347 صرفة و التطبيقيةمجلة إبن الهيثم للعلوم ال 2012 السنة 25 المجلد 2 العدد Ibn Al-Haitham Journal for Pure and Applied S cience No. 2 Vol. 25 Year 2012 The Commutator of Two Fuzzy Subsets L. N. M. Tawfiq , R. H. Shihab Department of Mathematics, College of Education Ibn Al-Haitham, University of Baghdad. Received in:2 March 2011 Accepted in: 12 April 2011 Abstract In this paper we introduce the idea of the commutator of two fuzzy subsets of a group and study the concept of the commutator of two fuzzy subsets of a group .We introduce and study some of its properties . Key ward: fuzzy set , fuzzy group, normal fuzzy subgroup. 1.Introduction Applying the concept of fuzzy sets of Zadeh to the group theory, Rosenfeld introduced the notion of a fuzzy group as early as 1971. The technique of generating a fuzzy group (the smallest fuzzy group) containing an arbitrarily chosen fuzzy set was developed only in 1992 by Malik , Mordeson and Nair, [1] . In this paper, we use our notion of commutator of two fuzzy subsets of a group. Now we introduce the following definitions which is necessary and needed in the next section : Definition 1.1 [1], [2] A mapping from a nonempty set X to the interval [0, 1] is called a fuzzy subset of X . Next, we shall give some definitions and concepts related to fuzzy subsets of G. Definition 1.2 Let v,m be fuzzy subsets of G, if ( ) ( )xvx £m for every Gx Œ , then we say that m is contained in v (or v contains m ) and we write vÕm (or mn   ). If vÕm and vπm , then m is said to be properly contained in v (or v properly contains m ) and we write vÃm ( or mn … ).[3] Note that: v=m if and only if ( ) ( )xvx =m for all Gx Œ .[4] Definition 1.3 [3] Let v,m be two fuzzy subsets of G. Then v and vm m» « are fuzzy subsets as follows: (i) ( ) { })(),(max)( xvxxv mm =» 348 صرفة و التطبيقيةمجلة إبن الهيثم للعلوم ال 2012 السنة 25 المجلد 2 العدد Ibn Al-Haitham Journal for Pure and Applied S cience No. 2 Vol. 25 Year 2012 (i) ( ) { })(),(min)( xvxxv mm =« , for all Gx Œ Then vandv «» mm are called the union and intersection of m and v , respectively. Definition 1.4[5] For v,m are two fuzzy subsets of G, we define the operation vom as follows: ( ) ( ) {{ } }v x sup min ( a ),v( b ) a,b G and x a* bm m= Œ =o For all Gx Œ . We call vom the product of m and v . Now, we are ready to give the definition of a fuzzy subgroup of a group. Definition 1.5[1], [6] A fuzzy subset m of a group G is a fuzzy subgroup of G if: (i) ( ) ( ){ } ( )min a , b a* bm m m£ (ii) ( ) ( )aa mm =-1 , for all Gba Œ, . Theorem 1.6 [3] If m is a fuzzy subset of G, then m is a fuzzy subgroup of G, if and only if, m satisfies the following conditions: (i) mmm Õo (ii) mm =-1 where m-1(x) = m(x), " x ŒG. Proposition 1.7 [6] Let m be a fuzzy group. Then ( ) ( ) Gaea Œ"£ mm . Definition 1.8 [7] If m is a fuzzy subgroup of G, then m is said to be abelian if Gyx Œ" , , ( ) ( ) 0,0 >> yx mm , then ( ) ( )yxxy mm = . Definition 1.9 [8] , [9] A fuzzy subgroup m of G is said to be normal fuzzy subgroup if ( ) ( )x* y y * x , x, y Gm m= " Œ . 