IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.23 (3) 2010 On Fuzzy Internal Direct Product NN.. MM.. NNaammaa.. DDee ppaarrttmmee nntt ooff MMaatthhee mmaattiiccss,, CCoollllee ggee ooff SScciiee nnccee ooff WWoommeenn,, UUnniivvee rrss iittyy ooff BBaagghhddaa dd.. Abstract The main aim of this paper is to introduce the concept of a Fuzzy Internal Direct Product of fuzzy subgroups of group . We study some properties and prove some theorems about this concept ,which is very important and interesting of fuzzy groups and very useful in applications of fuzzy mathematics in general and especially in fuzzy groups. 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 fuzzy inner product to generate Fuzzy Internal Direct Product of fuzzy subgroups of 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, be fuzzy 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 properly contained in v (or vproperly contains  ) and we write v ( or   ).[3] Note that: v if and only if    xvx  for all Gx .[4] Definition 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 , respectively. Now, we are ready to give the definition of a fuzzy subgroup of a group. Definition 1.4[1], [5]: A fuzzy 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. SCI. VOL.23 (3) 2010 Proposition 1.5 [6]: Let  be a fuzzy group. Then     Gaea   . Definition 1.6 [7]: If  is a fuzzy subgroup of G, then  is said to be abelian if Gyx  , ,     0,0  yx  , then    yxxy   . Definition 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 fuzzy internal direct p roduct of fuzzy sub groups of groups : Definition 2.1 Let A be a fuzzy subgroup of a group (G,.) and N1,N2,…,Nn be fuzzy normal subgroups 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 product of fuzzy normal subgroups : N1,N2 ,…, Nn , then for i  j, Ni  Nj = {et}, and if ni  Ni , nj  Nj then ni nj = nj ni . Proof: Suppose that ns  Ni  Nj then we can write ns as Viewing ns a an fuzzy singleton in Ni . similarly, we can write ns as       otherwise0 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. SCI. VOL.23 (3) 2010 Since the two decompositions 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. Suppose ni Ni, nj Nj , and ij then ninjni -1Nj since Nj is normal fuzzy; thus ninjni - 1 nj -1 Nj . Similarly, since ni -1 Ni , njni -1 nj -1 Ni, whence ninjni -1 nj -1  Ni then ninjni -1 nj -1  Ni Nj ={et}. Thus ninjni -1 nj -1 = et ; this gives the desired result ninj = njni . Remark One should point out that if k1,…,kn are normal fuzzy subgroups 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 product of k1,…,kn . A more stringent condition is needed. Clearly from the above theorem we can obtain the following corollary : Corollary 2.3 A fuzzy subgroup A of a group (G, .) is the fuzzy internal direct p roduct of the normal fuzzy subgroups 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 subgroups of a group (G, .) .The join Hk of H and k is the intersection of all fuzzy subgroups of G containing H.K . Clearly, this intersection will be the smallest possible fuzzy subgroup of G containing H.K , and if elements in H and K commute, in particular, 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 . Note 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 product of fuzzy subgroups 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 . ys = y s . xt for all xt  H and ys  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 regard A as isomorphic to the fuzzy internal direct p roduct of H and K under the map , with (xt, ys) = xt . ys = (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 corresponding assertions regarding in H  K, which are obvious. Conversely, let (1), (2) and (3) hold. We must show that the map  of the fuzzy internal direct product H  K in to A, given by : (xt, ys) = xt . ys = (x.y)r where r = min{t, s}, x, y  G, r, s  [0,1], is an isomorphism. The map  has already been defined suppose (xt1, ys1) = (xt2, ys2) . Then xt1 . ys1 = xt2 . ys2 ; consequently But x -1 t2 . xt1  H and ys2 . 