OnFiniteNeutroGroupsofType-NG[1,2,4]
A.A.A.Agboola
DepartmentofMathematics,FederalUniversityofAgriculture,Abeokuta,Nigeria. agboolaaaa@funaab.edu.ng Incommemorationofthe60thbirthdayoftheauthor
Abstract
TheNeutroGroupsasalternativestotheclassicalgroupsareofdifferenttypeswithdifferentalgebraicproperties.Inthispaper,wearegoingtostudyaclassofNeutroGroupsoftype-NG[1,2,4].InthisclassofNeutroGroups,theclosurelaw,theaxiomofassociativityandexistenceofinversearetakingtobeeitherpartially trueorpartiallyfalseforsomeelements;whiletheexistenceofidentityelementandaxiomofcommutativity aretakingtobetotallytrueforalltheelements.SeveralexamplesofNeutroGroupsoftype-NG[1,2,4]are presentedalongwiththeirbasicproperties.ItisshownthatLagrange’stheoremholdsforsomeNeutroSubgroupsofaNeutroGroupandfailedtoholdforsomeNeutroSubgroupsofthesameNeutroGroup.Itisalso shownthattheunionoftwoNeutroSubgroupsofaNeutroGroupcanbeaNeutroSubgroupevenifoneisnot containedintheother;andthattheintersectionoftwoNeutroSubgroupsmaynotbeaNeutroSubgroup.The conceptsofNeutroQuotientGroupsandNeutroGroupHomomorphismsarepresentedandstudied.Itisshown thatthefundamentalhomomorphismtheoremoftheclassicalgroupsisholdingintheclassofNeutroGroups oftype-NG[1,2,4].
Keywords: NeutroGroup;AntiGroup;NeutroSubgroup;NeutroQuotientGroup;NeutroGroupHomomorphism.
1IntroductionandPreliminaries
Inanyclassicalalgebraicstructure (X, ∗),thelawofcompositionoftheelementsof X otherwisecalleda binaryoperation ∗ iswelldefinedforalltheelementsof X thatis, x ∗ y ∈ X ∀x,y ∈ X;andaxiomslikeassociativity,commutativity,distributivity,etc.definedon X withrespectto ∗ aretotallytrueforalltheelements of X.Thecompositionsofelementsof X thiswayarerestrictiveanddonotreflectthereality.Itdoesnotgive roomforcompositionsthatareeitherpartiallydefined,partiallyundefined(indeterminate),andpartiallyouterdefinedortotallyouterdefinedwithrespectto ∗.Howeverinthedomainofknowledge,scienceandreality,the lawofcompositionandaxiomsdefinedon X mayeitherbeonlypartiallydefined(partiallytrue),orpartially undefined(partiallyfalse),ortotallyundefined(totallyfalse)withrespectto ∗.Inanattempttomodeltherealitybyallowingthelawofcompositionon X tobeeitherpartiallydefined,partiallyundefined(indeterminate), andpartiallyouterdefinedortotallyouterdefined,Smarandache[8]in2019introducedthenotionsofNetroDefinedandAntiDefinedlaws,aswellasthenotionsofNeutroAxiomandAntiAxiominspiredbyhisworkin[9], whichhasgivenbirthtonewfieldsofresearchcalledNeutroStructuresandAntiStructures.Foranyclassical algebraiclaworaxiomdefinedon X,therecorrespondneutrosophictriplets < Law,NeutroLaw,AntiLaw > and < Axiom,NeutroAxiom,AntiAxiom > respectively.Smarandachein[7]studiedNeutroAlgebrasand AntiAlgebrasandin[6],hestudiedPartialAlgebras,UniversalAlgebras,EffectAlgebrasandBoole’sPartial AlgebrasandheshowedthatNeutroAlgebrasaregeneralizationofPartialAlgebras.RezaeiandSmarandache [5]studiedNeutro-BE-algebrasandAnti-BE-algebrasandfundamentallytheyshowedthatanyclassicalalgebra S with n operations(lawsandaxioms)where n ≥ 1 willhave (2n 1) NeutroAlgebrasand (3n 2n) AntiAlgebras.Agboolaetal.in[1]studiedNeutroAlgebrasandAntiAlgebrasviz-a-viztheclassicalnumber systems N, Z, Q, R and C andin[2],hestudiedNeutroGroupsbyconsideringthreeNeutroAxioms(NeutroAssociativity,existenceofNeutroNeutralelementandexistenceofNeutroInverseelement).Inaddition,he studiedNeutroSubgroups,NeutroCyclicGroups,NeutroQuotientGroupsandNeutroGroupHomomorphisms. Heshowedthatgenerally,Lagrange’stheoremand1stisomorphismtheoremoftheclassicalgroupsdonot Doi:10.5281/zenodo.400660284
holdintheclassofNeutroGroups.In[3],AgboolastudiedNeutroRingsbyconsideringthreeNeutroAxioms (NeutroAbelianGroup(additive),NeutroSemigroup(multiplicative)andNeutroDistributivity(multiplication overaddition)).HepresentedSeveralresultsandexamplesonNeutroRings,NeutroSubgrings,NeutroIdeals, NeutroQuotientRingsandNeutroRingHomomorphisms.Heshowedthatthe1stisomorphismtheoremofthe classicalringsholdsintheclassofNeutroRings.MotivatedandinspiredbytheworkofRezaeiandSmarandachein[5],theworkonNeutroGroupspresentedin[2]isrevisitedandthepresentworkisdevotedtothe studyofaclassofNeutroGroupsoftype-NG[1,2,4].InthisclassofNeutroGroups,theclosurelaw,theaxiomofassociativityandexistenceofinversearetakingtobeeitherpartiallytrueorpartiallyfalseforsome elements;whiletheexistenceofidentityelementandaxiomofcommutativityaretakingtobetotallytruefor alltheelements.SeveralexamplesofNeutroGroupsoftype-NG[1,2,4]arepresentedalongwiththeirbasic properties.ItisshownthatLagrange’stheoremholdsforsomeNeutroSubgroupsofaNeutroGroupandfailed toholdforsomeNeutroSubgroupsofthesameNeutroGroup.ItisalsoshownthattheunionoftwoNeutroSubgroupsofaNeutroGroupcanbeaNeutroSubgroupevenifoneisnotcontainedintheother;andthatthe intersectionoftwoNeutroSubgroupsmaynotbeaNeutroSubgroup.TheconceptsofNeutroQuotientGroups andNeutroGroupHomomorphismsarepresentedandstudied.Itisshownthatthefundamentalhomomorphism theoremoftheclassicalgroupsisholdingintheclassofNeutroGroupsoftype-NG[1,2,4].
