Fuzzy Topological Transformation Groups

In this paper we have introduced the concept of Topological Transformation Groups in fuzzy setting as a natural transition from the corresponding crisp structure and study some properties thereof. Classical results on orbits, orbit closure, invariant subsets are investigated in this setting. Finally we have constructed some new fuzzy topological transformation groups from given ones


Introduction
A classical Topological Transformation Group is a structure (π, G, X,) where G is a topological group, X is a topological space and π is a continuous function from G × X → X satisfying π(0, x) = x and π(s, π(t, x)) = π(s + t, x), where 0 is the identity of G.In this paper we fuzzify the above concept as a natural transition from the corresponding crisp structure.For this fuzzification we will consider a fuzzy topological group (Chu-hai Yu, 1987), a fuzzy topological space and a fuzzy continuous map from G × X → X satisfying the above stated conditions.Throughout our discussion the fuzzy topology on any set will contain all the constant fuzzy subsets.In other words we will use Lowen (R. Lowen, 1976) definition of fuzzy topology.

Preliminaries
In this section we recall some preliminary definitions and results to be used in the sequel.
Let X be a non-empty set.A fuzzy set in X is an element of the set I X of all functions from X into the unit interval I.A fuzzy point of a set X is a fuzzy subset which takes non-zero value at a single point and zero at every other point.The fuzzy point which takes value α 0 at x ∈ X, and zero elsewhere is denoted by x α .If x ∈ X, then the fuzzy point x 1 will be denoted simply by x.Let λ be a fuzzy subset of X. Suppose λ(x) = α for x ∈ X.Then λ can be expressed as union of all its fuzzy points, i.e, λ = ∨ x∈X x α .Here ∨ denote union.We will use the same notation ∨ to denote supremum of a set of numbers.Similarly ∧ will be used to denote intersection of fuzzy sets as well as infimum of a set of real numbers.
Let λ and μ be fuzzy subsets of X, then we write λ ⊆ μ whenever λ(x) ≤ μ(x).Let λ be a fuzzy subset of a group (G, +).Then we define a fuzzy subset −λ as −λ For the definition of a fuzzy topology, we will use the one given by Lowen (1976) since his definition is more appropriate in our case.So, throughout this paper, by a fuzzy topology on a set X we will mean a sub-collection τ of I X satisfying the following conditions: (i) τ contains every constant fuzzy subset in X ; A fuzzy topological space is a set X on which there is given a fuzzy topology τ.The elements of τ are the open fuzzy sets in X.
Complement of an open fuzzy set is called a closed fuzzy set.Interior of a fuzzy set λ is the union of all the open fuzzy set contained in λ and the closure of λ is the intersection of all fuzzy set containing λ.The interior and closure of λ will be denoted by λ o and clλ respectively.A map f from a fuzzy topological space X to a fuzzy topological space Y, is called continuous if Let X be a fuzzy topological space and x ∈ X.A fuzzy set μ in X is called a neighborhood of the fuzzy point x α if there exists an open fuzzy set ρ with ρ ⊆ μ and x α ∈ ρ ⊆ μ.Given a crisp topological space (X, T ), the collection (T ), of all fuzzy sets in X which are lower semicontinuous, as functions from X to the unit interval I = [0, 1] equipped with the usual topology, is a fuzzy topology on X (Lowen, 1976).We will refer to the fuzzy topology (T ) as the fuzzy topology generated by the usual topology T. If (X, T j ) j∈J is a family of crisp topological spaces and T the product topology on X = j∈J X j , then (T ) is the product of the fuzzy topologies (T j ), j ∈ J (Lowen, 1977).
Proof.Let μ ∈ (τ 3 ).Then μ is a lower semicontinuous function from (X 3 , T 3 ) to the unit interval I. Since f is continuous with respect to the topologies T and T 3 , it follows that the function f −1 (μ) = μ • f is a lower semicontinuous function from (X, T) to the unit interval.Thus f −1 (μ) ∈ (T ) = δ.This completes the proof.
Result 2.2.Let ( π, G, X) be a classical topological transformation group.If we equip G and X with the induced fuzzy topologies and G × X, with the corresponding product fuzzy topology, then the mapping π: G × X → X is fuzzy continuous.
Proof.It follows from the previous result.
Result 2.4 : Let (X, δ) and (Y, μ) be two fuzzy topological spaces.If f : X → Y is fuzzy continuous, then for any subset A ⊆ X, f / A is fuzzy continuous.In particular an inclusion map is fuzzy continuous.
Definition 2.6 If σ is a fuzzy subset of X and η is a fuzzy subset of Y, then the fuzzy subset Result 2.9 (R. Lowen, 1976) (X, δ) and (Y, τ) be two fuzzy topological spaces and f : X → Y be a fuzzy continuous function.If λ is fuzzy compact subset of X, then f (λ) is fuzzy compact in Y.
Definition 2.11 (Rajesh Kumar, 1993).Let (G, +) be a group.Then a fuzzy subset λ is said to be a fuzzy subgroup of G if λ(x + y) ≥ min{λ(x), λ(y)) and λ(−x) = λ(x) Remark : If λ is a fuzzy subgroup of G then supp is a crisp subgroup of G.
Definition 2.12 (N.Palaniappan, 2005) A fuzzy topological space (X, τ) is said to be product related to another fuzzy topological space (Y, δ) if for any fuzzy set υ of X and ζ of Y whenever λ c υ and Palaniappan, 2005) Let (X, τ) be product related to (Y, δ).Then for any fuzzy subset λ of X and a fuzzy subset μ of Y, cl(λ × μ) = clλ × clμ.

