Table Of ContentInternationalJournalofNeutrosophicScience(IJNS) Vol. 12,No. 2,PP.71-80,2020
Introduction to AntiGroups
A.A.A.Agboola(cid:5)
DepartmentofMathematics,FederalUniversityofAgriculture,Abeokuta,Nigeria.
[email protected]
(cid:5)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.
1 Introduction
Neutrosophic logic (NL) introduced by Smarandache in 1995 is an alternative to the existing classical log-
icsandthegeneralizationoffuzzylogic(FL)ofZadeh[12]andintuitionisticfuzzylogic(IFL)ofAtanassov
[5]. Neutrosophic logic is a non-classical logic that can be used as a mathematical tool to model situations
characterized by uncertainty, vagueness, ambiguity, imprecision, undefined, unknown, incompleteness, in-
consistency, redundancy, contradiction etc. In neutrosophic logic [11], each proposition is estimated to have
percentage of truth in a subset T, the percentage of indeterminacy in a subset I, and the percentage of fal-
sityinasubsetF where(T,I,F)arestandardornon-standardsubsetsofthenon-standardinterval]−0,1+[,
wheren = infT +infI +infF ≥− 0, andn = supT +supI +supF ≤ 3+. Statically, (T,I,F)
inf sup
are subsets but dynamically, they a functions/operators depending on many known or unknown parameters.
In NL, if < A > is an idea, or proposition, theory, law, axiom, event, concept, entity etc., there correspond
<Non-A>=<Anti-A>whichistheoppositeof<A>and<Neut-A>whichstandsforwhatisneither
< A > nor < Anti-A >, that is neutrality in between the two extremes. Consequently in NL, it is possible
to have the triad (< A >,< Neut-A >,< Anti-A >). The non-restriction in NL allows for paraconsis-
tent, dialetheist, and incomplete information to be characterized. This special and unique feature of NL has
made it applicable in solving problems involving uncertainty, vagueness, ambiguity, imprecision, undefined,
unknown,incompleteness,inconsistency,redundancy,contradictionetc. arisingfromscience,socialscience,
engineering,technology,computerscience,artificialintelligence,ICT,roboticsetc.
TheindeterminacyfactorI inNLisfundamentalintheformulationandestablishmentofanyneutrosophic
algebraicstructure.Givenanyclassicalalgebraicstructure(X,∗),astructureX(I)=<X∪I >generatedby
X andI underthebinaryoperation∗ofX iscalledaneutrosophicalgebraicstructurewithitsnamederived
fromthenameofX. Forinstance,ifX isagroup,thenX(I)iscalledaneutrosophicgroup. SinceI−1,the
inverseofI doesnotexist,findingx−1,theinverseofanyneutrosophicelementx ∈ X(I)becomesdifficult
and impossible. Consequently, algebraic manipulations of the elements of X(I) become restrictive. The
recentintroductionoftheconceptsofNeutroStructuresandAntiStructureshavelessenedtherestrictivenessof
thealgebraicmanipulationsoftheelementsofneutrosophicalgebraicstructuresimposedbytheneutrosophic
elementI.
Doi:10.5281/zenodo.4274130 71
InternationalJournalofNeutrosophicScience(IJNS) Vol. 12,No. 2,PP.71-80,2020
In any classical algebraic structure (X,∗), the of composition of the elements with respect to the binary
operation ∗ is well defined for all the elements of X that is, x∗y ∈ X ∀x,y ∈ X. All the axioms like
associativity,commutativity,distributivity,monotonicityetc.definedonXwithrespectto∗aretotallytruefor
alltheelementsofX. ThecompositionsofelementsofX thiswayarerestrictiveanddonotreflectthereality.
