Applications of (Neutro/Anti)sophications to Semihypergroups

Journal of Mathematics

Volume 2021,Article ID 6649349, 7 pages https://doi.org/10.1155/2021/6649349

Applicationsof(Neutro/Anti)sophicationstoSemihypergroups

A.Rezaei , 1 F.Smarandache , 2 andS.Mirvakili 1

1DepartmentofMathematics,PayameNoorUniversity,P.O.Box19395-3697,Tehran,Iran

2DepartmentofMathematicsandScience,UniversityofNewMexico,Gallup,NewMexico87301,USA

CorrespondenceshouldbeaddressedtoA.Rezaei;rezaei@pnu.ac.ir

Received 17 December 2020; Revised 18 February 2021; Accepted 25 February 2021; Published 30 March 2021

Academic Editor:ParimalaMani

Copyright © 2021A.Rezaeietal. isisanopenaccessarticledistributedundertheCreativeCommonsAttributionLicense, whichpermitsunrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalworkisproperlycited.

Inthispaper,weextendthenotionofsemi-hypergroups(resp.hypergroups)toneutro-semihypergroups(resp.neutrohypergroups).Weinvestigatethepropertyofanti-semihypergroups(resp.anti-hypergroups).Wealsogiveanewalternativeof neutro-hyperoperations(resp.anti-hyperoperations),neutro-hyperoperation-sophications(resp.anti-hypersophications). Moreover,weshowthatthesenewconceptsaredifferentfromclassicalconceptsbyseveralexamples.

1.Introduction

Ahypergroup,asageneralizationofthenotionofagroup, wasintroducedbyF.Marty[1]in1934. efirstbookin hypergrouptheorywaspublishedbyCorsini[2].Nowadays, hypergroupshavefoundapplicationstomanysubjectsof pureandappliedmathematics,forexample,ingeometry, topology,cryptographyandcodingtheory,graphsand hypergraphs,probabilitytheory,binaryrelations,theoryof fuzzyandroughsetsandautomatatheory,physics,andalso inbiologicalinheritance[3–7]. efirstbookinsemihypergrouptheorywaspublishedbyDavvazin2016(see [8]).Inrecentyears,severalothervaluablebooksin hyperstructureshavebeenwrittenbyDavvazetal.[6,9,10].

M.Al-Tahanetal.introducedtheCorsinihypergroup andstudieditspropertiesasaspecialhypergroupthatwas definedbyCorsini. eyinvestigatedanecessaryandsufficientconditionfortheproductionalhypergrouptobea Corsinihypergroup,andtheycharacterizedallCorsini hypergroupsoforders2and3uptoisomorphism[3].Semihypergroup,hypergroup,andfuzzyhypergroupoforder2 areenumeratedin[7,11,12].S.Hoskova-Mayerovaetal. usedthefuzzymultisetstointroducetheconceptoffuzzy multi-hypergroupsasageneralizationoffuzzyhypergroups, definedthedifferentoperationsonfuzzymulti-hypergroups, andextendedthefuzzyhypergroupstofuzzymultihypergroups[13].

In2019and2020,withinthefieldofneutrosophy, Smarandache[14–16]generalizedtheclassicalalgebraic structurestoneutroalgebraicstructures(orneutroalgebras) (whoseoperationsandaxiomsarepartiallytrue,partially indeterminate,andpartiallyfalse)asextensionsofpartial algebraandtoantialgebraicstructures(orantialgebras) (whoseoperationsandaxiomsaretotallyfalse).Furthermore, heextendedanyclassicalstructure,nomatterwhatfieldof knowledge,toaneutrostructureandanantistructure. ese arenewfieldsofresearchwithinneutrosophy.Smarandache in[16]revisitedthenotionsofneutroalgebrasandantialgebras,wherehestudiedpartialalgebras,universalalgebras, effectalgebras,andBoole’spartialalgebrasandshowedthat neutroalgebrasarethegeneralizationofpartialalgebras.Also, withrespecttotheclassicalhypergraph(thatcontains hyperedges),Smarandacheaddedthesupervertices(agroup ofverticesputtogethertoformasupervertex),inorderto formasuper-hypergraph. en,heextendedthesuperhypergraphto n-super-hypergraph,byextendingthepower set P(V) to Pn (V) thatisthe n-powersetoftheset V (the nsuper-hypergraph,throughits n -super-hypergraph-vertices and n -superhypergraph-edgesthatbelongto Pn (V),canbe thebest(sofar)tomodelourcomplexandsophisticated reality).Furthermore,heextendedtheclassicalhyperalgebra to n-aryhyperalgebraanditsalternatives n -aryneutrohyperalgebraand n -aryanti-hyperalgebra[17]. enotionof neutrogroupwasdefinedandstudiedbyAgboolain[18].

