On Neutro-LA-semihypergroups and Neutro-Hv-LAsemigroups

On Neutro-LA-semihypergroups and Neutro-Hv-LAsemigroups

Saeid Mirvakili

Payame Noor University

Akbar Rezaei  (  rezaei@pnu.ac.ir )

Payame Noor University https://orcid.org/0000-0002-6003-3993

Omaima Al-Shanqiti

Umm Al-Qura University

Florentin Smarandache

University of New Mexico - Gallup

Bijan Davvaz

Yazd University

Research Article

Keywords: Hyperoperation, Neutrohyperoperation, Antihyperoperation, LA-semihypergroup, Neutro-LAsemihypergroup, Anti-LA-semihypergroup, Hv-LA-semigroup, Neutro-Hv-LA-semigroup, Anti-Hv-LAsemigroup

Posted Date: March 28th, 2023

DOI: https://doi.org/10.21203/rs.3.rs-2553188/v1

License:   This work is licensed under a Creative Commons Attribution 4.0 International License.   Read Full License

OnNeutro-LA-semihypergroupsand Neutro-Hv -LA-semigroups

S.Mirvakilia,A.Rezaeia,∗ ,O.Al-Shanqitib,F.Smarandachec,B.Davvazd

aDepartmentofMathematics,PayameNoorUniversity, P.O.Box19395–4697Tehran,Iran.

bDepartmentofMathematics,UmmAl-QuraUniversityMecca, P.O.Box24341,SaudiArabia.

cDepartmentofMathematicsandScience,UniversityofNewMexico,Gallup, NM87301,USA.

dDepartmentofMathematics,YazdUniversity,Yazd,Iran.

E-mail:saeed mirvakili@pnu.ac.ir

E-mail:rezaei@pnu.ac.ir

E-mail:omshanqiti@uqu.edu.sa

E-mail:smarand@unm.edu

E-mail:davvaz@yazd.ac.ir

Abstract. Inthispaper,weextendthenotionof LA-semihypergroups (resp. Hv -LA-semigroups)toneutro-LA-semihypergroups(respectively, neutro-Hv -LA-semigroups).Anti-LA-semihypergroups(respectively,antiHv -LA-semigroups)arestudiedandinvestigatedsomeoftheirproperties. Weshowthatthesenewconceptsaredifferentfromclassicalconcepts byseveralexamples.Theseareparticularcasesoftheclassicalalgebraicstructuresgeneralizedtoneutroalgebraicstructuresandantialgebraicstructures(Smarandache,2019).

Keywords: Hyperoperation,Neutrohyperoperation,Antihyperoperation, LAsemihypergroup,Neutro-LA-semihypergroup,Anti-LA-semihypergroup, HvLA-semigroup,Neutro-Hv -LA-semigroup,Anti-Hv -LA-semigroup.

2000Mathematicssubjectclassification: 20N20,20N99.

2S.Mirvakili,A.Rezaei,O,Al-Shanqiti,F.Smarandache,B. Davvaz

1. Introduction

KazimandNaseeruddin[13]providedtheconceptofleftalmostsemigroup (abbreviatedas LA-semigroup).Theygeneralizedsomeusefulresultsofsemigrouptheory.Later,Mushtaq[17]andothersfurtherinvestigatedthestructureandaddedmanyusefulresultstothetheoryof LA-semigroups;seealso [1,3,11,14,18,19,26].

Ahypergroupasageneralizationofthenotionofagroup,wasintroducedby F.Marty[16]in1934.Somevaluablebooksinhyperstructureshavepublished [4,5,6,7,27].In1990,Vougiouklisintroducedtheconceptof Hv -structuresin FourthAHACongressasageneralizationofthewell-knownalgebraichyperstructures.Twobooksonalgebraic Hv -structureorweakhyperstructurehave beenpublished[7,27].

HilaandDine[10]introducedthenotionof LA-semihypergroupsasageneralizationofsemigroups,semihypergroups,and LA-semigroups.Yaqoob,Corsini andYousafzai[28]extendedtheworkofHilaandDine.Gulistan,Yaqooband Shahzad,[9]introducedthenotionof Hv -LA-semigroupsas LA-semihypergroups. Theyshowedthatevery LA-semihypergroupisan Hv -LA-semigroupandeach LA-semigroupendowedwithanequivalencerelationcaninducedan Hv -LAsemigroupandtheyinvestigatedisomorphismtheoremwiththehelpofregular relations.

In2019and2020,withinthefieldofneutrosophy,Smarandache[21,22,23] generalizedtheclassicalalgebraicstructurestoneutroalgebraicstructures(or neutroalgebras)whoseoperationsandaxiomsarepartiallytrue,partiallyindeterminate,andpartiallyfalseasextensionsofpartialalgebra,andtoantialgebraicstructures(orantialgebras) {whoseoperationsandaxiomsaretotally false}.Andingeneral,heextendedanyclassicalstructure,innomatterwhat fieldofknowledge,toaneutrostructureandanantistructure.Thesearenew fieldsofresearchwithinneutrosophy.

Smarandachein[23]revisitedthenotionsofneutroalgebrasandantialgebras,wherehestudiedpartialalgebras,Universalalgebras,Effectalgebrasand Boole′ spartialalgebras,andshowedthatneutroalgebrasaregeneralization ofpartialalgebras.Further,heextendedtheclassicalhyperalgebrato n-ary hyperalgebraanditsalternatives n-aryneutrohyperalgebraand n-aryantihyperalgebra[25].

ThenotionofneutrogroupwasdefinedandstudiedbyA.A.A.Agboolain [2].A.Rezaeietal.introducedthenotionsofneutrosemihypergroupand antisemihypergroup[20].Recently,S.Mirvakilietal.extendthenotionof Hvsemigroupstoneutro-Hv -semigroupsandanti-Hv -semigroupsandinvestigated manyoftheirproperties[15].

Inthispaper,theconceptofneutro-LA-semihypergroups(resp.neutro-HvLA-semigroup)andanti-LA-semihypergroups(resp.anti-Hv -LA-semigroup)is

On[Neutro](Anti)LA-semihypergroupsand[Neutro](Anti)Hv -LA-semigroups3

formallypresented.Moreover,Wecharacterize LA-semihypergroups(resp. HvLA-semigroup),neutro-LA-semihypergroups(resp.neutro-Hv -LA-semigroup) andanti-LA-semihypergroups(resp.anti-Hv -LA-semigroup)oforder2.

2. Preliminaries

Inthissectionwerecallsomebasicnotionsandresultsregardingto LAsemigroups, LA-semihypergroupsand Hv -LA-semigroups.

Agroupoid(H, ◦)isanon-emptyset H togetherwithamap ◦ : H × H → H called(binary)operation.Thestructure(H, ◦)iscalledagroupoid.

Definition2.1. [13]Agroupoid(H, ◦)iscalledan LA-semigroup,if(a◦b)◦c = (c ◦ b) ◦ a,forall a,b,c ∈ H

Example 2.2 [17]Let(Z, +)denotethecommutativegroupofintegersunder addition.Defineabinaryoperation ◦ in Z asfollows:

a ◦ b = b a, ∀a,b ∈ Z, where denotestheordinarysubtractionofintegers.Then(Z, ◦)isan LAsemigroup.