Theydonotgiveroomsforcompositionsthatareeitherpartiallydefined,partiallyundefined(indeterminate),
and partially outerdefined or totally outerdefined with respect to ∗. However in the domain of knowledge,
scienceandreality,thelawofcompositionandaxiomsdefinedonX mayeitherbeonlypartiallydefined(par-
tiallytrue),orpartiallyundefined(partiallyfalse),ortotallyundefined(totallyfalse)withrespecttothebinary
operation∗. In2019,Smarandache[10]addressedtheproblemofallowingthelawofcompositiononX tobe
eitherpartiallydefinedandpartiallyundefinedortotallyundefinedbyintroducingthenotionsofNetroDefined
andAntiDefinedlaws,aswellasthenotionsofNeutroAxiomandAntiAxiominspiredbyNLheintroducedin
1995. TheworkofSmarandachein[10]hasgivenbirthtothenewfieldsofresearchcalledNeutroStructures
and AntiStructures. For any classical algebraic law or axiom defined on X, there correspond neutrosophic
triads(< Law >,< NeutroLaw >,< AntiLaw >)and(< Axiom >,< NeutroAxiom >,< AntiAxiom >)
respectively. In [9], Smarandache studied NeutroAlgebras and AntiAlgebras and in [8], he studied Partial
Algebras,UniversalAlgebras,EffectAlgebrasandBoole’sPartialAlgebrasandheshowedthatNeutroAlge-
brasaregeneralizationofPartialAlgebras. In[7],RezaeiandSmarandachestudiedNeutro-BE-algebrasand
Anti-BE-algebras and they showed that any classical algebra S with n operations (laws and axioms) where
n ≥ 1willhave(2n −1)NeutroAlgebrasand(3n −2n)AntiAlgebras. In[2], Agboolaetal. studiedNeu-
troAlgebrasandAntiAlgebrasviz-a-viztheclassicalnumbersystemsN, Z, Q, RandCandin[3], Agboola
studied NeutroGroups by considering three NeutroAxioms (NeutroAssociativity, existence of NeutroNeutral
element and existence of NeutroInverse element). In addition, he studied NeutroSubgroups, NeutroCyclic-
Groups, NeutroQuotientGroups and NeutroGroupHomomorphisms. He showed that generally, Lagrange’s
theorem and fundamental homomorphism theorem of the classical groups do not hold in the class of Neu-
troGroupsstudied. In[4], AgboolaintroducedandstudiedNeutroRingsbyconsideringthreeNeutroAxioms
(NeutroAbelianGroup (additive), NeutroSemigroup (multiplicative) and NeutroDistributivity (multiplication
overaddition)). HepresentedSeveralresultsandexamplesonNeutroRings,NeutroSubgrings,NeutroIdeals,
NeutroQuotientRingsandNeutroRingHomomorphisms. Heshowedthatthatthefundamentalhomomorphism
of the classical rings holds in the class of NeutroRings considered. Motivated and inspired by the work of
Rezaei and Smarandache in [7], Agboola in [1] revisited the NeutroGroups by studying a particular class of
NeutroGroupsandpresentedtheirbasicandelementaryproperties. Inthepresentpaperhowever,theconcept
ofAntiGroupsisformallypresented. AparticularclassofAntiGroupsisstudiedwithpresentationofseveral
examplesandbasicproperties. Itisshownthatsomealgebraicpropertiesoftheclassicalgroupsdonothold
in the class of AntiGroups studied. Specifically, it is shown that intersection of two AntiSubgroups is not
necessarilyanAntiSubgroupandtheunionoftwoAntiSubgroupsmaybeanAntiSubgroup. Also,itisshown
thatLagranges’theoremandfundamentaltheoremofhomomorphismsoftheclassicalgroupsdonotholdin
theclassofAntiGroupsstudiedinthispaper.
2 Preliminaries
Inthissection,wewillgivesomedefinitionsandresultsthatwillbeusedlaterinthepaper.
Definition2.1. [8]
(i) Aclassicaloperationisanoperationwelldefinedforalltheset’selements.
(ii) A NeutroOperation is an operation partially well defined, partially indeterminate, and partially outer
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
[degree of indeterminacy (I)], or false for the other set’s elements [degree of falsehood (F)], where
T,I,F ∈ [0,1],with(T,I,F) (cid:54)= (1,0,0)thatrepresentstheclassicalaxiom,and(T,I,F) (cid:54)= (0,0,1)
thatrepresentstheAntiAxiom.
Doi:10.5281/zenodo.4274130 72
InternationalJournalofNeutrosophicScience(IJNS) Vol. 12,No. 2,PP.71-80,2020
(vi) AnAntiLaw/AntiAxiomdefinedonanonemptysetisalaw/axiomthatisfalseforallset’selements.