Definition1.1. [6]
(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.
(vi) AnAntiLaw/AntiAxiomdefinedonanonemptysetisalaw/axiomthatisfalseforallset’selements.
(vii) ANeutroAlgebraisanalgebrathathasatleastoneNeutroOperationoroneNeutroAxiom(axiomthat istrueforsomeelements,indeterminateforotherelements,andfalseforotherelements),andnoAntiOperationorAntiAxiom.
(viii) AnAntiAlgebraisanalgebraendowedwithatleastoneAntiOperationoratleastoneAntiAxiom.
Theorem1.2. [5]Let U beanonemptyfiniteorinfiniteuniverseofdiscourseandlet S beafiniteorinfinite subsetof U.If n classicaloperations(lawsandaxioms)aredefinedon S where n ≥ 1,thentherewillbe (2n 1) NeutroAlgebrasand (3n 2n) AntiAlgebras.
2MainResults
(G5) x ∗ y = y ∗ x,then (G, ∗) iscalledanabeliangroup.
Definition2.2. [NeutroSophicationofthelawandaxiomsoftheclassicalgroup]
(NG1) Thereexistatleastthreeduplets (x,y), (u,v), (p,q), ∈ G suchthat x ∗ y ∈ G (degreeoftruthT)and [u ∗ v = outer-defined/indeterminate(degreeofindeterminacyI)or p ∗ q ∈ G] (degreeoffalsehoodF) [NeutroClosureLaw].
(NG2) Thereexistatleastthreetriplets (x,y,z), (p,q,r), (u,v,w) ∈ G suchthat x∗(y ∗z)=(x∗y)∗z (degree oftruthT)and[[p ∗ (q ∗ r)]or [(p ∗ q) ∗ r]= outer-defined/indeterminate(degreeofindeterminacyI)or u ∗ (v ∗ w) =(u ∗ v) ∗ w](degreeof,falsehoodF)[NeutroAxiomofassociativity(NeutroAssociativity)].
(NG3) Thereexistsanelement e ∈ G suchthat x ∗ e = e ∗ x = x (degreeoftruthT)and[[x ∗ e]or[e ∗ x]= outer-defined/indeterminate(degreeofindeterminacyI)or x ∗ e = x = e ∗ x](degreeoffalsehoodF) foratleastone x ∈ G [NeutroAxiomofexistenceofneutralelement(NeutroNeutralElement)].
(NG4) Thereexistsanelement u ∈ G suchthat x ∗ u = u ∗ x = e (degreeoftruthT)and[[x ∗ u]or[u ∗ x)]= outer-defined/indeterminate(degreeofindeterminacyI)or x ∗ u = e = u ∗ x]foratleastone x ∈ G (degreeoffalsehoodF)[NeutroAxiomofexistenceofinverseelement(NeutroInverseElement)]where e isaNeutroNeutralElementin G.
(NG5) Thereexistatleastthreeduplets (x,y), (u,v), (p,q) ∈ G suchthat x ∗ y = y ∗ x (degreeoftruthT)and [[u ∗ v]or[v ∗ u]= outer-defined/indeterminate(degreeofindeterminacyI)or p ∗ q = q ∗ p](degreeof falsehoodF)[NeutroAxiomofcommutativity(NeutroCommutativity)].
Definition2.3. [AntiSophicationofthelawandaxiomsoftheclassicalgroup]
(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)].
Definition2.4. [NeutroGroup]
ANeutroGroup NG isanalternativetotheclassicalgroup G thathasatleastoneNeutroLaworatleastone of {NG1,NG2,NG3,NG4} withnoAntiLaworAntiAxiom.
Definition2.5. [AntiGroup]
AnAntiGroup AG isanalternativetotheclassicalgroup G thathasatleastoneAntiLaworatleastoneof {AG1,AG2,AG3,AG4}.
Definition2.6. [NeutroAbelianGroup]
ANeutroAbelianGroup NG isanalternativetotheclassicalabeliangroup G thathasatleastoneNeutroLaw oratleastoneof {NG1,NG2,NG3,NG4} and NG5 withnoAntiLaworAntiAxiom.
