STRONG FUZZY TOPOLOGICAL GROUPS. V.L.G. Nayagam*, D. Gauld, G. Venkateshwari and G. Sivaraman (Received January 2008)

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1 NEW ZEALAND JOURNAL OF MATHEMATICS Volume 38 (2008), STRONG 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 fuzzy 2000 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

2 188 V.L.G. NAYAGAM, D. GAULD, G. VENKATESHWARI AND G. SIVARAMAN singleton 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, δ) and (Y, σ) 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 σ such that f(p) µ, there exists ν δ such that p ν and f(ν) 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 ν (p ν) iff p(x) < ν(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, δ) be a fuzzy topological space. The induced topology τ δ on the collection (G) of all fuzzy singletons of G is defined as the topology generated by σ = {V µ µ δ}, where V µ = {p (G) p µ} and hence ( (G), τ δ ) is called an induced topological space. Definition [3] 1.7. Let X be a group and let (X, δ) be a fully stratified fuzzy topological space. Then (X, δ) is a fuzzy topological group if it satisfies the following conditions: (I) The mapping f : (X, δ) (X, δ) (X, δ) defined by f((x, y)) = xy is fuzzy continuous. (II) The mapping g : (X, δ) (X, δ) defined by g(x) = x 1 is fuzzy continuous. Definition [9] 1.8. Let X be a group and (X, δ) be a fuzzy topological space. Then (X, δ) 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) λ there are Q-neighborhoods U of a λ and V of b λ such that UV W. (II) For all a X and any Q-neighborhood V of a 1 λ, there exists a Q- neighborhood U of a λ 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, δ) 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 > 1/2. Let δ be the collection of all fuzzy sets µ whose fuzzy values µ(z) [0, a] {1}, for every z in G. Clearly (G, δ) 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 (p 1, p 2 ) (G) (G), where supp p 1 = {t 1 } and supp p 2 = {t 2 }. Let V µ σ such that M((p 1, p 2 )) = p 1 p 2 V µ, where σ is a base for the induced topology τ δ. Clearly supp p 1 p 2 = {t 1 t 2 }. Case 1 : fuzzy value of p 1 p 2 > a.

3 STRONG FUZZY TOPOLOGICAL GROUPS 189 Hence µ(t 1 t 2 ) > min{p 1 (t 1 ), p 2 (t 2 )}, and hence µ(t 1 t 2 ) = 1. Let µ 1 (t) = 1 if t = t 1 and µ 1 (t) = 0 if t t 1 and µ 2 (t) = 1 if t = t 2 and µ 2 (t) = 0 if t t 2. Clearly µ 1, µ 2 δ and p i V µi, for i = 1, 2. Now we prove that V µ1 V µ2 V µ. Let p V µ1, q V µ2. Now p(t 1 ) 1 and q(t 2 ) 1 and hence pq(t 1 t 2 ) 1 = µ(t 1 t 2 ) and hence V µ1 V µ2 V µ. Case 2 : fuzzy value of p 1 p 2 a. Sub case 1 : one of the values of p 1 (t 1 ) or p 2 (t 2 ) > a, say p 1 (t 1 ) a and p 2 (t 2 ) > a, Clearly p 1 δ. Let µ 1 = p 1 and µ 2 (t) = 1 if t = t 2 and µ 2 (t) = 0 if t t 2. Clearly µ 2 δ and hence V µ1, V µ2 σ and clearly p i V µi, for i = 1, 2. Now we claim that V µ1 V µ2 V µ. Let p V µ1, q V µ2 and hence p µ 1 and q µ 2. Clearly the only possibility of supports of p, q are t 1, t 2 respectively. pq is a fuzzy singleton defined on t 1 t 2 with value min{p(t 1 ), q(t 2 )} p 1 (t 1 ). Since p 1 p 2 V µ, µ(t 1 t 2 ) p 1 p 2 (t 1 t 2 ) = p 1 (t 1 ) pq(t 1 t 2 ) and hence pq µ and hence pq V µ. Similarly we can prove the case when both fuzzy values p 1 (t 1 ) and p 2 (t 2 ) 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, δ) is a Hausdorff fuzzy topological space and hence (G, δ) 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) = sup z1z 2=z r(z 