Definition2.3. ([4,6])Ahypergroupoid(H, ◦)isanon-emptyset H together withamap ◦ : H × H → P ∗ (H)called(binary)hyperoperation,where P ∗ (H) denotesthesetofallnon-emptysubsetsof H.Thehyperstructure(H, ◦)is calledahypergroupoidandimageofthepair(x,y)isdenotedby x ◦ y

If A and B arenon-emptysubsetsof H and x ∈ H,thenby A ◦ B, A ◦ x, and x ◦ B wemean A ◦ B =

Definition2.4. ([4,6])(1)Ahypergroupoid(H, ◦)iscalledasemihypergroup ifitsatisfiesthefollowing:

(A)(∀a,b,c ∈ H)(a ◦ (b ◦ c)=(a ◦ b) ◦ c)

(2)Ahypergroupoid(H, ◦)iscalledan Hv -semigroupifitsatisfiesthefollowing:

(WA)(∀a,b,c ∈ H)(a ◦ (b ◦ c) ∩ (a ◦ b) ◦ c) =

Definition2.5. ([9,10])(1)Ahypergroupoid(H, ◦)iscalledaLeftAlmost semihypergrouporan LA-semihypergroupifitsatisfiesthefollowing:

(LA)(∀a,b,c ∈ H)(

(2)Ahypergroupoid(H, ◦)iscalledaLeftAlmost Hv -semigroupor Hv -LAsemigroupifitsatisfiesthefollowing:

(WLA)(∀a,b,c ∈ H)(

Example 2.6 ([4,6])Let H beanonemptysetandforall x,y ∈ H,wedefine x ◦ y = H.Then(H, ◦)isasemihypergroupandan LA-semihypergroup.

Wedefinethecommutativelawon(H, ◦)asfollows:

4S.Mirvakili,A.Rezaei,O,Al-Shanqiti,F.Smarandache,B. Davvaz

(C)(∀a,b ∈ H)(a ◦ b = b ◦ a).

Also,wedefinetheweakcommutativelawon(H, ◦)asfollows:

(WC)(∀a,b ∈ H)(a ◦ b ∩ b ◦ a = ∅).

Theorem2.7. Let (H, ◦) beacommutativehypergroupoid.Then (H, ◦) isan LA-semihypergroupifandonlyif (H, ◦) isasemihypergroup.

Proof. Let x,y,z ∈ H.Thenbycommutativitywehave

Example 2.8 Let H = {a,b}.Definethehyperoperation ◦ on H withthe followingCayleytable.

◦ ab

Then(H, ◦)isasemihypergroup,butisnotan LA-semihypergroup.

Example 2.9. Let H = {a,b}.Definethehyperoperation ◦ on H withthe followingCayleytable.

◦ ab

Then(H, ◦)isan LA-semihypergroup,butisnotasemihypergroup.

Theorem2.10. Let (H, ◦) beacommutativehypergroupoid.Then (H, ◦) is an Hv -LA-semigroupifandonlyif (H, ◦) isan Hv -semigroup

Proof. Let x,y,z ∈ H.Thenbycommutativitywehave

Example 2.11 Let H = {a,b}.Definethehyperoperation ◦ on H withthe followingCayleytable.

◦ ab

a aa

b bH

Then(H, ◦)isan Hv -semigroup,butisnotan Hv -LA-semigroup.

Example 2.12. Let H = {a,b}.Definethehyperoperation ◦ on H withthe followingCayleytable.

◦ ab

a ab

b Ha

Then(H, ◦)isan Hv -LA-semigroup,butisnotan Hv -semigroup.

3. OnNeutro-LA-semihypergroups,Neutro-Hv -LA-semigroups, Anti-LA-semihypergroupsandAnti-Hv -LA-semigroups

F.Smarandachegeneralizedtheclassicalalgebraicstructurestotheneutroalgebraicstructuresandantialgebraicstructures. neutro-sophicationof anitem C (thatmaybeaconcept,aspace,anidea,anoperation,anaxiom, atheorem,atheory,etc.)meanstosplit C intothreeparts(twopartsopposite toeachother,andanotherpartwhichistheneutral/indeterminacybetween theopposites),aspertinenttoneutrosophy {(<A>,<neutA>,<antiA>), orwithothernotation(T,I,F )},meaningcaseswhere C ispartiallytrue(T ), partiallyindeterminate(I),andpartiallyfalse(F ).While anti-sophication of C meanstototallydeny C (meaningthat C ismadefalseonitswhole domain)(fordetailseeSmarandache[21,22,24,25]).

Neutro-sophicationofanaxiom onagivenset X,meanstosplittheset X intothreeregionssuchthat:ononeregiontheaxiomistrue(wesaydegree oftruth T oftheaxiom),onanotherregiontheaxiomisindeterminate(wesay degreeofindeterminacy I oftheaxiom),andonthethirdregiontheaxiom isfalse(wesaydegreeoffalsehood F oftheaxiom),suchthattheunionof theregionscoversthewholeset,whiletheregionsmayormaynotbedisjoint, where(T,I,F )isdifferentfrom(1, 0, 0)andfrom(0, 0, 1).Anti-sophicationof anaxiomonagivenset X,meanstohavetheaxiomfalseonthewholeset X (wesaytotaldegreeoffalsehood F oftheaxiom),or(0, 0, 1).

Similarlyforthe neutro-sophicationofanoperation definedonagiven setX,meanstosplittheset X intothreeregionssuchthatononeregionthe operationiswell-defined(orinner-defined)(wesaydegreeoftruth T ofthe operation),onanotherregiontheoperationisindeterminate(wesaydegree ofindeterminacy I oftheoperation),andonthethirdregiontheoperationis outer-defined(wesaydegreeoffalsehood F oftheoperation),suchthatthe unionoftheregionscoversthewholeset,whiletheregionsmayormaynotbe disjoint,where(T,I,F )isdifferentfrom(1, 0, 0)andfrom(0, 0, 1).

Anti-sophicationofanoperation onagivenset X,meanstohavethe operationouter-definedonthewholeset X (wesaytotaldegreeoffalsehood F oftheaxiom),or(0, 0, 1).

Inthissectionwewilldefinethe neutro-LA-semihypergroups and antiLA-semihypergroups

Definition3.1.Neutrohyperoperation(Neutrohyperlaw) Aneutrohyperoperationisamap ◦ : H × H → P (U )where U isauniverseofdiscourse thatcontains H thatsatisfiesthebelowneutro-sophicationprocess.

Theneutro-sophication(degreeofwell-defined,degreeofindeterminacy,degreeofouter-defined)ofthehyperoperationisthefollowingneutrohyperoperation:

(NHA)(∃x,y ∈ H)(x ◦ y ∈ P ∗ (H))and(∃x,y ∈ H)(x ◦ y isanindeterminatesubset,or x ◦ y ̸∈ P ∗ (H)).

Theneutro-sophication(degreeoftruth,degreeofindeterminacy,degreeof falsehood)ofthe LA-semihypergroupaxiomisthefollowingneutrohyperLAsemihypergroup:

(NLA)(∃a,b,c ∈ H suchthat(a,b,c) =(x,x,x)or(a,b,c) =(x,y,x))

((a ◦ b) ◦ c =(c ◦ b) ◦ a)and(∃d,e,f ∈ H suchthat(d,e,f ) =(x,x,x)or

(d,e,f ) =(x,y,x))((d ◦ e) ◦ f =(f ◦ e) ◦ d or(d ◦ e) ◦ f =indeterminate,or (f ◦ e) ◦ d =indeterminate).

