On Some NeutroHyperstructures

OnSomeNeutroHyperstructures

MadeleineAl-Tahan 1 ,BijanDavvaz 2 ,FlorentinSmarandache 3,* andOsmanAnis 4

1 DepartmentofMathematics,LebaneseInternationalUniversity,Bekaa1803,Lebanon; madeline.tahan@liu.edu.lb

2

DepartmentofMathematics,YazdUniversity,Yazd89139,Iran;davvaz@yazd.ac.ir

3 MathematicsandScienceDivision,GallupCampus,UniversityofNewMexico,705GurleyAve., Gallup,NM87301,USA

4

DepartmentofMathematics,LebaneseInternationalUniversity,Akkar35111,Lebanon; osman.anis@liu.edu.lb

* Correspondence:smarand@unm.edu

Abstract: Neutrosophy,thestudyofneutralities,isanewbranchofPhilosophythathasapplications inmanydifferentfieldsofscience.InspiredbytheideaofNeutrosophy,Smarandacheintroduced NeutroAlgebraicStructures(orNeutroAlgebras)byallowingthepartialityandindeterminacytobe includedinthestructures’operationsand/oraxioms.Theaimofthispaperistocombinetheconcept ofNeutrosophywithhyperstructurestheory.Inthisregard,weintroduceNeutroSemihypergroupsas wellasNeutroHv-Semigroupsandstudytheirpropertiesbyprovidingseveralillustrativeexamples.

Keywords: NeutroHypergroupoid;NeutroSemihypergroup;NeutroHv-semigroup;NeutroHyperideal;NeutroStrongIsomorphism

MSC: 03A99;03G99;20N20

Citation: Al-Tahan,M.;Davvaz,B.; Smarandache,F.;Anis,O.OnSome NeutroHyperstructures. Symmetry 2021, 13,535. https://doi.org/ 10.3390/sym13040535

AcademicEditor:PalleE.T. Jorgensen

Received:20February2021 Accepted:23March2021 Published:25March2021

Publisher’sNote: MDPIstaysneutral withregardtojurisdictionalclaimsin publishedmapsandinstitutionalaffiliations.

1.Introduction

In1995andinspiredbytheexistenceofneutralities,SmarandacheintroducedNeutrosophyasanewbranchofPhilosophythatdealswithindeterminacy.Duringthepast, ideaswereviewedas“True”or“False”;however,ifweviewanideafromaneutrosophic pointofview,itwillbe“True”,“False”,or“Indeterminate”.Theindeterminacyisthe keythatdistinguishesNeutrosophyfromotherapproaches.Inthepasttwentyyears, thisfielddemonstratedimportantprogressinwhichitgrabbedtheattentionofmany researchersanddifferentworksweredonefrombothatheoreticalpointofviewandfrom anapplicativeview.Unlikeourrealworldthatisfullofimperfectionsandpartialities, abstractsystemsareconstructedonagivenperfectspace(set),wheretheoperationsare totallywell-definedandtheaxiomsaretotallytrueforallspacialelements.Startingfrom thelatteridea,Smarandache[1–3]introducedNeutroAlgebra,whoseoperationsarepartiallywell-defined,partiallyindeterminate,andpartiallyouter-defined,andtheaxiomsare partiallytrue,partiallyindeterminate,andpartiallyfalse.Manyresearchersworkedon specialtypesofNeutroAlgebrasbyapplyingthemtodifferenttypesofalgebraicstructures suchasgroups,rings, BE-Algebras, BCK-Algebras,etc.Formoredetails,wereferto[4–10].

Copyright: ©2021bytheauthors. LicenseeMDPI,Basel,Switzerland. Thisarticleisanopenaccessarticle distributedunderthetermsand conditionsoftheCreativeCommons Attribution(CCBY)license(https:// creativecommons.org/licenses/by/ 4.0/).

Ontheotherhand,hyperstructuretheoryisageneralizationofclassicalalgebraic structuresandwasintroducedin1934attheeighthCongressofScandinavianMathematiciansbyMarty[11].Martygeneralizedthenotionofgroupsbydefininghypergroups. Theclassofalgebraichyperstructuresislargerthanthatofalgebraicstructureswherethe operationontwoelementsinthelatterisagainanelement,whereasthehyperoperationof twoelementsinthefirstclassisanon-voidset.Fordetailsabouthyperstructuretheoryand itsapplications,werefertothearticles[12–15]andthebooks[16–18].Ageneralizationof algebraichyperstructures,knownasweakhyperstructures(Hv-structures),wasintroduced

in1994byVougiouklis[19].Theaxiomsinthelatterareweakerthanthatofalgebraic hyperstructures.Fordetailsabout Hv-structures,wereferto[19–22].

AsanaturalextensionofNeutroAlgebraicStructure,NeutroHyperstructurewasdefinedrecently[23,24]whereIbrahimandAgboola[23]definedNeutroHypergroupsand studiedaspecialtype.OurpaperisconcernedaboutsomeNeutroHyperstructuresandis organizedasfollows:Section 2 presentssomebasicpreliminariesrelatedtohyperstructure theory.Section 3 definesNeutroSemihypergroups,NeutroHv-Semigroups,andsomerelatednewconceptsandillustratesthesenewconceptsviaexamples.Moreover,westudy somepropertiesoftheirsubsetsunderNeutroStrongHomomorphism.

