Introduction to NeutroHyperGroups

NeutrosophicSetsandSystems,Vol.38,2020

IntroductiontoNeutroHyperGroups

M.A.Ibrahim1 andA.A.A.Agboola2

1 DepartmentofMathematics,FederalUniversityofAgriculture,PMB2240,Abeokuta,Nigeria; muritalaibrahim40@gmail.com

2 DepartmentofMathematics,FederalUniversityofAgriculture,PMB2240,Abeokuta,Nigeria; agboolaaaa@funaab.edu.ng

Incommemorationofthe60thbirthdayofthesecondauthor Correspondence:agboolaaaa@funaab.edu.ng

Abstract NeutroSophicationandAntiSophicationareprocessesthroughwhichNeutroAlgebraicandAntiAlgebraic structurescanbegeneratedfromanyclassicalstructures.Givenanyclassicalstructurewith m operations(lawsand axioms)where m ≥ 1wecangenerate(2m 1)NeutroStructuresand(3m 2m)AntiStructures.Inthispaper,we introduceforthefirsttimetheconceptofNeutroHyperGroups.Specifically,westudyaparticularclassofNeutroHyperGroupscalled[2, 3] NeutroHyperGroupsandpresenttheirbasicpropertiesandseveralexamples.Itisshownthat theintersectionoftwo[2, 3] NeutroSubHyperGroupsisnotnecessarilya[2, 3] NeutroSubHypergroupbuttheirunion mayproducea[2, 3] NeutroSubhypergroup.Also,thequotientofa[2, 3] NeutroHyperGroupfactoredbya[2, 3] NeutroSubHyperGroupisshowntobea[2, 3] NeutroHyperGroup.Examplesareprovidedtoshowthatintheneutrosophicenvironment,[2, 3] NeutroHyperGroupsareassociatedwithdismutationreactionsinsomechemicalreactions andbiologicalprocesses.

Keywords: NeutroHyperGroup,NeutroSubHyperGroup,NeutroHyperGroupHomomorphism,NeutroHyperGroupIsomomorphism.

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1. Introduction

In2013,AgboolaandDavvazestablishedtheconnectionsbetweenneutrosophicsetandalgebraic hyperstructures.In[1–3]theystudiedneutrosophichypergroup,neutrosophiccanonicalhypergroup andneutrosophichyperrings.Sincethenseveralneutrosophicalgebraicstructureshavebeenstudied andmanyresultshavebeenobtainedandpublished.Recently,IbrahimandAgboolain[13]studied NeutrosophicHypernearringsandpresentedsomeoftheirproperties.In2019,FlorentinSmarandache in[20]presentedtheconceptofNeutroAlgebraicStructuresandAntiAlgebraicStructureswhichcanbe generatedfromclassicalalgebraicstructuresthroughprocessescalledNeutroSophicationandAntiSophicationrespectively.Herecalled,improvedandextendedseveraldefinitionsandpropertiesofthesenew structuresin[19].Thesenewconceptshaveprovidednewmethodologiesforhandlingindeterminate, M.A.IbrahimandA.A.A.Agboola,IntroductiontoNeutroHyperGroups

incompleteandimpreciseinformationandprocesses.TheworkofSmarandachein[20]wasstudiedviza-viztheclassicalnumbersystems N, Z, Q, R and C byAgboolaetal.in[4].In[5,6],Agboolaformally presentedthenotionsofNeutroGroups,NeutroSubgroups,NeutroRings,NeutroSubrings,NeutroIdeal, NeutroQuotientRings,andheestablishedseveralpropertiesofthesestructuresandtheirsubstructures fortheclassesheconsidered.Recently,RezaeiandSmarandachein[17]introducedtheconceptsof Neutro-BE-algebrasandAnti-BE-algebrasandin[7,12]AgboolaandIbrahimintroducedtheconcept ofNeutroVectorSpacesandAntiRings.Thepresentpaperwillbeconcernedwiththeintroductionof theconceptofNeutroHyperGroupsandpresentationsoftheirbasicpropertiesandexamples.Formore detailsonNeutrosophyandapplications,thereadersshouldsee[8,9,14–16,18,22].

2. Preliminaries

Inthissection,wewillgivesomedefinitions,examplesandresultsthatwillbeusedinthesequel. Definition2.1. LetHbeanon-emptysetand ◦ : H × H −→ P ∗(H)beahyperoperation.Thecouple (H, ◦)iscalledahypergroupoid.Foranytwonon-emptysubsetsAandBofHand x ∈ H, wedefine

Definition2.2. Ahypergroupoid(H, ◦)iscalledasemihypergroupifforalla,b,cofHwehave (a ◦ b) ◦ c = a ◦ (b ◦ c), whichmeansthat u∈a◦b u ◦ c = v∈b◦c

a ◦ v.

Ahypergroupoid(H, ◦)iscalledaquasihypergroupifforallaofHwehave a ◦ H = H ◦ a = H. This conditionisalsocalledthereproductionaxiom.

Definition2.3. Ahypergroupoid(H, ◦)whichisbothasemihypergroupandaquasi-hypergroupis calledahypergroup.

