`NEW ZEALAND JOURNAL OF MATHEMATICS Volume 38 (2008), 187­195STRONG FUZZY TOPOLOGICAL GROUPS V.L.G. Nayagam* , D. Gauld, G. Venkateshwari and G. Sivaraman(Received January 2008)Abstract. Following the introduction of fuzzy sets in 1965, a notion of fuzzy topological group was proposed by Foster in 1979: essentially he took a group and furnished it with a fuzzy topological structure. An equivalent notion of fuzzy topological group was introduced by Ma and Yu in 1984 by replacing points by fuzzy points. Recently two of the coauthors have introduced the notion of the topology induced on the set of all fuzzy singletons by the fuzzy topology. In this paper we extend the notion of fuzzy topological group by allowing the points of our strong fuzzy topological groups to be fuzzy singletons of a given group and using the induced topology. We study properties of strong fuzzy topological groups, analysing such entities as its connection with the previous notions, subgroups, images and products of strong fuzzy topological groups.1. Introduction The concept of fuzzy sets was introduced in [21]. Rosenfeld [15] gave the idea of fuzzy subgroups. The notions of fuzzy cosets and results analogous to the results in crisp theory are studied in [8], [11] et.al. Different notions of fuzzy normal subgroups were introduced in [13], [10], [19]. The notion of fuzzy quotient semigroup was introduced in [10] and extended to the notion of generalised fuzzy quotient groups in [12]. The notion of induced topology on fuzzy singletons was introduced in [16]. The notion of translation invariant topology was studied in [19], [8] and it was extended to the notion of fuzzy translation invariant topology on a group in [17]. The notion of fuzzy topological groups was introduced in [3] and properties of fuzzy topological groups were studied in [9], [2], [5]. In this paper a new notion of strong fuzzy topological group is introduced and studied. Here we give a brief review of some preliminaries. Definition [21] 1.1. If S is any set, a mapping µ : S  [0, 1] is called a fuzzy subset of S. Definition [15] 1.2. A fuzzy subset µ of a group G is called a fuzzy subgroup of G if, for all x, y  G, the following conditions are satisfied: (i) µ(xy)  min(µ(x), µ(y)); and (ii) µ(x-1 )  µ(x) Definition [14] 1.3. Let S be any set. A fuzzy singleton p of S is a fuzzy set which has singleton support {x} with value p(x)  (0, 1]. Here we note that a fuzzy2000 Mathematics Subject Classification 54H11, 54A40, 20N25. Key words and phrases: fuzzy topological spaces, fuzzy subgroups, fuzzy left coset, fuzzy Hausdorff space, translation invariant topology, fuzzy topological group. *A part of the work was carried out in the Department of Mathematics, University of Auckland, New Zealand and was supported by TEQIP, NITT, INDIA188V.L.G. NAYAGAM, D. GAULD, G. VENKATESHWARI AND G. SIVARAMANsingleton p belongs to a fuzzy set µ (p  µ) iff p(x)  µ(x), where {x} is the support of p. Theorem [20] 1.4. Let (X, ) and (Y, ) be fuzzy topological spaces. A map f : X  Y is fuzzy continuous iff for every fuzzy point p and for every fuzzy open set µ   such that