2.AlgebraicHyperstructures

Inthissection,wepresentsomedefinitionsandexamplesabout(weak)algebraic hyperstructuresthatareusedthroughoutthepaper.Formoredetailsabouthyperstructure theory,wereferto[16–20].

Definition1 ([16]). Let H beanon-emptysetand P ∗(H) bethefamilyofallnon-emptysubsets of H.Then,amapping ◦ : H × H →P ∗(H) iscalledabinaryhyperoperationon H.Thecouple (H, ◦) iscalledahypergroupoid. IfAandBaretwonon-emptysubsetsofHandh ∈ H,thenwedefine: A ◦ B = a∈A b∈B

a ◦ b, h ◦ A = {h}◦ AandA ◦ h = A ◦{h}

Ahypergroupoid (H, ◦) iscalleda semihypergroup iftheassociativeaxiomissatisfied. i.e.,forevery x, y, z ∈ H, x ◦ (y ◦ z)=(x ◦ y) ◦ z.Inotherwords, u∈y◦z x ◦ u = v∈x◦y v ◦ z

Anelement h inahypergroupoid (H, ◦) iscalled idempotent if h ◦ h = h

Example1. Let H beanynon-emptysetanddefine“ ”on H asfollows.Forall x, y ∈ H, x y = {x, y}.Then (H, ) isasemihypergroup.

Example2. LetH0 = {e, b, c} and (H0, +) bedefinedbythefollowingtable. + ebc e e {e, b}{e, c} b e {e, b}{e, c} c e {e, b}{e, c}

Then (H0, +) isasemihypergroupandeisanidempotentelementinH0

Asageneralizationofalgebraichyperstructures,Vougiouklis[19,20]introduced Hvstructures.Weakaxiomsin Hv-structuresreplacesomeaxiomsofclassicalalgebraichyperstructures.

Definition2 ([19,20]). Ahypergroupoid (H, ◦) iscalledan Hv-semigroupiftheweakassociative axiomissatisfied.i.e., (x ◦ (y ◦ z)) ∩ ((x ◦ y) ◦ z) = ∅ forallx, y, z ∈ H.

Example3. Let H1 = {0,1,2,3} and“+”bethehyperoperationon H1 definedbythefollowingtable. + 0123 0 01 {0,2} 3 1 1230 2 2301 3 3012

Then (H1, +) isanHv-semigroup.

Remark1. Everysemigroupisasemihypergroupandeverysemihypergroupisan Hv-semigroup.

Definition3 ([17]). Let (H, ◦) beasemihypergroup(Hv-semigroup)and M = ∅ ⊆ H.Then M isa

1. subsemihypergroup(Hv-subsemigroup)ofHif (M, ◦) isasemihypergroup(Hv-semigroup).

2. lefthyperidealof H if M isasubsemihypergroup(Hv-subsemigroup)of H and h ◦ a ⊆ M for allh ∈ H.

3. righthyperidealof H if M isasubsemihypergroup(Hv-subsemigroup)of H and a ◦ h ⊆ M for allh ∈ H.

4. hyperidealofHifMisboth:alefthyperidealofHandarighthyperidealofH.

Remark2. Let (H, ◦) beasemihypergroup(Hv-semigroup)and M = ∅ ⊆ H.Toprovethat M issubsemihypergroup(Hv-subsemigroup)of H,itsufficestoshowthat a ◦ b ⊆ M forall a, b ∈ M

3.NeutroHyperstructures

Inthissection,wedefineNeutroSemihypergroupsandNeutroHv-Semigroups,present someillustrativeexamples,andstudyseveralpropertiesofsomeimportantsubsetsof NeutroSemihypergroupsandNeutroHv-Semigroups.

Definition4. Let A beanynon-emptysetand“ ”beahyperoperationon A.Then“ ”iscalled aNeutroHyperoperationon A ifsome(orall)ofthefollowingconditionsholdinawaythat (T, I, F) / ∈{(1,0,0), (0,0,1)}.

1. Thereexistx, y ∈ Awithx · y ⊆ A.(Thisconditioniscalleddegreeoftruth,“T”).

2. Thereexistx, y ∈ Awithx · y A.(Thisconditioniscalleddegreeoffalsity,“F”).

3. Thereexist x, y ∈ A with x · y isindeterminatein A.(Thisconditioniscalleddegreeof indeterminacy,“I”).

Definition5. Let A beanynon-emptysetand“·”beahyperoperationon A.Then“·”iscalledan AntiHyperoperationonAifx y Aforallx, y ∈ A.

Definition6. Let A beanynon-emptysetand“·”beahyperoperationon A.Then“·”iscalled NeutroAssociativeon A ifthereexist x, y, z, a, b, c, e, f , g ∈ A satisfyingsome(orall)ofthe followingconditionsinawaythat (T, I, F) / ∈{(1,0,0), (0,0,1)}

1. x (y z)=(x y) z;(Thisconditioniscalleddegreeoftruth,“T”).

2. a · (b · c) =(a · b) · c;(Thisconditioniscalleddegreeoffalsity,“F”).

3. e · ( f · g) isindeterminateor (e · f ) · g isindeterminateorwecannotfindif e · ( f · g) and (e · f ) · gareequal.(Thisconditioniscalleddegreeofindeterminacy,“I”).

Definition7. Let A beanynon-emptysetand“·”beahyperoperationon A.Then“·”iscalled AntiAssociativeonAifa · (b · c) =(a · b) · cforalla, b, c ∈ A.

