OnNeutroQuadrupleGroups F.SmarandacheA.RezaeiA.A.A.AgboolaY.B.Jun R.A.BorzooeiB.DavvazA.BroumandSaeidM.Akram M.HamidiS.Mirvakilii February16-19,2021,Kashan F.Smarandache,A.Rezaei,A.A.A.Agboola,Y.B.Jun,,,,R.A.Borzooei,B.Davvaz,A.BroumandSaeid,M.Akram,,,M.Hamidi,S.Mirvakilii OnNeutroQuadrupleGroups February16-19,2021,Kashan1/49
SectionName NeutroAlgebraicstructuresandAntiAlgebraicstructures AsgeneralizationsandalternativesofclassicalalgebraicstructuresFlorentin Smarandachehasintroducedin2019theNeutroAlgebraicstructures(orNeutroAlgebras)andAntiAlgebraicstructures(orAntiAlgebras).Unliketheclassicalalgebraicstructures,wherealloperationsarewell-definedandallaxioms aretotallytrue,inNeutroAlgebrasandAntiAlgebrastheoperationsmaybe partiallywell-definedandtheaxiomspartiallytrueorrespectivelytotallyouterdefinedandtheaxiomstotallyfalse. F.Smarandache,A.Rezaei,A.A.A.Agboola,Y.B.Jun,,,,R.A.Borzooei,B.Davvaz,A.BroumandSaeid,M.Akram,,,M.Hamidi,S.Mirvakilii OnNeutroQuadrupleGroups February16-19,2021,Kashan2/49
SectionName NeutroAlgebraicstructuresandAntiAlgebraicstructures TheseNeutroAlgebrasandAntiAlgebrasformanewfieldofresearch,which isinspiredfromourrealworld.Inthistalk,westudyneutrosophicquadruple algebraicstructuresandNeutroQuadrupleAlgebraicStructures.NeutroQuadrupleGroupisstudiedinparticularandseveralexamplesareprovided.Itisshown that (NQ(Z), ÷) isaNeutroQuadrupleGroup. F.Smarandache,A.Rezaei,A.A.A.Agboola,Y.B.Jun,,,,R.A.Borzooei,B.Davvaz,A.BroumandSaeid,M.Akram,,,M.Hamidi,S.Mirvakilii OnNeutroQuadrupleGroups February16-19,2021,Kashan3/49
SectionName Operation,NeutroOperation,AntiOperation Whenwedefineanoperationonagivenset,itdoesnotautomaticallymean thattheoperationiswell-defined.Therearethreepossibilities: Theoperationiswell-defined(orinner-defined)forallset’selements(as inclassicalalgebraicstructuresthisisclassicalOperation). Theoperationifwell-definedforsomeelements,indeterminateforother elements,andouter-definedforotherselements(thisis NeutroOperation). Theoperationisouter-definedforallset’selements(thisis AntiOperation). F.Smarandache,A.Rezaei,A.A.A.Agboola,Y.B.Jun,,,,R.A.Borzooei,B.Davvaz,A.BroumandSaeid,M.Akram,,,M.Hamidi,S.Mirvakilii OnNeutroQuadrupleGroups February16-19,2021,Kashan4/49
SectionName Axiom,NeutroAxiom,AntiAxiom Similarlyforanaxiom,definedonagivenset,endowedwithsomeoperation(s).Whenwedefineanaxiomonagivenset,itdoesnotautomatically meanthattheaxiomistrueforallset’selements.Wehavethreepossibilities again: Theaxiomistrueforallset’selements(totallytrue)(asinclassical algebraicstructures;thisisaclassicalAxiom). Theaxiomiftrueforsomeelements,indeterminateforotherelements, andfalseforotherelements(thisisNeutroAxiom). Theaxiomisfalseforallset’selements(thisisAntiAxiom). F.Smarandache,A.Rezaei,A.A.A.Agboola,Y.B.Jun,,,,R.A.Borzooei,B.Davvaz,A.BroumandSaeid,M.Akram,,,M.Hamidi,S.Mirvakilii OnNeutroQuadrupleGroups February16-19,2021,Kashan5/49