f (p)  µ, there exists    such that p   and f ()  µ. Here we note that a fuzzy point is a fuzzy subset which has a singleton support and fuzzy value in (0, 1) and a fuzzy point p with support {x} is said to lie in  (p  ) iff p(x) &lt; (x). Definition [10] 1.5. Let G be a group. Let µ be any fuzzy subset of G. Let p be any fuzzy singleton in G. Let supp p = {x}. Define a fuzzy left coset pµ of G by pµ(z) = min(p(x), µ(x-1 z)), z  G. Similarly a fuzzy right coset µp is defined by µp(z) = min{µ(zx-1 ), p(x)}. Definition [16] 1.6. Let (G, ) be a fuzzy topological space. The induced topology  on the collection (G) of all fuzzy singletons of G is defined as the topology generated by  = {Vµ | µ  }, where Vµ = {p  (G) | p  µ} and hence ((G),  ) is called an induced topological space. Definition [3] 1.7. Let X be a group and let (X, ) be a fully stratified fuzzy topological space. Then (X, ) is a fuzzy topological group if it satisfies the following conditions: (I) The mapping f : (X, ) × (X, )  (X, ) defined by f ((x, y)) = xy is fuzzy continuous. (II) The mapping g : (X, )  (X, ) defined by g(x) = x-1 is fuzzy continuous. Definition [9] 1.8. Let X be a group and (X, ) be a fuzzy topological space. Then (X, ) is a fuzzy topological group if it satisfies the following two conditions: (I) For all a, b  X and any Q-neighborhood W of fuzzy point (ab) there are Q-neighborhoods U of a and V of b such that U V  W . (II) For all a  X and any Q-neighborhood V of a -1 , there exists a Qneighborhood U of a such that U -1  V . Note 1.9. From the propositions 2.1, 2.2 of [2], the above definitions 1.7 and 1.8 are equivalent. 2. Strong fuzzy Topological Groups In this section, a new notion of strong fuzzy topological group is introduced and studied. Definition 2.1. Let G be any group. A fuzzy Hausdorff space (G, ) is said to be strong fuzzy topological group if i). M : (G) × (G)  (G) defined by M (p, q) = pq, for every (p, q)  (G) × (G), is continuous. ii). I : (G)  (G) defined by I(p) = p-1 , for every p  (G), is continuous. Example 2.2. Let G = {e, x, y, xy} is Klein's four group. Let a &gt; 1/2. Let  be the collection of all fuzzy sets µ whose fuzzy values µ(z)  [0, a]  {1}, for every z in G. Clearly (G, ) is a fuzzy topological space. We claim that M : (G) × (G)  (G) defined by M (p, q) = pq, for every (p, q)  (G) × (G), is continuous. Let (p1 , p2 )  (G) × (G), where supp p1 = {t1 } and supp p2 = {t2 }. Let Vµ   such that M ((p1 , p2 )) = p1 p2  Vµ , where  is a base for the induced topology  . Clearly supp p1 p2 = {t1 t2 }. Case 1 : fuzzy value of p1 p2 &gt; a.STRONG FUZZY TOPOLOGICAL GROUPS189Hence µ(t1 t2 ) &gt; min{p1 (t1 ), p2 (t2 )}, and hence µ(t1 t2 ) = 1. Let µ1 (t) = 1 if t = t1 and µ1 (t) = 0 if t = t1 and µ2 (t) = 1 if t = t2 and µ2 (t) = 0 if t = t2 . Clearly µ1 , µ2   and pi  Vµi , for i = 1, 2. Now we prove that Vµ1 Vµ2  Vµ . Let p  Vµ1 , q  Vµ2 . Now p(t1 )  1 and q(t2 )  1 and hence pq(t1 t2 )  1 = µ(t1 t2 ) and hence Vµ1 Vµ2  Vµ . Case 2 : fuzzy value of p1 p2  a. Sub case 1 : one of the values of p1 (t1 ) or p2 (t2 ) &gt; a, say p1 (t1 )  a and p2 (t2 ) &gt; a, Clearly p1  . Let µ1 = p1 and µ2 (t) = 1 if t = t2 and µ2 (t) = 0 if t = t2 . Clearly µ2   and hence Vµ1 , Vµ2   and clearly pi  Vµi , for i = 1, 2. Now we claim that Vµ1 Vµ2  Vµ . Let p  Vµ1 , q  Vµ2 and hence p  µ1 and q  µ2 . Clearly the only possibility of supports of p, q are t1 , t2 respectively. pq is a fuzzy singleton defined on t1 t2 with value min{p(t1 ), q(t2 )}  p1 (t1 ). Since p1 p2  Vµ , µ(t1 t2 )  p1 p2 (t1 t2 ) = p1 (t1 )  pq(t1 t2 ) and hence pq  µ and hence pq  Vµ . Similarly we can prove the case when both fuzzy values p1 (t1 ) and p2 (t2 )  a. Hence M is continuous. Now we prove I is continuous. In Klein's group, the inverse x-1 of every element x is itself. Hence p-1 = p. So I is the Identity map, which is continuous. Clearly (G, ) is a Hausdorff fuzzy topological space and hence (G, ) is a strong fuzzy topological group. Theorem 2.3. If M : (G) × (G)  (G) defined by M (p, q) = pq, for every (p, q)  (G) × (G) is continuous, then m : G × G  G defined by m(x, y) = xy is fuzzy continuous. Proof : Let M be continuous. To prove that m is fuzzy continuous, let r be a fuzzy point in G×G with support {(x, y)}. Let µ be a fuzzy open set in G containing m(r). Now m(r)(z) = supz1 z2 =z r(z1 , z2 ). Hence m(r)(xy) = supz1 z2 =xy r(z1 , z2 ) = r(x, y), so m(r) is a fuzzy point defined on xy with value r(x, y). Now define fuzzy singletons r1 , r2 defined on x, y respectively with fuzzy values µ((x, y)) and 1 respectively. Since r1 r2 = M (r1 , r2 ) is a fuzzy singleton defined on xy with fuzzy value µ((x, y)), r1 r2  µ. Since M is continuous and r1 r2  Vµ   , we have Vµ1 , Vµ2   such that r1  Vµ1 , r2  Vµ2 and Vµ1 Vµ2  Vµ . Since r1  µ1 , r2  µ2 , r(x, y) &lt; (r1 × r2 )(x, y)  µ1 × µ2 (x, y). Hence r  µ1 × µ2 . Now we prove that m(µ1 × µ2 )  µ. m(µ1 × µ2 )(t) = supt1 t2 =t (µ1 × µ2 )(t1 , t2 ). Vµ1 Vµ2  Vµ  min{µ1 (x), µ2 (y)}  µ(xy), x, y  G. So (µ1 × µ2 )(t1 , t2 )  µ(t1 t2 ) = µ(t). Hence m(µ1 × µ2 )(t)  µ(t). Hence by the above theorem 1.4, m is fuzzy continuous. Definition 2.4. Let X and Y be two nonempty sets. Let f : X  Y be any map. For any fuzzy singleton p defined on x  X, if the function if : (X)  (Y ) is defined by if (p) = q, where q is the fuzzy singleton defined on f (x)  Y with q(f (x)) = p(x), then if is called the induced function of f . Lemma 2.5. Let f : X  Y be any map and if : (X)  (Y ) be the induced map of f . i). For any fuzzy set µ of Y, Vf -1 (µ) = if -1 (Vµ ). ii). For any fuzzy set µ of X, if Vµ = µ F (X) Vµ , then µ = µ µ . But the converse need not be true.190V.L.G. NAYAGAM, D. GAULD, G. VENKATESHWARI AND G. SIVARAMANProof of the lemma : i). Now p  Vf -1 (µ)  p(x)  µf -1 (µ) (x)  p(x)  µ(f (x))  if (p)(f (x))  µf ((x))  if (p)  Vµ  p  if -1 (Vµ ) ii). Now we assume that Vµ = µ F (X) Vµ , for a fuzzy set µ. To prove that µ = µ µ , it is enough to prove that µ(x) = µ µ (x), x  X. Let x  X. Define a fuzzy singleton p on x such that p(x) = µ( x). Clearly p  Vµ . Hence p  µ F (X) Vµ  p  Vµ for some . So p(x)  µ (x). Hence µ(x)  µ (x). So µ(x)  supµ F (X) µ (x). If µ( x) &lt; supµ F (X) µ (x), there exists some µ  F (X) such that µ(x) &lt; µ (x)  supµ F (X) µ (x). By defining a fuzzy singleton q on x such that q(x) = µ (x), we have q  Vµ and hence q  µ F (X) Vµ . But q  Vµ , a contradiction to our assumption. So µ(x) = supµ F (X) µ (x). Hence / µ = µ µ . That the converse need not be true can be seen from the following example. Let µi = 1 - 1 . Clearly  µi = 1 and hence V µi = V1 = F (X). But i Vµi = F (X) = i V1 . Theorem 2.6. Let (X, ), (Y, ) be fuzzy topological spaces. A function f : (X, )  (Y, ) is a fuzzy continuous function if and only if the induced function if : ((X),  )  ((Y ),  ) is continuous. Proof : Let (X, ), (Y, ) be fuzzy topological spaces. Let f : (X, )  (Y, ) be any map and if : ((X),  )  ((Y ),  ) be the induced function of f . Let us assume that f is fuzzy continuous. To prove that if is continuous, let VA be a basic open set in  and hence µ  . By part i of the above lemma, if -1 (Vµ ) = Vf -1 (µ) . Since f is fuzzy continuous, f -1 (µ) is fuzzy open in  and hence if -1 (Vµ ) is a basic open set in  . Now we prove the converse. Suppose that if is continuous and let µ  . We prove that f -1 (µ)  . Since µ is fuzzy open, Vµ is basic open in  . By continuity of if , if -1 (Vµ ) is open in  and hence if -1 (Vµ ) = µ  Vµ . So by part i of the above lemma, Vf -1 (µ) = µ  Vµ . Now by part ii of the above lemma, f -1 (µ) = µ µ . Hence f -1 (µ)  . Theorem 2.7. The continuity of I : (G)  (G) defined by I(p) = p-1 , for every p  (G) and the fuzzy continuity of i : G  G defined by i(x) = x-1 are equivalent. Proof : Since I is the induced function of i, by the above theorem, the continuity of I and the fuzzy continuity of i are equivalent. Theorem 2.8. If (G, ) is a strong fuzzy topological group, then it is a fuzzy topological group. Proof : By theorem 2.3, the continuity of M implies the fuzzy continuity of m, the condition of the definition 1.7. By theorem 2.7, the continuity of I and the fuzzy continuity of i, the condition ii) of the definition 1.7 are equivalent. Theorem 2.9. Let (G, ) be a fuzzy topological space on a group G. Then (G, ) is a strong fuzzy topological group if and only if MI : (G) × (G)  (G) defined by MI (p, q) = pq -1 , for every (p, q)  (G) × (G) is continuous.STRONG FUZZY TOPOLOGICAL GROUPS191Proof :  part. Let (G, ) be a strong fuzzy topological group. Let f1 , f2 : (G)  (G) be defined by f1 (p) = p, f2 (q) = q -1 respectively. Clearly f1 and f2 are continuous. Hence the function f : (G) × (G)  (G) × (G) defined by f (p, q) = (p, q -1 ), for every (p, q)  (G) × (G), is continuous. Now MI (p, q) = pq -1 = M (f (p, q)) = (M  f )(p, q) and hence MI is continuous.  part. Let us assume that MI (p, q) = pq -1 , for every (p, q)  (G) × (G) is continuous. Clearly the function g : (G)  (G) × (G) defined by g(p) = (1e , p) is continuous and Hence I(p) = p-1 = MI (g(p)) = (MI  g)(p) is continuous. If f1 : (G)  (G) is defined by f1 (p) = p, then h : (G) × (G)  (G) × (G) defined by h(p, q) = (f1 , I)(p, q) = (p, q -1 ) is continuous. Now M (p, q) = MI (h(p, q)) = MI  h and hence M is continuous. Theorem 2.10. Let (G, ) be a strong fuzzy topological group. Every subgroup of G is a strong fuzzy topological