340 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 2 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 2 Vol. 25 Year 2012 On Fuzzy Groups and Group Homomorphism L. N. M. Tawfiq , M. M. Qa'aed Department of Mathematics, College of Education Ibn Al-Haitham, Universityof Baghdad. Received in: 9December 2009 Accepted in: 14 December 2010 Abstract In this paper, we study the effect of group homomorphism on the chain of level subgroups of fuzzy groups. We prove a necessary and sufficient conditions under which the chains of level subgroups of homomorphic images of an a arbitrary fuzzy group can be obtained from that of the fuzzy groups . Also, we find the chains of level subgroups of homomorphic images and pre-images of arbitrary fuzzy groups. Key ward:- Fuzzy Groups, Group Homomorphism. 1.Introduction If X is a non- empty set then a function ]1,0[: ®Xm is called a fuzzy subset of X [1] . A fuzzy subset m of G is said to be fuzzy subgroup of G if and only if )()()}(),(min{)( 1xxandyxxy -=³ mmmmm [2] . It is easy to see that ifm is fuzzy subgroup of G, then Gxxe Î"³ ),()( mm .[3] We say that m has the sup-property if every non- empty subset of Im( m ) has a maximal element.[4], [5] . If m is a fuzzy subset of G, then the subset })(;{ txGxt ³Î= mm , tÎ[0, 1] is called the level subset of m in G and )(;{ xGxt mm Î=* >t} is called the strong level subset of m in G when t = 0 the subset * 0m is called support of m in G and it will be denoted by *m [6], [7] . If l is a fuzzy subgroup of G, then the level subsets tl of l in G and the strong level subsets * tl of l in G , t )],(,0[ elÎ are subgroups of G and viseversa [8] . If Î21,tt Im( m ) such that 21 tt ¹ , then obviously, 21 tt mm ¹ . Further, if Im( m ) = { niti ,..,2,1: = } where ,...21 nttt >>> then the level subgroups of m form a chain of subgroups of G. C( m ) G nttt =ÌÌ̺ mmm ... 21 [2] . Let HGf ®: be a homomorphism of groups , l be a fuzzy subgroup of G , m a fuzzy subgroup of H . Then ,)),(()(1 Gxxff Î"=- mm =))(( yf l Sub{ )}();( 1 yfxx -Îl if ;)(1 F¹- yf 0 if ;)(1 F=- yf Hy Î" 341 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 2 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 2 Vol. 25 Year 2012 and the fuzzy sets )(lf and )(1 m-f are fuzzy subgroups of H and G respectively [7], [9] . Now let YXf ®: be a function and ][ml be a fuzzy subset of X[Y] . Then we say that l is f - invariant if )()( 21 xx ll = whenever .,),()( 2121 Xxxxfxf Î= [5] 2. Homomorphic pre-images of fuzzy groups In this section, we prove necessary and sufficient conditions under which the chains of level subgroups of homomorphic pre-images of an arbitrary fuzzy group can be obtained from that of the fuzzy group. Let ¦: G ® H is a group homomorphism and m is a fuzzy subgroup of H We shall denote by ¦ -1 ( C(m) ) the chain consisting of inverse images under ¦ of members of C(m). l is a fuzzy subgroup of G . Proposition (2.1) If m is a fuzzy subgroup of H and {mtj | jÎJ } is the collection of all level subgroups of m, then {¦ -1 (mtj ) | jÎJ } is the co llect ion o f a ll leve l subgroups o f ¦ -1 (m ) . Proof Let l = ¦ -1 (m) and tÎ[0,1]. Then : xÎlt Û¦ -1 (m ) ³ t Û m(¦(x)) ³ t Û ¦(x)Î mt Û x Î ¦ -1 (mt ). Hence lt = ¦ -1 (mt ) " t Î [0, 1]………………………(1) In particular, we have : ltj = ¦ -1 (mtj ) "jÎJ . If l has a level subgroup lt which does not belong to {¦ -1 (mtj ) | jÎJ } then m must have a level