Hindawi

Recently,M.Al-Tahanetal.studiedneutro-orderedalgebra andsomerelatedtermssuchasneutro-orderedsubalgebra andneutro-orderedhomomorphismin[19].

Inthispaper,theconceptofneutro-semihypergroupand anti-semihypergroupisformallypresented.And,newalternativesareintroduced,suchasneutro-hyperoperations (resp.anti-hyperoperations),neutro-hyperaxioms,andantihyperaxioms.Weshowthatthesedefinitionsaredifferent fromclassicaldefinitionsbypresentingseveralexamples. Also,weenumerateneutro-hypergroupandanti-hypergroupoforder2(seeTable1)andobtainsomeknownresults (seeTable2).

2.Preliminaries

Inthissection,werecallsomebasicnotionsandresults regardinghyperstructures.

Definition1 (see[2,8]).Ahypergroupoid (H, ∘ ) isa nonemptyset H togetherwithamap ∘: H × H ⟶ P∗ (H) called(binary)hyperoperation,where P∗ (H) denotestheset ofallnonemptysubsetsof H. ehyperstructure (H, ∘ ) is calledahypergroupoid,andtheimageofthepair (x,y) is denotedby x ∘ y. If A and B arenonemptysubsetsof H and x ∈ H,thenby A ∘ B, A ∘ x,and x ∘ B wemean A ∘ B � ∪ a∈A,b∈B a ∘ b, A ∘ x � A ∘ x {},and x ∘ B � x {} ∘ B

Definition2 (see[2,8]).Ahypergroupoid (H, ∘ ) iscalleda semi-hypergroupifitsatisfiesthefollowing:

(A) (∀a,b,c ∈ H)(a ∘ (b ∘ c)�(a ∘ b) ∘ c) (associativity).

Definition3 (see[2,8]).Ahypergroupoid (H, ∘ ) iscalleda quasi-hypergroupifreproductionaxiomisvalid. ismeans that,forall a of H,wehave (R) (∀a ∈ H)(H ∘ a � a ∘ H � H) (i.e. (∀a,b ∈ H) (∃c,d ∈ H) s.t. b ∈ c ∘ a,b ∈ a ∘ d).

Definition4 (see[2,8]).Ahypergroupoid (H, ∘ ) whichis bothasemi-hypergroupandaquasi-hypergroupiscalleda hypergroup.

Example1 (see[2,8])

(i)Let H beanonemptyset,andforall x,y ∈ H,we define x ∘ y � H. en, (H, ∘ ) isahypergroup, calledthetotalhypergroup.

(ii)Let G beagroupand H anormalsubgroupof G,and forall x,y ∈ G,wedefine x ∘ y � xyH. en, (G, ∘ ) isahypergroup.

Definition5 (see[2,12]).Let (H, ∘ ) beahypergroupoid. ecommutativelawon (H, ∘ ) isdefinedasfollows:

(C) (∀a,b ∈ H)(a ∘ b � b ∘ a). (H, ∘ ) iscalledacommutativehypergroupoid.

Table 1:Classificationofthehypergroupoidsoforder2. A NAAA

C R 6 4 NR— AR Etc. 3 2 NC R NR— AR Etc.— AC R 2 8 NR— AR Etc. 6 10 4

Table 2:Classificationofthesemi-hypergroupsoforder2.

ComNoncom N

Semigroup 3 2 5 Group 1 1 Semi-hypergroup 9 8 17 Hypergroup 6 2 8

Example2 (see[13]).Let Z bethesetofintegers,anddefine ° 1 on Z asfollows.Forall x,y ∈ Z, x ∘ 1 y � 2Z, if x,y havesamepartiy , 2Z + 1, otherwise � �. (1) en, (Z, ∘1 ) isacommutativehypergroup.

