340 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 2 العدد Ibn Al-Haitham Journal for Pure and Applied S cience 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. Re c e ive d i n : 9 De c e mbe r 2 0 0 9 Ac c e pte d i n : 1 4 De c e mbe r 2 0 1 0 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[: ÆXm 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 - =≥ mmmmm [2] . It is easy to see that if m is fuzzy subgroup of G, then Gxxe Œ"≥ ),()( 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 ≥Œ= mm , tŒ[0, 1] is called the level subset of m in G and )(;{ xGxt mm Œ= * >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[ elŒ are subgroups of G and viseversa [8] . If Œ21,tt Im( m ) such that 21 tt π , then obviously, 21 tt mm π . 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 =ÃÃÃ∫ mmm ... 21 [2] . Let HGf Æ: be a homomorphism of groups , l be a fuzzy subgroup of G , m a fuzzy subgroup of H . Then ,)),(()(1 Gxxff Œ"=- mm =))(( yf l Sub{ )}();( 1 yfxx -Œl if ;)(1 Fπ- yf 0 if ;)(1 F=- yf Hy Œ" 341 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 2 العدد Ibn Al-Haitham Journal for Pure and Applied S cience 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 Æ: 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 Œ= [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 Æ 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 | jŒJ } is the collection of all level subgroups of m, t he n {¶ - 1 ( mtj ) | jŒJ } is t he co llect io n o f a ll le ve l s ub gro ups o f ¶ - 1 (m ) . Proof Let l = ¶ -1 (m) and tŒ[0,1]. Then : xŒlt €¶ -1 (m ) ≥ t € m(¶(x)) ≥ t € ¶(x)Œ mt € x Œ ¶ -1 (mt ). Hence lt = ¶ -1 (mt ) " t Œ [0, 1]………………………(1) In particular, we have : ltj = ¶ -1 (mtj ) "jŒJ . If l has a level subgroup l t which does not belong to {¶ - 1 (mtj ) | jŒJ } then m must have a level subgroup mt which does not belong to {mtj | jŒJ } 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 Æ H by ¶(x) = e ," x Œ G. Then ¶ is a group homomorphism. Define m : H Æ [0, 1] by : m(e) = 1, m( (12)) = 0.5, m(x) = 0.3, " x Œ 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 xŒG. Hence, l1 = l0.5 = l0.3 = G. N ow, we proceed to derive a necessary and sufficient condition for the distinctness of all the ¶ - 1 (mtj ). For t Œ Im(m) , we define : Fm(t) = { xŒG | 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 S cience No. 2 Vol. 25 Year 2012 Let ¶: G Æ H be a group homomorphism and m is a fuzzy subgroup of H with Im(m)={tj | jŒJ } where J is a countable index set. Then ¶ -1 (mtj ) are all distinct if and only if : ¶(G)∩Fm(tj) π ∆, "jŒ J. Proof Assume that ¶ -1 (mtj ) , jŒJ, are all distinct. Let e* denote the identity element in H. Since ¶ is a homomorphism, e*Œ¶(G). Also , t0 ≥ tj for every jŒJ and hence m(e*) = t0. Hence , e*Œ Fm(t0) . Therefore, ¶(G ) ∩ Fm (tj ) π ∆ . Now, suppose ¶(G) ∩ Fm(tj ) π ∆ is empty for some p > 0 . Since tp-1 > tp , we have mtp-1à mtp and hence ¶ -1 (mtp-1 ) Õ ¶ -1 (mtp ) . Now, x Œ ¶ -1 (mtp ) fi ¶(x) Œ mp U Fm(tj ) fi ¶(x) Œ mtp-1 since ¶(G) ∩ Fm( tj ) = ∆ . fi x Œ ¶ -1 (mtp-1 ) . Hence, ¶ -1 (mtp ) Õ ¶ -1 (mtp-1 ) and therefore : ¶ -1 ( mtp) = ¶ -1 ( mtp-1 ). This contradicts the assumption that ¶ -1 ( mtj ) are all distinct. Hence, ¶(G ) ∩ Fm(tj ) π ∆, " j Œ J. Assume that ¶ -1 ( mtj )’s are not all distinct. Then we can find p,q Œ J such that tp π tq and ¶ -1 ( mtp ) = ¶ -1 ( mtq )……………..