Table Of ContentInternationalJournalofNeutrosophicScience(IJNS) Vol. 6,No. 1,PP.41-47,2020
Introduction to NeutroGroups
Agboola1A.A.A.
DepartmentofMathematics,FederalUniversityofAgriculture,Abeokuta,Nigeria.
[email protected]
Abstract
TheobjectiveofthispaperistoformallypresenttheconceptofNeutroGroupsbyconsideringthreeNeutroAx-
ioms(NeutroAssociativity,existenceofNeutroNeutralelementandexistenceofNeutroInverseelement). Sev-
eral interesting results and examples of NeutroGroups, NeutroSubgroups, NeutroCyclicGroups, NeutroQuo-
tientGroupsandNeutroGroupHomomorphismsarepresented. Itisshownthatgenerally,Lagrange’stheorem
and1stisomorphismtheoremoftheclassicalgroupsdonotholdintheclassofNeutroGroups.
Keywords: Neutrosophy, NeutroGroup, NeutroSubgroup, NeutroCyclicGroup, NeutroQuotientGroup and
NeutroGroupHomomorphism.
1 Introduction
In 2019, Florentin Smarandache2 introduced new fields of research in neutrosophy which he called Neu-
troStructures and AntiStructures respectively. The concepts of NeutroAlgebras and AntiAlgebras were re-
centlyintroducedbySmarandachein.3 Smarandachein4 revisitedthenotionsofNeutroAlgebrasandAntiAl-
gebras where he studied Partial Algebras, Universal Algebras, Effect Algebras and Boole’s Partial Algebras
and showed that NeutroAlgebras are generalization of Partial Algebras. In,1 Agboola et al examined Neu-
troAlgebrasandAntiAlgebrasviz-a-viztheclassicalnumbersystemsN,Z,Q,RandC. ThementionofNeu-
troGroupbySmarandachein2motivatedustowritethepresentpaper.TheconceptofNeutroGroupisformally
presentedinthispaperbyconsideringthreeNeutroAxioms(NeutroAssociativity,existenceofNeutroNeutral
elementandexistenceofNeutroInverseelement). WestudyNeutroSubgroups,NeutroCyclicGroups,Neutro-
QuotientGroupsandNeutroGroupHomomorphisms. Wepresentseveralinterestingresultsandexamples. Itis
shownthatgenerally,Lagrange’stheoremand1stisomorphismtheoremoftheclassicalgroupsdonotholdin
theclassofNeutroGroups.
For more details about NeutroAlgebras, AntiAlgebras, NeutroAlgebraic Structures and AntiAlgebraic
Structures,thereadersshouldsee.1–4
2 Formal Presentation of NeutroGroup and Properties
Inthissection,weformallypresenttheconceptofaNeutroGroupbyconsideringthreeNeutroAxioms(Neu-
troAssociativity,existenceofNeutroNeutralelementandexistenceofNeutroInverseelement)andwepresent
itsbasicproperties.
Definition 2.1. Let G be a nonempty set and let ∗ : G×G → G be a binary operation on G. The couple
(G,∗)iscalledaNeutroGroupifthefollowingconditionsaresatisfied:
(i) ∗isNeutroAssociativethatisthereexitsatleastonetriplet(a,b,c)∈Gsuchthat
a∗(b∗c)=(a∗b)∗c (1)
andthereexistsatleastonetriplet(x,y,z)∈Gsuchthat
x∗(y∗z)(cid:54)=(x∗y)∗z. (2)
1Correspondence:[email protected]
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(ii) ThereexistsaNeutroNeutralelementinGthatisthereexistsatleastanelementa∈Gthathasasingle
neutralelementthatiswehavee∈Gsuchthat
a∗e=e∗a=a (3)
andforb∈Gtheredoesnotexiste∈Gsuchthat
b∗e=e∗b=b (4)
orthereexiste ,e ∈Gsuchthat
1 2
b∗e = e ∗b=b or (5)
1 1
b∗e = e ∗b=b with e (cid:54)=e (6)
2 2 1 2
(iii) There exists a NeutroInverse element that is there exists an element a ∈ G that has an inverse b ∈ G
withrespecttoaunitelemente∈Gthatis
a∗b=b∗a=e (7)
orthereexistsatleastoneelementb ∈ Gthathastwoormoreinversesc,d ∈ Gwithrespecttosome
unitelementu∈Gthatis
b∗c = c∗b=u (8)
b∗d = d∗b=u. (9)
Inaddition,if∗isNeutroCommutativethatisthereexistsatleastaduplet(a,b)∈Gsuchthat
a∗b=b∗a (10)
andthereexistsatleastaduplet(c,d)∈Gsuchthat
c∗d(cid:54)=d∗c, (11)
then(G,∗)iscalledaNeutroCommutativeGrouporaNeutroAbelianGroup.