Definition8. Let A beanynon-emptysetand“·”beahyperoperationon A.Then“·”iscalleda NeutroWeakAssociativeon A ifthereexist x, y, z, a, b, c, e, f , g ∈ A satisfyingsome(orall)ofthe followingconditionsinawaythat (T, I, F) / ∈{(1,0,0), (0,0,1)}

1. [x (y z)] ∩ [(x y) z] = ∅;(Thisconditioniscalleddegreeoftruth,“T”).

2. [a (b c)] ∩ [(a b) c]= ∅;(Thisconditioniscalleddegreeoffalsity,“F”).

3. e ( f g) isindeterminateor (e f ) g isindeterminateorwecannotfindif e ( f g) and (e · f ) · ghavecommonelements.(Thisconditioniscalleddegreeofindeterminacy,“I”).

Definition9. Let A beanon-emptysetand“ ”beahyperoperationon A.Then (A, ) iscalleda 1. NeutroHypergroupoidif“ ”isaNeutroHyperoperation.

2. NeutroSemihypergroupif“ ”isNeutroAssociativebutnotanAntiHyperoperation.

3. NeutroHv-Semigroupif“ ”isNeutroWeakAssociativebutnotanAntiHyperoperation.

Example4. LetA = {0,1} and (A, +) bedefinedbythefollowingtable. + 01 0 {0,1} 0 1 10

Then (A, +) isaNeutroSemihypergroupandNeutroHv-Semigroup.Thisisclearas 0 +(0 + 0)= {0,1} =(0 + 0)+ 0 and (1 + 1)+ 1 = 0 = 1 = 1 +(1 + 1)





[x, y] ifx < y; [y, x] ify < x; 0 ifx = y = 0; 1 x ifx = y = 0

Then (R, ) isaNeutroSemihypergroup.Thisisclearas (1 1) 1 = 1 = 1 (1 1) and (1 2) 2 = { 1 2 }∪ [1,2] =[ 1 2 ,1]= 1 (2 2)

Example6. LetM = {m, a, d} and (M, ) bedefinedbythefollowingtable. mad m mmm a m {m, a} d d mdd

Then (M, ) isaNeutroSemihypergroup.Thisisclearas m (m m)= m =(m m) m and a · (a · d)= d = {m, d} =(a · a) · d.

Remark3. Itiswellknowninclassicalalgebraichyperstructuresthateverysemihypergroupis ahypergroupoid.ThismayfailtooccurinNeutroHyperstructures.InExample 6, (M, ·) isa NeutroSemihypergroupthatisnotaNeutroHypergroupoid.

Proposition1. Every Hv-semigroupthatisnotasemihypergroupandhasanidempotentelement isaNeutroSemihypergroup.

Proof. Let (H, ◦) bean Hv-semigroupwith h2 = h forsome h ∈ H.Then h ◦ (h ◦ h)= h = (h ◦ h) ◦ h.Since (H, ◦) isnotasemihypergroup,itfollowsthatthereexist x, y, z ∈ H with x ◦ (y ◦ z) =(x ◦ y) ◦ z.Therefore, (H, ◦) isaNeutroSemihypergroup.

Example7. LetM = {m, a, d} and (M, ) bedefinedbythefollowingtable. mad m m {a, d} d a {a, d} dm d dma

Then (M, ) isan Hv-semigrouphaving m asanidempotentelementandhence,itisa NeutroSemihypergroup.

Remark4. Itiswellknowninalgebraichyperstructuresthateverysemihypergroupisan Hvsemigroup.ThismaynotholdinNeutroHyperstructures.i.e.,ANeutroSemihypergroupmaynot beaNeutroHv-Semigroup.

The Hv-semigroup (M, ) inExample 7 isaNeutroSemihypergroupthatisnotNeutroHvSemigroup.

Example8. Let Z bethesetofintegersanddefine“⊕”on Z2 asfollows.Forallm, n, p, q ∈ Z,

(m,0) ⊕ (0,0)=(0,0) ⊕ (m,0)= {(0,0), (m,0)}, (0, n) ⊕ (0,0)=(0,0) ⊕ (0, n)= {(0,0), (0, n)}, andif (n, p, q) =(0,0,0), (m, p, q) =(0,0,0) (m, n) ⊕ (p, q)=( p, q) ⊕ (m, n)=(m + p, n + q)

Then (Z2 , ⊕) isaNeutroSemihypergroup.Thisisclearas [(1,2) ⊕ (1,3)] ⊕ (1,4)=(3,9)=(1,2) ⊕ [(1,3) ⊕ (1,4)] and [(1,0) ⊕ (1,0)] ⊕ (0,0)= {(2,0), (0,0)} = {(2,0), (1,0), (0,0)} =(1,0) ⊕ [(1,0) ⊕ (0,0)] Example9. Let Z bethesetofintegersanddefine“ ”on Z2 asfollows.Forallm, n, p, q ∈ Z, (m, n) (p, q)=      (mp, nq) if (m, n) =(1,1) and (p, q) =(1,1); {(p, q), (1,1)} if (m, n)=(1,1); {(m, n), (1,1)} if (p, q)=(1,1).