(vii) ANeutroAlgebraisanalgebrathathasatleastoneNeutroOperationoroneNeutroAxiom(axiomthat
istrueforsomeelements,indeterminateforotherelements,andfalseforotherelements),andnoAnti-
OperationorAntiAxiom.
(viii) AnAntiAlgebraisanalgebraendowedwithatleastoneAntiOperationoratleastoneAntiAxiom.
Theorem2.2. [7]LetUbeanonemptyfiniteorinfiniteuniverseofdiscourseandletS beafiniteorinfinite
subset of U. If n classical operations (laws and axioms) are defined on S where n ≥ 1, then there will be
(2n−1)NeutroAlgebrasand(3n−2n)AntiAlgebras.
Definition2.3. [Classicalgroup][6]
LetGbeanonemptysetandlet∗ : G×G → GbeabinaryoperationonG. 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) Thereexistse∈Gsuchthatx∗e=e∗x=x∀x∈G[axiomofexistenceofneutralelement].
(G4) Thereexistsy ∈Gsuchthatx∗y =y∗x=e∀x∈G[axiomofexistenceofinverseelement]where
eistheneutralelementofG.
Ifinaddition∀x,y ∈G,wehave
(G5) x∗y =y∗x,then(G,∗)iscalledanabeliangroup.
Definition2.4. [AntiSophicationofthelawandaxiomsoftheclassicalgroup][1]
(AG1) Foralltheduplets(x,y)∈G,x∗y (cid:54)∈G[AntiClosureLaw].
(AG2) Forallthetriplets(x,y,z)∈G,x∗(y∗z)(cid:54)=(x∗y)∗z[AntiAxiomofassociativity(AntiAssociativity)].
(AG3) Theredoesnotexistanelemente∈Gsuchthatx∗e=e∗x=x∀x∈G[AntiAxiomofexistenceof
neutralelement(AntiNeutralElement)].
(AG4) Theredoesnotexistu ∈ Gsuchthatx∗u = u∗x = e∀x ∈ G[AntiAxiomofexistenceofinverse
element(AntiInverseElement)]whereeisanAntiNeutralElementinG.
(AG5) Foralltheduplets(x,y)∈G,x∗y (cid:54)=y∗x[AntiAxiomofcommutativity(AntiCommutativity)].
Definition2.5. [AntiGroup][1]
An AntiGroup AG is an alternative to the classical group G that has at least one AntiLaw or at least one of
{AG1,AG2,AG3,AG4}.
Definition2.6. [AntiAbelianGroup][1]
AnAntiAbelianGroupAGisanalternativetotheclassicalabeliangroupGthathasatleastoneAntiLaworat
leastoneof{AG1,AG2,AG3,AG4}andAG5.
Theorem2.7. [1]Let(G,∗)beafiniteorinfiniteclassicalgroup. Thenthereare65typesofAntiGroups.
Theorem 2.8. [1] Let (G,∗) be a finite or infinite classical abelian group. Then there are 211 types of
AntiAbelianGroups.
Definition2.9. Let(AG,∗)beanAntiGroup. AGissaidtobefiniteofordernifthecardinalityofAGisn
thatiso(AG)=n. Otherwise,AGiscalledaninfiniteAntiGroupandwewriteo(AG)=∞.
SincetherearemanytypesofAntiGroups,inwhatfollows,AntiGroupswillbeclassifiedandnamedtype-
AG[,]accordingtowhichofAG1−AG5is(are)satisfied.IfonlyAG1issatisfied,theAntiGroupwillbecalled
oftype-AG[1],type-AG[3,4]ifonlyAG3andAG4aresatisfiedandsoon. AntiGroupsoftype-AG[1,2,3,4]
oroftype-AG[1,2,3,4,5]willbecalledtrivialAntiGroupsortrivialAntiAbelianGroupsrespectively.
Example 2.10. Let AG = Q∗ be the set of all irrational positive numbers and consider algebraic structure
+
(AG,.)where(cid:48)(cid:48).(cid:48)(cid:48)istheordinarymultiplicationofrealnumbers. Then(AG,.)isaninfinitetrivialAntiGroup.