Fuzzy topological transformation groups
In this section we will introduce the concept of fuzzy topological transformation group and prove some properties.
Definition 3.1 Let X be fuzzy topological space, G be a fuzzy topological group.If ) is called a fuzzy topological transformation group.Definition 3.2 Let t ∈ G, then the t-transition of (π, G, X, ) denoted by π t is the mapping : π t : X → X such that π t (x) = π(t, x).Result 3.3 (i) π 0 is the identity mapping of X.
(iii) π t is one-to-one mapping of X onto X and −(π t ) = π −t .
(iv) For t ∈ G, π t is a fuzzy homomorphism of X onto X. Proof.Straightforward.
Definition 3.4 The transition group of (π , G, X,) is the set G = {π t : t ∈ G}.The transition projection of (G, X, π) is the mapping θ : G → G defined as θ(t) = π t .Definition 3.5 (G, X, π) is said to be effective if t ∈ G with t 0 ⇒ π t (x) x for some x.
Result 3.8 π x is a fuzzy continuous mapping of G into X. Proof.Straightforward.
Result 3.9 Let X, Y, Z be fuzzy topological spaces and f : X × Y → Z be a fuzzy continuous map.If a α , b β be fuzzy points of X and Y respectively and γ be a fuzzy neighbourhood of f (a α , b β ) then there exists fuzzy neighbourhoods η and ρ of a α and b β respectively such that f (η × ρ) ⊆ γ.

March, 2009
Result 3.10 Let X, Y, Z be fuzzy topological spaces and f : X × Y → Z be a fuzzy continuous map.If λ and μ are fuzzy compact subsets of X and Y respectively and γ is a fuzzy neighbourhood of f (λ × μ), then for any ε > 0, there exists fuzzy open sets λ and μ such that λ ≥ λ − ε and μ ≥ μ − ε and f (λ × μ ) ⊆ γ.
Proof.Let ε > 0 be arbitrary.Let x ∈ X be arbitrarily fixed and suppose λ(x) = α.Then for any y ∈ Y with μ(y) = β, by previous result there exist fuzzy open sets λ y x α and μ y y β such that f (x α ×y β ) ∈ f (λ y ×μ y ) ⊆ γ.This is true for each y ∈ Y. Thus the collection C μ = {μ y : y ∈ Y} is an open cover of μ.As μ is compact there is a finite sub-collection say S μ of C μ satisfying ∨{μ y : μ y ∈ S } ≥ μ − ε.Let μ x denote the union of all members of S μ and λ x denote the intersection of the corresponding λ y s.Then (ii) Let G and X be product related, then for a fuzzy subset λ of G and a fuzzy subset μ of X, π(clλ (iii) If λ is a compact fuzzy subset of G and μ is a compact fuzzy subset of X, then π(λ × μ) is a compact fuzzy subset of X.
(iv) If λ is a compact fuzzy subset of G and μ is a compact fuzzy subset of X, and γ is a fuzzy neighbourhood of π(λ × μ), then for any ε > 0, there exists fuzzy open sets λ and μ such that (iii) λ and μ are fuzzy compact, so λ × μ is fuzzy compact.As continuous image of a fuzzy compact set is fuzzy compact π(λ × μ) is fuzzy compact.

Journal of Mathematics Research
March, 2009 subset, then fuzzy invariance reduces to crisp invariance.Result 4.2 (i) If μ is a fuzzy subset of X, and λ is a fuzzy subgroup of X satisfying λ(0) = 1, then the following statements are pairwise equivalent : (ii) 0 and 1 are invariant.
(v) If {μ i } is a collection of λ-invariant fuzzy subset, then ∨μ i and ∧μ i are λ-invariant.
(vi) Let A be a crisp subset of G and μ a fuzzy subset of X., then μ is χ A -invariant iff μ c is χ −A -invariant.

