IntroductiontoAntiGroups
A.A.A.Agboola DepartmentofMathematics,FederalUniversityofAgriculture,Abeokuta,Nigeria. agboolaaaa@funaab.edu.ng Incommemorationofthe60thbirthdayoftheauthor
Abstract
ThenotionofAntiGroupsisformallypresentedinthispaper.AparticularclassofAntiGroupsoftype-AG[4]is studiedwithseveralexamplesandbasicpropertiespresented.InAntiGroupsoftype-AG[4],theexistenceofan inverseistakingtobetotallyfalseforalltheelementswhiletheclosurelaw,theexistenceofidentityelement, theaxiomsofassociativityandcommutativityaretakingtobeeitherpartiallytrue,partiallyindeterminateor partiallyfalseforsomeelements.Itisshownthatsomealgebraicpropertiesoftheclassicalgroupsdonothold intheclassofAntiGroupsoftype-AG[4].Specifically,itisshownthatintersectionoftwoAntiSubgroupsisnot necessarilyanAntiSubgroupandtheunionoftwoAntiSubgroupsmaybeanAntiSubgroup.Also,itisshown thatdistinctleft(right)cosetsofAntiSubgroupsofAntiGroupsoftype-AG[4]donotpartitiontheAntiGroups; andthatLagranges’theoremandfundamentaltheoremofhomomorphismsoftheclassicalgroupsdonothold intheclassofAntiGroupsoftype-AG[4].
Keywords: NeutroGroup,AntiGroup,AntiSubgroup,AntiQuotientGroup,AntiGroupHomomorphism.
1Introduction
Neutrosophiclogic(NL)introducedbySmarandachein1995isanalternativetotheexistingclassicallogicsandthegeneralizationoffuzzylogic(FL)ofZadeh[12]andintuitionisticfuzzylogic(IFL)ofAtanassov [5].Neutrosophiclogicisanon-classicallogicthatcanbeusedasamathematicaltooltomodelsituations characterizedbyuncertainty,vagueness,ambiguity,imprecision,undefined,unknown,incompleteness,inconsistency,redundancy,contradictionetc.Inneutrosophiclogic[11],eachpropositionisestimatedtohave percentageoftruthinasubset T ,thepercentageofindeterminacyinasubset I,andthepercentageoffalsityinasubset F where (T,I,F ) arestandardornon-standardsubsetsofthenon-standardinterval ] 0, 1+[, where ninf =inf T +inf I +inf F ≥ 0,and nsup =sup T +sup I +sup F ≤ 3+.Statically, (T,I,F ) aresubsetsbutdynamically,theyafunctions/operatorsdependingonmanyknownorunknownparameters. InNL,if < A > isanidea,orproposition,theory,law,axiom,event,concept,entityetc.,therecorrespond < Non-A >=< Anti-A > whichistheoppositeof < A > and < Neut-A > whichstandsforwhatisneither < A > nor < Anti-A >,thatisneutralityinbetweenthetwoextremes.ConsequentlyinNL,itispossible tohavethetriad (< A >,< Neut-A >,< Anti-A >).Thenon-restrictioninNLallowsforparaconsistent,dialetheist,andincompleteinformationtobecharacterized.ThisspecialanduniquefeatureofNLhas madeitapplicableinsolvingproblemsinvolvinguncertainty,vagueness,ambiguity,imprecision,undefined, unknown,incompleteness,inconsistency,redundancy,contradictionetc.arisingfromscience,socialscience, engineering,technology,computerscience,artificialintelligence,ICT,roboticsetc.
