IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.23 (3) 2010 On Fuzzy Internal Direct Product NN.. MM.. NNaamm aa.. DDee ppaarrtt mm ee nntt ooff MMaatt hh ee mm aatt ii ccss,, CCooll llee ggee ooff SS ccii ee nn ccee ooff WWoomm eenn ,, UUnn ii vvee rrss ii tt yy ooff BBaagghh ddaa dd.. Abstract The main aim of this p aper is to introduce the concept of a Fuz zy Internal Direct Product of fuzzy subgroup s of group . We st udy some prop erties and p rove some theorems about t his concept ,which is very imp ortant and interesting of fuzzy group s and very useful in app lications of fuzzy mathematics in general and esp ecially in fuzz y group s. Introduction Ap p lying the concep t of fuzzy sets of Zadeh to the group theory, Rosenfeld introduced the notion of a fuzz y group as early as 1971. The technique of generating a fuzzy group (the smallest fuzzy group ) containing an arbitrarily chosen fuzz y set was develop ed only in 1992 by M alik , M ordeson and Nair, [1]. In this p aper, we use our notion of fuzzy inner p roduct t o generate Fuz zy Internal Direct Product of fuzzy subgroup s of group . Now we introduce the following definitions which is necessary and needed in the next section : De fini tion 1.1 [1], [2]: A mapp ing from a nonemp ty set X t o the interval [0, 1] is called a fuzz y subset of X . Next, we shall give some definitions and concepts related to fuz zy subsets of G. De fini tion 1.2: Let v, be fuzz y subsets of G, if    xvx  for every Gx , then we say that  is contained in v(or v contains  ) and we write v (or   ). If v and v , then  is said to be prop erly contained in v(or vp rop erly contains  ) and we write v ( or   ).[3] Not e that: v if and only if    xvx  for all Gx .[4] De fini tion 1.3 [3] : Let v, be two fuzzy subsets of G. Then v and v   are fuzzy subsets as follows: (i)    )(),(max)( xvxxv   (ii)    )(),(min)( xvxxv   , for all Gx Then vandv   are called the union and intersection of  and v , resp ectively. Now, we are ready to give the definition of a fuzzy subgroup of a group . De fini tion 1.4[1], [5]: A fuz zy subset  of a group G is a fuzzy subgroup of G if: (i)       min a , b a*b   (ii)    aa  1 , for all Gba , . IHJPAS IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.23 (3) 2010 Proposi tion 1.5 [6]: Let  be a fuzzy group . Then     Gaea   . De fini tion 1.6 [7]: If  is a fuzzy subgroup of G, then  is said to be abelian if Gyx  , ,     0,0  yx  , then    yxxy   . De fini tion 1.7 [8] , [9]: A fuzzy subgroup  of G is said to be normal fuzzy subgroup if    x* y y*x , x,y G    . Fuzzy Internal Direct Product Now we are ready to introduce the definition and some theorems about fuz zy internal direct p roduct of fuzzy sub group s of group s : Definition 2.1 Let A be a fuzzy subgroup of a group (G,.) and N1,N2,…,Nn be fuzzy normal subgroup s in A such that : 1- A = N1N2…Nn 2- Let xt  A, xG, t[0, 1], then : xt(y )= ni  Ni in unique way. Then A is said to be fuzzy internal direct p roduct of N1,N2,…,Nn. Now we introduce the following theorem : Theorem 2.2 If a fuzzy subgroup A of a group (G, .) is the fuzzy internal direct p roduct of fuzzy normal subgroup s : N1,N2 ,…, Nn , then for i  j, Ni  Nj = {et}, and if ni  Ni , nj  Nj then ni nj = nj ni . Proof: Sup p ose that ns  Ni  Nj then we can write ns as Viewing ns a an fuzz y singleton in Ni . similarly, we can write ns as       