3.OnNeutro-hypergroupsandAntihypergroups

F.Smarandachegeneralizedtheclassicalalgebraicstructures totheneutroalgebraicstructuresandantialgebraicstructures.Neutro-sophicationofanitem C (thatmaybea concept,aspace,anidea,ahyperoperation,anaxiom,a theorem,atheory,analgebra,etc.)meanstosplit C into threeparts(twopartsoppositetoeachother,andanother partwhichistheneutral/indeterminacybetweentheopposites),aspertinenttoneutrosophy((〈A〉, 〈neutA〉, 〈antiA〉),orwithothernotation (T,I,F)),meaningcases where C ispartiallytrue (T),partiallyindeterminate (I),and partiallyfalse (F),whileantisophicationof C meansto totallydeny C (meaningthat C ismadefalseonitswhole domain)(see[14,15,17,20]).

Neutrosophicationofanaxiomonagivenset X means tosplittheset X intothreeregionssuchthat,ononeregion, theaxiomistrue(wesaythedegreeoftruth T oftheaxiom), onanotherregion,theaxiomisindeterminate(wesaythe degreeofindeterminacy I oftheaxiom),andonthethird region,theaxiomisfalse(wesaythedegreeoffalsehood F of theaxiom),suchthattheunionoftheregionscoversthe wholeset,whiletheregionsmayormaynotbedisjoint, where (T,I,F) isdifferentfrom (1, 0, 0) andfrom (0, 0, 1)

Antisophicationofanaxiomonagivenset X meansto havetheaxiomfalseonthewholeset X (wesaytotaldegree offalsehood F oftheaxiom)or (0, 0, 1)

Neutrosophicationofahyperoperationdefinedona givenset X meanstosplittheset X intothreeregionssuch that,ononeregion,thehyperoperationiswell-defined(or inner-defined)(wesaythedegreeoftruth T ofthehyperoperation),onanotherregion,thehyperoperationisindeterminate(wesaythedegreeofindeterminacy I ofthe hyperoperation),andonthethirdregion,thehyperoperationisouter-defined(wesaythedegreeoffalsehood F ofthehyperoperation),suchthattheunionoftheregions coversthewholeset,whiletheregionsmayormaynotbe disjoint,where (T,I,F) isdifferentfrom (1, 0, 0) andfrom (0, 0, 1)

Antisophicationofahyperoperationonagivenset X meanstohavethehyperoperationouter-definedonthe wholeset X (wesaytotaldegreeoffalsehood F oftheaxiom) or (0, 0, 1).

Inthissection,wewilldefinetheneutro-hypergroups andanti-hypergroups.

Definition6. Aneutro-hyperoperationisamap ∘: H × H ⟶ P(U),where U isauniverseofdiscoursethat contains H thatsatisfiesthebelowneutrosophication process.

eneutrosophication(degreeofwell-defined,degreeof indeterminacy,anddegreeofouter-defined)ofthehyperoperationisthefollowingneutrohyperoperation(NH):

(NR) (∃x,y ∈ H)(x ∘ y ∈ P∗ (H)) and (∃x,y ∈ H)(x ∘ y isanindeterminatesubset,or x ∘ y ∉ P∗ (H)).

eneutrosophication(degreeoftruth,degreeofindeterminacy,anddegreeoffalsehood)ofthehypergroupaxiomofassociativityisthefollowing neutroassociativity(NA):

(NA) (∃a,b,c ∈ H)(a ∘ (b ∘ c)�(a ∘ b) ∘ c) and (∃d,e, f ∈ H)(d ∘ (e ∘ f) ≠ (d ∘ e) ∘ f or d ∘ (e ∘ f)� indeterminate,or (d ∘ e) ∘ f � indeterminate).

Neutroreproductionaxiom(NR):

(NR) (∃a ∈ H)(H ∘ a � a ∘ H � H) and (∃b ∈ H) (H ∘ b, b ∘ H,and H arenotallthreeequal,orsomeof themareindeterminate).

Also,wedefinetheneutrocommutativity(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).

Now,wedefineaneutro-hyperalgebraicsystem S �〈H,F,A〉,where H isasetorneutrosophicset, F isaset ofthehyperoperations,and A isthesetofhyperaxioms,such thatthereexistsatleastoneneutro-hyperoperationorat leastoneneutro-hyperaxiomandnoanti-hyperoperation andnoanti-hyperaxiom.

