ON FUZZY SUB IS-ALGEBRAS

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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-emptysetafuzzysubsetofGisafunctionμ:G[0,1][10]

Definition2.8:LetandbeafuzzysetsonG.Definethefuzzysetasfollows:

()()min{(),()} xxx 

forallxG.[5]

Definition2.9:LetandbeafuzzysetsonG.Definethefuzzysetasfollows:

()()max{(),()} xxx   forallxG.[5]

3.MAINRESULTS

Inthissection,wefindsomeresultsaboutfuzzysubIS-algebra,normalsubIS-algebra,fuzzynormalsubISalgebraandfuzzynormalsubIS-algebraoffuzzysubIS-algebra.



12 12 2)()min(),(), 1)(*)min(),(), 

Proposition3.2:LetandbefuzzyIS-algebraofG.ThenisafuzzyIS-algebraofG.

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*0abc 000bb aa0cb bbb00 ccba0 0abc 00000 a0a0a b00bb c0abc

Definition31:AfuzzysetdefinedonGiscalledafuzzysubIS-algebraofGifitsatisfiesthefollowingconditions: xxxxxxG xxxx 1212 12 

Proof:LetandarethefuzzysubIS-algebraandlet xy , then min()(),()().

Proposition3.3:LetandarefuzzysubIS-algebraofGthenisafuzzysubIS-algebraofGif

OnFuzzySubIS-Algebras 3
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()()min{(),(()} hypothesis] xy xxyy xyxyby xyxyxy      so,
()(*)min{(*),((*)} xy xxyy xyxyby xyxyxy     
min{min(),(),min(),()} min{min(),(),min(),()}[
min()(),()() min{min(),(),min(),()} min{min(),(),min(),()}[hypothesis]
HenceisafuzzysubIS-algebra.
 or Proof:
min{max(),(),max(),()}[] max{min(),(),min(),()}[] ()()max{(),(()} xy or xxyy xyxybyhypthoses xyxyxy      so, min()(),()(). min{max(),(),max(),()}[] max{min(),(),min(),()}[] ()(*)max{(*),((*)} xy xxyyor xyxybyhypthoses xyxyxy      Hence νisafuzzysubIS-algebra. Proposition3.4:LetGbeaIS-algebraandletν,beafuzzysubIS-algebrathen νisafuzzysubIS-algebraof G×G Proof:LetandarefuzzyIS-algebra xyxyGG (,),(,) 1122 then min()(,),()(,)} minmin(),(),min(),()} minmin(),(),min(),()} min(),() ()((,).(,))()((,)) 22 11 22 11 12 12 1212 1212 1122 xyxy xyxy xxyy xxyy xxyy xyxy      
LetandarethefuzzysubIS-algebra,andlet xy , then min()(),()().

HenceisafuzzysubIS-algebra.

Definition35:AfuzzysubIS-algebraofGissaidtobenormalfuzzysubIS-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:LetandarenormalfuzzysubIS-algebraofGthenbeanormalfuzzysubIS-algebraofGif or

Proof

LetandνarenormalfuzzysubIS-algebraofGsuchthatorthen

 isafuzzysubIS-algebraofG[byProposition(3.3)]

also(0)=1andν(0)=1so

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min()(,),()(,)} minmin(),(),min(),()} minmin(),(),min(),()} min(*),(*) ()((,)*(,))()((*,*)) 22 11 22 11 12 12 1212 1212 1122 xyxy xyxy xxyy xxyy xxyy xyxy      
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()(0)max{(0),(0)}1   thereforeisanormalfuzzysubIS-algebra.

Proposition3.9:LetandνbeanormalfuzzysubIS-algebrathenisanormalfuzzysubIS-algebra.

Proof:

LetandνarenormalfuzzysubIS-algebraofGthen, sinceandνarefuzzysubIS-algebra

so[byProposition(3.4)]isafuzzysubIS-algebra

Now, (,0)min{(0),(0)}min{1,1}1   [since arenormalfuzzysubIS-algebra]

HenceisnormalfuzzysubISalgebra.

Definition310:LetGbeaIS-algebraandafuzzysetonX.TheniscalledafuzzynormalsubIS-algebraofGifit satisfiesthefollowingconditions: 3)()),

1)isafuzzysubIS-algebraofG (*)(*),\{0}

Proposition3.11:LetandνarefuzzynormalsubIS-algebraofGthenbeafuzzynormalsubIS-algebra.

Proof

LetandνarefuzzynormalsubIS-algebraofG, thenisafuzzysubIS-algebraofG[byProposition(3.2)]

Now,

thereforeisafuzzynormalsubIS-algebra.