1, z 2 ). Hence m(r)(xy) = sup z1z 2=xy r(z 1, z 2 ) = r(x, y), so m(r) is a fuzzy point defined on xy with value r(x, y). Now define fuzzy singletons r 1, r 2 defined on x, y respectively with fuzzy values µ((x, y)) and 1 respectively. Since r 1 r 2 = M(r 1, r 2 ) is a fuzzy singleton defined on xy with fuzzy value µ((x, y)), r 1 r 2 µ. Since M is continuous and r 1 r 2 V µ τ δ, we have V µ1, V µ2 τ δ such that r 1 V µ1, r 2 V µ2 and V µ1 V µ2 V µ. Since r 1 µ 1, r 2 µ 2, r(x, y) < (r 1 r 2 )(x, y) µ 1 µ 2 (x, y). Hence r µ 1 µ 2. Now we prove that m(µ 1 µ 2 ) µ. m(µ 1 µ 2 )(t) = sup t1t 2=t(µ 1 µ 2 )(t 1, t 2 ). V µ1 V µ2 V µ min{µ 1 (x), µ 2 (y)} µ(xy), x, y G. So (µ 1 µ 2 )(t 1, t 2 ) µ(t 1 t 2 ) = µ(t). Hence m(µ 1 µ 2 )(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 i f : (X) (Y ) is defined by i f (p) = q, where q is the fuzzy singleton defined on f(x) Y with q(f(x)) = p(x), then i f is called the induced function of f. Lemma 2.5. Let f : X Y be any map and i f : (X) (Y ) be the induced map of f. i). For any fuzzy set µ of Y, V f 1 (µ) = i 1 f (V µ ). ii). For any fuzzy set µ of X, if V µ = µα F (X) V µα, then µ = µα µ α. But the converse need not be true.

4 190 V.L.G. NAYAGAM, D. GAULD, G. VENKATESHWARI AND G. SIVARAMAN Proof of the lemma : i). Now p V f 1 (µ) p(x) µ f 1 (µ)(x) p(x) µ(f(x)) i f (p)(f(x)) µf((x)) i f (p) V µ p i f 1 (V µ ) ii). Now we assume that V µ = µα F (X) V µα, for a fuzzy set µ. To prove that µ = µα µ α, it is enough to prove that µ(x) = µα µ α (x), x X. Let x X. Define a fuzzy singleton p on x such that p(x) = µ ( x). Clearly p V µ. Hence p µα F (X) V µα p V µα for some α. So p(x) µ α (x). Hence µ(x) µ α (x). So µ(x) sup µα F (X) µ α (x). If µ ( x) < sup µα F (X) µ α (x), there exists some µ β F (X) such that µ(x) < µ β (x) sup µα F (X) µ α (x). By defining a fuzzy singleton q on x such that q(x) = µ β (x), we have q V µβ and hence q µα F (X) V µα. But q / V µ, a contradiction to our assumption. So µ(x) = sup µα F (X) µ α (x). Hence µ = µα µ α. That the converse need not be true can be seen from the following example. Let µ i = 1 1 i. Clearly µ i = 1 and hence V µi = V 1 = F (X). But i V µi F (X) = V 1. Theorem 2.6. Let (X, δ), (Y, σ) be fuzzy topological spaces. A function f : (X, δ) (Y, σ) is a fuzzy continuous function if and only if the induced function i f : ( (X), τ δ ) ( (Y ), τ σ ) is continuous. Proof : Let (X, δ), (Y, σ) be fuzzy topological spaces. Let f : (X, δ) (Y, σ) be any map and i f : ( (X), τ δ ) ( (Y ), τ σ ) be the induced function of f. Let us assume that f is fuzzy continuous. To prove that i f is continuous, let V A be a basic open set in τ σ and hence µ σ. By part i of the above lemma, i 1 f (V µ ) = V f 1 (µ). Since f is fuzzy continuous, f 1 (µ) is fuzzy open in δ and hence i 1 f (V µ ) is a basic open set in τ δ. Now we prove the converse. Suppose that i f is continuous and let µ σ. We prove that f 1 (µ) δ. Since µ is fuzzy open, V µ is basic open in τ σ. By continuity of i f, i 1 f (V µ ) is open in τ δ and hence i 1 f (V µ ) = µα δ V µα. So by part i of the above lemma, V f 1 (µ) = µα δ V µα. Now by part ii of the above lemma, f 1 (µ) = µα µ α. Hence f 1 (µ) δ. 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, δ) 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, δ) be a fuzzy topological space on a group G. Then (G, δ) is a strong fuzzy topological group if and only if M I : (G) (G) (G) defined by M I (p, q) = pq 1, for every (p, q) (G) (G) is continuous.