Doi:10.5281/zenodo.4274130 73
InternationalJournalofNeutrosophicScience(IJNS) Vol. 12,No. 2,PP.71-80,2020
Example2.11. LetAG=N.
(i) Let∗beabinaryoperationonAGdefined∀x,y ∈AGby
x∗y =x+y+xy.
Then(AG,∗)isafiniteAntiGroupoftype-AG[3,4].
(ii) Let∗beabinaryoperationonAGdefined∀x,y ∈AGby
x∗y =x+y+1.
Then(AG,∗)isafiniteAntiGroupoftype-AG[3,4].
3 A Study of Finite AntiGroups of Type-AG[4]
In this section, we are going to study a particular class of AntiGroups (AG,∗) where G4 is totally false for
alltheelementsofAGwhileG1,G2,G3andG5areeitherpartiallytrue,partiallyindeterminateorpartially
falseforsomeelementsofAG.
Definition3.1. Let(AG,∗)and(AH,◦)beAntiGroupsoftype-AG[4]. ThedirectproductofAGandAH
denotedbyAG×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]andletg,x,y ∈AG. Then
(i) g∗x=g∗y (cid:54)=⇒ x=y.
(ii) x∗g =y∗g (cid:54)=⇒ x=y.
Proof. Sinceg−1doesnotexistand∗isNeutroAssociative,therequiredresultsfollow.
Proposition3.4. Let(AG,∗)beanAntiGroupoftype-AG[4],x,y ∈AGandletm,n∈N. Then
(i) xm+1 (cid:54)=xm∗x.
(ii) x−m (cid:54)=(cid:0)x−1(cid:1)m.
(iii) xm∗x−m (cid:54)=N whereN isaNeutroNeutralElementinAG.
e e
(iv) xm∗xn (cid:54)=xm+n.
(v) (xm)n (cid:54)=xmn.
(vi) (x∗y)m (cid:54)=xm∗ym.
Proof. Sincex−1doesnotexistand∗isNeutroAssociative,therequiredresultsfollow.
Corollary3.5. AnAntiGroup(AG,∗)oftype-AG[4]cannotbegeneratedbyanelementx ∈ AGandhence
cannotbycyclic.
Definition 3.6. Let (AG,∗) be an AntiGroup of type-AG[4]. A nonempty subset AH of AG is called an
AntiSubgroup of AG if (AH,∗) is also an AntiGroup of the same type as AG. Otherwise, if (AH,∗) is an
AntiGroupofatypedifferentfromthetypeofAG,thenAH iscalledaQuasiAntiSubgroupofAG.
Doi:10.5281/zenodo.4274130 74
InternationalJournalofNeutrosophicScience(IJNS) Vol. 12,No. 2,PP.71-80,2020
Definition3.7. Let(AG,∗)beanAntiGroupoftype-AG[4]andletAH andAK beAntiSubgroupsofAG.
ThesetA∗Bisdefinedby
A∗B ={x∈AG:x=h∗k forsome h∈AH,k ∈AK}.
Proposition3.8. LetAH,AK andALbeAntiSubgroupsofanAntiGroup(AG,∗)oftype-AG[4]. Then
(i) AH ∗AH (cid:54)=AH.
(ii) AH ∗AK (cid:54)=AK∗AH.
(iii) AH ∗(AK∗AL)(cid:54)=(AH ∗AK)∗AL.
Proof. Obvious.
Definition3.9. Let(AG,∗)beanAntiGroupoftype-AG[4]andleta∈AGbeafixedelement.
(i) AnAntiCenterofAGdenotedbyAZ(AG)isasetdefinedby
AZ(AG)={x∈AG:x∗g (cid:54)=g∗x ∀g ∈AG}.
(ii) AnAntiCentralizerofa∈AGdenotedbyAC isasetdefinedby
a
AC ={g ∈AG:g∗a(cid:54)=a∗g}.
a
Example3.10. LetU={a,b,c,d,e,f}beauniverseofdiscourseandletAG={a,b,c,e}beasubsetofU.
(i) Let∗beabinaryoperationdefinedonAGasshownintheCayleytablebelow.