otherw ise0 xyifGy,...,y,yandy...yyy )}y(n),...,y(n),y(nsup{min{ n21n21 nn2211       wiseother0 n...,,1iNn,nyif}}e,...,e...,e),y(n,e,...,e{{minsup (y)n iinj1is1i1 s       wiseother0 n,...1iNn,nyif}}e,...,e),y(n,e,...,e,...,e{{minsup (y)n iin1js1ji1 s IHJPAS IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.23 (3) 2010 Since the two decomp ositions in this form for ns must coincide, the entry from Ni in each must be equal. In the first decomposition this entry is ns, in the other it is et; hence ns= et. Thus Ni  Nj = {et} for i  j. Sup p ose ni Ni, nj Nj , and ij then ninj ni -1 Nj since Nj is normal fuzz y ; thus ninj ni - 1 nj -1 Nj . Similarly , since ni -1 Ni , nj ni -1 nj -1 Ni, whence ninj ni -1 nj -1  Ni then ninj ni -1 nj -1  Ni Nj ={et}. Thus ninj ni -1 nj -1 = et ; this gives t he desired result ninj = nj ni . Remark One should p oint out that if k1,…,kn are normal fuzzy subgroup s of A, such that A= k1k2…kn and ki kj ={et} for ij it need not be true that A is the fuzzy internal direct p roduct of k1,…,kn . A more stringent condition is needed. Clearly from the above theorem we can obtain the following corollary : Corollary 2.3 A fuz zy subgroup A of a group (G, .) is t he fuzz y internal direct p roduct of t he normal fuzzy subgroup s N1,…,Nn if and only if : (1) A = N1N2…Nn (2) Ni (N1N2…Ni-1Ni+1…Nn) = {e} for i= 1,…,n Definition 2.4 Let H and K are fuzzy subgroup s of a group (G, .) .The join Hk of H and k is the intersection of all fuzz y subgroup s of G containing H.K . Clearly , this intersection will be the smallest p ossible fuzz y subgroup of G containing H.K , and if elements in H and K commute, in p articular, if G is abelian, we have H  K = H.K ,since H . K  H  K (by definition 2.4) and since : H(x1) = sup {min {H(x1), K(e)} x1  G} and K(x2) = sup {min {H(e), K(x2)} x2  G}, then : H  H . K and K  H . K, so H  K  H . K , thus H  K = H.K . Not e that Clearly , H  K would be contained in any fuzzy subgroup containing both H and K . Thus we see that H K is the smallest fuzzy subgroup of G containing both H and K . Theorem 2.5 A fuzzy subgroup A of a group (G, .) is the fuzzy internal direct p roduct of fuzzy subgroup s H and K if and only if :      wiseother xifGxxxxxxKxH K 0 )Im(},,)}(),({{minsup .H 212121 IHJPAS IBN AL- HAITHAM J. FO R PURE & APPL. SC I. VO L.23 (3 ) 2010 1) A = H  K 2) xt . y s = y s . xt for all xt  H and y s  K , t , s  [0,1], x, y  G. 3) H  K = {e} Proof: Let A be the fuzzy internal direct p roduct of H and K. we claim that (1), (2) and (3) are obvious if one will r egard A as isomorphic to the fuzzy internal direct p roduct of H and K under the map , with ( xt, y s) = xt . y s = (x . y )r where r = min { t , s} , x, y  G, r, s  [0, 1]. Under this map Corresponds to H , and Corresponds to K . Then (1), (2) and (3) follow immediately from the corresp onding assertions regardin g in H  K, which are obv ious. Conversely , let (1), (2) and (3) hold. We must show that t he map  of the fuzz y internal direct p roduct H  K in to A, given by : (xt, y s) = xt . y s = (x.y )r where r = min{t , s}, x, y  G, r, s  [0,1], is an isomorphism. The map  has already been defined sup p ose (xt1, y s1) = (xt2, y s2) . Then xt1 . y s1 = xt2 . y s2 ; consequently But x -1 t2 . xt1  H and y s2 . y -1 s1  K and they are the same element and thus in H  K = {e} by (3). Therefore, x -1 t2 . xt1 = e