subgroup of G with its subspace topology. Proof : Let H be a subgroup of G. We have to prove that (H,  | H) is also a strong fuzzy topolgical group. By hypothesis, MG : (G) × (G)  (G) defined by MG (p, q) = pq, for every (p, q)  (G) × (G), is continuous. We have to prove that MH : (H) × (H)  (H) defined by MH (p, q) = pq, for every (p, q)  (H) × (H), is continuous. Let p, q  (H) and let Vµ  |H be a basic open set in the subspace (H, |H ) such that pq  Vµ . So µ   | H. Hence there exists    such that µ =  | H. So V   with pq  V . By the continuity of MG , there exists V1 , V2   such that MG (V1 × V2 )  V . Clearly p, q  V1 |H ×V2 |H and MH (V1 |H ×V2 |H )  V|H = Vµ . Hence MH is continuous. Clearly IH = IG | (H) is continuous. Hence the theorem. Theorem 2.11. Let (G1 , 1 , 1 ) and (G2 , 2 , 2 ) be strong fuzzy topological groups. Then (G1 × G2 , , 1 × 2 ) is a strong fuzzy topological group. Proof : We note that ((x1 , x2 ), (y1 , y2 )) = (x1 1 y1 , x2 2 y2 ). Clearly (G1 × G2 , 1 × 2 ) is fuzzy Hausdorff in the product topology. By theorem 7.3 of [3] and theorem 2.7, i is continuous. Now to prove that (G1 × G2 , , 1 × 2 ) is a strong fuzzy topological group, it is enough to prove that M : (G1 × G2 ) × (G1 × G2 )  (G1 × G2 ) defined by M (p, q) = pq is continuous, where p and q are fuzzy singletons defined on (x1 , x2 ) and (y1 , y2 ) respectively. To prove that M is continuous, let (p, q)  (G1 ×G2 ) and Vµ  1 ×2 with µ  1 ×2 such that M (p, q) = pq  Vµ . So pq(x1 1 y1 , x2 2 y2 )  µ(x1 1 y1 , x2 2 y2 ). Since µ  1 ×2 , µ = (µ × ) and hence pq(x1 1 y1 , x2 2 y2 )  min(µ (x1 1 y1 ),  (x2 2 y2 )). Let µ = µ1 and  = µ2 . We note that µi  i . Hence pq(x1 1 y1 , x2 2 y2 )  min(µ1 (x1 1 y1 ), µ2 (x2 2 y2 )). Now we define fuzzy singletons p1 , p2 on x1 , x2 with fuzzy value p(x1 , x2 ) and q1 , q2 on y1 , y2 with fuzzy value q(y1 , y2 ). Here we note that p1 , q1  (G1 ) and p2 , q2  (G2 ). Clearly p1 q1 (x1 1 y1 ) = pq(x1 1 y1 , x2 2 y2 )  µ1 (x1 1 y1 ) and hence p1 q1  Vµ1 . So we have (pi , qi )  (Gi ) × (Gi ) and pi qi  Vµi  i . Let Mi : (Gi × Gi )  (Gi ) be defined by Mi (ai , bi ) = ai i bi , where ai , bi  (Gi ). Since (Gi , i , i ) are strong fuzzy topological groups, Mi are continuous. Hence by continuity of Mi , there exists Vi × Vi such that (pi , qi )  Vi × Vi and Mi (Vi × Vi )  Vµi . Hence pi  i and qi  i . Hence p = p1 × p2  1 × 2 and q = q1 × q2  1 × 2 . Let  = 1 × 2 and  = 1 × 2 . Clearly (p, q)  V × V . To prove M is continuous, it is enough to prove that M (V × V )  Vµ . Let r  V and s  V , where r and192V.L.G. NAYAGAM, D. GAULD, G. VENKATESHWARI AND G. SIVARAMANs are fuzzy points defined on (z1 , z2 ) and (t1 , t2 ). We have to prove that rs  Vµ . rs(z1 1 t1 , z2 2 t2 ) =  = =  = min{r(z1 , z2 ), s(t1 , t2 )} min{(z1 , z2 ), (t1 , t2 )} min{1 × 2 (z1 , z2 ), 1 × 2 (t1 , t2 )} min{1 × 1 (z1 , t1 ), 2 × 2 (z2 , t2 )} min{µ1 (z1 1 t1 ), µ2 (z2 2 t2 )} µ1 × µ2 (z1 t1 , z2 t2 )Hence rs  Vµ1 ×µ2  Vµ and hence the theorem. Theorem 2.12. Let f : (G, )  (G , ) be an injective fuzzy continuous fuzzy open homomorphism. Then the image of a strong fuzzy topological subgroup H