SectionName Algebra,NeutroAlgebra,AntiAlgebra Analgebraicstructurewho’salloperationsarewell-definedandall axiomsaretotallytrueiscalledClassicalAlgebraicStructure(or Algebra). AnalgebraicstructurethathasatleastoneNeutroOperationorone NeutroAxiom(andnoAntiOperationandnoAntiAxiom)iscalled NeutroAlgebraicStructure(orNeutroAlgebra). AnalgebraicstructurethathasatleastoneAntiOperationorAntiAxiom iscalledAntiAlgebraicStructure(orAntiAlgebra). < Algebra, NeutroAlgebra, AntiAlgebra >. F.Smarandache,A.Rezaei,A.A.A.Agboola,Y.B.Jun,,,,R.A.Borzooei,B.Davvaz,A.BroumandSaeid,M.Akram,,,M.Hamidi,S.Mirvakilii OnNeutroQuadrupleGroups February16-19,2021,Kashan6/49
SectionName TheNeutrosophicQuadrupleNumbersandtheAbsorbanceLawwereintroducedbySmarandachein2015[1];theyhavethegeneralform: N = a + bT + cI + dF, where a,b,c,d maybenumbersofanytype(natural, integer,rational,irrational,real,complex,etc.),where“a"istheknownpart oftheneutrosophicquadruplenumber N ,while“bT + cI + dF "istheunknownpartoftheneutrosophicquadruplenumber N ;thentheunknownpart issplitintothreesubparts:degreeofconfidence(T ),degreeofindeterminacy ofconfidence-nonconfidence(I),anddegreeofnonconfidence(F ). N isa four-dimensionalvectorthatcanalsobewrittenas: N =(a,b,c,d). F.Smarandache,A.Rezaei,A.A.A.Agboola,Y.B.Jun,,,,R.A.Borzooei,B.Davvaz,A.BroumandSaeid,M.Akram,,,M.Hamidi,S.Mirvakilii OnNeutroQuadrupleGroups February16-19,2021,Kashan7/49
SectionName Therearetranscendental,irrationaletc.numbersthatarenotwellknown,they areonlypartiallyknownandpartiallyunknown,theymayhaveinfinitelymany decimals.Noteventhemostmodernsupercomputerscancomputemorethan afewthousandsdecimals,buttheinfinitelymanyleftdecimalsstillremain unknown.Therefore,suchnumbersareverylittleknown(becauseonlyafinite numberofdecimalsareknown),andinfinitelyunknown(becauseaninfinite numberofdecimalsareunknown).Takeforexample: √2=1.4142 .... F.Smarandache,A.Rezaei,A.A.A.Agboola,Y.B.Jun,,,,R.A.Borzooei,B.Davvaz,A.BroumandSaeid,M.Akram,,,M.Hamidi,S.Mirvakilii OnNeutroQuadrupleGroups February16-19,2021,Kashan8/49
SectionName
Aneutrosophicsetofquadruplenumbersdenotedby NQ(X) isasetdefined by NQ(X)= {(a,bT,cI,dF ): a,b,c,d ∈ R or C} where T,I,F havetheirusualneutrosophiclogicmeanings. F.Smarandache,A.Rezaei,A.A.A.Agboola,Y.B.Jun,,,,R.A.Borzooei,B.Davvaz,A.BroumandSaeid,M.Akram,,,M.Hamidi,S.Mirvakilii OnNeutroQuadrupleGroups February16-19,2021,Kashan9/49
Definition
SectionName Multiplicationoftwoneutrosophicquadruplenumbers Multiplicationoftwoneutrosophicquadruplenumberscannotbecarriedout likemultiplicationoftworealorcomplexnumbers.Inordertomultiplytwo neutrosophicquadruplenumberstheprevalenceorderof {T,I,F } isrequired. Considerthefollowingprevalenceorders:Supposeinanoptimisticwaywe considertheprevalenceorder T I F .Thenwehave: TI = IT =max{T,I} = T,TF = FT =max{T,F } = T, IF = FI =max{I,F } = I,TT = T 2 = T, II = I 2 = I,FF = F 2 = F. Orweconsidertheprevalenceorder T ≺ I ≺ F .Thenwehave: TI = IT =max{T,I} = I,TF = FT =max{T,F } = F, IF = FI =max{I,F } = F,TT = T 2 = T, II = I 2 = I,FF = F 2 = F. F.Smarandache,A.Rezaei,A.A.A.Agboola,Y.B.Jun,,,,R.A.Borzooei,B.Davvaz,A.BroumandSaeid,M.Akram,,,M.Hamidi,S.Mirvakilii OnNeutroQuadrupleGroups February16-19,2021,Kashan10/49