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OnFuzzySubIS-Algebras 5
xyyxxyG xyyxxyG   
2)
IS-algebra] min{(),()}[,
xy     so,
IS-algebra] min{(*),(*)}[,
xyxy xy     
()(),,.
()()min{(),()} yxxyG yxyxarefuzzynormalsub xyxy
()(*),\{0}
()(*)min{(*),(*)} yxxyG yxyxarefuzzynormalsub

Proposition3.12:LetandνarefuzzynormalsubIS-algebraofG.ThenbeafuzzynormalsubIS-algebraif

Proof:

Supposethatand

thenand



Proposition3.13:LetandarefuzzynormalsubIS-algebraofGthen

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 or
νarefuzzynormalsubIS-algebra
νarefuzzysubIS-algebrathen
()(), max{(),()} ()()max{(),()} [] yxxyG yxyx xyxyxy byhypothesis     so, ()(*),\{0}. max{(*),(*)} ()(*)max{(*),(*)} [] yxxyG yxyx xyxyxy byhypothesis     HenceisafuzzynormalsubIS-algebra.
beafuzzysubIS-algebra[byProposition(3.3)] Now,

GG Proof: Letλandμ
(x1,x2),(y1,y2) G×Gwherex1,x2,y1,y2 G x=(x1,x2),y=(y1,y2) thenλandμbeafuzzysubIS-algebraofGso λ×μisafuzzysubIS-algebra[byProposition(3.4)] now, ()() ()((,)(,)) ] min{(),()}[ min{(),()} ()(,) ()()()((,)(,)) 1212 1122 1122 1122 1212 IS-algebra , yx yyxx yxyx xyxy xyxy xyxxyy arefuzzynormalsub        andso,
isafuzzynormalsubIS-algebraof
beafuzzynormalsubIS-algebraofGandlet

letxxyyGGwherexxyyG

()(*)()((1,2)*(1,2)) IS-algebra

()(1*1,2*2)

thereforeisafuzzynormalsubIS-algebra.

Proposition3.14:LetGbeaIS-algebraand,betwofuzzysetsinGsuchthatisafuzzysubIS-algebraof G×G

1) eitherxorxforallxG ()(0)()(0) 

2) If()(0) xforallx theneither()(0)()(0)  xorx

3) If()(0) xforallx theneither()(0)()(0)  xorx

4) eitherorisafuzzysubIS-algebraofG.

Proposition3.15:LetbeafuzzynormalsubIS-algebraofGtheneitherorisafuzzynormalsubIS-algebra ofG.

Proof: LetbeafuzzynormalsubIS-algebraofG

sobeafuzzysubIS-algebraofG

thenbyuseProposition(3.14),eitherorisafuzzysubIS-algebraofG ifbeafuzzysubIS-algebraofG

so[by(3.14)]()(0) x toproveisanormal let12 , xx then

OnFuzzySubIS-Algebras 7
www.iaset.us editor@iaset.us suchthatxxxyyyGG
(,),(,) (,),(,),,\{0} 12 12 1212 1212
min{(1*1),(2*2)}[, min{(1*1),(2*2)}
()(*) ()((1,2)*(1,2)) ]
yx yyxx yxyxarefuzzynormalsubs xyxy xyxy xyxxyy       
.Then:

HenceisafuzzynormalsubIS-algebra.

Insimilarway.ifisafuzzynormalsubIS-algebraandisafuzzysubIS-algebra.

WecanprovethatisafuzzynormalsubIS-algebra.

Definition316:LetGbeaIS-algebra,andarefuzzysubIS-algebraofGsuchthattheniscalledfuzzy normalsubIS-algebraoffuzzysubIS-algebraif:

Proposition3.17:LetGbeaIS-algebraandletandbefuzzynormalsubIS-algebraoffuzzysubIS-algebra

ThenisafuzzynormalsubIS-algebraof

Proof:

LetandarefuzzynormalsubIS-algebraoffuzzysubIS-algebra.

ThenisafuzzysubIS-algebra[byProposition(3.2)] Now,let

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NAASRating3.45 () min{(0),()} ()(0,) ()((0,)(0,)) ()((0,)(0,)) ()(0,) ()min{(0),()} 21 21 21 21 12 12 12 12 xx xx xx xx xx xx xx xx         Now,let,/{0}12 xxG \{0} () min{(0),(*)} ()(0,*) ] ()((0,)*(0,))[ ()((0,)*(0,)) ()(0,*) ()min{(0),(*)} 12 1 * 2 21 21 1 2 2 1 12 12 2 * 1 IS-algebra xxG xx xx xx xxisafuzzynormalsub xx xx xx xx         
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(2)()min{(),()},, (1)(*)min{(*),()} yxxyyxy yxxyy    
(*)min{(*),()},(*)min{(*),()} yxxyyyxxyy    and ()min{(),()},()min{(),()} xyy xyyyx yx    therefore
xy , ,since

1)()()min{(),()}

min{min{(),()},min{(),()}

2)()(*)min{(*),(*)}

HenceisafuzzynormalsubIS-algebraof.