subgroup mt which does not belong to {mtj | jÎJ } such that (1) holds. This is a contradiction. Hence the result. We observe from the following example that some of the ¦ -1 ( mtj )’s may be equal so that C( ¦ -1 ( m ) ) has fewer components than C ( m ) . Example (2.2) Let G = {1, -1, i, -i } and H = { e, (12), (13), (23), (123), (132) }. Then G is a group w. r. t. the usual multiplication of numbers and H is the permutation group of degree three, with e as identity transformation. Define ¦ : G ® H by ¦(x) = e ," x Î G. Then ¦ is a group homomorphism. Define m : H ® [0, 1] by : m(e) = 1, m( (12)) = 0.5, m(x) = 0.3, " x Î H \ {e, (12)}. Then m is a fuzzy subgroup of H with level subgroups : m1= {e}, m0.5 = {e, (12)}, m0.3 = H . But l = ¦ -1 (m ) is defined by : l(x) = 1 for every xÎG. Hence, l1 = l0.5 = l0.3 = G. Now, we proceed to derive a necessary and sufficient condition for the distinctness of all the ¦ -1 (mtj ). For t Î Im(m) , we define : Fm(t) = { xÎG | m(x) = t }. Theorem (2.3) 342 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 2 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 2 Vol. 25 Year 2012 Let ¦: G ® H be a group homomorphism and m is a fuzzy subgroup of H with Im(m)={tj | jÎJ } where J is a countable index set. Then ¦ -1 (mtj ) are all distinct if and only if : ¦(G)∩Fm(tj) ¹ Æ, "jÎ J. Proof Assume that ¦ -1 (mtj ) , jÎJ, are all distinct. Let e* denote the identity element in H. Since ¦ is a homomorphism, e*Φ(G). Also , t0 ³ tj for every jÎJ and hence m(e*) = t0. Hence , e*Î Fm(t0) . Therefore, ¦(G ) ∩ Fm (tj ) ¹ Æ . Now, suppose ¦(G) ∩ Fm(tj ) ¹ Æ is empty for some p > 0 . Since tp-1 > tp , we have mtp-1Ì mtp and hence ¦ -1 (mtp-1 ) Í ¦ -1 (mtp ) . Now, x Î ¦ -1 (mtp ) Þ ¦(x) Î mp U Fm(tj ) Þ ¦(x) Î mtp-1 since ¦(G) ∩ Fm( tj ) = Æ . Þ x Î ¦ -1 (mtp-1 ) . Hence, ¦ -1 (mtp ) Í ¦ -1 (mtp-1 ) and therefore : ¦ -1 ( mtp) = ¦ -1 ( mtp-1 ). This contradicts the assumption that ¦ -1 ( mtj ) are all distinct. Hence, ¦(G ) ∩ Fm(tj ) ¹ Æ, " j Î J. Assume that ¦ -1 ( mtj )’s are not all distinct. Then we can find p,q Î J such that tp ¹ tq and ¦ -1 ( mtp ) = ¦ -1 ( mtq )……………..(2) We assume that tp < tq . Since ¦(G) ∩ Fm(tp ) is non-empty, there exists xÎG such that ¦(x) Î Fm( tp ). This implies that m( ¦(x) ) = tp . Since tp < tq , we have, ¦(x)Î mtp and ¦(x) Ï mtq . Therefore : x Φ -1 ( mtp ) and xϦ -1 (mtq). This contradicts (2). Therefore ¦ -1 (mtj) are all distinct. Remark (2.4) It can be observed from the proof that the second part of the proof in the above theorem hold even when J is uncountable. If ¦ is a surjection, then ¦(G )∩Fm(tj ) ¹ Æ, "jÎJ; and hence ¦ -1 ( mtj ) are all distinct. Corollary (2.5) If Im(m) = {tj | jÎJ } and ¦(G)∩Fm(tj) ¹ Æ, "jÎJ, then : C( ¦ -1 (m) ) º ¦ -1 ( C(m) ). In particular, if J ={1, 2, …, n} and t1 > t2 >…> tn then : C( ¦ -1 (m) ) º ¦ -1 ( mt1) Ì ¦ -1 ( mt2 ) Ì …̦ -1 ( mtn ). Proof : The result follows from theorem (2.3 ). 