∗ a b c e
a d a c e
b a d e c .
c b a f e
e a b c f
ItisevidentfromthetablethatG1,G2,G3,G5areeitherpartiallytrueorpartiallyfalsewithrespectto
∗butG4istotallyfalseforalltheelementsofAG. Hence(AG,∗)isafiniteAntiGroupoftype-AG[4].
(ii) Let∗beabinaryoperationdefinedonAGasshownintheCayleytablebelow.
∗ e a b c
e d a b c
a a f c b .
b b a ? c
c c b a ?
ItisevidentfromthetablethatG1,G2,G3,G5areeitherpartiallytrue,partiallyindeterminateorpar-
tiallyfalsewithrespectto∗butG4istotallyfalseforalltheelementsofAG. Hence(AG,∗)isafinite
AntiGroupoftype-AG[4].
Example 3.11. Let (AG,∗) be the AntiGroup of Example 3.10(i) and let AH = {a,b,e} and AH =
1 2
{b,c,e}betwosubsetsofAG. Let∗bedefinedonAH andAH asshownintheCayleytablesbelow:
1 2
∗ a b e ∗ b c e
a d a e b d e c
AH : , AH : .
1 b a d c 2 c a f e
e a b f e b c f
ItcaneasilybeseenfromthetablesthatAH isanAntiSubgroupofAGwhileAH isaQuasiAntiSubgroup
1 2
ofAG. ItisnotedthatLagranges’theoremdoesnothold. Itisalsonotedthat:
AH ∪AH = {a,b,c,e}=AG,
1 2
NH ∩NH = {b,e},
1 2
Doi:10.5281/zenodo.4274130 75
InternationalJournalofNeutrosophicScience(IJNS) Vol. 12,No. 2,PP.71-80,2020
fromwhichitisdeducedthatNH ∪NH isanAntiSubgroupofAGbutAH ∩AH isaQuasiAntiSubgroup
1 2 1 2
ofAGasitisevidentintheCayleytablebelow:
∗ b e
NH ∩NH : b d c .
1 2
e b f
AZ(AG)={a,b,c,e}=AGisanAntiSubgroupofAG.Also,AC =AC =AC =AC ={a,b,c,e}=
a b c e
AGareAntiSubgroupsofAG.
Example 3.12. Let (AG,∗) be the AntiGroup of Example 3.10(ii) and let AH = {e,a,b} and AH =
1 2
{e,b,c}betwosubsetsofAG. Let∗bedefinedonAH andAH asshownintheCayleytablesbelow:
1 2
∗ e a b ∗ e b c
e d a b e d b c
AH : , AH : .
1 a a f c 2 b b ? c
b b a ? c c a ?
ItcaneasilybeseenfromthetablesthatAH andAH areAntiSubgroupsofAG. ItisnotedthatLagranges’
1 2
theoremdoesnothold. Itisalsonotedthat:
AH ∪AH = {e,a,b,c}=AG,
1 2
NH ∩NH = {e,b},
1 2
fromwhichitisdeducedthatNH ∪NH isanAntiSubgroupofAGandAH ∩AH isanAntiSubgroup
1 2 1 2
ofAGasitisevidentintheCayleytablebelow:
∗ e b
NH ∩NH : e d f .
1 2
b b ?
AZ(AG) = {a,b,c} is a QuasiAntiSubgroup of AG. Also, AC = AC = {a,b,c}, AC = {b,c} are
a b c
QuasiAntiSubgroupsofAGandAC ={}=∅isneitheranAntiSubgroupnoraQuasiAntiSubgroupofAG.
e
Example3.13. (i) LetAG = Z = {0,1,2,3}andlet⊕beabinaryoperationonAGasdefinedinthe
4
Cayleytable.
⊕ 0 1 2 3
0 0or1 1 2 3
1 1 2 3 0or2 .
2 2 3 0or3 1
3 3 ? 1 2
Then(AG,⊕)isafiniteAntiGroupoftype-AG[4]andAH ={0,1,2}isanAntiSubgroupofAG.
(ii) LetAG={1,2,3,4}⊆Z andlet⊗beabinaryoperationonAGasdefinedintheCayleytable.
