InternationalJournalofAppliedMathematics& StatisticalSciences(IJAMSS); ISSN(P):2319–3972;ISSN(E):2319–3980 Vol.11,Issue2,Jul–Dec2022;1–12 ©IASET
ONFUZZYSUBIS-ALGEBRAS
SundusNajahJabir
FacultyofEducation,Kufa,University,Iraq
ABSTRACT
InthispaperwestudysubIS-,algebra,fuzzysubIS-,algebra,normalsubIS-algebra,fuzzynormalsubIS-algebra,fuzzy normalsubIS-algebraoffuzzysubIS-algebra.
KEYWORDS:BCI-Algebras,Semigroup,IS-Algebra,SubIS-Algebra,IS-AlgebraHomomorphism,TheCartesian Product,FuzzySubIS-Algebra,NormalSubIS-Algebra
ArticleHistory
Received:26Jul2022|Revised:27Jul2022|Accepted:28Jul2022
1.INTRODUCTION
In1996,K.IsekiintroducedthenotionofBCK/BCI-algebras.ForthegeneraldevelopmentofBCK/BCI-algebras[6],In [2]introducedanewclassofalgebrasrelatedtoBCI-algebrasandsemigroupscalledaBCI-semigroup.Inthispaperwe studyanewtypeoffuzzysubIS-algebraarenormalsubIS-algebra,fuzzynormalsubIS-algebraandfuzzynormalsubISalgebraoffuzzysubIS-algebra.
2.PRELIMINARY
Wereviewsomedefinitionsthatwillbeusefulinourresults.
Definition21:ASemigroupisanorderedpair(,) G,whereGisanon-emptysetand“”isanassociativebinary operationonG.[3]
Definition2.2ABCI-algebraistriple(G,*,0)whereGisanon-emptyset“*”isbinaryoperationonG,0 Gisan elementsuchthatthefollowingaxiomsaresatisfiedforalls,t,rG:
((st)(sr))(rt)=0,
(s(st)t=0,
ss=0,
st=0andts=0implies=t
If0s=0forallsGthenGiscalledBCK-algebra.[1]
Definition2.3:AnIS-algebraisanon-emptysetwithtwobinaryoperation“*”and“ . ”andconstant0satisfyingthe axioms:
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(G,,0)isaBCI-algebra.
(G,.)isaSemigroup, s.(tr)=(s.t)(s.r)and(st).r=(s.r)(t.r),foralls,t,r G.[6]
Example2.4:letG={0,a,b,c}define“*”operationandmultiplication“”bythefollowingtables:
ThenbyroutinecalculationswecanseethatGisanIS-algebra.[6]
Definition2.5:LetGandYbeIS-algebraamappingfG : iscalledanIS-algebrahomomorphism(briefly homomorphism)if(*)()*() fxyfxfyand()()() fxyfxfyforallxyG ,
LetfG : IS-algebrahomomorphism.Thentheset{:()0} xGfx iscalledthekerneloff,and denotebyKerf.Moreover,theset{():} fxxGiscalledtheimageoffanddenotebyImf.[4]
Definition2.6:LetλandμbethefuzzysubsetsinasetG,theCartesianproduct
λ×μ:G×G [0,1]isdefinedby(λ×μ)(x,y)=min{λ(x),μ(y)} forallx,y G.[9]
Definition2.7:LetGbeanon-emptysetafuzzysubsetofGisafunctionμ:G[0,1][10]
Definition2.8:LetandbeafuzzysetsonG.Definethefuzzysetasfollows:
()()min{(),()} xxx
forallxG.[5]
Definition2.9:LetandbeafuzzysetsonG.Definethefuzzysetasfollows:
()()max{(),()} xxx forallxG.[5]
3.MAINRESULTS
Inthissection,wefindsomeresultsaboutfuzzysubIS-algebra,normalsubIS-algebra,fuzzynormalsubISalgebraandfuzzynormalsubIS-algebraoffuzzysubIS-algebra.
12 12 2)()min(),(), 1)(*)min(),(),
Proposition3.2:LetandbefuzzyIS-algebraofG.ThenisafuzzyIS-algebraofG.
ImpactFactor(JCC):6.6810
NAASRating3.45
Proof:LetandarethefuzzysubIS-algebraandlet xy , then min()(),()().
Proposition3.3:LetandarefuzzysubIS-algebraofGthenisafuzzysubIS-algebraofGif
HenceisafuzzysubIS-algebra.
Definition35:AfuzzysubIS-algebraofGissaidtobenormalfuzzysubIS-algebraifthereexistsxGsuchthat ()1. x
Remark3.6:AfuzzysubIS-algebraofGissaidtobenormalfuzzysubIS-algebraifandonlyif(0)=1
Proof
LetbeanormalfuzzysubIS-algebraofGthen
thereexistsxGsuchthat )=1
since ) ) x G
so(0)1then(0)=1
Conversely,itisclear.
Proposition3.7:LetandνarenormalfuzzysubIS-algebraofGthen∩νbeanormalfuzzysubIS-algebraofG.
Proof:
LetandarenormalfuzzysubIS-algebraofGthen
∩isafuzzysubIS-algebraofG[byProposition(3.2)]
also(0)=1andν(0)=1so
()(0)min{(0),(0)}1
therefore() isanormalfuzzysubIS-algebra.
Proposition3.8:LetandarenormalfuzzysubIS-algebraofGthenbeanormalfuzzysubIS-algebraofGif or
Proof
LetandνarenormalfuzzysubIS-algebraofGsuchthatorthen
isafuzzysubIS-algebraofG[byProposition(3.3)]
also(0)=1andν(0)=1so
ImpactFactor(JCC):6.6810
()(0)max{(0),(0)}1 thereforeisanormalfuzzysubIS-algebra.