Definition2.7. [AntiAbelianGroup]
AnAntiAbelianGroup AG isanalternativetotheclassicalabeliangroup G thathasatleastoneAntiLaworat leastoneof {AG1,AG2,AG3,AG4} and AG5.
Proposition2.8. Let (G, ∗) beafiniteorinfiniteclassicalnonabeliangroup.Then:
(i) thereare15typesofNeutroNonAbelianGroups, (ii) thereare65typesofAntiNonAbelianGroups.
Proof. ItfollowsfromTheorem1.2.
Proposition2.9. Let (G, ∗) beafiniteorinfiniteclassicalabeliangroup.Then: (i) thereare31typesofNeutroAbelianGroups,
Doi:10.5281/zenodo.400660286
(ii) thereare211typesofAntiAbelianGroups.
Proof. ItfollowsfromTheorem1.2.
Remark2.10. ItisevidentfromTheorem2.8andTheorem2.9thattherearemanytypesofNeutroGroupsand NeutroAbelianGroups.ThetypeofNeutroGroupsstudiedbyAgboolain[2]isthatforwhich G1,G2,G3,G4 and G5 areeitherpartiallytrueorpartiallyfalse.
Definition2.11. Let (NG, ∗) beaNeutroGroup. NG issaidtobefiniteoforder n ifthecardinalityof NG is n thatis o(NG)= n.Otherwise, NG iscalledaninfiniteNeutroGroupandwewrite o(NG)= ∞.
Definition2.12. Let (AG, ∗) beanAntiGroup. AG issaidtobefiniteoforder n ifthecardinalityof AG is n thatis o(AG)= n.Otherwise, AG iscalledaninfiniteAntiGroupandwewrite o(AG)= ∞
Example2.13. Let NG = N = {1, 2, 3, 4 ··· , }.Then (NG,.) isafiniteNeutroGroupwhere isthe binaryoperationofordinarymultiplication.
Example2.14. Let AG = Q∗ + bethesetofallirrationalpositivenumbersandconsideralgebraicstructure (AG, ∗) where ∗ isordinarymultiplicationofnumbers.Itisclearthat ∗ isatotalAntiLawdefinedon AG Thebinaryoperation ∗ istotallyassociativeforallthetriplets (x,y,z) with x,y,z ∈ AG.Thereisnoneutral element(s)foralltheelements AG andhencenoelementof AG hasaninverse.Finally,theoperation ∗ iscommutativeforalltheduplets (x,y) with x,y ∈ AG.HencebyDefinition2.5, (AG, ∗) isaninfinite AntiGroup.
Example2.15. 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 .
Itisclearfromthetablethatexceptforthecompositions a ∗ a = d,b ∗ b = e,c ∗ c = f thatareouter-defined withthedegreeoffalsity 33%,therestcompositionsareinner-definedwith 66.7% degreeoftruth.Thisshows that G1 ispartiallytrueandpartiallyfalsesothat ∗ isaNeutroLaw.Also, G2 ispartialtrueandpartially false.Thereare 33 =27 possibletripletsoutofwhichonly 6 canverifyassociativityof ∗.Hencedegree ofassociativityof ∗ is 22 2% whilethedegreeofnon-associativityis 77 8% sothat ∗ isNeutroAssociative. However, G3 and G4 aretotallyfalseforalltheelementsof AG whichshowsthat AG3 and AG4 aresatisfied. Lastly, G5 ispartiallytruewiththedegreeoftruth 50% andpartiallyfalsewith 50% degreeoffalsitywhich showsthat ∗ isNeutroCommutative.HencebyDefinition2.5, (AG, ∗) isafiniteAntiGroup.
3CertainTypesofNeutroGroups
Inthissection,wearegoingtostudycertaintypesofNeutroGroups (NG, ∗).TheNeutroGroupswillbenamed accordingtowhichof NG1 NG5 is(are)satisfied.Inthesequel, x ∗ y willbewrittenas xy ∀x,y ∈ NG
Example3.1. Let U = {a,b,c,d,e,f } beauniverseofdiscourseandlet NG = {e,a,b,c} beasubsetof U Let ∗ beabinaryoperationdefinedon NG asshownintheCayleytablebelow: ∗ e a b c e e a b c a a b a b b b c f c c c d c e
Itisclearfromthetablethat G1,G2,G3,G4 and G5 arepartiallytrueandpartiallyfalsewithrespectto ∗ as shownbelow: (i) NeutroClosureLaw(NG1): Exceptforthecompositions b ∗ b = f,c ∗ a = d whicharefalsewith 12.5%degreeoffalsity,allothercompositionsaretruewith87.5%degreeoftruth. Doi:10.5281/zenodo.400660287
(ii) NeutroAssociativity(NG2):
a ∗ (b ∗ c)=(a ∗ b) ∗ c = b. a ∗ (a ∗ a)= a, but, (a ∗ a) ∗ a = c = a.
(iii) NeutroNeutralElement(NG3):
Ne = Na = Nb = e but Nc = e or b.
(iv) NeutroInverseElement(NG4):
Ie = e, Ia doesnotexist, Ib doesnotexist, Ic = e.
(v) NeutroCommutativity(NG5): b ∗ c = c ∗ b = c. a ∗ b = a but b ∗ a = c = a.
