Nnccl(A)c. Thus, (Nncint(A))c = Nnccl(A)c

Florentin Smarandache (author and editor) Collected Papers, IX 396

Proposition 2.9:

Let (X, Nncτ ) be any Nnc topological space. If A is a Nnc sets in (X, Nncτ), the following properties are true:

(i) Nnccl(A) = A iff A is a Nnc closed set.

(ii)Nncint(A) = A iff A is a Nnc-open set. (iii) Nnccl(A) is the smallest Nnc closed set containing A. (iv)Nncint(A) is the largest Nnc open set contained in A.

Proof: (i), (ii), (iii) and (iv) are obvious.

3.New open setes in Nnc Topological Spaces

Definition 3.1: Let (X, Nncτ) be any Nnc topological space. Let A be an Nnc set in (X, Nncτ). Then A is said to be:

(i)A Nnc preopen set (Nnc P OS) if A⊆ Nncint(Nnccl(A)). The complement of an Nnc preopen set is called an Nnc-preopen set in X. The family of all Nnc-P-OS (resp. NncP CS) of X is denoted by (NncPOS(X)) (resp. NncPCS ). (ii)An Nnc semiopen set (Nnc S OS) if A⊆ Nnccl(Nncint(A)). The complement of a Nnc semiopen set is called a Nnc semiopen set in X. The family of all Nnc S OS (resp. Nnc S CS) of X is denoted by (NncPOS(X)) (resp. NncPCS ). (iii) A Nnc  open set (Nnc  OS) if A⊆ Nncint (Nnccl(Nncint(A))). The complement of a Nnc  open set is called a Nnc  open set in X. The family of all Nnc  OS (resp. Nnc  CS) of X is denoted by (NncOS(X)) (resp. NncCS ).

Example 3.2: {,,,},Xabcd = 12{,,A},{,,B} nc NN N nc N XX    ==

Florentin Smarandache (author and editor) Collected Papers, IX 397

A{},{},{},{},{,},{}, abcBabdc==  then we have ,2 ,,} { NN nc XAB = which is a bi neutrosophic crisp topology on X. Then the pair (X, 2ncτ) is a bi neutrosophic crisp topological space on X. If H{,},{},{} abcd =  ,then H is a Nnc P OS but not Nnc 

OS It is clear that Hc is a Nnc P CS A is a Nnc S OS. It is clear that Ac is a Nnc S CS A is a Nnc  OS It is clear that Ac is a Nnc  CS

Definition 3.3: Let (X, Nncτ ) be a Nnc topological space on X and A be a Nnc set on X then

(i) Nnc P int(A) = ∪ {G: G ⊆ A and G is a Nnc P OS in X}.

(ii) Nnc P cl(A) = ∩ {F: A ⊆ F and F is a Nnc P CS in X}.

(iii) Nnc S int(A) = ∪ {G: G ⊆ A and G is a Nnc S OS in X}.

(iv) Nnc S cl(A) = ∩ {F: A ⊆ F and F is a Nnc S CS in X}.

(v) Nnc  int(A) = ∪ {G: G ⊆ A and G is a Nnc  OS in X}

(vi) Nnc  cl(A) = ∩ {F: A ⊆ F and F is a Nnc  CS in X}.

In Proposition 3.4 and Proposition 3.5, by the notion Nnc k cl(A)( Nnc k int(A)), we mean Nnc P cl(A)( Nnc P int(A)) (if k = p ), Nnc S cl(A)( Nnc S int(A)) (if k = S) and Nnc  cl(A)( Nnc  int(A)) ( if k = ).

Proposition 3.4: Let (X, Nncτ) be any Nnc topological space. If A and B are any two Nnc sets in (X, Nncτ), then the Nnc S closure operator satisfies the following properties:

(i) A ⊆ Nnc k cl(A).

(ii) Nnc k int(A) ⊆ A.

(iii) A ⊆ B ⇒ Nnc k cl(A)⊆ Nnc k cl(B)

Florentin Smarandache (author and editor) Collected Papers, IX 398

(iv) A ⊆ B ⇒ Nnc k int(A) ⊆ Nnc k int(B).

(v) Nnc k cl (A ∪ B) = Nnc k cl(A)∪Nnc k cl(B)

(vi) Nnc k int (A ∩ B) = Nnc k int(A) ∩ Nnc k int(B).

(vii) (Nnc k cl(A))c = Nnc k cl(A)c .

(viii) (Nnc k int(A))c = Nnc k int(A)c

Proposition 3.5:

Let (X, Nncτ) be any Nnc topological space. If A is an Nnc sets in (X, Nncτ). Then the following properties are true:

(i) Nnc k cl(A)= A iff A is a Nnc k closed set.

(ii) Nnc k int(A)= A iff A is a Nnc k open set.

(iii) Nnc k cl(A) is the smallest Nnc k closed set containing A. (iv) Nnc k int(A) is the largest Nnc k open set contained in A.

Proof: (i), (ii), (iii) and (iv) are obvious.

Proposition 3.6:

Let (X, Nncτ) be a Nnc-topological space on X Then the following statements hold in whcih the equality of each statement are not true:

(i)Every Nnc OS (resp. Nnc CS) is a Nnc  OS (resp. Nnc  CS).

(ii)Every Nnc  OS (resp. Nnc  CS) is a Nnc S OS (resp. Nnc S CS).

(iii) Every Nnc -OS (resp. Nnc -CS) is a Nnc P-OS (resp. Nnc P-CS).

Proposition 3.7:

Let (X, Nncτ) be a Nnc topological space on X, then the following statements hold, and the equality of each statement are not true:

(i)Every Nnc OS (resp. Nnc CS) is a Nnc S OS (resp. Nnc S CS).

(ii)Every Nnc-OS (resp. Nnc-CS) is a Nnc P-OS (resp. Nnc P-CS).

Florentin Smarandache (author and editor) Collected Papers, IX 399

Proof.

(i)Suppose that A is a Nnc OS. Then A=Nncint(A), and so A⊆Nnccl(A)= Nnccl(Nncint(A)). so that A is a Nnc S OS.

(ii)Suppose that A is a Nnc OS. Then A=Nncint(A), and since A⊆Nnccl(A) so A=Nncint(A) ⊆Nncint(Nnccl(A)). so that A is a Nnc P OS.

Proposition 3.8: Let (X, Nncτ) be a Nnc topological space on X and A a Nnc set on X. Then A is an Nnc  OS (resp. Nnc  CS) iff A is a Nnc S OS (resp. Nnc S CS) and Nnc P OS (resp. Nnc P CS).

Proof. The necessity condition follows from the Definition 3.1. Suppose that A is both a Nnc S OS and a Nnc P OS. Then A⊆ Nnccl(Nncint(A)) , and hence Nnccl(A)⊆ Nnccl(Nnccl(Nncint(A)))= Nnccl(Nncint(A)). It follows that A⊆ Nncint(Nnccl(A))⊆ Nncint(Nnccl(Nncint(A))) , so that A is a Nnc  OS.

Proposition 3.9: Let (X, Nncτ) be an Nnc topological space on X and A an Nnc set on X. Then A is an Nnc  CS iff A is an Nnc S CS and Nnc P CS.

Proof. The proof is straightforward.

Theorem 3.10: Let (X, Nncτ) be a Nnc topological space on X and A a Nnc set on X. If B is a Nnc S OS such that B ⊆A⊆ Nncint(Nnccl (A)), then A is a Nnc  OS .

Proof. Since B is a Nnc S OS, we have B ⊆ Nncint(Nnccl (A)) . Thus, A ⊆ Nncint(Nnccl (B))⊆ Nncint(Nnccl (Nnccl (Nncint( (B))))⊆ Nncint(Nnccl (Nncint( (B))))

Florentin Smarandache (author and editor) Collected Papers, IX 400

⊆ Nncint(Nnccl (Nncint( (A)))) and therefore A is a Nnc  OS.

Theorem 3.11:

Let (X, Nncτ ) be an Nnc topological space on X and A be an Nnc set on X. Then ANncOS(X)) iff there exists an Nnc OS H such that H⊆A⊆ Nncint (Nnccl (A)).

Proposition 3.12:

The union of any family of NncOS(X) is a NncOS(X)

Proof. The proof is straightforward.

Remark 3.13:

The following diagram shows the relations among the different types of weakly neutrosophic crisp open sets that were studied in this paper:

Conclusion

Nnc OS Nnc  OS

Nnc P OS

Nnc P OS Nnc S OS + Nnc S OS

In this work, we have introduced some new notions of N neutrosophic crisp open (closed) sets called Nnc semi (open) closed sets, Nnc preopen (closed) sets, and Nnc α open

Diagram (3.1) Florentin Smarandache (author and editor) Collected Papers, IX 401

(closed) sets and studied some of their basic properties in the context of neutrosophic crisp topological spaces. The neutrosophic crisp semi closed sets can be used to derive a new decomposition of neutrosophic crisp continuity.

References

[1] W. Al-Omeri, Neutrosophic crisp Sets via Neutrosophic crisp Topological Spaces NCTS, Neutrosophic Sets and Systems, Vol.13, 2016, pp.96-104

[2] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20(1986), 87-96.

[3] C. Chang., Fuzzy topological spaces, J. Math. Anal. Appl. 24(1968), 182-190.

[4] D. Coker., An introduction to intuitionistic fuzzy topological spaces, Fuzzy sets and systems, 88(1997),81-89.

[5] A. A. Salama and S. A. Alblowi, Neutrosophic Set and Neutrosophic Topological Spaces, IOSR Journal of Mathematics, Vol.3, Issue 4 (Sep-Oct. 2012), pp.31-35.

[6] A. A. Salama, F. Smarandache and V. Kroumov, Neutrosophic crisp Sets and Neutro sophic crisp Topological Spaces, Neutrosophic Sets and Systems, Vol.2, 2014, pp. 25-30

[7] F. Smarandache, A Unifying Field in Logics: Neutrosophic Logic. Neutrosophy, Neutro-sophic Set, Neutrosophic Probability. American Research Press, Rehoboth, NM, (1999). [8] F. Smarandache, Neutrosophy and Neutrosophic Logic, First International Conference on Neutrosophy, Neutrosophic Logic, Set, Probability, and Statistics, University of New Mexico, Gallup, NM 87301, USA (2002).

[9] M. L. Thivagar, S. Jafari, V. Antonysamy and V.Sutha Devi, The Ingenuity of Neutro-sophic Topology via N-Topology, Neutrosophic Sets and Systems, Vlo.19, 2018, pp. 91-100.

[10] M. Lellis Thivagar, V. Ramesh, M. D. Arockia, On new structure of N -topology, Co-gent Mathematics (Taylor and Francis),3, 2016:1204104.

[11] L. A. Zadeh., Fuzzy sets, Information and control, 8(1965),338-353.

Florentin Smarandache (author and editor) Collected Papers, IX 402

Neutrosophic Rare α-Continuity

R. Dhavaseelan, S. Jafari, R. M. Latif, F. Smarandache

R. Dhavaseelan, S. Jafari, R. M. Latif, Florentin Smarandache (2018). Neutrosophic Rare α-Continuity. New Trends in Neutrosophic Theory and Applications II, 336-344

ABSTRACT

In this paper, we introduce the concepts of neutrosophic rare α-continuous, neutrosophic rarely continuous, neutrosophic rarely pre-continuous, neutrosophic rarely semi-continuous are introduced and studied in light of the concept of rare set in neutrosophic setting.

KEYWORDS: Neutrosophic rare set; neutrosophic rarely α-continuous; neutrosophic rarely pre-continuous; neutrosophic almost α-continuous; neutrosophic weekly α-continuous; neutrosophic rarely semi-continuous.

1 INTRODUCTION AND PRELIMINARIES

The study of fuzzy sets was initiated by Zadeh (1965). Thereafter the paper of Chang (1968) paved the way for the subsequent tremendous growth of the numerous fuzzy topological concepts. Currently Fuzzy Topology has been observed to be very beneﬁcial in ﬁxing many realistic problems. Several mathematicians have tried almost all the pivotal concepts of General Topology for extension to the fuzzy settings. In 1981, Azad gave fuzzy version of the concepts given by Levine 1961; 1963 and thus initiated the study of weak forms of several notions in fuzzy topological spaces. Popa (1979) introduced the notion of rare continuity as a generalization of weak continuity (Levine, 1961) which has been further investigated by Long and Herrington (1982) and Jafari (1995; 1997). Noiri (1987) introduced and

Florentin Smarandache (author and editor) Collected Papers, IX 403

investigatedweakly α-continuityasageneralizationofweakcontinuity.Healsointroduced andinvestigatedalmost α-continuity(Noiri,1988).TheconceptsofRarely α-continuity wasintroducedbyJafari(2005).Theconceptsoffuzzyrare α-continuityandintuitionistic fuzzyrare α-continuitywereintroducedbyDhavaseelanandJafari(n.d.-b,n.d.-c).Afterthe adventoftheconceptsofneutrosophyandneutrosophicsetintroducedbySmarandachethe (1999;2002),theconceptsofneutrosophiccrispsetandneutrosophiccrisptopologicalspaces wereintroducedbySalamaandAlblowi(2012).

Thepurposeofthepresentpaperistointroduceandstudytheconceptsofneutrosophic rare α-continuousfunctions,neutrosophicrarelycontinuousfunctions,neutrosophicrarely pre-continuousfunctionsandneutrosophicrarelysemi-continuousfunctionsinlightofthe conceptofraresetinaneutrosophicsetting.

Definition1.1. LetXbeanonemptyfixedset.Aneutrosophicset[brieflyNS] A isanobject havingtheform A = { x,µA (x),σA (x),γA (x) : x ∈ X},where µA (x),σA (x)and γA (x)which representsthedegreeofmembershipfunction(µA (x)),thedegreeofindeterminacy(namely σA (x))andthedegreeofnonmembership(γA (x)),respectively,ofeachelement x ∈ X tothe set A Remark1.1. (1)Aneutrosophicset A = { x,µA (x),σA (x),γA (x) : x ∈ X} canbe identifiedtoanorderedtriple µA ,σA ,γA in]0 , 1+[on X. (2)Forthesakeofsimplicity,weshallusethesymbol A = µA ,σA ,γA fortheneutrosophic set A = { x,µA (x),σA (x),γA (x) : x ∈ X}.

Definition1.2. Let X beanonemptysetandtheneutrosophicsets A and B intheform A = { x,µA (x),σA (x),γA (x) : x ∈ X}, B = { x,µB (x),σB (x),γB (x) : x ∈ X}.Then (a) A ⊆ B iff µA (x) ≤ µB (x), σA (x) ≤ σB (x)and γA (x) ≥ γB (x)forall x ∈ X; (b) A = B iff A ⊆ B and B ⊆ A; (c) A = { x,γA (x),σA (x),µA (x) : x ∈ X};[complementofA] (d) A ∩ B = { x,µA (x) ∧ µB (x),σA (x) ∧ σB (x),γA (x) ∨ γB (

x) : x ∈ X}; (e) A ∪ B = { x,µA (x) ∨ µB (x),σA (x) ∨ σB (x),γA (x) ∧ γB (x) : x ∈ X}; (f)[]A = { x,µA (x),σA (x), 1 µA (x) : x ∈ X}; (g) A = { x, 1 γA (x),σA (x),γA (x) : x ∈ X}

Let {Ai : i ∈ J } beanarbitraryfamilyofneutrosophicsetsin

(a) Ai = { x, ∧µAi (x), ∧σAi (x), ∨γAi (x) : x ∈ X}; (b) Ai = { x, ∨µAi (x), ∨σAi (x), ∧γAi (x) : x ∈ X} Florentin Smarandache (author and editor) Collected Papers, IX 404

Deﬁnition1.3.

X.Then

Sinceourmainpurposeistoconstructthetoolsfordevelopingneutrosophictopological spaces,wemustintroducetheneutrosophicsets0N and1N in X asfollows:

Deﬁnition1.4. 0N = { x, 0, 0, 1 : x ∈ X} and1N = { x, 1, 1, 0 : x ∈ X}

Deﬁnition1.5. (Dhavaseelan&Jafari,n.d.-a)Aneutrosophictopology(brieﬂyNT)ona nonemptyset X isafamily T ofneutrosophicsetsin X satisfyingthefollowingaxioms: (i)0N , 1N ∈ T , (ii) G1 ∩ G2 ∈ T forany G1,G2 ∈ T , (iii) ∪Gi ∈ T forarbitraryfamily {Gi | i ∈ Λ}⊆ T

Inthiscasetheorderedpair(X,T )orsimply X iscalledaneutrosophictopologicalspace (brieflyNTS)andeachneutrosophicsetin T iscalledaneutrosophicopenset(brieflyNOS). Thecomplement A ofaNOS A in X iscalledaneutrosophicclosedset(brieflyNCS)in X. Definition1.6. (Dhavaseelan&Jafari,n.d.-a)Let A beaneutrosophicsetinaneutrosophictopologicalspace X.Then

Nint(A)= {G | G isaneutrosophicopensetin X and G ⊆ A} iscalledtheneutrosophicinteriorof A;

Ncl(A)= {G | G isaneutrosophicclosedsetin X and G ⊇ A} iscalledtheneutrosophicclosureof A Deﬁnition1.7. (Dhavaseelan&Jafari,n.d.-a)Let X beanonemptyset.If r,t,s be realstandardornonstandardsubsetsof]0 , 1+[,thentheneutrosophicset xr,t,s iscalleda neutrosophicpoint(brieﬂyNP)in X givenby xr,t,s(xp)=    (r,t,s), if x = xp (0, 0, 1), if x = xp for xp ∈ X iscalledthesupportof xr,t,s,where r denotesthedegreeofmembershipvalue, t thedegreeofindeterminacyand s thedegreeofnon-membershipvalueof xr,t,s.

Deﬁnition1.8. (Dhavaseelan&Jafari,n.d.-b)Anintuitionisticfuzzyset R iscalledintuitionisticfuzzyraresetif IFint(R)=0∼

Deﬁnition1.9. (Dhavaseelan&Jafari,n.d.-b)Anintuitionisticfuzzyset R iscalledintuitionisticfuzzynowheredensesetif IFint(IFcl(R))=0∼

2MAINRESULTS

Deﬁnition2.1. Aneutrosophicset A inaneutrosophictopologicalspace(X,T )iscalled

Florentin Smarandache (author and editor) Collected Papers, IX 405

1)aneutrosophicsemiopenset(brieﬂyNSOS)if A ⊆ Ncl(Nint(A)).

2)aneutrosophic α openset(brieﬂy NαOS)if A ⊆ Nint(Ncl(Nint(A))).

3)aneutrosophicpreopenset(brieﬂyNPOS)if A ⊆ Nint(Ncl(A)).

4)aneutrosophicregularopenset(brieﬂyNROS)if A = Nint(Ncl(A)).

5)aneutrosophicsemipreopenor β openset(brieﬂy NβOS)if A ⊆ Ncl(Nint(Ncl(A))). AneutrosophicsetAiscalledaneutrosophicsemiclosedset,neutrosophic α-closedset,neutrosophicpreclosedset,neutrosophicregularclosedsetandneutrosophic β-closedset(brieﬂy NSCS,NαCS,NPCS,NRCSandNβCS,resp.),ifthecomplementof A isaneutrosophic semiopenset,neutrosophic α-openset,neutrosophicpreopenset,neutrosophicregularopen set,andneutrosophic β-openset,respectively.

Definition2.2. LetaneutrosophicsetAofaneutrosophictopologicalspace(X,T ).Then neutrosophic α-closureof A (briefly Nclα(A))isdefinedas Nclα(A)= {K| Kisaneutrosophic α closedsetin X and A ⊆ K}.

Definition2.3. (Jun&Song,2005)Letaneutrosophicset A ofaneutrosophictopological space(X,T ).Thenneutrosophic α interiorof A (briefly Nintα(A))isdefinedas Nintα(A)= {K| K isaneutrosophic α opensetin X and K ⊆ A}

Deﬁnition2.4. Aneutrosophicset R iscalledneutrosophicraresetif Nint(R)=0N .

Deﬁnition2.5. Aneutrosophicset R iscalledneutrosophicnowheredensesetif Nint(Ncl(R))=0N

Deﬁnition2.6. Let(X,T )and(Y,S)betwoneutrosophictopologicalspaces.Afunction f :(X,T ) → (Y,S)iscalled

(i)neutrosophic α-continuousifforeachneutrosophicpoint xr,t,s in X andeachneutrosophicopenset G in Y containing f (xr,t,s),thereexistsaneutrosophic α openset U in X suchthat f (U ) ≤ G.

(ii)neutrosophicalmost α-continuousifforeachneutrosophicpoint xr,t,s in X andeach neutrosophicopenset G containing f (xr,t,s),thereexistsaneutrosophic α openset U suchthat f (U ) ≤ Nint(Ncl(G)).

(iii)neutrosophicweakly α-continuousifforeachneutrosophicpoint xr,t,s in X andeach neutrosophicopenset G containing f (xr,t,s),thereexistsaneutrosophic α openset U suchthat f (U ) ≤ Ncl(G).

Deﬁnition2.7. Let(X,T )and(Y,S)betwoneutrosophictopologicalspaces.Afunction f :(X,T ) → (Y,S)iscalled

Florentin Smarandache (author and editor) Collected Papers, IX 406

(i)neutrosophicrarely α-continuousifforeachneutrosophicpoint xr,t,s in X andeach neutrosophicopenset G in(Y,S)containing f (xr,t,s),thereexistaneutrosophicrare set R with G ∩ Ncl(R)=0N andneutrosophic α openset U in(X,T )suchthat f (U ) ≤ G ∪ R.

(ii)neutrosophicrarelycontinuousifforeachneutrosophicpoint xr,t,s in X andeach neutrosophicopenset G in(Y,S)containing f (xr,t,s),thereexistaneutrosophicrareset R with G∩Ncl(R)=0N andneutrosophicopenset U in(X,T )suchthat f (U ) ≤ G∪R.

(iii)neutrosophicrarelyprecontinuousifforeachneutrosophicpoint xr,t,s in X andeach neutrosophicopenset G in(Y,S)containing f (xr,t,s),thereexistaneutrosophicrare set R with G ∩ Ncl(R)=0N andneutrosophicpreopenset U in(X,T )suchthat f (U ) ≤ G ∪ R.

(iv)neutrosophicrarelysemi-continuousifforeachneutrosophicpoint xr,t,s in X andeach neutrosophicopenset G in(Y,S)containing f (xr,t,s),thereexistaneutrosophicrare setRwith G ∩ Ncl(R)=0N andneutrosophicsemiopenset U in(X,T )suchthat f (U ) ≤ G ∪ R.

Example2.1. Let X = {a,b,c}.Deﬁnetheneutrosophicsets A, B and C asfollows: A = x, ( a 0 , b 0 , c 1 ), ( a 0 , b 0 , c 1 ), ( a 1 , b 1 , c 0 ) , B = x, ( a 1 , b 0 , c 0 ), ( a 1 , b 0 , c 0 ), ( a 0 , b 1 , c 1 ) and C = x, ( a 0 , b 1 , c 0 ), ( a 0 , b 1 , c 0 ), ( a 1 , b 0 , c 1 ) .Then T = {0N , 1N ,C} and S = {0N , 1N ,A,B,A ∪ B} areneutrosophictopologiesonX.Let(X,T )and(X,S)beneutrosophictopologicalspaces. Deﬁne f :(X,T ) → (X,S)asaidentityfunction.Clearly f isneutrosophicrarely α continuous.

Proposition2.1. Let(X,T )and(Y,S)beanytwoneutrosophictopologicalspaces.Fora function f :(X,T ) → (Y,S)thefollowingstatementsareequivalents:

(i)Thefunction f isneutrosophicrarely α-continuousat xr,t,s in(X,T ).

(ii)Foreachneutrosophicopenset G containing f (xr,t,s),thereexistsaneutrosophic α openset U in(X,T )suchthat Nint(f (U ) ∩ G)=0N

(iii)Foreachneutrosophicopenset G containing f (xr,t,s),thereexistsaneutrosophic α openset U in(X,T )suchthat Nint(f (U )) ≤ Ncl(G).

(iv)Foreachneutrosophicopenset G in(Y,S)containing f (xr,t,s),thereexistsaneutrosophicrareset R with G ∩ Ncl(R)=0N suchthat xr,t,s ∈ Nintα(f 1(G ∪ R)).

(v)Foreachneutrosophicopenset G in(Y,S)containing f (xr,t,s),thereexistsaneutrosophicrareset R with Ncl(G) ∩ R =0N suchthat xr,t,s ∈ Nintα(f 1(Ncl(G) ∪ R))

(vi)Foreachneutrosophicregularopenset G in(Y,S)containing f (xr,t,s),thereexistsa neutrosophicrareset R with Ncl(G) ∩ R =0N suchthat xr,t,s ∈ Nintα(f 1(G ∪ R))

Florentin Smarandache (author and editor) Collected Papers, IX 407

Proof. (i) ⇒ (ii)Let G beaneutrosophicopensetin(Y,S)containing f (xr,t,s).By f (xr,t,s) ∈ G ≤ Nint(Ncl(G))and Nint(Ncl(G))containing f (xr,t,s),thereexistsaneutrosophicrareset R with Nint(Ncl(G)) ∩ Ncl(R)=0N andaneutrosophic α openset U in(X,T )containing xr,t,s suchthat f (U ) ≤ Nint(Ncl(G)) ∪ R.Wehave Nint(f (U ) ∩ G)Nint(G) ≤ Nint(Ncl(G) ∪ R) ∩ (Ncl(G)) ≤ Ncl(G) ∪ Nint(R) ∩ (Ncl(G))=0N (ii) ⇒ (iii)Obvious. (iii) ⇒ (i)Let G beaneutrosophicopensetin(Y,S)containing f (xr,t,s).Thenby(iii), thereexistsaneutrosophic α-openset U containing xr,t,s suchthat Nint(f (U ) ≤ Ncl(G). Wehave f (U )=(f (U ) ∩ (Nint(f (U )))) ∪ Nint(f (U )) < (f (U ) ∩ (Nint(f (U )))) ∪ Ncl(G)= (f (U )∩(Nint(f (U ))))∪G∪(Ncl(G)∩G)=(f (U )∩(Nint(f (U )))∩G)∪G∪(Ncl(G)∩G).Set R1 = f (U ) ∩ (Nint(f (U ))) ∩ G and R2 = Ncl(G) ∩ G.Then R1 and R2 areneutrosophicrare sets.More R = R1 ∪ R2 isaneutrosophicsetsuchthat Ncl(R) ∩ G =0N and f (U ) ≤ G ∪ R Thisshowthat f isneutrosophicrarely α-continuous.

(i) ⇒ (iv)Supposethat G beaneutrosophicopensetin(Y,S)containing f (xr,t,s).Then thereexistsaneutrosophicrareset R with G∩Ncl(R)=0N and U beaneutrosophic α-open setin(X,T )containing xr,t,s suchthat f (U ) ≤ G∪R.Itfollowsthat xr,t,s ∈ U ≤ f 1(G∪R). Thisimpliesthat xr,t,s ∈ Nintα(f 1(G ∪ R)).

(iv) ⇒ (v)Supposethat G beaneutrosophicopensetin(Y,S)containing f (xr,t,s).

Thenthereexistsaneutrosophicrareset R with G ∩ Ncl(R)=0N suchthat xr,t,s ∈ Nintα(f 1(G ∪ R)).Since G ∩ Ncl(R)=0N ,R ≤ G,where G = (Ncl(G)) ∪ (Ncl(G) ∩ G). Now,wehave R ≤ R ∪(Ncl(G))∪(Ncl(G)∩G).Now, R1 = R ∩(Ncl(G)).Itfollowsthat R1 isaneutrosophicraresetwith Ncl(G) ∩ R1 =0N .Therefore xr,t,s ∈ Nintα(f 1(G ∪ R)) ≤ Nintα(f 1(G ∪ R1)).

(v) ⇒ (vi)Assumethat G beaneutrosophicregularopensetin(Y,S)containing f (xr,t,s). Thenthereexistsaneutrosophicrareset R with Ncl(G) ∩ R =0N suchthat xr,t,s ∈ Nintα(f 1(Ncl(G) ∪ R)).Now R1 = R ∪ (Ncl(G) ∪ G).Itfollowsthat R1 isaneutrosophic raresetand(G ∩ Ncl(R1))=0N .Hence xr,t,s ∈ Nintα(f 1(Ncl(G) ∪ R))= Nintα(f 1(G ∪ (Ncl(G) ∩ G)) ∪ R)= Nintα(f 1(G ∪ R1)). Therefore xr,t,s ∈ Nintα(f 1(G ∪ R1)). (vi) ⇒ (ii)LetGbeaneutrosophicopensetin(Y,S)containing f (xr,t,s).By f (xr,t,s) ∈ G ≤ Nint(Ncl(G))andthefactthat Nint(Ncl(G))isaneutrosophicregularopenin(Y,S), thereexistsaneutrosophicrareset R and Nint(Ncl(G)) ∩ Ncl(R)=0N ,suchthat xr,t,s ∈ Nintα(f 1(Nint(Ncl(G)) ∪ R).Let U = Nintα(f 1(Nint(Ncl(G)) ∪ R) Hence U isa neutrosophic α-opensetin(X,T )containing xr,t,s andtherefore f (U ) ≤ Nint(Ncl(G)) ∪ R Hence,wehave Nint(f (U ) ∩ G)=0N . Proposition2.2. Let(X,T )and(Y,S)beanytwoneutrosophictopologicalspace.Then afunction f :(X,T ) → (Y,S)isaneutrosophicrarely α-continuousifandonlyif f 1(G) ≤ Nintα(f 1(G ∪ R))foreveryneutrosophicopenset G in(Y,S),where R isaneutrosophic raresetwith Ncl(R) ∩ G =0N .

Florentin Smarandache (author and editor) Collected Papers, IX 408

Proof. Supposethat G beaneutrosophicrarely α-opensetin(Y,S)containing f (xr,t,s). Then G ∩ Ncl(R)=0N and U beaneutrosophic α-opensetin(X,T )containing xr,t,s, suchthat f (U ) ≤ G ∪ R.Itfollowsthat xr,t,s ∈ U ≤ f 1(G ∪ R).Thisimpliesthat f 1(G) ≤ Nintα(f 1(G ∪ R)).

Deﬁnition2.8. Afunction f :(X,T ) → (Y,S)isneutrosophic Iα-continuousat xr,t,s in(X,T )ifforeachneutrosophicopenset G in(Y,S)containing f (xr,t,s),thereexistsa neutrosophic α-openset U containing xr,t,s,suchthat Nint(f (U )) ≤ G

If f hasthispropertyateachneutrosophicpoint xr,t,s in(X,T ),thenwesaythat f is neutrosophic Iα-continuouson(X,T ).

Example2.2. Let X = {a,b,c}.Deﬁnetheneutrosophicsets A and B asfollows: A = x, ( a 0 , b 1 , c 0 ), ( a 0 , b 1 , c 0 ), ( a 1 , b 0 , c 1 ) and B = x, ( a 1 , b 0 , c 0 ), ( a 1 , b 0 , c 0 ), ( a 0 , b 1 , c 1 ) .Then T = {0N , 1N ,A} and S = {0N , 1N ,B} areneutrosophictopologieson X.Let(X,T )and(X,S) beneutrosophictopologicalspaces.Let f :(X,T ) → (X,S)asdeﬁnedby f (a)= f (b)= b and f (c)= c isneutrosophic Iα-continuous.

Proposition2.3. Let(Y,S)beaneutrosophicregularspace.Thenthefunction f : (X,T ) → (Y,S)isneutrosophic Iα continuouson X ifandonlyif f isneutrosophicrarely α-continuouson X Proof. ⇒ Itisobvious. ⇐ Let f beneutrosophicrarely α-continuouson(X,T ).Supposethat f (xr,t,s) ∈ G,where G isaneutrosophicopensetin(Y,S)andaneutrosophicpoint xr,t,s in X.Bytheneutrosophicregularityof(Y,S),thereexistsaneutrosophicopenset G1 in(Y,S)suchthat G1 containing f (xr,t,s)and Ncl(G1) ≤ G.Since f isneutrosophicrarely α-continuous,then thereexistsaneutrosophic α opensetU,suchthat Nint(f (U )) ≤ Ncl(G1).Thisimplies that Nint(f (U )) ≤ G whichmeansthat f isneutrosophic Iα-continuouson X

Deﬁnition2.9. Afunction f :(X,T ) → (Y,S)iscalledneutrosophicpre-α-openiffor everyneutrosophic α-openset U in X suchthat f (U )isaneutrosophic α-openin Y

Proposition2.4. Ifafunction f :(X,T ) → (Y,S)isaneutrosophicpre-α-openand neutrosophicrarely α-continuousthen f isneutrosophicalmost α-continuous.

Proof. supposethataneutrosophicpoint xr,t,s in X andaneutrosophicopenset G in Y , containing f (xr,t,s).Since f isneutrosophicrarely α-continuousat xr,t,s,thenthereexistsa neutrosophic α-openset U in X suchthat Nint(f (U )) ⊂ Ncl(G).Since f isneutrosophic pre-α-open,wehave f (U )in Y .Thisimpliesthat f (U ) ⊂ Nint(Ncl(Nint(f (U )))) ⊂ Nint(Ncl(G)).Hence f isneutrosophicalmost α-continuous.

Forafunction f : X → Y ,thegraph g : X → X × Y offisdeﬁnedby g(x)=(x,f (x)), foreach x ∈ X

Florentin Smarandache (author and editor) Collected Papers, IX 409

Proposition2.5. Let f :(X,T ) → (Y,S)beanyfunction.Ifthe g : X → X × Y of f is neutrosophicrarely α-continuousthen f isalsoneutrosophicrarely α-continuous.

Proof. Supposethataneutrosophicpoint xr,t,s in X andaneutrosophicopenset W in Y ,containing g(xr,t,s).Itfollowsthatthereexistsneutrosophicopensets1X and V in X and Y respectively,suchthat(xr,t,s,f (xr,t,s)) ∈ 1X × V ⊂ W .Since f isneutrosophic rarely α-continuous,thereexistsaneutrosophic α-openset G suchthat Nint(f (G)) ⊂ Ncl(V ).Let E =1X ∩ G.Itfollowsthat E beaneutrosophic α-opensetin X andwehave Nint(g(E)) ⊂ Nint(1X × f (G)) ⊂ 1X × Ncl(V ) ⊂ Ncl(W ).Therefore g isneutrosophic rarely α-continuous.

REFERENCES

Azad,K.(1981).Onfuzzysemicontinuity,fuzzyalmostcontinuityandfuzzyweaklycontinuity. JournalofMathematicalAnalysisandApplications, 82 (1),14–32.

Chang,C.-L.(1968).Fuzzytopologicalspaces. JournalofmathematicalAnalysisand Applications, 24 (1),182–190.

Dhavaseelan,D.,&Jafari,S.(n.d.-a).Generalizedneutrosophiccontra-continuity. submitted .

Dhavaseelan,D.,&Jafari,S.(n.d.-b).Intuitionisticfuzzyrare α-continuity. submitted

Dhavaseelan,D.,&Jafari,S.(n.d.-c).Anoteonfuzzyrare α-continuity. submitted

Jafari,S.(1995).Anoteonrarelycontinuousfunctions,univ. Bacˆau.Stud.Cerc.St.Ser. Mat , 5 ,29–34.