SectionName Divisionoftwoneutrosophicquadruplenumbers Twoneutrosophicquadruplenumbers m =(a1 ,b1 T,c1 I,d1 F ) and n = (a2 ,b2 T,c2 I,d2 F ) cannotbedividedaswedoforrealandcomplexnumbers. Sincetheliteralneutrosophiccomponents T , I and F arenotinvertible,theinversionofaneutrosophicquadruplenumberorthedivisionofaneutrosophic quadruplenumberbyanotherneutrosophicquadruplenumbermustbecarried outasystematicway.Supposewearetoevaluate m/n.Thenwemustlookfor aneutrosophicquadruplenumber p =(x,yT,zI,wF ) equivalentto m/n.In thisway,wewrite m/n = p ⇒ (a1 ,b1 T,c1 I,d1 F ) (a2 ,b2 T,c2 I,d2 F ) =(x,yT,zI,wF ) ⇔ (a2 ,b2 T,c2 I,d2 F )(x,yT,zI,wF ) ≡ (a1 ,b1 T,c1 I,d1 F ). (1) F.Smarandache,A.Rezaei,A.A.A.Agboola,Y.B.Jun,,,,R.A.Borzooei,B.Davvaz,A.BroumandSaeid,M.Akram,,,M.Hamidi,S.Mirvakilii OnNeutroQuadrupleGroups February16-19,2021,Kashan11/49
SectionName Divisionoftwoneutrosophicquadruplenumbers Assumingtheprevalenceorder T I F andfromtheequalityoftwo neutrosophicquadruplenumbers,weobtainfromEquation(1) a2 x = a1 (2) b2 x +(a2 + b2 + c2 + d2 )y + b2 z + b2 w = b1 (3) c2 x +(a2 + c2 + d2 )z + c2 w = c1 (4) d2 x +(a2 + d2 )w = d1 (5) asystemoflinearequationsinunknowns x,y,z and w. F.Smarandache,A.Rezaei,A.A.A.Agboola,Y.B.Jun,,,,R.A.Borzooei,B.Davvaz,A.BroumandSaeid,M.Akram,,,M.Hamidi,S.Mirvakilii OnNeutroQuadrupleGroups February16-19,2021,Kashan12/49
SectionName Divisionoftwoneutrosophicquadruplenumbers Bysimilarlyassumingtheprevalenceorder T ≺ I ≺ F ,weobtainfrom Equation(1) a2 x = a1 (6) b2 x +(a2 + b2 )y = b1 (7) c2 x + c2 y +(a2 + b2 + c2 )z = c1 (8) d2 x + d2 y + d2 z +(a2 + b2 + c2 + d2 )w = d1 (9) asystemoflinearequationsinunknowns x,y,z and w. F.Smarandache,A.Rezaei,A.A.A.Agboola,Y.B.Jun,,,,R.A.Borzooei,B.Davvaz,A.BroumandSaeid,M.Akram,,,M.Hamidi,S.Mirvakilii OnNeutroQuadrupleGroups February16-19,2021,Kashan13/49
SectionName Example Let a =(2, −T,I, 2F ) and b =(1, 2T, −I,F ) betwoneutrosophicquadruple numbersin NQ(R). (i) Fortheprevalenceorder T I F ,weobtain (2, −T,I, 2F ) (1, 2T, I,F ) = 2, − 11 3 T, 3I, 0F . (ii) Fortheprevalenceorder T ≺ I ≺ F ,weobtain (2, −T,I, 2F ) (1, 2T, −I,F ) = 2, − 5 3 T, 2 3 I, 1 3 F . F.Smarandache,A.Rezaei,A.A.A.Agboola,Y.B.Jun,,,,R.A.Borzooei,B.Davvaz,A.BroumandSaeid,M.Akram,,,M.Hamidi,S.Mirvakilii OnNeutroQuadrupleGroups February16-19,2021,Kashan14/49
SectionName Neutrosophicquadrupleset Let NQ(X) beaneutrosophicquadruplesetandlet ∗ : NQ(X) × NQ(X) → NQ(X) beaclassicalbinaryoperationon NQ(X).Thecouple (NQ(X), ∗) iscalledaneutrosophicquadruplealgebraicstructure.The structure (NQ(X), ∗) isnamedaccordingtotheclassicallawsandaxioms satisfiedorobeyedby ∗. F.Smarandache,A.Rezaei,A.A.A.Agboola,Y.B.Jun,,,,R.A.Borzooei,B.Davvaz,A.BroumandSaeid,M.Akram,,,M.Hamidi,S.Mirvakilii OnNeutroQuadrupleGroups February16-19,2021,Kashan15/49
SectionName Neutrosophicquadruplehyperset If ∗ : NQ(X) × NQ(X) → P(NQ(X)) istheclassicalhyperoperation on NQ(X).Thenthecouple (NQ(X), ∗) iscalledaneutrosophicquadruplehyperalgebraicstructure;andthehyperstructure (NQ(X), ∗) isnamed accordingtotheclassicallawsandaxiomssatisfiedby ∗. F.Smarandache,A.Rezaei,A.A.A.Agboola,Y.B.Jun,,,,R.A.Borzooei,B.Davvaz,A.BroumandSaeid,M.Akram,,,M.Hamidi,S.Mirvakilii OnNeutroQuadrupleGroups February16-19,2021,Kashan16/49
