On 2-SuperHyperLeftAlmostSemihypergroups

Neutrosophic Sets and Systems, Vol. 51, 2022

On2-SuperHyperLeftAlmostSemihypergroups

PairoteYiarayong1,∗ 1DepartmentofMathematics,FacultyofScienceandTechnology,PibulsongkramRajabhatUniversity, Phitsanulok65000,Thailand;E-mail:pairote0027@hotmail.com ∗Correspondence:pairote0027@hotmail.com

Abstract TheaimofthispaperistoextendtheconceptofhyperidealstotheSuperHyperAlgebras.In thispaper,weintroducetheconceptof2-SuperHyperLeftAlmostSemihypergroupswhichisageneralizationof LA-semihypergroups.Furthermore,wedefineandstudy2-SuperHyper-LA-subsemihypergroups,SuperHyperLeft(Right)HyperIdealsandSuperHyperHyperIdealsof2-SuperHyperLeftAlmostSemihypergroups,andrelated propertiesareinvestigated.Wegiveanexampletoshowthatingeneralthesetwonotionsaredifferent.Finally, weshowthateverySuperHyperRightHyperIdealof2-SuperHyper-LA-semihypergroup S withpureleftidentity isSuperHyperHyperIdeal.

Keywords: SuperHyperAlgebra; LA-subsemihypergroup;2-SuperHyperLeftAlmostSemihypergroup;SuperHyperHyperIdeals;SuperHyperLeft(Right)HyperIdeal.

1. Introduction

Theconceptofleftalmostsemihypergroups(LA-semihypergroups),whichisageneralization of LA-semigroupsandsemihypergroups,wasintroducedbyHilaandDine[9]in2011.They definedtheconceptofhyperidealsandbi-hyperidealsin LA-semihypergroups.Untilnow, LA-semihypergroupshavebeenappliedtomanyfields[2, 4–6, 13, 16, 18].In2013,Yaqoobet al.[17]havecharacterizedintra-regular LA-semihypergroupsbyusingthepropertiesoftheir leftandrighthyperidealsandinvestigatedsomeusefulconditionsforan LA-semihypergroupto becomeanintra-regular LA-semihypergroup.In2014,Amjadetal.[1]generalizedtheconcepts oflocallyassociative LA-semigroupstohypergroupoidsandstudiedseveralproperties.They definedtheconceptoflocallyassociative LA-semihypergroupsandcharacterizedalocally associative LA-semihypergroupintermsof(m,n)-hyperideals.In2016,Khanetal.[10]proved thatan LA-semigroup S is0(0, 2)-bisimpleifandonlyif S isright0-simple.In2018,Azhar etal.[3]appliedthenotionof(∈, ∈∨qk)-fuzzysetsto LA-semihypergroups.Theyintroduced P.Yiarayong,OnSoftNeutrosophicHyperidealoverNeutrosophic LA-Semihypergroups

thenotionof(∈, ∈∨qk)-fuzzyhyperidealsinanordered LA-semihypergroupandthenderived theirbasicproperties.In2019,Gulistanetal.[8]presentedanewdefinitionofgeneralized fuzzyhyperideals,generalizedfuzzybi-hyperidealsandgeneralizedfuzzynormalbi-hyperideals inanordered LA-semihypergroup.Theycharacterizedordered LA-semihypergroupsbythe propertiesoftheir(∈γ , ∈γ ∨qδ)-fuzzyhyperideals,(∈γ , ∈γ ∨qδ)-fuzzybi-hyperidealsand(∈γ , ∈γ ∨qδ)-fuzzynormalbi-hyperideals.In2021,Suebsungetal.[12]haveintroducedthenotion ofleftalmosthyperideals,rightalmosthyperideals,almosthyperidealsandminimalalmost hyperidealsin LA-semihypergroups.In2022,Nakkhasen[11]characterizedintra-regular LAsemihyperringsbythepropertiesoftheirhyperideals.