Jafari,S.(1997).Onsomepropertiesofrarelycontinuousfunctions,univ. Bacˆau.Stud. Cerc.St.Ser.Mat , 7 ,65–73.

Jafari,S.(2005).Rare[α]-continuity. BulletinoftheMalaysianMathematicalSciences Society , 28 (2).

Jun,Y.B.,&Song,S.(2005).Intuitionisticfuzzysemi-preopensetsandintuitionistic fuzzysemi-precontinuousmappings. JournalofAppliedMathematicsandComputing , 19 (1/2),467.

Levine,N.(1961).Adecompositionofcontinuityintopologicalspaces. TheAmerican MathematicalMonthly , 68 (1),44–46.

Levine,N.(1963).Semi-opensetsandsemi-continuityintopologicalspaces. TheAmerican MathematicalMonthly , 70 (1),36–41.

Long,P.,&Herrington,L.(1982).Propertiesofrarelycontinuousfunctions. GlasnikMat , 17 (37),147–153.

Noiri,T.(1987).Weakly α-continuousfunctions. InternationalJournalofMathematicsand MathematicalSciences, 10 (3),483–490.

Noiri,T.(1988).Almost α-continuousfunctions. KyungpookMath.J., 28 (1),71–77.

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Popa,V.(1979).Surcertaindecompositiondelacontinuite’danslesespacestopologiques. GlasnikMat.SetrIII , 14 (34),359–362.

Salama,A.,&Alblowi,S.(2012).Neutrosophicsetandneutrosophictopologicalspaces. IOSRJournalofMathematics(IOSR-JM), 3 (4),31–35.

Smarandache,F.(1999). Aunifyingﬁeldinlogics.neutrosophy:Neutrosophicprobability, setandlogic.Rehoboth:AmericanResearchPress.

Smarandache,F.(2002).Neutrosophyandneutrosophiclogic.In Neutrosophyandneutrosophiclogic. UniversityofNewMexico,Gallup,NM87301,USA.

Zadeh,L.(1965).Fuzzysets. InformationandControl , 8 (3),338–353.

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Neutrosophic Semi-Continuous Multifunctions

R. Dhavaseelan, S. Jafari, N. Rajesh, F. Smarandache

R. Dhavaseelan, S. Jafari, N. Rajesh, Florentin Smarandache (2018). Neutrosophic SemiContinuous Multifunctions. New Trends in Neutrosophic Theory and Applications II, 345-354

ABSTRACT

In this paper we introduce the concepts of neutrosophic upper and neutrosophic lower semicontinuous multifunctions and study some of their basic properties.

KEYWORDS: Neutrosophic topological space, semi-continuous multifunctions.

1 INTRODUCTION

There is no doubt that the theory of multifunctions plays an important role in functional analysis and ﬁxed point theory. It also has a wide range of applications in economic theory, decision theory, non-cooperative games, artiﬁcial intelligence, medicine and information sci-ences. Inspired by the research works of Smarandache (1999; 2001; 2007), we introduce and study the notions of neutrosophic upper and neutrosophic lower semi-continuous mul-tifunctions in this paper. Further, we present some characterizations and properties of such notions.

2PRELIMINARIES

Throughoutthispaper,by(X,τ )orsimplyby X wewillmeanatopologicalspaceinthe classicalsense,and(Y,τ1)orsimply Y willstandforaneutrosophictopologicalspaceas deﬁnedbySalamaandAlblowi(2012).

Definition1. Smarandache(1999,2001,2007)Let X beanon-emptyfixedset.Aneutrosophicset A isanobjecthavingtheform A =<x,µA(x),σA(x),γA(x) >,where µA(x), σA(x) and γA(x) arerepresentthedegreeofmembershipfunction,thedegreeofindeterminacy,and the degreeofnon-membership,respectivelyofeachelement x ∈ X totheset A Definition2. (Salama&Alblowi,2012)Aneutrosophictopologyonanonemptyset X is afamily τ ofneutrosophicsubsetsof X whichsatisfiesthefollowingthreeconditions:

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1. 0, 1 ∈ τ ,

2.If g,h ∈ τ ,their g ∧ h ∈ τ ,

3.If fi ∈ τ foreach i ∈ I,then ∨i∈I fi ∈ τ

Thepair(X,τ )iscalledaneutrosophictopologicalspace.

Deﬁnition3. Membersof τ arecalledneutrosophicopensets,denotedby NO(X),andcomplementofneutrosophicopensetsarecalledneutrosophicclosedsets,wherethecomplement ofaneutrosophicset A,denotedby Ac,is 1 A.

Neutrosophicsetsin Y will bedenotedby λ,γ,δ,ρ,etc.,andalthoughsubsetsof X will bedenotedby A,B,U,V ,etc.Aneutrosophicpointin Y withsupport y ∈ Y andvalue α(0 <α ≤ 1)isdenotedby yα.Aneutrosophicset λ in Y issaidtobequasi-coincident (q-coincident)withaneutrosophicset µ,denotedby λqµ,ifandonlyifthereexists y ∈ Y suchthat λ(y)+ µ(y) > 1.Aneutrosophicset λ of Y iscalledaneutrosophicneighbourhood ofafuzypoint yα in Y ifthereexistsaneutrosophicopenset µ in Y suchthat yα ∈ µ ≤ λ. Theintersectionofallneutrosophicclosedsetsof Y containing λ iscalledtheneutrosophic closureof λ andisdenotedbyCl(λ).Theunionofallneutrosophicopensetscontained in λ iscalledthe neutrosophicinteriorof λ andisdenotedbyInt(λ).Thefamilyofall opensetsofatopologicalspace X isdenotedby O(X)and O(X,x)denotedthefamily {A ∈ O(X)|x ∈ A}, where x isapointof X

Deﬁnition4. Let (X,τ ) beatopologicalspaceintheclassicalsenseand (Y,τ1) beanneutrosophic topologicalspace. F :(X,τ ) → (Y,τ1) iscalledaneutrosophicmultifunctionifand onlyif foreach x ∈ X,F (x) isaneutrosophicsetin Y

Deﬁnition 5. For a neutrosophic multifunction F : (X, τ ) → (Y, τ1), the upper inverse F +(λ) and lower inverse F (λ) of a neutrosophic set λ in Y are deﬁned as follows: F +(λ) = {x ∈ X|F (x) ≤ λ} and F (λ) = {x ∈ X|F (x)qλ}

Lemma 1. For a neutrosophic multifunction F : (X, τ ) → (Y, τ1), we have F (1 λ) = X F +(λ), for any neutrosophic set λ in Y

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Deﬁnition5. Foraneutrosophicmultifunction F :(X,τ ) → (Y,τ1),theupperinverse F +(λ) andlowerinverse F (λ) ofaneutrosophicset λ in Y aredeﬁnedasfollows: F +(λ)= {x ∈ X|F (x) ≤ λ} and F (λ)= {x ∈ X|F (x)qλ}

Lemma1. Foraneutrosophicmultifunction F :(X,τ ) → (Y,τ1),wehave F (1 λ)= X F +(λ),foranyneutrosophicset λ in Y .

3NEUTROSOPHICSEMICONTINUOUSMULTI–FUNCTIONS

Deﬁnition6. Aneutrosophicmultifunction F :(X,τ ) → (Y,τ1) issaidtobe

1.neutrosophicuppersemicontinuousatapoint x ∈ X ifforeach λ ∈ NO(Y ) containing F (x) (therefore, F (x) ≤ λ),thereexists U ∈ O(X,x) suchthat F (U ) ≤ λ (therefore U ⊂ F +(λ)).

2.neutrosophiclowersemicontinuousatapoint x ∈ X ifforeach λ ∈ NO(Y ) with F (x)qλ,thereexists U ∈ O(X,x) suchthat U ⊆ F (λ)

3.neutrosophicuppersemicontinuous(neutrosophiclowersemicontinuous)ifitisneutrosophicuppersemicontinuous(neutrosophiclowersemicontinuous)ateachpoint x ∈ X.

Theorem1. Thefollowingassertionsareequivalentforaneutrosophicmultifunction F : (X,τ ) → (Y,τ1):

1. F isneutrosophicuppersemicontinuous;

2.Foreachpoint x of X andeachneutrosophicneighbourhood λ of F (x), F +(λ) isa neighbourhoodof x;

3.Foreachpoint x of X andeachneutrosophicneighbourhood λ of F (x),thereexistsa neighbourhood U of x suchthat F (U ) ≤ λ; 4.

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F +(λ) ∈ O(X) foroeach λ ∈ NO(Y );

F (δ) isaclosedsetin X foreachneutrosophicclosedset δ of Y ;

Cl(F (µ)) ⊆ F (Cl(µ)) foreachneutrosophicset µ of Y Proof. (1)⇒(2)Let x ∈ X and µ beaneutrosophicneighbourhoodof F (x).Thenthere exists λ ∈ NO(Y )suchthat F (x) ≤ λ ≤ µ,By(1),thereexists U ∈ O(X,x)suchthat F (U ) ≤ λ.Therefore x ∈ U ⊆ F +(µ)andhence F +(µ)isaneighbourhoodof x (2)⇒(3)Let x ∈ X and λ beaneutrosophicneighbourhoodof F (x).Put U = F +(λ).Then

by(2), U isneighbourhoodof x and F (U )= x∈U F (x) ≤ λ.

(3)⇒(4)Let λ ∈ NO(Y ),wewanttoshowthat F +(λ) ∈ O(X).Solet x ∈ F +(λ). Thenthereexistsaneighbourhood G of x suchthat F (G) ≤ λ.Thereforeforsome U ∈ O(X,x),U ⊆ G and F (U ) ≤ λ.Thereforeweget x ∈ U ⊆ F +(λ)andhence F +(λ) ∈ O(X). (4)⇒(5)Let δ beaneutrosophicclosedsetin Y .So,wehave X\F (δ)= F +(1 δ) ∈ O(X) andhence F (δ)isclosedsetin X (5)⇒(6)Let µ beanyneutrosophicsetin Y .SinceCl(µ)isneutrosophicclosedsetin Y , F (Cl(µ))isclosedsetin X and F (µ) ⊆ F (Cl(µ)).Therefore,weobtainCl(F (µ)) ⊆ F (Cl(µ)). (6)⇒(1)Let x ∈ X and λ ∈ NO(Y )with F (x) ≤ λ.Now F (1 λ)= {x ∈ X|F (x)q(1 λ)} So,for x notbelongsto F (1 λ).Then,wemusthave F (x) (1 λ)andthisimplies F (x) ≤ 1 (1 λ)= λ whichistrue.Therefore x/ ∈ F (1 λ)by(6), x/ ∈ Cl(F (1 λ))andthere exists U ∈ O(X,x)suchthat U ∩F (1 λ)= ∅.Therefore,weobtain F (U )= x∈U F (x) ≤ λ Thisproves F isneutrosophicuppersemicontinuous.

Theorem2. Thefollowingstatementsareequivalentforaneutrosophicmultifunction F : (X,τ ) → (Y,τ1):

1. F isneutrosophiclowersemicontinuous;

2.Foreach λ ∈ NO(Y ) andeach x ∈ F (λ),thereexists U ∈ O(X,x) suchthat U ⊆ F (λ);

3. F (λ) ∈ O(X) forevery λ ∈ NO(Y )

4. F +(δ) isaclosedsetin X foreveryneutrosophicclosedset δ of Y ;

5. Cl(F +(µ)) ⊆ F +(Cl(µ)) foreveryneutrosophicset µ of Y ;

6. F (Cl(A)) ≤ Cl(F (A)) foreverysubset A of X;

Proof. (1)⇒(2)Let λ ∈ NO(Y )and x ∈ F (λ)with F (x)qλ.Thenbyproperties–1,there exists U ∈ O(X,x)suchthat U ⊆ F (λ). (2)⇒(3)Let λ ∈ NO(Y )adn x ∈ F (λ).Thenby(2),thereexists U ∈ O(X,x)such that U ⊆ F (λ).Therefore,wehave x ∈ U ⊆ ClInt(U ) ⊆ ClInt(F (λ))andhence F (λ) ∈ O(X).

(3)⇒(4)Let δ beaneutrosophicclosedin Y .Sowehave X\F +(δ)= F (1 δ) ∈ O(X) andhence F +(δ)isclosedsetin X

(4)⇒(5)Let µ beanyneutrosophicsetin Y .SinceCl(µ)isneutrosophicclosedsetin Y , thenby(4),wehave F +(Cl(µ))isclosedsetin X and F +(µ) ⊆ F +(Cl(µ)).Therefore,we obtainCl(F +(µ)) ⊆ F +(Cl(µ)).

(5)⇒(6)Let A beanysubsetof X.By(5),Cl(A) ⊆ Cl F +(F (A)) ⊆ F +(Cl(F (A))).

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ThereforeweobtainCl(A) ⊆ F +(Cl F (A)).Thisimpliesthat F (Cl(A)) ≤ Cl F (A).

(6)⇒(5)Let µ beanyneutrosophicsetin Y .By(6), F (Cl F +(µ)) ≤ Cl(F (F +(µ)))and henceCl(F +(µ)) ⊆ F +(Cl(F (F +(µ)))) ⊆ F +(Cl(µ)).ThereforeCl(F +(µ)) ⊆ F +(Cl(µ)). (5)⇒(1)Let x ∈ X and λ ∈ NO(Y )with F (x)qλ.Now, F +(1 λ)= {x ∈ X|F (x) ≤ 1 λ} So,for x notbelongsto F +(1 λ),thenwehave F (x) 1 λ andthisimpliesthat F (x)qλ. Therefore, x/ ∈ F +(1 λ).Since1 λ isneutrosophicclosedsetin Y ,by(5), x/ ∈ Cl(F +(1 λ)) andthereexists U ∈ O(X,x)suchthat ∅ = U ∩ F +(1 λ)= U ∩ (X\F (λ)).Therefore, weobtain U ⊆ F (λ).Thisproves F isneutrosophiclowersemicontinuous.

Deﬁnition7. Foragivenneutrosophicmultifunction F :(X,τ ) → (Y,τ1),aneutrosophic multifunction Cl(F ):(X,τ ) → (Y,τ1) isdeﬁnedas (Cl F )(x)=Cl F (x) foreach x ∈ X.

WeuseCl F andthefollowingLemmatoobtainacharacterizationoflowerneutrosophic semicontinuousmultifunction.

Lemma2. If F :(X,τ ) → (Y,τ1) isaneutrosophicmultifunction,then (Cl F ) (λ)= F (λ) foreach λ ∈ NO(Y )

Proof. Let λ ∈ NO(Y )and x ∈ (Cl F ) (λ).Thismeansthat(Cl F )(x)qλ.Since λ ∈ NO(Y ),wehave F (x)qλ andhence x ∈ F (λ).Therefore(Cl F ) (λ) ⊆ F (λ) (∗). Conversely,let x ∈ F (λ)since λ ∈ NO(Y )then F (x)qλ ⊆ (Cl F )(x)qλ andhence x ∈ (Cl F ) (λ).Therefore F (λ) ⊆ (Cl F ) (λ) (∗∗). From(∗)and(∗∗),weget(Cl F ) (λ)= F (λ).

Theorem3. Aneutrosophicmultifunction F :(X,τ ) → (Y,τ1) isneutrosophiclowersemicontinuousifandonlyif Cl F :(X,τ ) → (Y,τ1) isneutrosophiclowersemicontinuous.

Proof. Suppose F isneutrosophiclowersemicontinuous.Let λ ∈ NO(Y )and F (x)qλ.This meansthat x ∈ F (λ).Thenthereexists U ∈ O(X,x)suchthat U ⊆ F (λ).Therefore,we have x ∈ U ⊆ Int(U ) ⊆ Int F (λ)andhence F (λ) ∈ O(X).ThenbyLemma2,wehave U ⊆ F (λ)=(Cl F ) (λ)and(Cl F ) (λ) ∈ O(X),andhence(Cl F )(x)qλ.ThereforeCl F is fuzylowersemicontinuous.Conversely,supposeCl F isneutrosophiclowersemicontinuous. Ifforeach λ ∈ NO(Y )with(Cl F )(x)qλ and x ∈ (Cl F ) (λ)thenthereexists U ∈ O(X,x) suchthat U ⊆ (Cl F ) (λ).ByLemma2andTheorem2,wehave U ⊆ (Cl F (λ))= F (λ) and F (λ) ∈ O(X).Therefore F isneutrosophiclowersemicontinuous. Deﬁnition8. Givenafamily {Fi :(X,τ ) → (Y,σ): i ∈ I} ofneutrosophicmultifunctions, wedeﬁnetheunion ∨ i∈I Fi andtheintersection ∧ i∈I Fi asfollows: ∨ i∈I Fi :(X,τ ) → (Y,σ), ( ∨ i∈I Fi)(x)= ∨ i∈I Fi(x) and ∧ i∈I Fi :(X,τ ) → (Y,σ), ( ∧ i∈I Fi)(x)= ∧ i∈I Fi(x) Theorem4. If Fi : X → Y areneutrosophicuppersemi-continuousmultifunctionsfor i =1, 2,...,n,then n ∨ i∈I Fi isaneutrosophicuppersemi-continuousmultifunction.

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Proof. Let A beaneutrosophicopensetof Y .Wewillshowthat( n ∨ i∈I Fi)+(A)= {x ∈ X : n ∨ i∈I Fi(x) ⊂ A} isopenin X.Let x ∈ ( n ∨ i∈I Fi)+(A).Then Fi(x) ⊂ A for i =1, 2,...,n.Since Fi : X → Y isneutrosophicuppersemi-continuousmultifunctionfor i =1, 2,...,n,then thereexistsanopenset Ux containing x suchthatforall z ∈ Ux, Fi(z) ⊂ A.Let U = n ∪ i∈I Ux Then U ⊂ ( n ∨ i∈I Fi)+(A).Thus,( n ∨ i∈I Fi)+(A)isopenandhence n ∨ i∈I Fi isaneutrosophicupper semi-continuousmultifunction.

Lemma3. Let {Ai}i∈I beafamilyofneutrosophicsetsinaneutrosophictopologicalspace X.Thenaneutrosophicpoint x isquasi-coincidentwith ∨Ai ifandonlyifthereexistsan i0 ∈ I suchthat xqAi0 .

Theorem5. If Fi : X → Y areneutrosophiclowersemi-continuousmultifunctionsfor i =1, 2,...,n,then n ∨ i∈I Fi isaneutrosophiclowersemi-continuousmultifunction.

Proof. Let A beaneutrosophicopensetof Y .Wewillshowthat( n ∨ i∈I Fi) (A)= {x ∈ X : ( n ∨ i∈I Fi)(x)qA} isopenin X.Let x ∈ ( n ∨ i∈I Fi) (A).Then( n ∨ i∈I Fi)(x)qA andhence Fi0(x)qA foran i0.Since Fi : X → Y isneutrosophiclowersemi-continuousmultifunction,there existsanopenset Ux containing x suchthatforall z ∈ U , Fi0(z)qA.Then( n ∨ i∈I Fi)(z)qA and hence U ⊂ ( n ∨ i∈I Fi) (A).Thus,( n ∨ i∈I Fi) (A)isopenandhence n ∨ i∈I Fi isaneutrosophiclower semi-continuousmultifunction.

Theorem6. Let F :(X,τ ) → (Y,σ) beaneutrosophicmultifunctionand {Ui : i ∈ I} bean opencoverfor X.Thenthefollowingareequivalent:

1. Fi = F|Ui isaneutrosophiclowersemi-continuousmultifunctionforall i ∈ I,

2. F isneutrosophiclowersemi-continuous.

Proof. (1) ⇒ (2):Let x ∈ X and A beaneutrosophicopensetin Y with x ∈ F (A).Since {Ui : i ∈ I} isanopencoverfor X,then x ∈ Ui0 foran i0 ∈ I.Wehave F (x)= Fi0(x)and hence x ∈ Fi0 (A).Since F|Ui 0 isneutrosophiclowersemi-continuous,thereexistsanopen set B = G ∩ Ui0 in Ui0 suchthat x ∈ B and F (A) ∩ Ui0 = F|Ui (A) ⊃ B = G ∩ Ui0,where G isopenin X.Wehave x ∈ B = G ∩ Ui0 ⊂ F|Ui 0(A)= F (A) ∩ Ui0 ⊂ F (A).Hence, F is neutrosophiclowersemi-continuous. (2) ⇒ (1):Let x ∈ X and x ∈ Ui.Let A beaneutrosophicopensetin Y with Fi(x)qA. Since F islowersemi-continuousand F (x)= Fi(x),thereexistsanopenset U containing x suchthat U ⊂ F (A).Take B = Ui ∩ U .Then B isopenin Ui containing x.Wehave B ⊂ F i(A).Thus Fi isaneutrosophiclowersemi-continuous. Theorem7. Let F :(X,τ ) → (Y,σ) beaneutrosophicmultifunctionand {Ui : i ∈ I} bean opencoverfor X.Thenthefollowingareequivalent:

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1. Fi = F|Ui isaneutrosophicuppersemi-continuousmultifunctionforall i ∈ I,

2. F isneutrosophicuppersemi-continuous.

Proof. ItissimilartothatofTheorem6. Remark8. Asubset A ofatopologicalspace (X,τ ) canbeconsideredasaneutrosophicset withcharacteristicfunctiondeﬁnedby A(x)= 1 ifx ∈ A 0 ifx / ∈ A.

Let (Y,σ) beaneutrosophictopologicalspace.Theneutrosophicsetsoftheform A × B with A ∈ τ and B ∈ σ formabasisfortheproductneutrosophictopology τ × σ on X × Y ,where forany (x,y) ∈ X × Y , (A × B)(x,y)= min{A(x),B(y)} Deﬁnition9. Foraneutrosophicmultifunction F :(X,τ ) → (Y,σ),theneutrosophicgraph multifunction GF : X → X × Y of F isdeﬁnedby GF (x)= x1 × F (x) forevery x ∈ X

Theorem9. Iftheneutrosophicgraphmultifunction GF ofaneutrosophicmultifunction F :(X,τ ) → (Y,σ) isneutrosophiclowersemi-continuous,then F isneutrosophiclower semi-continuous.

Proof. Supposethat GF isneutrosophiclowersemi-continuousand x ∈ X.Let A bea neutrosophicopensetin Y suchthat F (x)qA.Thenthereexists y ∈ Y suchthat(F (x))(y)+ A(y) > 1.Then(GF (x))(x,y)+(X × A)(x,y)=(F (x))(y)+ A(y) > 1.Hence, GF (x)q(X × A).Since GF isneutrosophiclowersemi-continuous,thereexistsanopenset B in X suchthat x ∈ B and GF (b)q(X × A)forall b ∈ B.Letthereexists b0 ∈ B suchthat F (b0)qA.Thenfor all y ∈ Y ,(F (b0))(y)+A(y) < 1.Forany(a,c) ∈ X ×Y ,wehave(GF (b0))(a,c) ⊂ (F (b0))(c) and(X × A)(a,c) ⊂ A(c).Sinceforall y ∈ Y ,(F (b0))(y)+ A(y) < 1,(GF (b0))(a,c)+ (X × A)(a,c) < 1.Thus, GF (b0)q(X × A),where b0 ∈ B.Thisisacontradictionsince GF (b)q(X × A)forall b ∈ B.Hence, F isneutrosophiclowersemi-continuous.

Theorem10. Iftheneutrosophicgraphmultifunction GF ofaneutrosophicmultifunction F : X → Y isneutrosophicuppersemi-continuous,then F isneutrosophicuppersemicontinuous.

Proof. Supposethat GF isneutrosophicuppersemi-continuousandlet x ∈ X.Let A be neutrosophicopenin Y with F (x) ⊂ A.Then GF (x) ⊂ X × A.Since GF isneutrosophic uppersemi-continuous,thereexistsanopenset B containing x suchthat GF (B) ⊂ X × A. Forany b ∈ B and y ∈ Y ,wehave(F (b))(y)=(GF (b))(b,y) ⊂ (X × A)(b,y)= A(y).Then (F (b))(y) ⊂ A(y)forall y ∈ Y .Thus, F (b) ⊂ A forany b ∈ B.Hence, F isneutrosophic uppersemi-continuous.

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Theorem11. Let F :(X,τ ) → (Y,σ) beaneutrosophicmultifunction.Thenthefollowing areequivalent:

1. F isneutrosophiclowersemi-continuous, 2.Forany x ∈ X andanynet (xi)i∈I convergingto x in X andeachneutrosophicopen set B in Y with x ∈ F (B),thenet (xi)i∈I iseventuallyin F (B)

Proof. (1) ⇒ (2):Let(xi)beanetconvergingto x in X and B beanyneutrosophicopen setin Y with x ∈ F (B).Since F isneutrosophiclowersemi-continuous,thereexistsan openset A ⊂ X containing x suchthat A ⊂ F (B).Since xi → x,thereexistsanindex i0 ∈ I suchthat xi ∈ A forevery i ≥ i0.Wehave xi ∈ A ⊂ F (B)forall i ≥ i0.Hence, (xi)i∈I iseventuallyin F (B). (2) ⇒ (1):Supposethat F isnotneutrosophiclowersemi-continuous.Thereexistsapoint x andaneutrosophicopenset A with x ∈ F (A)suchthat B F (A)foranyopenset B ⊂ X containing x.Let xi ∈ B and xi / ∈ F (A)foreachopenset B ⊂ X containing x Thentheneighborhoodnet(xi)convergesto x but(xi)i∈I isnoteventuallyin F (A).This isacontradiction.

Theorem12. Let F :(X,τ ) → (Y,σ) beaneutrosophicmultifunction.Thenthefollowing areequivalent:

1. F isneutrosophicuppersemi-continuous,

2.Forany x ∈ X andanynet (xi) convergingto x in X andanyneutrosophicopenset B in Y with x ∈ F +(B),thenet (xi) iseventuallyin F +(B)

Proof. TheproofissimilartothatofTheorem11.

Theorem13. Thesetofallpointsof X atwhichaneutrosophicmultifunction F :(X,τ ) → (Y,σ) isnotneutrosophicuppersemi-continuousisidenticalwiththeunionofthefrontier oftheupperinverseimageofneutrosophicopensetscontaining F (x)

Proof. Suppose F isnotneutrosophicuppersemi-continuousat x ∈ X.Thenthereexists aneutrosophicopenset A in Y containing F (x)suchthat A ∩ (X\F +(B)) = ∅ forevery openset A containing x.Wehave x ∈ Cl(X\F +(B))= X\ Int(F +(B))and x ∈ F +(B). Thus, x ∈ Fr(F +(B)).Conversely,let B beaneutrosophicopensetin Y containing F (x) with x ∈ Fr(F +(B)).Supposethat F isneutrosophicuppersemi-continuousat x.There existsanopenset A containing x suchthat A ⊂ F +(B).Wehave x ∈ Int(F +(B)).Thisis acontradiction.Thus, F isnotneutrosophicuppersemi-continuousat x.

Theorem14. ThesetofallpointsofXatwhichaneutrosophicmultifunction F :(X,τ ) → (Y,σ) isnotneutrosophiclowersemi-continuousisidenticalwiththeunionofthefrontier ofthelowerinverseimageofneutrosophicclosedsetswhicharequasi-coincidentwith F (x)

Florentin Smarandache (author and editor) Collected Papers, IX 419

Proof. ItissimilartothatofTheorem13.

Deﬁnition10. Aneutrosophicset λ ofaneutrosophictopologicalspace Y issaidtobe neutrosophiccompactrelativeto Y ifeverycover {λα}α∈∆ of λ byneutrosophicopensetsof Y hasaﬁnitesubcover {λi}n i=1 of λ

Deﬁnition11. Aneutrosophicset λ ofaneutrosophictopologicalspace Y issaidtobe neutrosophicLindelofrelativeto Y ifeverycover {λα}α∈∆ of λ byneutrosophicopensetsof Y hasacountablesubcover {λn}n∈N of λ.

Deﬁnition12. Aneutrosophictopologicalspace Y issaidtobeneutrosophiccompactif χY (characteristicfunctionof Y )isneutrosophiccompactrelativeto Y

Deﬁnition13. Aneutrosophictopologicalspace Y issaidtobeneutrosophicLindelofif χY (characteristicfunctionof Y )isneutrosophicLindelofrelativeto Y

Deﬁnition14. Aneutrosophicmultifunction F :(X,τ ) → (Y,τ1) issaidtobepunctually neutrosophiccompact(resp.punctuallyneutrosophicLindelof)ifforeach x ∈ X,F (x) is neutrosophiccompact(resp.neutrosophicLindelof).

Theorem15. Lettheneutrosophicmultifunction F :(X,τ ) → (Y,τ1) beaneutrosophic uppersemicontinuousand F ispunctuallyneutrosophiccompact.If A iscompactrelativeto X,then F (A) isneutrosophiccompactrelativeto Y Proof. Let {λα|α ∈ ∆} beanycoverof F (Z)byneutrosophiccopensetsof Y .Weclaim that F (A)isneutrosophiccompactrelativeto Y .Foreach x ∈ A,thereexistsaﬁnitesubset ∆(x)of∆suchthat F (x) ≤∪{λα|α ∈ ∆(x)}.Put λ(x)= ∪{λα|α ∈ ∆(x)}.Then F (x) ≤ λ(x) ∈ NO(Y )andthereexists U (x) ∈ O(X,x)suchthat F (U (x)) ≤ λ(x).Since {U (x)|x ∈ A} isanopencoverof A thereexistsaﬁnitenumberof A,say, x1,x2,..,xn suchthat A ⊆∪{U (xi)|i =1, 2,..,n}.Thereforeweobtain F (A) ≤ F ( n ∪ i=1 U (xi)) ≤ n ∪ i=1 F (U (xi)) ≤ n ∪ i=1 λ(xi) ≤ n ∪ i=1( ∪ α∈∆(xi ) λα).Thisshowsthat F (A)isneutrosophiccompactrelativeto Y Theorem16. Lettheneutrosophicmultifunction F :(X,τ ) → (Y,τ1) beaneutrosophic uppersemicontinuousand F ispunctuallyneutrosophicLindelof.If A isLindelofrelativeto X,then F (A) isneutrosophicLindelofrelativeto Y Proof. TheproofissimilartothatofTheorem15

Florentin Smarandache (author and editor) Collected Papers, IX 420

REFERENCES

Abdel-Basset, M., Mohamed, M., Smarandache, F., & Chang, V. (2018). Neutrosophic Association Rule Mining Algorithm for Big Data Analysis. Symmetry, 10(4), 106.

Abdel-Basset,M., & Mohamed, M. (2018). The Role of Single Valued Neutrosophic Sets and Rough Setsin Smart City: Imperfect and Incomplete Information Systems. Measurement. Volume 124, August 2018, Pages 47 55

Abdel-Basset, M., Gunasekaran, M., Mohamed, M., & Smarandache, F. A novel method for solving the fully neutrosophic linear programming problems. Neural Computing and Applications, 1-11.

Abdel-Basset, M., Manogaran, G., Gamal, A., & Smarandache, F. (2018). A hybrid approach of neutrosophic sets and DEMATEL method for developing supplier selection criteria. Design Automation for Embedded Systems, 1-22.

Abdel-Basset, M., Mohamed, M., & Chang, V. (2018). NMCDA: A framework for evaluating cloud computing services. Future Generation Computer Systems, 86, 12-29.

Abdel-Basset, M., Mohamed, M., Zhou, Y., & Hezam, I. (2017). Multi-criteria group decision making based on neutrosophic analytic hierarchy process. Journal of Intelligent & Fuzzy Systems, 33(6), 4055-4066.

Abdel-Basset, M.; Mohamed, M.; Smarandache, F. An Extension of Neutrosophic AHP–SWOT Analysis for Strategic Planning and Decision-Making. Symmetry 2018, 10, 116.

Salama,A.,&Alblowi,S.(2012).Neutrosophicsetandneutrosophictopologicalspaces. IOSRJournalofMathematics(IOSR-JM), 3 (4),31–35.

Smarandache,F.(1999). Neutrosophy:Aunifyingﬁeldinlogics:Neutrosophiclogic.neutrosophy,neutrosophicset,neutrosophicprobability.Rehoboth:AmericanResearch Press.

Smarandache,F.(Ed.).(2001). Neutrosophy,neutrosophiclogic,set,probability,and statistics. UniversityofNewMexico,Gallup,NM87301,USA:UniversityofNew Mexico,Gallup.

Smarandache,F.(2007). Aunifyingﬁeldinlogics:Neutrosophiclogic.neutrosophy,neutrosophicset,neutrosophicprobability.ProQuestInformation&Learning.

Florentin Smarandache (author and editor) Collected Papers, IX 421

On the Classification of Bol-Moufang Type of Some Varieties of Quasi Neutrosophic Triplet Loop

(Fenyves BCI-Algebras)

Tèmítópé Gbóláhàn Jaíyéolá, Emmanuel Ilojide, Memudu Olaposi Olatinwo, Florentin Smarandache

Tèmítópé Gbóláhàn Jaíyéolá, Emmanuel Ilojide, Memudu Olaposi Olatinwo, Florentin Smarandache (2018). On the Classification of Bol-Moufang Type of Some Varieties of Quasi Neutrosophic Triplet Loop (Fenyves BCI-Algebras). Symmetry, 10, 427; DOI: 10.3390/sym10100427

Abstract: In this paper, Bol-Moufang types of a particular quasi neutrosophic triplet loop (BCIalgebra), chritened Fenyves BCI-algebras are introduced and studied. 60 Fenyves BCI-algebras are introduced and classified. Amongst these 60 classes of algebras, 46 are found to be associative and 14 are found to be non-associative. The 46 associative algebras are shown to be Boolean groups. Moreover, necessary and sufficient conditions for 13 non-associative algebras to be associative are also obtained: p-semisimplicity is found to be necessary and sufficient for a F3, F5, F42 and F55 algebras to be associative while quasi-associativity is found to be necessary and sufficient for F19, F52, F56 and F59 algebras to be associative. Two pairs of the 14 non-associative algebras are found to be equivalent to associativity (F52 and F55, and F55 and F59). Every BCIalgebra is naturally an F54 BCI-algebra. The work is concluded with recommendations based on comparison between the behaviour of identities of Bol-Moufang (Fenyves’ identities) in quasigroups and loops and their behaviour in BCI-algebra. It is concluded that results of this work are an initiation into the study of the classification of finite Fenyves’ quasi neutrosophic triplet loops (FQNTLs) just like various types of finite loops have been classified. This research work has opened a new area of research finding in BCI-algebras, vis-a-vis the emergence of 540 varieties of Bol-Moufang type quasi neutrosophic triplet loops. A ‘Cycle of Algebraic Structures’ which portrays this fact is provided.