Also,Theneutro-sophication(degreeoftruth,degreeofindeterminacy,degree offalsehood)ofthe Hv -LA-semigroupaxiomisthefollowingneutrohyperHvLA-semigroup:

(NWLA)(∃a,b,c ∈ H suchthat(a,b,c) =(x,x,x)or(a,b,c) =(x,y,x))

((a ◦ b) ◦ c ∩ (c ◦ b) ◦ a = ∅)and(∃d,e,f ∈ H suchthat(d,e,f ) =(x,x,x)or (d,e,f ) =(x,y,x))((d ◦ e) ◦ f ∩ (f ◦ e) ◦ d = ∅ or(d ◦ e) ◦ f =indeterminate, or(f ◦ e) ◦ d =indeterminate).

Wedefinetheneutrohypercommutativity(NC)on(H, ◦)asfollows:

(NC)(∃a,b ∈ H)(a ◦ b = b ◦ a)and(∃c,d ∈ H)(c ◦ d = d ◦ c,or c ◦ d = indeterminate,or d ◦ c =indeterminate).

Also,wedefinetheneutrohyperweakcommutativity(NWC)on(H, ◦)asfollows:

(NWC)(∃a,b ∈ H)(a ◦ b ∩ b ◦ a = ∅)and(∃c,d ∈ H)(c ◦ d ∩ d ◦ c = ∅,or c ◦ d =indeterminate,or d ◦ c =indeterminate).

Now,wedefineaneutrohyperalgebraicsystem S =<H,F,A>,where H is asetorneutrosophicset, F isasetofthehyperoperations(hyperlaws),and A is thesetofhyperaxioms,suchthatthereexistsatleastoneneutrohyperoperation (neutrohyperlaw)oratleastoneneutrohyperaxiom,andnoantihyperoperation (antihyperlaw)andnoantihyperaxiom.

Definition3.2.Antihyperoperation {Antihyperlaw(AHL)}

Theantihyper-sophication(totallyouter-defined)ofthehyperoperation(hyperlaw)givesthedefinitionofantihyperoperationantihyperlaw(AHL):

(AHL)(∀x,y ∈ H)(x ◦ y ̸∈ P ∗ (H)).

Theantihyper-sophication(totallyfalse)ofthe LA-semihypergroup:

(ALA)(∀a,b,c ∈ Hsuchthat (a,b,c) =(x,x,x) or (a,b,c) =(x,y,x))((a◦

b) ◦ c) =(c ◦ b) ◦ a)

Also,theantihyper-sophication(totallyfalse)ofthe Hv -LA-semigroup:

(AWLA)(∀a,b,c ∈ Hsuchthat (a,b,c) =(x,x,x) or (a,b,c) =(x,y,x))((a◦ b) ◦ c) ∩ (c ◦ b) ◦ a = ∅)

Wedefinetheanticommutativity(AC)on(H, ◦)asfollows:

(AC)(∀a,b ∈ H with a = b)(a ◦ b = b ◦ a).

Also,wedefinetheantiweakcommutativity(AWC)on(H, ◦)asfollows:

6S.Mirvakili,A.Rezaei,O,Al-Shanqiti,F.Smarandache,B. Davvaz

On[Neutro](Anti)LA-semihypergroupsand[Neutro](Anti)Hv -LA-semigroups7

(AWC)(∀a,b ∈ H with a = b)(a ◦ b ∩ b ◦ a = ∅).

Definition3.3. (1)Aneutro-LA-semihypergroupisanalternativeof LAsemihypergroupthathasatleastone(NLA),withnoantihyperoperation.

(2)Aneutro-Hv -LA-semigroupisanalternativeof Hv -LA-semigroupthat hasatleastone(WNLA),withnoantihyperoperation.

(3)Ananti-LA-semihypergroupisanalternativeof LA-semihypergroupthat hasatleastone(ALA)oran(AHL)axiom.

(4)Ananti-Hv -LA-semigroupisanalternativeof Hv -LA-semigroupthat hasatleastone(WALA)oran(AHL)axiom.

Remark 3.4 Ifhyperoperation ◦ inDefinition3.3isoperation,thenwehave neutro-LA-semigroupandanti-LA-semigroup.

Example 3.5 (i)Let H = {a,b,c} and U = {a,b,c,d} auniverseofdiscourse thatcontains H.Definetheneutrohyperoperation ◦ on H withthefollowing Cayleytable.

◦ abc a aaa

b aa {a,b,d}

c c ? H

Then(H, ◦)isaneutrosemihypergroup.

Example 3.6 (i)Let N bethesetofnaturalnumbersexcept0.DefinehyperLow ◦ on N by x ◦ y = { x 2 x2 +1 ,y}.Then(N, ◦)isananti-LA-semihypergroup.

(AHL)isvalid,sinceforall x,y ∈ N, x ◦ y ̸∈ P ∗ (N).Thus,(AHL)holds.

(ii)Let H = {a,b}.Definethehyperoperation ◦ on H withthefollowing Cayleytable.

◦ ab

a ba

b aa

Then(H, ◦)isacommutativeanti-LA-semihypergroup.

(iii)Let H = {a,b} Definethehyperoperation ◦ onHwiththefollowing Cayleytable.

◦ ab a aa b bb

Then(H, ◦)isananticommutativeanti-LA-semihypergroup.

Theorem3.7. (1) Every LA-semihypergroupisan Hv -LA-semigroup.

(2) Everyanti-Hv -LA-semigroupisananti-LA-semihypergroup.

(3) Everyneutro-Hv -LA-semigroupisaneutro-LA-semihypergrouporan anti-LA-semihypergroup.

(4) Every Hv -LA-semigroupisan LA-semihypergrouporaneutro-LAsemihypergrouporananti-LA-semihypergroup.

8S.Mirvakili,A.Rezaei,O,Al-Shanqiti,F.Smarandache,B. Davvaz

Example 3.8. [9]TheConverseofpart(1)ofTheorem3.7isnottrue.Consider H = {x,y,z} anddefineahyperoperation ◦ on H bythefollowingtable: ◦ xyz x x {x,z} H y {x,z} xx z {x,y} x {x,z}

Then(H, ◦)isan Hv -LA-semigroupwhichisnotan LA-semihypergroupand notan Hv -semigroup.Indeed,wehave

Thus, z ◦ (y ◦ y) ∩ (z ◦ y) ◦ y = ∅.Therefore(H, ◦)isnotan Hv -semigroup. Also,

Thus,(H, ◦)isnotan LA-semihypergroup.

Example 3.9. TheConverseofpart(2)ofTheorem3.7isnottrue.Consider H = {x,y,z} anddefineahyperoperation ◦ on H bythefollowingtable:

◦ ab

a aH

b aa

Then(H, ◦)isacommutativeanti-LA-semihypergroupandisnotanti-Hv -LAsemigroup.

Let(H, ◦)isahypergroupoid.Thenthehyperoperation ∗ definedasfollows:

x ∗ y = y ◦ x, ∀x,y ∈ H. (H, ∗)iniscalleddualhypergroupoidof(H, ◦). Itiseasytoseethat:

Theorem3.10. (H, ◦) isasemihypergroupifandonlyif (H, ◦) isasemihypergroup.

