IntroductiontoNeutroGroups
Agboola1 A.A.A. DepartmentofMathematics,FederalUniversityofAgriculture,Abeokuta,Nigeria. agboolaaaa@funaab.edu.ng
Abstract
TheobjectiveofthispaperistoformallypresenttheconceptofNeutroGroupsbyconsideringthreeNeutroAxioms(NeutroAssociativity,existenceofNeutroNeutralelementandexistenceofNeutroInverseelement).SeveralinterestingresultsandexamplesofNeutroGroups,NeutroSubgroups,NeutroCyclicGroups,NeutroQuotientGroupsandNeutroGroupHomomorphismsarepresented.Itisshownthatgenerally,Lagrange’stheorem and1stisomorphismtheoremoftheclassicalgroupsdonotholdintheclassofNeutroGroups.
Keywords: Neutrosophy,NeutroGroup,NeutroSubgroup,NeutroCyclicGroup,NeutroQuotientGroupand NeutroGroupHomomorphism.
1Introduction
In2019,FlorentinSmarandache2 introducednewfieldsofresearchinneutrosophywhichhecalledNeutroStructuresandAntiStructuresrespectively.TheconceptsofNeutroAlgebrasandAntiAlgebraswererecentlyintroducedbySmarandachein.3 Smarandachein4 revisitedthenotionsofNeutroAlgebrasandAntiAlgebraswherehestudiedPartialAlgebras,UniversalAlgebras,EffectAlgebrasandBoole’sPartialAlgebras andshowedthatNeutroAlgebrasaregeneralizationofPartialAlgebras.In,1 AgboolaetalexaminedNeutroAlgebrasandAntiAlgebrasviz-a-viztheclassicalnumbersystems N, Z, Q, R and C.ThementionofNeutroGroupbySmarandachein2 motivatedustowritethepresentpaper.TheconceptofNeutroGroupisformally presentedinthispaperbyconsideringthreeNeutroAxioms(NeutroAssociativity,existenceofNeutroNeutral elementandexistenceofNeutroInverseelement).WestudyNeutroSubgroups,NeutroCyclicGroups,NeutroQuotientGroupsandNeutroGroupHomomorphisms.Wepresentseveralinterestingresultsandexamples.Itis shownthatgenerally,Lagrange’stheoremand1stisomorphismtheoremoftheclassicalgroupsdonotholdin theclassofNeutroGroups.
FormoredetailsaboutNeutroAlgebras,AntiAlgebras,NeutroAlgebraicStructuresandAntiAlgebraic Structures,thereadersshouldsee.1–4
2FormalPresentationofNeutroGroupandProperties
Inthissection,weformallypresenttheconceptofaNeutroGroupbyconsideringthreeNeutroAxioms(NeutroAssociativity,existenceofNeutroNeutralelementandexistenceofNeutroInverseelement)andwepresent itsbasicproperties.
(ii) ThereexistsaNeutroNeutralelementin G thatisthereexistsatleastanelement a ∈ G thathasasingle neutralelementthatiswehave e ∈ G suchthat
a ∗ e = e ∗ a = a (3) andfor b ∈ G theredoesnotexist e ∈ G suchthat b ∗ e = e ∗ b = b (4) orthereexist e1,e2 ∈ G suchthat b ∗ e1 = e1 ∗ b = b or (5) b ∗ e2 = e2 ∗ b = b with e1 = e2 (6)
(iii) ThereexistsaNeutroInverseelementthatisthereexistsanelement a ∈ G thathasaninverse b ∈ G withrespecttoaunitelement e ∈ G thatis
a ∗ b = b ∗ a = e (7) orthereexistsatleastoneelement b ∈ G thathastwoormoreinverses c,d ∈ G withrespecttosome unitelement u ∈ G thatis
b ∗ c = c ∗ b = u (8) b ∗ d = d ∗ b = u. (9)
Inaddition,if ∗ isNeutroCommutativethatisthereexistsatleastaduplet (a,b) ∈ G suchthat a ∗ b = b ∗ a (10) andthereexistsatleastaduplet (c,d) ∈ G suchthat c ∗ d = d ∗ c, (11) then (G, ∗) iscalledaNeutroCommutativeGrouporaNeutroAbelianGroup. Ifonlycondition(i)issatisfied,then (G, ∗) iscalledaNeutroSemiGroupandifonlyconditions(i)and(ii) aresatisfied,then (G, ∗) iscalledaNeutroMonoid.