Then (Z2 , ) isaNeutroSemihypergroup.Thisisclearas [(1,2) (1,3)] (1,4)=(1,24)=(1,2) [(1,3) (1,4)] and (1,1) [(2,2) (3,3)]= {(1,1), (6,6)} = {(1,1), (3,3), (6,6)} =[(1,1) (2,2)] (3,3) Example10. Let Z6 bethesetofintegersunderadditionmodulo6anddefine“ ”on Z6 asfollows. x y =(x + y) mod6 forall (x, y) / ∈{(0, 3), (0, 5)}, 0 3 = {0, 3}, and 0 5 = {0, 5}.

Then (Z6, ) isaNeutroSemihypergroup.Thisisclearas 0 (0 0)= 0 =(0 0) 0 and 0 (1 2)= {0, 3} = 3 =(0 1) 2

Example11. LetM = {m, a, d} and (M, •) bedefinedbythefollowingtable. • mad m aad a {m, a} md d ddm

Then (M, •) isaNeutroHv-Semigroup.Thisisclearas [m • (m • m)] ∩ [(m • m) • m]= {a}∩{m, a} = ∅

and [m • (d • d)] ∩ [(m • d) • d]= {a}∩{m} = ∅.

Moreover, (M, •) isaNeutroSemihypergroupasd • (d • d)=(d • d) • d.

Remark5. EveryNeutroSemigroupisboth:aNeutroSemihypergroupandaNeutroHv-Semigroup. So,theresultsrelatedtoNeutroSemihypergroups(NeutroHv-Semigroups)aremoregeneralthan thatrelatedtoNeutroSemigroupsandasaresult,wecandealwithNeutroSemigroupsasaspecial caseofNeutroSemihypergroups(NeutroHv-Semigroups).

Example12. LetS1 = {s, a, m} and (S1, 1) bedefinedbythefollowingtable. 1 sam s sms a mam m mmm

In[6],Al-Tahanetal.provedthat (S1, 1) isaNeutroSemigroup.Thus, (S1, 1) isaNeutroSemihypergroup.

Theorem1. Let (H, ◦) beaNeutroSemihypergroup(NeutroHv-Semigroup)and“ ”bedefined on H as x y = y ◦ x forall x, y ∈ H.Then (H, ) isaNeutroSemihypergroup(NeutroHvSemigroup).

Proof. Theproofisstraightforward.

Example13. Let M = {m, a, d} and (M, •) betheNeutroSemihypergroupdefinedinExample 11. ByapplyingTheorem 1,wegetthat (M, ) definedinthefollowingtableisaNeutroSemihypergroup andaNeutroHv-Semigroup.

mad m a {m, a} d a amd d ddm

Definition10. Let (H, ◦) beaNeutroSemihypergroup(NeutroHv-Semigroup)andS = ∅ ⊆ H. Then S isaNeutroSubsemihypergroup(NeutroHv-Subsemigroup)of H if (S, ◦) isaNeutroSemihypergroup(NeutroHv-Semigroup).

Remark6. Let (H, ◦) beaNeutroSemihypergroup(NeutroHv-Semigroup)and S = ∅ ⊆ H Unlikethecaseinalgebraichyperstructures(Remark 2),provingthat a ◦ b ⊆ S forall a, b ∈ S does notimplythatSisaNeutroSubsemihypergroup(NeutroHv-Subsemigroup)ofH.

AsanillustrationofRemark 6, 0 0 = {0}⊆{0} inExample 5 but {0} isnota NeutroSubsemihypergroupof R

Definition11. Let (H, ◦) beaNeutroSemihypergroup(NeutroHv-Semigroup)and S = ∅ ⊆ H beaNeutroSubsemihypergroup(NeutroHv-Subsemigroup).Then

(1) SisaNeutroLeftHyperidealofHifthereexistsx ∈ Ssuchthatr ◦ x ⊆ Sforallr ∈ H. (2) SisaNeutroRightHyperidealofSifthereexistsx ∈ Ssuchthatx ◦ r ⊆ Sforallr ∈ H. (3) S isaNeutroHyperidealof H ifthereexists x ∈ S suchthat r ◦ x ⊆ S and x ◦ r ⊆ S for allr ∈ H.

ANeutroSemihypergroup(NeutroHv-Semigroup)iscalledsimpleifithasnoproper NeutroSubsemihypergroups(NeutroHv-Subsemigroups).

Example14. Let (A, +) betheNeutroSemihypergroupdefinedinExample 4.Then A issimple. Thisisclearas {0} and {1} aretheonlyoptionsforanypossibleproperNeutroSubsemihypergroup and ({0}, +) and ({1}, +) areAntiHypergroupoids.

Example15. Let (M, •) betheNeutroSemihypergroupdefinedinExample 11.Then {m, a} isa NeutroSubsemihypergroupofM.

Example16. Let (Z2 , ⊕) betheNeutroSemihypergroupdefinedinExample 8, M1 = {(x,0) : x ∈ Z},andM2 = {(0, x) : x ∈ Z}.ThenM1, M2 areNeutroSubsemihypergroupsof Z2 .

Remark7. TheintersectionofNeutroSubsemihypergroupsmayfailtobeaNeutroSubsemihypergroup. ThisisclearfromExample 16 as {(0,0)} = M1 ∩ M2 isnotaNeutroSubsemihypergroupof Z2 .

Lemma1. Let (H, ◦) beaNeutroSemihypergroup(NeutroHv-Semigroup)and A, B behypergroupoids.If A, B areNeutroSubsemihypergroups(NeutroHv-Subsemigroups)of H then A ∪ B is aNeutroSubsemihypergroup(NeutroHv-Subsemigroup)ofH.

