347 صرفة و التطبيقيةمجلة إبن الهيثم للعلوم ال 2012 السنة 25 المجلد 2 العدد Ibn Al-Haitham Journal for Pure and Applied Science 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 Î , then we say that m is contained in v (or v contains m ) and we write vÍm (or mn Ê ). If vÍm and v¹m , then m is said to be properly contained in v (or v properly contains m ) and we write vÌm ( or mn É ).[3] Note that: v=m if and only if ( ) ( )xvx =m for all Gx Î .[4] Definition 1.3 [3] Let v,m be two fuzzy subsets of G. Then v and vm mÈ Ç are fuzzy subsets as follows: (i) ( ) { })(),(max)( xvxxv mm =È 348 صرفة و التطبيقيةمجلة إبن الهيثم للعلوم ال 2012 السنة 25 المجلد 2 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 2 Vol. 25 Year 2012 (i) ( ) { })(),(min)( xvxxv mm =Ç , for all Gx Î Then vandv ÇÈ 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= Î =o For all Gx Î . 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 Î, . 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 Ío (ii) mm =-1 where m-1(x) = m(x), " x ÎG. Proposition 1.7 [6] Let m be a fuzzy group. Then ( ) ( ) Gaea Î"£ mm . Definition 1.8 [7] If m is a fuzzy subgroup of G, then m is said to be abelian if Gyx Î" , , ( ) ( ) 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= " Î . 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 Î G: 349 صرفة و التطبيقيةمجلة إبن الهيثم للعلوم ال 2012 السنة 25 المجلد 2 العدد Ibn Al-Haitham Journal for Pure and Applied Science 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 úû ù êë é= BABA ,],[ ccc , where for all x Î G: , ( ) ,A 1 if a A a 0 if a A c Îì = í Ïî Proof: , ( , )B BA Ac c c cé ù =< >ë û and [ , ] { ( ) ^ ( )}, ( , ) , x a b Sup a b if x is acommutatorBA BA 0 otherwise c c c c = ìï= í ïî Then: On the other hand , { })()(sup ba ml Ù if x is a commutator x= [ ]ba, 0 otherwise, 1 if [ ]BAx ,Î 0 0 otherwise (2) 1 if [ ] [ ]BAbax ,, Î= 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 Science No. 2 Vol. 25 Year 2012 = úû ù êë é )(, xBAc From (1) and (2), we get úû ù êë é= 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 Î S3 : =)(xl =)(xm By definition ( 2.1 ) : ( ) =)(, xml { })()(sup ba ml Ù if x is a commutator ],[ bax = 0 otherwise 1 if }{ex = ½ if { }ex -AÎ 3 ¼ if 33 A-Î Sx 1 if { }ex = ⅓ if ( ) ( ) ( ){ }23,13,12Îx 0 otherwise 351 صرفة و التطبيقيةمجلة إبن الهيثم للعلوم ال 2012 السنة 25 المجلد 2 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 2 Vol. 25 Year 2012 = Hence [ ] ( ) =ñá= mlml ,, On the other hand : ( ) =)(, xlm Hence [ ] ( ) =ñá= lmlm ,, Thus [ ] [ ]lmml ,, = . Theorem 2. 5 If bml ,, and d are fuzzy subsets of G such that ml Í and db Í , then: [ ] [ ]dmbl ,, Í . Proof: [ ] ( )ñá= blbl ,, , by definition ( 2.1 ) for all Gx Î , ( ) =)(, xbl 1 if { }ex = ⅓ if { }ex -AÎ 3 0 otherwise 1 if { }ex = ⅓ if { }ex -AÎ 3 0 otherwise 1 if { }ex = ⅓ if { }ex -AÎ 3 0 otherwise 1 if { }ex = ⅓ if { }ex -AÎ 3 0 otherwise { })()(sup ba bl Ù if x is a commutator [ ]bax ,= { })()(sup ba dm Ù 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 Science No. 2 Vol. 25 Year 2012 £ ( ) )(, xdm= Thus, [ ] ( ) ( ) [ ]dmdmblbl ,,,, =ñá£ñá= Hence, [ ] [ ]dmbl ,, Í . . Corollary 2. 6 If ml, are fuzzy subsets of G such that ml Í , then [ ] [ ]dmdl ,, Í 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 Î)(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= Ù Let tip of { } LGxx =Î= /)(sup ll And, let tip of { } MGxx =Î= /)(sup mm Such that [ ]1,0, ÎML Now, Tip of ( )ml, ( ){ } { } [ ]{ } [ ]{ } { [ ] } { } { } ML GbbGaa Gbaandbaxba Gxbaxba Gxbaxba Gxx Ù= ÎÙÎ= Î=Ù= Î=Ù= Î=Ù= Î= /)(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 Science No. 2 Vol. 25 Year 2012 (< )> Since, [ ] ( )ñá= mlml ,, Therefore, tip of [ ] ML Ù=ml, That is tip of [ ], tip of tip ofl m l m= Ù . 