Definition2.4. Let(H, ◦)and(H, ◦ )betwohypergroupoids.Amap φ : H −→ H, iscalled (1) aninclusionhomomorphismifforall x,y of H, wehave φ(x ◦ y) ⊆ φ(x) ◦ φ(y); (2) agoodhomomorphismifforallx,yofH,wehave φ(x ◦ y)= φ(x) ◦ φ(y)

Definition2.5. LetHbeanon-emptysetandlet+beahyperoperationonH.Thecouple(H, +)is calledacanonicalhypergroupifthefollowingconditionshold: (1) x + y = y + x, forall x,y ∈ H, (2) x +(y + z)=(x + y)+ z, forall x,y,z ∈ H, (3) thereexistsaneutralelement0 ∈ H suchthat x +0= {x} =0+ x, forall x ∈ H, (4) forevery x ∈ H, thereexistsauniqueelement x ∈ H suchthat0 ∈ x +( x) ∩ ( x)+ x, (5) z ∈ x + y implies y ∈−x + z and x ∈ z y, forall x,y,z ∈ H. M.A.IbrahimandA.A.A.Agboola,IntroductiontoNeutroHyperGroups

Definition2.6. [21]

(i) Aclassicaloperationisanoperationwell-definedforalltheset’selements.

(ii) Aclassicalhyper-operationisahyper-operationwell-definedforalltheset’selements.

(iii) Aneutrooperationisanoperationpartiallywell-definedorpartiallyindeterminateorpartially outerdefinedonagivenset.

(iv) (Anti)Operationisanoperationthatisouterdefinedforallset’selements.

(v) Aclassicallaw/axiomdefinedonanonemptysetisalaw/axiomthatistotallytrue(i.e.,true forallset’selements).

(vi) ANeutroLaw/NeutroAxiomdefinedonanonemptysetisalaw/axiomthatistrueforsomeset’s element[degreeoftruth(T )],indeterminateforotherset’selements[degreeofindeterminacy (I)],orfalsefortheotherset’selements[degreeoffalsehood(F )],where T,I,F ∈ [0, 1], with (T,I,F ) =(1, 0, 0)thatrepresentstheclassicalaxiom,and(T,I,F ) =(0, 0, 1)thatrepresents theAntiAxiom.

(vii) AnAntiLaw/AntiAxiomdefinedonanonemptysetisalaw/axiomthatisfalseforallset’s elements.

(viii) NeutroHyperOperationisahyper-operationpartiallywell-defined,partiallyindeterminate,and partiallyouter-definedonagivenset.

(ix) AntiHyperOperationisahyper-operationouter-definedforallset’selements.

(x) ANeutroAlgebraisanalgebrathathasatleastoneNeutroOperationoroneNeutroAxiom (axiomthatistrueforsomeelements,indeterminateforotherelements,andfalseforother elements).

(xi) AnAntiAlgebraisanalgebraendowedwithatleastoneAntiOperationoratleastoneAntiAxiom.

Theorem2.7. [17]Let U beanonemptyfiniteorinfiniteuniverseofdiscourseandlet S beafinite orinfinitresubsetof U If n classicaloperations(lawsandaxioms)aredefinedon S where n ≥ 1, then therewillbe (2n 1) NeutroAlgebrasand (3n 2n) AntiAlgebras.

Definition2.8. [Classicalgroup]Let G beanonemptysetandlet ∗ : G×G → G beabinaryoperation on G.Thecouple(G, ∗)iscalledaclassicalgroupifthefollowingconditionshold:

(G1) x ∗ y ∈ G ∀x,y ∈ G [closurelaw].

(G2) x ∗ (y ∗ z)=(x ∗ y) ∗ z ∀x,y,z ∈ G [axiomofassociativity].

(G3) Thereexists e ∈ G suchthat x ∗ e = e ∗ x = x ∀x ∈ G [axiomofexistenceofneutralelement].

(G4) Thereexists y ∈ G suchthat x ∗ y = y ∗ x = e ∀x ∈ G [axiomofexistenceofinverseelement] where e istheneutralelementof G. Ifinaddition ∀x,y ∈ G,wehave

(G5) x ∗ y = y ∗ x,then(G, ∗)iscalledanabeliangroup. M.A.IbrahimandA.A.A.Agboola,IntroductiontoNeutroHyperGroups

Definition2.9. [NeutroSophicationofthelawandaxiomsoftheclassicalgroup][5]

(NG1) Thereexistsomeduplets(x,y), (u,v), (p,q), ∈ G suchthat x ∗ y ∈ G (inner-definedwithdegree oftruthT)and[u ∗ v =indeterminate(withdegreeofindeterminacyI)or p ∗ q ∈ G (outerdefined/falsehoodwithdegreeoffalsehoodF)][NeutroClosureLaw].

(NG2) Thereexistsometriplets(x,y,z), (p,q,r), (u,v,w) ∈ G suchthat x ∗ (y ∗ z)=(x ∗ y) ∗ z (innerdefinedwithdegreeoftruthT)and[[p ∗ (q ∗ r)]or[(p ∗ q) ∗ r]=indeterminate(withdegreeof indeterminacyI)or u ∗ (v ∗ w) =(u ∗ v) ∗ w (outer-defined/falsehoodwithdegreeoffalsehood F)][NeutroAxiomofassociativity(NeutroAssociativity)].

(NG3) Thereexistsanelement e ∈ G suchthat x ∗ e = e ∗ x = x (inner-definedwithdegreeoftruth T)and[[x ∗ e]or[e ∗ x]=indeterminate(withdegreeofindeterminacyI)or x ∗ e = x = e ∗ x (outer-defined/falsehoodwithdegreeoffalsehoodF)]foratleastone x ∈ G [NeutroAxiomof existenceofneutralelement(NeutroNeutralElement)].