Ifonlycondition(i)issatisfied,then(G,∗)iscalledaNeutroSemiGroupandifonlyconditions(i)and(ii)
aresatisfied,then(G,∗)iscalledaNeutroMonoid.
Definition2.2. Let(G,∗)beaNeutroGroup. GissaidtobefiniteofordernifthecardinalityofGisnthat
iso(G)=n. Otherwise,GiscalledaninfiniteNeutroGroupandwewriteo(G)=∞.
Example2.3. LetU = {a,b,c,d,e,f}beauniverseofdiscourseandletG = {a,b,c,d}beasubsetofU.
Let∗beabinaryoperationdefinedonGasshownintheCayleytablebelow:
∗ 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,letN andI representtheneutralelementandtheinverseelementrespectivelywithrespecttoany
x x
elementx∈G. Then
N = d,
a
I = c,
a
N ,N ,N donotexist,
b c d
I ,I ,I donotexist.
b c d
Thisinadditionto(G,∗)beingaNeutroSemiGroupimpliesthat(G,∗)isaNeutroGroup.
Itisalsoclearfromthetablethat∗isNeutroCommutative. Hence,(G,∗)isaNeutroAbelianGroup.
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Example2.4. LetG = Z andlet∗beabinaryoperationonGdefinedbyx∗y = x+2y forallx,y ∈ G
10
where+isadditionmodulo10. Then(G,∗)isaNeutroAbeliaGroup. Toseethis,forx,y,z ∈G,wehave
x∗(y∗z) = x+2y+4z, (12)
(x∗y)∗z = x+2y+2z. (13)
Equating (16) and (17) we obtain z = 0,5. Hence only the triplets (x,y,0),(x,y,5) can verify associativ-
ity of ∗ and not any other triplet (x,y,z) ∈ G. Hence, ∗ is NeutroAssociative and therefore, (G,∗) is a
NeutroSemigroup.
Next,lete∈Gsuchthatx∗e=x+2e=xande∗x=e+2x. Then,x+2e=e+2xfromwhichwe
obtaine = x. Butthen,only5∗5 = 5inG. ThisshowsthatGhasaNeutroNeutralelement. Itcanalsobe
shownthatGhasaNeutroInverseelement. Hence,(G,∗)isaNeutroGroup.
Lastly,
x∗y = x+2y, (14)
y∗x = y+2x. (15)
Equating(18)and(19),weobtainx=y. Henceonlytheduplet(x,x)∈Gcanverifycommutativityof∗and
notanyotherduplet(x,y)∈G. Hence,∗isNeutroCommutativeandthus(G,∗)isaNeutroAbelianGroup.
Remark2.5. GeneralNeutroGroupisaparticularcaseofgeneralNeutroAlgebrawhichisanalgebrawhich
has at least one NeutroOperation or one NeutroAxiom (axiom that is true for some elements, indeterminate
forotherelements,andfalsefortheotherelements). Therefore,aNeutroGroupisagroupthathaseitherone
NeutroOperation (partially well-defined, partially indeterminate, and partially outer-defined), or atleast one
NeutroAxiom(NeutroAssociativity,NeutroElement,orNeutroInverse).