Theindeterminacyfactor I inNLisfundamentalintheformulationandestablishmentofanyneutrosophic algebraicstructure.Givenanyclassicalalgebraicstructure (X, ∗),astructure X (I )=<X ∪ I> generatedby X and I underthebinaryoperation ∗ of X iscalledaneutrosophicalgebraicstructurewithitsnamederived fromthenameof X.Forinstance,if X isagroup,then X (I ) iscalledaneutrosophicgroup.Since I 1,the inverseof I doesnotexist,finding x 1,theinverseofanyneutrosophicelement x ∈ X (I ) becomesdifficult andimpossible.Consequently,algebraicmanipulationsoftheelementsof X (I ) becomerestrictive.The recentintroductionoftheconceptsofNeutroStructuresandAntiStructureshavelessenedtherestrictivenessof thealgebraicmanipulationsoftheelementsofneutrosophicalgebraicstructuresimposedbytheneutrosophic element I
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Inanyclassicalalgebraicstructure (X, ∗),theofcompositionoftheelementswithrespecttothebinary operation ∗ iswelldefinedforalltheelementsof X thatis, x ∗ y ∈ X ∀x,y ∈ X.Alltheaxiomslike associativity,commutativity,distributivity,monotonicityetc.definedon X withrespectto ∗ aretotallytruefor alltheelementsof X.Thecompositionsofelementsof X thiswayarerestrictiveanddonotreflectthereality. Theydonotgiveroomsforcompositionsthatareeitherpartiallydefined,partiallyundefined(indeterminate), andpartiallyouterdefinedortotallyouterdefinedwithrespectto ∗.Howeverinthedomainofknowledge, scienceandreality,thelawofcompositionandaxiomsdefinedon X mayeitherbeonlypartiallydefined(partiallytrue),orpartiallyundefined(partiallyfalse),ortotallyundefined(totallyfalse)withrespecttothebinary operation ∗.In2019,Smarandache[10]addressedtheproblemofallowingthelawofcompositionon X tobe eitherpartiallydefinedandpartiallyundefinedortotallyundefinedbyintroducingthenotionsofNetroDefined andAntiDefinedlaws,aswellasthenotionsofNeutroAxiomandAntiAxiominspiredbyNLheintroducedin 1995.TheworkofSmarandachein[10]hasgivenbirthtothenewfieldsofresearchcalledNeutroStructures andAntiStructures.Foranyclassicalalgebraiclaworaxiomdefinedon X,therecorrespondneutrosophic triads (< Law >,< NeutroLaw >,< AntiLaw >) and (< Axiom >,< NeutroAxiom >,< AntiAxiom >) respectively.In[9],SmarandachestudiedNeutroAlgebrasandAntiAlgebrasandin[8],hestudiedPartial Algebras,UniversalAlgebras,EffectAlgebrasandBoole’sPartialAlgebrasandheshowedthatNeutroAlgebrasaregeneralizationofPartialAlgebras.In[7],RezaeiandSmarandachestudiedNeutro-BE-algebrasand Anti-BE-algebrasandtheyshowedthatanyclassicalalgebra S with n operations(lawsandaxioms)where n ≥ 1 willhave (2n 1) NeutroAlgebrasand (3n 2n) AntiAlgebras.In[2],Agboolaetal.studiedNeutroAlgebrasandAntiAlgebrasviz-a-viztheclassicalnumbersystems N, Z, Q, R and C andin[3],Agboola studiedNeutroGroupsbyconsideringthreeNeutroAxioms(NeutroAssociativity,existenceofNeutroNeutral elementandexistenceofNeutroInverseelement).Inaddition,hestudiedNeutroSubgroups,NeutroCyclicGroups,NeutroQuotientGroupsandNeutroGroupHomomorphisms.Heshowedthatgenerally,Lagrange’s theoremandfundamentalhomomorphismtheoremoftheclassicalgroupsdonotholdintheclassofNeutroGroupsstudied.In[4],AgboolaintroducedandstudiedNeutroRingsbyconsideringthreeNeutroAxioms (NeutroAbelianGroup(additive),NeutroSemigroup(multiplicative)andNeutroDistributivity(multiplication overaddition)).HepresentedSeveralresultsandexamplesonNeutroRings,NeutroSubgrings,NeutroIdeals, NeutroQuotientRingsandNeutroRingHomomorphisms.Heshowedthatthatthefundamentalhomomorphism oftheclassicalringsholdsintheclassofNeutroRingsconsidered.Motivatedandinspiredbytheworkof RezaeiandSmarandachein[7],Agboolain[1]revisitedtheNeutroGroupsbystudyingaparticularclassof NeutroGroupsandpresentedtheirbasicandelementaryproperties.Inthepresentpaperhowever,theconcept ofAntiGroupsisformallypresented.AparticularclassofAntiGroupsisstudiedwithpresentationofseveral examplesandbasicproperties.Itisshownthatsomealgebraicpropertiesoftheclassicalgroupsdonothold intheclassofAntiGroupsstudied.Specifically,itisshownthatintersectionoftwoAntiSubgroupsisnot necessarilyanAntiSubgroupandtheunionoftwoAntiSubgroupsmaybeanAntiSubgroup.Also,itisshown thatLagranges’theoremandfundamentaltheoremofhomomorphismsoftheclassicalgroupsdonotholdin theclassofAntiGroupsstudiedinthispaper.
2Preliminaries
Inthissection,wewillgivesomedefinitionsandresultsthatwillbeusedlaterinthepaper.
Definition2.1. [8]
(i) Aclassicaloperationisanoperationwelldefinedforalltheset’selements.
(ii) ANeutroOperationisanoperationpartiallywelldefined,partiallyindeterminate,andpartiallyouter definedonthegivenset.
(iii) AnAntiOperationisanoperationthatisouterdefinedforallset’selements.
(iv) Aclassicallaw/axiomdefinedonanonemptysetisalaw/axiomthatistotallytrue(i.e.trueforallset’s elements).