of (G, ) is again a strong fuzzy topological subgroup of (G , ). Proof : We have to prove that (f (H),  | f (H)) is a strong fuzzy topological subgroup of (G , ). By theorem 2.9, it suffices to prove that MIf (H) : (f (H)) × (f (H))  (f (H)) defined by MIf (H) (q1 , q2 ) = q1 q2 -1 , for every (q1 , q2 )  (f (H)) × (f (H)), is continuous. Let (q1 , q2 )  (f (H)) × (f (H)) whose supports are y1 and y2 respectively and Vµ  |f (H) be a basic open set in f (H) with q1 q2 -1  Vµ . By definition, there exists    such that µ =  | f (H). Clearly q1 q2 -1  V . Since f is fuzzy continuous, f -1 ()  . Now define fuzzy singletons p1 , p2 on x1 and x2 with values q1 (y1 ) and q2 (y2 ) respectively, where x1 , x2  H with f (x1 ) = y1 and f (x2 ) = y2 . Since H is a subgroup and f is a homomorphism, p1 p2 -1  Vf -1 ()|H . Since H is a strong fuzzy topological group, there exists 1 , 2   | H such that (p1 , p2 )  V1 × V2 and MIH (V1 × V2 )  Vf -1 ()|H . Clearly there exist µ1 , µ2   such that µ1 | H = 1 , µ2 | H = 2 . Since f is fuzzy open, f (µ1 ), f (µ2 )   with f (µ1 ) | f (H) = f (1 ), f (µ2 ) | f (H) = f (2 ). Clearly (q1 , q2 )  Vf (1 ) × Vf (2 ) . Now we claim that MIf (H) (Vf (1 ) × Vf (2 ) )  Vµ . Let (q1 , q2 )  Vf (1 ) × Vf (2 ) . Since f is injective, there exists (p1 , p2 )  V1 × V2 with f (p1 ) = q1 , f (p2 ) = q2 . -1 -1 -1 Since MIH (V1 × V2 )  Vf -1 ()|H , p1 p2  Vf -1 ()|H . Hence q1 q2 (y1 y2 ) = -1 -1 -1 -1 -1 -1 p1 p2 (x1 x2 )  f ()(x1 x2 ) = (f (x1 )f (x2 ) ) = µ(y1 y2 ). Hence the theorem. Corollary 2.13. Let f : (G, )  (G , ) be a fuzzy continuous fuzzy open homomorphism such that every fuzzy open set of (G, ) is f -invariant. Then the image of a strong fuzzy topological subgroup H of (G, ) is again a strong fuzzy topological subgroup of (G , ). Proof : The proof is similar to the above theorem. In the above theorem, injectivity is used to claim for every pair (q1 , q2 )  Vf (1 ) × Vf (2 ) , there exists (p1 , p2 )  V1 × V2 with f (p1 ) = q1 , f (p2 ) = q2 . This can be claimed if 1 and 2 are f invariant. The proof of the following corollary is also similar and is left to the reader. Corollary 2.14. Let f : G  G be a homomorphism. Let (G, ) be a fuzzy topological space. Let  = {f (µ) | µ  }. Then the image of a strong fuzzy topological subgroup H of (G, ) is again a strong fuzzy topological subgroup of (G , ).STRONG FUZZY TOPOLOGICAL GROUPS193Theorem 2.15. Let f : (G, )  (G , ) be an injective fuzzy continuous fuzzy open homomorphism. Then the inverse image of a strong fuzzy topological subgroup H of (G , ) is again a strong fuzzy topological subgroup of (G , ). Proof : We have to prove that (f -1 (H),  | f -1 (H)) is a strong fuzzy topological subgroup of (G, ). By theorem 2.9, it suffices to prove that MIf -1 (H) : (f -1 (H))× (f -1 (H))  (f -1 (H)) defined by MIf -1 (H) (p1 , p2 ) = p1 p2 -1 , for every (p1 , p2 )  (f -1 (H)) × (f -1 (H)), is continuous. Let (p1 , p2 )  (f -1 (H))×(f -1 (H)) whose supports are x1 and x2 respectively and Vµ  |f -1 (H) be a basic open set in f -1 (H) with p1 p2 -1  Vµ . By definition, there exists    such that µ =  | f -1 (H). Clearly