343 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 2 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 2 Vol. 25 Year 2012 3.Homomorphic images of fuzzy groups. In this section, we study the relationship between C(l) and C(¦(l)) . And prove that if l is a fuzzy subgroup of G with Im(l) = {tj | j=1, 2, …, n} such that t1> t2 >…> tn and if ¦: G ®H is a surjective group homomorphism, then the chain ¦(lt1) Í ¦(lt2) ͦͅ(ltn ) contains all level subgroups of ¦(l). In the following proposition, we remove the restriction on the finiteness of | Im (l) | . Proposition (3.1) If ¦ is a surjection, l has sup-property and { ltj | jÎJ } is the collection of all level subgroups of l, then { ¦(ltj ) | jÎJ } is the collection of all level subgroups of ¦(l ) . Proof Let m = ¦(l) and t Î [0, 1]. Then u Î mt Þ m(u) ³ t Þ sup {l(x) | xΦ-1(u) }³ t. Since l has sup-property ,this implies that l(x0) ³ t , for some x0 Î ¦-1(u). Then x0 Î lt and hence ¦(x0 ) = u Î ¦( lt ). Therefore, we have mt Í ¦( lt ). Now , if uΦ(lt ) then u = ¦(x) for some xÎlt and hence. m(u) = sup{ l(z) | zÎ ¦-1(u)}= sup {l(z) | ¦(z) = ¦(x)} ³ l(x) ³ t (Since xÎlt). Therefore uÎmt and hence ¦( lt ) Í mt . Thus we have mt =¦(lt ) for every tÎ[0, 1]………...(3) In particular, mtj = ¦(ltj ), "jÎJ. Hence all ¦(ltj)’s are level subgroups of m = ¦(l) Also, it follows from (3) and the assumption that these are the only level subgroups of m . The following example shows that surjectiveness of ¦, in the above proposition, is essential. Example ( 3.2 ) Let G = {1, -1} and H = {1, -1, i, -i }. Define ¦ : G ® H by ¦(x) = x, "xÎG. Then ¦ is a non-surjective group homomorphism . Define l : G ® [0, 1] by l(1) = 0.3 and l(-1) = 0.1 . Then l is a fuzzy subgroup of G having sup-property. The level subgroups of l are l0.3 = {1} and l0.1= G. Now, m = ¦(l) is defined by : m(1) = 0.3, m(-1) = 0.1, m( i ) = m(-i ) = 0. Hence the level subgroups of m are m0.3 = ¦(l0.3) = {1}, m0.1 = ¦(l0.1) = {1, -1} and m0 = H. Therefore, {¦(l0.3) , ¦(l0.1)} does not contain all level subgroups of m . We observe from the following example that surjectiveness of ¦ does not guarantee the distinctness of all ¦( ltj ). Example ( 3.3 ) Let G = P3 and H be the subgroup {e, (12)} of P3, where P3 denotes the permutation group of degree three. 344 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 2 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 2 Vol. 25 Year 2012 Define ¦ : G ® H by : Then ¦ is a surjective group homomorphism. Define l: G ®[0, 1] by : l(x) Then l is a fuzzy subgroup of G having sup-property. The level subgroups of l are l0.9 = {e}, l0.5 = {e,(12)}, l0.2 = G. Now, ¦(l) is given by ¦(l)(e) = 0.9, ¦(l)((12)) = 0.5 and hence ¦(l0.9 ) ={e}, ¦(l0.5 ) = ¦(l0.2 ) = H. In the following theorem we obtain a necessary and sufficient condition for the distinctness of all ¦(ltj ). Theorem ( 3.4 ) If ¦:G ® H is a surjective group homomorphism and l is a fuzzy subgroup of G having sup-property and Im(l)={tj | jÎJ} where J is a countable index set. Then {¦(ltj) , jÎJ}, are all distinct if and only if l is ¦-invariant. Proof Suppose ¦(ltj )’s are all distinct . Since tj >tj+1 "jÎJ we have ltj Ì ltj+1 , and hence, ¦(ltj ) Ì ¦(ltj+1 ) . Let x, yÎG such that ¦(x) =¦(y). Let ¦(ltp ) be the smallest ¦(ltj ) which contains ¦(x). If p = 0 . Then ¦(x) = ¦(y) Î ¦(lt0 ) and hence l(x) = l(y) = l(e). If p ¹ 0. Then ¦(x), ¦(y) Î ¦(ltp ) and ¦(x), ¦(y) Ϧ(ltp-1). Hence x, y Îltp and x , yÏltp-1. Therefore l(x) = l(y) = tp . Thus, in both cases, we have , l(x) = l(y) , and hence l is ¦-invariant. Conversely, Assume that l is ¦-invariant. Then for any z Î H, ¦(l)(z) = l(x) , "x Î ¦-1(z)………………..