(v) ANeutroLaw/NeutroAxiom(orNeutrosophicLaw/NeutrosophicAxiom)definedonanonemptysetisa law/axiomthatistrueforsomeset’selements[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) thatrepresentstheAntiAxiom.
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(vi) AnAntiLaw/AntiAxiomdefinedonanonemptysetisalaw/axiomthatisfalseforallset’selements.
(vii) ANeutroAlgebraisanalgebrathathasatleastoneNeutroOperationoroneNeutroAxiom(axiomthat istrueforsomeelements,indeterminateforotherelements,andfalseforotherelements),andnoAntiOperationorAntiAxiom.
(viii) AnAntiAlgebraisanalgebraendowedwithatleastoneAntiOperationoratleastoneAntiAxiom.
Theorem2.2. [7]Let U beanonemptyfiniteorinfiniteuniverseofdiscourseandlet S beafiniteorinfinite subsetof U.If n classicaloperations(lawsandaxioms)aredefinedon S where n ≥ 1,thentherewillbe (2n 1) NeutroAlgebrasand (3n 2n) AntiAlgebras.
Definition2.3. [Classicalgroup][6] Let G beanonemptysetandlet ∗ : G × G → G beabinaryoperationon G.Thecouple (G, ∗) iscalleda classicalgroupifthefollowingconditionshold:
(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.
Definition2.4. [AntiSophicationofthelawandaxiomsoftheclassicalgroup][1]
(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 [AntiAxiomofexistenceof neutralelement(AntiNeutralElement)].
(AG4) Theredoesnotexist u ∈ G suchthat x ∗ u = u ∗ x = e ∀x ∈ G [AntiAxiomofexistenceofinverse element(AntiInverseElement)]where e isanAntiNeutralElementin G
(AG5) Foralltheduplets (x,y) ∈ G, x ∗ y = y ∗ x [AntiAxiomofcommutativity(AntiCommutativity)].
Definition2.5. [AntiGroup][1]
AnAntiGroup AG isanalternativetotheclassicalgroup G thathasatleastoneAntiLaworatleastoneof {AG1,AG2,AG3,AG4}.
Definition2.6. [AntiAbelianGroup][1]
AnAntiAbelianGroup AG isanalternativetotheclassicalabeliangroup G thathasatleastoneAntiLaworat leastoneof {AG1,AG2,AG3,AG4} and AG5.
Theorem2.7. [1]Let (G, ∗) beafiniteorinfiniteclassicalgroup.Thenthereare65typesofAntiGroups. Theorem2.8. [1]Let (G, ∗) beafiniteorinfiniteclassicalabeliangroup.Thenthereare211typesof AntiAbelianGroups.
Definition2.9. Let (AG, ∗) beanAntiGroup. AG issaidtobefiniteoforder n ifthecardinalityof AG is n thatis o(AG)= n.Otherwise, AG iscalledaninfiniteAntiGroupandwewrite o(AG)= ∞
SincetherearemanytypesofAntiGroups,inwhatfollows,AntiGroupswillbeclassifiedandnamedtypeAG[,]accordingtowhichof AG1 AG5 is(are)satisfied.Ifonly AG1 issatisfied,theAntiGroupwillbecalled oftype-AG[1],type-AG[3,4]ifonly AG3 and AG4 aresatisfiedandsoon.AntiGroupsoftype-AG[1,2,3,4] oroftype-AG[1,2,3,4,5]willbecalledtrivialAntiGroupsortrivialAntiAbelianGroupsrespectively.
Example2.10. Let AG = Q∗ + bethesetofallirrationalpositivenumbersandconsideralgebraicstructure (AG,.) where istheordinarymultiplicationofrealnumbers.Then (AG,.) isaninfinitetrivialAntiGroup.
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Example2.11. Let AG = N
(i) Let ∗ beabinaryoperationon AG defined ∀x,y ∈ AG by x ∗ y = x + y + xy.
Then (AG, ∗) isafiniteAntiGroupoftype-AG[3,4].
(ii) Let ∗ beabinaryoperationon AG defined ∀x,y ∈ AG by x ∗ y = x + y +1
Then (AG, ∗) isafiniteAntiGroupoftype-AG[3,4].
3AStudyofFiniteAntiGroupsofType-AG[4]
Inthissection,wearegoingtostudyaparticularclassofAntiGroups (AG, ∗) where G4 istotallyfalsefor alltheelementsof AG while G1,G2,G3 and G5 areeitherpartiallytrue,partiallyindeterminateorpartially falseforsomeelementsof AG
Definition3.1. Let (AG, ∗) and (AH, ◦) beAntiGroupsoftype-AG[4].Thedirectproductof AG and AH denotedby AG × AH isdefinedby
AG × AH = {(g,h): g ∈ AG,h ∈ AH}
Proposition3.2. Let (AG, ∗) and (AH, ◦) beAntiGroupsoftype-AG[4]andlet ⊗ beabinaryoperationon AG × AH definedby
(g,h) ⊗ (x,y)=(g ∗ x,h ◦ y) ∀ (g,h), (x,y) ∈ AG × AH.