5
⊗ 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]andAH ={1,2,3}isanAntiSubgroupofAG.
Definition3.14. LetAH beanAntiSubgroupoftheAntiGroup(AG,∗)oftype-AG[4]andletx∈AG.
(i) x∗AH theleftcosetofAH inAGisdefinedby
x∗AH ={x∗h:h∈AH}.
Doi:10.5281/zenodo.4274130 76
InternationalJournalofNeutrosophicScience(IJNS) Vol. 12,No. 2,PP.71-80,2020
(ii) AH ∗xtherightcosetofAH inAGisdefinedby
AH ∗x={h∗x:h∈AH}.
(iii) AH iscalledanormalAntiSubgroupifx∗AH =AH ∗xforatleastonex∈AG.
(iv) The number of distinct left or right cosets of AH in AG is called the index of AH in AG and it is
denotedby[AG:AH].
(vi) ThesetofalldistinctleftcosetsofAH inAGdenotedby(AG/AH) isdefinedby
L
(AG/AH) ={x∗AH :x∈AG}.
L
(vii) ThesetofalldistinctrightcosetsofAH inAGdenotedby(AG/AH) isdefinedby
R
(AG/AH) ={AH ∗x:x∈AG}.
R
SupposethatAG/AHisthesetofalldistinctleftcosetsofAHinAGandsupposethat(cid:12)isabinaryoperation
onAG/AH definedby
(x∗AH)(cid:12)(y∗AH)=(x∗y)∗AH ∀x∗AH,y∗AH ∈AG/AH.
Ifthecouple(AG/AH,(cid:12))isanAntiGroupoftype-AG[4],thenAG/AH iscalledanAntiQuotientGroupof
AGfactoredbyAH.
Lemma 3.15. Let AH be an AntiSubgroup of the AntiGroup (AG,∗) of type-AG[4] and let e ∈ AG be a
NeutroNeutralElement. Thenglobally,
e∗AH (cid:54)=AH.
Example3.16. Let(AG,⊕)beanAntiGroupofExample3.13(i)andletAH ={0,1,2}beitsAntiSubgroup.
TheleftandrightcosetsofAH inAGare:
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}.
ItisnotedthatAH isanormalAntiSubgroupofAGanddistinctleftandrightcosetsofAH donotpartition
AG.
NowconsidertheCayleytablebelow.
(cid:76)
0⊕AH 1⊕AH 2⊕AH 3⊕AH
0⊕AH 0⊕AH or1⊕AH 1⊕AH 2⊕AH 3⊕AH
1⊕AH 1⊕AH 2⊕AH 3⊕AH 0⊕AH or2⊕AH .
2⊕AH 2⊕AH 3⊕AH 0⊕AH or3⊕AH 1⊕AH
3⊕AH 3⊕AH ? 1⊕AH 2⊕AH
(cid:76)
Itisevidentfromthetablethat(AG/AH, )isanAntiGroupoftype-AG[4].
Example3.17. Let(AG,⊗)beanAntiGroupofExample3.13(ii)andlet AH = {1,2,3}beitsAntiSub-
group. TheleftcosetsofAH inAGare:
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}.
Doi:10.5281/zenodo.4274130 77
InternationalJournalofNeutrosophicScience(IJNS) Vol. 12,No. 2,PP.71-80,2020
ItisnotedthatAH isanormalAntiSubgroupofAGanddistinctleftandrightcosetsofAH donotpartition
AG.
NowconsidertheCayleytablebelow.
(cid:78)
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
(cid:78)
Itisevidentfromthetablethat(AG/AH, )isanAntiGroupoftype-AG[4].
Proposition3.18. LetAHbeanormalAntiSubgroupofanAntiGroup(AG,∗)oftype-AG[4]andletAG/AH
bethesetofdistinctleftcosetsofAH inAG. Forx∗AH,y∗AH ∈ AG/AH withx,y ∈ AG,let(cid:12)bea
binaryoperationdefinedonAG/AH by
(x∗AH)(cid:12)(y∗AH)=(x∗y)∗AH ∀x,y ∈AG.
Then,(AG/AH,(cid:12))isanAntiGroupoftype-AG[4].