(4) If ¦(ltj)’s are not distinct then there exists tp, tq Î Im(l) such that tp ¹ tq and ¦(ltp) = ¦(ltq) . Since tp, tqÎ Im(l), there exist x, yÎ G such that l(x) = tp , and l(y) = tq . Hence by (4) , we have : ¦(l)( ¦(x)) = tp and ¦(l)( ¦(y) ) = tq . Therefore tp, tqÎ Im( ¦(l) ) and hence it follows that : ¦(ltp) ¹¦(ltq). ¦(x) = (12) "xÎ{(12), (13), (23)} e "xÎ{e, (123), (132)} = 0.5 i f x = (12) 0.9 i f x = e 0.2 "xÎG\{e, (12)} 345 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 2 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 2 Vol. 25 Year 2012 This is a contradiction. Hence ¦(ltj ), jÎJ, are all distinct. We observe that the proof of the second part does not require the countability of J. Hence we have following result. Corollary ( 3.5 ) If ¦: G ® H is a surjective group homomorphism and l is an ¦-invariant fuzzy subgroup of G having sup-property then : C( ¦(l) ) º¦( C(l) ) . Proof The result follows from theorem ( 3.4 ) . Corollary ( 3.6 ) Let ¦: G ® H is a surjective group homomorphism and l be a fuzzy subgroup of G with Im(l) = {ti | i=1, 2, …, n} where t1> t2 >…> tn. Then : ( i ) {¦(lti )| i =1, 2, …, n} contains all level subgroups of ¦(l). ( ii ) { ¦(lti ), i =1,2, …,n}are all distinct if and only if l is¦-invariant. ( iii ) If l is ¦ -invariant then Im (¦(l) ) = Im(l) and C( ¦(l) ) º ¦( lt1 ) Í ¦( lt2 ) Í …ͦ( ltn ). Proof It is straight forward. Remark ( 3.7 ) Theorems (2.3) and (3.4) give us methods to obtain the chains of level subgroups of homomorphic images and pre-images of an arbitrary fuzzy group from that of the given fuzzy group . More specifically if ¦(G) Ç Fm(t) ¹ Æ for every t Î Im(m), then C( ¦-1(m) ) º ¦-1( C(m) ). Further, if ¦ is a surjection and l is ¦-invariant, then C( ¦(l) ) º ¦( C(l) ). References 1. Guptaa K.C and Sarmab B.K.,(1999) "nilpotent fuzzy groups", fuzzy set and systems, 101, : 167-176,. 2. Das.P.S.,1981 "Fuzzy groups and level subgroups", J. Math. Anal. And appl. 84: 264-269, 3. Mukherjee N.P. ,(1948) " Fuzzy normal subgroups and Fuzzy cosets ", Infor. Sci , 34: 225- 239. 4. Abou-Dareb, A, T.,(2000)" On Almost Quasi-Frobenius Fuzzy rings ", M. Sc. Thesis, University of Baghdad. 5. Malik D. S. and Mordeson J. N. (1991)," Fuzzy subgroups of abelian groups" , Chinese, J. Math., 19, No. 2. 6. Bhattacharya P., (1987)" Fuzzy subgroups:sume characterization", J. Math. Anal. And appl. 128: 241-252. 7.Mordeson J.N.,(1996) " L-subspaces and L-subfields " . 8. Martines L.,(1995) " Fuzzy subgroups of fuzzy groups and fuzzy ideals of fuzzy rings" , J. Math. Losangeles , 3 , No. 4. 346 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 2 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 2 Vol. 25 Year 2012 9. Ajmal N. ,1994 " homomorphism of fuzzy groups, correspondence theorem and fuzzy quotient groups ", Fuzzy sets and systems. 61 : 329-339 . الزمر الضبـابيـة و زمـر التشاكـلحـول مهيوب محمد قائد ¡محمد توفيقمى ناجي ل جامعة بغداد¡كلية التربية أبن الهيثم ¡قسم الرياضيات 2010 كانون االول14قبل البحث في: 2009 االولكانون 9استلم البحث في: الخالصة سالسـل الزمـر الجزئيـة المسـتوية مـن الزمـر الضـبابية وأثبتنـا الشـروط فـييهـتم هذا البحث بدراسة تأثير تشـاكل الزمـر الضرورية و الالزمة للحصول على سالسـل الزمـر الجزئيـة المسـتوية لصـور التشـاكل ( الصـور العكسـية ) ألي زمـرة ضـبابية ورة التشــاكل والصـــورة تلــك النظريــات مــن إيجــاد سالســل الزمــر الجزئيــة المســتوية لصــ ســاطةتمكنــا بو نفســه الوقــتباختياريــة العكسية لها في أي زمرة ضبابية اختيارية . الزمر الضبابية ، تشاكل الزمر-: الكلمات المفتاحية Proof لمى ناجي محمد توفيق, مهيوب محمد قائد قسم الرياضيات , كلية التربية أبن الهيثم ,جامعة بغداد الخلاصة