Proof. Let A, B beNeutroSubsemihypergroups.Since A and B arehypergroupoids,it followsthat“◦”isNeutroAssociativeonbothof A and B.Thelatterimpliesthatthereexist x, y, z, a, b, c, e, f , g ∈ A ⊆ A ∪ B satisfyingsome(orall)ofthefollowingconditionsina waythat (T, I, F) / ∈{(1,0,0), (0,0,1)}

1. T: x ◦ (y ◦ z)=(x ◦ y) ◦ z; 2. F: a ◦ (b ◦ c) =(a ◦ b) ◦ c; 3. I: e ◦ ( f ◦ g) isindeterminateor (e ◦ f ) ◦ g isindeterminateorwecannotfindif e ◦ ( f ◦ g) and (e ◦ f ) ◦ g areequal.

Therefore, A ∪ B isaNeutroSubsemihypergroupof H.Theproofof(NeutroHvSubsemigroupisdonesimilarly.

Example17. Let (Z2 , ) betheNeutroSemihypergroupdefinedinExample 9, N1 = {(x, y) ∈ Z2 : x, y ≥ 1}∪{(0,0)},and N2 = {(x, y) ∈ Z2 : x, y ≤ 1}∪{(0,0)}.Then N1, N2 are NeutroHyperidealsof Z2.Weshowthat N1 isaNeutroHyperidealof Z2 and N2 maybedone similarly.Since [(1,2) (1,3)] (1,4)=(1,24)=(1,2) [(1,3) (1,4)] and (1,1) [(2,2) (3,3)]= {(1,1), (6,6)} = {(1,1), (3,3), (6,6)} =[(1,1) (2,2)] (3,3), itfollowsthat N1 isaNeutroSubsemihypergroupof Z2.Having (0,0) ∈ N1 andforall (r, s) ∈ Z2 , (r, s) (0,0)=(0,0) (r, s)= (0,0) if (r, s) =(1,1); {(0,0), (1,1)} otherwise. ⊆ N1

impliesthatN1 isaNeutroHyperidealof Z2 .

Remark8. TheintersectionofNeutroHyperidealsmayfailtobeaNeutroHyperideal.Thisisclear fromExample 17 as {(0,0), (1,1)} = N1 ∩ N2 isnotaNeutroHyperidealof Z2

Lemma2. Let (H, ◦) beaNeutroSemihypergroup(NeutroHv-Semigroup)and A, B behypergroupoids.If A, B areNeutroLeftHyperideals(NeutroRightHyperidealsorNeutroHyperideals)of H.ThenA ∪ BisaNeutroLeftHyperideal(NeutroRightHyperidealorNeutroHyperideal)ofH.

Proof. Let A, B beNeutroLeftHyperidealsof H.Lemma 1 assertsthat A ∪ B isaNeutroSubsemihypergroup(NeutroHv-Subsemigroup)of H.Since A isaNeutroLeftHyperideal

of H,itfollowsthatthereexists a ∈ A suchthat r ◦ a ⊆ A forall r ∈ H.Thelatter impliesthatthereexists a ∈ A ∪ B suchthat r ◦ a ⊆ A ∪ B forall r ∈ H.Thus, A ∪ B isa NeutroLeftHyperidealof H.

Definition12. Let (H, ◦), (H , ) beNeutroSemihypergroups(NeutroHv-Semigroups)and φ : H → H beafunction.Then

(1) φ iscalledNeutroHomomorphismif φ(x ◦ y)= φ(x) φ(y) forsomex, y ∈ A.

(2) φ iscalledNeutroIsomomorphismif φ isabijectiveNeutroHomomorphism.

(3) φ iscalledNeutroStrongHomomorphismifforall x, y ∈ A, φ(x ◦ y)= φ(x) φ(y) when x ◦ y ⊆ H, φ(x) φ(y) H when x ◦ y H,and φ(x) φ(y) isindeterminatewhen x ◦ yisindeterminate.

(4) φ iscalledNeutroStrongIsomomorphismif φ isabijectiveNeutroOrderedStrongHomomorphism.Inthiscasewesaythat (H, ◦) ∼ =SI (H , ).

Example18. Let (M, •) and (M, ) betheNeutroSemihypergroupsdefinedin Examples 11 and 13,respectively.Then (M, •) ∼ =SI (M, ) as φ : (M, •) → (M, ) isa NeutroStongIsomorphism.Here, φ(m)= a, φ(a)= m, and φ(d)= d.

Theorem2. Therelation“∼ =SI ”isanequivalencerelationonthesetofNeutroSemihypergroups (NeutroHv-Semigroups).

Proof. Bytakingtheidentitymap,wecaneasilyprovethat“∼ =SI ”isareflexiverelation. Let A ∼ =SI B.ThenthereexistsaNeutroStrongIsomorphism φ : (A, ) → (B, ).We provethattheinversefunction φ 1 : B → A of φ isaNeutroStrongIsomorphism.Forall b1, b2 ∈ B,thereexist a1, a2 ∈ A with φ(a1)= b1 and φ(a2)= b2.Wehave φ 1(b1 b2)= φ 1(φ(a1) φ(a2))

Weconsiderthefollowingcasesfor φ(a1) φ(a2)

Case φ(a1) φ(a2) ⊆ B.Having φ aNeutroStrongIsomorphismand φ(a1) φ(a2) ⊆ B implythat a1 a2 ⊆ A andhence, φ 1(b1 b2)= φ 1(φ(a1) φ(a2))= φ 1(φ(a1 a2))= a1 a2 = φ 1(b1) φ 1(b2)

Case φ(a1) φ(a2) B.Suppose,togetcontradiction,that φ 1(φ(a1)) φ 1(φ(a2))= a1 a2 ⊆ A orindeterminate.Thenbyusingourhypothesisthat φ isNeutroStrongIsomorphism,wegetthat φ(a1) φ(a2) ⊆ B orindeterminate.