(NG4) Thereexistsanelement u ∈ G suchthat x ∗ u = u ∗ x = e (inner-definedwithdegreeoftruth T)and[[x ∗ u]or[u ∗ x)]=indeterminate(withdegreeofindeterminacyI)or x ∗ u = e = u ∗ x (outer-defined/falsehoodwithdegreoffalsehoodF)]foratleastone x ∈ G [NeutroAxiomof existenceofinverseelement(NeutroInverseElement)]where e isaNeutroNeutralElementin G (NG5) Thereexistsomeduplets(x,y), (u,v), (p,q) ∈ G suchthat x∗y = y∗x (inner-definedwithdegree oftruthT)and[[u ∗ v]or[v ∗ u]=indeterminate(withdegreeofindeterminacyI)or p ∗ q = q ∗ p (outer-defined/falsehoodwithdegreeoffalsehoodF)][NeutroAxiomofcommutativity (NeutroCommutativity)].

Definition2.10. [AntiSophicationofthelawandaxiomsoftheclassicalgroup][5]

(AG1) Foralltheduplets(x,y) ∈ G, x ∗ y ∈ G [AntiClosureLaw].

(AG2) Forallthetriplets(x,y,z) ∈ G, x ∗ (y ∗ z) =(x ∗ y) ∗ z [AntiAxiomofassociativity(AntiAssociativity)].

(AG3) Theredoesnotexistanelement e ∈ G suchthat x ∗ e = e ∗ x = x ∀x ∈ G [AntiAxiomof existenceofneutralelement(AntiNeutralElement)].

(AG4) Theredoesnotexist u ∈ G suchthat x ∗ u = u ∗ x = e ∀x ∈ G [AntiAxiomofexistenceof inverseelement(AntiInverseElement)]where e isanAntiNeutralElementin G.

(AG5) Foralltheduplets(x,y) ∈ G, x∗y = y ∗x [AntiAxiomofcommutativity(AntiCommutativity)].

Definition2.11. [5]ANeutroGroup NG isanalternativetotheclassicalgroup G thathasatleast oneNeutroLaworatleastoneof {NG1,NG2,NG3,NG4} withnoAntiLaworAntiAxiom.

Definition2.12. [5]AnAntiGroup AG isanalternativetotheclassicalgroup G thathasatleast oneAntiLaworatleastoneof {AG1,AG2,AG3,AG4} M.A.IbrahimandA.A.A.Agboola,IntroductiontoNeutroHyperGroups

Definition2.13. [5]ANeutroAbelianGroup NG isanalternativetotheclassicalabeliangroup G thathasatleastoneNeutroLaworatleastoneof {NG1,NG2,NG3,NG4} and NG5withnoAntiLaw orAntiAxiom.

Definition2.14. [5]AnAntiAbelianGroup AG isanalternativetotheclassicalabeliangroup G that hasatleastoneAntiLaworatleastoneof {AG1,AG2,AG3,AG4} and AG5.

Proposition2.15. [5]Let (G, ∗) beafiniteorinfiniteclassicalnonabeliangroup.Then: (i) thereare15typesofNeutroNonAbelianGroups, (ii) thereare65typesofAntiNonAbelianGroups.

Proposition2.16. [5]Let (G, ∗) beafiniteorinfiniteclassicalabeliangroup.Then: (i) thereare31typesofNeutroAbelianGroups, (ii) thereare211typesofAntiAbelianGroups.

Definition2.17. [5]Let(NG, ∗)beaNeutroGroup.Anonemptysubset NH of NG iscalleda NeutroSubgroupof NG if(NH, ∗)isalsoaNeutroGroupofthesametypeas NG.If(NH, ∗)isa NeutroGroupofatypedifferentfromthatof NG,then NH willbecalledaQuasiNeutroSubgroupof NG Example2.18. [5]

(i) Let NG = N = {1, 2, 3, 4 , }.Then(NG,.)isafiniteNeutroGroupwhere isthebinary operationofordinarymultiplication.

(ii) Let AG = Q∗ + bethesetofallirrationalpositivenumbers.Then(AG, ∗)isaninfiniteAntiGroup.

(iii) Let U = {a,b,c,d,e,f } beauniverseofdiscourseandlet AG = {a,b,c, } beasubsetof U.Let ∗ beabinaryoperationdefinedon AG asshownintheCayleytablebelow: ∗ a b c a d c b b c e a c b a f Then(AG, ∗)isafiniteAntiGroup.

3. FormulationofaNeutroHyperGroup

Definition3.1. [ClassicalHypergroup]

Let H beanon-emptysetand ◦ : H × H −→ P ∗(H)beahyperoperation.Then(H, ◦)isahypergroup ifthefollowingconditionshold: (H1) forall x,y ∈ H,x ◦ y ⊆ H (closurelaw), (H2) forall x,y,z ∈ H, (x ◦ y) ◦ z = x ◦ (y ◦ z)(associativeaxiom), M.A.IbrahimandA.A.A.Agboola,IntroductiontoNeutroHyperGroups

(H3) forall x ∈ H,x ◦ H = H ◦ x = H (reproductiveaxiom).

Definition3.2. [NeutroSophicationofthelawandaxiomsoftheclassicalhypergroup]

(NH1) Thereexistsomeduplets(u,v), (x,y), (p,q) ∈ H suchthat u ◦ v ⊆ H (inner-definedwiththe degreeoftruthT)and[x ◦ y =indeterminate(withthedegreeofindeterminacyI)or p ◦ q H (outer-defined/falsehoodwithdegreeoffalsehoodF)].

(NH2) Thereexistsometriplets(u,v,w), (x,y,z), (p,q,r) ∈ H suchthat u ◦ (v ◦ w)=(u ◦ v) ◦ w (inner-definedwiththedegreeoftruthT)and[x ◦ (y ◦ z)or(x ◦ y) ◦ z =indeterminate(with thedegreeofindeterminacyI)or(p ◦ q) ◦ r = p ◦ (q ◦ r)(outer-defined/falsehoodwithdegree offalsehoodF)].