2. The Commutator of Two Fuzzy Subsets of a Group In this section we introduce the idea of the commutator of two fuzzy subsets of a group and prove some of its properties. Definition 2.1 Let l and m be two fuzzy subsets of G. The commutator of l and m is the fuzzy subgroup [ ]ml, of G generated by the fuzzy subset ( )ml, of G which is defined as follows for any x Œ G: 349 صرفة و التطبيقيةمجلة إبن الهيثم للعلوم ال 2012 السنة 25 المجلد 2 العدد Ibn Al-Haitham Journal for Pure and Applied S cience No. 2 Vol. 25 Year 2012 ( ) =)(, xml Next, we will introduce some theorems about the commutator of two fuzzy subsets of a group which is useful in fuzzy mathematics . Theorem 2. 2 If A, B are subsets of G, then ˙̊ ˘ ÍÎ È = BABA ,],[ ccc , where for all x Œ G: , ( ) ,A 1 if a A a 0 if a A c ŒÏ = Ì œÓ Proof: , ( , )B BA Ac c c c È ˘ =< >Î ˚ and [ , ] { ( ) ^ ( )}, ( , ) , x a b Sup a b if x is acommutatorBA BA 0 otherwise c c c c = ÏÔ = Ì ÔÓ Then: On the other hand , { })()(sup ba ml Ÿ if x is a commutator x= [ ]ba, 0 otherwise, 1 if [ ]BAx ,Œ 0 0 otherwise (2) 1 if [ ] [ ]BAbax ,, Œ= 0 0 otherwise (1) =],[ BA cc 350 صرفة و التطبيقيةمجلة إبن الهيثم للعلوم ال 2012 السنة 25 المجلد 2 العدد Ibn Al-Haitham Journal for Pure and Applied S cience No. 2 Vol. 25 Year 2012 = ˙̊ ˘ ÍÎ È )(, xBAc From (1) and (2), we get ˙̊ ˘ ÍÎ È = BABA ,],[ ccc . Theorem 2. 3 For any two fuzzy subsets ml, of G , [ ] [ ]lmml ,, = Proof : The result follows from definition (2.1) and definition(1.5 ). For more explanation we give the following example: Example 2. 4 Let mland be two fuzzy subsets of 3S ( the group of all permutations on the set { },3,2,1 ) defined as follows, for any x Œ S3 : =)(xl =)(xm By definition ( 2.1 ) : ( ) =)(, xml { })()(sup ba ml Ÿ if x is a commutator ],[ bax = 0 otherwise 1 if }{ex = ½ if { }ex -AŒ 3 ¼ if 33 A-Œ Sx 1 if { }ex = ⅓ if ( ) ( ) ( ){ }23,13,12Œx 0 otherwise 351 صرفة و التطبيقيةمجلة إبن الهيثم للعلوم ال 2012 السنة 25 المجلد 2 العدد Ibn Al-Haitham Journal for Pure and Applied S cience No. 2 Vol. 25 Year 2012 = Hence [ ] ( ) =Ò·= mlml ,, On the other hand : ( ) =)(, xlm Hence [ ] ( ) =Ò·= lmlm ,, Thus [ ] [ ]lmml ,, = . Theorem 2. 5 If bml ,, and d are fuzzy subsets of G such that ml Õ and db Õ , then: [ ] [ ]dmbl ,, Õ . Proof: [ ] ( )Ò·= blbl ,, , by definition ( 2.1 ) for all Gx Œ , ( ) =)(, xbl 1 if { }ex = ⅓ if { }ex -AŒ 3 0 otherwise 1 if { }ex = ⅓ if { }ex -AŒ 3 0 otherwise 1 if { }ex = ⅓ if { }ex -AŒ 3 0 otherwise 1 if { }ex = ⅓ if { }ex -AŒ 3 0 otherwise { })()(sup ba bl Ÿ if x is a commutator [ ]bax ,= { })()(sup ba dm Ÿ if x is a commutator [ ]bax ,= 352 صرفة و التطبيقيةمجلة إبن الهيثم للعلوم ال 2012 السنة 25 المجلد 2 العدد Ibn Al-Haitham Journal for Pure and Applied S cience No. 2 Vol. 25 Year 2012 £ ( ) )(, xdm= Thus, [ ] ( ) ( ) [ ]dmdmblbl ,,,, =Ò·£Ò·= Hence, [ ] [ ]dmbl ,, Õ . . Corollary 2. 