Fuzzy Orbits
In this section we will introduce the notion of orbits in fuzzy setting and extend some classical results.
Definition 5.1 Let x ∈ X and λ a fuzzy subgroup of G. Then the fuzzy orbit of x under λ or the λorbit of x is defined to be fuzzy subset π(λ × x).The fuzzy orbit closure of x under λ or the λorbit closure of x is defined to be the fuzzy subset clπ(λ × x).
When λ = χ G , then the fuzzy orbit coincides with the crisp orbit.We will denote the orbit of x under λ by λ x .We assume that G and X are product related and that λ(0) = 1.
Proof.Since λ is fuzzy subgroup, suppλ is an ordinary group Since subgroup of a fuzzy topological group is a fuzzy topological group, suppλ is a fuzzy topological group.Since π(λ × μ) ⊆ μ, range of π/ suppλ×suppμ ) is contained in suppμ.Also restriction of fuzzy continuous function if fuzzy continuous.Hence the result.
Corollary 5.3 Let λ be a fuzzy subgroup of G then ( suppλ, X, π/ suppλ×X is fuzzy topological transformation group. Result 5.4 Consider the map f : X → X/ λ given by f x ∈ X} with the corresponding quotient topology.Then clearly this space is fuzzy homemorphic with the already introduced space X/ λ , where the corresponding homeomorphism is x → [λ x ].We can denote both the spaces by the same notation X/ λ .Consequently (φ, G, X/ λ ), where φ: G × X → X/ λ defined as φ(x) = g(π(t, x)) is a fuzzy topological transformation group.Here X/ λ stands for the quotient space {[x] : x ∈ X}.
Conclusion :.In this paper we have developed the notion of topological transformation group in fuzzy setting.We have attempted to extend most of the results of classical topological transformation group to this fuzzy setting.A topological transformation group is the basic structure in the study of topological dynamics.As the concept of orbit and orbit closure are fuzzified, it is expected that the other concepts of topological dynamics can be worked on in this setting.
Definition 2.7 (Liu Ying-Ming, 1997) Let (X, δ) and (Y, τ) be two fuzzy topological spaces.Then f : X → Y is fuzzy open (closed) if the image of every fuzzy open(closed) subset of X is fuzzy open(closed) in Y. Definition 2.8 (R. Lowen, 1976) Let X be a fuzzy topological space and λ a fuzzy subset of X.An open fuzzy cover of λ is a collection {λ α } of open fuzzy subsets of X such that λ ⊆ ∨λ α .If every open cover of λ and ε > 0 there exists a finite sub-collection {λ Proof.Without loss of generality we can assume that γ is fuzzy open.As f is continuous f −1 (γ) is a fuzzy open set containing a α × b β .So there exists basic fuzzy open sets say, η of a α and ρ of b λ x is a fuzzy open set containing x α and μ x is a fuzzy open set satisfying μ x ≥ μ − ε.But this is true for each x ∈ X.Thus we get a collection {μ x : x ∈ X} of fuzzy open sets each satisfying μ x ≥ μ−ε and another collection {λ x : x ∈ X} of fuzzy open sets such that x α ∈ λ x (α = λ(x)).Then C λ = {λ x : x ∈ X} is a cover of λ.As λ is compact there exists a finite collection S λ of C λ satisfying ∨{λ x : λ x ∈ S λ } ≥ λ − ε.Let λ denote the union of the members of S λ and μ denote the intersection of the corresponding μ x .Then λ is a fuzzy open set satisfying λ ≥ λ − ε and μ is a fuzzy open set satisfying μ ≥ μ − ε.Further then f (λ × μ ) ⊆ γ.Result 3.11 a fuzzy topological transformation group.Proof.f and π being continuous, we haveφ is continuous.Now φ(0, [λ x ]) = f (π(0, x)) = f (x) = [λ x ].Also φ(s, φ(t, [λ x ])) = φ{s, f (π(t, x))} = φ{s, [λ π(t, x) ]} = f {π(s, π(t, x)} = f {π(s + t, x)} = φ(s + t, [λ x ]).Thus (FGT1), (FGT2) and FGT3) are satisfied and hence (φ, G, X/ λ ) is a fuzzy topological transformation group.In X define a relation x ∼ y if [λ x ] = [λ y ].Then this relation is an equivalence.Denote the equivalence class of x by [x].Let g be the canonical mapping from X to {[x] : x ∈ X}, i.e, g(x) = [x].Equip {[x] :