Keywords: quasigroup; loop; BCI-algebra; Bol-Moufang; quasi neutrosophic loops; Fenyves identities

1.Introduction

BCK-algebrasandBCI-algebrasareabbreviatedastwoB-algebras.Theformerwasraisedin1966 byImaiandIseki[1],Japanesemathematicians,andthelatterwasputforwardinthesameyearby Iseki[2].Thetwoalgebrasoriginatedfromtwodifferentsources:settheoryandpropositionalcalculi. Therearesomesystemswhichcontaintheonlyimplicationalfunctoramonglogicalfunctors, suchas thesystemofweakpositiveimplicationalcalculus,BCK-systemandBCI-system.Undoubtedly, therearecommonpropertiesamongthosesystems.Weknowthattherearecloserelationships betweenthenotionsofthesetdifferenceinsettheoryandtheimplicationfunctorinlogicalsystems. Forexample,wehavethefollowingsimpleinclusionrelationsinsettheory:

(A B) (A C) ⊆ C B, A (A B) ⊆ B

Florentin Smarandache (author and editor) Collected Papers, IX 422

Thesearesimilartothepropositionalformulasinpropositionalcalculi: (p → q) → ((q → r) → (p → r)), p → ((p → q) → q),

whichraisethefollowingquestions:Whatarethemostessentialandfundamentalpropertiesofthese relationships?Canweformulateageneralalgebrafromtheaboveconsideration?Howwillweﬁnd anaxiomsystemtoestablishagoodtheoryofgeneralalgebras?Answeringthesequestions,K.Iseki formulatedthenotionsoftwoB-algebrasinwhichBCI-algebrasareawiderclassthanBCK-algebras. TheirnamesaretakenfromBCKandBCIsystemsincombinatorylogic.

BCI-Algebrasareveryinterestingalgebraicstructuresthathavegeneratedwideinterestamong puremathematicians.

1.1.BCI-algebra,Quasigroups,LoopsandtheFenyvesIdentities

Westartwithsomedeﬁnitionsandexamplesofsomevarietiesofquasineutrosophictripletloop.

Deﬁnition1. Atriple (X, ∗,0) iscalledaBCI-algebraifthefollowingconditionsaresatisﬁedforany x, y, z ∈ X: 1. ((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y)= 0; 2.x ∗ 0 = x; 3.x ∗ y = 0 andy ∗ x = 0 =⇒ x = y.

Wecallthebinaryoperation ∗ on X themultiplicationon X,andtheconstant0in X thezero elementof X.Weoftenwrite X insteadof (X, ∗,0) foraBCI-algebrainbrevity.Juxtaposition xy willat timesbeusedfor x ∗ y andwillhavepreferenceover ∗ i.e., xy ∗ z =(x ∗ y) ∗ z.

Example1. Let S beaset.Let 2S bethepowersetof S, thesetdifferenceand ∅ theemptyset.Then 2S , , ∅ isaBCI-algebra.

Example2. Suppose (G, · , e) isanabeliangroupwith e astheidentityelement.Deﬁneabinaryoperation ∗ on Gbyputtingx ∗ y = xy 1.Then (G, ∗, e) isaBCI-algebra.

Example3. (Z, ,0) and (R −{0}, ÷,1) areBCI-algebras.

Example4. Let S beaset.Let 2S bethepowersetof S, thesymmetricdifferenceand ∅ theemptyset.Then 2S , , ∅ isaBCI-algebra.

ThefollowingtheoremsgivenecessaryandsufﬁcientconditionsfortheexistenceofaBCI-algebra.

Theorem1. (Yisheng[3])

Let X beanon-emptyset, ∗ abinaryoperationon X and 0 aconstantelementof X.Then (X, ∗,0) isa BCI-algebraifandonlyifthefollowingconditionshold:

Deﬁnition2. ABCI-algebra (X, ∗,0) iscalledaBCK-algebraif 0 ∗ x = 0 forallx ∈ X.

Deﬁnition3. ABCI-algebra (X, ∗,0) iscalledaFenyvesBCI-algebraifitsatisﬁesanyoftheidentitiesof Bol-Moufangtype.

TheidentitiesofBol-Moufangtypearegivenbelow:

((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y

;

(x ∗ (x ∗ y)) ∗ y = 0;

∗ x =

;

∗

)= 0

3.x

4.x

y = 0 andy

x = 0 implyx = y.

Florentin Smarandache (author and editor) Collected Papers, IX 423

F1: xy ∗ zx = (xy ∗ z)x

F2: xy ∗ zx = (x ∗ yz)x (Moufang identity)

F3: xy ∗ zx = x(y ∗ zx)

F4: xy ∗ zx = x(yz ∗ x) (Moufang identity)

F5: (xy ∗ z)x = (x ∗ yz)x

F6: (xy ∗ z)x = x(y ∗ zx) (extra identity)

F7: (xy ∗ z)x = x(yz ∗ x)

F8: (x ∗ yz)x = x(y ∗ zx)

F9: (x ∗ yz)x = x(yz ∗ x)

F10: x(y ∗ zx)= x(yz ∗ x)

F11: xy xz =(xy ∗ x)z

F12: xy ∗ xz =(x ∗ yx)z

F13: xy ∗ xz = x(yx ∗ z) (extraidentity)

F14: xy ∗ xz = x(y ∗ xz)

F15: (xy ∗ x)z =(x ∗ yx)z

F16: (xy ∗ x)z = x(yx ∗ z)

F17: (xy ∗ x)z = x(y ∗ xz) (Moufangidentity)

F18: (x ∗ yx)z = x(yx ∗ z)

F19: (x ∗ yx)z = x(y ∗ xz) (leftBolidentity)

F20: x(yx ∗ z)= x(y ∗ xz)

F21: yx ∗ zx =(yx ∗ z)x

F22: yx ∗ zx =(y ∗ xz)x (extraidentity)

F23: yx ∗ zx = y(xz ∗ x)

F24: yx ∗ zx = y(x ∗ zx)

F25: (yx ∗ z)x =(y ∗ xz)x

F26: (yx ∗ z)x = y(xz ∗ x) (rightBolidentity)

F27: (yx ∗ z)x = y(x ∗ zx) (Moufangidentity)

F28: (y ∗ xz)x = y(xz ∗ x)

F29: (y ∗ xz)x = y(x ∗ zx)

F30: y(xz ∗ x)= y(x ∗ zx)

F31: yx ∗ xz =(yx ∗ x)z

F32: yx ∗ xz =(y ∗ xx)z

F33: yx ∗ xz = y(xx ∗ z)

F34: yx ∗ xz = y(x ∗ xz)

F35: (yx ∗ x)z =(y ∗ xx)z

F36: (yx ∗ x)z = y(xx ∗ z) (RCidentity)

F37: (yx ∗ x)z = y(x ∗ xz) (Cidentity)

F38: (y ∗ xx)z = y(xx ∗ z)

F39: (y ∗ xx)z = y(x ∗ xz) (LCidentity)

F40: y(xx ∗ z)= y(x ∗ xz)

F41: xx ∗ yz =(x ∗ xy)z (LCidentity)

F42: xx ∗ yz =(xx ∗ y)z

F43: xx ∗ yz = x(x ∗ yz)

F44: xx ∗ yz = x(xy ∗ z)

F45: (x ∗ xy)z =(xx ∗ y)z

F46: (x ∗ xy)z = x(x ∗ yz) (LCidentity)

F47: (x ∗ xy)z = x(xy ∗ z)

F48: (xx ∗ y)z = x(x ∗ yz) (LCidentity)

F49: (xx ∗ y)z = x(xy ∗ z)

F50: x(x ∗ yz)= x(xy ∗ z)

F51: yz ∗ xx =(yz ∗ x)x

F52: yz ∗ xx =(y ∗ zx)x

F53: yz ∗ xx = y(zx ∗ x) (RCidentity)

F54: yz ∗ xx = y(z ∗ xx)

F55: (yz ∗ x)x =(y ∗ zx)x

F56: (yz ∗ x)x = y(zx ∗ x) (RCidentity)

F57: (yz ∗ x)x = y(z ∗ xx) (RCidentity)

F58: (y ∗ zx)x = y(zx ∗ x)

F59: (y ∗ zx)x = y(z ∗ xx)

F60: y(zx ∗ x)= y(z ∗ xx)

Consequentuponthisdeﬁnition,thereare60varietiesofFenyvesBCI-algebras.Herearesome examplesofFenyves’BCI-algebras:

Example5. LetusassumetheBCI-algebra (G, ∗, e) inExample 2.Then (G, ∗, e) isan F8-algebra, F19-algebra, F29-algebra,F39-algebra,F46-algebra,F52-algebra,F54-algebra,F59-algebra.

Example6. LetusassumetheBCI-algebra 2S , , ∅ inExample 1.Then (2S , , ∅) isan F3-algebra, F5-algebra,F21-algebra,F29-algebra,F42-algebra,F46-algebra,F54-algebraandF55-algebra.

Example7. TheBCI-algebra (2S , , ∅) inExample 4 isassociative.

Example8. ByconsideringthedirectproductoftheBCI-algebras (G, ∗, e) and 2S , , ∅ ofExample 2 and Example 1 respectively,wehaveaBCI-algebra G × 2S , (∗, ), (e, ∅) whichisa F29-algebraanda F46-algebra.

Remark1. DirectproductsofsetsofBCI-algebraswillresultinBCI-algebraswhichare Fi-algebrafor distincti’s.

Deﬁnition4. ABCI-algebra (X, ∗,0) iscalledassociativeif (x ∗ y) ∗ z = x ∗ (y ∗ z) forallx, y, z ∈ X.

Deﬁnition5. ABCI-algebra (X, ∗,0) iscalledp-semisimpleif 0 ∗ (0 ∗ x)= xforallx ∈ X.

Florentin Smarandache (author and editor) Collected Papers, IX 424

Theorem2. (Yisheng[3])Supposethat (X, ∗,0) isaBCI-algebra.Deﬁneabinaryrelation on X bywhich x y ifandonlyif x ∗ y = 0 forany x, y ∈ X.Then (X, ) isapartiallyorderedsetwith 0 asaminimal element(meaningthatx 0 impliesx = 0 foranyx ∈ X).

Deﬁnition6. ABCI-algebra (X, ∗,0) iscalledquasi-associativeif (x ∗ y) ∗ z ≤ x ∗ (y ∗ z) forall x, y, z ∈ X

Thefollowingtheoremsgiveequivalentconditionsforassociativity,quasi-associativityand p-semisimplicityinaBCI-algebra:

Theorem3. (Yisheng[3])

GivenaBCI-algebraX,thefollowingareequivalentx, y, z ∈ X:

1.Xisassociative.

2. 0 ∗ x = x.

3.x ∗ y = y ∗ x ∀ x, y ∈ X.

Theorem4. (Yisheng[3])

LetXbeaBCI-algebra.Thenthefollowingconditionsareequivalentforanyx, y, z, u ∈ X:

1.Xisp-semisimple

2. (x ∗ y) ∗ (z ∗ u)=(x ∗ z) ∗ (y ∗ u).

3. 0 ∗ (y ∗ x)= x ∗ y.

4. (x ∗ y) ∗ (x ∗ z)= z ∗ y.

5.z ∗ x = z ∗ yimpliesx = y.(theleftcancellationlawi.e.,LCL)

6.x ∗ y = 0 impliesx = y.

Theorem5. (Yisheng[3])

GivenaBCI-algebraX,thefollowingareequivalentforallx, y ∈ X:

1.Xisquasi-associative.

2.x ∗ (0 ∗ y)= 0 impliesx ∗ y = 0.

3. 0 ∗ x = 0 ∗ (0 ∗ x).

4. (0 ∗ x) ∗ x = 0

Theorem6. (Yisheng[3])

Atriple (X, ∗,0) isaBCI-algebraifandonlyifthereisapartialordering on X suchthatthefollowing conditionsholdforanyx, y, z ∈ X:

1. (x ∗ y) ∗ (x ∗ z) z ∗ y;

2.x ∗ (x ∗ y) y;

3.x ∗ y = 0 ifandonlyifx y.

Theorem7. (Yisheng[3])

Let X beaBCI-algebra. X is p-semisimpleifandonlyifoneofthefollowingconditionsholdsforany x, y, z ∈ X:

1.x ∗ z = y ∗ zimpliesx = y.(therightcancellationlawi.e.,RCL)

2. (y ∗ x) ∗ (z ∗ x)= y ∗ z.

3. (x ∗ y) ∗ (x ∗ z)= 0 ∗ (y ∗ z).

Theorem8. (Yisheng[3])

Let X beaBCI-algebra. X is p-semisimpleifandonlyifoneofthefollowingconditionsholdsforany x, y ∈ X:

1.x ∗ (0 ∗ y)= y.

2. 0 ∗ x = 0 =⇒ x = 0

Florentin Smarandache (author and editor) Collected Papers, IX 425

Theorem9. (Yisheng[3])Supposethat (X, ∗,0) isaBCI-algebra. X isassociativeifandonlyif X is p-semisimpleandXisquasi-associative.

Theorem10. (Yisheng[3])Supposethat (X, ∗,0) isaBCI-algebra.Then (x ∗ y) ∗ z =(x ∗ z) ∗ y forall x, y, z ∈ X.

Remark2. InTheorem 9,quasi-associativityinBCI-algebraplaysasimilarroletothatwhichweakassociativity (i.e.,theFi identities)playsinquasigroupandlooptheory.

Wenowmoveontoquasigroupsandloops.

Deﬁnition7. Let L beanon-emptyset.Deﬁneabinaryoperation( )on L .If x y ∈ L forall x, y ∈ L, (L, ) iscalledagroupoid.Ifinagroupoid (L, ),theequations: a x = bandy a = b

haveuniquesolutionsfor x and y respectively,then (L, ) iscalledaquasigroup.Ifinaquasigroup (L, ),there existsauniqueelement e calledtheidentityelementsuchthatforall x ∈ L, x e = e x = x, (L, ) iscalled aloop.

Deﬁnition8. Let (L, ·) beagroupoid.

TheleftnucleusofListhesetNλ (L, ·)= Nλ (L)= {a ∈ L : ax · y = a · xy ∀ x, y ∈ L}

TherightnucleusofListhesetNρ (L, ·)= Nρ (L)= {a ∈ L : y · xa = yx · a ∀ x, y ∈ L}

ThemiddlenucleusofListhesetNµ (L, ·)= Nµ (L)= {a ∈ L : ya · x = y · ax ∀ x, y ∈ L}

ThenucleusofListhesetN(L, ·)= N(L)= Nλ (L, ·) ∩ Nρ (L, ·) ∩ Nµ (L, ·)

ThecentrumofListhesetC(L, )= C(L)= {a ∈ L : ax = xa ∀ x ∈ L}

ThecenterofListhesetZ(L, )= Z(L)= N(L, ) ∩ C(L, )

Intherecentpast,anduptonow,identitiesofBol-Moufangtypehavebeenstudiedonthe platformofquasigroupsandloopsbyFenyves[4],PhillipsandVojtechovsky[5],Jaiyeola[6 8], Robinson[9],Burn[10 12],KinyonandKunen[13]aswellasseveralotherauthors.

Sincethelate1970s,BCIandBCKalgebrashavebeengivenalotofattention.Inparticular, theparticipation intheresearchofpolishmathematiciansTadeuszTraczykandAndrzejWronski aswellasAustralianmathematicianWilliamH.Cornish,inadditiontoothers,iscausingthis branchofalgebratodeveloprapidly.Manyinterestingandimportantresultsareconstantly discovered.Now,thetheoryofBCI-algebrashasbeenwidelyspreadtoareassuchasgeneral theorywhichincludecongruences,quotientalgebras,BCI-Homomorphisms,directsumsanddirect products,commutativeBCK-algebras,positiveimplicativeandimplicativeBCK-algebras,derivations ofBCI-algebras,andidealtheoryofBCI-algebras([1,14 17]).

1.2.BCI-AlgebrasasaQuasiNeutrosophicTripletLoop

Considerthefollowingdeﬁnition.

Deﬁnition9. (QuasiNeutrosophicTripletLoops(QNTL),Zhangetal.[18])

Let (X, ∗) beagroupoid.

1. Ifthereexist b, c ∈ X suchthat a ∗ b = a and a ∗ c = b,then a iscalledanNT-elementwith(r-r)-property. Ifeverya ∈ XisanNT-elementwith(r-r)-property,then, (X, ∗) iscalleda(r-r)-quasiNTL.

2. Ifthereexist b, c ∈ X suchthat a ∗ b = a and c ∗ a = b,then a iscalledanNT-elementwith(r-l)-property. Ifeverya ∈ XisanNT-elementwith(r-l)-property,then, (X, ∗) iscalleda(r-l)-quasiNTL.

3. Ifthereexist b, c ∈ X suchthat b ∗ a = a and c ∗ a = b,then a iscalledanNT-elementwith(l-l)-property.

Ifeverya ∈ XisanNT-elementwith(l-l)-property,then, (X, ∗) iscalleda(l-l)-quasiNTL.

Florentin Smarandache (author and editor) Collected Papers, IX 426

4. Ifthereexist b, c ∈ X suchthat b ∗ a = a and a ∗ c = b,then a iscalledanNT-elementwith(l-r)-property. Ifeverya ∈ XisanNT-elementwith(l-r)-property,then, (X, ∗) iscalleda(l-r)-quasiNTL.

5. Ifthereexist b, c ∈ X suchthat a ∗ b = b ∗ a = a and a ∗ c = b,then a iscalledanNT-element with(lr-r)-property.Ifevery a ∈ X isanNT-elementwith(lr-r)-property,then, (X, ∗) iscalleda (lr-r)-quasiNTL.

6. Ifthereexist b, c ∈ X suchthat a ∗ b = b ∗ a = a and c ∗ a = b,then a iscalledanNT-element with(lr-l)-property.Ifevery a ∈ X isanNT-elementwith(lr-l)-property,then, (X, ∗) iscalleda (lr-l)-quasiNTL.

7. Ifthereexist b, c ∈ X suchthat a ∗ b = a and a ∗ c = c ∗ a = b,then a iscalledanNT-element with(r-lr)-property.Ifevery a ∈ X isanNT-elementwith(r-lr)-property,then, (X, ∗) iscalleda (r-lr)-quasiNTL.

8. Ifthereexist b, c ∈ X suchthat b ∗ a = a and a ∗ c = c ∗ a = b,then a iscalledanNT-element with(l-lr)-property.Ifevery a ∈ X isanNT-elementwith(l-lr)-property,then, (X, ∗) iscalleda (l-lr)-quasiNTL.

9. Ifthereexist b, c ∈ X suchthat a ∗ b = b ∗ a = a and a ∗ c = c ∗ a = b,then a iscalledanNT-element with(lr-lr)-property.Ifevery a ∈ X isanNT-elementwith(lr-lr)-property,then, (X, ∗) iscalleda (lr-lr)-quasiNTL.

ConsequentuponDeﬁnition 9 andthe60Fenyvesidentities Fi,1 ≤ i ≤ 60,thereare60 varietiesofFenyvesquasineutrosophictripletloops(FQNTLs)foreachoftheninevarietiesof QNTLsinDeﬁnition 9.Therebymakingit540varietiesofFenyvesquasineutrosophictripletloops (FQNTLs)inall.ABCI-algebraisa(r-r)-QNT,(r-l)-QNTLand(r-lr)-QNTL.Thus,any Fi BCI-algebra, 1 ≤ i ≤ 60belongstoatleastoneofthefollowingvarietiesofFenyvesquasineutrosophictriplet loops:(r-r)-QNTL,(r-l)-QNTLand(r-lr)-QNTLwhichwerefertoas(r-r)-FQNTL,(r-l)-FQNTL and(r-lr)-FQNTLrespectively.AnyassociativeQNTLwillbecalledquasineutrosophictriplet group(QNTG).

Thevarietyofquasineutrosophictripletloopisageneralizationofneutrosophictripletgroup (NTG)whichwasoriginallyintroducedbySmarandacheandAli[19].Neutrosophictripletset(NTS) isthefoundationofneutrosophictripletgroup.Newresultsanddevelopmentsonneutrosophictriplet groupsandneutrosophictripletloophavebeenreportedbyZhangetal.[18,20,21],andSmarandache andJaiyéolá[22,23].

Itmustbenotedthattripletsarenotconnectedatallwithintuitionisticfuzzyset.Neutrosophic set[24]isageneralizationofintuitionisticfuzzyset(ageneralizationoffuzzyset).InIntuitionistic fuzzyset,anelementhasadegreeofmembershipandadegreeofnon-membership,andthededuction ofthesumofthesetwofrom1isconsideredthehesitantdegreeoftheelement.Theseintuitionistic fuzzysetcomponentsaredependent(viz.[25 28]).Intheneutrosophicset,anelementhasthree independentdegrees:membership(truth-t),indeterminacy(i),andnon-membership(falsity-f), andtheirsumisupto3.However,thecurrentpaperutilizestheneutrosophictriplets,whichare notdeﬁnedinintuitionisticfuzzyset,sincethereisnoneutralelementinintuitionisticfuzzysets. Inaneutrosophictripletset (X, ∗),foreachelement x ∈ X thereexistsaneutralelementdenoted neut(x) ∈ X suchthat x ∗ neut(x)= neut(x) ∗ x = x,andanoppositeof x denoted anti(x) ∈ X suchthat anti(x) ∗ x = x ∗ anti(x)= neut(x).Thus,thetriple (x, neut(x), anti(x)) iscalleda neutrosophictripletwhichinthephilosophyof‘neutrosophy’,canbealgebraicallyharmonized with (t, i, f ) inneutrosophicsetandthenextendedforneutrosophichesitantfuzzy[29]setasproposed for (t, i, f )-neutrosophicstructures[30].Unfortunately,suchharmonizationisnotreadilydeﬁnedin intuitionisticfuzzysets.

Theorem11. (Zhangetal.[18])A(r-lr)-QNTGor(l-lr)-QNTGisaNTG.

ThispresentstudylooksatFenyvesidentitiesontheplatformofBCI-algebras.Themainobjective ofthisstudyistoclassifytheFenyvesBCI-algebrasintoassociativeandnon-associativetypes.Itwill

Florentin Smarandache (author and editor) Collected Papers, IX 427

alsobeshownthatsomeFenyvesidentitiesplaytherolesofquasi-associativityand p-semisimplicity, vis-a-visTheorem 9 inBCI-algebras.

2.MainResults

WeshallﬁrstclarifytherelationshipbetweenaBCI-algebra,aquasigroupandaloop.

Theorem12.

1.ABCIalgebraXisaquasigroupifandonlyifitisp-semisimple.

2.ABCIalgebraXisaloopifandonlyifitisassociative.

3.AnassociativeBCIalgebraXisaBooleangroup.

Proof. WeuseTheorem 3,Theorem 7 andTheorem 4

1. FromTheorems 7 and 4, p-semisimplicityisequivalenttotheleftandrightcancellationlaws, whichconsequentlyimpliesthat X isaquasigroupifandonlyifitis p-semisimple.

2. OneoftheaxiomsthataBCI-algebrasatisﬁesis x ∗ 0 = x forall x ∈ X.So,0isalreadytheright identityelement.Now,fromTheorem 3,associativityisequivalentto0 ∗ x = x forall x ∈ X.So, 0isalsotheleftidentityelementof X.Theconclusionfollows.

3. InaBCI-algebra, x ∗ x = 0forall x ∈ X.And0istheidentityelementof X.Hence,everyelement istheinverseofitself.

Lemma1. Let (X, ∗,0) beaBCI-algebra.

1. 0 ∈ Nρ (X)

2. 0 ∈ Nλ (X), Nµ (X) impliesXisquasi-associative.

3.If 0 ∈ Nλ (X),thenthefollowingareequivalent:

(a)Xisp-semisimple.

(b)xy = 0y · xforallx, y ∈ L. (c)xy = 0x yforallx, y ∈ L.

4.If 0 ∈ Nλ (X) or 0 ∈ Nµ (X),thenXisp-semisimpleifandonlyifXisassociative.

5.If 0 ∈ N(X),thenXisp-semisimpleifandonlyifXisassociative.

6.If (X, ∗,0) isaBCK-algebra,then

(a) 0 ∈ Nλ (X).

(b) 0 ∈ Nµ (X) impliesXisatrivialBCK-algebra.

7.Thefollowingareequivalent:

(a)Xisassociative.

(b)x ∈ Nλ (X) forallx ∈ X.

(c)x ∈ Nρ (X) forallx ∈ X.

(d)x ∈ Nµ (X) forallx ∈ X.

(e) 0 ∈ C(X)

(f)x ∈ C(X) forallx ∈ X.

(g)x ∈ Z(X) forallx ∈ X.

(h) 0 ∈ Z(X).

(i)Xisa(lr-r)-QNTL.

(j)Xisa(lr-l)-QNTL.

(k)Xisa(lr-lr)-QNTL

8.If (X, ∗,0) isaBCK-algebraand 0 ∈ C(X),thenXisatrivialBCK-algebra.

Proof. Thisisroutinebysimplyusingthedeﬁnitionsofnuclei,centrum,centerofaBCI-algebraand QNTLalongsideTheorems 3 10 appropriately.

Florentin Smarandache (author and editor) Collected Papers, IX 428

Remark3. BasedonTheorem 11,sinceanassociativeBCI-algebraisa(r-lr)-QNTG,then,anassociative BCI-algebraisaNTG.Thiscorroboratestheimportanceofthestudyofnon-associativeBCI-algebrai.e., weakassociativelaws(Fi-identities)inBCI-algebra,asmentionedearlierintheobjectiveofthiswork.

Theorem13. Let (X, ∗,0) beaBCI-algebra.If X isanyofthefollowingFenyvesBCI-algebras,then X isassociative.

1.F1-algebra

2.F2-algebra

3.F4-algebra

4.F6-algebra

5.F7-algebra

6.F9-algebra

7.F10-algebra

8.F11-algebra

9.F12-algebra

10.F13-algebra

Proof.

11.F14-algebra

12.F15-algebra

13.F16-algebra

14.F17-algebra

15.F18-algebra

16.F20-algebra

17.F22-algebra

18.F23-algebra

19.F24-algebra

20.F25-algebra

21.F26-algebra

22.F27-algebra

23.F28-algebra

24.F30-algebra

25.F31-algebra

26.F32-algebra

27.F33-algebra 28.F34-algebra 29.F35-algebra 30.F36-algebra

31.F37-algebra

32.F38-algebra 33.F40-algebra 34.F41-algebra

35.F43-algebra

36.F44-algebra

37.F45-algebra 38.F47-algebra 39.F48-algebra 40.F49-algebra

41.F50-algebra

42.F51-algebra

43.F53-algebra 44.F57-algebra 45.F58-algebra 46.F60-algebra

1. Let X bean F1-algebra.Then xy ∗ zx =(xy ∗ z)x.With z = y,wehave xy ∗ yx =(xy ∗ y)x which implies xy ∗ yx =(xy ∗ x)y =(xx ∗ y)y =(0 ∗ y)y = 0 ∗ (y ∗ y) (since0 ∈ Nλ (X);thisisachieved byputting y = x inthe F1 identity) = 0 ∗ 0 = 0.Thisimplies xy ∗ yx = 0.Nowreplacing x with y, and y with x inthelastequationgives yx ∗ xy = 0implyingthat x ∗ y = y ∗ x asrequired.

2. Let X bean F2-algebra.Then xy ∗ zx =(x ∗ yz)x.With y = z,wehave xz ∗ zx =(x ∗ zz)x = (x ∗ 0) ∗ x = x ∗ x = 0implyingthat xz ∗ zx = 0.Nowreplacing x with z,and z with x inthelast equationgives zx ∗ xz = 0implyingthat x ∗ z = z ∗ x asrequired.

3. Let X bea F4-algebra.Then, xy ∗ zx = x(yz ∗ x).Put y = x and z = 0,thenyouget0 ∗ 0x = x whichmeans X is p-semisimple.Put x = 0and y = 0toget0z = 0 ∗ 0z whichimpliesthat X is quasi-associative(Theorem 5).Thus,byTheorem 9, X isassociative.

4. Let X bean F6-algebra.Then, (xy ∗ z)x = x(y ∗ zx).Put x = y = 0toget0z = 0 ∗ 0z which impliesthat X isquasi-associative(Theorem 5).Put y = 0and z = x,thenwehave0 ∗ x = x. Thus, X isassociative.

5. Let X bean F7-algebra.Then (xy ∗ z)x = x(yz ∗ x).With z = 0,wehave xy ∗ x = x(y ∗ x) Put y = x inthelastequationtoget xx ∗ x =(x ∗ xx) implying0 ∗ x = x

6. Let X bean F9-algebra.Then (x ∗ yz)x = x(yz ∗ x).With z = 0,wehave (x ∗ y) ∗ x = x(y ∗ x). Put y = x inthelastequationtoget (x ∗ x)x = x(x ∗ x) implying0 ∗ x = x.

7. Let X bean F10-algebra.Then, x(y ∗ zx)= x(yz ∗ x).Put y = x = z,thenwehave x ∗ 0x = 0.So, 0x = 0 ⇒ x = 0.whichmeansthat X is p-semisimple(Theorem 8(2)).Hence, X hastheLCLby Theorem 4.Thence,the F10 identity x(y ∗ zx)= x(yz ∗ x) ⇒ y ∗ zx = yz ∗ x whichmeansthat X isassociative.

8. Let X bean F11-algebra.Then xy ∗ xz =(xy ∗ x)z.With y = 0,wehave x ∗ xz = xx ∗ z.Put z = x inthelastequationtoget x = 0 ∗ x asrequired.

9. Let X bean F12-algebra.Then xy ∗ xz =(x ∗ yx)z.With z = 0,wehave xy ∗ x = x ∗ yx.Put y = x inthelastequationtoget xx ∗ x = x ∗ xx implying0 ∗ x = x asrequired.

10. Let X bean F13-algebra.Then xy ∗ xz = x(yx ∗ z).With z = 0,wehave (x ∗ y)x = x ∗ yx which implies (x ∗ x)y = x ∗ yx whichimplies0 ∗ y = x ∗ yx.Put y = x inthelastequationtoget 0 ∗ x = x asrequired.

11. Let X bean F14-algebra.Then xy ∗ xz = x(y ∗ xz).With z = 0,wehave xy ∗ x = x ∗ yx.Put y = x inthelastequationtoget0 ∗ x = x asrequired.

12. Let X bean F15-algebra.Then (xy ∗ x)z =(x ∗ yx)z.With z = 0,wehave (xy ∗ x)=(x ∗ yx) Put y = x inthelastequationtoget0 ∗ x = x asrequired.

13. Let X bean F16-algebra.Then (xy ∗ x)z = x(yx ∗ z).With z = 0,wehave (xy ∗ x)=(x ∗ yx) Put y = x inthelastequationtoget0 ∗ x = x asrequired.

Florentin Smarandache (author and editor) Collected Papers, IX 429

14. Let X bean F17-algebra.Then (xy ∗ x)z = x(y ∗ xz).With z = 0,wehave (xy ∗ x)= x(y ∗ x). Put y = x inthelastequationtoget0 ∗ x = x asrequired.

15. Let X bean F18-algebra.Then (x ∗ yx)z = x(yx ∗ z).With y = 0,wehave (x ∗ 0x)z = x(0x ∗ z).

Since0 ∈ Nλ (X) and0 ∈ Nµ (X),(theseareobtainedbyputting x = 0and x = y respectively inthe F18-identity),thelastequationbecomes (x0 ∗ x)z = x(0 ∗ xz)= x0 ∗ xz = x ∗ xz which implies0 ∗ z = x ∗ xz.Put x = z inthelastequationtoget0 ∗ z = z asrequired.

16.Thisissimilartotheprooffor F10-algebra.

17. Let X bean F22-algebra.Then yx ∗ zx =(y ∗ xz)x.Put y = x, z = 0,then0x = 0 ∗ 0x which impliesthat X isquasi-associative.ByTheorem 10,the F22 identityimpliesthat yx ∗ zx = yx ∗ xz. Substitute x = 0toget yz = y ∗ 0z.Now,put y = z inthistoget z ∗ 0z = 0.So,0z = 0 ⇒ z = 0. Hence, X is p-semisimple(Theorem 8(2)).Thus,byTheorem 9, X isassociative.

18. Let X bean F23-algebra.Then yx ∗ zx = y(xz ∗ x).With z = 0,wehave yx ∗ 0x = y(x ∗ x) which implies yx ∗ 0x = y.Since0 ∈ Nµ (X),(thisisobtainedbyputting z = x inthe F23-identity), thelast equationbecomes (yx ∗ 0) ∗ x = y whichimplies (yx ∗ x)= y.Put x = y inthelast equationtoget0 ∗ y = y asrequired.

19. Let X bean F24-algebra.Then yx ∗ zx = y(x ∗ zx).With z = 0,wehave yx ∗ 0x = y(x ∗ 0x) Since0 ∈ Nµ (X),(thisisobtainedbyputting x = 0inthe F24-identity),thelastequationbecomes ((yx)0 ∗ x)= y(x0 ∗ x) whichimplies yx ∗ x = y.Put y = x inthelastequationtoget0 ∗ y = y asrequired.

20. Let X bean F25-algebra.Then (yx ∗ z)x =(y ∗ xz)x.Put x = 0,then yz = y ∗ 0z.Substitute z = y, then y ∗ 0y = 0.So,0y = 0 ⇒ y = 0.Hence, X is p-semisimple(Theorem 8(2)).Hence, X hasthe RCLbyTheorem 7.Thence,the F25 identity (yx ∗ z)x =(y ∗ xz)x implies yx ∗ z = y ∗ xz.Thus, X isassociative.

21. Let X bean F26-algebra.Then (yx ∗ z)x = y(xz ∗ x).With z = 0,wehave yx ∗ x = y.Put x = y in thelastequationtoget0 ∗ y = y asrequired.

22. Let X bean F27-algebra.Then (yx ∗ z)x = y(x ∗ zx).Put z = x = y,then0x ∗ x = 0whichimplies X isquasi-associative.Put x = 0and y = z toget z ∗ 0z = 0.So,0z = 0 ⇒ z = 0.Hence, X is p-semisimple(Theorem 8(2)).Thus,byTheorem 9, X isassociative.

23. Let X bean F28-algebra.Then (y ∗ xz)x = y(xz ∗ x).With z = 0,wehave yx ∗ x = y.Put x = y in thelastequationtoget0 ∗ y = y asrequired.

24.Theproofofthisissimilartotheprooffor F10-algebra.

25. Let X bean F31-algebra.Then yx ∗ xz =(yx ∗ x)z.ByTheorem 10,the F31 identitybecomes F25 identitywhichimpliesthat X isassociative.

26. Let X bean F32-algebra.Then yx ∗ xz =(y ∗ xx)z.With z = 0,wehave yx ∗ x = y.Put x = y in thelastequationtoget0 ∗ y = y asrequired.

27. Let X bean F33-algebra.Then yx ∗ xz = y(xx ∗ z).With z = 0,wehave yx ∗ x = y.Put x = y in thelastequationtoget0 ∗ y = y asrequired.

28. Let X bean F34-algebra.Then yx ∗ xz = y(x ∗ xz).With z = 0,wehave yx ∗ x = y.Put x = y in thelastequationtoget0 ∗ y = y asrequired.

29. Let X bean F35-algebra.Then (yx ∗ x)z =(y ∗ xx)z.With z = 0,wehave yx ∗ x = y.Put x = y in thelastequationtoget0 ∗ y = y asrequired.

30. Let X bean F36-algebra.Then (yx ∗ x)z = y(xx ∗ z).With z = 0,wehave yx ∗ x = y.Put x = y in thelastequationtoget0 ∗ y = y asrequired.