6 If ml, are fuzzy subsets of G such that ml Õ , then [ ] [ ]dmdl ,, Õ for every fuzzy subset d of G. Proof: The result follows from theorem (2. 5) by taking db = . Now, we introduce an important concept about the fuzzy subset. Definition 2. 7 Let l be a fuzzy subset of G. Then the tip of l is the supremum of the set { }Gxx Œ)(l . Theorem 2. 8 Let mland be fuzzy subsets of G. Then the tip of [ ]ml, is the minimum of tip of l and tip of m . Proof: We want to prove that the tip of [ ], tip of tip ofl m l m= Ÿ Let tip of { } LGxx =Œ= /)(sup ll And, let tip of { } MGxx =Œ= /)(sup mm Such that [ ]1,0, ŒML Now, Tip of ( )ml, ( ){ } { } [ ]{ } [ ]{ } { [ ] } { } { } ML GbbGaa Gbaandbaxba Gxbaxba Gxbaxba Gxx Ÿ= ŒŸŒ= Œ=Ÿ= Œ=Ÿ= Œ=Ÿ= Œ= /)(sup/)(sup ,,/)()(sup ,,/)()(supsup ,,/)()(supsup /)(,sup ml ml ml ml ml 353 صرفة و التطبيقيةمجلة إبن الهيثم للعلوم ال 2012 السنة 25 المجلد 2 العدد Ibn Al-Haitham Journal for Pure and Applied S cience No. 2 Vol. 25 Year 2012 (< )> Since, [ ] ( )Ò·= mlml ,, Therefore, tip of [ ] ML Ÿ=ml, That is tip of [ ], tip of tip ofl m l m= Ÿ . Theorem 2. 9 Let ml, be fuzzy subsets of G. If ( ) ( ) KSandS =H= ml , then : [ ]( ) [ ]KS ,, H=ml . Proof: First, we have ( ) { }| ( )S x G x 0 Hl l= Œ Ò = and ( ) { }| ( )S x G x 0 Km m= Œ Ò = . Then : [ ]KH , and, [ ]( )ml,S ( )( )Ò·= ml,S S= xx /{= : is a commutator } (2) From (1) and (2), we get [ ]( ) [ ]KHS ,, =ml . The following example illustrates theorems (2. 8) and (2. 9). Example 2. 10 Let mland be a fuzzy subsets of 3S which are defined as follows: =)(xl [ ]{ } [ ]{ } , / , , / ( ) ( ) a b a H b K a b a 0 b 0l m = · Ò Œ Œ = · Ò > Ò = { xx is a commutaltor } (1) 1 if { }ex = ⅓ if { }ex -AŒ 3 ¼ if 33 A-Œ Sx { })()(sup ba ml Ÿ if x is a commutator [ ]bax ,= 0 otherwise 354 صرفة و التطبيقيةمجلة إبن الهيثم للعلوم ال 2012 السنة 25 المجلد 2 العدد Ibn Al-Haitham Journal for Pure and Applied S cience No. 2 Vol. 25 Year 2012 and, =)(xm Then, tip of 1=l and tip of =m ½ ( ) =)(, xml = Since, [ ] ( )mlml ,, = . Then : [ ] =)(, xml Then tip of [ ] =ml, ½ tip of tip ofl m= Ÿ Also, ½ if 3AŒx 0 otherwise 3, Sx Œ { })()(sup ba ml Ÿ if x is a commutator [ ]bax ,= 0 otherwise ½ if { }ex = ⅓ if { }ex -AŒ 3 0 otherwise 3, Sx Œ ½ if { }ex = ⅓ if { }ex -AŒ 3 0 otherwise 355 صرفة و التطبيقيةمجلة إبن الهيثم للعلوم ال 2012 السنة 25 المجلد 2 العدد Ibn Al-Haitham Journal for Pure and Applied S cience No. 2 Vol. 25 Year 2012 ( ) 3SS =l and ( ) 3A=mS ( ) ( )[ ] [ ] ( ) ( ){ } 3,,, ASbSabaSS =ŒŒ= mlml Also, [ ]( ) 3, AS =ml Then, we get : [ ]( ) ( ) ( )[ ]mlml SSS ,, = . Next, we will give and prove the following propositions, which we will be needed later. Proposition 2. 