Inthispaper,weextendtheconceptofhyperidealstotheSuperHyperAlgebras.Inthis paper,weintroducetheconceptof2-SuperHyperLeftAlmostSemihypergroupswhichisageneralizationof LA-semihypergroups.Furthermore,wedefineandstudy2-SuperHyper-LAsubsemihypergroups,SuperHyperLeft(Right)HyperIdealsandSuperHyperHyperIdealsof2SuperHyperLeftAlmostSemihypergroups,andrelatedpropertiesareinvestigated.Wegivean exampletoshowthatingeneralthesetwonotionsaredifferent.Finally,weshowthatevery SuperHyperRightHyperIdealof2-SuperHyper-LA-semihypergroup S withpureleftidentityis SuperHyperHyperIdeal.

2. PreliminariesandBasicDefinitions

Inthissection,wegivesomebasicdefinitionsandpropertiesofleftalmostsemihypergroups andclassical-typeBinarySuperHyperOperationsthatarerequiredinthisstudy.

Recallthatamapping ◦ : S × S →P∗(S),where P∗(S)denotesthefamilyofallnonempty subsetsof S,iscalleda hyperoperation on S.Animageofthepair(x,y)isdenotedby x ◦ y.Thecouple(S, ◦)iscalleda hypergroupoid Let x beanelementsofanonemptysetof S andlet A,B betwononemptysubsetsof S Thenwedenote A ◦ B = ∪ a∈A,b∈B a ◦ b,x ◦ B = {x}◦ B and A ◦ x = A ◦{x}.

In2011,HilaandDine[9]introducedtheconceptandnotionofleftalmostsemihypergroup asageneralizationofsemigroups, LA-semigroupsandsemihypergroups. Definition2.1. [9]Ahypergroupoid(S, ◦)iscalleda leftalmostsemihypergroup(LAsemihypergroup) if ◦ isleftinvertivelaw,thatis(x ◦ y) ◦ z =(z ◦ y) ◦ x forevery x,y,z ∈ S

Clearly,every LA-semihypergroupis LA-semigroup.If(S, ◦)isan LA-semihypergroup, then ∪ a∈x◦y a ◦ z = ∪ b∈z◦y

b ◦ x forall x,y,z ∈ S.

Theconceptofclassical-typebinarySuperHyperOperationwasintroducedbySmarandache [14, 15].

P.Yiarayong,On2-SuperHyperLeftAlmostSemihypergroups

Definition2.2. [14, 15]Let Pn ∗ (S)bethe nth-powersetoftheset S suchthatnoneof P(S), P 2(S),..., P n(S)containtheemptyset.A classical-typebinarySuperHyperOperation •n isdefinedasfollows:

•n : S × S →P n ∗ (S) where P n ∗ (S)isthe nth-powersetoftheset S,withnoemptyset.

Animageofthepair(x,y)isdenotedby x •n y.Thecouple(S, •n)iscalleda2SuperHyperGroupoid.

ThefollowingisanexampleofExamplesofclassical-typebinarySuperHyperOperation(or 2-SuperHyperGroupoid).

Example2.3. [14]Let S = {a,b} beafinitediscreteset.Thenitspowerset,withouttheempty-set ∅,is: P(S)= {a,b,S} and P 2(S)= P 2 (P(S))= P 2 ({a,b,S})= {a,b,S, {a,S} , {b,S} , {a,b,S}}.Theclassical-typebinarySuperHyperOperationdefinedas follows, •2 : S × S →P 2 ∗ (S)

•2 a b a {a,S}{b,S} b a {a,b,S} Then(S, •2)isa2-SuperHyperGroupoidandisnotahypergroupoid.

3.2-SuperHyperLeftAlmostSemihypergroups

Inthissection,wegeneralizethisconceptinleftalmostsemihypergroupandintroduceSuperHyperLeft(Right)HyperIdealsof2-SuperHyper-LA-semihypergroupsandstudytheirproperties.