Theorem3.10for LA-semihypergroupsisnottrue.

Example 3.11. Let(H = {a,b}, ◦)bean LA-semihypergroupoforder2when thehyperoperation ◦ definedon H withthefollowingCayleytable.

◦ ab a HH

b aa

But(H, ∗)isnotan LA-semihypergroup.

Proposition3.12. Let (H, ◦H ) and (G, ◦G) betwoneutro-LA-semihypergroups (resp.anti-LA-semihypergroups).Then (H×G, ∗) isaneutro-LA-semihypergroup (resp.anti-LA-semihypergroups),where ∗ isdefinedon H × G by:forany

On[Neutro](Anti)LA-semihypergroupsand[Neutro](Anti)Hv -LA-semigroups9

(x1,y1), (x2,y2) ∈ H × G

(x1,y1) ∗ (x2,y2)=(x1 ◦H x2,y1 ◦G y2)

Notethatif(H, ◦)isaneutro-LA-semihypergroup,thenifthereisanonemptyset H1 ⊆ H, suchthat(H1, ◦)isan LA-semihypergroup,wecallit Smarandache LA-semihypergroup

Suppose(H, ◦H )and(G, ◦G)betwohypergroupoids.Afunction f : H → G iscalledahomomorphismif,forall a,b ∈ H,f (a ◦H b)= f (a) ◦G f (b). Proposition3.13. Let (H, ◦H ) bean LA-semihypergroup(Hv -LA-semigroup), (G, ◦G) beaneutro-LA-semihypergroup(neutro-Hv -LA-semigroup)and f : H → G beahomomorphism.Then (f (H), ◦G) isan LA-semihypergroup(HvLA-semigroup),where f (H)= {f (h): h ∈ H}

Proof. Assumethat(H, ◦H )isan LA-semihypergroup(Hv -LA-semigroup)and x,y,z ∈ f (H) Thenthereexist h1,h2,h3 ∈ f (H)suchthat f (h1)= x,f (h2)= y and f (h3)= z, andsowehave

y) ◦G z.

Definition3.14. Let(H, ◦H )and(G, ◦G)betwohypergroupoids.Abijection f : H → G isanisomorphismifitconservesthemultiplication(i.e. f (a ◦H b)= f (a) ◦G f (b))andwrite H ∼ = G. Abijection f : H → G isanantiIsomorphism

ifforall a,b ∈ H,f (a ◦H b) = f (b) ◦G f (a) Abijection f : H → G isa neutroIsomorphismifthereexist a,b ∈ H,f (a ◦H b)= f (b) ◦G f (a), i.e.degree oftruth(T ), thereexist c,d ∈ H,f (c◦H d)or f (c)◦G f (d)areindeterminate,i.e. degreeofIndeterminacy(I), andthereexist e,h ∈ H,f (e ◦H h) = f (e) ◦G f (h), i.e.degreeoffalsehood(F ), where(T,I,F )aredifferentfrom(1, 0, 0)and (0, 0, 1), and T,I,F ∈ [0, 1]

Proposition3.15. Let (Hi, ◦),where i ∈ Λ,beafamilyofneutro-LA-semihypergroups (neutro-Hv -LA-semigroups).Then ( i∈Λ Hi, ◦) isaneutro-LA-semihypergroup (neutro-Hv -LA-semigroup)orananti-LA-semihyperGgroup(anti-Hv -LA-semigroup) oran LA-semihypergroups(Hv -LA-semigroup).

Proof. Itistrivial.

10S.Mirvakili,A.Rezaei,O,Al-Shanqiti,F.Smarandache,B.Davvaz

Proposition3.16. Let (Hi, ◦) beafamilyofanti-LA-semihypergroupsn(antiHv -LA-semigroups),where i ∈ Λ.Then ( i∈Λ Hi, ◦) isananti-LA-semihypergroup (anti-Hv -LA-semigroup).

Proof. Itistrivial. □

Notethatif(H, ◦)isaneutro-LA-semihypergroup(neutro-Hv -LA-semigroup) and(G, ◦)isananti-LA-semihypergroup(anti-Hv -LA-semigroup),then(H ∩ G, ◦)isananti-LA-semihypergroupb(anti-Hv -LA-semigroup).Also,let(H, ◦H ) beaneutro-LA-semihypergroup(neutro-Hv -LA-semigroup)and(G, ◦G)bean anti-LA-semihypergroup(anti-Hv -LA-semigroup)and H ∩ G = ∅.Definehyperoperation ◦ on H G by:

x ◦ y =

x ◦H y if x,y ∈ H; x ◦G y if x,y ∈ G; {x,y} otherwise.

Then(H G, ◦)isaneutro-LA-semihypergroup(neutro-Hv -LA-semigroup), butitisnotananti-LA-semihypergroup(anti-Hv -LA-semigroup).

Proposition3.17. Let (H, ◦) beananti-LA-semihypergroup(anti-Hv -LAsemigroup)and e ∈ H.Then (H ∪{e}, ∗) isaneutro-LA-semihypergroup (neutro-Hv -LA-semigroup),where ∗ isdefinedon H ∪{e} by:

x ∗ y = x ◦H y if x,y ∈ H; {e,x,y} otherwise.

Proof. Itisstraightforward.

4. Characterizationofgroupoidsoforder2

Inthenextresultsweusetheoperation ◦ : H × H → H. Inthissection,let ◦ beanoperationon H = {a,b} and(A11,A12,A21,A22) insideofthebelowCayleytable:

◦ ab

a A11 A12

b A21 A22

Lemma4.1. Let (H = {a,b}, ◦H ) and (G = {a′ ,b′ }, ◦G) betwogroupoidswith theCayleytables (h11,h12,h21,h22) and (g11,g12,g21,g22) respectively.Then

H ∼ = G ifandonlyifforall i,j ∈{1, 2},

gij = h′ ij or

gij = G \ k′ ij , where k12 = h21,k21 = h12,k11 = h22 and k22 = h11

On[Neutro](Anti)LA-semihypergroupsand[Neutro](Anti)Hv -LA-semigroups11

Lemma4.2. Anycommutativesemigroupoforder2isan LA-semiGrouop.

Theorem4.3. Every LA-semigroupoforder2iscommutative.

Proof. Let(H = {a,b}, ◦)bean LA-semigroup.Wehave

(1)(a ◦ b) ◦ b =(b ◦ b) ◦ a,

(2)(a ◦ a) ◦ b =(b ◦ a) ◦ a.

Let a ◦ b = a.Thenby(1), a =(b ◦ b) ◦ a. If b ◦ b = b,thenweobtain b ◦ a = a, andso(H, ◦)iscommutative.Sowehave b ◦ b = a. Hence a ◦ a = a andby(2), a =(b ◦ a) ◦ a.If b ◦ a = b,thenweobtain a = b.andthisisacontradiction. Thus, b ◦ a = a,andso(H, ◦)iscommutative.

Now,let a◦b = a.Thenbythesimilarwayweobtain(H, ◦)iscommutative. □

Corollary4.4. Any LA-semigroupoforder2iscommutative.

Corollary4.5. Any LA-semigroupoforder2isasemigroup.

Corollary4.6. ThereisnoNon-commutative LA-semigroupoforder2.