Definition7. eanti-hypersophication(totallyouterdefined)ofthehyperoperationdefinesanti-hyperoperation (AH):(AH) (∀x,y ∈ H)(x ∘ y ∉ P∗ (H)). eanti-hypersophication(totallyfalse)ofthehypergroupisasfollows:

(AA) (∀x,y,z ∈ H)(x ∘ (y ∘ z) ≠ (x ∘ y) ∘ z) (antiassociativity)

(AR) (∀a ∈ H)(H ∘ a, a ∘ H,and H arenotequal) (antireproductionaxiom)

Also,wedefinetheanticommutativity(AC)on (H, ∘ ) asfollows: (AC)(∀a,b ∈ H with a ≠ b) (a ∘ b ≠ b ∘ a)

Definition8. Aneutro-semihypergroupisanalternativeof semi-hypergroupthathasatleast(NH)or(NA),whichdoes nothave(AA).

Example3

(i)Let H � a,b,c {} and U � a,b,c,d {} beauniverseof discoursethatcontains H.Definetheneutrohyperoperation ° 2 on H withCayley’stable.

◦2 abc aaaa bb {a, b}{a, b, d} cc ? H en, (H, ∘2 ) isaneutro-semihypergroup.Since a ∘2 b ∈ P∗ (H), b ∘2 c � a,b,d {} ∉ P∗ (H),and c ∘2 b � indeterminate,so(NH)holds.

(ii)Let H � a,b,c {}.Definethehyperoperation ∘3 on H withCayley’stable. ◦3 abc aaaa bb {a, b}{a, b} cc {b, c} H en, (H, ∘3 ) isaneutro-semihypergroup.(NA)isvalid, since (b ∘3 c) ∘3 a � a,b {} ∘3 a �(a ∘ 3 a) ∪ (b ∘ 3 a)� a {} ∪ b {} � a,b {} and b ∘3 (c ∘3 a)� b ∘3 c {} � b ∘3 c � a,b {}. Hence, (b ∘3 c) ∘3 a � b ∘3 (c ∘3 a).Also, b ∘3 a �� ∘3 c � b {} ∘3 c � b ∘3 c � a,b {} and b ∘3 (a ∘3 c)� b ∘3 a {} � b ∘3 a � b {},so (b ∘3 a) ∘ 3 c ≠ b ∘3 (a ∘3 c).

Definition9. Aneutrocommutativesemi-hypergroupisa semi-hypergroupthatsatisfies(NC).

Example4. Let H � a,b,c {}.Definethehyperoperation ∘4 on H withCayley’stable. ◦4 abc a {a, c} aa babc ca {b, c}{b, c} en, (H, ° 4 ) isasemi-hypergroup,butnotahypergroup,since a° 4 H � H° 4 a � a,c {} ≠ H.(NC)isvalid,since a ∘4 b � a {} � b ∘ 4 a and c ∘4 b � b,c {} ≠ b ∘4 c � c {}

Definition10. Aneutrocommutativehypergroupisa hypergroupthatsatisfies(NC).

Example5. Let H � a,b,c,d,e,f ��.Definetheoperation ° 5 on H withCayley’stable. ◦5 abcdef eeabcdf a abedfc b beafcd c cfdeba d dcfaeb ffdcbae

en, (H, ∘5 ,e) isagroupandsoisanaturalhypergroup.Also,itisaneutrocommutativehypergroup,since a ∘5 b � e � b ∘5 a and a ∘5 c � d ≠ c ∘5 a � f.

Definition11. Aneutrohypergroupisanalternativeof hypergroupthathasatleast(NH)or(NA)or(NR),which doesnothave(AA)and(AR).

Example6. Let H � a,b,c {}.Definethehyperoperation ∘6 on H withCayley’stable. ◦6 abc aabc bbbb ccca

en, (H, ∘6 ) isaneutrohypergroup. ehyperoperation ∘6 isassociative.(NR)isvalid,since a ∘6 H �(a ∘6 a) ∪ (a ∘6 b) ∪ (a ∘6 c)� H �(a ∘6 a) ∪ (b ∘6 a) ∪ (c ∘6 a)� H ∘6 a, b ∘6 H �(b ∘6 a) ∪ (b ∘6 b) ∪ (b ∘6 c)� b {} ≠ H ≠ c,b {} �(a ∘6 b) ∪ (b ∘6 b) ∪ (c ∘6 b)� H ∘6 b,and c ∘6 H �(c ∘6 a) ∪ (c ∘6 b) ∪ (c ∘6 c)� a,c {} ≠ H,but H ∘6 c � (a ∘6 c) ∪ (b ∘6 c) ∪ (c ∘6 c)� a,b,c {} � H.