Proposition3.9:LetandνbeanormalfuzzysubIS-algebrathenisanormalfuzzysubIS-algebra.
Proof:
LetandνarenormalfuzzysubIS-algebraofGthen, sinceandνarefuzzysubIS-algebra
so[byProposition(3.4)]isafuzzysubIS-algebra
Now, (,0)min{(0),(0)}min{1,1}1 [since arenormalfuzzysubIS-algebra]
HenceisnormalfuzzysubISalgebra.
Definition310:LetGbeaIS-algebraandafuzzysetonX.TheniscalledafuzzynormalsubIS-algebraofGifit satisfiesthefollowingconditions: 3)()),
1)isafuzzysubIS-algebraofG (*)(*),\{0}
Proposition3.11:LetandνarefuzzynormalsubIS-algebraofGthenbeafuzzynormalsubIS-algebra.
Proof
LetandνarefuzzynormalsubIS-algebraofG, thenisafuzzysubIS-algebraofG[byProposition(3.2)]
Now,
thereforeisafuzzynormalsubIS-algebra.
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Proposition3.12:LetandνarefuzzynormalsubIS-algebraofG.ThenbeafuzzynormalsubIS-algebraif
Proof:
Supposethatand
thenand
Proposition3.13:LetandarefuzzynormalsubIS-algebraofGthen
ImpactFactor(JCC):6.6810
letxxyyGGwherexxyyG
()(*)()((1,2)*(1,2)) IS-algebra
()(1*1,2*2)
thereforeisafuzzynormalsubIS-algebra.
Proposition3.14:LetGbeaIS-algebraand,betwofuzzysetsinGsuchthatisafuzzysubIS-algebraof G×G
1) eitherxorxforallxG ()(0)()(0)
2) If()(0) xforallx theneither()(0)()(0) xorx
3) If()(0) xforallx theneither()(0)()(0) xorx
4) eitherorisafuzzysubIS-algebraofG.
Proposition3.15:LetbeafuzzynormalsubIS-algebraofGtheneitherorisafuzzynormalsubIS-algebra ofG.
Proof: LetbeafuzzynormalsubIS-algebraofG
sobeafuzzysubIS-algebraofG
thenbyuseProposition(3.14),eitherorisafuzzysubIS-algebraofG ifbeafuzzysubIS-algebraofG
so[by(3.14)]()(0) x toproveisanormal let12 , xx then
HenceisafuzzynormalsubIS-algebra.
Insimilarway.ifisafuzzynormalsubIS-algebraandisafuzzysubIS-algebra.
WecanprovethatisafuzzynormalsubIS-algebra.
Definition316:LetGbeaIS-algebra,andarefuzzysubIS-algebraofGsuchthattheniscalledfuzzy normalsubIS-algebraoffuzzysubIS-algebraif:
Proposition3.17:LetGbeaIS-algebraandletandbefuzzynormalsubIS-algebraoffuzzysubIS-algebra
ThenisafuzzynormalsubIS-algebraof
Proof:
LetandarefuzzynormalsubIS-algebraoffuzzysubIS-algebra.
ThenisafuzzysubIS-algebra[byProposition(3.2)] Now,let
1)()()min{(),()}
min{min{(),()},min{(),()}
2)()(*)min{(*),(*)}
HenceisafuzzynormalsubIS-algebraof.
Proposition3.18:LetXbeaIS-algebraandletandarefuzzynormalsubIS-algebraoffuzzysubIS-algebrathen isafuzzynormalsubIS-algebraofifor
Proof:
LetandarefuzzynormalsubIS-algebraoffuzzysubIS-algebra
isafuzzysubIS-algebra[byProposition(3.3)]
Now,letxyG , then
Proposition3.19:IfandarefuzzynormalsubIS-algebraoffuzzysubIS-algebrathenisafuzzynormal subIS-algebraof.
Proof: Letand
arefuzzynormalsubIS-algebraof
let(x1,x2),(y1,y2) G×Gsuchthat x=(x1,x2),y=(y1,y2) so ,,arefuzzysubIS-algebraofG, thenisafuzzysubIS-algebra[byProposition(3.9)] thenisafuzzysubIS-algebraofG×G[byProposition(3.9)].
Proposition3.20:LetfG : beahomomorphismifisanormalfuzzysubIS-algebraofYthenf isanormal fuzzysubIS-algebraofG.
Proposition3.21:LetfG : beahomomorphismifisafuzzynormalsubIS-algebraofafuzzysubIS-algebra Thenf isafuzzynormalsubIS-algebraoff
Proof: LetisafuzzynormalsubIS-algebraof.Then
Now,toprovef isafuzzynormalsubIS-algebraoffthus
ImpactFactor(JCC):6.6810
NAASRating3.45
HencefisafuzzynormalsubIS-algebraoff.
Proposition3.22:LetfG : beepimorphismiff isanormalfuzzysubIS-algebraofGthenisanormalfuzzy subIS-algebraofY.
Proposition3.23:LetfG : epimorphismiff isafuzzynormalsubIS-algebraoff.Thenisafuzzy normalsubIS-algebraofV
Proof:
Letf isafuzzynormalsubIS-algebraoffthen sincefisanepimorphismifxasuchthatfax ()
()(())()()(())()()(). xfafaafaxsoxxx f andisafuzzysubIS-algebra[byProposition(3.22)]
Now,letxyabGsuchthatfaxfby (),() ,, then min{(),()} min{(()()),()} min{(()),(())} min{(),()}
andso,
f
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