Theorem4.7. [8] Thereexist5semigroups (H = {a,b}, ◦i), i =1, , 5,of order2bythefollowingCayleytable(uptoisomorphism).

Theorem4.8. Thereexist3 LA-semigroupsoforder2(uptoisomorphism).

Proof. ByCorollary5.2,theonly LA-semigroupsoforder2arecommutative semigroupsoforder2,andso(H, ◦i),for i =1, , 3,fromTheorem4.7are LA-semigroupsoforder2(uptoisomorphism).

Theorem4.9. Thereexist5anti-LA-semigroupsoforder2(uptoisomorphism).

Proof. Let(H = {a,b}, ◦)beananti-LA-semigroup.ThenWehave

(1)(a ◦ b) ◦ b =(b ◦ b) ◦ a,

(2)(a ◦ a) ◦ b =(b ◦ a) ◦ a.

Also,ThenwehaveoneofthefollowingCases:

(3)(a ◦ b = a & b ◦ a = a),

(4)(a ◦ b = a & b ◦ a = b),

(5)(a ◦ b = b & b ◦ a = a),

(6)(a ◦ b = b & b ◦ a = b).

Case3:By(1)and(2)wehave

(7) a =(b ◦ b) ◦ a,

(8)(a ◦ a) ◦ b = a ◦ a.

12S.Mirvakili,A.Rezaei,O,Al-Shanqiti,F.Smarandache,B.Davvaz

If a ◦ a = a,then(8)impliesthat a ◦ b = a,whichisacontradiction.If a ◦ a = b, then(8)impliesthat b ◦ b = b,andso b ◦ b = a.Weset ◦1 := ◦ andtherefore wewehaveananti-LA-semigroupasfollows:

◦1 ab a ba

b aa

Case4:By(1)and(2)wehave

(9) a =(b ◦ b) ◦ a, (10)(a ◦ a) ◦ b = b.

If a ◦ a = b,then(8)impliesthat b ◦ b = b,andso b ◦ b = a.Weset ◦2 := ◦ andthereforewewehaveananti-LA-semigroupasfollows:

◦2 ab

a ba

b ba

If a ◦ a = a,thenby(9) b ◦ b = a so b ◦ b = b.Weset ◦3 := ◦ andthereforewe wehaveananti-LA-semigroupasfollows:

◦3 ab

a aa

b bb

Case5:By(1)and(2)wehave

(11) b ◦ b =(b ◦ b) ◦ a, (12)(a ◦ a) ◦ b = a ◦ a.

If a ◦ a = a,then b ◦ b = a or b ◦ b = b.Let b ◦ b = a.Thenby(11) a ◦ a = b andthisisacontradiction.If b ◦ b = b,thenweset ◦4 := ◦ andthereforewe haveananti-LA-semigroupasfollows:

◦4 ab

a ab

b ab

If a ◦ a = b,then(12)impliesthat b ◦ b = a.Weset ◦5 := ◦ andthereforewe haveananti-LA-semigroupasfollows:

◦5 ab

a bb

b aa

Case6:By(1)and(2)wehave

(13) b ◦ b =(b ◦ b) ◦ a, (14)(a ◦ a) ◦ b = b.

On[Neutro](Anti)LA-semihypergroupsand[Neutro](Anti)Hv -LA-semigroups13

If a ◦ a = b,then(14)impliesthat b ◦ b = a.Weset ◦6 := ◦ andthereforewe haveananti-LA-semigroupasfollows:

◦6 ab

a bb

b ba

If a ◦ a = a,then(14)impliesthat a ◦ b = a,whichisacontradiction.(H, ◦1) ∼ = (H, ◦6),andsowehaveanti-LA-semigroups(H, ◦i),for i =1, , 5,oforder

2. □

Corollary4.10. Thereexists1commutativeanti-LA-semigroupoforder2 (uptoisomorphism).

Corollary4.11. Thereexist4Non-commutativeanti-LA-semigroupsoforder 2(uptoisomorphism).

Theorem4.12. Everyneutro-LA-semigroupoforder2isNon-commutative.

Proof. Let(H = {a,b}, ◦)beacommutativeneutro-LA-semigroup.ThenWe have

(1)(a ◦ b) ◦ b =(b ◦ b) ◦ a,

(2)(a ◦ a) ◦ b =(b ◦ a) ◦ a,

(3) a ◦ b = b ◦ a, or

(4)(a ◦ b) ◦ b =(b ◦ b) ◦ a,

(5)(a ◦ a) ◦ b =(b ◦ a) ◦ a,

(6) a ◦ b = b ◦ a.

Case2:If a ◦ b = a = b ◦ a,thenwehave

(7) a =(b ◦ b) ◦ a,

(8)(a ◦ a) ◦ b = a ◦ a, or

(9) a =(b ◦ b) ◦ a,

(10)(a ◦ a) ◦ b = a ◦ a. Now,(7)impliesthat b ◦ b = a and a ◦ a = b,whichisacontradictionwith(8). Also,(10)impliesthat a ◦ a = b and b ◦ b = a andthisisacontradictionwith

(9)

Case2:If a ◦ b = b = b ◦ a,thenbythesimilarwayofCase2weprovethat thereisnocommutativeneutro-LA-semigroupoforder2. □ Theorem4.13. Thereexist2neutro-LA-semigroupsoforder2(uptoisomorphism).

Proof. Let(H = {a,b}, ◦)beaneutro-LA-semigroup.ThenbyTheorem4.12 (H = {a,b}, ◦)isaNon-commutativeneutro-LA-semigroup.Now,wehave

14S.Mirvakili,A.Rezaei,O,Al-Shanqiti,F.Smarandache,B.Davvaz

(1)(a ◦ b) ◦ b =(b ◦ b) ◦ a,

(2)(a ◦ a) ◦ b =(b ◦ a) ◦ a, or

(3)(a ◦ b) ◦ b =(b ◦ b) ◦ a,

(4)(a ◦ a) ◦ b =(b ◦ a) ◦ a.

Case1:If a ◦ b = a and b ◦ a = b,thenby(1),(2),(3)and(4)weobtain

(5) a =(b ◦ b) ◦ a,

(6)(a ◦ a) ◦ b = b, or

(7) a =(b ◦ b) ◦ a, (8)(a ◦ a) ◦ b = b.

Let(5)and(6)betrue.If a ◦ a = a,then a ◦ b = a andthisisacontradiction.

If a ◦ a = b,thenusing(6), b ◦ b = b.Weset ◦1 := ◦ andthereforewehavea neutro-LA-semigroupasfollows:

◦1 ab a ba b bb

Let(7)and(8)betrue.If b ◦ b = b,then b ◦ a = a andthisisacontradiction.

If b ◦ b = a by(7), a ◦ a = a.Weset ◦2 := ◦ andthereforewehaveaneutroLA-semigroupasfollows:

◦2 ab a aa

b ba

Case2:If a ◦ b = b and b ◦ a = a,thenby(1),(2),(3)and(4)weobtain

(9) b ◦ b =(b ◦ b) ◦ a, (10)(a ◦ a) ◦ b = a ◦ a, or

(11) b ◦ b =(b ◦ b) ◦ a, (12)(a ◦ a) ◦ b = a ◦ a.