ItispossibletodefineNeutroGroupinanotherwaybyconsideringonlyoneNeutroAxiomorbyconsider-
ingtwoNeutroAxioms.
Theorem2.6. Let(G ,∗),i=1,2,··· ,nbeafamilyofNeutroGroups. Then
i
(i) G=(cid:84)n G isaNeutroGroup.
i=1 i
(ii) G=(cid:81)n G isaNeutroGroup.
i=1 i
Proof. Obvious.
Definition2.7. Let(G,∗)beaNeutroGroup. AnonemptysubsetH ofGiscalledaNeutroSubgroupofGif
(H,∗)isalsoaNeutroGroup.
TheonlytrivialNeutroSubgroupofGisG.
Example 2.8. Let (G,∗) be the NeutroGroup of Example 2.3 and let H = {a,c,d}. The compositions of
elementsofH 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(cid:54)=a.
N = d,
a
I = c,
a
N ,N donotexist,
c d
I ,I donotexist.
c d
a∗d = d∗a=a, but c∗d=d,d∗c=c(cid:54)=d.
Alltheseshowthat(H,∗)isaNeutroAbelianGroup. SinceH ⊂G,itfollowsthatH isaNeutroSubgroupof
G.
It should be observed that the order of H is not a divisor of the order of G. Hence, Lagrange’s theorem
doesnothold.
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Theorem2.9. Let(G,∗)beaNeutroGroupandlet(H ,∗),i = 1,2,··· ,nbeafamilyofNeutroSubgroups
i
ofG. Then
(i) H =(cid:84)n H isaNeutroSubgroupofG.
i=1 i
(ii) H =(cid:81)n H isaNeutroSubgroupofG.
i=1 i
Proof. Obvious.
Definition2.10. LetH beaNeutroSubgroupoftheNeutroGroup(G,∗)andletx∈G.
(i) xH theleftcosetofH inGisdefinedby
xH ={xh:h∈H}. (16)
(i) HxtherightcosetofH inGisdefinedby
Hx={hx:h∈H}. (17)
Example2.11. Let(G,∗)betheNeutroGroupofExample2.3andletH betheNeutroSubgroupofExample
2.8. H thesetofdistinctleftcosetsofH inGisgivenby
l
H ={{a,b,d},{a,c},{b,d}}
l
andH thesetofdistinctrightcosetsofH inGisgivenby
r
H ={{a,b,d},{a,b,c},{b,c,d},{a,d}}.
r
It should be observed that H and H do not partition G. This is different from what is obtainable in the
l r
classicalgroups. However,theorderofH is3whichisnotadivisoroftheorderofGandtherefore,[G:H]
l
the index of H in G is 3, that is [G : H] = 3. Also, the order of H is 4 which is a divisor of G. Hence,
r
[G:H]=4.
Example2.12. LetU = {a,b,c,d}beauniverseofdiscourseandletG = {a,b,c}beaNeutroGroupgiven
intheCayleytablebelow:
∗ a b c
a a c b
.
b c a c
c a c d
LetH ={a,b}beaNeutroSubgroupofGgivenintheCayleytablebelow:
∗ a b
a a c .
b c a
Then,thesetsofdistinctleftandrightcosetsofH inGarerespectivelyobtainedas:
H = {{a,c}},
l
H = {{a,},{c},{b,c}}.
r
Inthisexample,theorderofH thesetofdistinctleftcosetsofH inGis1whichisnotadivisoroftheorder
l
ofGandtherefore,[G:H]=1. However,theorderofH thesetofdistinctrightcosetsofH inGis3which
r
isadivisoroftheorderofGandtherefore,[G : H] = 3. Thisisalsodifferentfromwhatisobtainableinthe
classicalgroups.