Then (AG × AH, ⊗) isanAntiGroupoftype-AG[4].
Proof. TheprooffollowsfromthedefinitionofAntiGroupsoftype-AG[4]andthedefinitionofdirectproduct ofAntiGroupsoftype-AG[4].
Proposition3.3. Let (AG, ∗) beanAntiGroupoftype-AG[4]andlet g,x,y ∈ AG.Then
(i) g ∗ x = g ∗ y =⇒ x = y. (ii) x ∗ g = y ∗ g =⇒ x = y
Proof. Since g 1 doesnotexistand ∗ isNeutroAssociative,therequiredresultsfollow.
Proposition3.4. Let (AG, ∗) beanAntiGroupoftype-AG[4], x,y ∈ AG andlet m,n ∈ N.Then
(i) xm+1 = xm ∗ x (ii) x m = x 1 m (iii) xm ∗ x m = Ne where Ne isaNeutroNeutralElementin AG. (iv) xm ∗ xn = xm+n (v) (xm)n = xmn (vi) (x ∗ y)m = xm ∗ ym .
Proof. Since x 1 doesnotexistand ∗ isNeutroAssociative,therequiredresultsfollow.
Corollary3.5. AnAntiGroup (AG, ∗) oftype-AG[4]cannotbegeneratedbyanelement x ∈ AG andhence cannotbycyclic.
Definition3.6. Let (AG, ∗) beanAntiGroupoftype-AG[4].Anonemptysubset AH of AG iscalledan AntiSubgroupof AG if (AH, ∗) isalsoanAntiGroupofthesametypeas AG.Otherwise,if (AH, ∗) isan AntiGroupofatypedifferentfromthetypeof AG,then AH iscalledaQuasiAntiSubgroupof AG
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Definition3.7. Let (AG, ∗) beanAntiGroupoftype-AG[4]andlet AH and AK beAntiSubgroupsof AG Theset A ∗ B isdefinedby
A ∗ B = {x ∈ AG : x = h ∗ k forsome h ∈ AH,k ∈ AK}.
Proposition3.8. Let AH , AK and AL beAntiSubgroupsofanAntiGroup (AG, ∗) oftype-AG[4].Then
(i) AH ∗ AH = AH
(ii) AH ∗ AK = AK ∗ AH
(iii) AH ∗ (AK ∗ AL) =(AH ∗ AK) ∗ AL
Proof. Obvious.
Definition3.9. Let (AG, ∗) beanAntiGroupoftype-AG[4]andlet a ∈ AG beafixedelement.
(i) AnAntiCenterof AG denotedby AZ(AG) isasetdefinedby
AZ(AG)= {x ∈ AG : x ∗ g = g ∗ x ∀g ∈ AG}.
(ii) AnAntiCentralizerof a ∈ AG denotedby ACa isasetdefinedby
ACa = {g ∈ AG : g ∗ a = a ∗ g}.
Example3.10. Let U = {a,b,c,d,e,f } beauniverseofdiscourseandlet AG = {a,b,c,e} beasubsetof U
(i) Let ∗ beabinaryoperationdefinedon AG asshownintheCayleytablebelow.
∗ a b c e a d a c e b a d e c c b a f e e a b c f
Itisevidentfromthetablethat G1,G2,G3,G5 areeitherpartiallytrueorpartiallyfalsewithrespectto ∗ but G4 istotallyfalseforalltheelementsof AG.Hence (AG, ∗) isafiniteAntiGroupoftype-AG[4].
(ii) Let ∗ beabinaryoperationdefinedon AG asshownintheCayleytablebelow.
∗ e a b c e d a b c a a f c b b b a ? c c c b a ?
Itisevidentfromthetablethat G1,G2,G3,G5 areeitherpartiallytrue,partiallyindeterminateorpartiallyfalsewithrespectto ∗ but G4 istotallyfalseforalltheelementsof AG.Hence (AG, ∗) isafinite AntiGroupoftype-AG[4].
Example3.11. Let (AG, ∗) betheAntiGroupofExample3.10(i)andlet AH1 = {a,b,e} and AH2 = {b,c,e} betwosubsetsof AG.Let ∗ bedefinedon AH1 and AH2 asshownintheCayleytablesbelow:
AH1 :
∗ a b e a d a e b a d c e a b f ,AH2 :
∗ b c e b d e c c a f e e b c f
Itcaneasilybeseenfromthetablesthat AH1 isanAntiSubgroupof AG while AH2 isaQuasiAntiSubgroup of AG.ItisnotedthatLagranges’theoremdoesnothold.Itisalsonotedthat:
AH1 ∪ AH2 = {a,b,c,e} = AG, NH1 ∩ NH2 = {b,e},
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fromwhichitisdeducedthat NH1 ∪NH2 isanAntiSubgroupof AG but AH1 ∩AH2 isaQuasiAntiSubgroup of AG asitisevidentintheCayleytablebelow:
NH1 ∩ NH2 : ∗ b e b d c e b f .