SectionName Neutrosophicquadruplehomomorphism If (NQ(X), ∗) and (NQ(Y ), ◦) aretwoneutrosophicquadruplealgebraic structures.Themapping φ :(NQ(X), ∗) → (NQ(Y ), ◦) iscalledaneotrosophicquadruplehomomorphismif φ preserves ∗, ◦ andliteralneutrosophic components T,I and F thatisif: (i) φ(x ∗ y)= φ(x) ◦ φ(y) ∀ x,y ∈ NQ(X). (ii) φ(T )= T . (iii) φ(I)= I. (iv) φ(F )= F . F.Smarandache,A.Rezaei,A.A.A.Agboola,Y.B.Jun,,,,R.A.Borzooei,B.Davvaz,A.BroumandSaeid,M.Akram,,,M.Hamidi,S.Mirvakilii OnNeutroQuadrupleGroups February16-19,2021,Kashan17/49
SectionName
(i) (NQ(Z), +), (NQ(Q), +), (NQ(R), +) and (NQ(C), +) areabelian groups. (ii) (NQ(Z), +, ×), (NQ(Q), +, ×), (NQ(R), +, ×) and (NQ(C), +, ×) arecommutativerings. (iii) (NQ(Z), ×) isacommutativemonoid. (iv) (
Z
, ×) isnotagroup. (v)
Z
, ÷) isnotagroup. F.Smarandache,A.Rezaei,A.A.A.Agboola,Y.B.Jun,,,,R.A.Borzooei,B.Davvaz,A.BroumandSaeid,M.Akram,,,M.Hamidi,S.Mirvakilii OnNeutroQuadrupleGroups February16-19,2021,Kashan18/49
Theorem
NQ(
)
(NQ(
)
SectionName Neutrosophicquadruplegroups Let NQ(G) beanonemptysetandlet ∗ : NQ(G) × NQ(G) → NQ(G) bea binaryoperationon NQ(G).Thecouple (NQ(G), ∗) iscalledaneutrosophic quadruplegroupifthefollowingconditionshold: (QG1) x ∗ y ∈ G ∀x,y ∈ NQ(G) [closurelaw]. (QG2) x ∗ (y ∗ z)=(x ∗ y) ∗ z ∀x,y,z ∈ G [axiomofassociativity]. F.Smarandache,A.Rezaei,A.A.A.Agboola,Y.B.Jun,,,,R.A.Borzooei,B.Davvaz,A.BroumandSaeid,M.Akram,,,M.Hamidi,S.Mirvakilii OnNeutroQuadrupleGroups February16-19,2021,Kashan19/49
SectionName (QG3) Thereexists e ∈ NQ(G) suchthat x ∗ e = e ∗ x = x ∀x ∈ NQ(G) [axiomof existenceofneutralelement]. (QG4) Thereexists y ∈ NQ(G) suchthat x ∗ y = y ∗ x = e ∀x ∈ NQ(G) [axiomof existenceofinverseelement]where e istheneutralelementof NQ(G). Ifinaddition ∀x,y ∈ NQ(G),wehave (QG5) x ∗ y = y ∗ x,then (NQ(G), ∗) iscalledacommutativeneutrosophic quadruplegroup. F.Smarandache,A.Rezaei,A.A.A.Agboola,Y.B.Jun,,,,R.A.Borzooei,B.Davvaz,A.BroumandSaeid,M.Akram,,,M.Hamidi,S.Mirvakilii OnNeutroQuadrupleGroups February16-19,2021,Kashan20/49
SectionName NeutroSophicationofthelawandaxiomsoftheneutrosophicquadruple (NQ(G)1) Thereexistsomeduplets (x,y), (u,v), (p,q), ∈ NQ(G) suchthat x ∗ y ∈ G (inner-definedwithdegreeoftruthT)and[u ∗ v = indeterminate(withdegree ofindeterminacyI)or p ∗ q ∈ NQ(G) (outer-defined/falsehoodwithdegreeof falsehoodF)][NeutroClosureLaw]. (NQ(G)2) Thereexistsometriplets (x,y,z), (p,q,r), (u,v,w) ∈ NQ(G) suchthat x ∗ (y ∗z)=(x∗y)∗z (inner-definedwithdegreeoftruthT)and[[p∗(q ∗r)]or [(p∗ q) ∗ r]= indeterminate(withdegreeofindeterminacyI)or u ∗ (v ∗ w) = (u∗v)∗w (outer-defined/falsehoodwithdegreeoffalsehoodF)][NeutroAxiom ofassociativity(NeutroAssociativity)]. F.Smarandache,A.Rezaei,A.A.A.Agboola,Y.B.Jun,,,,R.A.Borzooei,B.Davvaz,A.BroumandSaeid,M.Akram,,,M.Hamidi,S.Mirvakilii OnNeutroQuadrupleGroups February16-19,2021,Kashan21/49