Wehavejustshownthat (NG, ∗) isafiniteNeutroAbelianGroup.ThisisanexampleoftheclassofNeutroGroupsstudiedbyAgboolain[2].ThisclassofNeutroGroupsarereferredtoasoftype-NG[1,2,3,4,5]. Example3.2. Let U = {a,b,c,d,e,f } beauniverseofdiscourseandlet NG = {a,b,c,e} beasubsetof U Let ∗ beabinaryoperationdefinedon NG asshownintheCayleytablebelow: ∗ a b c e a e c f a b c e d b c d a e c e a b c e
.
Itisclearfromthetablethat G3 and G4 aretotallytrueforalltheelementsof NG.However, G1, G2 and G5 arepartiallytrueandpartiallyfalsewithrespectto ∗ asshownbelow:
(i) NeutroClosureLaw(NG1): Exceptforthecompositions a ∗ c = f,b ∗ c = d,c ∗ a = d whichareouterdefinedwith18.75%degreeoffalsity,allothercompositionsareinner-definedwith81.25%degreeof truth.
(ii) NeutroAssociativity(NG2): c ∗ (b ∗ b)=(c ∗ b) ∗ b = c. a ∗ (b ∗ c)= a ∗ d = outer-defined, (a ∗ b) ∗ c = e.
(iii) NeutroCommutativity(NG5): a ∗ b = b ∗ a = c. a ∗ c = f but c ∗ a = d = f.
Wehavejustshownthat (NG, ∗) isafiniteNeutroAbelianGroupoftype-NG[1,2,5].
Example3.3. Let NG = Z andlet ∗ beabinaryoperationdefinedon NG by x ∗ y = x + xyx,y ∈ Z
Itisclearthatonly G1 istotallytrueforall x,y ∈ NG but G2,G3,G4 and G5 arepartiallytrueandpartially falsewithrespectto ∗ asshownbelow:
Doi:10.5281/zenodo.400660288
(i) NeutroAssociativity(NG2):
x ∗ (y ∗ z)= x + xy + xyz (x ∗ y) ∗ z = x + xy + xz + xyz byequating,wehave x + xy + xyz = x + xy + xz + xyz ⇒ xz =0 fromwhichweobtain x =0 or z =0.
Thisshowsthatonlythetriplets (0,y, 0), (0,y,z), (x,y, 0) canverifyassociativityof ∗.
(ii) NeutroNeutralElement(NG3): Itisclearthatonlytheelement 0 ∈ NG has 0 asitsneutralelementandnoneutral(s)forotherelements of NG.
(iii) NeutroInverseElement(NG4): Again,onlytheelement 0 ∈ NG has 0 astheinverseelementandnoinverse(s)forotherelementsof NG
(iv) NeutroCommutativity(NG5):
Onlytheduplet (0, 0) canverifythecommutativityof ∗ andnotanyotherduplet(s) (x,y).Hence, (NG, ∗) isaninfiniteNeutroAbelianGroupoftype-NG[2,3,4,5].
Definition3.4. Let (NG, ∗) beaNeutroGroup.Anonemptysubset NH of NG iscalledaNeutroSubgroup of NG if (NH, ∗) isalsoaNeutroGroupofthesametypeas NG.If (NH, ∗) isaNeutroGroupofatype differentfromthatof NG,then NH willbecalledaQuasiNeutroSubgroupof NG Example3.5. Let (NG, ∗) betheNeutroGroupofExample3.2andlet NH1 = {a,c,e} and NH2 = {b,c,e} betwosubsetsof NG.Let ∗ bedefinedon NH1 and NH2 asshownintheCayleytablesbelow:
NH1 :
∗ a c e a e f a c d e c e a c e
NH2 :
∗ b c e b e d b c a e c e b c e
Itcaneasilybeshownthat (NH1, ∗) and (NH2, ∗) areNeutroGroupsoftype-NG[1,2,5]andtherefore NH1 and NH2 areNeutroSubgroupsof NG.Wenotethat o(NG)=4,o(NH1)=3= o(NH2).Since 3 doesnotdivide 4,itfollowsthatLagranges’theoremdoesnothold.Nowconsiderthefollowing:
NH1 ∪ NH2 = {a,b,c,e} = NG. NH1 ∩ NH2 = {c,e}
Theseshowthat NH1 ∪ NH2 isaNeutroSubgroupof NG but NH1 ∩ NH2 isnotaNeutrosubgroupof NG Itishoweverobservedthat NH1 ∩ NH2 isagroupascanbeseenintheCayleytablebelow:
NH1 ∩ NH2 : ∗ c e c e c e c e
Definition3.6. Let (NG, ∗) beaNeutroGroupandlet a ∈ NG beafixedelement.
(i) Thecenterof NG denotedby Z(NG) isasetdefinedby Z(NG)= {x ∈ NG : xg = gx foratleastone g ∈ NG}
(ii) Thecentralizerof a ∈ G denotedby NCa isasetdefinedby NCa = {g ∈ NG : ga = ag}
Example3.7. Let (NG, ∗) betheNeutroGroupofExample3.2.Then:
(i) Z(NG)= {a,b,c,e} = NG.Thisshowsthat Z(NG) isaNeutroSubgroupof NG
(ii) NCa = {a,b,e}, NCb = {a,b,e}, NCc = {c,e} and NCe = {a,b,c,e}.Wehavethat NCa and NCb arenotNeutroSubgroupsof NG, NCe isagroupand NCc isaNeutroSubgroupof NG Doi:10.5281/zenodo.400660289
4CharacterizationofFiniteNeutroGroupsoftype-NG[1,2,4]
Inthissection,wearegoingtostudyfiniteNeutroGroupsoftype-NG[1,2,4]thatisNeutroGroups (NG, ∗) where G3 and G5 aretotallytrueforalltheelementsof NG andwhere G1, G2 and G4 areeitherpartially trueorpartiallyfalseforsomeelementsof NG.