Proof. SupposethatAH isanormalAntiSubgroupofanAntiGroup(AG,∗)oftype-AG[4]andsupposethat
thecompositionofelementsinAG/AH isgivenby(x∗AH)(cid:12)(y∗AH) = (x∗y)∗AH ∀x,y ∈ AG.
Thenthereexistsomeduplets(x,y),(u,v),(p,q) ∈ AGsuchthatx∗y ∈ AG(inner-defined)and[u∗v =
indeterminate or p∗q (cid:54)∈ AG (outer-defined/falsehood)]. Hence, (cid:12) satisfies the NeutroClosureLaw. Next,
thereexistsometriplets(x,y,z),(p,q,r),(u,v,w) ∈ AGsuchthatx∗(y∗z) = (x∗y)∗z (inner-defined)
and [[p∗(q ∗r)]or[(p∗q)∗r] = indeterminate or u∗(v ∗w) (cid:54)= (u∗v)∗w (outer-defined/falsehood)].
This again shows that (cid:12) satisfies the NeutroAssociativityAxiom. Also, there exists an element e ∈ AG
such that x∗e = e∗x = x (inner-defined) and [[x∗e]or[e∗x] = indeterminate or x∗e (cid:54)= x (cid:54)= e∗x
(outer-defined/falsehood)] for at least one x ∈ AG. This shows the existence of NeutroNeutralElement in
AGandhencethereexistsaNeutroNeutralElemente∗AH ∈ AG/AH. Againforallx ∈ AG, theredoes
not exist u ∈ AG such that x ∗ u = u ∗ x = e. This is an AntiAxiom of existence of inverse element
in AG and consequently, no element x∗AH ∈ AG/AH has an inverse. Lastly, there exist some duplets
(x,y),(u,v),(p,q) ∈ AG such that x ∗ y = y ∗ x (inner-defined) and [[u ∗ v]or[v ∗ u] = indeterminate
or p ∗ q (cid:54)= q ∗ p (outer-defined/falsehood)]. This shows that (cid:12) satisfies the NeutroCommutativityAxiom.
Hence,(AG/AH,(cid:12))isanAntiGroupoftype-AG[4].
Definition3.19. Let(AG,∗)and(AH,◦)beanytwoAntiGroupsoftype-AG[4]. Themappingφ : AG →
AH iscalledanAntiGroupHomomorphismifφdoesnotpreservethebinaryoperations∗and◦thatisforall
theduplet(x,y)∈AG,wehave
φ(x∗y)(cid:54)=φ(x)◦φ(y).
ThekernelofφdenotedbyKerφisdefinedby
Kerφ={x:φ(x)=e foratleastone e ∈AH}
AH AH
wheree isaNeutroNeutralElementinAH.
AH
TheimageofφdenotedbyImφisdefinedby
Imφ={y ∈AH :y =φ(x) forsome x∈AG}.
If in addition φ is an AntiBijection, then φ is called an AntiGroupIsomorphism. AntiGroupEpimorphism,
AntiGroupMonomorphism,AntiGroupEndomorphism,andAntiGroupAutomorphismaredefinedsimilarly.
Example3.20. (i) Let(AG,⊕)betheAntiGroupofExample3.13(i)andletφ:AG→AGbeamapping
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.4274130 78
InternationalJournalofNeutrosophicScience(IJNS) Vol. 12,No. 2,PP.71-80,2020
fromwhichweobtainthatφ(x⊕y)(cid:54)=φ(x)⊕yforallx,y ∈AG. Accordingly,φisanAntiGroupHo-
momorphism. Imφ={1,2,3}whichisanAntiSubgroupofAG. Kerφ={}=∅.
(ii) Let(AG,⊗)betheAntiGroupofExample3.13(ii)andletψ :AG→AGbeamappingdefinedby
ψ(x)=x⊗4 ∀x∈AG.
Then
ψ(1) = 4,
ψ(2) = 3,
ψ(3) = 2,
ψ(4) = 0,
fromwhichweobtainthatψ(x⊗y)(cid:54)=ψ(x)⊗yforallx,y ∈AG. Accordingly,ψisanAntiGroupHo-
momorphism. Imφ={0,2,3,4}whichisnotanAntiSubgroupofAG. Kerφ={}=∅.