Let(9)and(10)betrue.If a ◦ a = a,then a ◦ b = a andthisisacontradiction.

If a ◦ a = b by(10), b ◦ b = b.Weset ◦3 := ◦ andthereforewehavea neutro-LA-semigroupasfollows:

◦3 ab

a bb

b ab

Let(11)and(12)betrue.If b ◦ b = b,then b ◦ a = b andthisisacontradiction.

If b ◦ b = a by(11), a ◦ a = a.Weset ◦4 := ◦ andthereforewehavea

On[Neutro](Anti)LA-semihypergroupsand[Neutro](Anti)Hv -LA-semigroups15

neutroLA-semigroupasfollows:

◦4 ab a ab b aa Itisnottodifficulttoseethat(H, ◦1) ∼ = (H, ◦2)and(H, ◦3) ∼ = (H, ◦4) Thereforethereexist2neutro-LA-semigroupsoforder2uptoisomorphism. □

Now,bytheaboveresultsinthissection,weobtainthenumberofanti-LAsemigroups,neutro-LA-semigroupsand LA-semigroupsoforder2(classesup toisomorphism).

Table1. Classificationofthegroupoidsoforder2

CAC

LA-semigroups30

neutro-LA-semigroups02 Anti-LA-semigroup14

5. Characterizationofhypergroupoidsoforder2

Inthenextresultsweusethehyperoperationinsteadofneutrohyperoperation.

Inthissection,let ◦ beahyperoperationon H = {a,b} and(A11,A12,A21,A22) insideofthebelowCayleytable:

◦ ab a A11 A12

b A21 A22

Lemma5.1. Let (H = {a,b}, ◦H ) and (G = {a′ ,b′ }, ◦G) betwohypergroupoids withtheCayleytables (H11,H12,H21,H22) and (G11,G12,G21,G22) respectively.Then H ∼ = G ifandonlyifforall i,j ∈{1, 2}, Gij = H ′ ij or

Gij = K ′ ij if Kij = H; G \ K ′ ij if Kij = H.

where K11 = H22,K12 = H12,K21 = H21 and K22 = H11

Proof. Itisstraightforward. □

Lemma5.2. Let (H, ◦) beagroupoidoforder2. (H, ◦) isan LA-semihypergroup ifandonlyifitisacommutativesemigroup.

Theorem5.3. If (H, ◦) isananti-Hv -LA-semigroupoforder2,then ◦ isan operationand (H, ◦) isanti-LA-semigroup.

16S.Mirvakili,A.Rezaei,O,Al-Shanqiti,F.Smarandache,B.Davvaz

Theorem5.4. Thereexistsoneantiweakcommutative LA-semihypergroupof order2,uptoisomorphism.

Proof. Let(H = {a,b}, ◦)beanantiweakcommutative LA-semihypergroup.

Thenwehave(a ◦ b = a & b ◦ a = b)or(a ◦ b = b & b ◦ a = a) Suppose a ◦ b = a and b ◦ a = b. Since(H = {a,b}, ◦)isan LA-semihypergroup,weget

(1)(a ◦ b) ◦ b =(b ◦ b) ◦ a,

(2)(a ◦ a) ◦ b =(b ◦ a) ◦ a. Thus,by(1)and(2)weobtain

(3) a =(b ◦ b) ◦ a,

(4)(a ◦ a) ◦ b = b.

Now,if b ◦ b = a,then(3)impliesthat a ◦ a = a. So(4)impliesthat a ◦ b = b andthisisacontradiction.If b ◦ b = b,then(3)impliesthat b ◦ a = a andthis isacontradiction.Finally,if b ◦ b = H,then(3)impliesthat a = H andthis isacontradiction.

Let a ◦ b = b and b ◦ a = a. Thenusing(1)and(2)wehave

(5) b ◦ b =(b ◦ b) ◦ a,

(6)(a ◦ a) ◦ b = a ◦ a.

If a ◦ b = b and b ◦ a = a,thenusing(5)and(6),weget b ◦ b = a, b ◦ b = b, a ◦ a = a or a ◦ a = a.Therefore b ◦ b = H = a ◦ a. □

Theorem5.5. Thereexist3anticommutative LA-semihypergroupsoforder2, uptoisomorphism.

Proof. Let(H = {a,b}, ◦)beanantiweakcommutative LA-semihypergroup. Thenwehave(a ◦ b = b ◦ a. Thenwehave

(1)(a ◦ b = a & b ◦ a = b),

(2)(a ◦ b = b & b ◦ a = a),

(3)(a ◦ b = a & b ◦ a = H),

(4)(a ◦ b = b & b ◦ a = H),

(5)(a ◦ b = H & b ◦ a = a),

(6)(a ◦ b = H & b ◦ a = b).

ByproofofTheorem5.4,case(1)cannotadmittingan LA-semihypergroup. ByproofofTheorem5.4,case(2)admittinga LA-semihypergroup(H, ◦1)with thefollowingCayleytable: ◦1 ab

Hb b aH

Case3:Let a ◦ b = a and b ◦ a = H. Since(H = {a,b}, ◦)isan LAsemihypergroup,so

(7)(a ◦ b) ◦ b =(b ◦ b) ◦ a,

(8)(a ◦ a) ◦ b =(b ◦ a) ◦ a.

On[Neutro](Anti)LA-semihypergroupsand[Neutro](Anti)Hv -LA-semigroups17

Thus,by(3),(7)and(8)weobtain

(9) a =(b ◦ b) ◦ a, (10)(a ◦ a) ◦ b = H.

Now,if b ◦ b = a,then(9)impliesthat a ◦ a = a. So(10)impliesthat a ◦ b = H, whichisacontradiction.If b ◦ b = b,then(9)impliesthat b ◦ a = a andthisis acontradiction.Finally,if b ◦ b = H,then(9)impliesthat a = H,whichisa contradiction.

Case4:Let a ◦ b = b and b ◦ a = H. Thusby(4),(7)and(8)weobtain

(11) b ◦ b =(b ◦ b) ◦ a, (12)(a ◦ a) ◦ b = H.

Now,if b ◦ b = a,then(11)impliesthat a ◦ a = a. So(12)impliesthat a ◦ b = H, whichisacontradiction.If b ◦ b = b,then(11)impliesthat b ◦ a = b andthis isacontradiction.Sowehave b ◦ b = H.By(4)and(12)weobtain a ◦ a = b or a ◦ a = H.Thereforeweobtaintwohyperoperationsandwecallthesetwo hyperoperation ◦2 and ◦3,respectivelyasfollows:

◦2 ab

Case5:Let a ◦ b = H and b ◦ a = a. Thus,by(4),(7)and(8)weobtain

(13) H =(b ◦ b) ◦ a, (14)(a ◦ a) ◦ b = a ◦ a.

Now,if a ◦ a = b,then(14)impliesthat b ◦ b = b. So(13)impliesthat b ◦ a = H, whichisacontradiction.If a ◦ a = a,then(14)impliesthat a ◦ b = a andthis isacontradiction.Sowehave a ◦ a = H.By(5)and(13)weobtain(b ◦ b = a or b ◦ b = H.Thereforeweobtaintwohyperoperationsandwecallthesetwo hyperoperation ◦4 and ◦5,respectivelyasfollows:

◦4 ab a HH b aH

◦5 ab a HH b aa

Case6:Let a ◦ b = H & b ◦ a = b. Thusby(4),(7)and(8)weobtain

(15) H =(b ◦ b) ◦ a,

(16)(a ◦ a) ◦ b = b.