Definition2.2. Let (G, ∗) beaNeutroGroup. G issaidtobefiniteoforder n ifthecardinalityof G is n that is o(G)= n.Otherwise, G iscalledaninfiniteNeutroGroupandwewrite o(G)= ∞ Example2.3. Let U = {a,b,c,d,e,f } beauniverseofdiscourseandlet G = {a,b,c,d} beasubsetof U Let ∗ beabinaryoperationdefinedon G asshownintheCayleytablebelow: ∗ a b c d a b c d a b c d a c c d a b d d a b c a
Itisclearfromthetablethat: a ∗ (b ∗ c)=(a ∗ b) ∗ c = d, b ∗ (d ∗ c)= a, but (b ∗ d) ∗ c = b.
Thisshowsthat ∗ isNeutroAssociativeandhence (G, ∗) isaNeutroSemiGroup. Next,let Nx and Ix representtheneutralelementandtheinverseelementrespectivelywithrespecttoany element x ∈ G.Then
Na = d, Ia = c, Nb,Nc,Nd donotexist, Ib,Ic,Id donotexist
Thisinadditionto (G, ∗) beingaNeutroSemiGroupimpliesthat (G, ∗) isaNeutroGroup. Itisalsoclearfromthetablethat ∗ isNeutroCommutative.Hence, (G, ∗) isaNeutroAbelianGroup.
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Example2.4. Let G = Z10 andlet ∗ beabinaryoperationon G definedby x ∗ y = x +2y forall x,y ∈ G where + isadditionmodulo10.Then (G, ∗) isaNeutroAbeliaGroup.Toseethis,for x,y,z ∈ G,wehave x ∗ (y ∗ z)= x +2y +4z, (12) (x ∗ y) ∗ z = x +2y +2z. (13)
Equating(16)and(17)weobtain z =0, 5.Henceonlythetriplets (x,y, 0), (x,y, 5) canverifyassociativityof ∗ andnotanyothertriplet (x,y,z) ∈ G.Hence, ∗ isNeutroAssociativeandtherefore, (G, ∗) isa NeutroSemigroup.
Next,let e ∈ G suchthat x ∗ e = x +2e = x and e ∗ x = e +2x.Then, x +2e = e +2x fromwhichwe obtain e = x.Butthen,only 5 ∗ 5=5 in G.Thisshowsthat G hasaNeutroNeutralelement.Itcanalsobe shownthat G hasaNeutroInverseelement.Hence,(G, ∗) isaNeutroGroup.
Lastly, x ∗ y = x +2y, (14) y ∗ x = y +2x. (15)
Equating(18)and(19),weobtain x = y.Henceonlytheduplet (x,x) ∈ G canverifycommutativityof ∗ and notanyotherduplet (x,y) ∈ G.Hence, ∗ isNeutroCommutativeandthus (G, ∗) isaNeutroAbelianGroup.
Remark2.5. GeneralNeutroGroupisaparticularcaseofgeneralNeutroAlgebrawhichisanalgebrawhich hasatleastoneNeutroOperationoroneNeutroAxiom(axiomthatistrueforsomeelements,indeterminate forotherelements,andfalsefortheotherelements).Therefore,aNeutroGroupisagroupthathaseitherone NeutroOperation(partiallywell-defined,partiallyindeterminate,andpartiallyouter-defined),oratleastone NeutroAxiom(NeutroAssociativity,NeutroElement,orNeutroInverse).
ItispossibletodefineNeutroGroupinanotherwaybyconsideringonlyoneNeutroAxiomorbyconsideringtwoNeutroAxioms.
Theorem2.6. Let (Gi, ∗),i =1, 2, ,n beafamilyofNeutroGroups.Then
(i) G = n i=1 Gi isaNeutroGroup. (ii) G = n i=1 Gi isaNeutroGroup.
Proof. Obvious.