Proposition3.18:LetXbeaIS-algebraandletandarefuzzynormalsubIS-algebraoffuzzysubIS-algebrathen isafuzzynormalsubIS-algebraofifor

Proof:

LetandarefuzzynormalsubIS-algebraoffuzzysubIS-algebra

isafuzzysubIS-algebra[byProposition(3.3)]

Now,letxyG , then

Proposition3.19:IfandarefuzzynormalsubIS-algebraoffuzzysubIS-algebrathenisafuzzynormal subIS-algebraof.

Proof: Letand

arefuzzynormalsubIS-algebraof

OnFuzzySubIS-Algebras 9
min{()(),()}
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min{min{(),()},min{(),()}}
xyy xyxyyy xyyxyy
      
yxyxyx
and, min{()(*),()} min{min{(*),(*)},min{(),()} min{min{(*),()},min{(*),()}}
xyy xyxyyy xyyxyy
      
yxyxyx
min{()(),()} ] min{max{(),()},max{(),()}[sin max{min{(),()},min{(),()}} 1)()()max{(),()} xyy xyxyyyceor xyyxyy yxyxyx         andso, min{()(*),()} min{max{(*),(*)},max{(),()}[] max{min{(*),()},min{(*),()}} 2)()(*)max{(*),(*)} xyy xyxyyyor xyyxyy yxyx yx        HenceisafuzzynormalsubIS-algebraof

let(x1,x2),(y1,y2) G×Gsuchthat x=(x1,x2),y=(y1,y2) so ,,arefuzzysubIS-algebraofG, thenisafuzzysubIS-algebra[byProposition(3.9)] thenisafuzzysubIS-algebraofG×G[byProposition(3.9)].

Proposition3.20:LetfG : beahomomorphismifisanormalfuzzysubIS-algebraofYthenf isanormal fuzzysubIS-algebraofG.

Proposition3.21:LetfG : beahomomorphismifisafuzzynormalsubIS-algebraofafuzzysubIS-algebra Thenf isafuzzynormalsubIS-algebraoff

Proof: LetisafuzzynormalsubIS-algebraof.Then

Now,toprovef isafuzzynormalsubIS-algebraoffthus

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min{()(),()} min{()((,)(,)),(,)} min{min{(),()},min{(),()}} min{min{(),()},min{(),()}} min{(),()} ()(,) ()()()((,)(,)) 12 1212 12 1122 222 111 1122 1122 1212 xyy xxyyyy xyxyyy xyyxyy yxyx yxyx yxyyxx           andso, min{()(*),()} min{()((,)*(,)),(,)} min{min{(*),(*)},min{(),()}} min{min{(*),()},min{(*),()}} min{(*),(*)} ()(*,*) ()(*)()((,)*(,)) 12 1212 12 1122 222 111 1122 1122 1212 xyy xxyyyy xyxyyy xyyxyy yxyx yxyx yxyyxx            Hence

Now,toproveisafuzzynormalsubIS-algebraof
isafuzzynormalsubIS-algebraof
()(())(())()[sin]  xfxfxxce f f andf isafuzzysubIS-algebra[byProposition(3.20)] fisafuzzysubIS-algebra

HencefisafuzzynormalsubIS-algebraoff.

Proposition3.22:LetfG : beepimorphismiff isanormalfuzzysubIS-algebraofGthenisanormalfuzzy subIS-algebraofY.

Proposition3.23:LetfG : epimorphismiff isafuzzynormalsubIS-algebraoff.Thenisafuzzy normalsubIS-algebraofV

Proof:

Letf isafuzzynormalsubIS-algebraoffthen sincefisanepimorphismifxasuchthatfax ()

()(())()()(())()()(). xfafaafaxsoxxx f  andisafuzzysubIS-algebra[byProposition(3.22)]

Now,letxyabGsuchthatfaxfby (),() ,, then min{(),()} min{(()()),()} min{(()),(())} min{(),()}

andso,

f

OnFuzzySubIS-Algebras 11
editor@iaset.us min{(),()} min{(()),()} min{(()()),(())} (()()) ()(()) xyy fxyy fxfyfy fyfx yxfyx f f f f       so, min{(*),()} min{((*)),()} min{(()*()),(())} (()*()) (*)((*)) xyy fxyy fxfyfy fyfx yxfyx f f f f        
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()(()())(())() xyy fafby fabfb abb yxfbfafbaba f f       

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12 SundusNajahJabir ImpactFactor(JCC):6.6810 NAASRating3.45 min{(*),()} min{(()*()),()} min{((*)),(())} min{(*),()} (*) ((*)) (*)(()*()) xyy fafby fabfb abb ba fba yxfbfa f f f         HenceisafuzzynormalsubIS-algebraof
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