Notethateveryneutro-semihypergroup,neutrohypergroup,neutrocommutativesemi-hypergroup,and neutrocommutativehypergroupareneutro-hyperalgebraic systems.

Definition12. Ananti-semihypergroupisanalternativeof semi-hypergroupthathasatleast(AH)or(AA).

Example7

(i)Let N bethesetofnaturalnumbersexcept0.Define hyperoperation ∘7 on N by x ∘7 y �(x2/x2 + 1),y � � en, (N, ∘7 ) isananti-semihypergroup.(AH)is valid,since,forall x,y ∈ N, x ∘7 y ∉ P∗ (N). us, (AH)holds.

(ii)Let H � a,b {}.Definethehyperoperation ∘8 on H withCayley’stable.

◦8 ab aba bba

en, (H, ∘8 ) isananti-semihypergroup.(AA)is valid,since,forall x,y,z ∈ H, x ∘8 (y ∘8 z) ≠ (x ∘8 y) ∘8 z

(iii)Let H � a,b {}.Definethehyperoperation ∘9 on H withCayley’stable. ◦9 ab abH baa en, (H, ∘9 ) isananticommutativesemi-hypergroup. (AC)isvalid,since a ∘9 b � H ≠ b ∘9 a � a {}

Definition13. Ananti-hypergroupisanantisemihypergroup,oritsatisfies(AR).

Example8

(i)Let R bethesetofrealnumbers.Definehyperoperation ∘10 on R by x ∘10 y � x2 + 1,x2 1 � � en, (R, ∘10 ) isananti-semihypergroup,since,for all x,y,z ∈ R, x ∘10 (y ∘10 z) ≠ (x ∘10 y) ∘10 z.Because x ∘10 (y ∘10 z)� x ∘10 y2 + 1,y2 1 � � � x ∘10 (y2 + � 1),x ∘10 (y2 1)}� x2 + 1,x2 1 � �,but (x ∘10 y) ∘10 z � x2 + 1,x2 1 � � ∘10 z �((x2 + 1) ∘10 z) ∪ ((x2 1) ∘10 z)�(x2 + 1)2 + 1, � (x2 1)2 + 1}.Hence,(AA)isvalid.

(ii)Let H � a,b,c {}.Definethehyperoperation ∘11 on H withCayley’stable.

◦11 abc aaab b aa a cccc

en, (H, ∘11 ) isananti-semihypergroup. e hyperoperation ∘11 isassociative.Also,(AR)holds, since a ∘11 H �(a ∘11 a) ∪ (a ∘11 b) ∪ (a ∘11 c)� c {} ≠ H ≠ b,c {} �(a ∘11 a) ∪ (b ∘11 a) ∪ (c ∘11 a)� H ∘11 a, b ∘11 H �(b ∘11 a) ∪ (b ∘11 b) ∪ (b ∘11 c)� b {} ≠ H ≠ b,c {} �(a ∘11 b) ∪ (b ∘11 b) ∪ (c ∘11 b)� H ∘11 b,and c ∘11 H �(c ∘11 a) ∪ (c ∘11 b) ∪ (c ∘11 c)� c {} ≠ H ≠ b,c {} �(a ∘11 c) ∪ (b ∘11 c) ∪ (c ∘11 c)� H ∘11 c.

(iii)Let R bethesetofrealnumbers.Definehyperoperation ∘12 on R by x ∘12 y � x, 1 {}. en, (R, ∘12 ) isananti-semihypergroup. ehyperoperation ∘12 isassociative,since,forall x,y,z ∈ R, wehave x ∘12 (y ∘12 z)� x ∘12 y, 1 �� �(x ∘12 y) ∪ (x ∘12 1)� x, 1 {} ∪ x, 1 {} � x, 1 {} and (x ∘12 y) ∘ 12 z � x, 1 {} ∘12 z �(x ∘12 z) ∪ (1 ∘12 z)� x, 1 {} ∪ 1, 1 {} � x, 1 {},so x ∘12 (y ∘12 z)�(x ∘12 y) ∘ 12 z.However, for a ∈ R,wehave a ∘ 12 R � ∪ x∈R a ∘ 12 x � ∪ x∈R

a, 1 {} � a, 1 {} ≠ R and R ∘ 12 a � ∪ x∈R x ∘ 12 a � ∪ x∈R x, 1 {} � R. us, a ∘ 12 R ≠ R ∘ 12 a

Definition14. Ananticommutativesemi-hypergroupisa semi-hypergroupthatsatisfies(AC).