AZ(AG)= {a,b,c,e} = AG isanAntiSubgroupof AG.Also, ACa = ACb = ACc = ACe = {a,b,c,e} = AG areAntiSubgroupsof AG
Example3.12. Let (AG, ∗) betheAntiGroupofExample3.10(ii)andlet AH1 = {e,a,b} and AH2 = {e,b,c} betwosubsetsof AG.Let ∗ bedefinedon AH1 and AH2 asshownintheCayleytablesbelow:
AH1 :
∗ e a b e d a b a a f c b b a ? ,AH2 :
∗ e b c e d b c b b ? c c c a ?
Itcaneasilybeseenfromthetablesthat AH1 and AH2 areAntiSubgroupsof AG.ItisnotedthatLagranges’ theoremdoesnothold.Itisalsonotedthat:
AH1 ∪ AH2 = {e,a,b,c} = AG, NH1 ∩ NH2 = {e,b},
fromwhichitisdeducedthat NH1 ∪ NH2 isanAntiSubgroupof AG and AH1 ∩ AH2 isanAntiSubgroup of AG asitisevidentintheCayleytablebelow:
NH1 ∩ NH2 : ∗ e b e d f b b ?
AZ(AG)= {a,b,c} isaQuasiAntiSubgroupof AG.Also, ACa = ACb = {a,b,c}, ACc = {b,c} are QuasiAntiSubgroupsof AG and ACe = {} = ∅ isneitheranAntiSubgroupnoraQuasiAntiSubgroupof AG
Example3.13. (i) Let AG = Z4 = {0, 1, 2, 3} andlet ⊕ beabinaryoperationon AG asdefinedinthe Cayleytable. ⊕ 0 1 2 3 0 0 or 1 1 2 3 1 1 2 3 0 or 2 2 2 3 0 or 3 1 3 3 ? 1 2
Then (AG, ⊕) isafiniteAntiGroupoftype-AG[4]and AH = {0, 1, 2} isanAntiSubgroupof AG
(ii) Let AG = {1, 2, 3, 4}⊆ Z5 andlet ⊗ beabinaryoperationon AG asdefinedintheCayleytable. ⊗ 1 2 3 4 1 ? 2 3 4 2 2 4 0 3 3 3 ? 4 2 4 4 3 2 0
Then (AG, ⊗) isafiniteAntiGroupoftype-AG[4]and AH = {1, 2, 3} isanAntiSubgroupof AG
Definition3.14. Let AH beanAntiSubgroupoftheAntiGroup (AG, ∗) oftype-AG[4]andlet x ∈ AG (i) x ∗ AH theleftcosetof AH in AG isdefinedby x ∗ AH = {x ∗ h : h ∈ AH}
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(ii) AH ∗ x therightcosetof AH in AG isdefinedby AH ∗ x = {h ∗ x : h ∈ AH}
(iii) AH iscalledanormalAntiSubgroupif x ∗ AH = AH ∗ x foratleastone x ∈ AG.
(iv) Thenumberofdistinctleftorrightcosetsof AH in AG iscalledtheindexof AH in AG anditis denotedby [AG : AH ].
(vi) Thesetofalldistinctleftcosetsof AH in AG denotedby (AG/AH )L isdefinedby (AG/AH )L = {x ∗ AH : x ∈ AG}
(vii) Thesetofalldistinctrightcosetsof AH in AG denotedby (AG/AH )R isdefinedby (AG/AH )R = {AH ∗ x : x ∈ AG}
Supposethat AG/AH isthesetofalldistinctleftcosetsof AH in AG andsupposethat isabinaryoperation on AG/AH definedby
(x ∗ AH ) (y ∗ AH )=(x ∗ y) ∗ AH ∀ x ∗ AH,y ∗ AH ∈ AG/AH. Ifthecouple (AG/AH, ) isanAntiGroupoftype-AG[4],then AG/AH iscalledanAntiQuotientGroupof AG factoredby AH
Lemma3.15. Let AH beanAntiSubgroupoftheAntiGroup (AG, ∗) oftype-AG[4]andlet e ∈ AG bea NeutroNeutralElement.Thenglobally, e ∗ AH = AH.