5 STRONG FUZZY TOPOLOGICAL GROUPS 191 Proof : part. Let (G, δ) be a strong fuzzy topological group. Let f 1, f 2 : (G) (G) be defined by f 1 (p) = p, f 2 (q) = q 1 respectively. Clearly f 1 and f 2 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 M I (p, q) = pq 1 = M(f(p, q)) = (M f)(p, q) and hence M I is continuous. part. Let us assume that M I (p, q) = pq 1, for every (p, q) (G) (G) is continuous. Clearly the function g : (G) (G) (G) defined by g(p) = (1 e, p) is continuous and Hence I(p) = p 1 = M I (g(p)) = (M I g)(p) is continuous. If f 1 : (G) (G) is defined by f 1 (p) = p, then h : (G) (G) (G) (G) defined by h(p, q) = (f 1, I)(p, q) = (p, q 1 ) is continuous. Now M(p, q) = M I (h(p, q)) = M I h and hence M is continuous. Theorem Let (G, δ) 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, δ H) is also a strong fuzzy topolgical group. By hypothesis, M G : (G) (G) (G) defined by M G (p, q) = pq, for every (p, q) (G) (G), is continuous. We have to prove that M H : (H) (H) (H) defined by M H (p, q) = pq, for every (p, q) (H) (H), is continuous. Let p, q (H) and let V µ τ δ H be a basic open set in the subspace (H, τ δ H ) such that pq V µ. So µ δ H. Hence there exists ν δ such that µ = ν H. So V ν τ δ with pq V ν. By the continuity of M G, there exists V ν1, V ν2 τ δ such that M G (V ν1 V ν2 ) V ν. Clearly p, q V ν1 H V ν2 H and M H (V ν1 H V ν2 H) V ν H = V µ. Hence M H is continuous. Clearly I H = I G (H) is continuous. Hence the theorem. Theorem Let (G 1, 1, δ 1 ) and (G 2, 2, δ 2 ) be strong fuzzy topological groups. Then (G 1 G 2,, δ 1 δ 2 ) is a strong fuzzy topological group. Proof : We note that ((x 1, x 2 ), (y 1, y 2 )) = (x 1 1 y 1, x 2 2 y 2 ). Clearly (G 1 G 2, δ 1 δ 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 (G 1 G 2,, δ 1 δ 2 ) is a strong fuzzy topological group, it is enough to prove that M : (G 1 G 2 ) (G 1 G 2 ) (G 1 G 2 ) defined by M(p, q) = pq is continuous, where p and q are fuzzy singletons defined on (x 1, x 2 ) and (y 1, y 2 ) respectively. To prove that M is continuous, let (p, q) (G 1 G 2 ) and V µ τ δ1 δ 2 with µ δ 1 δ 2 such that M(p, q) = pq V µ. So pq(x 1 1 y 1, x 2 2 y 2 ) µ(x 1 1 y 1, x 2 2 y 2 ). Since µ δ 1 δ 2, µ = (µ α ν α ) and hence pq(x 1 1 y 1, x 2 2 y 2 ) min( µ α (x 1 1 y 1 ), ν α (x 2 2 y 2 )). Let µ α = µ 1 and ν α = µ 2. We note that µ i δ i. Hence pq(x 1 1 y 1, x 2 2 y 2 ) min(µ 1 (x 1 1 y 1 ), µ 2 (x 2 2 y 2 )). Now we define fuzzy singletons p 1, p 2 on x 1, x 2 with fuzzy value p(x 1, x 2 ) and q 1, q 2 on y 1, y 2 with fuzzy value q(y 1, y 2 ). Here we note that p 1, q 1 (G 1 ) and p 2, q 2 (G 2 ). Clearly p 1 q 1 (x 1 1 y 1 ) = pq(x 1 1 y 1, x 2 2 y 2 ) µ 1 (x 1 1 y 1 ) and hence p 1 q 1 V µ1. So we have (p i, q i ) (G i ) (G i ) and p i q i V µi τ δi. Let M i : (G i G i ) (G i ) be defined by M i (a i, b i ) = a i i b i, where a i, b i (G i ). Since (G i, i, δ i ) are strong fuzzy topological groups, M i are continuous. Hence by continuity of M i, there exists V νi V γi such that (p i, q i ) V νi V γi and M i (V νi V γi ) V µi. Hence p i ν i and q i γ i. Hence p = p 1 p 2 ν 1 ν 2 and q = q 1 q 2 γ 1 γ 2. Let ν = ν 1 ν 2 and γ = γ 1 γ 2. Clearly (p, q) V ν V γ. To prove M is continuous, it is enough to prove that M(V ν V γ ) V µ. Let r V ν and s V γ, where r and