Example9

(i)Let H � a,b {}.Definethehyperoperation ∘ 13 on H withCayley’stable. ◦13 ab aaa bHb en, (H, ∘ 13 ) isasemi-hypergroupand(AC)is valid,since a ∘ 13 b � a {} ≠ b ∘ 13 a � H. us, (H, ∘ 13 ) isananticommutativesemi-hypergroup.

(ii)Let H � a,b {}.Definethehyperoperation ∘ 14 on H withCayley’stable. ◦14 ab aba bba en, (H, ∘14 ) isananticommutativesemi-hypergroup, andthehyperoperation ∘ 14 isnotassociative,since (a ∘ 14 a) ∘ 14 a � b {} ∘ 14 a � b {} ≠ a ∘ 14 (a ∘ 14 a)� a ∘ 14 b {} � a {}. (AC)isvalid,since a ∘ 14 b � a {} ≠ b ∘ 14 a � b {}

Definition15. Ananticommutativehypergroupisa hypergroupthatsatisfies(AR).

Example10

(i)Let H � a,b {}.Definethehyperoperation ∘ 15 on H withCayley’stable. ◦15 ab aHa bHH

en, (H, ∘15 ) isananticommutativehypergroup. (AC)isvalid,since a ∘ 15 b � a {} ≠ b ∘ 15 a � H

(ii)Let H � a,b,c {}.Definethehyperoperation ∘ 16 on H withCayley’stable.

◦16 abc a aa H bbbH cccH

en, (H, ∘16 ) isananticommutativehypergroup. ehyperoperation ° 16 isassociative.Also,(AC) holds,since a ∘ 16 b � a {} ≠ b ∘ 16 a � b {}, a ∘ 16 c � H ≠ c ∘ 16 a � c {},and b ∘ 16 c � H ≠ c ∘ 16 b � c {}

(iii)Let H � a,b,c {}.Definethehyperoperation ∘ 17 on H withCayley’stable.

◦17 abc aabc babc cHHH

en, (H, ∘ 17 ) isananticommutativehypergroup,(AC) holds,since a ∘ 17 b � b {} ≠ b ∘ 17 a � a {}, a ∘ 17 c � c {} ≠ c ∘ 17 a � H,and b ∘ 17 c � c {} ≠ c ∘ 17 b � H

Notethateveryanti-semihypergroup,antihypergroup, anticommutativesemi-hypergroup,andanticommutative hypergroupareanti-hyperalgebraicsystems.

Inthefollowingresults,weusehyperoperationinsteadof neutro-hyperoperation.

Notethatif (H, ∘ ) isaneutro-semihypergroupand (G, ∘ ) isananti-semihypergroup,then (H ∩ G, ∘ ) isnota neutro-semihypergroup,butitisananti-semihypergroup. Also,let (H, ∘ H ) beaneutro-semihypergroup, (G, ∘ G ) be ananti-semihypergroup,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.

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ (2) en, (H⊎G, ∘ ) isaneutro-semihypergroup,butitis notananti-semihypergroup.

Proposition1. Let (H, ∘ ) beanantisemihypergroupand e ∈ H. en, (H ∪ e {}, ∗ ) isaneutrosemihypergroup,where ∗ isdefinedon H ∪ e {} by x ∗ y � x ∘ H y, if x,y ∈ H, e,x,y ��, otherwise � � (3)

Proof. Itisstraightforward.

Proposition2. Let (H, ∘ ) beacommutativehypergroupoid. en, (H, ∘ ) cannotbeananti-semihypergroup.

Proof. Let a ∈ H. en, a ∘ (a ∘ a)�(a ∘ a) ∘ a,so (H, ∘ ) cannotbeananti-semihypergroup.

Corollary1. Let (H, ∘ ) beahypergroupoid,andthereexists a ∈ H suchthat a ° a commutedwith a. en, (H, ∘ ) cannot beananti-semihypergroup.

Corollary2. Let (H, ∘ ) beahypergroupoidwithascalar idempotent,i.e.,thereexists a ∈ H suchthat a ° a � a. en, (H, ∘ ) cannotbeananti-semihypergroup.