31. Let X bean F37-algebra.Then (yx ∗ x)z = y(x ∗ xz).With z = 0,wehave yx ∗ x = y.Put x = y in thelastequationtoget0 ∗ y = y asrequired.

32. Let X bean F38-algebra.Then, yz = y ∗ 0z.Put z = y,then y ∗ 0y = 0.So,0y = 0 ⇒ y = 0.Hence, X is p-semisimple(Theorem 8(2)).Now,put y = x,then xz = x ∗ 0z.Now,substitute x = 0toget 0z = 0 ∗ 0z whichmeansthat X isquasi-associative.Thus,byTheorem 9, X isassociative.

33. Let X bean F40-algebra.Bythe F40 identity, y ∗ 0z = y(x ∗ xz).Put z = x = y toget0 ∗ 0x = 0.So, 0x = 0 ⇒ x = 0.Hence, X is p-semisimple(Theorem 8(2)).Thus, X hastheLCLbyTheorem 4. Thence,the F40 identity y(xx ∗ z)= y(x ∗ xz) becomes0 ∗ z = x ∗ xz.Substituting z = x,weget 0x = x whichmeansthat X isassociative.

Florentin Smarandache (author and editor) Collected Papers, IX 430

34. Let X bean F41-algebra.Then xx ∗ yz =(x ∗ xy)z.With z = 0,wehave0 ∗ y = x ∗ xy.Put y = x inthelastequationtoget0 ∗ x = x asrequired.

35. Let X bean F43-algebra.Then xx ∗ yz = x(x ∗ yz).With z = 0,wehave0 ∗ y = x(x ∗ y).Put x = y inthelastequationtoget0 ∗ y = y asrequired.

36. Let X bean F44-algebra.Then xx ∗ yz = x(xy ∗ z).With z = 0,wehave0 ∗ y = x(x ∗ y).Put x = y inthelastequationtoget0 ∗ y = y asrequired.

37. Let X bean F45-algebra.Then (x ∗ xy)z =(xx ∗ y)z.With z = 0,wehave x ∗ xy = 0 ∗ y.Put x = y inthelastequationtoget0 ∗ y = y asrequired.

38. Let X bean F47-algebra.Then (x ∗ xy)z = x(xy ∗ z).With y = 0,wehave0 ∗ z = x(x ∗ z).Put x = z inthelastequationtoget0 ∗ z = z asrequired.

39. Let X bean F48-algebra.Then (xx ∗ y)z = x(x ∗ yz).With z = 0,wehave0 ∗ y = x ∗ xy.Put x = y inthelastequationtoget0 ∗ y = y asrequired.

40. Let X bean F49-algebra.Then (xx ∗ y)z = x(xy ∗ z).With y = 0,wehave0 ∗ z = x ∗ xz.Put x = z inthelastequationtoget0 ∗ z = z asrequired.

41.Thisissimilartotheprooffor F10-algebra.

42. Let X bean F51-algebra.Then yz ∗ xx =(yz ∗ x)x.With z = 0,wehave y =(y ∗ x)x.Put x = y in thelastequationtoget0 ∗ y = y asrequired.

43. Let X bean F53-algebra.Then yz ∗ xx = y(zx ∗ x) whichbecomes yz = y(zx ∗ x).Put z = x to get yx = y ∗ 0x.Substituting y = x,weget x ∗ 0x = 0.So,0x = 0 ⇒ x = 0,whichmeansthat X is p-semisimple(Theorem 8(2)).Now,put y = 0in yx = y ∗ 0x toget0x = 0 ∗ 0x.Hence, X is quasi-associative.Thus, X isassociative.

44. Let X bean F57-algebra.Then (yz ∗ x)x = y(z ∗ xx).With z = 0,wehave yx ∗ x = y.Put x = y in thelastequationtoget0 ∗ y = y asrequired.

45. Let X bean F58-algebra.Then (y ∗ zx)x = y(zx ∗ x).Put y = x = z toget x ∗ 0x = 0.So, 0x = 0 ⇒ x = 0,whichmeansthat X is p-semisimple(Theorem 8(2)).Now,put z = x, y = 0to get0x = 0 ∗ 0x.Hence, X isquasi-associative.Thus, X isassociative.

46. Let X bean F60-algebra.Then y(zx ∗ x)= y(z ∗ xx).Put y = x = z toget x ∗ 0x = 0.So, 0x = 0 ⇒ x = 0,whichmeansthat X is p-semisimple(Theorem 8(2)).Hence, X hastheLCLby Theorem 4.Thence,the F10 identitybecomes zx ∗ x = z ∗ xx.Now,substitute z = x toget0x = x Thus, X isassociative.

Corollary1. Let (X, ∗,0) beaBCI-algebra.If X isanyofthefollowingFenyves’BCI-algebras,then (X, ∗) is aBooleangroup.

1.F1-algebra

2.F2-algebra

3.F4-algebra

4.F6-algebra

5.F7-algebra

6.F9-algebra

7.F10-algebra

8.F11-algebra

9.F12-algebra

10.F13-algebra

11.F14-algebra

12.F15-algebra

13.F16-algebra

14.F17-algebra

15.F18-algebra

16.F20-algebra

17.F22-algebra

18.F23-algebra

19.F24-algebra

20.F25-algebra

21.F26-algebra

22.F27-algebra

23.F28-algebra

24.F30-algebra

25.F31-algebra

26.F32-algebra

27.F33-algebra

28.F34-algebra

29.F35-algebra

30.F36-algebra

Proof. ThisfollowsfromTheorems 12 and 13.

Theorem14. Let (X, ∗,0) beaBCI-algebra.

31.F37-algebra

32.F38-algebra

33.F40-algebra

34.F41-algebra

35.F43-algebra

36.F44-algebra

37.F45-algebra

38.F47-algebra

39.F48-algebra

40.F49-algebra

41.F50-algebra

42.F51-algebra

43.F53-algebra

44.F57-algebra

45.F58-algebra

46.F60-algebra

1.LetXbeanF3-algebra.Xisassociativeifandonlyifx(x ∗ zx)= xzifandonlyifXisp-semisimple.

2.LetXbeanF5-algebra.Xisassociativeifandonlyif (xy ∗ x)x = yx.

3.LetXbeanF21-algebra.Xisassociativeifandonlyif (yx ∗ x)x = x ∗ y.

4.LetXbeanF42-algebra.XisassociativeifandonlyifXisp-semisimple.

Florentin Smarandache (author and editor) Collected Papers, IX 431

5.LetXbeanF55-algebra.Xisassociativeifandonlyif [(y ∗ x) ∗ x] ∗ x = x ∗ y.

6.(a)XisanF5-algebraandp-semisimpleifandonlyifXisassociative.

(b)LetXbeanF8-algebra.Xisassociativeifandonlyifx(y ∗ zx)= yz.

7.LetXbeanF19-algebra.Xisassociativeifandonlyifquasi-associative.

8.XisanF39-algebraandobeysy(x ∗ xz)= zyifandonlyifXisassociative.

9.LetXbeaF46-algebra.Xisassociativeifandonlyif 0(0 ∗ 0x)= x.

10.(a)XisanF52-algebraandF55-algebraifandonlyifXisassociative.

(b)XisanF52-algebraandobeys (y ∗ zx)x = zyifandonlyifXisassociative.

(c)XisanF55-algebraandp-semisimpleifandonlyifXisassociative.

(d)LetXbeanF52-algebra.XisassociativeifandonlyifXisquasi-associative.

11.(a)XisanF59-algebraandF55-algebraifandonlyifXisassociative. (b)XisanF52-algebraandobeys (y ∗ zx)x = zyifandonlyifXisassociative.

(c)LetXbeaF56-algebra.XisassociativeifandonlyifXisquasi-associative.

(d)LetXbeanF59-algebra.XisassociativeifandonlyifXisquasi-associative.

Proof.

1. Suppose X isa F3-algebra.Then, xy ∗ zx = x(y ∗ zx).Put y = x toget0 ∗ zx = x(x ∗ zx). Substituting x = 0,wehave0z = 0 ∗ 0z whichmeans X isquasi-associative.GoingbyTheorem 9, X isassociativeifandonlyif X is p-semisimple.Furthermore,byTheorem 4(3)and0 ∗ zx = x(x ∗ zx),an F3-algebra X isassociativeifandonlyif xy = x(x ∗ zx).

2. Suppose X isassociative.Then0 ∗ x = x. X is F5 implies (xy ∗ z)x =(x ∗ yz)x.With z = x, wehave (xy ∗ x)x =(x ∗ yx)x ⇒ (xy ∗ x)x =(x ∗ x)yx ⇒ (xy ∗ x)x = 0 ∗ yx ⇒ (xy ∗ x)x = yx asrequired.Conversely,suppose (xy ∗ x)x = yx.Put z = x in (xy ∗ z)x =(x ∗ yz)x toget (xy ∗ x)x =(x ∗ yx)x ⇒ (xy ∗ x)x =(x ∗ x)yx ⇒ (xy ∗ x)x = 0 ∗ yx ⇒ yx = 0 ∗ yx (since (xy ∗ x)x = yx).So, X isassociative.

3. Suppose X isassociative.Then x ∗ y = y ∗ x X is F21 implies yx ∗ zx =(yx ∗ z)x.With z = x, wehave (yx ∗ x)x = y ∗ x = x ∗ y asrequired.Conversely,suppose (yx ∗ x)x = x ∗ y.Put z = x in F21 toget (yx ∗ x)x = y ∗ x.So, x ∗ y = y ∗ x asrequired.

4. Suppose X isassociative.Then0 ∗ z = z X is F42 implies xx ∗ yz =(xx ∗ y)z.With y = 0, wehave 0 ∗ 0z = 0 ∗ z = z asrequired.Conversely,suppose0 ∗ 0z = z.Put y = 0in F42 toget 0 ∗ 0z = 0 ∗ z.So,0 ∗ z = z asrequired.

5. Suppose X isassociative.Then x ∗ y = y ∗ x X is F55 implies [(y ∗ z) ∗ x] ∗ x =[y ∗ (z ∗ x)] ∗ x With z = x,wehave [(y ∗ x) ∗ x] ∗ x = y ∗ x = x ∗ y asrequired.Conversely,suppose [(y ∗ x) ∗ x] ∗ x = x ∗ y.Put z = x in F55 toget y ∗ x =[(y ∗ x) ∗ x] ∗ x = x ∗ y.So, y ∗ x = x ∗ y asrequired. Theproofsof6to11followbyusingtheconcerned Fi and Fj identities(plusp-simplicitybyTheorem 12 insomecases)togetan Fk whichisequivalenttoassociativitybyTheorem 13 orwhichisnotequivalent toassociativityby1to5ofTheorem 14

3.Summary,ConclusionsandRecommendations

Inthiswork,wehavebeenabletoconstructexamplesofFenyves’BCI-algebras.Wehavealso obtainedthebasicalgebraicpropertiesofFenyves’BCI-algebras.Furthermore,wehavecategorized theFenyves’BCI-algebrasintoa46memberassociativeclass(ascapturedinTheorem 13).Members ofthisclassinclude F1, F2, F4, F6, F7, F9, F10, F11, F12, F13, F14, F15, F16, F17, F18, F20, F22, F23, F24, F25, F26, F27, F28, F30, F31, F32, F33, F34, F35, F36, F37, F38, F40, F41, F43, F44, F45, F47, F48, F49, F50, F51, F53, F57, F58, F60-algebras;anda14membernon-associativeclass.ThoseFenyvesidentitiesthatareequivalentto associativityinBCI-algebrasaredenotedby inthefifthcolumnofTable 1.Forthosethatbelong tothenon-associativeclass,wehavebeenabletoobtainconditionsunderwhichtheywouldbe associative(asreflectedinTheorem 14).Thisclassincludes F3, F5, F8, F19, F21,F29, F39 , F42, F46, F52, F54, F55, F56, F59-algebras.InTable 1 whichsummarizestheresults,membersofthisclassareidentifiedby thesymbol‘‡’.

OtherresearcherswhohavestudiedFenyves’identitiesontheplatformofloops,namelyPhillips andVojtechovsky[5],Jaiyeola[6],KinyonandKunen(2004)foundMoufang(F2, F4, F17, F27),extra

Florentin Smarandache (author and editor) Collected Papers, IX 432

(F6, F13, F22), F9, F15,leftBol(F19),rightBol(F26),Moufang(F4, F27), F30, F35, F36,C(F37), F38, F39, F40, LC(F39, F41, F46, F48), F42, F43, F45, F51,RC(F36, F53, F56, F57), F54,and F60 Fenyves’identitiesnottobe equivalenttoassociativityinloops.Interestingly,inourstudy,someoftheseidentities,particularly theextraidentity(F6, F13, F22), F7, F9, F15, F17,rightBol(F26),Moufang(F4, F27), F30, F35, F38, F40, RC(F36, F53, F57),C(F37),LC(F41, F48), F43, F45, F51 and F60 havebeenfoundtobeequivalentto associativityinBCI-algebras.Inaddition,theaforementionedresearchersfound F1, F3, F5, F7, F8, F10, F11, F12, F14, F16, F18, F20, F21, F23, F24, F25, F28, F29, F31, F32, F33, F34, F44, F47, F49, F50, F52, F55, F58 and F59 identitiestobeequivalenttoassociativityinloops.Wehavealsofoundsome(F7, F10, F11, F12, F14, F16, F18,F20, F23, F24, F25, F28, F31, F32, F33, F44, F47, F49, F50, F58)oftheseidentitiesto beequivalenttoassociativityinBCI-algebraswhilesomeothers(F3, F5, F8, F20, F21, F29,F55, F59) werenotequivalenttoassociativityinBCI-algebras.

Inlooptheory,itiswellknownthat:

• AloopisanextraloopifandonlyiftheloopisbothaMoufangloopandaC-loop.

• AloopisaMoufangloopifandonlyiftheloopisbotharightBolloopandaleftBol-loop.

• AloopisaC-loopifandonlyiftheloopisbothaRC-loopandaLC-loop.

Inthiswork,wehavebeenabletoestablish(asstatedbelow)somewhatsimilarresultsforafew oftheFenyves’identitiesinaBCI-algebra X:

• X isan Fi-algebraand Fj-algebraifandonlyif X isassociative,forthepairs: i = 52, j = 55, i = 59, j = 55.

Fenyves[31],andPhillipsandVojtˇechovský[32,33]foundsomeofthe60 Fi identitiestobe equivalenttoassociativityinquasigroupsandloops(i.e.,groups),andotherstodescribeweak associativelawssuchasextra,Bol,Moufang,central,ﬂexiblelawsinquasigroupsandloops.Their resultsaresummarisedinthesecond,thirdandfourthcolumnsofTable 1 withtheuseof .Inthis paper,wewentfurthertoestablishthat46Fenyves’identitiesareequivalenttoassociativityin BCI-algebraswhile14Fenyves’identitiesarenotequivalenttoassociativityinBCI-algebras.These twocategoriesaredenotedby and‡intheﬁfthcolumnofTable 1

Aftertheworksof[31 33],theauthorsin[34 38]didanextensionbyinvestigatingandclassifying variousgeneralizedformsoftheidentitiesofBol-Moufangtypesinquasigroupsandonesided/two sidedloopsintoassociativeandnon-associativecategories.Thisansweredaquestionoriginallyposed in[39]andalsoledtothestudyofoneofthenewlydiscoveredgeneralizedBol-Moufangtypesofloop inJaiyéo . láetal.[40].WhilealltheearliermentionedresearchworksonBol-Moufangtypeidentities focusedonquasigroupsandloop,thispaperfocusedonthestudyofBol-Moufangtypeidentities (Fenyves’identities)inspecialtypesofgroupoids(BCI-algebraandquasineutrosophictripletloops) whicharenotnecessarilyquasigroupsorloops(asprovedinTheorem 12).Examplesofsuchwell knownvarietiesofgroupoidswereconstructedbyIlojideetal.[41],e.g.,Abel-Grassmann’sgroupoid. TheresultsofthisworkareaninitiationintothestudyoftheclassificationoffiniteFenyves’quasi neutrosophictripletloops(FQNTLs)justlikevarioustypesoffiniteloopshavebeenclassified(e.g., Bolloops,MoufangloopsandFRUTEloops).Infact,alibraryoffiniteMoufangloopsofsmallorderis availableintheGAPS-LOOPSpackage[42].ItwillbeintriguingtohavesuchalibraryofFQNTLs. Overall,thisresearchwork(especiallyforthenon-associative Fi’s)hasopenedanewareaof researchfindingsinBCI-algebrasandBol-Moufangtypequasineutrosophictripletloopsasshownin Figure 1

Florentin Smarandache (author and editor) Collected Papers, IX 433

Table1. CharacterizationofFenyvesIdentitiesinQuasigroups,LoopsandBCI-AlgebrasbyAssociativity.

Fenyves Fi ≡ ASSFi ≡ ASS Quassigroup Fi + BCI IdentityInaloopInaloop ⇒ Loop ⇒ ASS F1 F2 F3 ‡ F4 F5 ‡ F6 F7 F8 ‡ F9 F10 F11 F12 F13 F14 F15 F16 F17 F18 F19 ‡ F20 F21 ‡ F22 F23 F24 F25 F26 F27 F28 F29 ‡ F30 F31 F32 F33 F34 F35 F36 F37 F38

Florentin Smarandache (author and editor) Collected Papers, IX 434

Table1. Cont.

Fenyves Fi ≡ ASSFi ≡ ASS Quassigroup Fi + BCI IdentityInaloopInaloop ⇒ Loop ⇒ ASS F39 ‡ F40 F41 F42 ‡ F43 F44 F45 F46 ‡ F47 F48 F49 F50 F51 F52 ‡ F53 F54 ‡ F55 ‡ F56 ‡ F57 F58 F59 ‡ F60

Figure1. NewCycleofAlgebraicStructures.

Florentin Smarandache (author and editor) Collected Papers, IX 435

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10.Burn,R.P.FiniteBolloops. Math.Proc.Camb.Phil.Soc. 1978, 84,377–385.[CrossRef]

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13.Kinyon,M.K.;Kunen,K.Thestructureofextraloops. QuasigroupsRelat.Syst. 2004, 12,39–60.

14.Hwang,Y.S.;Ahn,S.S.Soft q-idealsofsoftBCI-algebras. J.Comput.Anal.Appl. 2014, 16,571–582.

15.Iseki,K.OnBCK-Algebraswithcondition(S). Math.Semin.Note 1977, 5,215–222.

16.Lee,K.J.AnewkindofderivationsinBCI-algebras. Appl.Math.Sci. 2013, 7,81–84.[CrossRef]

17.Walendziak,A.Pseudo-BCH-Algebras. DiscussionesMath.Gen.Alg.Appl. 2015, 35,5–19.[CrossRef]

18. Zhang,X.;Wu,X.;Smarandache,F.;Hu,M.Left(Right)-QuasiNeutrosophicTripletLoops(Groups)and GeneralizedBE-Algebras. Symmetry 2018, 10,241.[CrossRef]

19.Smarandache,F.;Ali,M.Neutrosophictripletgroup. NeuralComput.Appl. 2018, 29,595–601.[CrossRef]

20. Zhang,X.;Smarandache,F.;Liang,X.NeutrosophicDupletSemi-GroupandCancellableNeutrosophicTriplet Groups. Symmetry 2017, 9,275.[CrossRef]

21. Zhang,X.;Hu,Q.;Smarandache,F.;An,X.OnNeutrosophicTripletGroups:BasicProperties,NT-Subgroups, andSomeNotes. Symmetry 2018, 10,289.[CrossRef]

22. Jaiyéolá,T.G.;Smarandache,F.InversePropertiesinNeutrosophicTripletLoopandtheirApplicationto Cryptography. Algorithms 2018, 11,32.[CrossRef] 23. Jaiyéolá,T.G.;Smarandache,F.SomeResultsonNeutrosophicTripletGroupandTheirApplications. Symmetry 2017, 10,202.[CrossRef] 24. Smarandache,F. AUnifyingFieldInLogics:NeutrosophicLogic.Neutrosophy,NeutrosophicSet,Neutrosophic ProbabilityAndStatistics;InfoLearnQuest:AnnArbor,MI,USA,2007. 25.Zadeh,L.A.Fuzzysets. Inform.Control. 1965, 28,338–353.[CrossRef]

26.Atanassov,K.Intuitionisticfuzzysets. FuzzySetsSyst 1986, 20,87–96.[CrossRef] 27. Ejegwa,P.A.;Akowe,S.O.;Otene,P.M.;Ikyule,J.M.AnOverviewOnIntuitionisticFuzzySets. Int.J.Sci. Technol.Res. 2014, 3,142–145.

28.Atanassov,K.T.Type-1FuzzySetsandIntuitionisticFuzzySets. Algorithms 2017, 10,106.[CrossRef]

29. Shao,S.;Zhang,X.;Bo,C.;Smarandache,F.NeutrosophicHesitantFuzzySubalgebrasandFiltersin Pseudo-BCIAlgebras. Symmetry 2018, 10,174.[CrossRef]

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31.Fenyves,F.ExtraloopsII. Publ.Math.Debrecen 1969, 16,187–192.

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32. Phillips,J.D.;Vojtecovsky,P.ThevarietiesofloopsofBol-Moufangtype. Alg.Univ. 2005, 54,259–271. [CrossRef]

33. Phillips,J.D.;Vojtecovsky,P.ThevarietiesofquasigroupsofBol-Moufangtype:Anequationalreasoning approach. J.Alg. 2005, 293,17–33.[CrossRef] 34. Cote,B.;Harvill,B.;Huhn,M.;Kirchman,A.ClassiﬁcationofloopsofgeneralizedBol-Moufangtype. QuasigroupsRelat.Syst. 2011, 19,193–206.

35. Akhtar,R.;Arp,A.;Kaminski,M.;VanExel,J.;Vernon,D.;Washington,C.ThevarietiesofBol-Moufang quasigroupsdeﬁnedbyasingleoperation. QuasigroupsRelat.Syst. 2012, 20,1–10.

36. Hernandez,B.;Morey,L.;Velek,A.GeneralizedBol-MoufangIdentitiesofLoopsandQuasigroups. Unpublished.Availableonline: www.units.miamioh.edu/sumsri/sumj/2012/algebra_hmv.pdf (accessedon 16August2012). 37. Hoganson,H.;Tapia,M.IdentitiesinQuasigroupsandLoops.Unpublished.Availableonline: www.units. miamioh.edu/sumsri/sumj/2012/algebra_ht.pdf (accessedon16August2012). 38. Aldrich,R.R.;Drummond,S.J.TheVarietiesofOne-SidedLoopsofBol-MoufangType.Unpublished. Availableonline: www.units.miamioh.edu/sumsri/sumj/2012/algebra_da.pdf (accessedon 16August2012).

39. Drapal,A.;Jedlicka,P.Onloopidentitiesthatcanbeobtainedbyanuclearidentiﬁcation. EuropeanJ.Combin. 2010, 31,1907–1923.[CrossRef]

40. Jaiyéolá,T.G.;Adeniregun,A.A.;Asiru,M.A.FiniteFRUTEloops. J.AlgebraAppl. 2017, 16,1750040.[CrossRef]

41. Ilojide,E.;Jaiyéolá,T.G.;Owojori,O.O.Varietiesofgroupoidsandquasigroupsgeneratedbylinear-bivariate polynomialsoverring Zn Int.J.Math.Comb. 2011, 2,79–97.

42. Nagy,G.P.;Vojtechovsky,P.LOOPS:APackageforGAP4.2017.Availableonline: http://www.math.du. edu/loops (accessedon27October2017).

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New Soft Set Based Class of Linear Algebraic Codes

Mumtaz Ali, Huma Khan, Le Hoang Son, Florentin Smarandache, W. B. Vasantha Kandasamy

Mumtaz Ali, Huma Khan, Le Hoang Son, Florentin Smarandache, W. B. Vasantha Kandasamy (2018). New Soft Set Based Class of Linear Algebraic Codes. Symmetry 10, 510; DOI: 10.3390/sym10100510

Abstract: In this paper, we design and develop a new class of linear algebraic codes defined as soft linear algebraic codes using soft sets. The advantage of using these codes is that they have the ability to transmit m-distinct messages to m-set of receivers simultaneously. The methods of generating and decoding these new classes of soft linear algebraic codes have been developed. The notion of soft canonical generator matrix, soft canonical parity check matrix, and soft syndrome are defined to aid in construction and decoding of these codes. Error detection and correction of these codes are developed and illustrated by an example.

Keywords: linear algebraic code; soft set theory; soft linear algebraic code; soft communication; soft syndrome; soft codewords; soft generator matrix

1.Introduction

Shannon[1,2]publishedanhistoricpaperthatmarkedthebeginningofbotherrorcorrecting codesandinformationtheory.Sincethen,severalresearchershavedevelopedanddesignedcodes likeBCHcodes[3,4],self-dualcodes[5],maximumdistancecodes[6],Hammingdistanceoflinear codes[7],andcodesoverZm [8,9].Howeverfuzzycodesanddistancepropertieswasdeveloped by[10].Forliteratureusedinthispaperoncodingtheory,seeReference[11].

Inthispaper,wedeﬁnesoftlinearcodesusingsoftsets.Softsets[12]aregeneralizationoffuzzy setsintroducedin[13].FuzzysetsworkonmembershipdegreewhoserangevariesfromReference [0,1]andsoftsetsdealwithuncertaintyinaparametricway.Thus,asoftsetisaparameterizedfamily ofsetsandtheboundaryofthesetdependsontheparameters.Sincethen,softsets[14]havebeen developedtoneutrosophicsoftsets[15],softneutrosophicgroups[16],softneutrosophicalgebraic structures,andtheirgeneralization[17 20].Relationshipamongsoftsetsandfuzzysetswasstudiedin Reference[20,21].Here,fortheﬁrsttime,softsettheoryhasbeenusedintheconstructionofalgebraic codes,whichwechoosetocallassoftalgebraiclinearcodes.

Thispaperisorganizedintosixsections.Section 1 isintroductoryinnature.Allbasicconcepts tomakethispaperaself-containedonearegiveninSection 2.Section 3 introducesthenewnotion ofalgebraicsoftcodesanddeﬁnesanddescribessomerelatedpropertiesofthem.Softparitycheck matrixandsoftgeneratormatrixareintroducedinSection 3.Section 4 describesdecoding,error detectionanderrorcorrectionofthesoftlinearalgebraiccodes.Section 5 givesthesoftcommunication

Collected Papers, IX 438

Florentin

Smarandache (author and editor)

modelandbringsoutthedifferencebetweenthelinearalgebraiccodesandsoftlinearalgebraiccodes. Section 6 givestheconclusionsbasedonourstudyandprobablefutureresearchforanyresearcher.

2.FundamentalNotions

Inthissectionthebasicconceptsneededtomakethispaperaself-containedoneisgiven. Thissectionisdividedintotwosubsections.Section 2.1 describesthebasicconceptsaboutthe linearalgebraiccodesandtheirrelatedpropertiesandSection 2.2 givesthedeﬁnitionandafew propertiesofsoftsets.

2.1.AlgebraicLinearCodesandTheirProperties

Allthebasicconcepts,definitionandpropertiesofalgebraiclinearcodesaretakenfrom Reference[11].Thefundamentalalgebraicstructureusedinthedefinitionoflinearalgebraiccodes arevectorspacesandvectorsubspacesdefinedoverafinitefield F.Throughoutthispaper,weonly considerthefinitefield Z2 ={0,1},thefinitefieldofcharacteristictwo.Weuse F todenote Z2

Definition1. LetVbeasetofelementsonwhichabinaryoperationcalledaddition,‘+’isdefined.LetFbea field.Anoperationproductormultiplication,denotedby‘.’,betweentheelementsinFandtheelementsinVis defined.ThesetViscalledavectorspaceoverthefieldFifitsatisfiesthefollowingconditions:

1 Visacommutativegroupunderaddition.

2. ForanyelementainFandanyelementvinV,a.v=v.aisinV.

3. Distributivelaw:ForanyuandvinVandforanya,b ∈ F a.(u+v)=a.u+a.v;(a+b).v=a.v+b.v.

4. Associativelaw:ForanyvinVandanyaandbinF;(a.b).v=a.(b.v).

5 Let1betheunitelementofF.ThenforanyvinV,1.v=vand0.v=0for0 ∈ Fand‘0’isthezerovectorof V.WecallapropersubsetUofV(U ⊂ V)tobeavectorsubspaceofVoverFifUitselfisavectorspace overF.

Deﬁnition2. Ablockcodeoflengthnwith2k codewordsiscalledalinearcode,denotedbyC(n,k),ifandonly ifits2k codewordsformak-dimensionalsubspaceofthevectorspaceVn ofallthentuplesovertheﬁeldGF(2). ThemethodforgeneratingtheseC(n,k)codesusingthegeneratormatrixGisasfollows.Gisgiveninthefollowing:

gi,j ∈ Z2 =F;for0 ≤ i ≤ k 1and0 ≤ j ≤ n 1.Consideru=(u0 u1 ... uk 1),themessagetobeencoded, thecorrespondingcodewordvisgivenbyv=u.G.EverycodewordvinC(n,k)isalinearcombinationofkcodewords.

TheerrordetectionanderrorcorrectionofthesecodesisgiveninReference[11].Ifthegenerator matrix G inthestandardformis G =(A; Ik × k),thenparitycheckmatrix H canbegotinthestandard formas H =(In k × n k; AT) Thegeneratormatrixcanbeinanyotherform,andthentheparitycheck matrixcanbefoundoutbytheusualmethodsgiveninReference[11].

Thesyndromeofthereceivedcodeword y,denotedby s(y) =yHT isobtainedfromtheparitycheck matrix H.Thus,theparitycheckmatrix H ofacodehelpstodetecttheerrorfromthereceivedword. Theerrorcorrectingcapacityofacodedependsonthemetricthatisusedoverthecode.Themost basicmetric,namelytheHammingmetricofthecodeisdeﬁnedasfollows:

G =       g00

. . gk

     

g01 g02 g0,n 1

10 g11 g12 g1,n 1

1,0 gk 1,1 gk 1,2 gk 1,n 1

Florentin Smarandache (author and editor) Collected Papers, IX 439

Definition3. Foranytwovectorsx=(x1 ... xn)andy=(y1 ... yn)inVn,thendimensionalvectorspace overthefieldF=Z2,theHammingdistanced(x,y)andtheHammingweightw(x)aredefinedasfollows:

d(x,y)=|{xi:xi = yi;xi ∈ x;yi ∈ y}| w(x)=|{xi:xi = 0;xi ∈ x}|.

Deﬁnition4. Theminimumdistancedmin ofacodeC(n,k)isdeﬁnedas dmin = min x, y ∈ C x = y

d(x, y)

Thecosetleadermethodusedforerrorcorrection,makesuseofthestandardarrayforsyndrome decodingasdescribedinReference[11].

2.2.SoftSetTheory

ThesoftsettheorywhichisageneralizationoffuzzysettheorywasproposedbyReference[12]. Whilethispart X concernstoaninceptivedomain, P(X)isthepowersetof X, V iscalledasetof parameters,or D ⊂ V.ThesoftsettheorydeﬁnedbyReference[12]isgivenbelow.

Deﬁnition5. Theset(f,D)issaidtobeasoftsetofXwhereamappingoffisgivenbyf:D → P(X). Inotherwords,asoftsetoverXisaparameterizedfamilyofsubsetsoftheuniverseX.Ford ∈ D,f(d)can beconsideredasthesetofd-elementsofthesoftset(f,D),orasthesetofd-approximateelementsofthesoftset. Let(f,D)and(g,E)betwosoftsetsoverX,(f,D)iscalledasoftsubsetof(g,E)ifD ⊆ Eandf(s) ⊆ g(s), foralls ∈ D.Thisrelationshipisdenotedby(f,D) ⊂ (g,E).Similarly,(f,D)iscalledasoftsupersetof(g,E) if(g,E)isasoftsubsetof(f,D)whichisdenotedby(f,D) ⊃ (g,E).If(f,D) ⊆ (g,E)and(g,E) ⊆ (f,D), thetwosoftsetsaresaidtobeequal.

3.AlgebraicSoftLinearCodesandTheirProperties

Inthissectiontheconceptofsoftlinearcodeandalgebraicsoftlinearcodeoftype1areproposed andnotionofsoftgeneratormatrixandsoftparitycheckmatrixareintroduced.

Definition6. LetF=Z2;bethefieldofcharacteristictwo.LetW=F × ... × F=Fm,beavectorspaceover thefieldFofdimensionm.P(W)bethepowersetofW.(f,D)issaidtobeasoftalgebraiclinearcodeoverFif andonlyiff(d)isalinearalgebraiccodeofWforalld ∈ D;D ⊂ V,whereVisthesetofparameters.

Itistobenotedthatnotallvectorsubspacesof W,formsalinearalgebraiccode.Further,thesoft algebraiclinearcodedoesnotingeneralincludealllinearalgebraiccodesof W

Example1. LetW=F3 beavectorspaceovertheﬁeldF.(f,D)isasoftlinearcodeoverWwheref(D)={f(d1), f(d2)}with f(d1)={000,111}andf(d2)={000,110,101,011}.

Clearly{000,111}and{000,110,101,011}arelinearalgebraiccodes.{{000,000},{000,110},{000,101}, {000,011},{111,000},{111,110},{111,101},and{111,011}}isthesetofsoftcodewordsof(f,D).Thereare8 softcodewordsforthesoftcode(f,D).

Inviewofthisexamplewedeﬁnesoftcodewordasfollows:

Florentin Smarandache (author and editor) Collected Papers, IX 440

Definition7. LetW=F × ... × F=Fm,beavectorspaceoverthefieldFofdimensionm.P(W)bethepower setofW.(f,D)beasoftalgebraiclinearcodeoverF.Letf(D)={f(d1), ... ,f(dt)}whereeachf(di);1 ≤ i ≤ tisa linearalgebraiccodeofW.Eacht-tuple{x1,x2, ... ,xt};xi ∈ f(di);1 ≤ i ≤ tisdefinedasthesoftcodewordofthe softalgebraiccode(f,D).Wehave|f(d1)| × |f(d2)|× ... × |f(dt)|numberofsoftcodewordsforthis(f,D).

Intheaboveexample,thesoftdimension(f, D)={1,2},thatisthenumberoflinearlyindependent codewordsofthelinearalgebraiccodeassociatedwith f (d1)and f (d2),respectively. Wehavethefollowingdeﬁnitioninviewofthis.

Deﬁnition8. LetW=Fm,beavectorspaceovertheﬁeldFofdimensionm.(f,D)beasoftalgebraiclinear codeoverF.Letf(D)={f(d1), ,f(dt)}whereeachf(di);1 ≤ i ≤ tisalinearalgebraiccodeofW.Hereeach f(di) ∈ f(D)isalinearalgebraiccodeanddimensionoff(di)isni whereni isthenumberoflinearindependent elementsoff(di).Thesoftdimensionof(f,D)={n1, ... ,nt}andthenumberofsoftcodewordsof(f,D)is|f(d1)| × |f(d2)| × ... × |f(dt)|where1 ≤ i ≤ t.