Example3.16. Let (AG, ⊕) beanAntiGroupofExample3.13(i)andlet AH = {0, 1, 2} beitsAntiSubgroup. Theleftandrightcosetsof AH in AG are: 0 ⊕ AH = {0, 1, 2} or {1, 2} = AH ⊕ 0, 1 ⊕ AH = {1, 2, 3}, = AH ⊕ 1 2 ⊕ AH = {0, 2, 3} or {2, 3} = AH ⊕ 2, 3 ⊕ AH = {1, 3, ?} = AH ⊕ 3, ∴ AG/AH = {0 ⊕ AH, 1 ⊕ AH, 2 ⊕ AH, 3 ⊕ AH}
Itisnotedthat AH isanormalAntiSubgroupof AG anddistinctleftandrightcosetsof AH donotpartition AG. NowconsidertheCayleytablebelow. 0 ⊕ AH 1 ⊕ AH 2 ⊕ AH 3 ⊕ AH 0 ⊕ AH 0 ⊕ AH or 1 ⊕ AH 1 ⊕ AH 2 ⊕ AH 3 ⊕ AH 1 ⊕ AH 1 ⊕ AH 2 ⊕ AH 3 ⊕ AH 0 ⊕ AH or 2 ⊕ AH 2 ⊕ AH 2 ⊕ AH 3 ⊕ AH 0 ⊕ AH or 3 ⊕ AH 1 ⊕ AH 3 ⊕ AH 3 ⊕ AH ? 1 ⊕ AH 2 ⊕ AH
Itisevidentfromthetablethat (AG/AH, ) isanAntiGroupoftype-AG[4]. Example3.17. Let (AG, ⊗) beanAntiGroupofExample3.13(ii)andlet AH = {1, 2, 3} beitsAntiSubgroup.Theleftcosetsof AH in AG are: 1 ⊗ AH = {?, 2, 3} = AH ⊗ 1, 2 ⊗ AH = {0, 2, 4} = AH ⊗ 2, 3 ⊗ AH = {3, 4, ?} = AH ⊗ 3, 4 ⊗ AH = {2, 3, 4} = AH ⊗ 4, ∴ AG/AH = {1 ⊗ AH, 2 ⊗ AH, 3 ⊗ AH, 4 ⊗ AH}.
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Itisnotedthat AH isanormalAntiSubgroupof AG anddistinctleftandrightcosetsof AH donotpartition AG NowconsidertheCayleytablebelow. 1 ⊗ AH 2 ⊗ AH 3 ⊗ AH 4 ⊗ AH 1 ⊗ AH ? 2 ⊗ AH 3 ⊗ AH 4 ⊗ AH 2 ⊗ AH 2 ⊗ AH 4 ⊗ AH 0 ⊗ AH 3 ⊗ AH 3 ⊗ AH 3 ⊗ AH ? 4 ⊗ AH 2 ⊗ AH 4 ⊗ AH 4 ⊗ AH 3 ⊗ AH 2 ⊗ AH 0 ⊗ AH
Itisevidentfromthetablethat (AG/AH, ) isanAntiGroupoftype-AG[4].
Proposition3.18. Let AH beanormalAntiSubgroupofanAntiGroup (AG, ∗) oftype-AG[4]andlet AG/AH bethesetofdistinctleftcosetsof AH in AG.For x ∗ AH,y ∗ AH ∈ AG/AH with x,y ∈ AG,let bea binaryoperationdefinedon AG/AH by (x ∗ AH ) (y ∗ AH )=(x ∗ y) ∗ AH ∀ x,y ∈ AG.
Then, (AG/AH, ) isanAntiGroupoftype-AG[4].
Proof. Supposethat AH isanormalAntiSubgroupofanAntiGroup (AG, ∗) oftype-AG[4]andsupposethat thecompositionofelementsin AG/AH isgivenby (x ∗ AH ) (y ∗ AH )=(x ∗ y) ∗ AH ∀ x,y ∈ AG Thenthereexistsomeduplets (x,y), (u,v), (p,q) ∈ AG suchthat x ∗ y ∈ AG (inner-defined)and[u ∗ v = indeterminateor p ∗ q ∈ AG (outer-defined/falsehood)].Hence, satisfiestheNeutroClosureLaw.Next, thereexistsometriplets (x,y,z), (p,q,r), (u,v,w) ∈ AG suchthat x ∗ (y ∗ z)=(x ∗ y) ∗ z (inner-defined) and[[p ∗ (q ∗ r)]or[(p ∗ q) ∗ r]= indeterminateor u ∗ (v ∗ w) =(u ∗ v) ∗ w (outer-defined/falsehood)]. Thisagainshowsthat satisfiestheNeutroAssociativityAxiom.Also,thereexistsanelement e ∈ AG suchthat x ∗ e = e ∗ x = x (inner-defined)and[[x ∗ e]or[e ∗ x]= indeterminateor x ∗ e = x = e ∗ x (outer-defined/falsehood)]foratleastone x ∈ AG.ThisshowstheexistenceofNeutroNeutralElementin AG andhencethereexistsaNeutroNeutralElement e ∗ AH ∈ AG/AH.Againforall x ∈ AG,theredoes notexist u ∈ AG suchthat x ∗ u = u ∗ x = e.ThisisanAntiAxiomofexistenceofinverseelement in AG andconsequently,noelement x ∗ AH ∈ AG/AH hasaninverse.Lastly,thereexistsomeduplets (x,y), (u,v), (p,q) ∈ AG suchthat x ∗ y = y ∗ x (inner-defined)and[[u ∗ v]or[v ∗ u]= indeterminate or p ∗ q = q ∗ p (outer-defined/falsehood)].Thisshowsthat satisfiestheNeutroCommutativityAxiom. Hence,(AG/AH, ) isanAntiGroupoftype-AG[4].