Now,if a ◦ a = b,then(16)impliesthat b ◦ b = b. So(15)impliesthat b ◦ a = H andthisisacontradiction.If a ◦ a = a,then(16)impliesthat a ◦ b = a andthis isacontradiction.Sowehave a ◦ a = H.By(5)and(16)weobtain H = b, whichisacontradiction.

Itiseasytoseethat(H, ◦2) ∼ = (H, ◦4)and(H, ◦3) ∼ = (H, ◦5). Therefore (H, ◦1),(H, ◦2)and(H, ◦4)areanticommutative LA-semihypergroupsoforder 2. □

18S.Mirvakili,A.Rezaei,O,Al-Shanqiti,F.Smarandache,B.Davvaz

Corollary5.6. Thereisnoanticommutative LA-semigroupoforder2. Theorem5.7. Thereisnoproperanti-Hv -LA-semigroupoforder2.

Proof. Let(H = {a,b}, ◦)beaproperanti-Hv -LA-semigroup.ThenWehave

(1)(a ◦ b) ◦ b ∩ (b ◦ b) ◦ a = ∅,

(2)(a ◦ a) ◦ b ∩ (b ◦ a) ◦ a = ∅.

Sinceanti-Hv -LA-semigroup(H = {a,b}, ◦)isproper,wehaveoneofthe followingcases:

(3) a ◦ a = H,

(4) a ◦ b = H,

(5) b ◦ a = H,

(6) b ◦ b = H.

Case3:If b◦a = a or b◦a = H,then(b◦a)◦a = H andthisisacontradiction with(2).If b ◦ a = b,then(b ◦ a) ◦ a = b and(a ◦ a) ◦ b = H ◦ b.Thus,by(2) wehave a ◦ b = a and b ◦ b = a. Then(a ◦ b) ◦ b = a and(b ◦ b) ◦ a = H,which isacontradictionwith(1).

Case4:Then(a ◦ b) ◦ b = H andthisisacontradictionwith(1).

Case5:Then(b ◦ a) ◦ a = H,whichisacontradictionwith(2).

Case6:If a◦b = b or a◦b = H,then(a◦b)◦b = H andthisisacontradiction with(2).If a ◦ b = a,then(a ◦ b) ◦ b = a and(b ◦ b) ◦ a = H ◦ a.Soby(1)we have a ◦ a = b and b ◦ a = b. Then(b ◦ a) ◦ a = b and(a ◦ a) ◦ b = H,whichis acontradictionwith(2).

Theorem5.8. Thereexist4commutativeanti-LA-semihypergroupsoforder 2.

Proof. Let(H = {a,b}, ◦)beacommutativeanti-LA-semihypergroup.Then Wehave

(1)(a ◦ b) ◦ b =(b ◦ b) ◦ a,

(2)(a ◦ a) ◦ b =(b ◦ a) ◦ a,

(3) a ◦ b = b ◦ a.

Now,wehave3Cases:

Case1:Let a ◦ b = a = b ◦ a. then

(4)(a =(b ◦ b) ◦ a,

(5)(a ◦ a) ◦ b = a ◦ a.

◦2 ab a ba b aa

On[Neutro](Anti)LA-semihypergroupsand[Neutro](Anti)Hv -LA-semigroups19

If a ◦ a = H,thenby(5)wehave

(6) H ◦ b = H, since a ◦ b = a, using(6),wehave H ◦ b = a.Thus, b ◦ b = b. Thenwehavea commutativeanti-LA-semihypergroupwiththefollowingCayleytable.

◦3 ab

a Ha

b ab

Case2:Let a ◦ b = b = b ◦ a. Inthesimilarwayweobtain3commutative anti-LA-semihypergroupsisomorphismwithanti-LA-semihypergroupsinCase 1.

Case3:Let a ◦ b = H = b ◦ a, by(1)and(2)wehave

(7) H =(b ◦ b) ◦ a,

(8)(a ◦ a) ◦ b = H.

Thus,(7)and(8)implythat a◦a = H = b◦b, a◦a = a and b◦b = b.So a◦a = b and b ◦ b = a.Thereforewehaveacommutativeanti-LA-semihypergroupwith thefollowingCayleytable.

◦4 ab

a bH

b Ha

Thenwehave4commutativeanti-LA-semihypergroupsoforder2uptoisomorphism. □

Now,wehaveageneralizationofTheorem4.12.

Theorem5.9. Thereisnocommutativeneutro-Hv -LA-semigroupsoforder

2.

Proof. Let(H = {a,b}, ◦)beaweakcommutativeneutro-Hv -LA-semigroup. Thenwehave

(1)(a ◦ b) ◦ b ∩ (b ◦ b

20S.Mirvakili,A.Rezaei,O,Al-Shanqiti,F.Smarandache,B.Davvaz

(9) a ∩ (b ◦ b) ◦ a = ∅,

(10)(a ◦ a) ◦ b ∩ a ◦ a = ∅

Now,(7)impliesthat b ◦ b = a and a ◦ a = b andthisisacontradictionwith

(8) Also,(10)impliesthat a ◦ a = b and b ◦ b = a,whichisacontradiction with(9)

Case3:If a ◦ b = b = b ◦ a,thenbythesimilarwayofCase2wecanprove thatthereisnoaweakcommutativeneutro-Hv -LA-semigroupoforder2. □

Theorem5.10. Thereexist2weakcommutativeneutro-Hv -LA-semigroupsof order2.

Proof. Let(H = {a,b}, ◦)beaweakcommutativeneutro-Hv -LA-semigroup. Thenwehave

(1)(a ◦ b) ◦ b ∩ (b ◦ b) ◦ a = ∅,

(2)(a ◦ a) ◦ b ∩ (b ◦ a) ◦ a = ∅

Theorem5.9andweakcommutativityimplythat

b ◦ a = a ◦ b = H, or

(6) a ◦ b = b ◦ a = H.

Now,wehavethefollowingCases:

Case1:Let(1),(2)and(5)betrue.Then H ∩ (b ◦ b) ◦ a = ∅ andthisisa contradiction.

Case2:Let(1),(2)and(6)betrue.By(6)and(1)wehave b ◦ b = a. Using(6)wehave a ◦ b = a or a ◦ b = b.If a ◦ b = a,thenby(1)weobtain a ◦ a = b. Thus,wehaveaweakcommutativeneutro-Hv -LA-semigroupwith thefollowingCayleytable.

◦ ab

a ba

b Ha

If a ◦ b = b,thenby(1)weobtain a ◦ a = b. Sowehaveaweakcommutative neutro-Hv -LA-semigroupwiththefollowingCayleytable.

◦ ab

a bb

b Ha

Case3:Let(3),(4)and(5)aretrue.By(5)and(4)wehave a ◦ a = b. by(5) wehave b ◦ a = a or b ◦ a = b.If b ◦ a = a,thenby(4)weobtain b ◦ b = a. Sowe

On[Neutro](Anti)LA-semihypergroupsand[Neutro](Anti)Hv -LA-semigroups21

haveaweakcommutativeneutro-Hv -LA-semigroupwiththefollowingCayley table.