Deﬁnition9. Let(f,D)bethesameasinaboveDeﬁnition8.(f,D)iscalledsoftcodeoftype1,ifthedimension of(f,D)={n1,n2, ,nt}issuchthatn1 =n2 = =nt

Inthefollowingwegiveanexampleofsoftcodeoftype1.

Example2. Let(f,D)beasoftcodeinW=F × × F=F5 overtheﬁeldF.Consider

f(d1)={00000,11111,10110,01001}, f(d2)={00000,11111,11001,00110}, f(d3)={00000,11111,00111,11000},and f(d4)={00000,11111,11100,00011}.

Thesoftdimensionof(f,D)is{2,2,2,2}.Hence(f,D)isasoftcodeoftype1.

Theorem1. Everysoftalgebraiclinearcodeoftype1istriviallyasoftalgebraiclinearcodebuttheconverseis nottrue.

Proof. Theresultfollowsfromthedeﬁnitionofsoftcodeoftype1.Fortheconverse,resultfollows fromExample1,wherethedimensionsof f (d1)and f (d2)aredifferent.

Nowweproceedontodeﬁnethesoftgeneratormatrixforsoftlinearalgebraiccode(f, D).

Definition10. Let(f,D)beasoftlinearalgebraiccodeasinDefinition8,wheref(D)={f(d1), ... ,f(dt)}. Weknowthatassociatedwitheachf(di)wehaveanalgebraiccodeofdimensionni.LetGi;1 ≤ i ≤ tbethe generatormatrixassociatedwiththisalgebraiccodeassociatedwithf(di).Thenwedefinethesoftgenerator matrixGs asthet-matrixgivenbyGs =[G1|G2| ... |Gt].IftheeachgeneratormatrixGi ofthesoftgenerator matrixGs isrepresentedinthestandardformthenthesoftgeneratormatrixGs isknownassoftcanonical generatormatrixandisdenotedbyGs*.

Example3. Thesoftgeneratormatrixofthesoftlinearcodeoftype1giveninExample2isasfollows:

whereGi isthegeneratormatrixofthealgebraiccodeassociatedwithf(di);i=1,2,3,4;clearlythisGS isnotthe softcanonicalgeneratormatrix.

Gs = [G1| G2|G3|G4] = 101 10 01001 11001 00110 00111 11000 11100 00011

Florentin Smarandache (author and editor) Collected Papers, IX 441

The following example gives the soft canonical generator matrix for the soft linear code.

Example 4. Suppose (f, D) be a soft code over W = F5 , where f(d1)={00000,10010,01001,00110,11011,10100,01111,11101}; f(d2)={00000,11111,10110,01001}

arealgebraiclinearcodeswithstandardgeneratormatrixG1 andG2 where G1 = 10110 01001 , G2 =

Thesoftcanonicalgeneratorbi-matrixof(f,D)is:

Nowweproceedontotodeﬁnesoftparitycheckmatrixandsoftcanonicalparitycheckmatrix forasoftlinearalgebraiccode.

Deﬁnition11. Consider(f,D)asinDeﬁnition8.Letf(D)={f(d1), ... ,f(dt)}whereeachf(di)islinearalgebraic code,letHi (1 ≤ i ≤ t)betheparitycheckmatrixassociatedwitheachlinearalgebraiccode.ThenHS ={H1|H2| ... |Ht}isthesoftparitycheckmatrixassociatedwiththesoftlinearalgebraiccode.

IfeachHi istakeninthestandardformthenthecorrespondingsoftparitycheckmatrix H∗ s isdeﬁnedasthe softcanonicalparitycheckmatrixofthesoftalgebraiclinearcode.

Now,inthefollowingsection,wegiveamethodtodeterminesofterrorsinreceivedcodewords andhowthesofterrorcorrectionsarecarriedout.

4.SoftLinearAlgebraicDecodingAlgorithms

Duringtransmissionoveranymedium,thetransmittedcodewordcangetcorruptedwitherrors. Theprocessofidentifyingtheseerrorsfromthereceivedcodewordisknownaserrordetectionandthe processofcorrectingtheerrorsandobtainingthecorrectcodewordisknownaserrorcorrection.Inthis section,weintroducethenotionthesoftdecodingalgorithm,errordetection,anderrorcorrectionfor softlinearalgebraiccodes.Themethodofsoftsyndromedecodingisproposed.

First,weproceedontodeﬁnethenotionofcosetandsoftcosetleader.Thedeﬁnitionofcosetand cosetleaderforanylinearalgebraiccodecanbehadfromReference[11].

Wenowdeﬁnethecosetleadersaselementsineachofthecosetswiththeleastweight.Forany code, C=C(n, k)isasfollowsasthealgebraiccodeisasubspaceof W soisasubgroupof W

 

  



10010 01001 00110

∗ s =   

  

     

10110 01001

10010 01001 00110

e0 = 0 x1,..., xm s = 0 e1 e1 + x1,..., e1 + xm e1 HT e2 e2 + x1,..., e2 + xm e2 HT . . . . . . ep ep + x1,..., ep + xm ep

Florentin Smarandache (author and editor) Collected Papers, IX 442

CosetLeadersCodewordsascosetsofCSyndromes

HT where ei’sarecosetleaders.Syndromeof

i, s(ei)= eiHT;0 ≤ i ≤ t.

H istheparitycheckmatrixofthelinearalgebraiccode C.Thecosetleadermethodisusedfor errorcorrectionbymakinguseofthestandardarrayforsyndromedecoding[11].

Definition12. Let(f,D)bethesoftlinearcodeasgiveninDefinition8.LetHs =(H1|H2| |Ht)bethe softparitycheckmatrixof(f,D).Supposeyisthereceivedsoftmessage,thesoftsyndromeofyisdefinedass(y) =yHT s ;ifs(y) = (o)thenwesaythesoftcodewordhassofterror.

Now,weproceedontoanalogouslydescribethesyndromedecodingmethodforsoftlinear algebraiccodes.

Let W=Fm beavectorspaceofdimension n over F=Z2.Let(f, D)beasoftalgebraiccodewith f (D)=(f (d1), , f (dt))whereeach f (di);1 ≤ i ≤ t;isalinearalgebraiccodeover W.Anysoftcodeword in(f, D)willbeoftheform x= (x1, x2, ... , xt)where xi ∈ f (di)and xi = yi 1,..., yi m a m-tupleforwhich itwillhave ki messagesymbols;1 ≤ i ≤ t.

If z isareceivedmessagewehavetofirstfindoutif z hasanyerrorandifzhaserrorwehaveto correctit.Nowtocheckforerrorwefindthesoftsyndrome s(z)=zHT = z HT 1 HT 2 ... HT t where each Hi istheparitycheckmatrixofthelinearalgebraiccodeassociatedwith f (di);1 ≤ i ≤ t.

If s(z) = (0)wehaveanerror.Thiserrorisdefinedasthesofterrorand s(z)isdefinedassoft syndromeofthesoftcodeword z received.Thisprocedureoffindingoutwhetherthereceivedsoft codewordiscorrectornot;itistermedassofterrordetection.

Now,weproceedontocorrectthesofterroras s(z) = (0);somesofterrorhasoccurredduring transmission.Wecanbuildananalogoustableforerrorcorrectionorstandardarrayforsoftsyndrome decoding.Softcosetleadersinthecaseofsoftcodeswillbecarriedoutinananalogousway,which willbedescribedbyanexample.

Example5. Let(f,D)beasoftcodedeﬁnedinExample1.Thesoftparitycheckmatrixof(f,D)be

Andthesoftcosetleadersof(f,D)are

e0 ={000,000}, e1 ={100,100}, e2 ={010,010}and e3 ={001,001}

TheTable 1 ofsoftsyndromedecodingisasfollows:

Table1. SoftSyndromeCoding.

SoftCosetLeadersSoftCodewordsasCosetsof(f, D)SoftSyndromes

e0 ={000,000}

{000,000},(000,110},{000,101},{000,011}, {111,000},{111,110},{111,101},{111,011}

e1HT ={10,1}

e0HT ={00,0} e1 ={100,100} {100,100},(100,010},{100,001},{100,111}, {011,100},{111,010},{011,001},{011,111}

e2 ={010,010} {010,010},(010,100},{010,111},{010,001}, {101,010},{101,100},{101,111},{101,001}

e3HT ={01,1}

e2HT ={11,1} e3 ={001,001} {001,001},(001,111},{001,100},{001,010}, {110,001},{110,111},{110,100},{110,010}

=       10 11 01       1 1 1      

H = [H1| H2] = 110 011 111 . ThetransposeofHisasfollows, HT

Florentin Smarandache (author and editor) Collected Papers, IX 443

Theorem2. Suppose(f,D)beasoftlinearalgebraiccodeoveraﬁeldF,giveninDeﬁnition8,anyelement receivedcodeword,whichhassomeerrory=(y1, ... ,yt);yi ∈ W=Fm;1 ≤ i ≤ t;thenthereisasoftcodeword nearesttoygivenbyx=y+softcosetleaderei ofthesoftcode(f,D).

Proof. Let(f, D)beasoftlinearalgebraiccodeoverafield F with H asthesoftparitycheckmatrix. Let y= (y1,..., yt)bethereceivedcodeword,wefindthesoftsyndrome; s(y) = yHT = (y1 yt ) HT 1 HT 2 HT t where s(y)=(0)impliesthatthereisnoerror,so y isthecorrectcodeword.If s(y) = (0),thenwework asfollows:First,wefindallthesoftlinearalgebraiccosetofthesoftlinearalgebraiccode(f, D)for softset-basedsyndromedecoding,andthenfindtheappropriatesoftlinearalgebraiccosetleaders ei fromthecollectionofcosetleadersusingtheoneanaloguesTable 1.Then,forallsoftcosetleaders wecalculatethesoftset-basedsyndromeandmakeatableofsoftlinearalgebraiccosetleaderswith theirsoftset-basedsyndromes.Fordecodingasoftlinearalgebraiccodeword y,wecanmerelyfind thesoftset-basedsyndromeofthesoftlinearalgebraiccodewordandthencomparesoftcosetleader syndromewiththeirsoftset-basedsyndrome.Afterthecomparison,weaddthesoftdecodedwordto thesoftlinearalgebraiccosetleader.Thus, y issoftdecodedas x=y+ei; ei isthesoftcosetleaderand xisthecorrectedword.

5.SoftSet-BasedCommunicationTransmissionandComparisonofSoftLinearAlgebraicCodes andLinearAlgebraicCodes

Inthissection,weproposeasoftset-basedcommunicationtransmission.Thefollowingproposed modelcomprisesofasoftlinearalgebraicencoderthatisanapproximatedcollectionofencoders. Hence,if(f, D)isasoftcode;in D correspondingtoeachparameter d,inthesoftencoderwehavean encoder.Moreover,wehaveasoftlinearalgebraicdecoderthatisthecollectionofdecoders;hence, toeachparameterin D,wehaveadecoderinthesoftlinearalgebraicdecoder.Inparameterset A= (a1, , am),thereare m parameters.Asoftset-basedcommunicationtransmissionreducesto classicalcommunicationtransmissionifwehave m =1.Themodelofsoftset-basedcommunication transmissionisgiveninthefollowingFigure 1

Florentin Smarandache (author and editor) Collected Papers, IX 444

Figure 1. Soft Communication transmission model.

The major difference among linear code and soft linear code is that for the soft linear code every soft code word has some attributes or concept, i.e., each soft code word is distinguished by some attributes, but the linear algebraic codes do not enjoy this property. Thus, one can work on the attributes of soft code words, for example an attribute “di ” can have some attribute that can trick the hackers. Therefore, the soft linear codes can be more secure as compared to the classical linear codes due to the parameterization. The soft linear codes have a different distinct structure. Soft linear code is a collection of subspaces, whereas a linear code is only one subspace. Each subspace relies on the set of parameters that are used. Hence, soft linear codes are more generalized in comparison to the linear codes.

Linear codes can transfer only one message to a receiver whereas soft linear codes can simultaneously transmit m-well deﬁned messages to m-set of receivers. The time taken for transmitting m-messages to m-receivers will take at least m unit of time in case of linear algebraic codes, whereas in case of soft algebraic codes the time taken will be only the time taken to transmit a single message, since the m-messages are transmitted simultaneously. The latest methodology makes use of bi-matrices and is more generalized uses with the perception of m-matrices. Clearly, this concept of soft algebraic code saves time. In soft decoding procedure, one can decode a set of code words (soft code word) at a time while it is not feasible in case of linear algebraic codewords decoding procedure.

6. Conclusions

There is an important role of algebraic codes in the minimization of data delinquency, which is generated by deficiencies, i.e., inference, noise channel, and crosstalk. In this paper, we have proposed the latest notions of soft linear algebraic codes for the first time by using the soft set. This latest class of codes can remit simultaneously m-messages to the m-people. Therefore, these new codes can save both time and economy. Soft parity check matrix (parity check m-matrix) and soft generator matrix (generator m-matrix) were defined. Decoding of soft linear codes was done using soft syndrome decoding techniques. The channel transmission is also illustrated. Finally, the major difference and comparison of soft linear codes with classical linear codes are presented.

Even though the proposed code has some advantages over the classical ones, it still has limitations in dealing with the multichannel coding problem, rank metrics, etc. Therefore, for future study, we wish to implement neutrosophic soft sets in algebraic linear codes. Further introduction of soft code with rank metric [22] and construction of T-direct soft codes [23] may be helpful to tackle the multichannel coding problem, which is left for researchers in coding theory. The general case based on N-soft sets and others [24 36] will be developed as well.

Smarandache (author and editor) Collected Papers, IX 445

Florentin

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Fatimah,F.;Rosadi,D.;Hakim,R.F.;Alcantud,J.C.R.N-softsetsandtheirdecisionmakingalgorithms. SoftComput. 2018, 22,3829–3842.[CrossRef]

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Dey,A.;Son,L.;Kumar,P.;Selvachandran,G.;Quek,S.NewConceptsonVertexandEdgeColoringof SimpleVagueGraphs. Symmetry 2018, 10,373.[CrossRef]

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28. Son,L.H.;Fujita,H.Neural-fuzzywithrepresentativesetsforpredictionofstudentperformance. Appl.Intell. 2018,1–16.[CrossRef] 29. Jha,S.;Kumar,R.;Chatterjee,J.M.;Khari,M.;Yadav,N.;Smarandache,F.Neutrosophicsoftsetdecision makingforstocktrendinganalysis. Evol.Syst. 2018,1–7.[CrossRef] 30.

Ngan,R.T.;Son,L.H.;Cuong,B.C.;Ali,M.H-maxdistancemeasureofintuitionisticfuzzysetsindecision making. Appl.SoftComput. 2018, 69,393–425.[CrossRef] 31. Ali,M.;Thanh,N.D.;VanMinh,N.Aneutrosophicrecommendersystemformedicaldiagnosisbasedon algebraicneutrosophicmeasures. Appl.SoftComput. 2018, 71,1054–1071.[CrossRef] 32.

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Extension of Soft Set to Hypersoft Set, and then to Plithogenic Hypersoft Set

Florentin Smarandache

Florentin Smarandache (2018). Extension of Soft Set to Hypersoft Set, and then to Plithogenic Hypersoft Set. Neutrosophic Sets and Systems, 22, 168-170

Abstract In this paper, we generalize the soft set to the hypersoft set by transforming the function F into a multi attribute function. Then we introduce the hybrids of Crisp, Fuzzy, Intuitionistic Fuzzy, Neutrosophic, and Plithogenic Hypersoft Set.

Keywords: Plithogeny; Plithogenic Set; Soft Set; Hypersoft Set; Plithogenic Hypersoft Set; Multi argument Function

1 Introduction

Wegeneralizethesoftsettothe hypersoftset bytransformingthefunction F intoamulti argumentfunction Then we make the distinction between the types of Universes of Discourse: crisp, fuzzy, intuitionistic fuzzy, neutrosophic, and respectively plithogenic Similarly,weshowthatahypersoftsetcanbecrisp,fuzzy,intuitionisticfuzzy,neutrosophic,orplithogenic. A detailed numerical example is presented for all types

2 Definition of Soft Set [1]

Let �� be a universe of discourse, ��(��) the power set of ��, and A a set of attributes. Then, the pair (F, ��), where ��:��⟶��(��) (1) is called a Soft Set over ��

3 Definition of Hypersoft Set

Let �� be a universe of discourse, ��(��) the power set of �� Let ��1,��2,…,��, for �� ≥1, be n distinct attributes, whose corresponding attribute values are respectively the sets ��1,��2, ,��, with �� ∩�� =∅, for �� ≠��, and ��,�� ∈{1,2, ,��} Then the pair (��,��1 ×��2 ×…×��), where: ��:��1 ×��2 × ×�� ⟶��(��) (2) is called a Hypersoft Set over ��

4 Particular case

For �� =2, we obtain the Γ Soft Set [2]

5 Types of Universes of Discourses

5.1. A Universe of Discourse �� is called Crisp if ∀�� ∈��, x belongs 100% to ��, or x’s membership (Tx) with respect to �� is 1. Let’s denote it x(1).

5.2. A Universe of Discourse �� is called Fuzzy if ∀�� ∈��, x partially belongs to ��, or �� ⊆[0,1], where �� may be a subset, an interval, a hesitant set, a single value, etc. Let’s denote it by ��(��)

5.3 A Universe of Discourse �� is called Intuitionistic Fuzzy if ∀�� ∈��, x partially belongs (��) and partially doesn’t belong (��) to ��, or ��,�� ⊆[0,1], where �� and �� may be subsets, intervals, hesitant sets, single values, etc. Let’s denote it by ��(��,��)

5.4. A Universe of Discourse �� is called Neutrosophic if ∀�� ∈��, x partially belongs (��), partially its membership is indeterminate (��), and partially it doesn’t belong (��) to ��, where ��,��,�� ⊆[0,1], may be subsets, intervals, hesitant sets, single values, etc. Let’s denote it by ��(��,��,��)

5.5. A Universe of Discourse �� over a set V of attributes’ values, where �� ={��1,��2, ,��},�� ≥1, is called Plithogenic, if ∀�� ∈��, x belongs to �� in the degree �� 0(��) with respect to the attribute value ��, for all

Florentin Smarandache (author and editor) Collected Papers, IX 448

�� ∈{1,2, ,��}. Since the degree of membership �� 0(��) may be crisp, fuzzy, intuitionistic fuzzy, or neutrosophic, the Plithogenic Universe of Discourse can be Crisp, Fuzzy, Intuitionistic Fuzzy, or respectively Neutrosophic. Consequently, a Hypersoft Set over a Crisp / Fuzzy / Intuitionistic Fuzzy / Neutrosophic / or Plithogenic Universe of Discourse is respectively called Crisp / Fuzzy / Intuitionistic Fuzzy / Neutrosophic / or Plithogenic Hypersoft Set

6 Numerical Example

Let �� ={��1,��2,��3,��4} and a set ℳ ={��1,��3}⊂��

Let the attributes be: ��1= size, ��2= color, ��3= gender, ��4= nationality, and their attributes’ values respectively:

Size = ��1 ={small, medium, tall}, Color = ��2 ={white, yellow, red, black}, Gender = ��3 ={male, female}, Nationality = ��4 ={American, French, Spanish, Italian, Chinese}.

Let the function be: ��:��1 ×��2 ×��3 ×��4 ⟶��(��). (3) Let’s assume: ��({tall,white,female,Italian})={��1,��3}.

With respect to the set ℳ, one has:

6.1 Crisp Hypersoft Set

��({tall,white,female,Italian})={��1(1),��3(1)}, (4) which means that, with respect to the attributes’ values {tall,white,female,Italian} all together, ��1 belongs 100% to the set ℳ; similarly ��3.

6.2 Fuzzy Hypersoft Set

��({tall,white,female,Italian})={��1(0.6),��3(0.7)}, (5) which means that, with respect to the attributes’ values {tall,white,female,Italian} all together, ��1 belongs 60% to the set ℳ; similarly, ��3 belongs 70% to the set ℳ

6.3 Intuitionistic Fuzzy Hypersoft Set

��({tall,white,female,Italian})={��1(06,01),��3(07,02)}, (6) which means that, with respect to the attributes’ values {tall,white,female,Italian} all together, ��1 belongs 60% and 10% it does not belong to the set ℳ; similarly, ��3 belongs 70% and 20% it does not belong to the set ℳ

6.4 Neutrosophic Hypersoft Set

��({tall,white,female,Italian})={��1(06,02,01),��3(07,03,02)}, (7) which means that, with respect to the attributes’ values {tall,white,female,Italian} all together, ��1 belongs 60% and its indeterminate belongness is 20% and it doesn’t belong 10% to the set ℳ; similarly, ��3 belongs 70% and its indeterminate belongness is 30% and it doesn’t belong 20%.

6.5

Plithogenic Hypersoft Set

��({tall,white,female,Italian})={��1 (��1 0 (tall),��1 0 (white),��1 0 (female),��1 0 (Italian)), ��2 (��2 0 (tall),��2 0 (white),��2 0 (female),��2 0 (Italian))}, (8) where ��1 0 (��) means the degree of appurtenance of element ��1 to the set ℳ with respect to the attribute value α; and similarly ��2 0 (��) means the degree of appurtenance of element ��2 to the set ℳ with respect to the attribute value α; where �� ∈{tall,white,female,Italian}

Unlike the Crisp / Fuzzy / Intuitionistic Fuzzy / Neutrosophic Hypersoft Sets [where the degree of appurtenance of an element x to the set ℳ is with respect to all attribute values tall, white, female, Italian together (as a whole), therefore a degree of appurtenance with respect to a set of attribute values], the Plithogenic Hypersoft Set is a refinement of Crisp / Fuzzy / Intuitionistic Fuzzy / Neutrosophic Hypersoft Sets [since the degree of appurtenance of an element x to the set ℳ is with respect to each single attribute value]. But the Plithogenic Hypersoft St is also combined with each of the above, since the degree of degree of appurtenance of an element x to the set ℳ with respect to each single attribute value may be: crisp, fuzzy, intuitionistic fuzzy, or neutrosophic.

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7

Classification of Plithogenic Hypersoft Sets

7.1 Plithogenic Crisp Hypersoft Set

It is a plithogenic hypersoft set, such that the degree of appurtenance of an element x to the set ℳ, with respect to each attribute value, is crisp: �� 0(��)=0 (nonappurtenance), or 1 (appurtenance)

In our example: ��({tall,white,female,Italian})={��1(1,1,1,1),��3(1,1,1,1)} (9)

7.2 Plithogenic Fuzzy Hypersoft Set

It is a plithogenic hypersoft set, such that the degree of appurtenance of an element x to the set ℳ, with respect to each attribute value, is fuzzy: �� 0(��)∈��([0,1]), power set of [0,1], where �� 0() may be a subset, an interval, a hesitant set, a single valued number, etc. In our example, for a single valued number: ��({tall,white,female,Italian})={��1(0.4,0.7,0.6,0.5),��3(0.8,0.2,0.7,0.7)} (10)

7.3 Plithogenic Intuitionistic Fuzzy Hypersoft Set

It is a plithogenic hypersoft set, such that the degree of appurtenance of an element x to the set ℳ, with respect to each attribute value, is intuitionistic fuzzy: �� 0(��)∈��([0,1]2), power set of [0,1]2 , where similarly �� 0(��) may be: a Cartesian product of subsets, of intervals, of hesitant sets, of single valued numbers, etc.

In our example, for single valued numbers: ��({tall,white,female,Italian})={��1(0.4,0.3)(0.7,0.2)(0.6,0.0)(0.5,0.1) ��3(0.8,0.1)(0.2,0.5)(0.7,0.0)(0.7,0.4)}. (11)

7.4 Plithogenic Neutrosophic Hypersoft Set

It is a plithogenic hypersoft set, such that the degree of appurtenance of an element x to the set ℳ, with respect to each attribute value, is neutrosophic: �� 0(��)∈��([0,1]3), power set of [0,1]3 , where �� 0(��) may be: a triple Cartesian product of subsets, of intervals, of hesitant sets, of single valued numbers, etc.

In our example, for single valued numbers: ��({tall,white,female,Italian})={�� 1[(04,01,03)(07,00,02)(06,03,00)(05,02,01)] �� 3[(08,01,01)(02,04,05)(07,01,00)(07,05,04)]} (12)

Conclusion & Future Research

For all types of plithogenic hypersoft sets, the aggregation operators (union, intersection, complement, inclusion, equality) have to be defined and their properties found. Applications in various engineering, technical, medical, social science, administrative, decision making and so on, fields of knowledge of these types of plithogenic hypersoft sets should be investigated.

References

[1] D. Molodtsov (1999). Soft Set Theory First Results. Computer Math. Applic. 37, 19 31.

[2] T. Srinivasa Rao, B. Srinivasa Kumar, S. Hanumanth Rao.AStudy onNeutrosophicSoftSet in Decision Making Problem. Journal of Engineering and Applied Sciences, Asian Research Publishing Network (ARPN), vol. 13, no. 7, April 2018.

[3] Florentin Smarandache. Plithogeny, Plithogenic Set, Logic, Probability, and Statistics. Brussels: Pons Editions, 2017.

[4] Florentin Smarandache. Plithogenic Set, an Extension of Crisp, Fuzzy, Intuitionistic Fuzzy, and Neutrosophic Sets Revisited. Neutrosophic Sets and Systems, vol. 21, 2018, pp. 153 166. https://doi.org/10.5281/zenodo.1408740.

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Algebraic Structures of Neutrosophic Triplets, Neutrosophic Duplets, or Neutrosophic Multisets

Florentin Smarandache, Xiaohong Zhang, Mumtaz Ali

Florentin Smarandache, Xiaohong Zhang, Mumtaz Ali (2019). Algebraic Structures of Neutrosophic Triplets, Neutrosophic Duplets, or Neutrosophic Multisets.

Symmetry 11, 171. DOI: 10.3390/sym11020171

Neutrosophy(1995)isanewbranchofphilosophythatstudiestriadsoftheform(<A>,<neutA>, <antiA>),where<A>isanentity(i.e.,element,concept,idea,theory,logicalproposition,etc.),<antiA> istheoppositeof<A>,while<neutA>istheneutral(orindeterminate)betweenthem,i.e.,neither <A>nor<antiA>[1].

Based on neutrosophy, the neutrosophic triplets were founded; they have a similar form: (x, neut(x), anti(x), that satisfy some axioms, for each element x in a given set [2 4].

The book Algebraic Structures of Neutrosophic Triplets, Neutrosophic Duplets, or Neutrosophic Multisets contains the successful invited submissions [5 56] to a special issue of Symmetry , reporting on state-of-the-art and recent advancements of neutrosophic triplets, neutrosophic duplets, neutrosophic multisets, and their algebraic structures—that have been defined recently in 2016, but have gained interest from world researchers, and several papers have been published in first rank international journals.

The topics approached in the 52 papers included in this book are: neutrosophic sets; neutrosophic logic; generalized neutrosophic set; neutrosophic rough set; multigranulation neutrosophic rough set (MNRS); neutrosophic cubic sets; triangular fuzzy neutrosophic sets (TFNSs); probabilistic single-valued (interval) neutrosophic hesitant fuzzy set; neutro-homomorphism; neutrosophic computation; quantum computation; neutrosophic association rule; data mining; big data; oracle Turing machines; recursive enumerability; oracle computation; interval number; dependent degree; possibility degree; power aggregation operators; multi-criteria group decision-making (MCGDM); expert set; soft sets; LA-semihypergroups; single valued trapezoidal neutrosophic number; inclusion relation; Q-linguistic neutrosophic variable set; vector similarity measure; cosine measure; Dice measure; Jaccard measure; VIKOR model; potential evaluation; emerging technology commercialization; 2-tuple linguistic neutrosophic sets (2TLNSs); TODIM model; Bonferroni mean; aggregation operator; NC power dual MM (NCPDMM) operator; fault diagnosis; defuzzification; simplified neutrosophic weighted averaging operator; linear and non-linear neutrosophic number; de-neutrosophication methods; neutro-monomorphism; neutro-epimorphism; neutro-automorphism; fundamental neutro-homomorphism theorem; neutro-isomorphism theorem; quasi neutrosophic triplet loop; quasi neutrosophic triplet group; BE-algebra; cloud model; Maclaurin symmetric mean; pseudo-BCI algebra; hesitant fuzzy set; photovoltaic plan; decision-making trial and evaluation laboratory (DEMATEL); Choquet integral; fuzzy measure; clustering algorithm; and many more.

In the opening paper [5] of this book, the authors introduce refined concepts for neutrosophic quantum computing such as neutrosophic quantum states and transformation gates, neutrosophic Hadamard matrix, coherent and decoherent superposition states, entanglement and measurement

Smarandache (author and

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notionsbasedonneutrosophicquantumstates.Theyalsogivesomeobservationsusingthese principles,andpresentanumberofquantumcomputationalmatrixtransformationsbasedon neutrosophiclogic,clarifyingquantummechanicalnotionsrelyingonneutrosophicstates.Thepaper isintendedtoextendtheworkofSmarandache[57 59]byintroducingamathematicalframeworkfor neutrosophicquantumcomputingandpresentingsomeresults.

Thesecondpaper[6]introducesoracleTuringmachineswithneutrosophicvaluesallowedinthe oracleinformationandthengivesomeresultswhenoneispermittedtouseneutrosophicsetsand logicinrelativecomputation.Theauthorsalsointroduceamethodtoenumeratetheelementsofa neutrosophicsubsetofnaturalnumbers.

Inthethirdpaper[7],anewapproachandframeworkbasedontheintervaldependentdegree forMCGDMproblemswithSNSsisproposed.Firstly,thesimplifieddependentfunctionand distributionfunctionaredefined.Then,theyareintegratedintotheintervaldependentfunction whichcontainsintervalcomputinganddistributioninformationoftheintervals.Subsequently,the intervaltransformationoperatorisdefinedtoconvertSNNsintointervals,andthentheinterval dependentfunctionforSNNsisdeduced.Finally,anexampleisprovidedtoverifythefeasibilityand effectivenessoftheproposedmethod,togetherwithitscomparativeanalysis.Inaddition,uncertainty analysis,whichcanreflectthedynamicchangeofthefinalresultcausedbychangesinthedecision makers’preferences,isperformedindifferentdistributionfunctionsituations.Thatincreasesthe reliabilityandaccuracyoftheresult.

Neutrosophictripletstructureyieldsasymmetricpropertyoftruthmembershipontheleft, indeterminacymembershipinthecenterandfalsemembershipontheright,asdopointsofobject, centerandimageofreflection.Asanextensionofaneutrosophicset,theQ-neutrosophicsetis introducedinthesubsequentpaper[8]tohandletwo-dimensionaluncertainandinconsistentsituations. TheauthorsextendthesoftexpertsettothegeneralizedQ-neutrosophicsoftexpertsetbyincorporating theideaofasoftexpertsettotheconceptofaQ-neutrosophicsetandattachingtheparameter offuzzysetwhiledefiningaQ-neutrosophicsoftexpertset.Thispatterncarriesthebenefitsof Q-neutrosophicsetsandsoftsets,enablingdecisionmakerstorecognizetheviewsofspecialists withnorequirementforextralumberingtasks,thusmakingitexceedinglyreasonableforusein decision-makingissuesthatincludeimprecise,indeterminateandinconsistenttwo-dimensionaldata. Someessentialoperations,namelysubset,equal,complement,union,intersection,ANDandOR operations,andadditionallyseveralpropertiesrelatingtothenotionofageneralizedQ-neutrosophic softexpertsetarecharacterized.Finally,analgorithmonageneralizedQ-neutrosophicsoftexpert setisproposedandappliedtoareal-lifeexampletoshowtheefficiencyofthisnotioninhandling suchproblems.

Inthefollowingpaper[9],theauthorsextendtheideaofaneutrosophictripletsetto non-associativesemihypergroupsanddeﬁneneutrosophictripletLA-semihypergroup.Theydiscuss somebasicresultsandproperties,andprovideanapplicationoftheproposedstructureinfootball.

Singlevaluedtrapezoidalneutrosophicnumbers(SVTNNs)areveryusefultoolsfordescribing complexinformation,becauseoftheiradvantageindescribingtheinformationcompletely,accurately andcomprehensivelyfordecision-makingproblems[60].Inthenextpaper[10],amethodbasedon SVTNNsisproposedfordealingwithMCGDMproblems.Firstly,thenewoperationSVTNNsare developedforavoidingevaluationinformationaggregationlossanddistortion.Thenthepossibility degreesandcomparisonofSVTNNsareproposedfromtheprobabilityviewpointforranking andcomparingthesinglevaluedtrapezoidalneutrosophicinformationreasonablyandaccurately. BasedonthenewoperationsandpossibilitydegreesofSVTNNs,thesinglevaluedtrapezoidal neutrosophicpoweraverage(SVTNPA)andsinglevaluedtrapezoidalneutrosophicpowergeometric (SVTNPG)operatorsareproposedtoaggregatethesinglevaluedtrapezoidalneutrosophicinformation. Furthermore,basedonthedevelopedaggregationoperators,asinglevaluedtrapezoidalneutrosophic MCGDMmethodisdeveloped.Finally,theproposedmethodisappliedtosolvethepracticalproblem

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ofthemostappropriategreensupplierselectionandtherankresultscomparedwiththeprevious approachdemonstratetheproposedmethod’seffectiveness.

Aftertheneutrosophicset(NS)wasproposed[58],NSwasusedinmanyuncertaintyproblems. Thesingle-valuedneutrosophicset(SVNS)isaspecialcaseofNSthatcanbeusedtosolvereal-word problems.Thenextpaper[11]mainlystudiesmultigranulationneutrosophicroughsets(MNRSs) andtheirapplicationsinmulti-attributegroupdecision-making.Firstly,theexistingdeﬁnitionof neutrosophicroughset(theauthorscallittype-Ineutrosophicroughset(NRSI)inthispaper)is analyzed,andthenthedeﬁnitionoftype-IIneutrosophicroughset(NRSII),whichissimilarto NRSI,isgivenanditspropertiesarestudied.Secondly,atype-IIIneutrosophicroughset(NRSIII)is proposedanditsdifferencesfromNRSIandNRSIIareprovided.Thirdly,singlegranulationNRSsare extendedtomultigranulationNRSs,andthetype-Imultigranulationneutrosophicroughset(MNRSI)is studied.Thetype-IImultigranulationneutrosophicroughset(MNRSII)andtype-IIImultigranulation neutrosophicroughset(MNRSIII)areproposedandtheirdifferentpropertiesareoutlined.Finally, MNRSIIIintwouniversesisproposedandanalgorithmfordecision-makingbasedonMNRSIIIis provided.Acarrankingexampleisstudiedtoexplaintheapplicationoftheproposedmodel.

Sincelanguageisusedforthinkingandexpressinghabitsofhumansinreallife,the linguisticevaluationforanobjectivethingisexpressedeasilyinlinguisticterms/values.However, existinglinguisticconceptscannotdescribelinguisticargumentsregardinganevaluatedobjectin two-dimensionaluniversalsets(TDUSs).Todescribelinguisticneutrosophicargumentsindecision makingproblemsregardingTDUSs,thenextarticle[12]proposesaQ-linguisticneutrosophicvariable set(Q-LNVS)fortheﬁrsttime,whichdepictsitstruth,indeterminacy,andfalsitylinguisticvalues independentlycorrespondingtoTDUSs,andvectorsimilaritymeasuresofQ-LNVSs.Thereafter,a linguisticneutrosophicMADMapproachbyusingthepresentedsimilaritymeasures,includingthe cosine,Dice,andJaccardmeasures,isdevelopedunderQ-linguisticneutrosophicsetting.Lastly, theapplicabilityandeffectivenessofthepresentedMADMapproachispresentedbyanillustrative exampleunderQ-linguisticneutrosophicsetting.