Case φ(a1) φ(a2) isindeterminate.Suppose,togetcontradiction,that φ 1(φ(a1)) φ 1(φ(a2))= a1 a2 ⊆ A or a1 a2 A.Thenbyusingourhypothesisthat φ isNeutroStrongIsomorphism,wegetthat φ(a1) φ(a2) ⊆ B or φ(a1) φ(a2) B

Thus, B ∼ =SI A andhence,“∼ =SI ”isasymmetricrelation.Let A ∼ =SI B and B ∼ =SI C ThenthereexistNeutroStrongIsomorphisms φ : A → B and ψ : B → C.Onecaneasilysee thatthecompositionfunction ψ ◦ φ : A → C of ψ and φ isaNeutroStrongIsomorphism. Thus, A ∼ =SI C andhence,“∼ =SI ”isatransitiverelation.

Lemma3. Let (H, ◦), (H , ) beNeutroSemihypergroups(NeutroHv-Semigroups)and φ : H → H beaninjectiveNeutroStrongHomomorphism.If M ⊂ H isaNeutroSubsemihypergroup(NeutroHv-Subsemigroup)of H then φ(M) isaNeutroSubsemihypergroup(NeutroHvSubsemigroup)ofH .

Proof. Let M beaNeutroSubsemihypergroupof H.If“◦”isNeutroHyperoperationon M thenitisclearthat“ ”isNeutroHyperoperationon φ(M).If“◦”isNeutroAssociative

thenthereexist x, y, z, a, b, c, d, e, f ∈ Msatisfyingsome(orall)ofthefollowingconditions inawaythat (T, I, F) / ∈{(1,0,0), (0,0,1)}.

1. T: x ◦ (y ◦ z)=(x ◦ y) ◦ z;

2. F: a ◦ (b ◦ c) =(a ◦ b) ◦ c;

3. I: e ◦ ( f ◦ g) isindeterminateor (e ◦ f ) ◦ g isindeterminateorwecannotfindif e ◦ ( f ◦ g) and (e ◦ f ) ◦ g areequal.

Thelatterandhaving φ aninjectiveNeutroStrongHomomorphismimplythatsome (orall)ofthefollowingconditionsaresatisfiedinawaythat (T, I, F) / ∈{(1,0,0), (0,0,1)}

1. T: φ(x) (φ(y) φ(z))=(φ(x) φ(y)) φ(z);

2. F: φ(a) (φ(b) φ(c)) =(φ(a) φ(b)) φ(c);

3. I: φ(e) (φ( f ) φ(g)) isindeterminateor (φ(e) φ( f )) φ(g) isindeterminateorwe cannotfindif φ(e) (φ( f ) φ(g)) and (φ(e) φ( f )) φ(g) areequal.

Thus, φ(M) isaNeutroSubsemihypergroup.Theproofthat φ(M) isaNeutroHvSubsemigroupof H isdonesimilarly.

Example19. Let (M, •) and (M, ) betheNeutroSemihypergroupsdefinedin Examples 11 and 13,respectively.Example 15 assertsthat {m, a} isaNeutroSubsemihypergroupof (M, •).UsingExample 18 andLemma 3,wegetthat {a, m} = {φ(m), φ(a)} isa NeutroSubsemihypergroupof (M, ).

Lemma4. Let (H, ◦), (H , ) beNeutroSemihypergroups(NeutroHv-Semigroups)and φ : H → H beaNeutroStrongIsomomorphism.If N ⊆ H isaNeutroSubsemihypergroup(NeutroHvSubsemigroup)of H then φ 1(N) isaNeutroSubsemihypergroup(NeutroHv-Subsemigroup) ofH.

Proof. Let N ⊂ H beaNeutroSubsemihypergroupof H .If“ ”isNeutroHyperoperationon N thenitisclearthat“◦”isNeutroHyperoperationon φ 1(N).Let“ ”be NeutroAssociative.Having φ isanontoNeutroStrongHomomorphismimpliesthatthere exist φ(x), φ(y), φ(z), φ(a), φ(b), φ(c), φ(d), φ(e), φ( f ) ∈ N satisfyingsome(orall)ofthe followingconditionsinawaythat (T, I, F) / ∈{(1,0,0), (0,0,1)}

1. T: φ(x) (φ(y) φ(z))=(φ(x) φ(y)) φ(z);

2. F: φ(a) (φ(b) φ(c)) =(φ(a) φ(b)) φ(c);

3. I: φ(e) (φ( f ) φ(g)) isindeterminateor (φ(e) φ( f )) φ(g) isindeterminateorwe cannotfindif φ(e) (φ( f ) φ(g)) and (φ(e) φ( f )) φ(g) areequal.

Having φ beaninjectiveNeutroStrongHomomorphismimpliesthatthereexist x, y, z, a, b, c, d, e, f ∈ φ 1(N) satisfyingsome(orall)ofthefollowingconditionsinawaythat (T, I, F) / ∈{(1,0,0), (0,0,1)}.