The2-SuperHyperLeftAlmostSemihypergroupsisgeneratedwiththehelpofleftalmost semihypergroupsandclassical-typebinarySuperHyperOperations.Sowecansaythat2SuperHyperLeftAlmostSemihypergroupisthegeneralizationofpreviouslydefinedconcepts relatedtobinarySuperHyperOperations.WeconsidertheSuperHyperLeftAlmostSemihypergroupasfollows.

Definition3.1. A2-SuperHyperGroupoid(S, •n)iscalleda nSuperHyperLeftAlmostSemihypergroup(2-SuperHyper-LA-semihypergroup) ifit satisfiestheSuperHyperLeftInvertivelaw;(x •n y) •n z =(z •n y) •n x forall x,y,z ∈ S

Thefollowingisanexampleofa2-SuperHyper-LA-semihypergroup S.

Example3.2. Let S = {a,b} beafinitediscreteset.Theclassical-typebinarySuperHyperOperationdefinedasfollows, •2 : S × S →P 2 ∗ (S) P.Yiarayong,On2-SuperHyperLeftAlmostSemihypergroups

•2 a b a {a,S} b b {b,S}{a,b,S}

Then,asiseasilyseen,(S, •2)isa2-SuperHyper-LA-semihypergroup.Since

(a •2 a) •2 b = {a,S}•2 a =(a •2 a) ∪ (S •2 a) = {a,S}∪ ∪ x∈S x •2 a = {a,S}∪ (a •2 a) ∪ (b •2 a) = {a,S}∪{a,S}∪{b,S} = {a,b,S} = b = a •2 b = a •2 (a •2 b) , wehave •2 isnotStrongSuperHyperAssociativity.

Theorem3.3. Every 2-SuperHyper-LA-semihypergroup S satisfiesthe SuperHyperMedial law,thatis,forall a,b,c,d ∈ S, (a •n b) •n (c •n d)=(a •n c) •n (b •n d).

Proof. Let a,b,c and d beanyelementsof S.Thenwehave (a •n b) •n (c •n d)=((c •n d) •n b) •n a =((b •n d) •n c) •n a =(a •n c) •n (b •n d) Thiscompletestheproof.

Theorem3.4. If S isa 2-SuperHyper-LA-semihypergroup,then (a •n b)2 = a2 •n b2 forall a,b ∈ S

Proof. Let a and b beanyelementsof S.ThenbyTheorem 3.3, (a •n b)2 =(a •n b) •n (a •n b) =(a •n a) •n (b •n b) = a2 •n b2

Anelement e ofa2-SuperHyper-LA-semihypergroup S iscalled leftidentity(resp.,pure leftidentity) ifforall a ∈N (S),a ∈ e •n a (resp., a = e •n a).Thefollowingisanexample ofapureleftidentityelementin2-SuperHyper-LA-semihypergroups. P.Yiarayong,On2-SuperHyperLeftAlmostSemihypergroups

Example3.5. 1.Let S = {a,b} beafinitediscreteset.Theclassical-typebinarySuperHyperOperationdefinedasfollows, •2 : S × S →P 2 ∗ (S) •2 a b a a {a,b,S} b {b,S} S

Then,asiseasilyseen,(S, •2)isa2-SuperHyper-LA-semihypergroupwithleftidentity a. 2.Let S = {a,b} beafinitediscreteset.Theclassical-typebinarySuperHyperOperation definedasfollows, •2 : S × S →P 2 ∗ (S) •2 ab a ab b bS

Then,asiseasilyseen,(S, •2)isa2-SuperHyper-LA-semihypergroupwithpureleftidentity a

Theorem3.6. A 2-SuperHyper-LA-semihypergroup S withpureleftidentity e satisfiesthe SuperHyperParamediallaw,thatis,forall a,b,c,d ∈ S, (a •n b) •n (c •n d)=(d •n c) •n (b •n a)

Proof. Let a,b,c and d beanyelementsof S.Thenwehave

(a •n b) •n (c •n d)=[(e •n a) •n b] •n (c •n d) =[(b •n a) •n e] •n (c •n d) =[(c •n d) •n e] •n (b •n a) =[(e •n d) •n c] •n (b •n a) =(d •n c) •n (b •n a) .