Inthefollowingarticle[13],theauthorscombinetheoriginalVIKORmodelwithatriangularfuzzy neutrosophicset[61]toproposethetriangularfuzzyneutrosophicVIKORmethod.Intheextended method,theyusethetriangularfuzzyneutrosophicnumbers(TFNNs)topresentthecriteriavaluesin MCGDMproblems.Firstly,theysummarilyintroducethefundamentalconcepts,operationformulas anddistancecalculatingmethodofTFNNs.ThentheyreviewsomeaggregationoperatorsofTFNNs. Thereafter,theyextendtheoriginalVIKORmodeltothetriangularfuzzyneutrosophicenvironment andintroducethecalculatingstepsoftheTFNNsVIKORmethod,theproposedmethodwhichismore reasonableandscientiﬁcforconsideringtheconﬂictingcriteria.Furthermore,anumericalexample forpotentialevaluationofemergingtechnologycommercializationispresentedtoillustratethenew method,andsomecomparisonsarealsoconductedtofurtherillustrateadvantagesofthenewmethod.

Anotherpaper[14]inthisbookaimstoextendtheoriginalTODIM(Portugueseacronym forinteractivemulti-criteriadecisionmaking)methodtothe2-tuplelinguisticneutrosophicfuzzy environment[62]toproposethe2TLNNsTODIMmethod.Intheextendedmethod,theauthors use2-tuplelinguisticneutrosophicnumbers(2TLNNs)topresentthecriteriavaluesinmultiple attributegroupdecisionmaking(MAGDM)problems.Firstly,theybrieflyintroducethedefinition, operationallaws,someaggregationoperators,andthedistancecalculatingmethodof2TLNNs.Then, thecalculationstepsoftheoriginalTODIMmodelarepresentedinsimplifiedform.Thereafter,they extendtheoriginalTODIMmodeltothe2TLNNsenvironmenttobuildthe2TLNNsTODIMmodel, theproposedmethod,whichismorereasonableandscientificinconsideringthesubjectivityofthe decisionmakers’(DMs’)behaviorsandthedominanceofeachalternativeoverothers.Finally,a numericalexampleforthesafetyassessmentofaconstructionprojectisproposedtoillustratethe newmethod,andsomecomparisonsarealsoconductedtofurtherillustratetheadvantagesofthe newmethod.

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ThepowerBonferronimean(PBM)operatorisahybridstructureandcantaketheadvantage ofapoweraverage(PA)operator,whichcanreducetheimpactofinappropriatedatagivenbythe prejudiceddecisionmakers(DMs)andBonferronimean(BM)operator,whichcantakeintoaccount thecorrelationbetweentwoattributes.Inrecentyears,manyresearchershaveextendedthePBM operatortohandlefuzzyinformation.TheDombioperationsofT-conorm(TCN)andT-norm(TN), proposedbyDombi,havethesupremacyofoutstandingﬂexibilitywithgeneralparameters.However, intheexistingliterature,PBMandtheDombioperationshavenotbeencombinedfortheabove advantagesforinterval-neutrosophicsets(INSs)[63].Inthefollowingpaper[15],theauthorsdeﬁne someoperationallawsforintervalneutrosophicnumbers(INNs)basedonDombiTNandTCNand discussseveraldesirablepropertiesoftheseoperationalrules.Secondly,theyextendthePBMoperator basedonDombioperationstodevelopaninterval-neutrosophicDombiPBM(INDPBM)operator,an interval-neutrosophicweightedDombiPBM(INWDPBM)operator,aninterval-neutrosophicDombi powergeometricBonferronimean(INDPGBM)operatorandaninterval-neutrosophicweightedDombi powergeometricBonferronimean(INWDPGBM)operator,anddiscussseveralpropertiesofthese aggregationoperators.ThentheydevelopaMADMmethod,basedontheseproposedaggregation operators,todealwithintervalneutrosophic(IN)information.Anillustrativeexampleisprovidedto showtheusefulnessandrealismoftheproposedMADMmethod.

Theneutrosophiccubicset(NCS)isahybridstructure[64],whichconsistsofINS[63](associated withtheundeterminedpartofinformationassociatedwithentropy)andSVNS[60](associatedwith thedeterminedpartofinformation).NCSisabettertooltohandlecomplexDMproblemswithINS andSVNS.Themainpurposeofthenextarticle[16]istodevelopsomenewaggregationoperators forcubicneutrosophicnumbers(NCNs),whichisabasicmemberofNCS.Takingtheadvantages ofMuirheadmean(MM)operatorandPAoperator,thepowerMuirheadmean(PMM)operatoris developedandisscrutinizedunderNCinformation.Tomanagetheproblemsupstretched,somenew NCaggregationoperators,suchastheNCpowerMuirheadmean(NCPMM)operator,weightedNC powerMuirheadmean(WNCPMM)operator,NCpowerdualMuirheadmean(NCPMM)operator andweightedNCpowerdualMuirheadmean(WNCPDMM)operatorareproposedandrelated propertiesoftheseproposedaggregationoperatorsareconferred.Theimportantadvantageofthe developedaggregationoperatoristhatitcanremovetheeffectofawkwarddataanditconsidersthe interrelationshipamongaggregatedvaluesatthesametime.Finally,anumericalexampleisgivento showtheeffectivenessofthedevelopedapproach.

Smarandachedefinedaneutrosophicset[57]tohandleproblemsinvolvingincompleteness, indeterminacy,andawarenessofinconsistencyknowledge,andhavefurtherdevelopedneutrosophic softexpertsets.Inthenextpaper[17]ofthisbook,thisconceptisfurtherexpandedto generalizedneutrosophicsoftexpertset(GNSES).Theauthorsthendefineitsbasicoperationsof complement,union,intersection,AND,OR,andstudysomerelatedproperties,withsupporting proofs.Subsequently,theydefineaGNSES-aggregationoperatortoconstructanalgorithmfora GNSESdecision-makingmethod,whichallowsforamoreefficientdecisionprocess.Finally,they applythealgorithmtoadecision-makingproblem,toillustratetheeffectivenessandpracticalityofthe proposedconcept.Acomparativeanalysiswithexistingmethodsisdoneandtheresultaffirmsthe flexibilityandprecisionoftheproposedmethod.

Inthenextpaper[18],theauthorsdefinetheneutrosophicvalued(andgeneralizedorG)metric spacesforthefirsttime.Besides,theydetermineamathematicalmodelforclusteringtheneutrosophic bigdatasetsusingG-metric.Furthermore,relativeweightedneutrosophic-valueddistanceand weightedcohesionmeasurearedefinedforneutrosophicbigdataset[65].Averypracticalmethodfor dataanalysisofneutrosophicbigdataisoffered,althoughneutrosophicdatatype(neutrosophicbig data)areinmassiveanddetailedformwhencomparedwithotherdatatypes.

Bol-Moufangtypesofaparticularquasineutrosophictripletloop(BCI-algebra),christened FenyvesBCI-algebras,areintroducedandstudiedinanotherpaper[19]ofthisbook.60Fenyves BCI-algebrasareintroducedandclassiﬁed.Amongstthese60classesofalgebras,46arefoundto

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beassociativeand14arefoundtobenon-associative.The46associativealgebrasareshowntobe Booleangroups.Moreover,necessaryandsufficientconditionsfor13non-associativealgebrastobe associativearealsoobtained:p-semisimplicityisfoundtobenecessaryandsufficientforaF3,F5,F42, andF55algebrastobeassociativewhilequasi-associativityisfoundtobenecessaryandsufficient forF19,F52,F56,andF59algebrastobeassociative.Twopairsofthe14non-associativealgebrasare foundtobeequivalenttoassociativity(F52andF55,andF55andF59).EveryBCI-algebraisnaturally aF54BCI-algebra.Theworkisconcludedwithrecommendationsbasedoncomparisonbetween thebehaviorofidentitiesofBol-Moufang(Fenyves’identities)inquasigroupsandloopsandtheir behaviorinBCI-algebra.Itisconcludedthatresultsofthisworkareaninitiationintothestudyof theclassificationoffiniteFenyves’quasineutrosophictripletloops(FQNTLs)justlikevarioustypes offiniteloopshavebeenclassified.Thisresearchworkhasopenedanewareaofresearchfindingin BCI-algebras,vis-a-vistheemergenceof540varietiesofBol-Moufangtypequasineutrosophictriplet loops.A‘cycleofalgebraicstructures’whichportraysthisfactisprovided.

Theuncertaintyandconcurrenceofrandomnessareconsideredwhenmanypracticalproblems aredealtwith.Todescribethealeatoryuncertaintyandimprecisioninaneutrosophicenvironment andpreventtheobliterationofmoredata,theconceptoftheprobabilisticsingle-valued(interval) neutrosophichesitantfuzzysetisintroducedinthenextpaper[20].Bydeﬁnition,theprobabilistic single-valuedneutrosophichesitantfuzzyset(PSVNHFS)isaspecialcaseoftheprobabilisticinterval neutrosophichesitantfuzzyset(PINHFS).PSVNHFSscansatisfyallthepropertiesofPINHFSs. AnexampleisgiventoillustratethatPINHFScomparedtoPSVNHFSismoregeneral.Then, PINHFSisthemainresearchobject.ThebasicoperationalrelationsofPINHFSarestudied,and thecomparisonmethodofprobabilisticintervalneutrosophichesitantfuzzynumbers(PINHFNs)is proposed.Then,theprobabilisticintervalneutrosophichesitantfuzzyweightedaveraging(PINHFWA) andtheprobabilityintervalneutrosophichesitantfuzzyweightedgeometric(PINHFWG)operators arepresented.Somebasicpropertiesareinvestigated.Next,basedonthePINHFWAandPINHFWG operators,adecision-makingmethodunderaprobabilisticintervalneutrosophichesitantfuzzy circumstanceisestablished.Finally,theauthorsapplythismethodtotheissueofinvestmentoptions. Thevalidityandapplicationofthenewapproachisdemonstrated.

Competitionamongdifferentuniversitiesdependslargelyonthecompetitionfortalent.Talent evaluationandselectionisoneofthemainactivitiesinhumanresourcemanagement(HRM)whichis criticalforuniversitydevelopment[21].Firstly,linguisticneutrosophicsets(LNSs)areintroducedto betterexpressmultipleuncertaininformationduringtheevaluationprocedure.Theauthorsfurther mergethepoweraveragingoperatorwithLNSsforinformationaggregationandproposeaLN-power weightedaveraging(LNPWA)operatorandaLN-powerweightedgeometric(LNPWG)operator. Then,anextendedtechniquefororderpreferencebysimilaritytoidealsolution(TOPSIS)method isdevelopedtosolveacaseofuniversityHRMevaluationproblem.Themaincontributionand noveltyoftheproposedmethodrelyonthatitallowstheinformationprovidedbydifferentDMsto supportandreinforceeachotherwhichismoreconsistentwiththeactualsituationofuniversityHRM evaluation.Inaddition,itseffectivenessandadvantagesoverexistingmethodsareveriﬁedthrough sensitivityandcomparativeanalysis.Theresultsshowthattheproposaliscapableinthedomainof universityHRMevaluationandmaycontributetothetalentintroductioninuniversities.

TheconceptofacommutativegeneralizedneutrosophicidealinaBCK-algebraisproposed,and relatedpropertiesareprovedinanotherpaper[22]ofthisbook.Characterizationsofacommutative generalizedneutrosophicidealareconsidered.Also,someequivalencerelationsonthefamilyofall commutativegeneralizedneutrosophicidealsinBCK-algebrasareintroduced,andsomeproperties areinvestigated.

Faultdiagnosisisanimportantissueinvariousﬁeldsandaimstodetectandidentifythefaultsof systems,products,andprocesses.Thecauseofafaultiscomplicatedduetotheuncertaintyofthe actualenvironment.Nevertheless,itisdifﬁculttoconsideruncertainfactorsadequatelywithmany traditionalmethods.Inaddition,thesamefaultmayshowmultiplefeaturesandthesamefeature

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mightbecausedbydifferentfaults.Inthenextpaper[23],aneutrosophicsetbasedfaultdiagnosis methodbasedonmulti-stagefaulttemplatedataisproposedtosolvethisproblem.Foranunknown faultsamplewhosefaulttypeisunknownandneedstobediagnosed,theneutrosophicsetbasedon multi-stagefaulttemplatedataisgenerated,andthenthegeneratedneutrosophicsetisfusedviathe simpliﬁedneutrosophicweightedaveraging(SNWA)operator.Afterwards,thefaultdiagnosisresults canbedeterminedbytheapplicationofdefuzziﬁcationmethodforadefuzzyingneutrosophicset. Mostkindsofuncertainproblemsintheprocessoffaultdiagnosis,includinguncertaininformation andinconsistentinformation,couldbehandledwellwiththeintegrationofmulti-stagefaulttemplate dataandtheneutrosophicset.Finally,thepracticalityandeffectivenessoftheproposedmethodare demonstratedviaanillustrativeexample.

Thenotionsofneutrosophy,neutrosophicalgebraicstructures,neutrosophicdupletand neutrosophictripletwereintroducedbyFlorentinSmarandache[57].Inanotherpaper[24]ofthis book,someneutrosophicdupletsarestudied.Aparticularcaseisconsidered,andthecomplete characterizationofneutrosophicdupletsaregiven.Someopenproblemsrelatedtoneutrosophic dupletsareproposed.

Inthenextpaper[25],theauthorsprovideanapplicationofneutrosophicbipolarfuzzysets appliedtodailylife’sproblemrelatedwiththeHOPEfoundation,whichisplanningtobuild achildren’shospital.Theydevelopthetheoryofneutrosophicbipolarfuzzysets,whichisa generalizationofbipolarfuzzysets.Aftergivingthedeﬁnitiontheyintroducesomebasicoperationof neutrosophicbipolarfuzzysetsandfocusonweightedaggregationoperatorsintermsofneutrosophic bipolarfuzzysets.Theydeﬁneneutrosophicbipolarfuzzyweightedaveraging(NBFWA)and neutrosophicbipolarfuzzyorderedweightedaveraging(NBFOWA)operators.Nexttheyintroduce differentkindsofsimilaritymeasuresofneutrosophicbipolarfuzzysets.Finally,asanapplication,the authorsgiveanalgorithmforthemultipleattributedecisionmakingproblemsundertheneutrosophic bipolarfuzzyenvironmentbyusingthedifferentkindsofneutrosophicbipolarfuzzyweighted/fuzzy orderedweightedaggregationoperatorswithanumericalexamplerelatedwithHOPEfoundation.

Inthefollowingpaper[26],theauthorsintroducetheconceptofneutrosophicnumbersfrom differentviewpoints[57 65].Theydeﬁnedifferenttypesoflinearandnon-lineargeneralized triangularneutrosophicnumberswhichareveryimportantforuncertaintytheory.Theyintroducethe de-neutrosophicationconceptforneutrosophicnumberfortriangularneutrosophicnumbers.This concepthelpstoconvertaneutrosophicnumberintoacrispnumber.Theconceptsarefollowedbytwo applications,namelyinanimpreciseprojectevaluationreviewtechniqueandarouteselectionproblem.

Inclassicalgrouptheory,homomorphismandisomorphismaresignificanttostudythe relationbetweentwoalgebraicsystems.Throughthenextarticle[27],theauthorspropose neutro-homomorphismandneutro-isomorphismfortheneutrosophicextendedtripletgroup(NETG) whichplaysasignificantroleinthetheoryofneutrosophictripletalgebraicstructures.Then,they defineneutro-monomorphism,neutro-epimorphism,andneutro-automorphism.Theygiveandprove sometheoremsrelatedtothesestructures.Furthermore,theFundamentalhomomorphismtheorem fortheNETGisgivenandsomespecialcasesarediscussed.Firstandsecondneutro-isomorphism theoremsarestated.Finally,byapplyinghomomorphismtheoremstoneutrosophicextendedtriplet algebraicstructures,theauthorshaveexaminedhowcloselydifferentsystemsarerelated.

Itisaninterestingdirectiontostudyroughsetsfromamulti-granularityperspective.Inroughset theory,themulti-particlestructurewasrepresentedbyabinaryrelation.Thenextpaper[28]considers anewneutrosophicroughsetmodel,multi-granulationneutrosophicroughset(MGNRS).First,the conceptofMGNRSonasingledomainanddualdomainswasproposed.Then,theirpropertiesand operatorswereconsidered.TheauthorsobtainedthatMGNRSondualdomainswilldegenerateinto MGNRSonasingledomainwhenthetwodomainsarethesame.Finally,akindofspecialmulti-criteria groupdecisionmaking(MCGDM)problemwassolvedbasedonMGNRSondualdomains,andan examplewasgiventoshowitsfeasibility.

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Asanewgeneralizationofthenotionofthestandardgroup,thenotionoftheNTGisderived fromthebasicideaoftheneutrosophicsetandcanberegardedasamathematicalstructuredescribing generalizedsymmetry.Inthenextpaper[29],thepropertiesandstructuralfeaturesofNTGarestudied indepthbyusingtheoreticalanalysisandsoftwarecalculations(infact,someimportantexamplesin thepaperarecalculatedandveriﬁedbymathematicssoftware,buttherelatedprogramsareomitted).

Themainresultsareobtainedasfollows:(1)byconstructingcounterexamples,somemistakesinthe someliteraturesarepointedout;(2)somenewpropertiesofNTGsareobtained,anditisproved thateveryelementhasauniqueneutralelementinanyneutrosophictripletgroup;(3)thenotionsof NT-subgroups,strongNT-subgroups,andweakcommutativeneutrosophictripletgroups(WCNTGs) areintroduced,thequotientstructuresareconstructedbystrongNT-subgroups,andahomomorphism theoremisprovedinweakcommutativeneutrosophictripletgroups.

Theaimofthefollowingpaper[30]istointroducesomenewoperatorsforaggregating single-valuedneutrosophic(SVN)informationandtoapplythemtosolvethemulti-criteria decision-making(MCDM)problems.Thesingle-valuedneutrosophicset,asanextensionand generalizationofanintuitionisticfuzzyset,isapowerfultooltodescribethefuzzinessand uncertainty[60],andMMisawell-knownaggregationoperatorwhichcanconsiderinterrelationships amonganynumberofargumentsassignedbyavariablevector.Inordertomakefulluseofthe advantagesofboth,theauthorsintroducetwonewprioritizedMMaggregationoperators,suchas theSVNprioritizedMM(SVNPMM)andSVNprioritizeddualMM(SVNPDMM)underanSVNset environment.Inaddition,somepropertiesofthesenewaggregationoperatorsareinvestigatedand somespecialcasesarediscussed.Furthermore,theauthorsproposeanewmethodbasedonthese operatorsforsolvingtheMCDMproblems.Finally,anillustrativeexampleispresentedtotestifythe efﬁciencyandsuperiorityoftheproposedmethodbycomparingitwiththeexistingmethod.

Makingpredictionsaccordingtohistoricalvalueshaslongbeenregardedascommonpractice bymanyresearchers.However,forecastingsolelybasedonhistoricalvaluescouldleadtoinevitable over-complexityanduncertaintyduetotheuncertaintiesinside,andtherandominfluenceoutside, ofthedata.Consequently,findingtheinherentrulesandpatternsofatimeseriesbyeliminating disturbanceswithoutlosingimportantdetailshaslongbeenaresearchhotspot.Inthefollowing paper[31],theauthorsproposeanovelforecastingmodelbasedonmulti-valuedneutrosophicsets tofindfluctuationrulesandpatternsofatimeseries.Thecontributionsoftheproposedmodel are:(1)usingamulti-valuedneutrosophicset(MVNS)todescribethefluctuationpatternsofatime series,themodelcouldrepresentthefluctuationtrendofup,equal,anddownwithdegreesoftruth, indeterminacy,andfalsitywhichsignificantlypreservedetailsofthehistoricalvalues;(2)measuring thesimilaritiesofdifferentfluctuationpatternsbytheHammingdistancecouldavoidtheconfusion causedbyincompleteinformationfromlimitedsamples;and(3)introducinganotherrelatedtime seriesasasecondaryfactortoavoidwarpanddeviationininferringinherentrulesofhistoricalvalues, whichcouldleadtomorecomprehensiverulesforfurtherforecasting.Toevaluatetheperformance ofthemodel,theauthorsexploretheTaiwanStockExchangeCapitalizationWeightedStockIndex (TAIEX)asthemajorfactor,andtheDowJonesIndexasthesecondaryfactortofacilitatethepredicting oftheTAIEX.Toshowtheuniversalityofthemodel,theyapplytheproposedmodeltoforecastthe ShanghaiStockExchangeCompositeIndex(SHSECI)aswell.

Thenewnotionofaneutrosophictripletgroup(NTG)proposedbySmarandacheisanew algebraicstructuredifferentfromtheclassicalgroup.Theaimofthenextpaper[32]istofurther expandthisnewconceptandtostudyitsapplicationinrelatedlogicalgebrasystems.Somenew notionsofleft(right)-quasineutrosophictripletloopsandleft(right)-quasineutrosophictripletgroups areintroduced,andsomepropertiesarepresented.Asacorollaryoftheseproperties,thefollowing importantresultareproved:foranycommutativeneutrosophictripletgroup,itseveryelementhasa uniqueneutralelement.Moreover,someleft(right)-quasineutrosophictripletstructuresinBE-algebras andgeneralizedBE-algebras(includingCI-algebrasandpseudoCI-algebras)areestablished,andthe adjointsemigroupsoftheBE-algebrasandgeneralizedBE-algebrasareinvestigatedfortheﬁrsttime.

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Inaneutrosophictripletset,thereisaneutralelementandantielementforeachelement.Inthe followingstudy[33],theconceptofneutrosophictripletpartialmetricspace(NTPMS)isgivenand thepropertiesofNTPMSarestudied.Theauthorsshowthatbothclassicalmetricandneutrosophic tripletmetric(NTM)aredifferentfromNTPM.Also,theyshowthatNTPMScanbedefinedwitheach NTMS.Furthermore,theauthorsdefineacontractionforNTPMSandgiveafixedpointtheory(FPT) forNTPMS.TheFPThasbeenrevealedasaverypowerfultoolinthestudyofnonlinearphenomena. Anotherpaper[34]ofthisbookpresentsamodifiedTechniqueforOrderPreferencebySimilarity toanIdealSolution(TOPSIS)withmaximizingdeviationmethodbasedontheSVNSmodel[60]. ASVNSisaspecialcaseofaneutrosophicsetwhichischaracterizedbyatruth,indeterminacy, andfalsitymembershipfunction,eachofwhichliesinthestandardintervalof[0,1].Anintegrated weightmeasureapproachthattakesintoconsiderationboththeobjectiveandsubjectiveweightsofthe attributesisused.Themaximizingdeviationmethodisusedtocomputetheobjectiveweightofthe attributes,andthenon-linearweightedcomprehensivemethodisusedtodeterminethecombined weightsforeachattributes.Theuseofthemaximizingdeviationmethodallowsourproposedmethod tohandlesituationsinwhichinformationpertainingtotheweightcoefficientsoftheattributesare completelyunknownoronlypartiallyknown.Theproposedmethodisthenappliedtoamulti-attribute decision-making(MADM)problem.Lastly,acomprehensivecomparativestudiesispresented,in whichtheperformanceofourproposedalgorithmiscomparedandcontrastedwithotherrecent approachesinvolvingSVNSsinliterature.

Oneofthemostsignificantcompetitivestrategiesfororganizationsissustainablesupplychain management(SSCM).Thevitalpartintheadministrationofasustainablesupplychainisthe sustainablesupplierselection,whichisamulti-criteriadecision-makingissue,includingmany conflictingcriteria.Thevaluationandselectionofsustainablesuppliersaredifficultproblemsdue tovague,inconsistent,andimpreciseknowledgeofdecisionmakers.Intheliteratureonsupply chainmanagementformeasuringgreenperformance,therequirementformethodologicalanalysisof howsustainablevariablesaffecteachother,andhowtoconsidervague,impreciseandinconsistent knowledge,isstillunresolved.Thenextresearch[35]providesanincorporatedmulti-criteria decision-makingprocedureforsustainablesupplierselectionproblems(SSSPs).Anintegrated frameworkispresentedviainterval-valuedneutrosophicsetstodealwithvague,impreciseand inconsistentinformationthatexistsusuallyinrealworld.Theanalyticnetworkprocess(ANP)is employedtocalculateweightsofselectedcriteriabyconsideringtheirinterdependencies.Forranking alternativesandavoidingadditionalcomparisonsofanalyticnetworkprocesses,theTOPSISisused. Theproposedframeworkisturnedtoaccountforanalyzingandselectingtheoptimalsupplier. AnactualcasestudyofadairycompanyinEgyptisexaminedwithintheproposedframework. Comparisonwithotherexistingmethodsisimplementedtoconfirmtheeffectivenessandefficiencyof theproposedapproach.

Theconceptofintervalneutrosophicsetshasbeenstudied[63]andtheintroductionofanew kindofsetintopologicalspacescalledtheintervalvaluedneutrosophicsupportsoftsetissuggestedin thenextpaper[36].Theauthorsalsostudysomeofitsbasicproperties.Themainpurposeofthepaper istogivetheoptimumsolutiontodecision-makinginreallifeproblemstheusingintervalvalued neutrosophicsupportsoftset.

Ininconsistentandindeterminatesettings,asausualtool,theNCScontainingsingle-valued neutrosophicnumbers[60]andintervalneutrosophicnumbers[64]canbeappliedindecision-making topresentitspartialindeterminateandpartialdeterminateinformation.However,afewresearchers havestudiedneutrosophiccubicdecision-makingproblems,wherethesimilaritymeasureofNCSsis oneoftheusefulmeasuremethods.Forthefollowingwork[37]inthisbook,theauthorsproposethe Dice,cotangent,andJaccardmeasuresbetweenNCSs,andindicatetheirproperties.Then,underan NCSenvironment,thesimilaritymeasures-baseddecision-makingmethodofmultipleattributesis developed.Inthedecision-makingprocess,allthealternativesarerankedbythesimilaritymeasure

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ofeachalternativeandtheidealsolutiontoobtainthebestone.Finally,twopracticalexamplesare appliedtoindicatethefeasibilityandeffectivenessofthedevelopedmethod.

Inreal-worlddiagnosticprocedures,duetothelimitationofhumancognitivecompetence,a medicalexpertmaynotconvenientlyusesomecrispnumberstoexpressthediagnosticinformation, andplentyofresearchhasindicatedthatgeneralizedfuzzynumbersplayasigniﬁcantroleindescribing complexdiagnosticinformation.Todealwithmedicaldiagnosisproblemsbasedongeneralizedfuzzy sets(FSs),thenotionofsingle-valuedneutrosophicmultisets(SVNMs)[60]isﬁrstlyusedtoexpressthe diagnosticinformation[38].Thenthemodelofprobabilisticroughsets(PRSs)overtwouniversesis appliedtoanalyzeSVNMs,andtheconceptsofsingle-valuedneutrosophicroughmultisets(SVNRMs) overtwouniversesandprobabilisticroughsingle-valuedneutrosophicmultisets(PRSVNMs)overtwo universesareintroduced.BasedonSVNRMsovertwouniversesandPRSVNMsovertwouniverses, single-valuedneutrosophicprobabilisticroughmultisets(SVNPRMs)overtwouniversesarefurther established.Next,athree-waydecisionmodelbyvirtueofSVNPRMsovertwouniversesinthecontext ofmedicaldiagnosisisconstructed.Finally,apracticalcasestudyalongwithacomparativestudyare carriedouttorevealtheaccuracyandreliabilityoftheconstructedthree-waydecisionsmodel.

Thenextarticle[39]isbasedonnewdevelopmentsonaNTGandapplicationsearlierintroduced in2016bySmarandacheandAli.NTGsprangupfromneutrosophictripletsetX:acollectionoftriplets (b,neut(b),anti(b))foranb∈Xthatobeyscertainaxioms(existenceofneutral(s)andopposite(s)).Some resultsthataretrueinclassicalgroupsareinvestigatedinNTGandshowntobeeitheruniversally trueinNTGortrueinsomepeculiartypesofNTG.DistinguishingfeaturesbetweenanNTGand someotheralgebraicstructuressuchas:generalizedgroup(GG),quasigroup,loop,andgroupare investigated.Someneutrosophictripletsubgroups(NTSGs)ofaneutrosophictripletgrouparestudied. Applicationsoftheneutrosophictripletset,andourresultsonNTGinrelationtomanagementand sports,arehighlightedanddiscussed.

Neutrosophiccubicsets[64]arethemoregeneralizedtoolbywhichonecanhandleimprecise informationinamoreeffectivewayascomparedtofuzzysetsandallotherversionsoffuzzysets. Neutrosophiccubicsetshavethemoreﬂexibility,precisionandcompatibilitytothesystemascompared topreviousexistingfuzzymodels.Ontheotherhand,thegraphsrepresentaproblemphysicallyin theformofdiagramsandmatrices,etc.,whichisveryeasytounderstandandhandle.Therefore,the authorsofthesubsequentpaper[40]applytheneutrosophiccubicsetstographtheoryinorderto developamoregeneralapproachwheretheycanmodelimpreciseinformationthroughgraphs.Oneof veryimportantfuturesoftwoneutrosophiccubicsetsistheR-unionthatR-unionoftwoneutrosophic cubicsetsisagainaneutrosophiccubicset.Sincethepurposeofthisnewmodelistocapturethe uncertainty,theauthorsprovideapplicationsinindustriestotesttheapplicabilityofthedeﬁnedmodel basedonpresenttimeandfuturepredictionwhichisthemainadvantageofneutrosophiccubicsets.

Thereafter,anotherpaper[41]presentsadecidingtechniqueforroboticdexteroushand configurations.Thisalgorithmcanbeusedtodecideonhowtoconfigurearobotichandsoitcangrasp objectsindifferentscenarios.Receivingasinputfromseveralsensorsignalsthatprovideinformation ontheobject’sshape,theDSmTdecision-makingalgorithmpassestheinformationthroughseveral stepsbeforedecidingwhathandconfigurationshouldbeusedforacertainobjectandtask.The proposeddecision-makingmethodforrealtimecontrolwilldecreasethefeedbacktimebetween thecommandandgraspedobject,andcanbesuccessfullyappliedonrobotdexteroushands.For this,theauthorshaveusedtheDezert–Smarandachetheorywhichcanprovideinformationevenon contradictoryoruncertainsystems.

Thestudy[42]thatfollowsintroducessimplifiedneutrosophiclinguisticnumbers(SNLNs)to describeonlineconsumerreviewsinanappropriatemanner.Consideringthedefectsofstudieson SNLNsinhandlinglinguisticinformation,thecloudmodelisusedtoconvertlinguistictermsin SNLNstothreenumericalcharacteristics.Then,anovelsimplifiedneutrosophiccloud(SNC)concept ispresented,anditsoperationsanddistancearedefined.Next,aseriesofsimplifiedneutrosophic cloudaggregationoperatorsareinvestigated,includingthesimplifiedneutrosophiccloudsMaclaurin

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symmetricmean(SNCMSM)operator,weightedSNCMSMoperator,andgeneralizedweighted SNCMSMoperator.Subsequently,aMCDMmodelisconstructedbasedontheproposedaggregation operators.Finally,ahotelselectionproblemispresentedtoverifytheeffectivenessandvalidityofour developedapproach.

Inrecentyears,typhoondisastershaveoccurredfrequentlyandtheeconomiclossescausedby themhavereceivedincreasingattention.Thenextstudy[43]focusesontheevaluationoftyphoon disastersbasedontheintervalneutrosophicsettheory.Anintervalneutrosophicset(INS)[63]isa subclassofaNS[57].However,theexistingexponentialoperationsandtheiraggregationmethodsare primarilyfortheintuitionisticfuzzyset.So,thispapermainlyfocusontheresearchoftheexponential operationallawsofINNsinwhichthebasesarepositiverealnumbersandtheexponentsareinterval neutrosophicnumbers.Severalpropertiesbasedontheexponentialoperationallawarediscussedin detail.Then,theintervalneutrosophicweightedexponentialaggregation(INWEA)operatorisusedto aggregateassessmentinformationtoobtainthecomprehensiveriskassessment.Finally,amultiple attributedecisionmaking(MADM)approachbasedontheINWEAoperatorisintroducedandapplied totheevaluationoftyphoondisastersinFujianProvince,China.Resultsshowthattheproposednew approachisfeasibleandeffectiveinpracticalapplications.

Inthecomingpaper[44]ofthisbook,theauthorsstudytheneutrosophictripletgroupsfora∈Z2p andprovethiscollectionoftriplets(a,neut(a),anti(a))iftrivialformsasemigroupunderproduct,and semi-neutrosophictripletsareincludedinthatcollection.Otherwise,theyformagroupunderproduct, anditisoforder(p 1),with(p+1,p+1,p+1)asthemultiplicativeidentity.Thenewnotionofpseudo primitiveelementisintroducedinZ2panalogoustoprimitiveelementsinZp,wherepisaprime. Openproblemsbasedonthepseudoprimitiveelementsareproposed.ThestudyisrestrictedtoZ2p andtakeonlytheusualproductmodulo2p.

Fuzzygraphtheoryplaysanimportantroleinthestudyofthesymmetryandasymmetry propertiesoffuzzygraphs.Withthisinmind,inthenextpaper[45],theauthorsintroducenew neutrosophicgraphscalledcomplexneutrosophicgraphsoftype1(abbr.CNG1).Theythenpresenta matrixrepresentationforitandstudysomepropertiesofthisnewconcept.TheconceptofCNG1isan extensionofthegeneralizedfuzzygraphsoftype1(GFG1)andgeneralizedsingle-valuedneutrosophic graphsoftype1(GSVNG1).TheutilityoftheCNG1introducedhereisappliedtoamulti-attribute decisionmakingproblemrelatedtoInternetserverselection.

Thepurposeofthesubsequentpaper[46]istostudynewalgebraicoperationsand fundamentalpropertiesoftotallydependent-neutrosophicsetsandtotallydependent-neutrosophic softsets.Firstly,thein-coordinationrelationshipsamongtheoriginalinclusionrelations oftotallydependent-neutrosophicsets(calledtype-1andtyp-2inclusionrelationsinthis paper)andunion(intersection)operationsareanalyzed,andthentype-3inclusionrelationof totallydependent-neutrosophicsetsandcorrespondingtype-3union,type-3intersection,and complementoperationsareintroduced.Secondly,thefollowingtheoremisproved:alltotally dependent-neutrosophicsets(basedonacertainuniverse)determinedageneralizedDeMorgan algebrawithrespecttotype-3union,type-3intersection,andcomplementoperations.Thirdly, therelationshipsamongthetype-3orderrelation,scorefunction,andaccuracyfunctionoftotally dependent-neutrosophicsetsarediscussed.Finally,somenewoperationsandpropertiesoftotally dependent-neutrosophicsoftsetsareinvestigated,andanothergeneralizedDeMorganalgebrainduced bytotallydependent-neutrosophicsoftsetsisobtained.