(NH3) Thereexistsatleastatriplet(u,v,x) ∈ H suchthat u ◦ H = H ◦ u = H (inner-definedwith thedegreeoftruthT)and[v ◦ H or H ◦ v =indeterminate(withthedegreeofindeterminacy I)or x ◦ H = H = H ◦ x (outer-defined/falsehoodwithdegreeoffalsehoodF)].

Definition3.3. [AntiSophicationofthelawandaxiomsoftheclassicalhypergroup] (AH1) u ◦ v H ∀u,v ∈ H (anticlosurelaw). (AH2) u ◦ (v ◦ w) =(u ◦ v) ◦ w ∀u,v,w ∈ H (antiassociativeaxiom) (AH3) x ◦ H = H and H ◦ x = H ∀x ∈ H (antireproductiveaxiom).

Definition3.4. ANeutroHyperGroup(NH, ◦)isanalternativetotheclassicalhypergroup(H, ◦) thathasaNeutroLaworatleastoneof NH2and NH3withnoAntilaworAntiAxiom.

Definition3.5. AnAntiHyperGroup(AH, ◦)isanalternativetotheclassicalhypergroup(H, ◦)that hasanAntiLaworatleastoneof AH2and AH3

Theorem3.6. Let (H, ◦) beaclassicalhypergroup.Then, (1) thereare7classesofNeutroHyperGroup. (2) thereare19classesofAntiHyperGroup.

Proof. TheprooffollowseasilyfromTheorem2.7.

Theorem3.6showsthatthereare7classesofNeutroHypergroups.Theclasseswhere NH1 NH3 holdarecalledthetrivialNeutroHyperGroups.ExamplesofNeutroHyperGroupsinthisclassare presentedbelow.

Example3.7. Let V = {u,v,w,s,t,z} beauniverseofdiscourseandlet NH = {v,w,s,z} beasubset of V. Defineon NH thebinaryOperation ◦ asshowninthetablebelow.

Itcaneasilybededucedfromthetablethat(NH, ◦)isatrivialNeutroGroup.Thesubset NK = {v,s} of NH isalsoatrivialNeutroGroupandhenceaNeutroSubgroupof NH. M.A.IbrahimandA.A.A.Agboola,IntroductiontoNeutroHyperGroups

◦ vwsz vstzv wzuvs svwsz zzuvs

Now,considertheNeutroSubgroup NK = {v,s}.Definedon NK ahyperoperation NH asfollows : x NK y =      x ◦ NK ◦ y = {x ◦ z ◦ y : z ∈ NK} ifx = y, {x,x ◦ y} ifx = y.

Fromthisdefinitionweconstructthetablebelow:

Table1. Cayleytableforthehyperoperation NK NK vwsz v {v,z}{t,v}{v,z}{v} w {w,z}{t,u}{v,w}{s,w} s {s,v}{s,w}{s,z}{s,z} z {z}{u,z}{v,z}{s,v}

ItcanbeseenfromTable1that NK satisfies:

(1) NeutroClosureLaw(NH1):Exceptforthecomposition

v NK w = {t,v},w NK w = {t,u} and z NK w = {u,z} whicharefalsewith18 75%degree offalsehood,allothercompositionaretruewith81 25%degreeoftruth.

(2) NeutroAssociative(NH2):

s NK (v NK v)=(s NK v) NK v = {s,v,z}. s NK (w NK v)= {s,w,z} but(s NK w) NK v = {s,v,w,z} = {s,w,z}

(3) NeutroReproductionAxiom(NH3):

s NK NH = NH NK s = {v,w,s,z} = NH. w NK NH = NH NK w = {u,v,w,s,t,z} = NH. Hence,(NH, NK )isatrivialNeutroHyperGroup.

Example3.8. Let V = {u,v,w,s,t,z} beauniverseofdiscourseandlet NH = {u,v,w,z} bea subsetof V. Defineon NH thebinaryoperation ◦ asshowninthetablebelow.

Itcanbeshownfromthetablethat(NH, ◦)isaNeutroGroupandthesubset NK = {u,v} of NH is aclassicalgroupwithrespectto ◦ M.A.IbrahimandA.A.A.Agboola,IntroductiontoNeutroHyperGroups

◦ uvsz uuvsz vvutw sswut zztwu

Now,definedon NH ahyperoperation NK asfollows: x NK y =

x ◦ NK ◦ y = {x ◦ z ◦ y : z ∈ NK} ifx = u,y = uandx = y, x ◦ yifx = y, {x,y} otherwise

Note,if x ◦ y isindeterminate,wewrite x ◦ y = I. Fromthisdefinitionweconstructthetablebelow:

Table2. Cayleytableforthehyperoperation NK NK uvsz uu {u,v}{u,s}{u,z} v {u,v} u {s,t}{w,z} s {u,s}{w,s} u {v,t} z {u,z}{t,I }{w,I } u

Itcanbeseenfromthetablethat NK satisfies:

(1) NeutroClosureLaw(NH1):Exceptforthecompositions

v NK s = {s,t},v NH z = {w,z},s NK v = {w,s} and s NH z = {v,t} whicharefalse with25 0%degreeoffalsehood,andthecompositions z NK v = {t,I} and z NK s = {w,I} whichareindeterminatewith12 5%degreeofindeterminacyallothercompositionsaretrue with62.5%degreeoftruth.