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, ys1)= (xt2, ys2). This shows that  is one to one. The fact that xt . ys = y s . xt by (2) for all xt  H and ys  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, ys1)(xt2, ys2)] = (xt1.xt2, ys1.ys2) = xt1.xt2.ys1.ys2, while [(xt1,ys1)][(xt2,ys2)] = xt1.ys1.xt2.ys2 . But by (2) we have ys1 . xt2 = xt2 . ys1 . Thus : [(xt1,ys1)(xt2,ys2)] = [(xt1,ys1)][(xt2, ys2)] . Remark: Not every fuzzy subgroup of abelian group is the fuzzy internal direct p roduct of two proper fuzzy subgroups. 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 ys be fuzzy singletons of A which commute and are of relatively prime orders r and s and , are fuzzy subgroups of . Then xt.ys is of order r.s . Fuzzy Invariants Of Fuzzy Subgroup In this section we introduce the following definition and Lemma about fuzzy invariants of fuzzy subgroup : Definition 3.1 Let A be fuzzy subgroup of abelian group (G, .) of order pn, p a prime, and A = A1  A2  …. Ak where each Ai is fuzzy generating set of order pni 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 possible to assign different sets of fuzzy invariants to A . We shall soon show that the fuzzy invariants of A are indeed unique and completely describe A. Note one other thing about the fuzzy 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 = Pn1Pn2 …Pnk = Pn1+n2+…+nk, whence n = n1+n2+ … + nk In other words, n1, n2, …, nk give us a partition 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 fuzzy subgroups A1, …, Ak which they generate, which a rose above to give the decomposition of A in to a fuzzy internal direct product of fuzzy generating subgroups, are not unique. Let’s see this in a very simple example 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 subgroups of order 2. But we have another decomposition of A as a fuzzy internal direct product, namely A = N  K where N = and K = . So, even in this fuzzy subgroup of very small order, we can get distinct decompositions of the fuzzy subgroup as the internal direct product of fuzzy generating subgroups. Lemma 3.2 Let A be fuzzy subgroup of abelian group (G, .) of order pn, p a prime. Suppose that A = A1  A2 …. Ak, where each Ai = is fuzzy generating of order pni, 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 pm, generated by , for i  t. The order of A(p m ) is p u , where mni P si a  IHJPAS IBN AL- HAITHAM J. FOR PURE & APPL. SCI. 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(pm) and so the fuzzy subgroup it generates, Bi, is also in A(p m). Since B1, …, Bt, At+1, …, Ak all in A(p m), their product (which is fuzzy direct, since the product A1  A2  …  Ak is fuzzy direct) is in A(p m ). Hence A(p m)  B1  … Bt  At+1  …  Ak. On the other hand, if : is in A(p m), since it then satisfies we set : However, the product of the fuzzy subgroups A1, …, Ak is fuzzy direct, so we get : Thus the 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 | pm , i  t+1. However, for i  t, we get from pni | ip m that p ni-m| i , therefore i = vi Pni-m for some integer vi . Putting all this in formation in to the values of the i's in the expression for yr as : We see that This says that y r  B1  …  Bt  At+1  …  Ak. Now since each Bi is of order pm and since o(Ai) = pni and since A = B1  …  Bt  At+1  …  Ak , o(A) = o(B1) o (B2) …. o(Bt) o (At+1) … o (Ak) = pmpm … pm pnt+1 …pnk Thus, if we write o (A) = p u, then: The lemma is proved. Corollary 3.3 If A is a fuzzy subgroup of a group in lemma (3.2), then o(A(p)) = p k . Proof: Apply 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 njmnjm pp j p j   ea)a( nimni p i P i   mniP 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 mntm1n        k 1ti i nmtu IHJPAS IBN AL- HAITHAM J. FO R PURE & APPL. SC I. VO L.23 (3) 2010 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 ,. 5. Abou-Zaid. , (1988).” On normal fuzzy subgroups ” , J.Facu.Edu.,.13. 6. Seselja .B and Tepavcevic A.,(1997).“Anote on fuzzy groups” , J.Yugoslav.Oper.Rese ,.7,.1,.49-54, . 7. Gupta K.c and Sarma B.K., (1999) . “nilpotent fuzzy groups” ,fuzzy set and systems ,.101,.167-176 ,. 8. Seselja . B . and Tepavcevic A. ,(1996)“ Fuzzy groups and collections of subgroups ” , fuzzy sets and systems ,.83 ,.85-91 ,. 9. Bandler. W. and Kohout. L. , (2000) Semantics of implication operators and fuzzy relational products, Internat. J. Man- Machine studies 12. 89-116 . IHJPAS للعلوم الصر 2010) 3( 23فة والتطبیقیة المجلدمجلة ابن الھیثم 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