Definition2.7. Let (G, ∗) beaNeutroGroup.Anonemptysubset H of G iscalledaNeutroSubgroupof G if (H, ∗) isalsoaNeutroGroup.
TheonlytrivialNeutroSubgroupof G is G Example2.8. Let (G, ∗) betheNeutroGroupof Example2.3 andlet H = {a,c,d}.Thecompositionsof elementsof H aregivenintheCayleytablebelow. ∗ a c d a b d a c d b d d a c a
.
Itisclearfromthetablethat: a ∗ (c ∗ d)=(a ∗ c) ∗ d = a, d ∗ (c ∗ a)= a, but (d ∗ c) ∗ a = d = a. Na = d, Ia = c, Nc,Nd donotexist, Ic,Id donotexist. a ∗ d = d ∗ a = a, but c ∗ d = d,d ∗ c = c = d.
Alltheseshowthat (H, ∗) isaNeutroAbelianGroup.Since H ⊂ G,itfollowsthat H isaNeutroSubgroupof G.
Itshouldbeobservedthattheorderof H isnotadivisoroftheorderof G.Hence,Lagrange’stheorem doesnothold.
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Theorem2.9. Let (G, ∗) beaNeutroGroupandlet (Hi, ∗),i =1, 2, ,n beafamilyofNeutroSubgroups of G.Then
(i) H = n i=1 Hi isaNeutroSubgroupof G
(ii) H = n i=1 Hi isaNeutroSubgroupof G
Proof. Obvious. Definition2.10. Let H beaNeutroSubgroupoftheNeutroGroup (G, ∗) andlet x ∈ G
(i) xH theleftcosetof H in G isdefinedby
xH = {xh : h ∈ H}. (16)
(i) Hx therightcosetof H in G isdefinedby
Hx = {hx : h ∈ H}. (17)
Example2.11. Let (G, ∗) betheNeutroGroupof Example2.3 andlet H betheNeutroSubgroupof Example 2.8 Hl thesetofdistinctleftcosetsof H in G isgivenby
Hl = {{a,b,d}, {a,c}, {b,d}} and Hr thesetofdistinctrightcosetsof H in G isgivenby
Hr = {{a,b,d}, {a,b,c}, {b,c,d}, {a,d}}
Itshouldbeobservedthat Hl and Hr donotpartition G.Thisisdifferentfromwhatisobtainableinthe classicalgroups.However,theorderof Hl is 3 whichisnotadivisoroftheorderof G andtherefore, [G : H ] theindexof H in G is3,thatis [G : H ]=3.Also,theorderof Hr is 4 whichisadivisorof G.Hence, [G : H ]=4
Example2.12. Let U = {a,b,c,d} beauniverseofdiscourseandlet G = {a,b,c} beaNeutroGroupgiven intheCayleytablebelow:
∗ a b c a a c b b c a c c a c d
Let H = {a,b} beaNeutroSubgroupof G givenintheCayleytablebelow: ∗ a b a a c b c a
Then,thesetsofdistinctleftandrightcosetsof H in G arerespectivelyobtainedas: Hl = {{a,c}}, Hr = {{a, }, {c}, {b,c}}
Inthisexample,theorderof Hl thesetofdistinctleftcosetsof H in G is 1 whichisnotadivisoroftheorder of G andtherefore, [G : H ]=1.However,theorderof Hr thesetofdistinctrightcosetsof H in G is 3 which isadivisoroftheorderof G andtherefore, [G : H ]=3.Thisisalsodifferentfromwhatisobtainableinthe classicalgroups.
Consequenton Examples2.8,2.11and2.12,westatethefollowingtheorem:
Theorem2.13. Let H beaNeutroSubgroupofthefiniteNeutroGroup (G, ∗).Thengenerally:
(i) o(H ) isnotadivisorof o(G).
(ii) Thereisno1-1correspondencebetweenanytwoleft(right)cosetsof H in G Doi:10.5281/zenodo.384076144
(iii) Thereisno1-1correspondencebetweenanyleft(right)cosetof H in G and H
(iv) If Nx = e thatis xe = ex = x forany x ∈ G,then eH = H , He = H and {e} isnotaNeutroSubgroup of G.