Proposition3. Let (H, ∘ H ) and (G, ∘ G ) betwoneutrosemihypergroups(resp.anti-semihypergroups). en, (H ×

G, ∗ ) isaneutro-semihypergroup(resp.anti-semihypergroups),where ∗ isdefinedon H × G.Forany (x1 ,y1 ), (x2 ,y2 ) ∈ H × G, x1 ,y1 � ∗ x2 ,y2 � � x1 ∘ H x2 ,y1 ∘ G y2 � (4)

Notethatif (H, ∘ ) isaneutro-semihypergroup,thenif thereisanonemptyset H1 ⊆H,suchthat (H1 , ∘ ) isasemihypergroup,wecallitSmarandachesemi-hypergroup. Suppose (H, ∘ H ) and (G, ∘ G ) aretwohypergroupoids. Afunction f: H ⟶ G iscalledahomomorphismif,forall a,b ∈ H, f(a ∘ H b)� f(a) ∘ G f(b) (see[21,22],fordetails).

Proposition4. Let (H, ∘ H ) beasemi-hypergroup, (G, ∘ G ) beaneutro-hypergroup,and f: H ⟶ G beahomomorphism. en, (f(H), ∘ G ) isasemi-hypergroup,where f(H)� f(h): h ∈ H � �.

Proof. Assumethat (H, ∘ H ) isasemi-hypergroupand x,y,z ∈ f(H). en,thereexist h1 ,h2 ,h3 ∈ f(H) suchthat f(h1 )� x, f(h2 )� y,and f(h3 )� z,sowehave x ∘ G y ∘ G z� � fh1 � ∘ G fh2 � ∘ G h3 �� � fh1 � ∘ G fh2 ∘ H h3 � � fh1 ∘ H h2 ∘ H h3 �� � fh1 ∘ H h2 � ∘ H h3 � � fh1 ∘ H h2 � ∘ G fh3 � � fh1 � ∘ G fh2 �� ∘ G fh3 � � x ∘ G y� ∘ G z. (5)

en, (f(H), ∘ G ) isasemi-hypergroup. □

Definition16. Let (H, ∘ H ) and (G, ∘ G ) betwohypergroupoids.Abijection f: H ⟶ G isanisomorphismifit conservesthemultiplication(i.e., f(a ∘ H b)� f(a) ∘ G f(b)) andwrite H � G.Abijection f: H ⟶ G isanantiisomorphismifforall a,b ∈ H, f(a ∘ H b) ≠ f(b) ∘ G f(a).A bijection f: H ⟶ G isaneutroisomorphismifthereexist a,b ∈ H, f(a ∘ H b)� f(b) ∘ G f(a),i.e.,degreeoftruth (T), thereexist c,d ∈ H and f(c ∘ H d) or f(c) ∘ G f(d) areindeterminate,i.e.,degreeofindeterminacy (I),andthere exist e,h ∈ H, f(e ∘ H h) ≠ f(e) ∘ G f(h),i.e.,degreeof falsehood (F),where (T,I,F) aredifferentfrom (1, 0, 0) and (0, 0, 1),and T,I,F ∈ [0, 1]. Let ° beahyperoperationon H � a,b {} and (A11 ,A12 ,A21 ,A22 ) insideofCayley’stable.

◦ ab a A11 A12 b A21 A22

where Ad11 � A22 , Ad12 � A12 , Ad21 � A21 ,and Ad22 � A11 .

Lemma2 (see[6]). If (H, ∘ ) isahypergroupoid,then (H, ∗ ) isahypergroupoidwhen x ∗ y � y ∘ x forall x,y ∈ H (H, ∗ ) inLemma2iscalleddualhypergroupoidof (H, ∘ )

Theorem1. Let (H � a,b {}, ∘ ). en, (H, ∘ )�(H, ∗ ) if andonlyif (H, ∘ ) isanticommutative.

Lemma3. ereexist4anticommutativeanti-semihypergroupoforder2(uptoisomorphism).

Proof. Let (H, ∘ ) beananticommutativeantisemihypergroup.ByCorollary2,wehave a ∘ a ≠ a and b ∘ b ≠ b.Also, a ∘ b ≠ b ∘ a.Considerthefollowing.