ConsequentonExamples2.8,2.11and2.12,westatethefollowingtheorem:
Theorem2.13. LetH beaNeutroSubgroupofthefiniteNeutroGroup(G,∗). Thengenerally:
(i) o(H)isnotadivisorofo(G).
(ii) Thereisno1-1correspondencebetweenanytwoleft(right)cosetsofH inG.
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(iii) Thereisno1-1correspondencebetweenanyleft(right)cosetofH inGandH.
(iv) IfN =ethatisxe=ex=xforanyx∈G,theneH (cid:54)=H,He(cid:54)=Hand{e}isnotaNeutroSubgroup
x
ofG.
(v) o(G)(cid:54)=[G:H]o(H).
(vi) Thesetofdistinctleft(right)cosetsofH inGisnotapartitionofG.
Definition2.14. Let(G,∗)beaNeuroGroup. Since∗isassociativeforatleastonetriplet(x,x,x) ∈ G,the
powersofxaredefinedasfollows:
x1 = x
x2 = xx
x3 = xxx
. .
. .
. .
xn = xxx···x nfactors ∀n∈N.
Theorem2.15. Let(G,∗)beaNeutroGroupandletx∈G. Thenforanym,n∈N,wehave:
(i) xmxn =xm+n.
(ii) (xm)n =xmn.
Definition 2.16. Let (G,∗) be a NeutroGroup. G is said to be cyclic if G can be generated by an element
x∈Gthatis
G=<x>={xn :n∈N}. (18)
Example2.17. Let(G,∗)betheNeutroGroupgiveninExample2.3andconsiderthefollowing:
a1 = a,a2 =b,a3 =c,a4 =d.
b1 = b,b2 =d,b3 =c,b4 =a.
c1 = c,c2 =b,c3 =a,c4 =d.
d1 = d,d2 =a,d3 =a,d4 =a.
∴ G = <a>=<b>=<c>,G(cid:54)=<d>.
TheseshowthatGiscyclicwiththegeneratorsa,b,c. Theelementd∈GdoesnotgenerateG.
Definition2.18. LetH beaNeutroSubgroupoftheNeutroGroup(G,∗). Thesets(G/H) and(G/H) are
l r
definedby:
(G/H) = {xH :x∈G} (19)
l
(G/H) = {Hx:x∈G}. (20)
r
LetxH,yH ∈(G/H) andlet(cid:12) beabinaryoperationdefinedon(G/H) by
l l l
xH (cid:12) yH =x∗yH ∀x,y ∈G. (21)
l
Also,letxH,yH ∈(G/H) andlet(cid:12) beabinaryoperationdefinedon(G/H) by
r r r
Hx(cid:12) Hy =Hx∗y ∀x,y ∈G. (22)
r
Itcanbeshownthatthecouples((G/H) ,(cid:12) )and((G/H) ,(cid:12) )areNeutroGroups.
l l r r
Example2.19. LetGandH beasgiveninExample2.12andconsider
(G/H) ={aH,bH,cH}={aH}={{a,c}}.
l
Then((G/H) ,(cid:12) )isaNeutroGroup.
l l
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Definition 2.20. Let (G,∗) and (H,◦) be any two NeutroGroups. The mapping φ : G → H is called a
NeutroGroupHomomorphismifφpreservesthebinaryoperations∗and◦thatisifforallx,y ∈G,wehave
φ(x∗y)=φ(x)◦φ(y). (23)
ThekernelofφdenotedbyKerφisdefinedas
Kerφ={x:φ(x)=e } (24)
H
wheree ∈H issuchthatN =e foratleastoneh∈H.
H h H
TheimageofφdenotedbyImφisdefinedas
Imφ={y ∈H :y =φ(x) forsome h∈H}. (25)
If in addition φ is a bijection, then φ is called a NeutroGroupIsomorphism and we write G ∼= H. Neu-
troGroupEpimorphism, NeutroGroupMonomorphism, NeutroGroupEndomorphism, and NeutroGroupAuto-
morphismaresimilarlydefined.