11 If l is a fuzzy subgroup of G, then : [ ] lll Õ, . Proof: For all ,Gx Œ ( ) =)(, xll = £ )(xl= { })()(sup ba ll Ÿ if x is a commutator [ ]bax ,= 0 otherwise { })()()()(sup 11 -- ŸŸŸ baba llll if x is a commutator [ ]bax ,= 0 otherwise ( ){ }11sup -- babal if x is a commutator [ ]bax ,= 0 otherwise 356 صرفة و التطبيقيةمجلة إبن الهيثم للعلوم ال 2012 السنة 25 المجلد 2 العدد Ibn Al-Haitham Journal for Pure and Applied S cience No. 2 Vol. 25 Year 2012 That is ( ) lll Õ, . Hence [l, l] Õ l. From theorem (2. 5) and proposition (2. 11) we obtain the following corollary : Corollary 2. 12 Let ndbml ,,,, and a be fuzzy subsets of G. If [ ] [ ]mldb ,, Õ and [ ] [ ]mlan ,, Õ . Then [ ] [ ][ ] [ ]mlandb ,,,, Õ . Proof: Since [ ] [ ]mldb ,, Õ and [ ] [ ]mla ,, Õv Then : [ ] [ ][ ] [ ] [ ][ ] [ ]mlmlmladb ,,,,,,, ÕÕv Hence : [ ] [ ][ ] [ ]mlandb ,,,, Õ . Proposition 2. 13 Let ml, be fuzzy subsets of G. Then: [ ] [ ] [ ]mlmlml ,,, =o . Proof: From definition (2. 1), [ ]ml, is fuzzy subgroup of G and by theorem (1.6) : [ ] [ ] [ ]mlmlml ,,, Õo (1) Now, let Gx Œ [ ] [ ]( ) [ ] [ ]{ }, , ( ) sup , ( ) , ( ), *x a b x a bl m l m l m l m= Ÿ =o [ ] [ ]{ }xexex =Ÿ≥ ),(,)(, mlml [ ] )(, xml= That is [ ] [ ] [ ]mlmlml ,,, oÕ (2) From (1) and (2), we get : [ ] [ ] [ ]mlmlml ,,, =o . Proposition 2. 14 Let ndbml ,,,, and a be fuzzy subsets of G, such that [ ] [ ]mldb ,, Õ and [ ] [ ]mlan ,, Õ . Then [ ] [ ] [ ]mladb ,,, Õvo . Proof: For all Gx Œ , [ ] [ ]( ) [ ] [ ]{ }, , ( ) sup , ( ) , ( ), *x a b x a bb d n a b d n a= Ÿ =o [ ] [ ]{ }sup , ( ) , ( ), *a b x a bl m l m£ Ÿ = [ ] [ ]( ) )(,, xmlml o= [ ] )(, xml= ( by proposition (2. 13) ) Hence, [ ] [ ] [ ]mladb ,,, Õvo . Now, we can give the following corollary: Corollary 2. 15 If ndbml ,,,, and a are fuzzy subsets of G, such that [ ] [ ]dmbl ,, Õ , then : (i) [ ] [ ] [ ] [ ]andmanbl ,,,, oo Õ (ii) [ ] [ ] [ ]dmdmbl ,,, Õo (iii) If [ ] [ ]anbl ,, Õ , then [ ] [ ] [ ]andmbl ,,, oÕ . Proof: The result follows from proposition (2. 14). 357 صرفة و التطبيقيةمجلة إبن الهيثم للعلوم ال 2012 السنة 25 المجلد 2 العدد Ibn Al-Haitham Journal for Pure and Applied S cience No. 2 Vol. 25 Year 2012 Next, we will give and prove the following proposition : Proposition 2. 16 Let [ ] [ ]dmbl ,,, be two fuzzy subgroups of G. Then [ ] [ ] )(,)(, ee dmbl = if and only if, [ ] [ ] [ ]dmblbl ,,, oÕ and [ ] [ ] [ ]dmbldm ,,, oÕ . Proof: First, if [l, b ](e) = [m, d](e), we prove : [ ] [ ] [ ]dmblbl ,,, oÕ and [ ] [ ] [ ]dmbldm ,,, oÕ . Let Gx Œ , [ ] [ ]( ) [ ] [ ]{ }, , ( ) sup , ( ) , ( ), *x a b x a bl b m d l b m d= Ÿ =o [ ] [ ]{ }, ( ) , ( ), *x e x x el b m d≥ Ÿ = [ ] [ ]{ }, ( ) , ( ), *x e x x el b l b= Ÿ = [ ] )(, xbl= That is, [ ] [ ] [ ]( ) )(,,)(, xx dmblbl o£ for all Gx Œ . Hence, [ ] [ ] [ ]dmblbl ,,, oÕ Also, [ ] [ ]( ) [ ] [ ]{ }, , ( ) sup , ( ) , ( ), *x a b x a bl b m d l b m d= Ÿ =o [ ] [ ]{ }, ( ) , ( ), *e x x e xl b m d≥ Ÿ = [ ] [ ]{ }, ( ) , ( ), *e x x e xm d m d= Ÿ = [ ] )(, xdm= That is [ ] [ ] [ ]( ) Gxallforxx Œ£ )(,,)(, dmbldm