6 192 V.L.G. NAYAGAM, D. GAULD, G. VENKATESHWARI AND G. SIVARAMAN s are fuzzy points defined on (z 1, z 2 ) and (t 1, t 2 ). We have to prove that rs V µ. rs(z 1 1 t 1, z 2 2 t 2 ) = min{r(z 1, z 2 ), s(t 1, t 2 )} min{ν(z 1, z 2 ), γ(t 1, t 2 )} = min{ν 1 ν 2 (z 1, z 2 ), γ 1 γ 2 (t 1, t 2 )} = min{ν 1 γ 1 (z 1, t 1 ), ν 2 γ 2 (z 2, t 2 )} min{µ 1 (z 1 1 t 1 ), µ 2 (z 2 2 t 2 )} = µ 1 µ 2 (z 1 t 1, z 2 t 2 ) Hence rs V µ1 µ 2 V µ and hence the theorem. Theorem Let f : (G, δ) (G, σ) be an injective fuzzy continuous fuzzy open homomorphism. Then the image of a strong fuzzy topological subgroup H of (G, δ) is again a strong fuzzy topological subgroup of (G, σ). Proof : We have to prove that (f(h), σ f(h)) is a strong fuzzy topological subgroup of (G, σ). By theorem 2.9, it suffices to prove that M If(H) : (f(h)) (f(h)) (f(h)) defined by M If(H) (q 1, q 2 ) = q 1 q 1 2, for every (q 1, q 2 ) (f(h)) (f(h)), is continuous. Let (q 1, q 2 ) (f(h)) (f(h)) whose supports are y 1 and y 2 respectively and V µ τ σ f(h) be a basic open set in f(h) with q 1 q 1 2 V µ. By definition, there exists ν σ such that µ = ν f(h). Clearly q 1 q 1 2 V ν. Since f is fuzzy continuous, f 1 (ν) δ. Now define fuzzy singletons p 1, p 2 on x 1 and x 2 with values q 1 (y 1 ) and q 2 (y 2 ) respectively, where x 1, x 2 H with f(x 1 ) = y 1 and f(x 2 ) = y 2. Since H is a subgroup and f is a homomorphism, p 1 p 1 2 V f 1 (ν) H. Since H is a strong fuzzy topological group, there exists ν 1, ν 2 δ H such that (p 1, p 2 ) V ν1 V ν2 and M IH (V ν1 V ν2 ) V f 1 (ν) H. Clearly there exist µ 1, µ 2 δ such that µ 1 H = ν 1, µ 2 H = ν 2. Since f is fuzzy open, f(µ 1 ), f(µ 2 ) σ with f(µ 1 ) f(h) = f(ν 1 ), f(µ 2 ) f(h) = f(ν 2 ). Clearly (q 1, q 2 ) V f(ν1) V f(ν2). Now we claim that M If(H) (V f(ν1) V f(ν2)) V µ. Let (q 1, q 2) V f(ν1) V f(ν2). Since f is injective, there exists (p 1, p 2) V ν1 V ν2 with f(p 1) = q 1, f(p 2) = q 2. Since M IH (V ν1 V ν2 ) V f 1 (ν) H, p 1p 2 1 V f 1 (ν) H. Hence q 1q 2 1 (y 1 y2 1 ) = p 1p 1 2 (x 1 x 1 2 ) f 1 (ν)(x 1 x 1 2 ) = ν(f(x 1)f(x 2 ) 1 ) = µ(y 1 y2 1 ). Hence the theorem. Corollary Let f : (G, δ) (G, σ) be a fuzzy continuous fuzzy open homomorphism such that every fuzzy open set of (G, δ) is f-invariant. Then the image of a strong fuzzy topological subgroup H of (G, δ) is again a strong fuzzy topological subgroup of (G, σ). Proof : The proof is similar to the above theorem. In the above theorem, injectivity is used to claim for every pair (q 1, q 2) V f(ν1) V f(ν2), there exists (p 1, p 2) V ν1 V ν2 with f(p 1) = q 1, f(p 2) = q 2. This can be claimed if ν 1 and ν 2 are f invariant. The proof of the following corollary is also similar and is left to the reader. Corollary Let f : G G be a homomorphism. Let (G, δ) be a fuzzy topological space. Let σ = {f(µ) µ δ}. Then the image of a strong fuzzy topological subgroup H of (G, δ) is again a strong fuzzy topological subgroup of (G, δ).