1. T: x ◦ (y ◦ z)=(x ◦ y) ◦ z;

2. F: a ◦ (b ◦ c) =(a ◦ b) ◦ c;

3. I: e ◦ ( f ◦ g) isindeterminateor (e ◦ f ) ◦ g isindeterminateorwecannotfindif e ◦ ( f ◦ g) and (e ◦ f ) ◦ g areequal.

Thus, φ 1(N) isaNeutroSubsemihypergroupof H.Theproofthat φ 1(N) isa NeutroHv-Subsemigroupof H maybedonesimilarly.

Theorem3. Let (H, ◦), (H , ) beNeutroSemihypergroups(NeutroHv-Semigroups)and φ : H → H beaNeutroStrongIsomorphism.Then M ⊆ H isaNeutroSubsemihypergroup(NeutroHvSubsemigroup)of H ifandonlyif φ(M) isaNeutroSubsemihypergroup(NeutroHv-Subsemigroup) ofH

Proof. TheprooffollowsfromTheorem 2 andLemmas 3 and 4

Corollary1. Let (H, ◦), (H , ) beNeutroSemihypergroups(NeutroHv-Semigroups)and φ : H → H beaNeutroStrongIsomorphism.ThenHissimpleifandonlyifH issimple.

Proof. TheprooffollowsfromTheorem 3.

Lemma5. Let (H, ◦), (H , ) beNeutroSemihypergroups(NeutroHv-Semigroups)and φ : H → H beaNeutroStrongIsomorphism.If M ⊆ H isaNeutroLeftHyperideal(NeutroRightHyperideal orNeutroHyperideal)of H then φ(M) isaNeutroLeftHyperideal(NeutroRightHyperidealor NeutroHyperideal)ofH .

Proof. Let M ⊆ H beaNeutroLeftHyperidealof H.Lemma 3 assertsthat φ(M) isa NeutroSubsemihypergroup(NeutroHv-Subsemigroup)of H .Having M aNeutroLeftHyperidealof H impliesthatthereexists x ∈ M suchthat r ◦ x ⊆ M forall r ∈ H.Having φ an ontoNeutroStrongHomomorphismimpliesthat φ(r) φ(x) ⊆ φ(M) forall s = φ(r) ∈ H Thus, φ(M) isaNeutroLeftHyperidealof H .TheproofsofNeutroRightHyperidealand NeutroHyperidealaredonesimilarly.

Lemma6. Let (H, ◦), (H , ) beNeutroSemihypergroups(NeutroHv-Semigroups)and φ : H → H beaNeutroStrongIsomorphism.If N ⊆ H isaNeutroLeftHyperideal(NeutroRightHyperideal orNeutroHyperideal)of H then φ 1(N) isaNeutroLeftHyperideal(NeutroRightHyperidealor NeutroHyperideal)ofH.

Proof. Let N ⊆ H beaNeutroLeftHyperidealof H.Lemma 3 assertsthat φ 1(N) isa NeutroSubsemihypergroup(NeutroHv-Subsemigroup)of H.Having N aNeutroLeftHyperidealof H impliesthatthereexists y ∈ N suchthat s y ⊆ N forall s ∈ H .Since φ isanNeutroStrongHomomorphism,itfollowsthat φ(r ◦ x) ⊆ N forall r ∈ H where y = φ(x).Thelatterimpliesthatthereexists x ∈ φ 1(N) with r ◦ x ⊆ φ 1(N) forall r ∈ H. Thus, φ 1(N) isaNeutroLeftHyperidealof H.TheproofsofNeutroRightHyperidealand NeutroHyperidealaredonesimilarly.

Theorem4. Let (H, ◦), (H , ) beNeutroSemihypergroups(NeutroHv-Semigroups)and φ : H → H beaNeutroStrongIsomorphism.Then M ⊆ H isaNeutroLeftHyperideal(NeutroRightHyperidealorNeutroHyperideal)of H ifandonlyif φ(M) isaNeutroLeftHyperideal (NeutroRightHyperidealorNeutroHyperideal)ofH .

Proof. TheprooffollowsfromTheorem 2,Lemmas 5 and 6.

Let Hα beanynon-emptysetforall α ∈ Γ and“·α ”beahyperoperationon Hα .We define“◦”on ∏α∈Γ Hα asfollows:Forall (xα ), (yα ) ∈ ∏α∈Γ Hα , (xα ) ◦ (yα )= {(tα ) : tα ∈ xα α yα }.

Theorem5. Let (H1, ◦1) and (H2, ◦2) behypergroupoids.Then (H1 × H2, ◦) isaNeutroSemihypergroup(NeutroHv-Semigroup)ifandonlyifeither (H1, ◦1) isaNeutroSemihypergroup (NeutroHv-Semigroup)or (H2, ◦2) isaNeutroSemihypergroup(NeutroHv-Semigroup)orbothare NeutroSemihypergroups(NeutroHv-Semigroups).

Proof. Theproofisstraightforward.

Example20. Let (R, ) bethesemihypergroupdefinedas: x y = {x, y} forall x, y ∈ R and (M, ) betheNeutroSemihypergroupdefinedinExample 6.Thenthefollowingaretrue. 1. (R × M, ◦) isaNeutroSemihypergroup, 2. (M × R, ◦) isaNeutroSemihypergroup,and 3. (M × M, ◦) isaNeutroSemihypergroup.