o Hence [ ] [ ] [ ]dmbldm ,,, oÕ . Conversely, we prove [ ] [ ] )(,)(, ee dmbl = . Suppose [ ] [ ] )(,)(, ee dmbl π , then if [ ] [ ] )(,)(, ee dmbl ≥ . [ ] [ ] [ ]( ) )(,,)(, ee dmblbl o£ [ ] [ ]{ }sup , ( ) , ( ), *a b e a bl b m d= Ÿ = [ ] [ ]{ }, ( ) , ( ), *e b e e el b m d£ Ÿ = [ ] )(, edm= That is [ ] [ ] )(,)(, ee dmbl £ , which is a contradiction . Now, if [ ] [ ] )(,)(, ee bldm ≥ ,then in the same way we get a contradiction. Therefore [ ] [ ] )(,)(, ee dmbl = . References 1 .Malik . D.S. , Mordeson . J. N. and Nair. P. S. ,(1992) ” Fuzzy Generators and Fuzzy Direct Sums of Abelian Groups”, Fuzzy sets and systems,.50, 193-199, 2.Majeed.S.N., (1999)”On fuzzy subgroups of abelian groups”,M.Sc. Thesis, University of Baghdad,. 3.Mordesn J.N.,( 1996)”L-subspaces and L-subfields” , . 4.Hussein. R. W., (1999)”Some results of fuzzy rings” , M.Sc. Thesis , University of Baghdad ,. 358 صرفة و التطبيقيةمجلة إبن الهيثم للعلوم ال 2012 السنة 25 المجلد 2 العدد Ibn Al-Haitham Journal for Pure and Applied S cience No. 2 Vol. 25 Year 2012 5.Liu. W.J., (1982) ”Fuzzy invariant subgroups and fuzzy ideals”, fuzzy sets and systems . l.8. 133-139,. 6.Abou-Zaid. , (1988)” On normal fuzzy subgroups ” , J.Facu.Edu.,.13, . 7. Seselja .B and Tepavcevic A. , (1997) “Anote on fuzzy groups” , J.Yugoslav. Oper. Rese ,.7, No.1, pp.49-54, . 8.Gupta K.c and Sarma B.K., (1999) “nilpotent fuzzy groups” ,fuzzy set and systems ,.101,.167-176 ,. 9.Seselja . B . and Tepavcevic A. ,(1996) “ Fuzzy groups and collections of subgroups ” , fuzzy sets and systems ,.83 ,.85-91 ,. 10.Bandler. W. and Kohout. L. , (2000) Semantics of implication operators and fuzzy relational products, Internat. J. Man- Machine studies 12 89 -116 . المبادل لمجموعتان جزئيتان ضبابيتان رجاء حامد شهاب ¡ لمى ناجي محمد توفيق .جامعة بغداد –كلية التربية أبن الهيثم –قسم الرياضيات .2011نيسان 12قبل البحث في ÑÇÐÇ2011 2:استلم البحث الخالصة يتضمن البحث تقديم فكرة المبادل لمجموعتان جزئيتان ضبابيتان ودراسة مفهوم المبادل لمجموعتان جزئيتان ضبابيتان من زمرة و دراسة خواصها وتقديم البراهين المهمة حول المفهوم . المجموعات الضبابية ، الزمر الضبابية الكلمات المفتاحية : The Commutator of Two Fuzzy Subsets Applying the concept of fuzzy sets of Zadeh to the group theory, Rosenfeld introduced the notion of a fuzzy group as early as 1971. The technique of generating a fuzzy group (the smallest fuzzy group) containing an arbitrarily chosen fuzzy set was developed only in 1992 by Malik , Mordeson and Nair, [1]. In this paper, we use our notion of commutator of two fuzzy subsets of a group. For are two fuzzy subsets of G, we define the operation as follows: If is a fuzzy subgroup of G, then is said to be abelian if , , then . Next, we will give and prove the following proposition : First, if [(, (](e) ( [(, (](e), we prove : References لمى ناجي محمد توفيق ,رجاء حامد شهاب قسم الرياضيات – كلية التربية أبن الهيثم – جامعة بغداد. الخلاصة