Definition3.19. Let (AG, ∗) and (AH, ◦) beanytwoAntiGroupsoftype-AG[4].Themapping φ : AG → AH iscalledanAntiGroupHomomorphismif φ doesnotpreservethebinaryoperations ∗ and ◦ thatisforall theduplet (x,y) ∈ AG,wehave φ(x ∗ y) = φ(x) ◦ φ(y).
Thekernelof φ denotedby Kerφ isdefinedby
Kerφ = {x : φ(x)= eAH foratleastone eAH ∈ AH}
where eAH isaNeutroNeutralElementin AH. Theimageof φ denotedby Imφ isdefinedby
Imφ = {y ∈ AH : y = φ(x) forsome x ∈ AG}.
Ifinaddition φ isanAntiBijection,then φ iscalledanAntiGroupIsomorphism.AntiGroupEpimorphism, AntiGroupMonomorphism,AntiGroupEndomorphism,andAntiGroupAutomorphismaredefinedsimilarly.
Example3.20. (i) Let (AG, ⊕) betheAntiGroupofExample3.13(i)andlet φ : AG → AG beamapping definedby
φ(x)=2 ⊕ x ∀ x ∈ AG. Then φ(0)=2, φ(1)=3, φ(2)=0 or 3, φ(3)=1,
Doi:10.5281/zenodo.427413078
fromwhichweobtainthat φ(x ⊕ y) = φ(x) ⊕ y forall x,y ∈ AG.Accordingly, φ isanAntiGroupHomomorphism. Imφ = {1, 2, 3} whichisanAntiSubgroupof AG Kerφ = {} = ∅
(ii) Let (AG, ⊗) betheAntiGroupofExample3.13(ii)andlet ψ : AG → AG beamappingdefinedby
ψ(x)= x ⊗ 4 ∀ x ∈ AG.
Then ψ(1)=4, ψ(2)=3, ψ(3)=2, ψ(4)=0,
fromwhichweobtainthat ψ(x ⊗ y) = ψ(x) ⊗ y forall x,y ∈ AG.Accordingly, ψ isanAntiGroupHomomorphism. Imφ = {0, 2, 3, 4} whichisnotanAntiSubgroupof AG. Kerφ = {} = ∅. Example3.21. (i) Let (AG, ∗) betheAntiGroupofExample3.10(i)andlet φ : AG × AG → AG bea projectiondefinedby
φ((x,y))= x ∀ x,y ∈ AG.
Then φ isnotanAntiGroupHomomorphismbecause φ((a,b) ⊗ (b,c))= φ((a,b)) ∗ φ((b,c))= a. However, Imφ = {a,b,c,e} = AG. (ii) Let (AG, ∗) betheAntiGroupofExample3.10(ii)andlet ψ : AG × AG → AG beaprojectiondefined by
ψ((x,y))= y ∀ x,y ∈ AG.
Then ψ isnotanAntiGroupHomomorphismbecause ψ((a,b) ⊗ (b,c))= ψ((a,b)) ∗ ψ((b,c))= c. However, Imψ = {a,b,c,e} = AG Example3.22. (i) Let (AG/AH, ) betheAntiQuotientGroupofExample3.16andlet φ : AG → AG/AH beamappingdefinedby
φ(x)= x ⊕ AH ∀ x ∈ AG.
Then
fromwhichweobtain
φ(0)=0 ⊕ AH, φ(1)=1 ⊕ AH, φ(2)=2 ⊕ AH, φ(3)=3 ⊕ AH,
φ(1 ⊕ 2)= φ(1) φ(2)=3 ⊕ AH.
Thisshowsthat φ isnotanAntiGroupHomomorphism.
(ii) Let (AG/AH, ) betheAntiQuotientGroupofExample3.17andlet ψ : AG → AG/AH bea mappingdefinedby ψ(x)= x ⊗ AH ∀ x ∈ AG.