Example2.21. Let(G,∗)beaNeutroGroupofExample2.12andletψ :G×G→Gbeaprojectiongiven
by
ψ((x,y))=x ∀x,y ∈G.
Thenψ isaNeutroGroupHomomorphism. TheKerψ = {(a,a),(a,b),a,c)}whichisaNeutroSubgroupof
G×GasshownintheCayleytablebelow.
∗ (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)
andImψ ={a,b,c}=G.
ConsequentonExample2.21,westatethefollowingtheorem:
Theorem2.22. Let(G,∗)and(H,◦)beNeutroGroupsandletN = e suchthate ∗x = x∗e = xfor
x G G G
atleastonex ∈ GandletN = e suchthate ∗y = y∗e = y foratleastoney ∈ H. Supposethat
y H H H
φ:G→H isaNeutroGroupHomomorphism. Then:
(i) φ(e )=e .
G H
(ii) KerφisaNeutroSubgroupofG.
(iii) ImφisaNeutroSubgroupofH.
(iv) φisinjectiveifandonlyifKerφ={e }.
G
Example2.23. ConsideringExample2.19,letφ:G→G/Hbeamappingdefinedbyφ(x)=xHforallx∈
G. Then,φ(a)=φ(b)=φ(c)=aH ={a,c}fromwhichwehavethatφisaNeutroGroupHomomorphism.
Kerφ={x∈G:φ(x)=e }={x∈G:xH =e =e }=(cid:54) H.
G/H G/H {{a,c}}
ConsequentonExample2.23,westatethefollowingtheorem:
Theorem2.24. LetH beaNeutroSubgroupofaNeutroGroup(G,∗). Themappingψ : G → G/H defined
by
ψ(x)=xH ∀x∈G
isaNeutroGroupHomomorphismandtheKerψ (cid:54)=H.
Theorem 2.25. Let φ : G → H be a NeutroGroupHomomorpism and let K = Kerφ. Then the mapping
ψ :G/K →Imφdefinedby
ψ(xK)=φ(x) ∀x∈G
isaNeutroGroupEpimorphismandnotaNeutroGroupIsomorphism.
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Proof. Thatψisawelldefinedsurjectivemappingisclear. LetxK,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)=e }
φ(x)
= {xK ∈G/K :φ(x)=e }
φ(x)
(cid:54)= {e }.
G/K
ThisshowsthatψisasurjectiveNeutroGroupHomomorphismandthereforeitisaNeutroGroupEpimorphism.
Sinceψisnotinjective,itfollowsthatG/K (cid:54)∼=Imφwhichisdifferentfromwhatisobtainableintheclassical
groups.
Theorem2.26. NeutroGroupIsomorphismofNeutrogroupsisanequivalencerelation.
Proof. Thesameastheclassicalgroups.
3 Conclusion
We have formally presented the concept of NeutroGroup in this paper by considering three NeutroAxioms
(NeutroAssociativity, existence of NeutroNeutral element and existence of NeutroInverse element). We pre-
sentedandstudiedseveralinterestingresultsandexamplesonNeutroSubgroups, NeutroCyclicGroups, Neu-
troQuotientGroups and NeutroGroupHomomorphisms. We have shown that generally, Lagrange’s theorem
and1stisomorphismtheoremoftheclassicalgroupsdonotholdintheclassofNeutroGroups. Furtherstudies
of NeutroGroups will be presented in our future papers. Other NeutroAlgebraicStructures such as NeutroR-
ings,NeutroModules,NeutroVectorSpacesetcareopenedtostudiesforNeutrosophicResearchers.
4 Appreciation
Theauthorisverygratefultoallanonymousreviewersforvaluablecommentsandsuggestionswhichwehave
foundveryusefulintheimprovementofthepaper.
References
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AntiAlgebras viz-a-viz the Classical Number Systems”, International Journal of Neutrosophic Sci-
ence,Volume4,Issue1,pp.16-19,2020.
[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”,InternationalJournalofNeu-
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