(2) NeutroAssociative(NH2):

u NK (v NK u)=(u NK v) NK u = {u,v} s NK (u NK v)= {u,w,s} but(s NK u) NK v = {u,v,w,s} = {u,w,s}

(3) NeutroReproductionAxiom(NH3):

u NK NH = NH NK s = {u,v,s,z} = NH.

z NK NH = {u,w,t,z,I} = NH and NH NK z = {u,v,w,t,z} = NH.

Hence,(NH, NK )isatrivialNeutroHyperGroup. M.A.IbrahimandA.A.A.Agboola,IntroductiontoNeutroHyperGroups

4.

StudyofaClassofNeutroHyperGroup

Inthissection,wearegoingtoconsideraparticularclassofNeutroHyperGroups(NH, )where (i) (NH, )isaclassicalhypergroupoid, (ii) thehypergroupoid(NH, )isaNeutroSemiHyperGroupand (iii) thehypergroupoid(NH, )isaNeutroQuasiHyperGroup. WewillrefertothisclassofNeutroHyperGroupsas[2, 3]-NeutroHyperGroup(i.e.,H2andH3of Definition3.1areNeutroAxioms).

Example4.1. Let NH = {u,v,s,t} beanonemptysetandlet · beabinaryoperationsdefinedon NH asshowninthetablebelow. uvst uvtsu vvutu sstvu tuuuu

Nowconsiderthesubset NK = {u,v} Definedon NH ahyperoperation NK asfollows:

Fromthisdefinitionweconstructthetablebelow. Table3. Cayleytableforthehyperoperation NK NK uvst uvtsu vv {u,t}{s,t} u ss {u,t}{u,v} u tutsu

ItcanbeseenfromTable3that:

(1) (NH, NK )isahypergroupoid. (2) NK isNeutroAssociative,since (t NK v) NK t = t NK (v NK t)= {u} (v NK s) NK t = {u} but v NK (s NK t)= {v} = {u}.

Hence,thehypergroupoid(NH, NK )isNeutroSemiHyperGroup. M.A.IbrahimandA.A.A.Agboola,IntroductiontoNeutroHyperGroups

(3) NK satisfiesNeutroReproductiveAxiom,since

s NK NH = NH NK s = {u,v,s,t} = NG.

v NK NH = {u,v,s,t} = {u,t} = NH NK v.

Hence,thehypergroupoid(NH, NK )isaNeutroQuasiHyperGroup. Accordingly,(NH, NK )isa[2, 3]-NeutroHyperGroup.

Example4.2. Let NH = {α,β,γ,φ,ψ} andlet beahyperoperationdefinedon NH asshownin thetablebelow; Table4. Cayleytableforthehyperoperation αβγφψ αααααα

βα {γ,φ}{φ,ψ}{β,γ }{α,ψ} γα {α,γ }{α,γ } γ {α,γ } φα {α,φ} φ {α,φ}{α,φ} ψα {γ,φ}{β,φ}{γ,ψ}{α,β }

Then,(NH, )isa[2, 3] NeutroHyperGroup. ItcanbeseenfromTable4that:

(1) (NH, )isahypergroupoid. (2) isNeutroAssociative,since (φ β) ψ = φ (β ψ)= {α,φ}. (β φ) γ = {α,γ,φ,ψ} but β (φ γ)= {β,γ} = {α,γ,φ,ψ}

Hence,thehypergroupoid(NH, )isNeutroSemiHyperGroup. (3) satisfiesNeutroReproductiveAxiom,since

ψ NH = NH ψ = {α,β,γ,φ,ψ} = NH.

φ NH = {α,φ} = {α,β,γ,φ,ψ} = NH φ.

Hence,thehypergroupoid(NH, )isaNeutroQuasiHyperGroup. Accordingly,(NH, )isa[2, 3]-NeutroHyperGroup.

Example4.3. Let NH = {m,n,p,q} andlet ◦ beahyperoperationdefinedon NH asshownintable 5below.

Then,(NH, ◦)isa[2, 3] NeutroHyperGroup.ItcanbeseenfromtheTable5that:

(1) (NH, ◦)isaHypergroupoid. M.A.IbrahimandA.A.A.Agboola,IntroductiontoNeutroHyperGroups

Table5. Cayleytableforthehyperoperation ◦ ◦ mnpq mm {m,n}{m,p}{m,q} n {m,n} pn {n,p} pp {p,q} p {p,q} qq {m,q} q {m,q}

(2) ◦ isNeutroAssociative,since (m ◦ m) ◦ n = m ◦ (m ◦ n)= {m,n}. (m ◦ n) ◦ q = {m,n,p,q} but m ◦ (n ◦ q)= {m,n,p} = {m,n,p,q}

Hence,thehypergroupoid(NH, ◦)isNeutroSemiHyperGroup.

(3) ◦ satisfiesNeutroReproductiveAxiom,since m ◦ NH = NH ◦ m = {m,n,p,q} = NH. p ◦ NH = {p,q} = {m,n,p,q} = NH ◦ p.

Hence,thehypergroupoid(NH, ◦)isaNeutroQuasiHyperGroup. Accordingly,(NH, ◦)isa[2, 3]-NeutroHyperGroup.

Example4.4. Let NH = {1, 2, 3, 4, 5, 6} andlet beahyperoperationdefinedon NH asshownin thetablebelow; Table6. Cayleytableforthehyperoperation 123456 1111111 21 {1, 2} 2 {2, 4}{1, 2}{2, 4} 313 {1, 3} 3 {1, 3}{1, 3} 41 {1, 4} 4 {2, 4}{1, 4}{2, 4} 51 {3, 5}{2, 5}{5, 6}{1, 2, 3, 5}{2, 4, 5, 6} 61 {3, 6}{4, 6}{5, 6}{1, 3, 4, 6}{2, 4, 5, 6}

ItcanbeshownfromTable6that,(NH, )isa[2, 3] NeutroHyperGroup.