p1 p2 -1  V . Since f is fuzzy open, f ()  . Now define fuzzy singletons q1 , q2 on y1 and y2 with values p1 (x1 ) and p2 (x2 ) respectively, where y1 , y2  H with f (x1 ) = y1 and f (x2 ) = y2 . Since f is a homomorphism, q1 q2 -1  Vf ()|H . Since H is a strong fuzzy topological group, there exists 1 , 2   | H such that (q1 , q2 )  V1 × V2 and MIH (V1 × V2 )  Vf ()|H . Clearly there exist µ1 , µ2   such that µ1 | H = 1 , µ2 | H = 2 . Since f is fuzzy continuous, f -1 (µ1 ), f -1 (µ2 )   with f -1 (µ1 ) | f -1 (H) = f -1 (1 ), f -1 (2 ) | f -1 (H) = f -1 (2 ). Clearly (p1 , p2 )  Vf -1 (1 ) × Vf -1 (2 ) . Now we claim that MIf -1 (H) (Vf -1 (1 ) × Vf -1 (2 ) )  Vµ . Let (p1 , p2 )  Vf -1 (1 ) × Vf -1 (2 ) . Hence there exists (q1 , q2 )  V1 × V2 with f (p1 ) = q1 , f (p2 ) = q2 ). Since -1 MIH (V1 × V2 )  Vf ()|H , q1 q2  Vf ()|H . -1 -1 -1 -1 -1 So, by injectivity of f , p1 p2 (x1 x2 ) = q1 q2 (y1 y2 )  f ()(y1 y2 ) = -1 -1 (x1 x2 ) = µ(x1 x2 ). Hence the theorem. The proofs of following corollaries are similar to the proofs of corollaries 2.13 and 2.14. Corollary 2.16. Let f : (G, )  (G , ) be a fuzzy continuous fuzzy open homomorphism such that every fuzzy open set of (G, ) is f -invariant. Then the inverse image of a strong fuzzy topological subgroup H of (G , ) is again a strong fuzzy topological subgroup of (G , ). Corollary 2.17. Let f : G  G be a homomorphism. Let (G , ) be a fuzzy topological space. Let  = {f -1 () |   }. 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Mashinchi and M. Mukaidono, Generalised fuzzy quotient groups, Fuzzy sets and systems 74 (1995), 245-257. [13] N.P. Mukhergee and P. Bhattacharya, Fuzzy normal subgroups and fuzzy cosets, Information Sciences 34 (1984), 225-239. [14] Pu Pao-Ming and Liu Ying-Ming, Fuzzy Topology I, Neighbourhood structure of a Fuzzy point and Moore-smith Convergence, J. Math. Anal. Appl. 76 (1980) 571-599 . [15] A. Rosenfeld, Fuzzy groups, J.Math. Anal. Appl. 35 (1971),512-517. [16] V. Lakshmana Gomathi Nayagam, Geetha Sivaraman, Induced topology on fuzzy singletons, Far East Journal of Applied Mathematics volume 32 issue 2, (2008), 189-198. [17] V. Lakshmana Gomathi Nayagam, G. Venkateshwari, Geetha Sivaraman, Fuzzy Translation Invariant Spaces (Communicated). [18] Rekha Srivastava, S.N. Lal and Arun K. Srivastava, Fuzzy Hausdorff Topological Spaces, J. Math. Anal. Appl. 81 (1981) 497-506. [19] Wang-Jin Liu, Fuzzy Invariant subgroups and Fuzzy ideals, Fuzzy Sets and Systems 8 (1982) 133-139. [20] C.K. Wong, Fuzzy Points and Local properties of Fuzzy Topology, J. Math. Anal. Appl. 46 (1974) 316-328. [21] L.A. Zadeh, Fuzzy Sets, Information and Control 8 (1965) 338-353.V.L.G. Nayagam Department of Mathematics National Institute of Technology Tiruchirapalli INDIA [email protected]D. Gauld Department of Mathematics University of Auckland Auckland NEW ZEALAND [email protected]STRONG FUZZY TOPOLOGICAL GROUPS195G. Venkateshwari Department of Mathematics Sacs MAVMM Engineering College Madurai INDIAG. Sivaraman Department of Mathematics Anna University Chennai INDIA`

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