Thiscompletestheproof.

Thefollowingmaybenotedfromtheabovedefinitions.

Lemma3.7. If S isa 2-SuperHyper-LA-semihypergroupwithpureleftidentity,then a •n (b •n c)= b •n (a •n c) holdsforall a,b,c ∈ S.

Proof. Let a,b and c beanyelementsof S.ThenbyTheorem 3.3, a • (b •n c)=(e •n a) • (b •n c) =(e •n b) • (a •n c) = b •n (a •n c) .

Thiscompletestheproof.

Now,wegivetheconceptof2-SuperHyperLeftAlmostSemihypergroups(2-SuperHyper-LAsubsemihypergroup)of2-SuperHyper-LA-semihypergroups. P.Yiarayong,On2-SuperHyperLeftAlmostSemihypergroups

Definition3.8. Anonemptysubset A ofa2-SuperHyper-LA-semihypergroup S iscalled2SuperHyperLeftAlmostSemihypergroup(2-SuperHyper-LA-subsemihypergroup) if A •n A ⊆ A

Thefollowingmaybenotedfromtheabovedefinitions.

Proposition3.9. Let A and B betwo 2-SuperHyper-LA-subsemihypergroupsofa 2SuperHyper-LA-semihypergroup S.If A ∩ B = ∅,then A ∩ B isa 2-SuperHyper-LAsubsemihypergroupof S.

Proof. Let A and B betwo2-SuperHyper-LA-subsemihypergroupsof S suchthat A ∩ B = ∅. Thenhavethat (A ∩ B) •2 (A ∩ B)=[A •n (A ∩ B)] ∩ [B •n (A ∩ B)] =(A •n A) ∩ (A •n B) ∩ (B •n A) ∩ (B •n B) ⊆ (A •n A) ∩ (B •n B) ⊆ A ∩ B, andso A ∩ B isa2-SuperHyper-LA-subsemihypergroupof S.

Nowwementionsomespecialclassof2-SuperHyper-LA-subsemihypergroupsina2SuperHyper-LA-semihypergroup.

Definition3.10. Anonemptysubset L ofa2-SuperHyper-LA-semihypergroup S iscalled SuperHyperLeft(Right)HyperIdeal if S •n L ⊆ L (R •n S ⊆ R).

Anonemptysubset I of S iscalleda SuperHyperHyperIdeal of S ifitisbothaSuperHyperLeftandaSuperHyperRightHyperIdealof S.

Proposition3.11. Let N (S) bea 2-SuperHyper-LA-semihypergroupwithpureleftidentity. Thenthefollowingpropertieshold.

(1) If L isaSuperHyperLeftHyperIdealof S,then S •n L = L.

(2) If N (R) isaSuperHyperRightHyperIdealof S,then R •n S = R.

(3) S •n S = S

Proof. 1.Since L isaSuperHyperLeftHyperIdealof S,wehave S •n L ⊆ L.Ontheother hand,let a beanelementof S suchthat a ∈ L.Thenwehave a = e •n a ∈ S •n L andhence S •n L = L P.Yiarayong,On2-SuperHyperLeftAlmostSemihypergroups

2.Since R isaSuperHyperRightHyperIdealof S,wehave R •n S ⊆ R.Ontheotherhand, let a beanelementof S suchthat a ∈ R.Thenwehave a = e •n a =(e •n e) •n a =(a •n e) •n e ⊆ (R •n S) •n S ⊆ R •n S.

Thereforeweobtainthat R ⊆ R •n S andhence R •n S = R 3.Theproofissimilartotheproofof(2). Byapplyingtheabovedefinition,westatethefollowingresult.

Theorem3.12. Let S bea 2-SuperHyper-LA-semihypergroupwithpureleftidentity.Then thefollowingpropertieshold.