Proposition4.5. Let (NH1, 1) and (NH2, 2) beanytwo [2, 3] NeutroHyperGroups.Let NH1 × NH2 = {(v,k): v ∈ NH1 and k ∈ NH2}, for x =(v1,k1),y =(v2,k2) ∈ NH1 × NH2 define: x y =((v1 1 v2), (k1 2 k2))

Then (NH1 × NH2, ) isa [2, 3] NeutroHyperGroup. M.A.IbrahimandA.A.A.Agboola,IntroductiontoNeutroHyperGroups

Proof. (1) Let x =(v1,k1),y =(v2,k2) ∈ NH1 × NH2, then x y =(v1,k1) (v2,k2)=(v1 1 v2,k1 2 k2) ⊆ NH1 ×NH2 ∵ v1 1 v2 ⊆ NH1 and k1 2 k2 ⊆ NH2

Hence,(NH1 × NH2, )isahypergroupoid.

(2) Thereexistsatleastatriplet((v1,k1), (v2,k2), (v3,k3)) ∈ NH1 × NH2 suchthat ((v1,k1) (v2,k2)) (v3,k3)=((v1 1 v2), (k1 2 k2)) (v3,k3) =((v1 1 v2) 1 v3, (k1 2 k2) 2 k3) =(v1 1 (v2 1 v3),k1 2 (k2 2 k3)) ∵ NH2holdsin NH1and NH2 =(v1,k1) ((v2 1 v3), (k2 2 k3)) =(v1,k1) ((v2,k2) (v3,k3))

Also,thereexistsatleastatriplet((a1,b1), (a2,b2), (a3,b3)) ∈ NH1 × NH2 suchthat ((a1,b1) (a2,b2)) (a3,b3)=((a1 1 a2), (b1 2 b2)) (a3,b3) =((a1 1 a2) 1 a3, (b1 2 b2) 2 b3) =(a1 1 (a2 1 a3),b1 2 (b2 2 b3)) ∵ NH2holdsin NH1and NH2 =(a1,b1) ((a2 1 a3), (b2 2 b3)) =(a1,b1) ((a2,b2) (a3,b3)).

Hence, NH2holdsin NH1 × NH2. (3) Thereexistsatleasta(v,k) ∈ NH1 × NH2 with v ∈ NH1 and k ∈ NH2 suchthat (v,k) NH1 × NH2 =(v,k) {(v1,k1): v1 ∈ NH1,k1 ∈ NH2} = {(v,k) (v1,k1): v1 ∈ NH1,k1 ∈ NH2} = {((v 1 v1), (k 2 k1)): v1 ∈ NH1,k1 ∈ NH2} =(v 1 NH1,k 2 NH2) =(NH1 1 v,NH2 2 k)(∵ NH3holdsin NH1 and NH2) = NH1 × NH2 (v,k) = NH1 × NH2

Also,thereexistsatleasta(u,q) ∈ NH1 × NH2 with u ∈ NH1 and q ∈ NH2 suchthat (u,q) NH1 × NH2 =(u,q) {(v1,k1): v1 ∈ NH1,k1 ∈ NH2} = {(u,q) (v1,k1): v1 ∈ NH1,k1 ∈ NH2} = {((u 1 v1), (q 2 k1)): v1 ∈ NH1,k1 ∈ NH2} =(u 1 NH1,q 2 NH2) =(NH1 1 u,NH2 2 q)(∵ NH3holdsin NH1 and NH2) = NH1 × NH2 (u,q)

Accordingly,(NH1 × NH2, )isa[2, 3] NeutroHyperGroup. Proposition4.6. Let (NV, 1) bea [2, 3] NeutroHyperGroupand (H, 2) beanyhypergroup.Let NV × H = {(v,h): v ∈ NV and h ∈ H}, for x =(v1,h1),y =(v2,h2) ∈ NV × H define: x y =((v1 1 v2), (h1 2 h2)

Then (NV × H, ) isa [2, 3] NeutroHyperGroup. M.A.IbrahimandA.A.A.Agboola,IntroductiontoNeutroHyperGroups

Proof. Theprooffollowssimilarapproachastheproofof4.5.

Definition4.7. Let NH bea[2, 3] NeutroHyperGroup,anon-emptysubset NK of NH iscalleda [2, 3] NeutroSubHyperGroupof NH if NK isitselfa[2, 3] NeutroHyperGroup.

Example4.8. Let(NH, ◦)bethe[2, 3] NeutroHyperGroupdefinedinExample4.3andlet NK = {m,p,q} beasubsetof NH. Let ◦ bedefinedasshowninthetablebelow.

Table7. Cayleytableforthehyperoperation ◦ ◦ mpq mm {m,p}{m,q} ppp {p,q} qqq {m,q}

ItcanbeshownfromTable7that(NK, ◦)isa[2, 3] NeutroHyperGroup.Then,(NK, ◦)isa [2, 3] NeutroSubHyperGroupof NH.

Example4.9. Let(NH, ◦)bethe[2, 3] NeutroHyperGroupdefinedinExample4.4andlet NK = {1, 2, 3, 4} and NW = {1, 2, 3, 5} besubsetsof NH.