Intherecentyears,schooladministratorsoftencomeacrossvariousproblemswhileteaching, counseling,andpromotingandprovidingotherserviceswhichengenderdisagreementsand interpersonalconflictsbetweenstudents,theadministrativestaff,andothers.Actionlearningis aneffectivewaytotrainschooladministratorsinordertoimprovetheirconflict-handlingstyles.In thenextpaper[47],anovelapproachisusedtodeterminetheeffectivenessoftraininginschool administratorswhoattendedanactionlearningcoursebasedontheirconflict-handlingstyles.To thisend,aRahimOrganizationConflictInventoryII(ROCI-II)instrumentisusedthatconsistsof

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boththedemographicinformationandtheconflict-handlingstylesoftheschooladministrators.The proposedmethodusestheneutrosophicset(NS)andsupportvectormachines(SVMs)toconstruct anefficientclassificationschemeneutrosophicsupportvectormachine(NS-SVM).Theneutrosophic c-means(NCM)clusteringalgorithmisusedtodeterminetheneutrosophicmembershipsandthena weightingparameteriscalculatedfromtheneutrosophicmemberships.Thecalculatedweightvalue isthenusedinSVMashandledinthefuzzySVM(FSVM)approach.Variousexperimentalworks arecarriedinacomputerenvironmentouttovalidatetheproposedidea.Allexperimentalworksare simulatedinaMATLABenvironmentwithafive-foldcross-validationtechnique.Theclassification performanceismeasuredbyaccuracycriteria.Thepredictionexperimentsareconductedbasedon twoscenarios.Inthefirstone,allstatementsareusedtopredictifaschooladministratoristrainedor notafterattendinganactionlearningprogram.Inthesecondscenario,fiveindependentdimensions areusedindividuallytopredictifaschooladministratoristrainedornotafterattendinganaction learningprogram.Accordingtotheobtainedresults,theproposedNS-SVMoutperformsforall experimentalworks.

Thenotionsoftheneutrosophichesitantfuzzysubalgebraandneutrosophichesitantfuzzyfilter inpseudo-BCIalgebrasareintroduced,andsomepropertiesandequivalentconditionsareinvestigated inthenextpaper[48].Therelationshipsbetweenneutrosophichesitantfuzzysubalgebras(filters) andhesitantfuzzysubalgebras(filters)arediscussed.Fivekindsofspecialsetsareconstructedby aneutrosophichesitantfuzzyset,andtheconditionsforthetwokindsofsetstobefiltersaregiven. Moreover,theconditionsfortwokindsofspecialneutrosophichesitantfuzzysetstobeneutrosophic hesitantfuzzyfiltersareproved.

Tosolvetheproblemsrelatedtoinhomogeneousconnectionsamongtheattributes,theauthors ofthefollowingpaper[49]introduceanovelmultipleattributegroupdecision-making(MAGDM) methodbasedontheintroducedlinguisticneutrosophicgeneralizedweightedpartitionedBonferroni meanoperator(LNGWPBM)forlinguisticneutrosophicnumbers(LNNs).Firstofall,inspiredbythe meritsofthegeneralizedpartitionedBonferronimean(GPBM)operatorandLNNs,theycombine theGPBMoperatorandLNNstoproposethelinguisticneutrosophicGPBM(LNGPBM)operator, whichsupposesthattherelationshipsareheterogeneousamongtheattributesinMAGDM.Inaddition, aimedatthedifferentimportanceofeachattribute,theweightedformoftheLNGPBMoperator isinvestigated.Then,theauthorsdiscusssomeofitsdesirablepropertiesandspecialexamples accordingly.Finally,theyproposeanovelMAGDMmethodonthebasisoftheintroducedLNGWPBM operator,andillustrateitsvalidityandmeritbycomparingitwiththeexistingmethods.

Basedonthemultiplicityevaluationinsomerealsituations,thenextpaper[50]ﬁrstlyintroduces asingle-valuedneutrosophicmultiset(SVNM)asasubclassofneutrosophicmultiset(NM)toexpress themultiplicityinformationandtheoperationalrelationsofSVNMs.Then,acosinemeasurebetween SVNMsandweightedcosinemeasurebetweenSVNMsarepresentedtomeasurethecosinedegree betweenSVNMs,andtheirpropertiesareinvestigated.Basedontheweightedcosinemeasureof SVNMs,amultipleattributedecision-makingmethodunderaSVNMenvironmentisproposed,in whichtheevaluatedvaluesofalternativesaretakenintheformofSVNMs.Therankingorderof allalternativesandthebestonecanbedeterminedbytheweightedcosinemeasurebetweenevery alternativeandtheidealalternative.Finally,anactualapplicationontheselectingproblemillustrates theeffectivenessandapplicationoftheproposedmethod.

RooftopdistributedphotovoltaicprojectshavebeenquicklyproposedinChinabecauseofpolicy promotion.Before,therooftopsoftheshoppingmallhadnotbeenoccupied,anditwasurgedto haveadecision-makingframeworktoselectsuitableshoppingmallphotovoltaicplans.However,a traditionalMCDMmethodfailedtosolvethisissueatthesametime,duetothefollowingthreedefects: theinteractionsproblemsbetweenthecriteria,thelossofevaluationinformationintheconversion process,andthecompensationproblemsbetweendiversecriteria.Inthesubsequentpaper[51],an integratedMCDMframeworkisproposedtoaddresstheseproblems.Firstofall,thecompositive evaluationindexisconstructed,andtheapplicationofDEMATELmethodhelpedanalyzetheinternal

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influenceandconnectionbehindeachcriterion.Then,theinterval-valuedneutrosophicsetisutilized toexpresstheimperfectknowledgeofexpertsgroupandavoidtheinformationloss.Next,anextended eliminationetchoicetranslationreality(ELECTRE)IIImethodisapplied,anditsucceedinavoiding thecompensationproblemandobtainingthescientificresult.Theintegratedmethodusedmaintained symmetryinthesolarphotovoltaic(PV)investment.Lastbutnotleast,acomparativeanalysisusing TechniqueforOrderPreferencebySimilaritytoanIdealSolution(TOPSIS)methodandVIKORmethod iscarriedout,andalternativeplanX1ranksfirstatthesame.Theoutcomecertifiedthecorrectness andrationalityoftheresultsobtainedinthisstudy.

Inthenextpaper[52],byutilizingtheconceptofaneutrosophicextendedtriplet(NET),the authorsdeﬁnetheneutrosophicimage,neutrosophicinverse-image,neutrosophickernel,andthe NETsubgroup.Thenotionoftheneutrosophictripletcosetanditsrelationwiththeclassicalcosetare deﬁnedandthepropertiesoftheneutrosophictripletcosetsaregiven.Furthermore,theneutrosophic tripletnormalsubgroups,andneutrosophictripletquotientgroupsarestudied.

Thefollowingpaper[53]inthebookproposesnovelskinlesiondetectionbasedonneutrosophic clusteringandadaptiveregiongrowingalgorithmsappliedtodermoscopicimages,calledNCARG. First,thedermoscopicimagesaremappedintoaneutrosophicsetdomainusingtheshearlettransform resultsfortheimages.Theimagesaredescribedviathreememberships:true,indeterminate,and falsememberships.Anindeterminatefilteristhendefinedintheneutrosophicsetforreducingthe indeterminacyoftheimages.Aneutrosophicc-meansclusteringalgorithmisappliedtosegmentthe dermoscopicimages.Withtheclusteringresults,skinlesionsareidentifiedpreciselyusinganadaptive regiongrowingmethod.Toevaluatetheperformanceofthisalgorithm,apublicdataset(ISIC2017)is employedtotrainandtesttheproposedmethod.Fiftyimagesarerandomlyselectedfortrainingand 500imagesfortesting.Severalmetricsaremeasuredforquantitativelyevaluatingtheperformance ofNCARG.Theresultsestablishthattheproposedapproachhastheabilitytodetectalesionwith highaccuracy,95.3%averagevalue,comparedtotheobtainedaverageaccuracy,80.6%,foundwhen employingtheneutrosophicsimilarityscoreandlevelset(NSSLS)segmentationapproach.

Everyorganizationseekstosetstrategiesforitsdevelopmentandgrowthandtodothis,itmust takeintoaccountthefactorsthataffectitssuccessorfailure.Themostwidelyusedtechniquein strategicplanningisSWOTanalysis.SWOTexaminesstrengths(S),weaknesses(W),opportunities (O),andthreats(T),toselectandimplementthebeststrategytoachieveorganizationalgoals.The chosenstrategyshouldharnesstheadvantagesofstrengthsandopportunities,handleweaknesses, andavoidormitigatethreats.SWOTanalysisdoesnotquantifyfactors(i.e.,strengths,weaknesses, opportunities,andthreats)anditfailstorankavailablealternatives.Toovercomethisdrawback, theauthorsofthenextpaper[54]integrateitwiththeanalytichierarchyprocess(AHP).TheAHPis abletodeterminebothquantitativeandthequalitativeelementsbyweightingandrankingthemvia comparisonmatrices.Duetothevagueandinconsistentinformationthatexistsintherealworld,they applytheproposedmodelinaneutrosophicenvironment.ArealcasestudyofStarbucksCompanyis presentedtovalidatethemodel.

BigDataisalarge-sizedandcomplexdataset,whichcannotbemanagedusingtraditionaldata processingtools.Theminingprocessofbigdataistheabilitytoextractvaluableinformationfrom theselargedatasets.Associationruleminingisatypeofdataminingprocess,whichisintendedto determineinterestingassociationsbetweenitemsandtoestablishasetofassociationruleswhose supportisgreaterthanaspeciﬁcthreshold.Theclassicalassociationrulescanonlybeextractedfrom binarydatawhereanitemexistsinatransaction,butitfailstodealeffectivelywithquantitative attributes,throughdecreasingthequalityofgeneratedassociationrulesduetosharpboundary problems.Inordertoovercomethedrawbacksofclassicalassociationrulemining,theauthorsofthe followingresearch[55]proposeanewneutrosophicassociationrulealgorithm.Thealgorithmuses anewapproachforgeneratingassociationrulesbydealingwithmembership,indeterminacy,and non-membershipfunctionsofitems,conductingtoanefﬁcientdecision-makingsystembyconsidering allvagueassociationrules.Toprovethevalidityofthemethod,theycomparethefuzzyminingand

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the neutrosophic mining [65]. The results show that the proposed approach increases the number of generated association rules.

The INS is a subclass of the NS and a generalization of the interval-valued intuitionistic fuzzy set (IVIFS), which can be used in real engineering and scientiﬁc applications. The last paper [56] in the book aims at developing new generalized Choquet aggregation operators for INSs, including the generalized interval neutrosophic Choquet ordered averaging (G-INCOA) operator and generalized interval neutrosophic Choquet ordered geometric (G-INCOG) operator. The main advantages of the proposed operators can be described as follows: (i) during decision-making or analyzing process, the positive interaction, negative interaction or non-interaction among attributes can be considered by the G-INCOA and G-INCOG operators; (ii) each generalized Choquet aggregation operator presents a unique comprehensive framework for INSs, which comprises a bunch of existing interval neutrosophic aggregation operators; (iii) new multi-attribute decision making (MADM) approaches for INSs are established based on these operators, and decision makers may determine the value of λ by different MADM problems or their preferences, which makes the decision-making process more ﬂexible; (iv) a new clustering algorithm for INSs are introduced based on the G-INCOA and G-INCOG operators, which proves that they have the potential to be applied to many new fields in the future.

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30. Garg,H.;Nancy.Multi-CriteriaDecision-MakingMethodBasedonPrioritizedMuirheadMeanAggregation OperatorunderNeutrosophicSetEnvironment. Symmetry 2018, 10,280.[CrossRef]

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44. WB,V.K.;Kandasamy,I.;Smarandache,F.AClassicalGroupofNeutrosophicTripletGroupsUsing{Z2p, ×}. Symmetry 2018, 10,194.

45. Quek,S.G.;Broumi,S.;Selvachandran,G.;Bakali,A.;Talea,M.;Smarandache,F.SomeResultsontheGraph TheoryforComplexNeutrosophicSets. Symmetry 2018, 10,190.[CrossRef]

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47. Turhan,M.;¸Sengür,D.;Karabatak,S.;Guo,Y.;Smarandache,F.NeutrosophicWeightedSupportVector MachinesfortheDeterminationofSchoolAdministratorsWhoAttendedanActionLearningCourseBased onTheirConﬂict-HandlingStyles. Symmetry 2018, 10,176.[CrossRef]

48. Shao,S.;Zhang,X.;Bo,C.;Smarandache,F.NeutrosophicHesitantFuzzySubalgebrasandFiltersin Pseudo-BCIAlgebras. Symmetry 2018, 10,174.[CrossRef]

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53. Guo,Y.;Ashour,A.S.;Smarandache,F.ANovelSkinLesionDetectionApproachUsingNeutrosophic ClusteringandAdaptiveRegionGrowinginDermoscopyImages. Symmetry 2018, 10,119.[CrossRef]

54. Abdel-Basset,M.;Mohamed,M.;Smarandache,F.AnExtensionofNeutrosophicAHP–SWOTAnalysisfor StrategicPlanningandDecision-Making. Symmetry 2018, 10,116.[CrossRef]

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Florentin Smarandache (author and editor) Collected Papers, IX 466

Neutrosophic Hedge Algebras

Florentin Smarandache

Florentin Smarandache (2019). Neutrosophic Hedge Algebras. Broad Research in Artificial Intelligence and Neuroscience 10(3), 117-123

Abstract

We introduce now for the first time the neutrosophic hedge algebras as an extension of classical hedge algebras, together with an application of neutrosophic hedge algebras.

1.Introduction

The classical hedge algebras deal with linguistic variables. In neutrosophic environment we have introduced the neutrosophic linguistic variables. We have defined neutrosophic partial relationships between single-valued neutrosophic numbers. Neutrosophic operations are used in ordertoaggregatetheneutrosophiclinguisticvalues.

2.MaterialsandMethods

We introduce now, for the first time, the Neutrosophic Hedge Algebras, as extension of classicalHedgeAlgebras.

Let'sconsideraLinguisticVariable: with ��(��) as the word domain of ��, whose each element is a word (label), or string of words.

Let�� beanattributethatdescribesthevalueofeachelement��∈��(��),asfollows: ��:��(��)→[0,1] . (1) ��(��)istheneutrosophicvalueof�� withrespecttothisattribute: ��(��)=〈�� ,�� ,��〉, (2) where�� ,�� ,�� ∈[0,1],suchthat �� meansthedegreeofvalueof��; �� meanstheindeterminatedegreeofvalueof��; �� meansthedegreeofnon-valueof��

Wemayalsousethenotation:��〈�� ,�� ,��〉 Aneutrosophicpartialrelationship≤ on��(��),definedasfollows: ��〈�� ,�� ,��〉≤ ��〈�� ,�� ,��〉, (3) ifandonlyif�� ≤�� ,and�� ≥�� ,�� ≥�� Therefore, (��(��),≤ )becomes a neutros-ophic partial order set (or neutrosophic poset), and≤ iscalledaneutrosophicinequality. Let��={0,��,1}beasetofconstants,��⊂��(��),where: 0=theleastelement,or0〈 , , 〉; w=theneutral(middle)element,or��〈 . , . , . 〉; and1=thegreatestelement,or1〈 , , 〉.

Let �� be a word-set of two neutrosophic generators, ��⊂��(��), qualitatively a negative primary neutrosophic term (denoted �� ), and the other one that is qualitatively a positive primary neutrosophicterm(denoted�� ),suchthat:

Florentin Smarandache (author and editor) Collected Papers, IX 467

BRAIN – Broad Research in Artificial Intelligence and Neuroscience

Volume 10, Issue 3 (September, 2019), ISSN 2067-3957

0≤ �� ≤ ��≤ �� ≤ 1, (4) ortranscribedusingtheneutrosophiccom-ponents:

0〈 , , 〉 ≤ �� 〈 , , 〉 ≤ ��〈 . , . , . 〉 ≤ �� 〈 , , 〉 ≤ 1〈 , , 〉, where

0≤�� ≤0.5≤�� ≤1(herethereareclassicalinequalities) 1≥�� ≥0.5≥�� ≥0,and 1≥�� ≥0.5≥�� ≥0

Let ��⊂��(��) be the set of neutrosophic hedges, regarded as unary operations. Each hedge h∈H is a functor, or comparative particle for adjectives and adverbs as in the natural language(English). h:Dom(x)→Dom(x) x→h(x). (5)

Insteadofh(x)oneeasilywriteshxtobeclosertothenaturallanguage. Byassociatingtheneutrosophiccomponents,onehas: h_〈t_h,i_h,f_h〉 x_〈t_x,i_x,f_x〉

A hedge applied to x may increase, decrease, or approximate the neutrosophic value of the elementx.

There also exists a neutrosophic identity I∈Dom(x), denoted I_〈0,0,0〉 that does not hange ontheelements:

I_〈0,0,0〉 x_〈t_x,i_x,f_x〉

In most cases, if a hedge increases / decreases the neutrosophic value of an element x situated above the neutral element w, the same hedge does the opposite, decreases / increases the neutrosophicvalueofanelement ysituatedbelowtheneutralelementw.

Andreciprocally.

If a hedge approximates the neutrosophic value, by diminishing it, of an element x situated above the neutral element w, then it approximates the neutrosophic value, by enlarging it, of an element ysituatedbelowtheneutralelementw.

Let's refer the hedges with respect to the upperpart (⊔), above the neutralelement, since for thelowerpart(L)itwillautomaticallybetheoppositeeffect.

Wesplitdesetofhedgesintothreedisjointsubsets:

H_⊔^+=thehedgesthatincreasetheneutrosophicvalueoftheupperelements;

H_⊔^-=thehedgesthatdecreasetheneutrosophicvalueoftheupperelements;

H_⊔^∼=thehedgesthatapproximatetheneutrosophicvalueoftheupperelements.

Notations: Let ��=��⊔∪��∪�� , where ��⊔ cons-titutes the upper element set, while �� the lower elementsubset,�� theneutralelement.��⊔ and�� aredisjointtwobytwo.

3.OperationsonNeutrosophicComponents

Let〈�� ,�� ,��〉,〈�� ,�� ,��〉neutrosophicnumbers. Then: �� +�� = �� +�� ,if�� +�� ≤1; 1,if�� +�� >1; (6) and

Florentin Smarandache (author and editor) Collected Papers, IX 468

�� −�� = 0,if�� −�� <0; �� −�� ,if�� −�� ≥0. (7) Similarlyfor�� and�� : �� +�� = �� +�� ,if�� +�� ≤1; 1,if�� +�� >1; (8) �� −�� = 0,if�� −�� <0; �� −�� ,if�� −�� ≥0. (9) and �� +�� = �� +��,if�� +�� ≤1; 1,if�� +�� >1; (10) �� −�� = 0,if�� −�� <0; �� −��,if�� −�� ≥0. (11)

4.NeutrosophicHedge-ElementOperators

Wedefinethefollowingoperators:

4.1.Neutrosophic

Increment

Hedge ↑ Element=〈�� ,�� ,��〉 ↑ 〈�� ,�� ,��〉=〈�� +�� ,�� −�� ,�� −��〉, (12) meaningthatthefirsttripletincreasesthesecond.

4.2.NeutrosophicDecrement

Hedge ↓ Element=〈�� ,�� ,��〉 ↓ 〈�� ,�� ,��〉=〈�� −�� ,�� +�� ,�� +��〉, (13) meaningthatthefirsttripletdecreasesthesecond.

4.3.Theorem1

Theneutrosophicincrementanddecrementoperatorsarenon-commutattive.

5.NeutrosophicHedge-HedgeOperators

Hedge ↑ Hedge=〈�� ,�� ,��〉 ↑ 〈�� ,�� ,��〉=〈�� +�� ,�� +�� ,�� +��〉 (14) Hedge ↓ Hedge=〈�� ,�� ,��〉 ↓ 〈�� ,�� ,��〉=〈�� −�� ,�� −�� ,�� −��〉 (15)

6.NeutrosophicHedgeOperators

Let

i.e.��⊔ isanupperelementof��(��),and ℎ

⊔〈�� ⊔

⊔,��⊔〉

⊔

⊔,�� ⊔〉∈��⊔, ℎ⊔ ∽〈�� ⊔ ∽,�� ⊔ ∽,�� ⊔ ∽〉∈��⊔ ∽ , thenℎ⊔ appliedto��⊔ gives (ℎ⊔��⊔)〈�� ⊔,�� ⊔,�� ⊔〉 ↑ 〈�� ⊔,�� ⊔,�� ⊔〉,

ℎ⊔ appliedto��⊔ gives

⊔��⊔)〈�� ⊔,�� ⊔,�� ⊔〉 ↓ 〈�� ⊔,�� ⊔,�� ⊔〉, andℎ⊔ appliedto��⊔ gives

⊔ ∼ ��⊔)〈�� ⊔,�� ⊔,�� ⊔〉 ↓ 〈�� ⊔,�� ⊔

⊔

Florentin Smarandache (author and editor) Collected Papers, IX 469

��

,��

∈��(��)

〈��

,��

〉∈��⊔, ℎ

〈��

,��

and

(ℎ

,��

〉

Now, let �� 〈�� ,�� ,�� 〉∈��(�� ), i.e. �� is a lower element of ��(��). Then, ℎ⊔ applied to �� gives: ℎ⊔�� 〈�� ,�� ,�� 〉 ↓ 〈�� ⊔,�� ⊔,�� ⊔〉, andℎ⊔ appliedto�� gives: ℎ⊔�� 〈�� ,�� ,�� 〉 ↑ 〈�� ⊔,�� ⊔,�� ⊔〉, andℎ⊔ appliedto�� gives: ℎ⊔�� 〈�� ,�� ,�� 〉 ↑ 〈�� ⊔ ∽,�� ⊔ ∽,�� ⊔ ∽〉.

In the same way, we may apply many increasing, decreasing, approximate or other type of hedgestothesameupperorlowerelement ℎ⊔ ℎ⊔ ℎ⊔…ℎ⊔ ��, generatingnewelementsin��(��) Thehedgesmaybeappliedtotheconstantsaswell.

6.1.Theorem2

Ahedgeappliedtoanotherhedgewekeansorstengthensorapproximatesit.

6.2.

Theorem3

If ℎ⊔ ∈��⊔ and��⊔ ∈��(��⊔),thenℎ⊔��⊔ ≥��⊔.

If ℎ⊔ ∈��⊔ and�� ∈��(�� ),thenℎ⊔�� ≤ ��

If ℎ⊔ ∈��⊔ and�� ∈��(�� ),thenℎ⊔�� ≥ �� .

6.3.ConverseHedges

Two hedges ℎ and ℎ ∈�� are converse to each other, if ∀��∈��(��), ℎ ��≤ �� is equivalenttoℎ ��≥ ��.

6.4.CompatibleHedges

Two hedges ℎ and ℎ ∈�� are compatible, if ∀��∈��(��), ℎ ��≤ �� is equivalent to ℎ ��≤ ��.

6.5.CommutativeHedges

Two hedges ℎ and ℎ ∈�� are commutative, if ∀��∈��(��), ℎ ℎ ��=ℎ ℎ ��. Otherwise theyarecallednon-commutative.

6.6.CumulativeHedges

If ℎ ⊔ andℎ ⊔ ∈�� ,thentwoneutrosophicedgescanbecumulatedintoone: ℎ ⊔ 〈�� ⊔ ,�� ⊔ ,�� ⊔ 〉ℎ ⊔ 〈�� ⊔ ,�� ⊔ ,�� ⊔ 〉=ℎ ⊔ 〈�� ⊔ ,�� ⊔ ,�� ⊔ 〉 ↑ 〈�� ⊔ ,�� ⊔ ,�� ⊔ 〉 (16) Similarly,ifℎ ⊔ andℎ ⊔ ∈�� ,thenwecancumulatethemintoone: ℎ ⊔ 〈�� ⊔ ,�� ⊔,�� ⊔ 〉ℎ ⊔ 〈�� ⊔ ,�� ⊔,�� ⊔ 〉=ℎ ⊔ 〈�� ⊔ ,�� ⊔,�� ⊔ 〉 ↑ 〈�� ⊔ ,�� ⊔,�� ⊔ 〉. (17)

Florentin Smarandache (author and editor) Collected Papers, IX 470

Now, if the two hedges are converse, ℎ ⊔ and ℎ ⊔, but the neutrosophic components of the first (which is actually a neutrosophic number) are greater than the second, we cumulate them into one as follows: ℎ ⊔ = ℎ ⊔ ℎ ⊔ 〈�� ⊔ ,�� ⊔ ,�� ⊔ 〉 ↓ 〈�� ⊔ ,�� ⊔,�� ⊔ 〉. (18)

But, if the neutrosophic components of the second are greater, and the hedges are commutative,wecumulatethemintooneasfollows: ℎ ⊔ = ℎ ⊔ ℎ ⊔ 〈�� ⊔ ,�� ⊔,�� ⊔ 〉 ↓ 〈�� ⊔ ,�� ⊔ ,�� ⊔ 〉 (19)

7.NeutrosophicHedgeAlgebra

��=(��,��,��,��∪��,≤ ) constitutes an abstract algebra, called Neutrosophic Hedge Algebra.

7.1.ExampleofaNeutrosophicHedgeAlgebra��

Let ��={��,��} the set of generators, repres-ented as neutrosophic generators as follows: ��〈 . , . , . 〉,��〈 . , . , . 〉. Let��={��,��}thesetofhedges,repres-entedasneutrosophichedgesasfollows: ��〈 . , . , . 〉,��〈 . , . , . 〉, where��∈��⊔ and��∈��⊔ �� is a neutrosophic linguistic variable whose domain is �� at the beginning, but extended by generators. Theneutrosophicconstantsare ��= 0〈 , , 〉,��〈 . , . , . 〉,1〈 , , 〉 . Theneutrosophicidentityis��〈 , , 〉. We use the neutrosophic inequality ≤ , and the neutrosophic increment / decrement operatorspreviouslydefined.

Let's apply the neutrosophic hedges in order to generate new neutrosophic elements of the neutrosophiclinguisticvariable�� appliedto��[upperelement]hasapositiveeffect: ��〈 . , . , . 〉��〈 . , . , . 〉 =(��)〈 . . , . . , . . 〉 =(��)〈 . , . , . 〉. Then: ��〈 . , . , . 〉(��)〈 . , . , . 〉 =(��)〈 . , , . 〉. ��appliedto�� [lowerelement]hasanegativeeffect: ��〈 . , . , . 〉��〈 . , . , . 〉 =(��)〈 . . , . . , . . 〉 = (��)〈 . , . , . 〉. Ifwecompute(��)first,whichisaneutrosophichedge-hedgeoperator: ��〈 . , . , . 〉��〈 . , . , . 〉 =(��)〈 . . , . . , . . 〉 = (��)〈 . , . , . 〉, andweapplyittoBig,weget: (��)〈 . , . , . 〉��〈 . , . , . 〉 =(��)〈 . . , . . , . . 〉 =(��)〈 . , , . 〉, so,wegetthesameresult. �� appliedto��hasanegativeeffect:

Florentin Smarandache (author and editor) Collected Papers, IX 471

��〈 . , . , . 〉��〈 . , . , . 〉 =(��)〈 . . , . . , . 〉 =(��)〈 . , . , . 〉.

�� appliedto�� hasapositiveeffect: ��〈 . , . , . 〉��〈 . , . , . 〉 =(��)〈 . . , . . , . . 〉 = (��)〈 . , . , . 〉.

The set of neutrosophic hedges H is enriched through the generation of new neutrosophic hedgesbycombiningahedgewithanotheroneusingtheneutrosophichedge-hedgeoperators. Further, the newly generated neutrosophic hedges are applied to the elements of the linguisticvariable,andmorenewelementsaregenerated.

Let'scomputemoreneutrosophicelements: ��=��〈 . , . , . 〉��〈 . , . , . 〉��〈 . , . , . 〉 =(��)〈 . , . , . 〉↑〈 . , . , . 〉 ↓〈 . , . , . 〉 =(��)〈 . . , . . , . . 〉↓〈 . , . , . 〉 =(��)〈 . . , . . , . . 〉 =(��)〈 . , , 〉 ��=��〈 . , . , . 〉��〈 . , . , . 〉 =(��)〈 . , . , . 〉↑〈 . , . , . 〉 =(��)〈 . , . , . 〉 ��=��〈 . , . , . 〉��〈 . , . , . 〉 =(��)〈 . , . , . 〉↓〈 . , . , . 〉 =(��)〈 . , . , . 〉 ��=��〈 . , . , . 〉��〈 . , . , . 〉��〈 . , . , . 〉 =(��)〈 . , . , . 〉��〈 . , . , . 〉 =(��)〈 . , . , . 〉 ��=��〈 . , . , . 〉��〈 . , . , . 〉��〈 . , . , . 〉 =��〈 . , . , . 〉(��)〈 . , . , . 〉 =(��)〈 . , . , . 〉 ��=��〈 . , . , . 〉��〈 , , 〉 =(��)〈 . , . , . 〉↓〈 , , 〉 =(��)〈 . , . , . 〉 ��=��〈 . , . , . 〉��〈 , , 〉 =(��)〈 . , . , . 〉↑〈 , , 〉 =(��)〈 . , . , . 〉

7.2.Theorem4

Any increasing hedge ℎ〈 ,, 〉 applied to the absolute maximum cannot overpass the absolute maximum.

Proof: ℎ〈 ,, 〉 ↑ 1〈 , , 〉 =(ℎ1)〈 , , 〉 =(ℎ1)〈 , , 〉 =1〈 , , 〉

7.3.

Theorem5

Any decreasing hedge ℎ〈 ,, 〉 applied to the absolute minimum cannot pass below the absoluteminimum.

Proof: ℎ〈 ,, 〉 ↓ 0〈 , , 〉 =(ℎ��)〈 , , 〉 =(ℎ��)〈 , , 〉 =0〈 , , 〉

8.DiagramoftheNeutrosophicHedgeAlgebraτ 1〈 , , 〉 ABSOLUTEMAXIMUM ��〈 . , , . 〉 VeryVeryBig

Florentin Smarandache (author and editor) Collected Papers, IX 472

��〈 . , . , . 〉 LessAbsoluteMaximum

��〈 . , . , . 〉 VeryBig

��〈 . , . , . 〉

��〈 . , . , . 〉 VeryMedium

��〈 . , . , . 〉 LessBig

��〈 . , , 〉 VeryLessBig

��〈 . , . , . 〉 VeryLessSmall

��〈 . , . , . 〉 MEDIUM

��〈 . , . , . 〉 LessMedium

��〈 . , . , . 〉 LessSmall

��〈 . , . , . 〉

��〈 . , . , . 〉 VerySmall

��〈 . , . , . 〉 LessAbsoluteMinimum

��〈 . , . , . 〉 VeryVerySmall

0〈 , , 〉

9.Conclusions

ABSOLUTEMINIMUM

In this paper, the classical hedge algebras have been extended for the first time to neutrosophic hedge algebras. With respect to an attribute, we have inserted the neutrosophic degrees of membership / indeterminacy / nonmembership of each generator, hedge, and constant. More than in the classical hedge algebras, we have introduced several numerical hedge operators: for hedge applied to element, and for hedge combined with hedge. An extensive example of a neutrosophichedgealgebraisgiven,andimportantpropertiesrelatedtoitarepresented.

References

Cat Ho, N.; Wechler,W. Hedge Algebras: An algebraic Approach to Structure of Sets of Linguistic Truth Values.FuzzySetsandSystems1990,281-293.

Lakoff, G. Hedges, a study in meaning criteria and the logic of fuzzy concepts. 8th Regional MeetingoftheChicagoLinguisticSociety,1972.

Zadeh, L.A. A fuzzy-set theoretic interpretation of linguistic hedges. Journal of Cybernetics 1972, Volume2,04-34.

Florentin Smarandache (author and editor) Collected Papers, IX 473

Neutrosophic quadruple ideals in neutrosophic quadruple BCI-algebras

G.Muhiuddin, Florentin Smarandache, Young Bae Jun

G. Muhiuddin, Florentin Smarandache, Young Bae Jun (2019). Neutrosophic quadruple ideals in neutrosophic quadruple BCI-algebras. Neutrosophic Sets and Systems 25, 161-173

Abstract: In the present paper, we discuss the Neutrosophic quadruple q-ideals and (regular) neutrosophic quadruple ideals and investigate their related properties. Also, for any two nonempty subsets U and V of a BCI-algebra S, conditions for the set N Q(U, V ) to be a (regular) neutrosophic quadruple ideal and a neutrosophic quadruple q-ideal of a neutrosophic quadruple BCI-algebra N Q(S) are discussed. Furthermore, we prove that let U, V, I and J be ideals of a BCI-algebra S such that I ⊆ U and J ⊆ V . If I and J are q-ideals of S, then the neutrosophic quadruple (U, V ) set N Q(U, V ) is a neutrosophic quadruple q-ideal of N Q(S)

Keywords: neutrosophic quadruple BCK/BCI-number, neutrosophic quadruple BCK/BCI-algebra, (regular) neutrosophic quadruple ideal, neutrosophic quadruple q-ideal.

1Introduction

Todealwithincomplete,inconsistentandindeterminateinformation,Smarandacheintroducedthenotionof neutrosophicsets(see([1],[2]and[3]).Infact,neutrosophicsetisausefulmathematicaltoolwhichextends thenotionsof classicset,(intuitionistic)fuzzysetandintervalvalued(intuitionistic)fuzzyset.Neutrosophic settheoryhasusefulapplicationsinseveralbranches(seefore.g.,[4],[5],[6]and[7]).

In[8],Smarandacheconsideredanentry(i.e.,anumber,anidea,anobjectetc.)whichisrepresentedbya knownpart (a) andanunknownpart (bT ,cI,dF ) where T,I,F havetheirusualneutrosophiclogicmeanings and a,b,c,d arerealorcomplexnumbers,andthenheintroducedtheconceptofneutrosophicquadruplenumbers.Neutrosophicquadruplealgebraicstructuresandhyperstructuresarediscussedin[9]and[10].Recently, neutrosophicsettheory hasbeenappliedtotheBCK/BCI-algebrasonvariousaspects(seefore.g.,[11],[12] [13],[14],[15],[16],[17],[18],[19]and[20].)Usingthenotionofneutrosophicquadruplenumbersbasedon aset,Junetal.[21]constructedneutrosophicquadrupleBCK/BCI-algebras.Theyinvestigatedseveralproperties,andconsideredidealandpositiveimplicativeidealinneutrosophicquadrupleBCK-algebra,andclosed

Collected Papers, IX 474

Florentin

Smarandache (author and editor)

idealinneutrosophicquadrupleBCI-algebra.Givensubsets A and B ofaneutrosophicquadrupleBCK/BCIalgebra,theyconsideredsets NQ(U,V ) whichconsistsofneutrosophicquadrupleBCK/BCI-numberswitha condition.Theyprovidedconditionsfortheset NQ(U,V ) tobea(positiveimplicative)idealofaneutrosophic quadrupleBCK-algebra,andtheset NQ(U,V ) tobea(closed)idealofaneutrosophicquadrupleBCI-algebra. Theygaveanexampletoshowthattheset { ˜ 0} isnotapositiveimplicativeidealinaneutrosophicquadrupleBCK-algebra,andthentheyconsideredconditionsfortheset {0} tobeapositiveimplicativeidealina neutrosophicquadrupleBCK-algebra.Muhiuddinetal.[22]discussedseveralpropertiesand(implicative) neutrosophicquadrupleidealsin(implicative)neutrosophicquadruple BCK-algebras.