(1) If x isanelementof S,then x •n S isaSuperHyperLeftHyperIdealof S. (2) If x isanelementof S,then S •n x isaSuperHyperLeftHyperidealof S (3) If x isanelementof S,then S •n x ∪ x •n S isaSuperHyperRightHyperIdealof S.

Proof. 1.Let x beanelementof S.ByLemma 3.7 andProposition 3.11 (3),wehave S •n [x •n S]= x •n [S •n S] = x •n S.

Thereforeweobtainthat x •n S isaSuperHyperLeftHyperIdealof S 2.Let x beanelementof S.ByTheorem 3.6 andProposition 3.11 (3),wehave S •n (S •n x)=(S •n S) •n (S •n x) =(x •n S) •n (S •n S) =[(S •n S) •n S] •n x = S •n x.

Thereforeweobtainthat S •n x isaSuperHyperLeftHyperIdealof S 3.Let x beanelementof S.ByTheorem 3.6,Lemma 3.7 andProposition 3.11 (3),we have (S •n x ∪ x •n S) •n S =[(S •n x) •n S] ∪ [(x •n S) •n S] =[(S •n x) •n (S •n S)] ∪ [(S •n S) •n x] =[(S •n S) •n (x •n S)] ∪ (S •n x) =[x •n ((S •n S) •n S)] ∪ (S •n x) = S •n x ∪ x •n S.

Thereforeweobtainthat S •n x ∪ x •n S isaSuperHyperRightHyperIdealof S.

Forthat,weneedthefollowingtheorem. P.Yiarayong,On2-SuperHyperLeftAlmostSemihypergroups

Theorem3.13. Let S bea 2-SuperHyper-LA-semihypergroupwithpureleftidentity.Then thefollowingpropertieshold.

(1) If x isanelementof S,then x2 •n S isaSuperHyperHyperIdealof S.

(2) If x isanelementof S,then S •n x2 isaSuperHyperHyperIdealof S.

(3) If x isanelementof S,then S •n x ∪ x •n S isaSuperHyperHyperIdealof S

Proof. 1.Let x beanelementof N (S).ByTheorem 3.12 (1),wehavethat x2 •n S isa SuperHyperLeftHyperIdealof N (S).Since (x2 •n S) •n S =(S •n S) •n x2 = x2 •n (S •n S) = x2 •n S, wehave x2 •n S isaSuperHyperRightHyperIdealof S andso x2 •n S isaSuperHyperHyperIdeal of S.

2.Theproofissimilartotheproofof(1).

3.Let x beanelementof S.ByTheorem 3.12 (3),wehavethat S •n x ∪ x •n S isa SuperHyperRightHyperIdealof N (S).ByTheorem 3.6,Lemma 3.7 andProposition 3.11 (3), wehave S •n (S •n x ∪ x •n S)=[S •n (S •n x)] ∪ [S •n (x •n S)] =[(S •n S) •n (S •n x)] ∪ [x •n (S •n S)] =[(x •n S) •n (S •n S)] ∪ (x •n S) =[((S •n S) •n S) •n x] ∪ (x •n S) = S •n x ∪ x •n S.

Thereforeweobtainthat S •n x ∪ x •n S isaSuperHyperLeftHyperIdealof S andhence S •n x ∪ x •n S isaSuperHyperHyperIdealof S

Theorem3.14. EverySuperHyperRightHyperIdealof 2-SuperHyper-LA-semihypergroup S withpureleftidentityisSuperHyperHyperIdeal.

Proof. Let R beaSuperHyperRightHyperIdealof S.ByTheorem 3.6,Lemma 3.7 andProposition 3.11 (3),wehave S •n R =(S •n S) •n R =(R •n S) •n S ⊆ R •n S ⊆ R.

Thereforeweobtainthat R isaSuperHyperLeftHyperIdealof S andhence R isaSuperHyperHyperIdealof S

P.Yiarayong,On2-SuperHyperLeftAlmostSemihypergroups

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Received: June 7, 2022. Accepted: September 23, 2022.

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