Example4.1. Let NG = {1, 2, 3, 4}⊆ Z5 andlet ∗ beabinaryoperationon NG definedas
x ∗ y = x + y +4 ∀x,y ∈ NG.
Then (NG, ∗) isafiniteNeutroGroupoftype-NG[1,2,4]ascanbeseenintheCayleytable
∗ 1 2 3 4 1 1 2 3 4 2 2 3 4 0 3 3 4 0 1 4 4 0 1 2
Example4.2. Let NG = {1, 2, 3}⊆ Z4 andlet • beabinaryoperationdefinedon NG asshowninthe Cayleytable
1 2 3 1 1 2 3 2 2 0 2 3 3 2 1
Then (NG, •) isafiniteNeutroGroupoftype-NG[1,2,4].
Example4.3. Let NK = {1, 3, 5}⊆ Z8 andlet ◦ beabinaryoperationdefineon NK asshownintheCayley table
◦ 1 3 5 1 1 3 5 3 3 1 7 5 5 7 1
Then (NK, ◦) isafiniteNeutroGroupoftype-NG[1,2,4].
Example4.4. Let NG = {1, 2, 3, 4, 5}⊆ Z10 andlet ∗ beabinaryoperationdefineon NG asshowninthe Cayleytable ∗ 1 2 3 4 5 1 1 2 3 4 5 2 2 4 6 8 0 3 3 6 9 2 5 4 4 8 2 6 0 5 5 0 5 0 5
Then (NG, ∗) isafiniteNeutroGroupoftype-NG[1,2,4].
.
Example4.5. Let (NG, •) and (NK, ◦) beNeutroGroupsofExamples4.2and4.3respectively.Then NG × NG = {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}, NK × NK = {(1, 1), (1, 3), (1, 5), (3, 1), (3, 3), (3, 5), (5, 1), (5, 3), (5, 5)}, NG × NK = {(1, 1), (1, 3), (1, 5), (2, 1), (2, 3), (2, 5), (3, 1), (3, 3), (3, 5)}
Itcaneasilybeshownthat (NG × NG, •), (NK × NK, ◦) and (NG × NK, ✷) areNeutroGroupsoftypeNG[1,2,4].
Proposition4.6. Let (NG, •) and (NK, ◦) beanytwoNeutroGroupsofthesametype-NG[1,2,4].Then (NG × NG, •), (NK × NK, ◦) and (NG × NK, ✷) areNeutroGroupsoftype-NG[1,2,4].
Proof. Theproofiseasyandsoomitted.
Doi:10.5281/zenodo.400660290
Proposition4.7. Let (NG, ∗) beafiniteNeutroGroupoftype-NG[1,2,4].Theclassicallawsofindicesdonot hold.
Proof. Since ∗ isaNeutroLawand ∗ isNeutroAssociativeover NG,theresultfollows.
Corollary4.8. NoNeutroGroup (NG, ∗) oftype-NG[1,2,4]canbecyclicthatis,generatedbyanelement x ∈ NG.
Proposition4.9. Let (NG, ∗) beafiniteNeutroGroupoftype-NG[1,2,4].If x and y areinvertibleelements of NG,then
(i) x 1 1 = x (ii) (xy) 1 = x 1y 1 .
Proof. Obvious.
Example4.10. Let (NG, ∗) betheNeutroGroupofExample4.1andconsiderthefollowingsubsetsof NG NH1 = {1, 2},NH2 = {1, 3},NH3 = {1, 4},NH4 = {1, 2, 3},NH5 = {1, 2, 4},NH6 = {1, 3, 4}
Itcanbeshownthat (NHi, ∗) ,i =1, 2, 3, 4, 5, 6 areNeutroSubgroupsof NG.Nextconsiderthefollowing:
NH1 ∪ NH2 = NH1 ∪ NH4 = {1, 2, 3} [NeutroSubgroupsofNG]
NH1 ∪ NH3 = NH1 ∪ NH5 = NH3 ∪ NH5 = {1, 2, 4} [NeutroSubgroupsofNG] NH2 ∪ NH3 = NH2 ∪ NH6 = {1, 3, 4} [NeutroSubgroupsofNG].
NH3 ∪ NH4 = NH5 ∪ NH6 = {1, 2, 3, 4} [trivialNeutroSubgroupsofNG]
NH1 ∩ NH4 = NH1 ∩ NH5 = NH4 ∩ NH5 = {1, 2} [NeutroSubgroupsofNG]
NH2 ∩ NH4 = NH2 ∩ NH6 = NH4 ∩ NH6 = {1, 3} NeutroSubgroupsofNG] NH3 ∩ NH5 = NH3 ∩ NH6 = {1, 4} [NeutroSubgroupsofNG].
NH1 ∩ NH2 = NH2 ∩ NH3 = {1} [notNeutroSubgroupsofNG]
Example4.11. Let (NG, ∗) betheNeutroGroupofExample4.4andconsiderthefollowingsubsetsof NG.
NH1 = {1, 2, 4, 5},NH2 = {1, 2, 3, 5}.