7 STRONG FUZZY TOPOLOGICAL GROUPS 193 Theorem Let f : (G, δ) (G, σ) be an injective fuzzy continuous fuzzy open homomorphism. Then the inverse image of a strong fuzzy topological subgroup H of (G, σ) is again a strong fuzzy topological subgroup of (G, δ). Proof : We have to prove that (f 1 (H), δ f 1 (H)) is a strong fuzzy topological subgroup of (G, δ). By theorem 2.9, it suffices to prove that M If 1 (H) : (f 1 (H)) (f 1 (H)) (f 1 (H)) defined by M If 1 (H) (p 1, p 2 ) = p 1 p 1 2, for every (p 1, p 2 ) (f 1 (H)) (f 1 (H)), is continuous. Let (p 1, p 2 ) (f 1 (H)) (f 1 (H)) whose supports are x 1 and x 2 respectively and V µ τ δ f 1 (H) be a basic open set in f 1 (H) with p 1 p 1 2 V µ. By definition, there exists ν δ such that µ = ν f 1 (H). Clearly p 1 p 1 2 V ν. Since f is fuzzy open, f(ν) σ. Now define fuzzy singletons q 1, q 2 on y 1 and y 2 with values p 1 (x 1 ) and p 2 (x 2 ) respectively, where y 1, y 2 H with f(x 1 ) = y 1 and f(x 2 ) = y 2. Since f is a homomorphism, q 1 q 1 2 V f(ν) H. Since H is a strong fuzzy topological group, there exists ν 1, ν 2 σ H such that (q 1, q 2 ) V ν1 V ν2 and M IH (V ν1 V ν2 ) V f(ν) H. Clearly there exist µ 1, µ 2 σ such that µ 1 H = ν 1, µ 2 H = ν 2. Since f is fuzzy continuous, f 1 (µ 1 ), f 1 (µ 2 ) δ with f 1 (µ 1 ) f 1 (H) = f 1 (ν 1 ), f 1 (ν 2 ) f 1 (H) = f 1 (ν 2 ). Clearly (p 1, p 2 ) V f 1 (ν 1) V f 1 (ν 2). Now we claim that M If 1 (H) (V f 1 (ν 1) V f 1 (ν 2)) V µ. Let (p 1, p 2) V f 1 (ν 1) V f 1 (ν 2). Hence there exists (q 1, q 2) V ν1 V ν2 with f(p 1) = q 1, f(p 2) = q 2). Since M IH (V ν1 V ν2 ) V f(ν) H, q 1q 2 1 V f(ν) H. So, by injectivity of f, p 1p 2 1 (x 1x 2 1 ) = q 1q 2 1 (y 1y 2 1 ) f(ν)(y 1y 2 1 ) = ν(x 1x 2 1 ) = µ(x 1x 2 1 ). Hence the theorem. The proofs of following corollaries are similar to the proofs of corollaries 2.13 and Corollary Let f : (G, δ) (G, σ) be a fuzzy continuous fuzzy open homomorphism such that every fuzzy open set of (G, δ) is f-invariant. Then the inverse image of a strong fuzzy topological subgroup H of (G, σ) is again a strong fuzzy topological subgroup of (G, δ). Corollary Let f : G G be a homomorphism. Let (G, σ) be a fuzzy topological space. Let δ = {f 1 (ν) ν σ}. Then the inverse image of a strong fuzzy topological subgroup H of (G, σ) is again a strong fuzzy topological subgroup of (G, δ). 3. Conclusions In this paper, a new notion of strong fuzzy topological groups is defined and their properties are studied. In future we can extend this notion to the intuitionistic fuzzy set up. Using this notion, one can study the question of a fuzzy quotient semigroup becoming a topological fuzzy quotient semigroup. So it opens a new area of study in fuzzy topological algebraic structures. References [1] C.L. Chang, Fuzzy Topological Spaces, J. Math. Anal. Appl. 24 (1968) [2] Chun Hai YU and MA Ji Liang, On Fuzzy Topological Groups, Fuzzy Sets and Systems 23 (1987) [3] David H. Foster, Fuzzy Topological Groups, J. Math. Anal. Appl. 67, (1979)