SectionName NeutroSophicationofthelawandaxiomsoftheneutrosophicquadruple (NQ(G)3) Thereexistsanelement e ∈ NQ(G) suchthat x ∗ e = e ∗ x = x (inner-defined withdegreeoftruthT)and[[x ∗ e]or[e ∗ x]= indeterminate(withdegreeof indeterminacyI)or x ∗ e = x = e ∗ x (outer-defined/falsehoodwithdegree offalsehoodF)]foratleastone x ∈ NQ(G) [NeutroAxiomofexistenceof neutralelement(NeutroNeutralElement)]. F.Smarandache,A.Rezaei,A.A.A.Agboola,Y.B.Jun,,,,R.A.Borzooei,B.Davvaz,A.BroumandSaeid,M.Akram,,,M.Hamidi,S.Mirvakilii OnNeutroQuadrupleGroups February16-19,2021,Kashan22/49
SectionName NeutroSophicationofthelawandaxiomsoftheneutrosophicquadruple (NQ(G)4) Thereexistsanelement u ∈ NQ(G) suchthat x ∗ u = u ∗ x = e (innerdefinedwithdegreeoftruthT)and[[x ∗ u]or[u ∗ x)]= indeterminate(with degreeofindeterminacyI)or x ∗ u = e = u ∗ x (outer-defined/falsehoodwith degreoffalsehoodF)]foratleastone x ∈ G [NeutroAxiomofexistenceof inverseelement(NeutroInverseElement)]where e isaNeutroNeutralElement in NQ(G). F.Smarandache,A.Rezaei,A.A.A.Agboola,Y.B.Jun,,,,R.A.Borzooei,B.Davvaz,A.BroumandSaeid,M.Akram,,,M.Hamidi,S.Mirvakilii OnNeutroQuadrupleGroups February16-19,2021,Kashan23/49
SectionName NeutroSophicationofthelawandaxiomsoftheneutrosophicquadruple (NQ(G)5) Thereexistsomeduplets (x,y), (u,v), (p,q) ∈ NQ(G) suchthat x ∗ y = y ∗ x (inner-definedwithdegreeoftruthT)and[[u∗v]or[v∗u]= indeterminate(with degreeofindeterminacyI)or p∗q = q ∗p (outer-defined/falsehoodwithdegree offalsehoodF)][NeutroAxiomofcommutativity(NeutroCommutativity)]. F.Smarandache,A.Rezaei,A.A.A.Agboola,Y.B.Jun,,,,R.A.Borzooei,B.Davvaz,A.BroumandSaeid,M.Akram,,,M.Hamidi,S.Mirvakilii OnNeutroQuadrupleGroups February16-19,2021,Kashan24/49
SectionName NeutroQuadrupleGroup ANeutroQuadrupleGroup NQ(G) isanalternativetotheneutrosophic quadruplegroup Q(G) thathasatleastoneNeutroLaworatleastoneof {NQ(G)1,NQ(G)2,NQ(G)3,NQ(G)4} withnoAntiLaworAntiAxiom. F.Smarandache,A.Rezaei,A.A.A.Agboola,Y.B.Jun,,,,R.A.Borzooei,B.Davvaz,A.BroumandSaeid,M.Akram,,,M.Hamidi,S.Mirvakilii OnNeutroQuadrupleGroups February16-19,2021,Kashan25/49
SectionName NeutroCommutativeQuadrupleGroup ANeutroCommutativeQuadrupleGroup NQ(G) isanalternativetothecommutativeneutrosophicquadruplegroup Q(G) thathasatleastoneNeutroLaw oratleastoneof {NQ(G)1,NQ(G)2,NQ(G)3,NQ(G)4} and NQ(G)5 withnoAntiLaworAntiAxiom. F.Smarandache,A.Rezaei,A.A.A.Agboola,Y.B.Jun,,,,R.A.Borzooei,B.Davvaz,A.BroumandSaeid,M.Akram,,,M.Hamidi,S.Mirvakilii OnNeutroQuadrupleGroups February16-19,2021,Kashan26/49
SectionName Theorem Let U beanonemptyfiniteorinfiniteuniverseofdiscourseandlet S beafinite orinfinitesubsetof U.If n classicaloperations(lawsandaxioms)aredefined on S where n ≥ 1,thentherewillbe (2n − 1) NeutroAlgebrasand (3n − 2n) AntiAlgebras(see[2]). F.Smarandache,A.Rezaei,A.A.A.Agboola,Y.B.Jun,,,,R.A.Borzooei,B.Davvaz,A.BroumandSaeid,M.Akram,,,M.Hamidi,S.Mirvakilii OnNeutroQuadrupleGroups February16-19,2021,Kashan27/49
SectionName
Let (NQ
beaneutrosophicquadruplegroup.Then: (i)
F.Smarandache,A.Rezaei,A.A.A.Agboola,Y.B.Jun,,,,R.A.Borzooei,B.Davvaz,A.BroumandSaeid,M.Akram,,,M.Hamidi,S.Mirvakilii OnNeutroQuadrupleGroups February16-19,2021,Kashan28/49
Theorem
(G), ∗)
thereare15typesofNeutroQuadrupleGroups, (ii) thereare31typesofNeutroCommutativeQuadrupleGroups.