Inthispaper,weintroducethenotionsof(regular)neutrosophicquadrupleidealandneutrosophicquadruple q-idealinneutrosophicquadrupleBCI-algebras,andinvestigaterelatedproperties.Givennonemptysubsets A and B ofaBCI-algebra S,weconsiderconditionsfortheset NQ(U,V ) tobea(regular)neutrosophic quadrupleidealof NQ(S) andaneutrosophicquadruple q-idealof NQ(S)

2Preliminaries

Webeginwiththefollowingdeﬁnitionsandpropertiesthatwillbeneededinthesequel. Anonemptyset S withaconstant0andabinaryoperation ∗ iscalledaBCI-algebraifforall x,y,z ∈ S thefollowingconditionshold([23]and[24]):

(I) (((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y)=0),

(II) ((x ∗ (x ∗ y)) ∗ y =0),

(III) (x ∗ x =0), (IV) (x ∗ y =0,y ∗ x =0 ⇒ x = y).

IfaBCI-algebra S satisﬁesthefollowingidentity:

(V) (∀x ∈ S)(0 ∗ x =0),

then S iscalleda BCK-algebra. Deﬁneabinaryrelation ≤ on X byletting x ∗ y =0 ifandonlyif x ≤ y. Then (S, ≤) isapartiallyorderedset.

Theorem2.1. Let S beaBCK/BCI-algebra.Thenfollowingconditionsarehold:

(∀x ∈ S)(x ∗ 0= x) , (2.1)

(∀x,y,z ∈ S)(x ≤ y ⇒ x ∗ z ≤ y ∗ z,z ∗ y ≤ z ∗ x) , (2.2)

(∀x,y,z ∈ S)((x ∗ y) ∗ z =(x ∗ z) ∗ y) , (2.3)

(∀x,y,z ∈ S)((x ∗ z) ∗ (y ∗ z) ≤ x ∗ y) (2.4) where x ≤ y ifandonlyif x ∗ y =0.

AnyBCI-algebra S satisﬁesthefollowingconditions(see[25]):

(∀x,y ∈ S)(x ∗ (x ∗ (x ∗ y))= x ∗ y), (2.5)

(∀x,y ∈ S)(0 ∗ (x ∗ y)=(0 ∗ x) ∗ (0 ∗ y)), (2.6)

(∀x,y ∈ S)(0 ∗ (0 ∗ (x ∗ y))=(0 ∗ y) ∗ (0 ∗ x)) (2.7)

Florentin Smarandache (author and editor) Collected Papers, IX 475

Anonemptysubset A ofaBCK/BCI-algebra S iscalleda subalgebra of S if x ∗ y ∈ A forall x,y ∈ A. A subset I ofaBCK/BCI-algebra S iscalledan ideal of S ifitsatisﬁes: 0 ∈ I, (2.8) (∀x ∈ S)(∀y ∈ I )(x ∗ y ∈ I ⇒ x ∈ I ) (2.9)

Anideal I ofaBCI-algebra S issaidtobe regular (see[26])ifitisalsoasubalgebraof S. ItisclearthateveryidealofaBCK-algebraisregular(see[26]).

Asubset I ofaBCI-algebra S iscalleda q-ideal of S (see[27])ifitsatisﬁes(2.8)and (∀x,y,z ∈ S)(x ∗ (y ∗ z) ∈ I,y ∈ I ⇒ x ∗ z ∈ I ) (2.10)

Wereferthereadertothebooks[25, 28]forfurtherinformationregardingBCK/BCI-algebras,andtothe site“http://fs.gallup.unm.edu/neutrosophy.htm”forfurtherinformationregardingneutrosophicsettheory. Weconsiderneutrosophicquadruplenumbersbasedonasetinsteadofrealorcomplexnumbers.

Deﬁnition2.2 ([21]). Let S beaset.A neutrosophicquadruple S-number isanorderedquadruple (a,xT,yI, zF ) where a,x,y,z ∈ S and T,I,F havetheirusualneutrosophiclogicmeanings.

Thesetofallneutrosophicquadruple S-numbersisdenotedby NQ(S),thatis, NQ(S):= {(a,xT,yI,zF ) | a,x,y,z ∈ S}, anditiscalledthe neutrosophicquadrupleset basedon S.If S isaBCK/BCI-algebra,aneutrosophicquadruple S-numberiscalleda neutrosophicquadrupleBCK/BCI-number andwesaythat NQ(S) isthe neutrosophic quadrupleBCK/BCI-set. Let S beaBCK/BCI-algebra.Wedeﬁneabinaryoperation on NQ(S) by (a,xT,yI,zF ) (b,uT,vI,wF )=(a ∗ b, (x ∗ u)T, (y ∗ v)I, (z ∗ w)F ) forall (a,xT,yI,zF ), (b,uT,vI,wF ) ∈ NQ(S).Given a1 ,a2 ,a3 ,a4 ∈ S,theneutrosophicquadruple BCK/BCI-number (a1 ,a2 T,a3 I,a4 F ) isdenotedby a,thatis, a =(a1 ,a2 T,a3 I,a4 F ), andthezeroneutrosophicquadrupleBCK/BCI-number (0, 0T, 0I, 0F ) isdenotedby 0,thatis, ˜ 0=(0, 0T, 0I, 0F ) Wedeﬁneanorderrelation“ ”andtheequality“=”on NQ(S) asfollows: x y ⇔ xi ≤ yi for i =1, 2, 3, 4, x =˜ y ⇔ xi = yi for i =1, 2, 3, 4 forall x, y ∈ NQ(S).Itiseasytoverifythat“ ”isanequivalencerelationon NQ(S)

Theorem2.3 ([21]). If S isaBCK/BCI-algebra,then (NQ(S); , ˜ 0) isaBCK/BCI-algebra.

Florentin Smarandache (author and editor) Collected Papers, IX 476

Wesaythat (NQ(S); , ˜ 0) isa neutrosophicquadrupleBCK/BCI-algebra, anditissimplydenotedby NQ(S).

Let S beaBCK/BCI-algebra.Givennonemptysubsets A and B of S,considertheset NQ(U,V ):= {(a,xT,yI,zF ) ∈ NQ(S) | a,x ∈ U & y,z ∈ V },

whichiscalledthe neutrosophicquadruple (U,V )-set Theset NQ(U,U ) isdenotedby NQ(U ),anditiscalledthe neutrosophicquadruple U -set.

3(Regular)neutrosophicquadrupleideals

Deﬁnition3.1. Givennonemptysubsets U and V ofaBCI-algebra S,iftheneutrosophicquadruple (U,V ) set NQ(U,V ) isa(regular)idealofaneutrosophicquadrupleBCI-algebra NQ(S),wesay NQ(U,V ) isa (regular)neutrosophicquadrupleideal of NQ(S).

Question1. If U and V aresubalgebrasofaBCI-algebra S,thenistheneutrosophicquadruple (U,V )-set NQ(U,V ) aneutrosophicquadrupleidealof NQ(S)?

TheanswertoQuestion 1 isnegativeasseeninthefollowingexample.

Example3.2. ConsideraBCI-algebra S = {0, 1,a,b,c} withthebinaryoperation ∗,whichisgiveninTable 1. ThentheneutrosophicquadrupleBCI-algebra NQ(S) has625elements.Notethat U = {0,a} and V = {0,b}

Table1:Cayleytableforthebinaryoperation“∗” ∗ 01 abc 000 abc 110 abc aaa 0 cb bbbc 0 a cccba 0 aresubalgebrasof S.Theneutrosophicquadruple (U,V )-set NQ(U,V ) consistsofthefollowingelements: NQ(U,V )= {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} where

0=(0, 0T, 0I, 0F ), 1=(0, 0T, 0I,bF ), 2=(0, 0T,bI, 0F ), 3=(0, 0T,bI,bF ), 4=(0,aT, 0I, 0F ), 5=(0,aT, 0I,bF ), 6=(0,aT,bI, 0F ), 7=(0,aT,bI,bF ), 8=(a, 0T, 0I, 0F ), ˜ 9=(a, 0T, 0I,bF ), ˜ 10=(a, 0T,bI, 0F ), ˜ 11=(a, 0T,bI,bF ), 12=(a,aT, 0I, 0F ), 13=(a,aT, 0I,bF ), ˜ 14=(a,aT,bI, 0F ), ˜ 15=(a,aT,bI,bF ).

Florentin Smarandache (author and editor) Collected Papers, IX 477

Ifwetake (1,aT,bI, 0F ) ∈ NQ(S),then (1,aT,bI, 0F ) / ∈ NQ(U,V ) and (1,aT,bI, 0F ) 9= 15 ∈ NQ(U,V )

Hencetheneutrosophicquadruple (U,V )-set NQ(U,V ) isnotaneutrosophicquadrupleidealof NQ(S)

Weconsiderconditionsfortheneutrosophicquadruple (U,V )-set NQ(U,V ) tobearegularneutrosophic quadrupleidealof NQ(S).

Lemma3.3 ([21]). If U and V aresubalgebras(resp.,ideals)ofaBCI-algebra S,thentheneutrosophic quadruple (U,V )-set NQ(U,V ) isaneutrosophicquadruplesubalgebra(resp.,ideal)of NQ(S).

Theorem3.4. Let U and V besubalgebrasofaBCI-algebra S suchthat (∀x,y ∈ S)(x ∈ U (resp., V ),y/ ∈ U (resp., V ) ⇒ y ∗ x/ ∈ U (resp., V )) (3.1) Thentheneutrosophicquadruple (U,V )-set NQ(U,V ) isaregularneutrosophicquadrupleidealof NQ(S).

Proof. ByLemma 3.3, NQ(U,V ) isaneutrosophicquadruplesubalgebraof NQ(S).Henceitisclearthat 0 ∈ NQ(U,V ).Let x =(x1 ,x2 T,x3 I,x4 F ) ∈ NQ(S) and y =(y1 ,y2 T,y3 I,y4 F ) ∈ NQ(S) besuchthat ˜ y ˜ x ∈ NQ(U,V ) and ˜ x ∈ NQ(U,V ).Then xi ∈ U and xj ∈ V for i =1, 2 and j =3, 4.Also, y x =(y1 ,y2 T,y3 I,y4 F ) (x1 ,x2 T,x3 I,x4 F ) =(y1 ∗ x1 , (y2 ∗ x2 )T, (y3 ∗ x3 )I, (y4 ∗ x4 )F ) ∈ NQ(U,V ), andso y1 ∗ x1 ∈ U , y2 ∗ x2 ∈ U , y3 ∗ x3 ∈ V and y4 ∗ x4 ∈ V .If y/ ∈ NQ(U,V ),then yi / ∈ A or yj / ∈ B for some i =1, 2 and j =3, 4.Itfollowsfrom(3.1)that yi ∗ xi / ∈ U or yj ∗ xj / ∈ V forsome i =1, 2 and j =3, 4 Thisisacontradiction,andso ˜ y ∈ NQ(U,V ).Thus NQ(U,V ) isaneutrosophicquadrupleidealof NQ(S), andtherefore NQ(U,V ) isaregularneutrosophicquadrupleidealof NQ(S). Corollary3.5. Let U beasubalgebraofaBCI-algebra S suchthat (∀x,y ∈ S)(x ∈ U,y/ ∈ U ⇒ y ∗ x/ ∈ U ) (3.2)

Thentheneutrosophicquadruple U -set NQ(U ) isaregularneutrosophicquadrupleidealof NQ(S). Theorem3.6. Let U and V besubsetsofaBCI-algebra S.Ifanyneutrosophicquadrupleideal NQ(U,V ) of NQ(S) satisﬁes 0 ˜ x ∈ NQ(U,V ) forall ˜ x ∈ NQ(U,V ),then NQ(U,V ) isaregularneutrosophic quadrupleidealof NQ(S). Proof. Forany x, y ∈ NQ(U,V ),wehave (˜ x y) x =(˜ x x) y = 0 y ∈ NQ(U,V ) Since NQ(U,V ) isanidealof NQ(S),itfollowsthat ˜ x ˜ y ∈ NQ(U,V ).Hence NQ(U,V ) isaneutrosophic quadruplesubalgebraof NQ(S),andtherefore NQ(U,V ) isaregularneutrosophicquadrupleidealof NQ(S).

Corollary3.7. Let U beasubsetofaBCI-algebra S.Ifanyneutrosophicquadrupleideal NQ(U ) of NQ(S) satisﬁes ˜ 0 ˜ x ∈ NQ(U ) forall ˜ x ∈ NQ(U ),then NQ(U ) isaregularneutrosophicquadrupleidealof NQ(S).

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Florentin

Theorem3.8. If U and V areidealsofafiniteBCI-algebra S,thentheneutrosophicquadruple (U,V )-set NQ(U,V ) isaregularneutrosophicquadrupleidealof NQ(S). Proof. ByLemma 3.3, NQ(U,V ) isaneutrosophicquadrupleidealof NQ(S).Since S isfinite, NQ(S) is alsofinite.Assumethat |NQ(S)| = n.Foranyelement x ∈ NQ(U,V ),considerthefollowing n+1 elements: 0, 0 x, (0 x) x, ··· , (··· ((0 x) x) ··· ) x n-times

Thenthereexistnaturalnumbers p and q with p>q suchthat (··· ((0 x) x) ··· ) x p-times =(··· ((0 x) x) ··· ) x q-times

Hence 0=((··· ((0 x) x) ··· ) x p times ) ((··· ((0 x) x) ··· ) x q times ) =((··· ((0 x) x) ··· ) x q times ) x) ··· ) x p q times ) ((··· ((0 x) x) ··· ) x q times ) =( ((0 ˜ x) ˜ x) ) ˜ x p q times

∈ NQ(U,V )

Since NQ(U,V ) isanidealof NQ(S),itfollowsthat 0 x ∈ NQ(U,V ).Therefore NQ(U,V ) isaregular neutrosophicquadrupleidealof NQ(S) byTheorem 3.6

Corollary3.9. If U isanidealofaﬁniteBCI-algebra S,thentheneutrosophicquadruple U -set NQ(U ) isa regularneutrosophicquadrupleidealof NQ(S).

4Neutrosophicquadruple q-ideals

Deﬁnition4.1. Givennonemptysubsets U and V of S,iftheneutrosophicquadruple (U,V )-set NQ(U,V ) isa q-idealofaneutrosophicquadrupleBCI-algebra NQ(S),wesay NQ(U,V ) isa neutrosophicquadruple q-ideal of NQ(S)

Example4.2. ConsideraBCI-algebra S = {0, 1,a} withthebinaryoperation ∗,whichisgiveninTable 2. ThentheneutrosophicquadrupleBCI-algebra NQ(S) has81elements.Ifwetake U = {0, 1} and V = {0, 1}, then NQ(U,V )= { ˜ 0, ˜ 1, ˜ 2, ˜ 3, ˜ 4, ˜ 5, ˜ 6, ˜ 7, ˜ 8, ˜ 9, ˜ 10, ˜ 11, ˜ 12, ˜ 13, ˜ 14, ˜ 15} isaneutrosophicquadruple q-idealof NQ(S) where 0=(0, 0T, 0I, 0F ), 1=(0, 0T, 0I, 1F ), 2=(0, 0T, 1I, 0F ), 3=(0, 0T, 1I, 1F ), 4=(0, 1T, 0I, 0F ), 5=(0, 1T, 0I, 1F ), ˜ 6=(0, 1T, 1I, 0F ), ˜ 7=(0, 1T, 1I, 1F ), ˜ 8=(1, 0T, 0I, 0F ), ˜ 9=(1, 0T, 0I, 1F ), ˜ 10=(1, 0T, 1I, 0F ), ˜ 11=(1, 0T, 1I, 1F ),

Florentin Smarandache (author and editor) Collected Papers, IX 479

∗ 01 a 000 a 110 a aaa 0 ˜ 12=(1, 1T, 0I, 0F ), ˜ 13=(1, 1T, 0I, 1F ), 14=(1, 1T, 1I, 0F ), 15=(1, 1T, 1I, 1F ).

Theorem4.3. Foranynonemptysubsets U and V ofaBCI-algebra S,iftheneutrosophicquadruple (U,V )-set NQ(U,V ) isaneutrosophicquadruple q-idealof NQ(S),thenitisbothaneutrosophicquadruplesubalgebra andaneutrosophicquadrupleidealof NQ(S).

Proof. Assumethat NQ(U,V ) isaneutrosophicquadruple q-idealof NQ(S).Since 0 ∈ NQ(U,V ),we have 0 ∈ U and 0 ∈ V .Let x,y,z ∈ S besuchthat x ∗ (y ∗ z) ∈ U ∩ V and y ∈ U ∩ V .Then (y,yT,yI,yF ) ∈ NQ(U,V ) and (x,xT,xI,xF ) ((y,yT,yI,yF ) (z,zT,zI,zF )) =(x,xT,xI,xF ) (y ∗ z, (y ∗ z)T, (y ∗ z)I, (y ∗ z)F ) =(x ∗ (y ∗ z), (x ∗ (y ∗ z))T, (x ∗ (y ∗ z))I, (x ∗ (y ∗ z))F ) ∈ NQ(U,V ).

Since NQ(U,V ) isaneutrosophicquadruple q-idealof NQ(S),itfollowsthat (x ∗ z, (x ∗ z)T, (x ∗ z)I, (x ∗ z)F )=(x,xT,xI,xF ) (z,zT,zI,zF ) ∈ NQ(U,V ).

Hence x ∗ z ∈ U ∩ V ,andtherefore U and V are q-idealsof S.Sinceevery q-idealisbothasubalgebra andanideal,itfollowsfromLemma 3.3 that NQ(U,V ) isbothaneutrosophicquadruplesubalgebraanda neutrosophicquadrupleidealof NQ(S).

TheconverseofTheorem 4.3 isnottrueasseeninthefollowingexample. Example4.4. ConsideraBCI-algebra S = {0,a,b,c} withthebinaryoperation ∗,whichisgiveninTable 3.

Table3:Cayleytableforthebinaryoperation“∗” ∗ 0 abc 00 cba aa 0 cb bba 0 c ccba 0

Table2:Cayleytableforthebinaryoperation“

∗”

Florentin Smarandache (author and editor) Collected Papers, IX 480

ThentheneutrosophicquadrupleBCI-algebra NQ(S) has256elements.Ifwetake A = {0} and B = {0}, then NQ(U,V )= {0} isbothaneutrosophicquadruplesubalgebraandaneutrosophicquadrupleidealof NQ(S).Ifwetake x :=(c,bT, 0I,aF ), z :=(a,bT, 0I,cF ) ∈ NQ(S),then ˜ x ( ˜ 0 ˜ z)=(c,bT, 0I,aF ) ( ˜ 0 (a,bT, 0I,cF )) =(c,bT, 0I,aF ) (c,bT, 0I,aF )= 0 ∈ NQ(U,V )

But x z =(c,bT, 0I,aF ) (a,bT, 0I,cF ) =(c ∗ a, (b ∗ b)T, (0 ∗ 0)I, (a ∗ c)F ) =(b, 0T, 0I,bF ) / ∈ NQ(U,V ).

Therefore NQ(U,V ) isnotaneutrosophicquadruple q-idealof NQ(S)

Weprovideconditionsfortheneutrosophicquadruple (U,V )-set NQ(U,V ) tobeaneutrosophicquadruple q-ideal.

Theorem4.5. If U and V are q-idealsofaBCI-algebra S,thentheneutrosophicquadruple (U,V )-set NQ(U,V ) isaneutrosophicquadruple q-idealof NQ(S).

Proof. Supposethat U and V are q-idealsofaBCI-algebra S.Obviously, ˜ 0 ∈ NQ(U,V ).Let ˜ x = (x1 ,x2 T,x3 I,x4 F ), y =(y1 ,y2 T,y3 I,y4 F ) and z =(z1 ,z2 T,z3 I,z4 F ) beelementsof NQ(S) besuch that x (˜ y z) ∈ NQ(U,V ) and y ∈ NQ(U,V ).Then yi ∈ A, yj ∈ B for i =1, 2 and j =3, 4,and ˜ x (˜ y ˜ z)=(x1 ,x2 T,x3 I,x4 F ) ((y1 ,y2 T,y3 I,y4 F ) (z1 ,z2 T,z3 I,z4 F )) =(x1 ,x2 T,x3 I,x4 F ) (y1 ∗ z1 , (y2 ∗ z2 )T, (y3 ∗ z3 )I, (y4 ∗ z4 )F ) =(x1 ∗ (y1 ∗ z1 ), (x2 ∗ (y2 ∗ z2 ))T, (x3 ∗ (y3 ∗ z3 ))I, (x4 ∗ (y4 ∗ z4 ))F ) ∈ NQ(U,V ), thatis, xi ∗ (yi ∗ zi) ∈ U and xj ∗ (yj ∗ zj ) ∈ B for i =1, 2 and j =3, 4.Itfollowsfrom(2.10)that xi ∗ zi ∈ U and xj ∗ zj ∈ V for i =1, 2 and j =3, 4.Thus ˜ x ˜ z =(x1 ∗ z1 , (x2 ∗ z2 )T, (x3 ∗ z3 )I, (x4 ∗ z4 )F ) ∈ NQ(U,V ), (4.1) andtherefore NQ(U,V ) isaneutrosophicquadruple q-idealof NQ(S).

Corollary4.6. If A isa q-idealofaBCI-algebra S,thentheneutrosophicquadruple U -set NQ(U ) isa neutrosophicquadruple q-idealof NQ(S).

Corollary4.7. If {0} isa q-idealofaBCI-algebra S,thentheneutrosophicquadruple (U,V )-set NQ(U,V ) isaneutrosophicquadruple q-idealof NQ(S) foranyideals U and V of S

Corollary4.8. If {0} isa q-idealofaBCI-algebra S,thentheneutrosophicquadruple U -set NQ(U ) isa neutrosophicquadruple q-idealof NQ(S) foranyideal U of S

Smarandache (author and editor) Collected Papers, IX 481

Florentin

Theorem4.9. Let U and V beidealsofaBCI-algebra S suchthat (∀x,y,z ∈ S)(x ∗ (y ∗ z) ∈ U ∩ V ⇒ (x ∗ y) ∗ z ∈ U ∩ V ) (4.2) Thentheneutrosophicquadruple (U,V )-set NQ(U,V ) isaneutrosophicquadruple q-idealof NQ(S) Proof. Itisclearthat 0 ∈ NQ(U,V ).Let ˜ x =(x1 ,x2 T,x3 I,x4 F ), ˜ y =(y1 ,y2 T,y3 I,y4 F ) and ˜ z = (z1 ,z2 T,z3 I,z4 F ) beelementsof NQ(S) besuchthat x (˜ y z) ∈ NQ(U,V ) and y ∈ NQ(U,V ).Then y1 ,y2 ∈ U , y3 ,y4 ∈ V and x (˜ y z)=(x1 ,x2 T,x3 I,x4 F ) ((y1 ,y2 T,y3 I,y4 F ) (z1 ,z2 T,z3 I,z4 F )) =(x1 ,x2 T,x3 I,x4 F ) (y1 ∗ z1 , (y2 ∗ z2 )T, (y3 ∗ z3 )I, (y4 ∗ z4 )F ) =(x1 ∗ (y1 ∗ z1 ), (x2 ∗ (y2 ∗ z2 ))T, (x3 ∗ (y3 ∗ z3 ))I, (x4 ∗ (y4 ∗ z4 ))F ) ∈ NQ(U,V ), thatis, xi ∗ (yi ∗ zi) ∈ U and xj ∗ (yj ∗ zj ) ∈ V for i =1, 2 and j =3, 4.Itfollowsfrom(2.3)and(4.2)that (xi ∗ zi) ∗ yi =(xi ∗ yi) ∗ zi ∈ U and (xj ∗ zj ) ∗ yj =(xj ∗ yj ) ∗ zj ∈ V for i =1, 2 and j =3, 4.Since U and V areidealsof S,wehave xi ∗ zi ∈ U and xj ∗ zj ∈ V for i =1, 2 and j =3, 4.Thus ˜ x ˜ z =(x1 ∗ z1 , (x2 ∗ z2 )T, (x3 ∗ z3 )I, (x4 ∗ z4 )F ) ∈ NQ(U,V ), (4.3) andtherefore NQ(U,V ) isaneutrosophicquadruple q-idealof NQ(S).

Corollary4.10. Let U beanidealofaBCI-algebra S suchthat (∀x,y,z ∈ S)(x ∗ (y ∗ z) ∈ U ⇒ (x ∗ y) ∗ z ∈ U ) (4.4)

Thentheneutrosophicquadruple U -set NQ(U ) isaneutrosophicquadruple q-idealof NQ(S).

Theorem4.11. Let U and V beidealsofaBCI-algebra S suchthat (∀x,y ∈ S)(x ∗ (0 ∗ y) ∈ U ∩ V ⇒ x ∗ y ∈ U ∩ V ). (4.5)

Thentheneutrosophicquadruple (U,V )-set NQ(U,V ) isaneutrosophicquadruple q-idealof NQ(S) Proof. Assumethat x ∗ (y ∗ z) ∈ U ∩ V forall x,y,z ∈ S.Notethat ((x ∗ y)) ∗ (0 ∗ z)) ∗ (x ∗ (y ∗ z))=((x ∗ y) ∗ (x ∗ (y ∗ z))) ∗ (0 ∗ z) ≤ ((y ∗ z) ∗ y) ∗ (0 ∗ z) =(0 ∗ z) ∗ (0 ∗ z)=0 ∈ U ∩ V

Thus (x ∗ y) ∗ (0 ∗ z) ∈ U ∩ V since U and V areidealsof S.Itfollowsfrom(4.9)that (x ∗ y) ∗ z ∈ U ∩ V . UsingTheorem 4.9, NQ(U,V ) isaneutrosophicquadruple q-idealof NQ(S) Corollary4.12. Let U beanidealofaBCI-algebra S suchthat (∀x,y ∈ S)(x ∗ (0 ∗ y) ∈ U ⇒ x ∗ y ∈ U ) (4.6)

Thentheneutrosophicquadruple U -set NQ(U ) isaneutrosophicquadruple q-idealof NQ(S).

Florentin Smarandache (author and editor) Collected Papers, IX 482

Theorem4.13. Let U and V beidealsofaBCI-algebra S suchthat (∀x,y ∈ S)(x ∈ U ∩ U ⇒ x ∗ y ∈ U ∩ V ) (4.7)

Thentheneutrosophicquadruple (U,V )-set NQ(U,V ) isaneutrosophicquadruple q-idealof NQ(S) Proof. Assumethat x ∗ (y ∗ z) ∈ U ∩ V and y ∈ U ∩ V forall x,y,z ∈ S.Using(2.3)and(4.7),weget (x ∗ z) ∗ (y ∗ z)=(x ∗ (y ∗ z)) ∗ z ∈ U ∩ V and y ∗ z ∈ U ∩ V .Since U and V areidealsof S,itfollowsthat x ∗ z ∈ U ∩ V .Hence U and V are q-idealsof S,andtherefore NQ(U,V ) isaneutrosophicquadruple q-ideal of NQ(S) byTheorem 4.5 Corollary4.14. Let U beanidealofaBCI-algebra S suchthat (∀x,y ∈ S)(x ∈ U ⇒ x ∗ y ∈ U ). (4.8)

Thentheneutrosophicquadruple U -set NQ(U ) isaneutrosophicquadruple q-idealof NQ(S)

Theorem4.15. Let U,V,I and J beidealsofaBCI-algebra S suchthat I ⊆ U and J ⊆ V .If I and J are q-idealsof S,thentheneutrosophicquadruple (U,V )-set NQ(U,V ) isaneutrosophicquadruple q-idealof NQ(S)

Proof. Let x,y,z ∈ S besuchthat x ∗ (0 ∗ y) ∈ U ∩ V .Then (x ∗ (x ∗ (0 ∗ y))) ∗ (0 ∗ y)=(x ∗ (0 ∗ y)) ∗ (x ∗ (0 ∗ y))=0 ∈ I ∩ J by(2.3)and(III).Since I and J are q-idealsof S,itfollowsfrom(2.3)and(2.10)that (x ∗ y) ∗ (x ∗ (0 ∗ y))=(x ∗ (x ∗ (0 ∗ y))) ∗ y ∈ I ∩ J ⊆ U ∩ V

Since U and V areidealsof S,wehave x ∗ y ∈ U ∩ V .Therefore NQ(U,V ) isaneutrosophicquadruple q-idealof NQ(S) byTheorem 4.11.

Corollary4.16. Let U and I beidealsofaBCI-algebra S suchthat I ⊆ U .If I isaq-idealof S,thenthe neutrosophicquadruple U -set NQ(U ) isaneutrosophicquadruple q-idealof NQ(S). Theorem4.17. Let U,V,I and J beidealsofaBCI-algebra S suchthat I ⊆ U , J ⊆ V and (∀x,y,z ∈ S)(x ∗ (y ∗ z) ∈ I ∩ J ⇒ (x ∗ y) ∗ z ∈ I ∩ J ) (4.9)

Thentheneutrosophicquadruple (U,V )-set NQ(U,V ) isaneutrosophicquadruple q-idealof NQ(S).

Proof. Let x,y,z ∈ S besuchthat x ∗ (y ∗ z) ∈ I ∩ J and y ∈ I ∩ J .Then (x ∗ z) ∗ y =(x ∗ y) ∗ z ∈ I ∩ J by(2.3)and(4.9).Since I and J areidealsof S,itfollowsthat x ∗ z ∈ I ∩ J .Thisshowsthat I and J are q-idealsof S.Therefore NQ(U,V ) isaneutrosophicquadruple q-idealof NQ(S) byTheorem 4.15. Corollary4.18. Let U and I beidealsofaBCI-algebra S suchthat I ⊆ U and (∀x,y,z ∈ S)(x ∗ (y ∗ z) ∈ I ⇒ (x ∗ y) ∗ z ∈ I ) (4.10)

Florentin Smarandache (author and editor) Collected Papers, IX 483

Thentheneutrosophicquadruple U -set NQ(U ) isaneutrosophicquadruple q-idealof NQ(S).

Theorem4.19. Let U,V,I and J beidealsofaBCI-algebra S suchthat I ⊆ U , J ⊆ V and (∀x,y ∈ S)(x ∈ I ∩ J ⇒ x ∗ y ∈ I ∩ J ) (4.11)

Thentheneutrosophicquadruple (U,V )-set NQ(U,V ) isaneutrosophicquadruple q-idealof NQ(S).

Proof. BytheproofofTheorem 4.13,weknowthat I and J are q-idealsof S.Hence NQ(U,V ) isaneutrosophicquadruple q-idealof NQ(S) byTheorem 4.15.

Corollary4.20. Let U and I beidealsofaBCI-algebra S suchthat I ⊆ U and (∀x,y ∈ S)(x ∈ I ⇒ x ∗ y ∈ I ) (4.12)

Thentheneutrosophicquadruple A-set NQ(U ) isaneutrosophicquadruple q-idealof NQ(S).

Theorem4.21. Let U,V,I and J beidealsofaBCI-algebra S suchthat I ⊆ U , J ⊆ V and (∀x,y ∈ S)(x ∗ (0 ∗ y) ∈ I ∩ J ⇒ x ∗ y ∈ I ∩ J ) (4.13)

Thentheneutrosophicquadruple (U,V )-set NQ(U,V ) isaneutrosophicquadruple q-idealof NQ(S).

Proof. Assumethat x∗(y ∗z) ∈ I ∩J Forall x,y,z ∈ S.Then (x∗y)∗z ∈ I ∩J bytheproofofTheorem 4.11 ItfollowsfromTheorem 4.17 thatneutrosophicquadruple (U,V )-set NQ(U,V ) isaneutrosophicquadruple q-idealof NQ(S).

Corollary4.22. Let U and I beidealsofaBCI-algebra S suchthat I ⊆ U and (∀x,y ∈ S)(x ∗ (0 ∗ y) ∈ I ⇒ x ∗ y ∈ I ). (4.14)

Thentheneutrosophicquadruple U -set NQ(U ) isaneutrosophicquadruple q-idealof NQ(S).

FutureWork

:Usingtheresultsofthispaper,wewillaplyittoanotheralgebraicstructures,forexample, MV-algebras,BL-algebras,MTL-algebras, R0 -algebras,hoops,(ordered)semigroupsand(semi,near)rings etc.

Acknowledgements

:Weareverythankfultothereviewer(s)forcarefuldetailedreadingandhelpfulcomments/suggestionsthatimprovetheoverallpresentationofthispaper.

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Florentin Smarandache (author and editor) Collected Papers, IX 486

This ninth volume of Collected Papers includes 87 papers comprising 982 pages on (theoretic and applied) neutrosophics, written between 2014-2022 by the author alone or in collaboration with the following 81 co-authors (alphabetically ordered): E.O. Adeleke, A.A.A. Agboola, Ahmed B. Al Nafee, Ahmed Mostafa Khalil, Akbar Rezaei, S.A. Akinleye, Ali Hassan, Mumtaz Ali, Rajab Ali Borzooei , Assia Bakali, Cenap Özel, Victor Christianto, Chunxin Bo, Rakhal Das, Bijan Davvaz, R. Dhavaseelan, B. Elavarasan, Fahad Alsharari, T. Gharibah, Hina Gulzar, Hashem Bordbar, Le Hoang Son, Emmanuel Ilojide, Tèmítópé Gbóláhàn Jaíyéolá, M. Karthika, Ilanthenral Kandasamy, W.B. Vasantha Kandasamy, Huma Khan, Madad Khan, Mohsin Khan, Hee Sik Kim, Seon Jeong Kim, Valeri Kromov, R. M. Latif, Madeleine Al-Tahan, Mehmat Ali Ozturk, Minghao Hu, S. Mirvakili, Mohammad Abobala, Mohammad Hamidi, Mohammed Abdel-Sattar, Mohammed A. Al Shumrani, Mohamed Talea, Muhammad Akram, Muhammad Aslam, Muhammad Aslam Malik, Muhammad Gulistan, Muhammad Shabir, G. Muhiuddin, Memudu Olaposi Olatinwo, Osman Anis, Choonkil Park, M. Parimala, Ping Li, K. Porselvi, D. Preethi, S. Rajareega, N. Rajesh, Udhayakumar Ramalingam, Riad K. Al Hamido, Yaser Saber, Arsham Borumand Saeid, Saeid Jafari, Said Broumi, A.A. Salama, Ganeshsree Selvachandran, Songtao Shao, Seok Zun Song, Tahsin Oner, M. Mohseni Takallo, Binod Chandra Tripathy, Tugce Katican, J. Vimala, Xiaohong Zhang, Xiaoyan Mao, Xiaoying Wu, Xingliang Liang, Xin Zhou, Yingcang Ma, Young Bae Jun, Juanjuan Zhang