Example3.21. (i) Let(AG,∗)betheAntiGroupofExample3.10(i)andletφ : AG×AG → AGbea
projectiondefinedby
φ((x,y))=x ∀x,y ∈AG.
Then φ is not an AntiGroupHomomorphism because 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)andletψ :AG×AG→AGbeaprojectiondefined
by
ψ((x,y))=y ∀x,y ∈AG.
Then ψ is not an AntiGroupHomomorphism because ψ((a,b)⊗(b,c)) = ψ((a,b))∗ψ((b,c)) = c.
However,Imψ ={a,b,c,e}=AG.
(cid:76)
Example3.22. (i) Let (AG/AH, ) be the AntiQuotientGroup of Example 3.16 and let φ : AG →
AG/AH beamappingdefinedby
φ(x)=x⊕AH ∀x∈AG.
Then
φ(0) = 0⊕AH,
φ(1) = 1⊕AH,
φ(2) = 2⊕AH,
φ(3) = 3⊕AH,
fromwhichweobtain
(cid:77)
φ(1⊕2)=φ(1) φ(2)=3⊕AH.
ThisshowsthatφisnotanAntiGroupHomomorphism.
(cid:78)
(ii) Let (AG/AH, ) be the AntiQuotientGroup of Example 3.17 and let ψ : AG → AG/AH be a
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
(cid:79)
ψ(2⊗4)=ψ(2) ψ(4)=3⊗AH.
ThisshowsthatψisnotanAntiGroupHomomorphism.
Remark3.23. Thefundamentaltheoremofhomomorphismsoftheclassicalgroupscannotholdintheclass
ofAntiGroupsoftype-AG[4]asdemonstratedinExamples3.22(i)and(ii).
Doi:10.5281/zenodo.4274130 79
InternationalJournalofNeutrosophicScience(IJNS) Vol. 12,No. 2,PP.71-80,2020
4 Conclusion
The notion of AntiGroups was formally presented in this paper. A particular class of AntiGroups of type-
AG[4]wasstudied. InAntiGroupsoftype-AG[4],theexistenceofaninverseelementwastakingtobetotally
falseforalltheelementswhiletheclosurelaw,theexistenceofidentityelement,theaxiomsofassociativity
and commutativity were taking to be either partially true, partially indeterminate or partially false for some
elements. It was shown that some algebraic properties of the classical groups do not hold in the class of
AntiGroupsoftype-AG[4].Specifically,itwasshownthatintersectionoftwoAntiSubgroupsisnotnecessarily
an AntiSubgroup and the union of two AntiSubgroups may be an AntiSubgroup. Also, it was shown that
distinctleft(right)cosetsofAntiSubgroupsofAntiGroupsoftype-AG[4]donotpartitiontheAntiGroups;and
thatLagranges’theoremandfundamentaltheoremofhomomorphismsoftheclassicalgroupsdonotholdin
theclassofAntiGroupsoftype-AG[4]. MoreclassesofAntiGroupswillbestudiedinourfuturepapers.
5 Acknowledgment
The contributions of Professor Florentin Smarandache towards the improvement of this paper are acknowl-
edged. Thecommentsandsuggestionsofalltheanonymousreviewersareequallyacknowledged.
References
[1] Agboola, A.A.A., On Finite NeutroGroups of Type-NG[1,2,4], International Journal of Neutrosophic
Science(IJNS),Vol.10(2),pp.84-95,2020.(Doi:10.5281/zenodo.4006602)
[2] Agboola, A.A.A., Ibrahim, M.A. and Adeleke, E.O., Elementary Examination of NeutroAlgebras and
AntiAlgebrasviz-a-viztheClassicalNumberSystems(IJNS),InternationalJournalofNeutrosophicSci-
ence,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., Introduction to NeutroRings, International Journal of Neutrosophic Science (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,InternationalJournalofNeutro-
sophicScience,vol.2(1),pp.08-17,2020.
[9] Smarandache, F., Introduction to NeutroAlgebraic Structures and AntiAlgebraic Structures (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&Learn-
ing,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.4274130 80