SectionName
F.Smarandache,A.Rezaei,A.A.A.Agboola,Y.B.Jun,,,,R.A.Borzooei,B.Davvaz,A.BroumandSaeid,M.Akram,,,M.Hamidi,S.Mirvakilii OnNeutroQuadrupleGroups February16-19,2021,Kashan29/49
Theorem Forpositiveintegers n =2, 3, 4, ··· , (i) (NQ(Zn), −) isaNeutroQuadrupleGroup. (ii) (NQ(Zn), ×) isaNeutroCommutativeQuadrupleGroup.
Theorem
(i) (NQ(Z), −) isaNeutroQuadrupleGroup. (ii) (NQ(Z), ×) isaNeutroCommutativeQuadrupleGroup. (iii) (NQ(Z), ÷) isaNeutroCommutativeQuadrupleGroup.
SectionName
F.Smarandache,A.Rezaei,A.A.A.Agboola,Y.B.Jun,,,,R.A.Borzooei,B.Davvaz,A.BroumandSaeid,M.Akram,,,M.Hamidi,S.Mirvakilii OnNeutroQuadrupleGroups February16-19,2021,Kashan30/49
SectionName NeutroClosureof ÷ over NQ(Z) Forthedegreeoftruth,let a =(0, 0T,I, 0F ) ∈ NQ(Z).Then a ÷ a = (0, 0T,I, 0F ) (0, 0T,I, 0F ) =(1 − k1 − k2 , 0T,k1 I,k2 F ) ∈ NQ(Z),k1 ,k2 ∈ Z. F.Smarandache,A.Rezaei,A.A.A.Agboola,Y.B.Jun,,,,R.A.Borzooei,B.Davvaz,A.BroumandSaeid,M.Akram,,,M.Hamidi,S.Mirvakilii OnNeutroQuadrupleGroups February16-19,2021,Kashan31/49
SectionName NeutroAssociativityof ÷ over NQ(Z) Forthedegreeofindeterminacy,let a =(4, 5T, −2I, −7F ), b =(0, −6T,I, 3F ) ∈ NQ(Z).Then a ÷ b = (4, 5T, −2I, −7F ) (0, 6T,I, 3F ) = 4 0 , ?T, ?I, ?F ∈ NQ(Z). F.Smarandache,A.Rezaei,A.A.A.Agboola,Y.B.Jun,,,,R.A.Borzooei,B.Davvaz,A.BroumandSaeid,M.Akram,,,M.Hamidi,S.Mirvakilii OnNeutroQuadrupleGroups February16-19,2021,Kashan32/49
SectionName NeutroAssociativityof ÷ over NQ(Z) Forthedegreeoffalsehood,let a =(0, 0T, 0I,F ), b =(0, 0T, 0I, 2F ) ∈ NQ(Z).Then a ÷ b = (0, 0T, 0I,F ) (0, 0T, 0I, 2F ) = 1 2 − k, 0T, 0I,kF ∈ NQ(Z),k ∈ Z. F.Smarandache,A.Rezaei,A.A.A.Agboola,Y.B.Jun,,,,R.A.Borzooei,B.Davvaz,A.BroumandSaeid,M.Akram,,,M.Hamidi,S.Mirvakilii OnNeutroQuadrupleGroups February16-19,2021,Kashan33/49
SectionName NeutroAssociativityof ÷ over NQ(Z) Forthedegreeoftruth,let a =(6, 6T, 6I, 6F ),b =(2, 2T, 2I, 2F ), c =( 1, 0T, 0I, 0F ) ∈ NQ(Z).Then a ÷ (b ÷ c)=(6, 6T, 6I, 6F ) ÷ ((2, 2T, 2I, 2F ) ÷ ( 1, 0T, 0I, 0F )) =(6, 6T, 6I, 6F ) ÷ (−2, 0T, 0I, 0F ) =(−3, 0T, 0I, 0F ). (a ÷ b) ÷ c =((6, 6T, 6I, 6F ) ÷ (2, 2T, 2I, 2F )) ÷ (−1, 0T, 0I, 0F ) =(3, 0T, 0I, 0F ) ÷ ( 1, 0T, 0I, 0F ) =( 3, 0T, 0I, 0F ). F.Smarandache,A.Rezaei,A.A.A.Agboola,Y.B.Jun,,,,R.A.Borzooei,B.Davvaz,A.BroumandSaeid,M.Akram,,,M.Hamidi,S.Mirvakilii OnNeutroQuadrupleGroups February16-19,2021,Kashan34/49