Let bedefinedasshowninthetablesbelow:

Table8. Cayleytableforthehyperoperation 1234 11111 21 {1, 2} 2 {2, 4} 313 {1, 3} 3 41 {1, 4} 4 {2, 4}

Table9. Cayleytablefor thehyperoperation 1235 11111 21 {1, 2} 2 {1, 2} 313 {1, 3}{1, 3} 51 {3, 5}{2, 5}{1, 2, 3, 5}

WecanseefromTables8and9that(NK, )and(NW, )are[2, 3] NeutroHyperGroups.Then, (NK, )and(NW, )are[2, 3] NeutroSubHyperGroupsof NH. Now,considerthefollowing:

(1) NK ∪ NW = {1, 2, 3, 4, 5}

(2) NK ∩ NW = {1, 2, 3}.

ItcanbeshownfromTable6that NK ∪ NW isa[2, 3] NeutroSubHyperGroupof NR but NK ∩ NW isanon-trivialNeutroSemiHyperGroupof NH.

TheseobservationsarerecordedinRemark4.10

Remark4.10. Let NK and NW betwo[2, 3] NeutroSubHyperGroupsofa[2, 3] NeutroHyperGroup NH.Then

M.A.IbrahimandA.A.A.Agboola,IntroductiontoNeutroHyperGroups

(1) NK ∪ NW canbea[2, 3] NeutroSubHyperGroupof NH.

(2) NK ∩ NW isnotnecessarilya[2, 3] NeutroSubHyperGroupof NH.

Proposition4.11. Let NH bea [2, 3] NeutroHyperGroupandlet NW bea [2, 3] NeutroSubHyperGroupof NH. For aNW,bNW ∈ NH/NW with a,b ∈ NH, let beahyperoperationdefinedon NH/NW by aNW bNW = {cd | c ∈ aNW,d ∈ bNW }

Then, (NH/NW, ) isa [2, 3] NeutroHyperGroupwhichisknownasa [2, 3] NeutroQuotientHyperGroup.

Proof. TheproofofthisPropositionwillbebyaconstructedexampleasgiveninExample4.12.

Example4.12. Let(NH, ◦)bethe[2, 3] NeutroHyperGroupdefinedinExample4.4andlet NW bethe[2, 3] NeutroSubHyperGroupofExample4.9.Thenwehave NH/NW = {NW,p ◦ NW,n ◦ NW,q ◦ NW }. Defineon NH/NW ahyperoperation asshowninthetablebelow Table10. Cayleytableforthehyperoperation NWpNWnNWqNW NW {NW,pNW,qNW }{NW,pNW,qNW }{NW,pNW,nNW }{NW,pNW,qNW } pNW {pNW,qNW }{pNW,qNW }{pNW,qNW }{pNW,qNW } nNW {NW,pNW,nNW }{pNW,nNW }{NW,nNW }{pNW,qNW,nNW } qNW {NW,qNW }{NW,qNW }{nNW,qNW } qNW

Then,itcanbeseenfromthetablethat:

(1) (NH/NW, )isahypergroupoid.

(2) thereexistsatleastatriplet(pNW,nNW,qNW ) ∈ NH/NW suchthat pNW (nNW qNW )=(pNW nNW ) qNW = {pNW,qNW }. And,thereexistsatleastatriplet(qNW,nNW,qNW ) ∈ NH/NW suchthat (qNW nNW ) qNW = {pNW,nNW,qNW } = {NW,nNW,qNW } = qNW (nNW qNW )

(3) thereexists nNW ∈ NH/NW suchthat nNW NH/NW = NH/NW nNW = NH/NW.

And,thereexists pNW ∈ NH/NW suchthat pNW NH/NW = {pNW,qNW } = {NW,pNW,nNW,qNW } = NH/NW pNW. Accordingly,(NH/NW, )isa[2, 3] NeutroHyperGroup. M.A.IbrahimandA.A.A.Agboola,IntroductiontoNeutroHyperGroups

Systems,Vol.38,2020

Definition4.13. Let(NH, )and(NW, ◦)beanytwo[2, 3] NeutroHyperGroups.Themapping φ : NH −→ NW

(1) iscalledaNeutroHyperGroupHomomorphismif φ(a b) ⊆ φ(a) ◦ φ(b)foratleastaduplet (x,y) ∈ NH.

(2) iscalledagoodNeutroHyperGroupHomomorphismif φ(a b)= φ(a) ◦ φ(b)foratleastaduplet (x,y) ∈ NH.

(3) iscalledNeutroHyperGroupIsomorphismif φ isaNeutroHyperGroupHomomorphismand φ 1 isalsoaNeutroHyperGroupHomomorphism.

Definition4.14. Let(NH, )and(NW, )beanytwo[2, 3] NeutroHyperGroupswithNeutroNeutralElements eNH and eNW respectively.

Let φ : NH −→ NW beagoodNeutroHyperGroupHomomorphism.

Thekernelof φ denotedby NHKerφ isdefinedas

NHKerφ = {x : φ(x)= eNW }

Theimageof φ denotedby NHImφ isdefinedas NHImφ = {y ∈ NW : y = φ(x) foratleastoney ∈ NW }

Example4.15. Let(NK, ◦)bethe[2, 3] NeutroHyperGroupofExample4.8andlet φ : NK × NK −→ P ∗(NK)

begivenby φ(k1,k2)= k1 ◦ k2 forall k1,k2 ∈ NK. Then φ isagoodNeutroHyperGroupHomomorphism. Wehave NHKerφ = {(m,m), (p,m), (p,p), (q,m), (q,p)} and NHImφ = {m,p,q, {m,p}, {m,q}, {p,q}}

Itcanbeshownthat NHKerφ isa[2, 3] NeutroSubHyperGroupof(NK × NK, ◦)and NHImφ is a[2, 3] NeutroSubHyperGroupof(P ∗(NK), ◦)

Proposition4.16. Let (NH, ) and (NW, ) beanytwo [2, 3] NeutroHypergroups. Let φ : NH −→ NW beagoodNeutroHperGroupHomomorphism.Then:

(1) NHKerφ isaNeutroSubHyperGroupof NH.