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= Î ñ = and ( ) { }| ( )S x G x 0 Km m= Î ñ = . Then : [ ]KH , and, [ ]( )ml,S ( )( )ñá= 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 = á ñ Î Î = á ñ > ñ = { xx is a commutaltor } (1) 1 if { }ex = ⅓ if { }ex -AÎ 3 ¼ if 33 A-Î Sx { })()(sup ba ml Ù 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 Science 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= Ù Also, ½ if 3AÎx 0 otherwise 3, Sx Î { })()(sup ba ml Ù if x is a commutator [ ]bax ,= 0 otherwise ½ if { }ex = ⅓ if { }ex -AÎ 3 0 otherwise 3, Sx Î ½ if { }ex = ⅓ if { }ex -AÎ 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 Science No. 2 Vol. 25 Year 2012 ( ) 3SS =l and ( ) 3A=mS ( ) ( )[ ] [ ] ( ) ( ){ } 3,,, ASbSabaSS =ÎÎ= 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 Í, . Proof: For all ,Gx Î ( ) =)(, xll = £ )(xl= { })()(sup ba ll Ù if x is a commutator [ ]bax ,= 0 otherwise { })()()()(sup 11 -- ÙÙÙ 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 Science No. 2 Vol. 25 Year 2012 That is ( ) lll Í, . Hence [l, l] Í 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 ,, Í and [ ] [ ]mlan ,, Í . Then [ ] [ ][ ] [ ]mlandb ,,,, Í . Proof: Since [ ] [ ]mldb ,, Í and [ ] [ ]mla ,, Ív Then : [ ] [ ][ ] [ ] [ ][ ] [ ]mlmlmladb ,,,,,,, ÍÍv Hence : [ ] [ ][ ] [ ]mlandb ,,,, Í . 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 ,,, Ío (1) Now, let Gx Î [ ] [ ]( ) [ ] [ ]{ }, , ( ) sup , ( ) , ( ), *x a b x a bl m l m l m l m= Ù =o [ ] [ ]{ }xexex =Ù³ ),(,)(, mlml [ ] )(, xml= That is [ ] [ ] [ ]mlmlml ,,, oÍ (2) From (1) and (2), we get : [ ] [ ] [ ]mlmlml ,,, =o . Proposition 2. 14 Let ndbml ,,,, and a be fuzzy subsets of G, such that [ ] [ ]mldb ,, Í and [ ] [ ]mlan ,, Í . Then [ ] [ ] [ ]mladb ,,, Ívo . Proof: For all Gx Î , [ ] [ ]( ) [ ] [ ]{ }, , ( ) sup , ( ) , ( ), *x a b x a bb d n a b d n a= Ù =o [ ] [ ]{ }sup , ( ) , ( ), *a b x a bl m l m£ Ù = [ ] [ ]( ) )(,, xmlml o= [ ] )(, xml= ( by proposition (2. 13) ) Hence, [ ] [ ] [ ]mladb ,,, Ívo . Now, we can give the following corollary: Corollary 2. 15 If ndbml ,,,, and a are fuzzy subsets of G, such that [ ] [ ]dmbl ,, Í , then : (i) [ ] [ ] [ ] [ ]andmanbl ,,,, oo Í (ii) [ ] [ ] [ ]dmdmbl ,,, Ío (iii) If [ ] [ ]anbl ,, Í , then [ ] [ ] [ ]andmbl ,,, oÍ . 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 Science 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 ,,, oÍ and [ ] [ ] [ ]dmbldm ,,, oÍ . Proof: First, if [l, b](e) = [m, d](e), we prove : [ ] [ ] [ ]dmblbl ,,, oÍ and [ ] [ ] [ ]dmbldm ,,, oÍ . Let Gx Î , [ ] [ ]( ) [ ] [ ]{ }, , ( ) sup , ( ) , ( ), *x a b x a bl b m d l b m d= Ù =o [ ] [ ]{ }, ( ) , ( ), *x e x x el b m d³ Ù = [ ] [ ]{ }, ( ) , ( ), *x e x x el b l b= Ù = [ ] )(, xbl= That is, [ ] [ ] [ ]( ) )(,,)(, xx dmblbl o£ for all Gx Î . Hence, [ ] [ ] [ ]dmblbl ,,, oÍ Also, [ ] [ ]( ) [ ] [ ]{ }, , ( ) sup , ( ) , ( ), *x a b x a bl b m d l b m d= Ù =o [ ] [ ]{ }, ( ) , ( ), *e x x e xl b m d³ Ù = [ ] [ ]{ }, ( ) , ( ), *e x x e xm d m d= Ù = [ ] )(, xdm= That is [ ] [ ] [ ]( ) Gxallforxx Σ )(,,)(, dmbldm o Hence [ ] [ ] [ ]dmbldm ,,, oÍ . Conversely, we prove [ ] [ ] )(,)(, ee dmbl = . Suppose [ ] [ ] )(,)(, ee dmbl ¹ , then if [ ] [ ] )(,)(, ee dmbl ³ . [ ] [ ] [ ]( ) )(,,)(, ee dmblbl o£ [ ] [ ]{ }sup , ( ) , ( ), *a b e a bl b m d= Ù = [ ] [ ]{ }, ( ) , ( ), *e b e e el b m d£ Ù = [ ] )(, edm= That is [ ] [ ] )(,)(, ee dmbl £ , which is a contradiction . Now, if [ ] [ ] )(,)(, ee bldm ³ ,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 Science 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