Itcanbeshownthat (NHi, ∗) ,i =1, 2 areNeutroSubgroupsof NG.Nextconsiderthefollowing:
NH1 ∪ NH2 == {1, 2, 5} [aNeutroSubgroupofNG]. NH1 ∩ NH2 = {1, 2, 3, 4, 5} [atrivialNeutroSubgroupofNG]
Remark4.12. Examples4.10and4.11haveshownthatintheNeutroGroupsoftype-NG[1,2,4],wecanhave thefollowing:
(i) Lagrange’stheoremmayholdforsomeNeutroSubgroupsoftheNeutroGroupsandfailtoholdforsome NeutroSubgroups.
(ii) TheunionoftwoNeutroSubgroupsoftheNeutroGroupscanbeNeutroSubgroupsevenifoneisnot containedintheother.
(iii) TheintersectionoftwoNeutroSubgroupsoftheNeutroGroupscanbeNeutroSubgroups.
Definition4.13. Let NH beaNeutroSubgroupoftheNeutroGroup (NG, ∗) andlet x ∈ NG.
(i) xNH theleftcosetof NH in NG isdefinedby xNH = {xh : h ∈ NH}
(ii) Thenumberofdistinctleftcosetsof NH in NG iscalledtheindexof NH in NG denotedby [NG : NH]
Doi:10.5281/zenodo.400660291
(iii) Thesetofalldistinctleftcosetsof NH in NG denotedby NG/NH isdefinedby NG/NH = {xNH : x ∈ NG}
Example4.14. Let (NG, ∗) betheNeutroGroupofExample4.1andlet (NHi, ∗) ,i =1, 2, 3, 4, 5, 6 bethe NeutroSubgroupsofExample4.10.Theleftcosetsof NHi arecomputedasfollows.
1NH1 = {1, 2}, 2NH1 = {2, 3}, 3NH1 = {3, 4}, 4NH1 = {0, 4}
1NH2 = {1, 3}, 2NH2 = {2, 4}, 3NH2 = {0, 3}, 4NH2 = {1, 4}
1NH3 = {1, 4}, 2NH3 = {0, 2}, 3NH3 = {1, 3}, 4NH3 = {2, 4}.
1NH4 = {1, 2, 3}, 2NH4 = {2, 3, 4}, 3NH4 = {0, 3, 4}, 4NH4 = {0, 1, 4}
1NH5 = {1, 2, 4}, 2NH5 = {0, 2, 3}, 3NH5 = {1, 3, 4}, 4NH5 = {0, 2, 4}
1NH6 = {1, 3, 4}, 2NH6 = {0, 2, 4}, 3NH6 = {0, 1, 3}, 4NH6 = {1, 2, 4} ∴ NG/NHi = {1NHi, 2NHi, 3NHi, 4NHi},i =1, 2, 3, 4, 5, 6.
Example4.15. Let (NG, ∗) betheNeutroGroupofExample4.4andlet (NHi, ∗) ,i =1, 2 betheNeutroSubgroupsofExample4.11.Theleftcosetsof NHi arecomputedasfollows.
1NH1 = {1, 2, 4, 5}, 2NH1 = {0, 2, 4, 8}, 3NH1 = {2, 3, 5, 6}, 4NH1 = {0, 4, 6, 8}, 5NH1 = {0, 5} 1NH2 = {1, 2, 3, 5}, 2NH2 = {0, 2, 4, 6}, 3NH2 = {3, 5, 6, 9}, 4NH2 = {0, 2, 4, 8}, 5NH1 = {0, 5}. ∴ NG/NHi = {1NHi, 2NHi, 3NHi, 4NHi, 5NHi},i =1, 2
Lemma4.16. Let NH beaNeutroSubgroupoftheNeutroGroup (NG, ∗) oftype-NG[1,2,4]andlet x ∈ NG. Then, xNH = NH ifandonlyif x = e where e istheidentityelementin NG.
Proof. Obvious.
Remark4.17. Examples4.14and4.15haveshownthatintheNeutroGroupsoftype-NG[1,2,4]distinctleft cosetsofNeutroSubgroupsintheNeutroGroupsdonotnecessarilypartitiontheNeutroGroups. Let NH beaNeutroSubgroupofaNeutroGroup (NG, ∗) oftype-NG[1,2,4]andlet NG/NH bethesetof distinctleftcosetsof NH in NG.For xNH,yNH ∈ NG/NH with x,y ∈ NG,let beabinaryoperation definedon NG/NH by xNH yNH = xyNH ∀ x,y ∈ NG.
Wewanttoinvestigateifthecouple (NG/NH, ) isaNeutroGroupoftype-NG[1,2,4]usingthefollowing examples.
Example4.18. Let NG/NHi = {1NHi, 2NHi, 3NHi, 4NHi},i =1, 2, 3, 4, 5, 6 beasgiveninExample 4.14.For i =1,wehave
1NH1 2NH1 3NH1 4NH1 1NH1 1NH1 2NH1 3NH1 4NH1 2NH1 2NH1 3NH1 4NH1 0NH1 3NH1 3NH1 4NH1 0NH1 1NH1 4NH1 4NH1 0NH1 1NH1 2NH1
ItisclearfromtheCayleytablethat (NG/NH1, ) isaNeutroGroupoftype-NG[1,2,4]with 1NH1 asthe identityelement.Thisisalsotruefor i =2, 3, 4, 5, 6.