(2) NHImφ isaNeutroSubHyperGroupof NW.

Proof. TheprooffollowsfromExample4.15.

M.A.IbrahimandA.A.A.Agboola,IntroductiontoNeutroHyperGroups

5. Applicationsof [2, 3] NeutroHyperGroupsinBiologicalandChemicalSciences

In[10],Davvazetal.providedsomeexamplesofhyperstructuresandweakhyperstruturesassociated withdismutationreactions.Inwhatfollows,wewillprovideexamplestoshowthatwhendismutation reactionstakeplaceintheneutrosophicenvironment,theyareassociatedwith[2, 3] NeutroHyperGroups.

Example5.1. Let NX = {x0 = Sn,x2 = Sn2+,x4 = Sn4+} beasetofTinindifferentoxidation state.Defineon NX, ahyperoperation asshowninthetablebelow,where isacomproportionation reaction(withoutenergy).Then,itcanbeseenfromTable11that: Table11. Cayleytableforthehyperoperation x0 x2 x4 x0 x0 {x0 ,x2 } x2 x2 {x0 ,x2 } x2 {x2 ,x4 } x4 x2 {x2 ,x4 } x4

(1) (NX, )isahypergroupoid.

(2) Forthetriplet(x2,x4,x2) ∈ NX,wehave x2 (x4 x2)=(x2 x4) x2 = {x2,x4} andforthetriplet(x0,x2,x4) ∈ NX,wehavehave (x0 x2) x4 = {x2,x4} = {x0,x2} = x0 (x2 x4)

(3) For x2 ∈ NX,wehave x2 NX = NX x2 = {x0,x2,x4} andfor x4 ∈ NX,wehave x4 NX = NX x4 = {x2,x4} = NX. Accordingly,(NX, )isa[2, 3] NeutroHyperGroup.

Example5.2. Let BG = {a0 = AA,a1 = AS,a3 = SS} beasetofbloodgroup.Defineon BG, a hyperoperation asshowninthetablebelow,where denotemating. Then,itcanbeseenfromtable12that:

(1) (BG, )isahypergroupoid.

AComproportionationisachemicalreactionwheretworeactantseachcontainingthesameelementbutwithadifferent oxidationnumber,willgiveaproductwithoxidationnumberintermediateofthetworeactant. M.A.IbrahimandA.A.A.Agboola,IntroductiontoNeutroHyperGroups

Table12. Cayleytableforthehyperoperation

a0 a1 a3 a0 a0 {a0 ,a1 } a1 a1 {a0 ,a1 }{a0 ,a1 ,a3 }{a1 ,a3 } a3 a1 {a1 ,a3 } a3

(2) Forthetriplet(a1,a3,a1) ∈ BG,weobtain

a1 (a3 a1)=(a1 a3) a1 = {a0,a1,a3}

andforthetriplet(a0,a1,a3) ∈ BG,weobtain (a0 a1) a3 = {a1,a3} = {a0,a1} = a0 (a1 a3)

(3) Foranelement a1 ∈ BG,weobtain

a1 BG = BG a1 = {a0,a1,a3} = BG andforananelement a3 ∈ BG,weobtain a3 BG = BG a3 = {a1,a3} = BG.

Accordingly,(BG, )isa[2, 3] NeutroHyperGroup.

Remark5.3. ItisevidentfromExamples5.1and5.2thatintheneutrosophicenvironment,[2, 3] NeutroHyperGroupsareassociatedwithdismutationreactionsinsomechemicalreactionsandbiological processes.

6. Conclusions

Inthispaper,wehaveforthefirsttimeintroducedtheconceptofNeutroHyperGroups.Specifically,aclassofNeutroHyperGroupscalled[2, 3] NeutroHyperGroupwasinvestigatedandsome oftheirelementarypropertiesandseveralexampleswerepresented.Itwasshownthattheintersectionoftwo[2, 3] NeutroSubHyperGroupsisnotnecessarilya[2, 3] NuetroSubHyperGroupbut theirunionmayproducea[2, 3] NeutroSubHyperGroup.Also,itwasshownthatthequotientofa [2, 3] NeutroHyperGroupfactoredbya[2, 3] NeutroSubHyperGroupisa[2, 3] NeutroHyperGroup. Exampleswereprovidedtoshowthatintheneutrosophicenvironment,[2, 3] NeutroHyperGroups areassociatedwithdismutationreactionsinsomechemicalreactionsandbiologicalprocesses.Inour futurework,wehopetousethealgebraicpropertiesofNeutroHyperGroupstoanalyzesomechemical reactionsandbiologicalprocesses.

Funding: Thisresearchreceivednoexternalfunding.

Acknowledgments: TheauthorsacknowledgethesuggestionsofProfessorFlorentinSmarandacheon theNeutroSophicationofthehypergroupconsideredinthispaper.

ConflictsofInterest: Theauthorsdeclarenoconflictofinterest. M.A.IbrahimandA.A.A.Agboola,IntroductiontoNeutroHyperGroups

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Received: June 10, 2020. Accepted: Nov 20, 2020

M.A.IbrahimandA.A.A.Agboola,IntroductiontoNeutroHyperGroups