If a ∘ a � H,then a ∘ (a ∘ a)� a ∘ H � H � H ∘ a � (a ∘ a) ∘ a,acontradiction. en,weget a ∘ a � b and b ∘ b � a

Now,wehave

Case1.If a ∘ b � a,then b ∘ a � H or b ∘ a � b,soweget (b,a,b,a) and (b,a,H,a) aretwoantisemihypergroups

Case2.If a ∘ b � b,then b ∘ a � H or b ∘ a � a,soweget (b,b,a,a) and (b,b,H,a) aretwoantisemihypergroups

Case3.If a ∘ b � H,then b ∘ a � a or b ∘ a � b,soweget (b,H,a,a) and (b,H,b,a) aretwoantisemihypergroups

Itcanbeseethat (b,a,H,a)�(b,H,b,a) and (b,H,a,a)�(b,b,H,a). erefore, (b,b,a,a), (b,a,b,a), (b,a,H,a),and (b,H,a,a) are4nonisomorphicantisemihypergroupsoforder2. □

Corollary3. ereexiststwononisomorphicantisemigroupsoforder2: (b,b,a,a) and (b,a,b,a).Antisemigroup (b,b,a,a) isthedualformoftheanti-semigroup (b,a,b,a).

Corollary4. ereexiststwononisomorphicantisemihypergroupsoforder2: (b,a,H,a) and (b,H,a,a). Anti-semihypergroup (b,a,H,a) isthedualformoftheantisemihypergroup (b,H,a,a)

Theorem2. Let (H, ∘ ) beahypergroupoidoforder2. en, (H, ∘ ) doesnothave(NR)or(AR).

Lemma1 (see[5]). Let (H � a,b {}, ∘ H ) and (G � a ′ ,b′ ��, ∘ G ) behypergroupoidswithCayley’stables (A,B,C,D) and (A′ ,B′ ,C′ ,D′ ),respectively. en, H � G if andonlyif,forall i,j ∈ 1, 2 {}, Aij � Aij ′ or Aij ′ � Adij , if Aij � H, G∖Aij ′ , if Aij ≠ H,

⎧ ⎪ ⎨ ⎪ ⎩ ⎫ ⎪ ⎬ ⎪ ⎭ , (6)

Proof. Let H � a,b {}.Suppose Ha ≠ H, aH ≠ H,and Ha ≠ aH.Hence, Ha � a {} or Ha � b {}.First,give Ha � a {}, then aH ≠ H and Ha ≠ aH impliesthat aH � b {}. en, a ∘ a⊆Ha � b {} and a ∘ a⊆Ha � a {}. erefore, b {} � a ∘ a ∘ a � a {},andthisisacontradiction.Inthesimilarway,we obtain Hb ≠ H, bH ≠ H,and Hb ≠ bH,acontradiction. UsingLemmas1and2and eorem1,wecanfind45 nonisomorphicclasseshypergroupoidsoftheorder2.We characterizethese45classesinTable1.

Notethatsemi-hypergroups,hypergroups,andfuzzy hypergroupsoforder2areenumeratedin[7,11,12].

Weobtainanti-semihypergroupsandneutrosemihypergroupsoforder2andtheclassificationofthe hypergroupoidsoforder2(classesuptoisomorphism).

R, NR, AR, A, NA, AA, C, NC, andACinTable1are denotedinSections2and3.

AresultfromTable1confirmstheenumerationofthe hyperstructureoforder2[11,23,24],whichissummarized asfollows. □

4.ConclusionandFutureWork

Inthispaper,wehavestudiedseveralspecialtypesofhypergroups,neutro-semihypergroups,anti-semihypergroups,neutro-hypergroups,andanti-hypergroups.Newresultsand examplesonthesenewalgebraicstructureshavebeeninvestigated.Also,wecharacterizeallneutro-hypergroupsandantihypergroupsofordertwouptoisomorphism. eseconcepts canfurtherbegeneralized.

Futureresearchtobedonerelatedtothistopicare

(a)Defineneutro-quasihypergroup,anti-quasihypergroup,neutrocommutativequasi-hypergroup, andanticommutativequasi-hypergroup

(b)Defineneutro-hypergroups,anti-hypergroups, neutrocommutativehypergroups,andanticommutativehypergroups

(c)DefineandinvestigateneutroHv-groups,antiHvgroups,neutroHv-rings,andantiHv-rings

(d)Itwillbeinterestingtocharacterizeinfiniteneutrohypergroupsandanti-hypergroupsupto isomorphism

(e) eseresultscanbeappliedtootherhyperalgebraicstructures,suchashyper-rings,hyperspaces,hyper-BCK-algebra,hyper-BE-algebras, andhyper-K-algebras.

DataAvailability

edatausedtosupportthefindingsofthisstudyareincludedwithinthearticle.

ConflictsofInterest

eauthorsdeclarethattheyhavenoconflictsofinterest.

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