Example4.19. Let NG/NHi = {1NHi, 2NHi, 3NHi, 4NHi, 5NHi},i =1, 2 beasgiveninExample 4.15.For i =1,wehave
1NH1 2NH1 3NH1 4NH1 5NH1 1NH1 1NH1 2NH1 3NH1 4NH1 5NH1 2NH1 2NH1 4NH1 6NH1 8NH1 0NH1 3NH1 3NH1 6NH1 9NH1 2NH1 5NH1 4NH1 4NH1 8NH1 2NH1 6NH1 0NH1 5NH1 5NH1 0NH1 5NH1 0NH1 5NH1
.
ItisevidentfromtheCayleytablethat (NG/NH1, ) isaNeutroGroupoftype-NG[1,2,4]with 1NH1 asthe identityelement..Thisisalsotruefor i =2
Doi:10.5281/zenodo.400660292
Remark4.20. ItisevidentfromExamples4.18and4.19thatif NH isaNeutroSubgroupoftheNeutroGroup oftype-NG[1,2,4],then NG/NH thesetofdistinctleftcosetscanbemadeaNeutroGroupoftype-NG[1,2,4] bydefiningappropriatebinaryoperation on NG/NH.TheNeutroGroup NG/NH iscalledtheQuotientNeutroGroupof NG factoredby NH.
Definition4.21. Let (NG, ∗) and (NH, ◦) beanytwoNeutroGroupsoftype-NG[1,2,4].Themapping φ : NG → NH iscalledaNeutroGroupHomomorphismif φ preservesthebinaryoperations ∗ and ◦ thatisiffor atleastaduplet (x,y) ∈ G,wehave
φ(x ∗ y)= φ(x) ◦ φ(y)
TheKernelof φ denotedby Kerφ isdefinedby
Kerφ = {x : φ(x)= eNH }
where eNH istheidentityelementin NH.
TheImageof φ denotedby Imφ isdefinedby
Imφ = {y ∈ H : y = φ(x) forsome h ∈ NH}
Ifinaddition φ isaNeutroBijection,then φ iscalledaNeutroGroupIsomorphismandwewrite NG ∼ = NH NeutroGroupEpimorphism,NeutroGroupMonomorphism,NeutroGroupEndomorphism,andNeutroGroupAutomorphismaresimilarlydefined.
Example4.22. Let (NG, •) betheNeutroGroupofExample4.2andlet φ : NG×NG → NG beaprojection givenby ψ(x,y)= y ∀x,y ∈ NG.
Then φ(1, 1)= φ(2, 1)= φ(3, 1)=1,φ(1, 2)= φ(2, 2)= φ(3, 2)=2,φ(1, 3)= φ(2, 3)= φ(3, 3)=3
Since φ((2, 1)(3, 3))= φ(2, 3)=3 and φ(2, 1)φ(3, 3)=1 • 3=3 but φ((2, 2)(2, 3))= φ(0, 2)=? and φ(2, 2)φ(2, 3)=2 • 3=2,itfollowsthat φ isaNeutroGroupHomomorphism. Imφ = {1, 2, 3} = NG and Kerφ = {(1, 1), (2, 1), (3, 1)}.The Kerφ isaNeutroSubgroupof NG × NG ascanbeseeninthefollowing Cayleytable
• (1, 1) (2, 1) (3, 1) (1, 1) (1, 1) (2, 1) (3, 1) (2, 1) (2, 1) (0, 1) (2, 1) (3, 1) (3, 1) (2, 1) (1, 1)
Itisevidentfromthetablethat (Kerφ, •) isaNeutroGroupoftype-NG[1,2,4]andsince Kerφ ⊆ NG × NG, itfollowsthat ker φ isaNeuroSubgroup.
Example4.23. Let (NK, ◦) betheNeutroGroupofExample4.3andlet ψ : NK × NK → NK bea projectiongivenby ψ(x,y)= x ∀x,y ∈ NK. Then
ψ(1, 1)= ψ(1, 3)= ψ(1, 5)=1,ψ(3, 1)= ψ(3, 3)= ψ(3, 5)=3,ψ(5, 1)= ψ(5, 3)= ψ(5, 5)=5.
Since ψ((1, 1)(1, 3))= ψ(1, 3)=1 and ψ(1, 1)ψ(1, 3)=1 ◦ 1=1 but ψ((1, 5)(5, 3))= ψ(5, 7)=? and ψ(1, 5)ψ(5, 3)=1 ◦ 5=5,itfollowsthat ψ isaNeutroGroupHomomorphism. Imψ = {1, 3, 5} = NK and Kerψ = {(1, 1), (1, 3), (1, 5)}.The Kerψ isaNeutroSubgroupof NK × NK ascanbeseeninthe followingCayleytable ◦ (1, 1) (1, 3) (1, 5) (1, 1) (1, 1) (1, 3) (1, 5) (1, 3) (1, 3) (1, 1) (1, 7) (1, 5) (1, 5) (1, 7) (1, 1) . Itisevidentfromthetablethat (Kerψ, ◦) isaNeutroGroupoftype-NG[1,2,4]andsince Kerψ ⊆ NK ×NK, itfollowsthat ker ψ isaNeuroSubgroup. Doi:10.5281/zenodo.400660293
Example4.24. Let NG/NHi = {1NHi, 2NHi, 3NHi, 4NHi},i =1, 2, 3, 4, 5, 6 betheNeutroQuotientGroupofExample4.18.For i =1,let ψ : NG → NG/NH1 beamappingdefinedby ψ(x)= xNH1 ∀x ∈ NG.