(2) We assume that tp < tq . Since ¶(G) ∩ Fm(tp ) is non-empty, there exists xŒG such that ¶(x) Œ Fm( tp ). This implies that m( ¶(x) ) = tp . Since tp < tq , we have, ¶(x)Œ mtp and ¶(x) œ mtq . Therefore : x Œ¶ -1 ( mtp ) and xœ¶ -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 ) π ∆, "jŒJ; and hence ¶ -1 ( mtj ) are all distinct. Corollary (2.5) If Im(m) = {tj | jŒJ } and ¶(G)∩Fm(tj) π ∆, "jŒJ, then : C( ¶ -1 (m) ) ∫ ¶ -1 ( C(m) ). In particular, if J ={1, 2, …, n} and t1 > t2 >…> tn then : C( ¶ -1 (m) ) ∫ ¶ -1 ( mt1) à ¶ -1 ( mt2 ) à …ö -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 S cience 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 Æ H is a surjective group homomorphism, then the chain ¶(l t1) Õ ¶(l t2) Õ …Õ ¶(l tn ) 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 | jŒJ } is the collection of all level subgroups of l, then { ¶(ltj ) | jŒJ } is the collection of all level subgroups of ¶(l ) . Proof Let m = ¶(l) and t Œ [0, 1]. Then u Œ mt fi m(u) ≥ t fi sup {l(x) | xŒ¶ -1(u) }≥ t. Since l has sup-property ,this implies that l(x0) ≥ t , for some x0 Œ ¶-1(u). Then x0 Œ lt and hence ¶(x0 ) = u Œ ¶( lt ). Therefore, we have mt Õ ¶( lt ). Now , if uŒ¶(lt ) then u = ¶(x) for some xŒlt and hence. m(u) = sup{ l(z) | zŒ ¶-1(u)}= sup {l(z) | ¶(z) = ¶(x)} ≥ l(x) ≥ t (Since xŒlt). Therefore uŒmt and hence ¶( lt ) Õ mt . Thus we have mt =¶(l t ) for every tŒ[0, 1]………...(3) In particular, mtj = ¶(ltj ), "jŒJ. 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 Æ H by ¶(x) = x, "xŒG. Then ¶ is a non-surjective group homomorphism . Define l : G Æ [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 S cience No. 2 Vol. 25 Year 2012 Define ¶ : G Æ H by : Then ¶ is a surjective group homomorphism. Define l : G Æ[0, 1] by : l (x) Then l is a fuzzy subgroup of G having sup- property. The level subgroups of l are l 0. 9 = {e} , l 0. 5 = {e,(12)} , l 0. 2 = G. N ow, ¶(l ) is given by ¶(l )(e) = 0.9, ¶(l )((12)) = 0.5 and hence ¶(l 0. 9 ) ={e} , ¶(l 0. 5 ) = ¶(l 0. 2 ) = H. In the following theorem we obtain a necessary and sufficient condition for the distinctness of all ¶(ltj ). Theorem ( 3.4 ) If ¶:G Æ H is a surjective group homomorphism and l is a fuzzy subgroup of G having sup-property and Im(l)={tj | jŒJ} where J is a countable index set. Then {¶(ltj) , jŒJ}, are all distinct if and only if l is ¶-invariant. Proof Suppose ¶(ltj )’s are all distinct . Since tj >tj+1 "jŒJ we have ltj à ltj+1 , and hence, ¶(ltj ) à ¶(ltj+1 ) . Let x, yŒG such that ¶(x) =¶(y). Let ¶(ltp ) be the smallest ¶(ltj ) which contains ¶(x). If p = 0 . Then ¶(x) = ¶(y) Œ ¶(lt0 ) and hence l(x) = l(y) = l(e). If p π 0. Then ¶(x), ¶(y) Œ ¶(ltp ) and ¶(x), ¶(y) œ¶(ltp-1). Hence x, y Œltp and x , yœltp-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 Œ H, ¶(l)(z) = l(x) , "x Œ ¶-1(z)………………..(4) If ¶(ltj)’s are not distinct then there exists tp, tq Œ Im(l) such that tp π tq and ¶(ltp) = ¶(ltq) . Since tp, tqŒ Im(l), there exist x, yŒ G such that l(x) = tp , and l(y) = tq . Hence by (4) , we have : ¶(l)( ¶(x)) = tp and ¶(l)( ¶(y) ) = tq . Therefore tp, tqŒ Im( ¶(l) ) and hence it follows that : ¶(ltp) π¶(ltq). ¶(x) = (12) "xŒ{(12), (13), (23)} e "xŒ{e, (123), (132)} = 0.5 i f x = ( 12) 0.9 i f x = e 0.2 "xŒG\{e, (12)} 345 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 2 العدد Ibn Al-Haitham Journal for Pure and Applied S cience No. 2 Vol. 25 Year 2012 This is a contradiction. Hence ¶(ltj ), jŒJ, 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 Æ H is a surjective group homomorphism and l is an ¶-invariant fuzzy subgroup of G having sup-property then : C( ¶(l) ) ∫¶( C(l) ) . Proof The result follows from theorem ( 3.4 ) . Corollary ( 3.6 ) Let ¶: G Æ 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) ) ∫ ¶( lt1 ) Õ ¶( lt2 ) Õ …Õ¶( 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) « Fm(t) π ∆ for every t Œ Im(m), then C( ¶-1(m) ) ∫ ¶-1( C(m) ). Further, if ¶ is a surjection and l is ¶-invariant, then C( ¶(l) ) ∫ ¶( 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 S cience 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