◦ ab

a bH

b aa

If b ◦ a = b,thenby(4)weobtain b ◦ b = a. Thenwehaveaweakcommutative neutro-Hv -LA-semigroupwiththefollowingCayleytable.

◦ ab

a bH

b aa

Case1:Let(3),(4)and(6)aretrue.Then(a ◦ a) ◦ H ◦ a = ∅ Thisisa contradiction.

Theorem5.11. Thereexistsoneantiweakcommutative LA-semihypergroupof order2,uptoisomorphism.

Proof. Let(H = {a,b}, ◦)beanantiweakcommutative LA-semihypergroup. ThenWehave

Using(3)weget a ◦ b = H and b ◦ a = H. Then

(4) a ◦ b = a,b ◦ a = b, or

(5) a ◦ b = b,b ◦ a = a. If(4)istrue,then

(6) a =(b ◦ b) ◦ a, (7)(a ◦ a) ◦ b = b.

If b ◦ b = a or b ◦ b = b or b ◦ b = H,thenwehaveacontradiction. So,let(5)betrue.Then

(8) b ◦ b =(b ◦ b) ◦ a,

(9)(a ◦ a) ◦ b = a ◦ a. So,if b ◦ b = a,then a ◦ a = b. By(9)wehave b = a,whichisacontradiction.If b ◦ b = b,then b ◦ a = b thisisacontradictionwith(5).If a ◦ a = b or a ◦ a = a, thenbythesimilarwaywehaveacontradiction.Thus, b ◦ b = H,andso a ◦ a = H. Thereforewehaveanantiweakcommutative LA-semihypergroup withthefollowingCayleytable.

◦ ab a Hb b aH

22S.Mirvakili,A.Rezaei,O,Al-Shanqiti,F.Smarandache,B.Davvaz

Usingtheaboveresultsinthesections4and5,wecancharacterize45 non-isomorphicclasseshypergroupoidsoftheorder2.Weobtainanti-LAsemihypergroups,neutro-LA-semihypergroups, LA-semihypergroups,anti-HvLA-semigroups,neutro-Hv -LA-semigroups, Hv -LA-semigroupsoforder2(classes uptoisomorphism).

Table2. Classificationofthehypergroupoidsoforder2

CWCACAWC

LA-semihypergroups91131

Hv -LA-semigroups1330203

Neutro-LA-semihypergroup29103

Neutro-Hv -LA-semigroup0275

Anti-LA-semihypergroup313188

Anti-Hv -LA-semigroup1144

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Humanparticipantsand/oranimals Thisarticledoesnotcontainany studieswithhumanparticipantsoranimalsperformedbytheauthors.

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References

1.M.Akram,N.YaqoobandM.Khan, On(m,n)-idealsin LA-semigroups,AppliedMathematicalSciences,7(44)(2013)2187–2191.

2.A.A.A.Agboola, Introductiontoneutrogroups,InternationalJournalofNeutrosophic Science,6(1)(2020)41–47.

3.J.R.Cho,J.JezekandT.Kepka, Paramedialgroupoids,CzechoslovakMathematical Journal,49(2)(1999)277–290.

4.P.Corsini, Prolegomenaofhypergrouptheory,AvianiEditor,(1993).

5.P.Corsini,V.Leoreanu-Fotea, Applicationsofhyperstruturetheory,KluwerAcademic Publishers,(2003).

6.B.Davvaz, Semihypergrouptheory,Elsevier,(2016).

7.B.DavvazandT.Vougiouklis, Awalkthroughweakhyperstructures; Hv -structure,World ScientificPublishingCo.Pte.Ltd.,Hackensack,NJ,(2019)

8.A.Distler, Classificationandeumerationoffinitesemigroups,Ph.D.Thesis,University ofStAndrews,(2010).

9.M.Gulistan,N.YaqoobandM.Shahzad, Anoteon Hv -LA-semigroups,U.P.B.Sci. Bull.,SeriesA,77(33)(2015)93–106.

10.K.HilaandJ.Dine, Onhyperidealsinleftalmostsemihypergroups,ISRNAlgebra, ArticleID953124(2011)8pages.

11.P.Holgate, Groupoidssatisfyingasimpleinvertivelaw,TheMathematicsStudent,61(14)(1992)101–106.

On[Neutro](Anti)LA-semihypergroupsand[Neutro](Anti)Hv -LA-semigroups23

12.J.Jantosciak, Transpositionhypergroups:Noncommutativejoinspaces,J.Algebra,187 (1997)97–119.

13.M.A.KazimandM.Naseeruddin, Onalmostsemigroups,TheAligarhBulletinofMathematics,2(1972)1–7.

14.M.KhanandN.Ahmad, Characterizationsofleftalmostsemigroupsbytheirideals, JournalofAdvancedResearchinPureMathematics,2(3)(2010)61–73.

15.S.Mirvakili,F.SmarandacheandA.Rezaei, Onneutro-Hv -semigroups,JournalofIntelligent&FuzzySystems,42(2022)1289–299.

16.F.Marty, Surunegeneralizationdelanotiondegroupe,HuitiemecongressdeMathematiciens,Scandinaves,Stockholm,(1934)45–49.

17.Q.MushtaqandS.M.Yusuf, On LA-semigroups,TheAligarhBulletinofMathematics, 8(1978)65–70.

18.Q.MushtaqandS.M.Yusuf, Onlocallyassociative LA-semigroups,TheJournalofNaturalSciencesandMathematics,19(1)(1979)57–62.

19.P.V.ProticandN.Stevanovic, AG-testandsomegeneralpropertiesofAbel-Grassmann’s groupoids,PureMathematicsandApplications,6(4)(1995)371–383.

20.A.Rezaei,F.Smarandache,S.Mirvakili, Applicationsof(neutro/anti)sophicationsto semihypergroups,JournalofMathematicsVolume2021,ArticleID6649349,7pages.

21.F.Smarandache, Neutroalgebraisageneralizationofpartialalgebra,InternationalJournalofNeutrosophicScience(IJNS),2(1)(2020)8–17.

22.F.Smarandache, Introductiontoneutroalgebraicstructuresandantialgebraic structures, inAdvancesofStandardandNonstandardNeutrosophicTheories,PonsPublishing HouseBrussels,Belgium,Ch.6,(2019)240–265.

23.F.Smarandache, Introductiontoneutroalgebraicstructuresandantialgebraic structures (revisited),NeutrosophicSetsandSystems,31(2020)1–16.

24.F.Smarandache, Neutrosophy/Neutrosophicprobability,set,andlogic,AmericanResearchPress,(1998).

25.F.Smarandache, Extensionofhypergraphton-superhypergraphandtoplithogenicnsuperhypergraph,andextensionofhyperalgebrato n-ary(classical-/neutro-/anti-) hyperalgebra,neutrosophicsetsandsystems,33(2020)290–296.

26.N.StevanovicandP.V.Protic, CompositionofAbel-Grassmann’s3-bands,NoviSad JournalofMathematics,34(2)(2004)175–182.

27.T.Vougiouklis, Hyperstructuresandtheirrepresentations,HadronicPress,Inc.(1994) (seepage31).

28.N.Yaqoob,P.CorsiniandF.Yousafzai, Onintra-regularleftalmostsemihypergroups withpureleftidentity,JournalofMathematics,ArticleID510790(2013)10pages.