and xt1 = xt2 . Likewise, ys1 = y s2, so (xt1, y s1)= (xt2, y s2). This shows t hat  is one to one. The fact that xt . y s = y s . xt by (2) for all xt  H and y s  k means that H.k(x) = {sup { min{H(x1), k(x2)}}; x = x1. x2 , x1, x2  G} is a fuzzy subgroup , for we have seen that this is the case if fuzzy singletons of H commute with those of K . Thus by (1), H. K = H  K = A, so  is on to A . Finally , [(xt1, y s1)(xt2, y s2)] = (xt1.xt2, y s1.y s2) = xt1.xt2.y s1.y s2, while [(xt1,y s1)][(xt2,y s2)] = xt1.y s1.xt2.y s2 . But by (2) we have y s1 . xt2 = xt2 . y s1 . Thus : [(xt1,y s1)(xt2,y s2)] = [(xt1,y s1)][(xt2, y s2)] . Remark: Not every fuzzy subgroup of abelian group is the fuzz y internal direct p roduct of two p rop er fuzz y subgroup s. The following corollary is immediate consequences of the above theorem. }Hx|)e,x{(H tt  }|),{( KyyeK ss  KandH 1 1s2s1t 1 2t yyxx   IHJPAS IBN AL- HAITHAM J. FO R PURE & APPL. SC I. VO L.23 (3 ) 2010 Corollary 2.6 Let A be fuzzy subgroup of a group (G, .). Let xt and y s be fuzzy singletons of A which commute and are of relatively p rime orders r and s and , are fuzz y subgroup s of . Then xt.y s is of order r.s . Fuzzy Invariants Of Fuzzy Subgroup In this section we introduce the following definition and Lemma about fuzzy invariants of fuz zy subgroup : Definition 3.1 Let A be fuzz y subgroup of abelian group (G, .) of order p n , p a prime, and A = A1  A2  …. Ak where each Ai is fuzzy generating set of order p ni with n1  n2 … nk > 0, then the integers n1, n2, …, nk are called the fuzzy invariants of A just because we called the integers above the fuzzy invariants of A does not mean that they are really the fuzzy invariants of A. That is, it is p ossible to assign different sets of fuzzy invariants t o A . We shall soon show that the fuzzy invariants of A are indeed unique and comp letely describe A. Not e one other thing about the fuzz y invariants of A. If A = A1  … Ak, where Ai is fuzzy generating set of order P ni , n1  n2  … nk > 0, then o(A) = o(A1)o(A2) …o(Ak), hence P n = P n1 P n2 …P nk = P n1+n2+ …+nk , whence n = n1+n2+ … + nk In other words, n1, n2, …, nk give us a p artition of n . Before discussing the uniqueness of the fuzzy invariants of A, one thing should be made absolutely clear the singleton fuzzy a1,…, ak and the fuzz y subgroup s A1, …, Ak which they generate, which a rose above to give the decomp osition of A in to a fuzzy internal direct p roduct of fuz zy generating subgroup s, are not unique. Let’s see this in a very simple examp le Let G = {e,a,b,a.b} be an abelian group of order 4 where a 2 = b 2 = e, ab = ba and A(e) = A(a) = 1, A(b) = A(a.b) = 3/4. Then A= H  K where H = , K = are fuzzy generating subgroup s of order 2. But we have another decomposition of A as a fuzz y internal direct p roduct, namely A = N  K where N = and K = . So, even in this fuz zy subgroup of very small order, we can get distinct decompositions of the fuzzy subgroup as the internal direct p roduct of fuzzy generating subgroup s. Lemma 3.2 Let A be fuzzy subgroup of abelian group (G, .) of order p n , p a prime. Sup p ose that A = A1  A2 …. Ak, where each Ai = is fuzzy generating of order p ni , and n1  n2  …  nk > 0. If m is an integer such that nt > m  nt+1 then : A(p m ) = B1  … Bt  At+1  …  Ak where Bi is fuzzy generating of order p m , generated by , for i  t. The