SectionName NeutroAssociativityof ÷ over NQ(Z) Forthedegreeofindeterminacy,let a =(4, T, 2I, 7F ), b =(0,T, 0I, −8F ),c =(0, 0T, 9I, −F ) ∈ NQ(Z).Then a ÷ (b ÷ c)=(4, −T, 2I, −7F ) ÷ ((0,T, 0I, −8F ) ÷ (0, 0T, 9I, −F )) =(4, −T, 2I, −7F ) ÷ 8 − k, 1 8 T, −9I,kF ,k ∈ Z =(?, ?T, ?I, ?F ). (a ÷ b) ÷ c =((4, T, 2I, 7F ) ÷ (0,T, 0I, 8F )) ÷ (0, 0T, 9I, F ) = 4 0 , ?T, ?I, ?F ÷ (0, 0T, 9I, F ) =(?, ?T, ?I, ?F ). F.Smarandache,A.Rezaei,A.A.A.Agboola,Y.B.Jun,,,,R.A.Borzooei,B.Davvaz,A.BroumandSaeid,M.Akram,,,M.Hamidi,S.Mirvakilii OnNeutroQuadrupleGroups February16-19,2021,Kashan35/49
SectionName NeutroAssociativityof ÷ over NQ(Z) Forthedegreeoffalsehood,let a =(0, 5T, 0I, 0F ),b =(0,T, 0I, 0F ), c =(5, 0T, 0I, 0F ) ∈ NQ(Z).Then a ÷ (b ÷ c)=(0, 5T, 0I, 0F ) ÷ ((0,T, 0I, 0F ) ÷ (5, 0T, 0I, 0F )) =(0, 5T, 0I, 0F ) ÷ 0, 1 5 T, 0I, 0F =(25 k1 k2 k3 ,k1 T,k2 I,k3 F ) ∈ NQ(Z), (a ÷ b) ÷ c =((0, 5T, 0I, 0F ) ÷ (0,T, 0I, 0F )) ÷ (5, 0T, 0I, 0F ) =(5 − k1 − k2 − k3 ,k1 T,k2 I,k3 F ) ÷ (5, 0T, 0I, 0F ), = 1 5 (5 k1 k2 k3 ), 1 5 k1 T, 1 5 k2 I, 1 5 k3 F ∈ NQ(Z) k1 ,k2 ,k3 ∈ Z. F.Smarandache,A.Rezaei,A.A.A.Agboola,Y.B.Jun,,,,R.A.Borzooei,B.Davvaz,A.BroumandSaeid,M.Akram,,,M.Hamidi,S.Mirvakilii OnNeutroQuadrupleGroups February16-19,2021,Kashan36/49
SectionName ExistenceofNeutroUnitaryElementandNeutroInverseElementin NQ(Z) Let a =(0,T, 0I, 0F ),b =(0, 0T,I, 0F ),c =(0, 0T, 0I,F ) ∈ NQ(Z). Then a ÷ a = (0,T, 0I, 0F ) (0,T, 0I, 0F ) =(1 − k1 − k2 − k3 ,k1 T,k2 I,k3 F ) (10) b ÷ b = (0, 0T,I, 0F ) (0, 0T,I, 0F ) =(1 k1 k2 , 0T,k1 I,k2 F ) (11) F.Smarandache,A.Rezaei,A.A.A.Agboola,Y.B.Jun,,,,R.A.Borzooei,B.Davvaz,A.BroumandSaeid,M.Akram,,,M.Hamidi,S.Mirvakilii OnNeutroQuadrupleGroups February16-19,2021,Kashan37/49
c ÷ c = (0, 0T, 0I,F ) (0, 0T, 0I,F ) =(1 k, 0T, 0I,kF ) (12) a ÷ b = (0,T, 0I, 0F ) (0, 0T,I, 0F ) =(−(k1 + k2 ),T,k1 I,k2 F ) (13) b ÷ a = (0, 0T,I, 0F ) (0,T, 0I, 0F ) =( (k1 + k2 + k3 ),k1 T,k2 I,k3 F ) (14)
SectionName
F.Smarandache,A.Rezaei,A.A.A.Agboola,Y.B.Jun,,,,R.A.Borzooei,B.Davvaz,A.BroumandSaeid,M.Akram,,,M.Hamidi,S.Mirvakilii OnNeutroQuadrupleGroups February16-19,2021,Kashan38/49
k,k1 ,k2 ,k3 ∈ Z.
