On NeutroQuadrupleGroups

2021·0.4 MB·

by  F. Smarandache

On NeutroQuadrupleGroups
F.Smarandache A.Rezaei A.A.A.Agboola Y.B.Jun
R.A.Borzooei B.Davvaz A.BroumandSaeid M.Akram
M.Hamidi S.Mirvakilii
February 16-19, 2021,Kashan
F.Smarandache,A.Rezaei,A.A.A.Agboola,Y.B.Jun,,,,R.AO.BnoNrzeouotreoi,QBu.aDdrauvpvlaezG,rAou.pBsroumandSaeid,MFe.bArukarraym,1,,6M-1.9H,a2m02id1i,,KSa.sMhairnvakilii1/49
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NeutroAlgebraicstructuresandAntiAlgebraicstructures
As generalizations and alternatives of classical algebraic structures Florentin
Smarandache has introduced in 2019 the NeutroAlgebraic structures (or Neu-
troAlgebras)andAntiAlgebraicstructures(orAntiAlgebras). Unliketheclas-
sicalalgebraicstructures,wherealloperationsarewell-definedandallaxioms
are totally true, in NeutroAlgebras and AntiAlgebras the operations may be
partiallywell-definedandtheaxiomspartiallytrueorrespectivelytotallyouter-
definedandtheaxiomstotallyfalse.
F.Smarandache,A.Rezaei,A.A.A.Agboola,Y.B.Jun,,,,R.AO.BnoNrzeouotreoi,QBu.aDdrauvpvlaezG,rAou.pBsroumandSaeid,MFe.bArukarraym,1,,6M-1.9H,a2m02id1i,,KSa.sMhairnvakilii2/49
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NeutroAlgebraicstructuresandAntiAlgebraicstructures
These NeutroAlgebras and AntiAlgebras form a new field of research, which
is inspired from our real world. In this talk, we study neutrosophic quadruple
algebraicstructuresandNeutroQuadrupleAlgebraicStructures. NeutroQuadru-
pleGroupisstudiedinparticularandseveralexamplesareprovided. Itisshown
that(NQ(Z),÷)isaNeutroQuadrupleGroup.
F.Smarandache,A.Rezaei,A.A.A.Agboola,Y.B.Jun,,,,R.AO.BnoNrzeouotreoi,QBu.aDdrauvpvlaezG,rAou.pBsroumandSaeid,MFe.bArukarraym,1,,6M-1.9H,a2m02id1i,,KSa.sMhairnvakilii3/49
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Operation,NeutroOperation,AntiOperation
When we define an operation on a given set, it does not automatically mean
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.AO.BnoNrzeouotreoi,QBu.aDdrauvpvlaezG,rAou.pBsroumandSaeid,MFe.bArukarraym,1,,6M-1.9H,a2m02id1i,,KSa.sMhairnvakilii4/49
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Axiom,NeutroAxiom,AntiAxiom
Similarly for an axiom, defined on a given set, endowed with some opera-
tion(s). When we define an axiom on a given set, it does not automatically
mean that the axiom is true for all set’s elements. We have three possibilities
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.AO.BnoNrzeouotreoi,QBu.aDdrauvpvlaezG,rAou.pBsroumandSaeid,MFe.bArukarraym,1,,6M-1.9H,a2m02id1i,,KSa.sMhairnvakilii5/49
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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.AO.BnoNrzeouotreoi,QBu.aDdrauvpvlaezG,rAou.pBsroumandSaeid,MFe.bArukarraym,1,,6M-1.9H,a2m02id1i,,KSa.sMhairnvakilii6/49
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The Neutrosophic Quadruple Numbers and the Absorbance Law were intro-
ducedbySmarandachein2015[1];theyhavethegeneralform:
N = a+bT +cI +dF,wherea,b,c,dmaybenumbersofanytype(natural,
integer, rational, irrational, real, complex, etc.), where “a" is the known part
of the neutrosophic quadruple number N, while “bT + cI + dF" is the un-
known part of the neutrosophic quadruple number N; then the unknown part
issplitintothreesubparts: degreeofconfidence(T), degreeofindeterminacy
of confidence-nonconfidence (I), and degree of nonconfidence (F). N is a
four-dimensionalvectorthatcanalsobewrittenas: N = (a,b,c,d).
F.Smarandache,A.Rezaei,A.A.A.Agboola,Y.B.Jun,,,,R.AO.BnoNrzeouotreoi,QBu.aDdrauvpvlaezG,rAou.pBsroumandSaeid,MFe.bArukarraym,1,,6M-1.9H,a2m02id1i,,KSa.sMhairnvakilii7/49
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Therearetranscendental,irrationaletc. numbersthatarenotwellknown,they
areonlypartiallyknownandpartiallyunknown,theymayhaveinfinitelymany
decimals. Not even the most modern supercomputers can compute more than
a few thousands decimals, but the infinitely many left decimals still remain
unknown. Therefore,suchnumbersareverylittleknown(becauseonlyafinite
number of decimals are known), and infinitely unknown (because an infinite
√
numberofdecimalsareunknown). Takeforexample: 2 = 1.4142....
F.Smarandache,A.Rezaei,A.A.A.Agboola,Y.B.Jun,,,,R.AO.BnoNrzeouotreoi,QBu.aDdrauvpvlaezG,rAou.pBsroumandSaeid,MFe.bArukarraym,1,,6M-1.9H,a2m02id1i,,KSa.sMhairnvakilii8/49
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Definition
AneutrosophicsetofquadruplenumbersdenotedbyNQ(X)isasetdefined
by
NQ(X) = {(a,bT,cI,dF) : a,b,c,d ∈ RorC}
whereT,I,F havetheirusualneutrosophiclogicmeanings.
F.Smarandache,A.Rezaei,A.A.A.Agboola,Y.B.Jun,,,,R.AO.BnoNrzeouotreoi,QBu.aDdrauvpvlaezG,rAou.pBsroumandSaeid,MFe.bArukarraym,1,,6M-1.9H,a2m02id1i,,KSa.sMhairnvakilii9/49
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Multiplicationoftwoneutrosophicquadruplenumbers
Multiplication of two neutrosophic quadruple numbers cannot be carried out
like multiplication of two real or complex numbers. In order to multiply two
neutrosophicquadruplenumberstheprevalenceorderof{T,I,F}isrequired.
Consider the following prevalence orders: Suppose in an optimistic way we
considertheprevalenceorderT (cid:31) I (cid:31) F. Thenwehave:
TI = IT = max{T,I} = T, TF = FT = max{T,F} = T,
IF = FI = max{I,F} = I, TT = T2 = T,
II = I2 = I, FF = F2 = F.
OrweconsidertheprevalenceorderT ≺ I ≺ F. Thenwehave:
TI = IT = max{T,I} = I, TF = FT = max{T,F} = F,
IF = FI = max{I,F} = F, TT = T2 = T,
II = I2 = I, FF = F2 = F.
F.Smarandache,A.Rezaei,A.A.A.Agboola,Y.B.Jun,,,,R.AO.BnoNrzeouotreoi,QBu.aDdrauvpvlaezG,rAou.pBsroumandSaeid,FMeb.rAuakrryam1,6,,-1M9.,H2a0m21id,iK,aSs.hManirvakil1ii0/49