(v) o(G) =[G : H ]o(H )
(vi) Thesetofdistinctleft(right)cosetsof H in G isnotapartitionof G
Definition2.14. Let (G, ∗) beaNeuroGroup.Since ∗ isassociativeforatleastonetriplet (x,x,x) ∈ G,the powersof x aredefinedasfollows:
x 1 = x x 2 = xx x 3 = xxx . . xn = xxx xn factors ∀n ∈ N.
Theorem2.15. Let (G, ∗) beaNeutroGroupandlet x ∈ G.Thenforany m,n ∈ N,wehave:
(i) xmxn = xm+n (ii) (xm)n = xmn
Definition2.16. Let (G, ∗) beaNeutroGroup. G issaidtobecyclicif G canbegeneratedbyanelement x ∈ G thatis
G =<x>= {xn : n ∈ N} (18)
Example2.17. Let (G, ∗) betheNeutroGroupgivenin Example2.3 andconsiderthefollowing:
a 1 = a,a 2 = b,a3 = c,a 4 = d. b1 = b,b2 = d,b3 = c,b4 = a. c 1 = c,c 2 = b,c3 = a,c 4 = d. d1 = d,d2 = a,d3 = a,d4 = a. ∴ G = <a>=<b>=<c>,G =<d>.
Theseshowthat G iscyclicwiththegenerators a,b,c.Theelement d ∈ G doesnotgenerate G Definition2.18. Let H beaNeutroSubgroupoftheNeutroGroup (G, ∗).Thesets (G/H )l and (G/H )r are definedby: (G/H )l = {xH : x ∈ G} (19) (G/H )r = {Hx : x ∈ G}. (20)
Let xH,yH ∈ (G/H )l andlet l beabinaryoperationdefinedon (G/H )l by xH l yH = x ∗ yH ∀ x,y ∈ G. (21)
Also,let xH,yH ∈ (G/H )r andlet r beabinaryoperationdefinedon (G/H )r by Hx r Hy = Hx ∗ y ∀ x,y ∈ G. (22)
Itcanbeshownthatthecouples ((G/H )l, l) and ((G/H )r , r ) areNeutroGroups. Example2.19. Let G and H beasgivenin Example2.12 andconsider (G/H )l = {aH,bH,cH} = {aH} = {{a,c}}.
Then ((G/H )l, l) isaNeutroGroup.
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Definition2.20. Let (G, ∗) and (H, ◦) beanytwoNeutroGroups.Themapping φ : G → H iscalleda NeutroGroupHomomorphismif φ preservesthebinaryoperations ∗ and ◦ thatisifforall x,y ∈ G,wehave
φ(x ∗ y)= φ(x) ◦ φ(y) (23)
Thekernelof φ denotedby Kerφ isdefinedas
Kerφ = {x : φ(x)= eH } (24)
where eH ∈ H issuchthat Nh = eH foratleastone h ∈ H Theimageof φ denotedby Imφ isdefinedas
Imφ = {y ∈ H : y = φ(x) forsome h ∈ H} (25)
Ifinaddition φ isabijection,then φ iscalledaNeutroGroupIsomorphismandwewrite G ∼ = H.NeutroGroupEpimorphism,NeutroGroupMonomorphism,NeutroGroupEndomorphism,andNeutroGroupAutomorphismaresimilarlydefined.
Example2.21. Let (G, ∗) beaNeutroGroupof Example2.12 andlet ψ : G × G → G beaprojectiongiven by ψ((x,y))= x ∀x,y ∈ G.
Then ψ isaNeutroGroupHomomorphism.The Kerψ = {(a,a), (a,b),a,c)} whichisaNeutroSubgroupof G × G asshownintheCayleytablebelow.