8 194 V.L.G. NAYAGAM, D. GAULD, G. VENKATESHWARI AND G. SIVARAMAN [4] N.R. Das, Prabin Das, Neighborhood systems in fuzzy topological group, Fuzzy Sets and Systems 116 (2000) [5] M. Ganster, D.N. Georgiouy and S. Jafariz, On Fuzzy Topological Groups and Fuzzy Continuous Functions, Hacettepe Journal of Mathematics and Statistics, 34 S (2005), [6] M.H. Ghanim and E.E. Kerre and A.S. Mashhour, Separation Axioms, Subspaces and Sums in Fuzzy Topology, J. Math. Anal. Appl. 102 (1984) [7] Inheung Chon, Some properties of fuzzy topological groups, Fuzzy Sets and Systems 123 (2001) [8] A.K. Katsaras and D.B. Liu, Fuzzy vector spaces and fuzzy topological vector spaces, J. Math. Anal. Appl. 58 (1977), [9] MA Ji Liang and YU Chun Hai, Fuzzy Topological Groups, Fuzzy Sets and Systems 12 (1984) [10] D.S. Malik, John N. Mordeson and P.S. Nair, Fuzzy normal subgroups in fuzzy subgroups, J.Korean Math. Soc. 29 (1992), No. 1, 1-8. [11] D.S. Malik, John N. Mordeson and M.K. Sen, Fuzzy normal sub- groups, solvablity of fuzzy subgroups and congruence relations, The Journal of fuzzy mathematics Vol. 2, No. 2, (1994), [12] M. Mashinchi and M. Mukaidono, Generalised fuzzy quotient groups, Fuzzy sets and systems 74 (1995), [13] N.P. Mukhergee and P. Bhattacharya, Fuzzy normal subgroups and fuzzy cosets, Information Sciences 34 (1984), [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) [15] A. Rosenfeld, Fuzzy groups, J.Math. Anal. Appl. 35 (1971), [16] V. Lakshmana Gomathi Nayagam, Geetha Sivaraman, Induced topology on fuzzy singletons, Far East Journal of Applied Mathematics volume 32 issue 2, (2008), [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) [19] Wang-Jin Liu, Fuzzy Invariant subgroups and Fuzzy ideals, Fuzzy Sets and Systems 8 (1982) [20] C.K. Wong, Fuzzy Points and Local properties of Fuzzy Topology, J. Math. Anal. Appl. 46 (1974) [21] L.A. Zadeh, Fuzzy Sets, Information and Control 8 (1965) V.L.G. Nayagam Department of Mathematics National Institute of Technology Tiruchirapalli INDIA velulakshmanan@nitt.edu D. Gauld Department of Mathematics University of Auckland Auckland NEW ZEALAND d.gauld@auckland.ac.nz

9 STRONG FUZZY TOPOLOGICAL GROUPS 195 G. Venkateshwari Department of Mathematics Sacs MAVMM Engineering College Madurai INDIA G. Sivaraman Department of Mathematics Anna University Chennai INDIA