FromExample4.14wehave
ψ(1)=1NH1 = {1, 2},ψ(2)=2NH1 = {2, 3},ψ(3)=3NH1 = {3, 4},ψ(4)=4NH1 = {0, 4} Next, ψ(2 ∗ 3)= ψ(4)=4NH1 = {0, 4}, ψ(2) ψ(3)=2NH1 3NH1 =2 ∗ 3NH1 =4NH1 = {0, 4} butthen, ψ(2 ∗ 4)= ψ(0)=? ψ(2) ψ(4)=2NH1 4NH1 =2 ∗ 4NH1 =0NH1 =?
Thisshowsthat ψ isaNeutroGroupHomomorphism.The Kerψ =1NH1 = {1, 2} theidentityelementof NG/NH1 Example4.25. Let NG/NHi = {1NHi, 2NHi, 3NHi, 4NHi, 5NHi},i =1, 2 betheNeutroQuotientGroupofExample4.19.For i =1,let φ : NG → NG/NH1 beamappingdefinedby φ(x)= xNH1 ∀x ∈ NG.
FromExample4.15wehave
ψ(1)=1NH1 = {1, 2, 4, 5},φ(2)=2NH1 = {0, 2, 4, 8},φ(3)=3NH1 = {2, 3, 5, 6}, φ(4)=4NH1 = {0, 4, 6, 8},φ(5)=5NH1 = {0, 5} Next, φ(3 ∗ 5)= φ(5)=5NH1 = {0, 5}, φ(3) φ(5)=3NH1 5NH1 =3 ∗ 5NH1 =5NH1 = {0, 5} butthen, φ(3 ∗ 2)= φ(6)=? φ(3) φ(2)=3NH1 2NH1 =3 ∗ 2NH1 =6NH1 =?. Thisshowsthat ψ isaNeutroGroupHomomorphism.The Kerψ =1NH1 = {1, 2, 4, 5} theidentityelement of NG/NH1 Proposition4.26. Let (NG, ∗) and (NH, ◦) beNeutroGroupsoftype-NG[1,2,4]andlet eNG and eNH be identityelementsin NG and NH respectively.Supposethat φ : NG → NH isaNeutroGroupHomomorphism.Then:
(i) φ(eNG)= eNH
(ii) φ x 1 =(φ(x)) 1 foreveryinvertibleelement x ∈ NG
(iii) Kerφ isaNeutroSubgroupof NG (iv) Imφ isaNeutroSubgroupof NH .
(v) φ isNeutroInjectiveifandonlyif Kerφ = {eNG}
Proof. Thesameasfortheclassicalgroupsandsoomitted.
Proposition4.27. Let NH beaNeutroSubgroupofaNeutroGroup (NG, ∗) oftype-NG[1,2,4].Themapping ψ : NG → NG/NH definedby ψ(x)= xNH ∀ x ∈ NG
isaNeutroGroupHomomorphismandthe Kerψ = NH
Proof. Thesameasfortheclassicalgroupsandsoomitted.
Proposition4.28. Let φ : NG → NH beaNeutroGroupHomomorpismandlet NK = Kerφ.Thenthe mapping ψ : NG/NK → Imφ definedby ψ(xNK)= φ(x) ∀ x ∈ NG
isaNeutroGroupIsomorphism.
Proof. Thesameasfortheclassicalgroupsandsoomitted.
Doi:10.5281/zenodo.400660294
5Conclusion
WehaveinthisworkstudiedaclassofNeutroGroups (NG, ∗) oftype-NG[1,2,4].InthisclassofNeutroGroups,theclosurelaw,theaxiomofassociativityandexistenceofinverseweretakingtobeeitherpartiallytrueorpartiallyfalseforsomeelementsof NG;whiletheexistenceofidentityelementandaxiomof commutativityweretakingtobetotallytrueforalltheelementsof NG.SeveralexamplesofNeutroGroups oftype-NG[1,2,4]werepresentedalongwiththeirbasicproperties.ItwasshownthatLagrange’stheorem holdsforsomeNeutroSubgroupsofaNeutroGroupandfailedtoholdforsomeNeutroSubgroupsofthesame NeutroGroup.ItwasalsoshownthattheunionoftwoNeutroSubgroupsofaNeutroGroupcanbeaNeutroSubgroupevenifoneisnotcontainedintheother;andthattheintersectionoftwoNeutroSubgroupsmay notbeaNeutroSubgroup.TheconceptsofNeutroQuotientGroupsandNeutroGroupHomomorphismswere presentedandstudied.Itwasshownthatthefundamentalhomomorphismtheoremoftheclassicalgroupsis holdingintheclassofNeutroGroupsoftype-NG[1,2,4].WehopetostudyAntiGroups,revisitNeutroRings, studyAntiRings,NeutroVectorSpaces,AntiVectorSpaces,NeutroModules,AntiModules,NeutroHypergroups, AntiHypergroups,NeutroHyperrings,AntiHyperrings,NeutroHypervectorSpacesandAntiHypervectorSpaces inourfuturepapers.
6Acknowledgment
TheauthorisverygratefultoProfessorFlorentinSmarandacheforhisprivatediscussions,commentsand suggestionsduringthepreparationofthiswork.
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