∗ (a,a) (a,b) (a,c) (a,a) (a,a) (a,c) (a,b) (a,b) (a,c) (a,a) (a,c) (a,c) (a,a) (a,c) (a,d) and Imψ = {a,b,c} = G
Consequenton Example2.21,westatethefollowingtheorem:
Theorem2.22. Let (G, ∗) and (H, ◦) beNeutroGroupsandlet Nx = eG suchthat eG ∗ x = x ∗ eG = x for atleastone x ∈ G andlet Ny = eH suchthat eH ∗ y = y ∗ eH = y foratleastone y ∈ H .Supposethat φ : G → H isaNeutroGroupHomomorphism.Then:
(i) φ(eG)= eH
(ii) Kerφ isaNeutroSubgroupof G
(iii) Imφ isaNeutroSubgroupofH.
(iv) φ isinjectiveifandonlyif Kerφ = {eG}
Example2.23. Considering Example2.19,let φ : G → G/H beamappingdefinedby φ(x)= xH forall x ∈ G.Then, φ(a)= φ(b)= φ(c)= aH = {a,c} fromwhichwehavethat φ isaNeutroGroupHomomorphism. Kerφ = {x ∈ G : φ(x)= eG/H } = {x ∈ G : xH = eG/H = e{{a,c}} } = H. Consequenton Example2.23,westatethefollowingtheorem:
Theorem2.24. Let H beaNeutroSubgroupofaNeutroGroup (G, ∗).Themapping ψ : G → G/H defined by ψ(x)= xH ∀ x ∈ G isaNeutroGroupHomomorphismandthe Kerψ = H .
Theorem2.25. Let φ : G → H beaNeutroGroupHomomorpismandlet K = Kerφ.Thenthemapping ψ : G/K → Imφ definedby ψ(xK)= φ(x) ∀ x ∈ G isaNeutroGroupEpimorphismandnotaNeutroGroupIsomorphism. Doi:10.5281/zenodo.384076146
Proof. That ψ isawelldefinedsurjectivemappingisclear.Let xK,yK ∈ G/K bearbitrary.Then
ψ(xKyK)= ψ(xyK)
= φ(xy)
= φ(x)φ(y)
= ψ(xK)ψ(yK)
Kerψ = {xK ∈ G/K : ψ(xK)= eφ(x)}
= {xK ∈ G/K : φ(x)= eφ(x)}
= {eG/K }
Thisshowsthat ψ isasurjectiveNeutroGroupHomomorphismandthereforeitisaNeutroGroupEpimorphism. Since ψ isnotinjective,itfollowsthat G/K ∼ = Imφ whichisdifferentfromwhatisobtainableintheclassical groups.
Theorem2.26. NeutroGroupIsomorphismofNeutrogroupsisanequivalencerelation.
Proof. Thesameastheclassicalgroups.
3Conclusion
WehaveformallypresentedtheconceptofNeutroGroupinthispaperbyconsideringthreeNeutroAxioms (NeutroAssociativity,existenceofNeutroNeutralelementandexistenceofNeutroInverseelement).WepresentedandstudiedseveralinterestingresultsandexamplesonNeutroSubgroups,NeutroCyclicGroups,NeutroQuotientGroupsandNeutroGroupHomomorphisms.Wehaveshownthatgenerally,Lagrange’stheorem and1stisomorphismtheoremoftheclassicalgroupsdonotholdintheclassofNeutroGroups.Furtherstudies ofNeutroGroupswillbepresentedinourfuturepapers.OtherNeutroAlgebraicStructuressuchasNeutroRings,NeutroModules,NeutroVectorSpacesetcareopenedtostudiesforNeutrosophicResearchers.
4Appreciation
Theauthorisverygratefultoallanonymousreviewersforvaluablecommentsandsuggestionswhichwehave foundveryusefulintheimprovementofthepaper.
References
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[2] Smarandache,F.,”IntroductiontoNeutroAlgebraicStructures,inAdvancesofStandardandNonstandard NeutrosophicTheories,”PonsPublishingHouseBrussels,Belgium,Ch.6,pp.240-265,2019.
[3] Smarandache,F.,”IntroductiontoNeutroAlgebraicStructuresandAntiAlgebraicStructures(revisited)”, NeutrosophicSetsandSystems,vol.31,pp.1-16,2020.DOI:10.5281/zenodo.3638232.
[4] Smarandache,F.,”NeutroAlgebraisaGeneralizationofPartialAlgebra”,InternationalJournalofNeutrosophicScience,vol.2(1),pp.08-17,2020.
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