Then
ψ(1)=1 ⊗ AH, ψ(2)=2 ⊗ AH, ψ(3)=3 ⊗ AH, ψ(4)=4 ⊗ AH,
fromwhichweobtain
ψ(2 ⊗ 4)= ψ(2) ψ(4)=3 ⊗ AH. Thisshowsthat ψ isnotanAntiGroupHomomorphism. Remark3.23. Thefundamentaltheoremofhomomorphismsoftheclassicalgroupscannotholdintheclass ofAntiGroupsoftype-AG[4]asdemonstratedinExamples3.22(i)and(ii).
Doi:10.5281/zenodo.427413079
InternationalJournalofNeutrosophicScience(IJNS)Vol.12,No.2,PP.71-80,2020
4Conclusion
ThenotionofAntiGroupswasformallypresentedinthispaper.AparticularclassofAntiGroupsoftypeAG[4]wasstudied.InAntiGroupsoftype-AG[4],theexistenceofaninverseelementwastakingtobetotally falseforalltheelementswhiletheclosurelaw,theexistenceofidentityelement,theaxiomsofassociativity andcommutativityweretakingtobeeitherpartiallytrue,partiallyindeterminateorpartiallyfalseforsome elements.Itwasshownthatsomealgebraicpropertiesoftheclassicalgroupsdonotholdintheclassof AntiGroupsoftype-AG[4].Specifically,itwasshownthatintersectionoftwoAntiSubgroupsisnotnecessarily anAntiSubgroupandtheunionoftwoAntiSubgroupsmaybeanAntiSubgroup.Also,itwasshownthat distinctleft(right)cosetsofAntiSubgroupsofAntiGroupsoftype-AG[4]donotpartitiontheAntiGroups;and thatLagranges’theoremandfundamentaltheoremofhomomorphismsoftheclassicalgroupsdonotholdin theclassofAntiGroupsoftype-AG[4].MoreclassesofAntiGroupswillbestudiedinourfuturepapers.
5Acknowledgment
ThecontributionsofProfessorFlorentinSmarandachetowardstheimprovementofthispaperareacknowledged.Thecommentsandsuggestionsofalltheanonymousreviewersareequallyacknowledged.
References
[1] Agboola,A.A.A.,OnFiniteNeutroGroupsofType-NG[1,2,4],InternationalJournalofNeutrosophic Science(IJNS),Vol.10(2),pp.84-95,2020.(Doi:10.5281/zenodo.4006602)
[2] Agboola,A.A.A.,Ibrahim,M.A.andAdeleke,E.O.,ElementaryExaminationofNeutroAlgebrasand AntiAlgebrasviz-a-viztheClassicalNumberSystems(IJNS),InternationalJournalofNeutrosophicScience,Vol.4(1),pp.16-19,2020.DOI:10.5281/zenodo.3752896.
[3] Agboola,A.A.A.,IntroductiontoNeutroGroups,InternationalJournalofNeutrosophicScience(IJNS), Vol.6(1),pp.41-47,2020.(DOI:10.5281/zenodo.3840761).
[4] Agboola,A.A.A.,IntroductiontoNeutroRings,InternationalJournalofNeutrosophicScience(IJNS), Vol.7(2),pp.62-73,2020.(DOI:10.5281/zenodo.3877121).
[5] Atanassov,K.,Intuitionisticfuzzysets,FuzzySetsandSystems,Vol.20,pp.87-96,1986.
[6] Gilbert,L.andGilbert,J.,ElementsofModernAlgebra,EighthEdition,CengageLearning,USA,2015.
[7] Rezaei,A.andSmarandache,F.,OnNeutro-BE-algebrasandAnti-BE-algebras(revisited),International JournalofNeutrosophicScience(IJNS),Vol.4(1),pp.08-15,2020.(DOI:10.5281/zenodo.3751862).
[8] Smarandache,F.,NeutroAlgebraisaGeneralizationofPartialAlgebra,InternationalJournalofNeutrosophicScience,vol.2(1),pp.08-17,2020.
[9] Smarandache,F.,IntroductiontoNeutroAlgebraicStructuresandAntiAlgebraicStructures(revisited), NeutrosophicSetsandSystems(NSS),vol.31,pp.1-16,2020.DOI:10.5281/zenodo.3638232.
[10] Smarandache,F.,IntroductiontoNeutroAlgebraicStructures,inAdvancesofStandardandNonstandard NeutrosophicTheories,PonsPublishingHouseBrussels,Belgium,Ch.6,pp.240-265,2019.
[11] Smarandache,F.,Neutrosophy,NeutrosophicProbability,Set,andLogic,ProQuestInformation & Learning,AnnArbor,Michigan,USA,105p.,1998.
http://fs.unm.edu/eBook-Neutrosophic6.pdf (editiononline).
[12] Zadeh,L.A.,FuzzySets,InformationandControl,Vol.8,pp.338-353,1965.
Doi:10.5281/zenodo.427413080