order of A(p m ) is p u , where mn iP si a  IHJPAS IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.23 (3) 2010 Proof: First of all, we claim that At+1, …., Ak are all in A(p m ), since m  nt+1  …  nk > 0, if j  t+1, Hence Aj , for j  t+1 lies in A(p m ). Secondly, if i  t then ni > m and whence each such is in A(p m ) and so t he fuzzy subgroup it generates, Bi, is also in A(p m ). Since B1, …, Bt, At+1, …, Ak all in A(p m ), their p roduct (which is fuzzy direct, since the p roduct A1  A2  …  Ak is fuzzy direct) is in A(p m ). Hence A(p m )  B1  … Bt  At+1  …  Ak. On t he other hand, if : is in A(p m ), since it t hen satisfies we set : However, the product of t he fuzz y subgroup s A1, …, Ak is fuzzy direct, so we get : Thus t he order of ai, that is, p ni must divide ip m for i = 1, 2, …, k. If i  t+ 1 this is automatically true whatever be the choice of t+1, ..., k since m  nt+1  …  nk , Hence p ni | p m , i  t+1. However, for i  t, we get from p ni | ip m that p ni-m | i , therefore i = vi P ni-m for some integer vi . Put ting all this in formation in to the values of t he i's in the exp ression for y r as : We see that This say s t hat y r  B1  …  Bt  At+1  …  Ak. Now since each Bi is of order p m and since o(Ai) = p ni and since A = B1  …  Bt  At+1  …  Ak , o(A) = o(B1) o (B2) …. o(Bt) o (At+1) … o (Ak) = p m p m … p m p nt+1 …p nk Thus, if we write o (A) = p u , then: The lemma is p roved. Corollary 3.3 If A is a fuzzy subgroup of a group in lemma (3.2), then o(A(p )) = p k . Proof: Ap p ly the above lemma to the case m = 1. Then t = k, hence u = l. k = k and so o (A) = p k .    k 1ti i nmtu e)a(a n jmn jm pp j p j   ea)a( n imn i p i P i   mn i P i a  k k 2 2 1 1r a....a.ay  ,ey mp r  mmm kp k p1 1 p r a....aye  ea,....,ea mm kp k p1 1   k k 1 1r a...ay  k k 1t 1t vtp t P1v 1r a....aa...ay mn tm1n        k 1ti i nmtu IHJPAS IBN AL- HAITHAM J. FO R PURE & APPL. SC I. VO L.23 (3 ) 2010 Re ferences 1. M alik . D. s. , M ordeson . J. N. and Nair. P. S. , (1992) ” Fuz zy Generators and Fuzzy Direct Sums of Abelian Group s”, Fuzzy sets and sy stems,.50,.193-199,. 2. M ajeed.S.N., (1999)”On fuz zy subgroups of abelian group s”,M .Sc. Thesis, University of Baghdad,. 3. M ordesn J.N.,( 1996) ”L-subspaces and L-subfields”. 4. Hussein. R. W., (1999) ”Some results of fuz zy rings” ,M .Sc. Thesis ,University of Baghd ad ,. 5. Abou-Z aid. , (1988).” On normal fuzzy subgroup s ” , J.Facu.Edu.,.13. 6. Seselja .B and Tepavcevic A.,(1997).“ Anot e on fuzzy groups” , J.Yugosl av.Op er.Rese ,.7,.1,.49-54, . 7. Gupt a K.c and Sarma B.K., (1999) . “nilp otent fuzzy groups” ,fuzzy set and systems ,.101,.167-176 ,. 8. Seselja . B . and Tepavcevic A. ,(1996)“ Fuzzy group s and collections of subgroup s ” , fuzzy sets and sy stems ,.83 ,.85-91 ,. 9. Bandler. W. and Kohout. L. , (2000) Semantics of implication operators and fuzzy relational p roducts, Internat. J. M an- M achine st udies 12. 89-116 . IHJPAS للعلوم الصر قیة المجلدمجلة ابن الھیثم 2010) 3( 23فة والتطبی الجداء المباشر الداخلي الضبابيحول نغم موسى نعمة جامعة بغداد ، العلوم للبناتكلیة ،قسم الریاضیات الخالصة یتضـــمن البحـــث تقــــدیم تعریـــف الجــــداء المباشـــر الــــداخلي الضـــبابي وبعــــض الخـــواص والمبرهنــــات فـي مجـال الریاضـیات الضـبابیة بشـكل تطبیقـات واسـعة اوذ اً ضـروریو اً مـمهـ اً موضوع دالذي یع المتعلقة به .عام وفي مجال الزمرة الضبابیة بشكل خاص IHJPAS