SectionName ExistenceofNeutroUnitaryElementandNeutroInverseElementin NQ(Z) Forthedegreeoftruth,putting k1 =1,k2 = k3 =0 inEquation(10), k1 =1, k2 =0 inEquation(11)and k =1 inEquation(12)wewillobtain a ÷ a = a,b ÷ b = b and c ÷ c = c.Theseshowthat a,b,c arerespectively NeutroUnitaryElementsandNeutroInverseElementsin NQ(Z). F.Smarandache,A.Rezaei,A.A.A.Agboola,Y.B.Jun,,,,R.A.Borzooei,B.Davvaz,A.BroumandSaeid,M.Akram,,,M.Hamidi,S.Mirvakilii OnNeutroQuadrupleGroups February16-19,2021,Kashan39/49
SectionName ExistenceofNeutroUnitaryElementandNeutroInverseElementin NQ(Z) Forthedegreeoffalsehood,putting k1 =1,k2 = k3 =0 inEquation(10), k1 =1,k2 =0 inEquation(11)and k =1 inEquation(12)wewillobtain a ÷ a = a,b ÷ b = b and c ÷ c = c.Theseshowthat a,b,c arerespectively notNeutroUnitaryElementsandNeutroInverseElementsin NQ(Z). F.Smarandache,A.Rezaei,A.A.A.Agboola,Y.B.Jun,,,,R.A.Borzooei,B.Davvaz,A.BroumandSaeid,M.Akram,,,M.Hamidi,S.Mirvakilii OnNeutroQuadrupleGroups February16-19,2021,Kashan40/49
SectionName NeutroCommtativityof ÷ over NQ(Z) Forthedegreeoftruth,putting k1 =1,k2 = k3 =0 inEquation(10), k1 =1, k2 =0 inEquation(11)and k =1 inEquation(12)wewillobtain a ÷ a = a,b ÷ b = b and c ÷ c = c.Theseshowthecommutativityof ÷ wrt a,b and c NQ(Z). Forthedegreeoffalsehood,putting k1 = k2 = k3 =1 inEquation(13) andEquation(14),wewillobtain a ÷ b =(−2,T,I,F ) and b ÷ a = (−3,T,I,F ) = a ÷ b.Hence, ÷ isNeutroCommutativein NQ(Z). F.Smarandache,A.Rezaei,A.A.A.Agboola,Y.B.Jun,,,,R.A.Borzooei,B.Davvaz,A.BroumandSaeid,M.Akram,,,M.Hamidi,S.Mirvakilii OnNeutroQuadrupleGroups February16-19,2021,Kashan41/49
SectionName NeutroQuadrupleSubgroup Let (NQ(G), ∗) beaneutrosophicquadruplegroup.Anonemptysubset NQ(H) of NQ(G) iscalledaNeutroQuadrupleSubgroupof NQ(G) if (NQ(H), ∗) isaneutrosophicquadruplegroupofthesametypeas (NQ(G), ∗). F.Smarandache,A.Rezaei,A.A.A.Agboola,Y.B.Jun,,,,R.A.Borzooei,B.Davvaz,A.BroumandSaeid,M.Akram,,,M.Hamidi,S.Mirvakilii OnNeutroQuadrupleGroups February16-19,2021,Kashan42/49
SectionName
Example
F.Smarandache,A.Rezaei,A.A.A.Agboola,Y.B.Jun,,,,R.A.Borzooei,B.Davvaz,A.BroumandSaeid,M.Akram,,,M.Hamidi,S.Mirvakilii OnNeutroQuadrupleGroups February16-19,2021,Kashan43/49
(i) For n =2, 3, 4, ··· (NQ(nZ), ) isaNeutroQuadrupleSubgroupof (NQ(Z), −). (ii) For n =2, 3, 4, ··· (NQ(nZ), ×) isaNeutroQuadrupleSubgroupof (NQ(Z), ×).
SectionName Example (i) Let NQ(H)= {(a,bT,cI,dF ): a,b,c,d ∈{1, 2, 3}} beasubsetofthe NeutroQuadrupleGroup (NQ(Z4 ), ).Then (NQ(H), ) isa NeutroQuadrupleSubgroupof (NQ(Z4 ), −). (ii) Let NQ(K)= {(w,xT,yI,zF ): a,b,c,d ∈{1, 3, 5}} beasubsetof theNeutroQuadrupleGroup (NQ(Z6 ), ×).Then (NQ(H), ×) isa NeutroQuadrupleSubgroupof (NQ(Z6 ), ×). F.Smarandache,A.Rezaei,A.A.A.Agboola,Y.B.Jun,,,,R.A.Borzooei,B.Davvaz,A.BroumandSaeid,M.Akram,,,M.Hamidi,S.Mirvakilii OnNeutroQuadrupleGroups February16-19,2021,Kashan44/49
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A.A.A.Agboola,M.A.Ibrahim, IntroductiontoAntiRings,Neutrosophic SetsandSystems,36(2020),293–307.
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M.Akram,H.Gulzar,F.Smarandache,S.Broumi, ApplicationofneutrosophicsoftsetstoK-Algebras,Axioms7(4)(2018),83.
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M.Hamidi,F.Smarandache, Neutro-BCK-Algebra,InternationalJournal ofNeutrosophicScience,8(2020),110–117.
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