Inwhatfollows,wepresentawaytoconstructanewNeutroSemihypergroup(NeutroHvSemigroup)fromanexistingone.Thistoolisofgreatimportancetoprovethatforanypositiveinteger n ≥ 2,thereexistsatleastoneNeutroSemihypergroup(NeutroHv-Semigroup) oforder n

Let (H, ◦) beaNeutroSemihypergroup(NeutroHv-Semigroup)and J beanynonemptysetsuchthat H ∩ J = ∅ and (H ◦ H) ∩ J = ∅.Theextension H[ J] of H by J isgiven as H[ J]= H ∪ J.Wedefinethehyperoperation“ ”on H[ J] asfollows.

x y = x ◦ y if x, y ∈ H; H ∪ J otherwise.

Theorem6. Let (H, ◦) beaNeutroSemihypergroup(NeutroHv-Semigroup)and J beanynonemptysetsuchthat H ∩ J = ∅ and (H ◦ H) ∩ J = ∅.Then (H[ J], ) isaNeutroSemihypergroup (NeutroHv-Semigroup).

Proof. Let (H, ◦) beaNeutroSemihypergroup.If“◦”isaNeutroHyperoperationthen thereexist u, v, w, x, y, z ∈ H with u ◦ v ⊆ H representing“T”, w ◦ x H representing “F”, y ◦ z isindeterminaterepresenting“I”.Where (T, I, F) / ∈{(1,0,0), (0,0,1)}.Since (H ◦ H) ∩ J = ∅,itfollowsthatthereexist u, v, w, x, y, z ∈ H with u ◦ v ⊆ H[ J] representing “T”, w ◦ x H[ J] representing“F”(as w ◦ x H and w ◦ x J), y ◦ z isindeterminate representing“I”.Where (T, I, F) / ∈{(1,0,0), (0,0,1)}.Thus,“ ”isNeutroHyperoperation on H[ J].If“◦”isNeutroAssociativeon H thenitisclearthat“ ”isNeutroAssociativeon H[ J].Therefore, (H[ J], ) isaNeutroSemihypergroup.Thecase (H[ J], ) isaNeutroHvSemigroupisdonesimilarly.

Example21. Let (M, ·) betheNeutroSemihypergroupdefinedinExample 6 and N = {n}.Then M[N]= {m, a, d, n} and (M[N], ) istheNeutroSemihypergroupdefinedbythefollowingtable.

madn m mmm {m, a, d, n} a m {m, a} d {m, a, d, n} d mdd {m, a, d, n} n {m, a, d, n}{m, a, d, n}{m, a, d, n}{m, a, d, n}

Theorem7. Let n ≥ 2 beaninteger.ThenthereisatleastoneNeutroSemihypergroupoforder n Proof. TheprooffollowsfromExample 4 andTheorem 6

Corollary2. ThereareinfinitelymanyNeutroSemihypergroupsuptoNeutroStrongIsomorphism. Proof. TheprooffollowsfromTheorem 7

Theorem8. Letn ≥ 2 beanyinteger.ThenthereisatleastoneNeutroHv-Semigroupofordern. Proof. TheprooffollowsfromExample 4 andTheorem 6.

Corollary3. ThereareinfinitelymanyNeutroHv-SemigroupsuptoNeutroStrongIsomorphism. Proof. TheprooffollowsfromTheorem 8.

4.Conclusions

Inthispaper,wediscussedthepropertiesofsomeNeutroHyperstructures.More precisely,weintroducedNeutroSemihypergroups(NeutroHv-Semigroups),constructed severalexamples,andstudiedsomeoftheirimportantsubsetsunderNeutroStrongIsomorphism.Itwasshownthroughexamplesthatsomeofthewellknownresultsforalgebraic hyperstructuresdonotholdforNeutroHyperstructures.Moreover,itwasprovedthat thereisatleastoneNeutroSemihypergroup(NeutroHv-Semigroups)oforder n where n is anyintegergreaterthanone.Theresultsinthispapermaybeconsideredasabaseforany possiblestudyinthefieldofNeutroHyperstructures.

Forfutureresearch,weraisethefollowingideas.

1. FindallNeutroSemihypergroups(NeutroHv-Semigroups)ofsmallorder(uptoNeutroStrongIsomorphism).

2. FindboundsforthenumberoffiniteNeutroSemihypergroups(NeutroHv-Semigroups) ofarbitraryorder n (uptoNeutroStrongIsomorphism).

3. ClassifysimpleNeutroSemihypergroups(NeutroHv-Semigroups)uptoNeutroStrongIsomorphism.

4. DefineotherNeutroHyperstructuressuchasNeutroPolygroup,NeutroHyperring,etc.

5. FindapplicationsofNeutroHyperstructuresinsomefieldslikeBiology,Physics, Chemistry,etc.

AuthorContributions: Conceptualization,M.A.-T.,B.D.,F.S.,andO.A.;methodology,M.A.-T.,B.D., F.S.,andO.A.;writing—originaldraftpreparation,M.A.-T.;writing—reviewandediting,M.A.-T., B.D.,F.S.,andO.A.Allauthorshavereadandagreedtothepublishedversionofthemanuscript.

Funding: Thisresearchreceivednoexternalfunding.

InstitutionalReviewBoardStatement: Notapplicable.

InformedConsentStatement: Notapplicable.

DataAvailabilityStatement: Notapplicable.

ConflictsofInterest: Theauthorsdeclarenoconflictofinterest.

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