INITIAL SOFT L-FUZZY QUASI-UNIFORM SPACES

Hájek in [5] introduced a complete residuated lattice which is an algebraic structure for many valued logic. It is an important mathematical tool for algebraic structure of fuzzy contexts, see [2,6,9,10]. Recently, Molodtsov in [13] introduced the soft set as a mathematical tool for dealing information as the uncertainty of data in engineering, physics, computer sciences and many other diverse field. Presently, the soft set theory is making progress rapidly, see [1,4]. Pawlak’s rough set (see [14,15]) can be viewed as a special case of soft rough sets, see [4]. The topological structures of soft sets have been developed by many researchers, see [3,9,10,16,19,20].


Introduction
Hájek in [5] introduced a complete residuated lattice which is an algebraic structure for many valued logic.It is an important mathematical tool for algebraic structure of fuzzy contexts, see [2,6,9,10].Recently, Molodtsov in [13] introduced the soft set as a mathematical tool for dealing information as the uncertainty of data in engineering, physics, computer sciences and many other diverse field.Presently, the soft set theory is making progress rapidly, see [1,4].Pawlak's rough set (see [14,15]) can be viewed as a special case of soft rough sets, see [4].The topological structures of soft sets have been developed by many researchers, see [3,9,10,16,19,20].
Kim in [9] introduced a fuzzy soft F : A → L U as an extension as the soft F : A → P (U ) where L is a complete residuated lattice.Kim [9,10] introduced the soft topological structures, L-fuzzy quasi-uniformities and soft L-fuzzy topogenous orders in complete residuated lattices.
In this paper, we prove the existences of initial soft L-fuzzy (quasi-) uniformities in a complete residuated lattice.From this fact, we define subspaces and product spaces for soft L-fuzzy (quasi-)uniformities.Moreover, we give their examples.
Lemma 2. (see [2,6]) For each x, y, z, x i , y i , w ∈ L, we have the following properties. (1) Definition 3. (see [9]) Let X be an initial universe of objects and E the set of parameters (attributes) in X.A pair (F, A) is called a fuzzy soft set over X, where A ⊂ E and F : A → L X is a mapping.We denote S(X, A) as the family of all fuzzy soft sets under the parameter A. Definition 4. (see [9,10]) Let (F, A) and (G, A) be two fuzzy soft sets over a common universe X.
(1) (F, A) is a fuzzy soft subset of (G, A), denoted by (F, Definition 5. (see [9,10]) Let S(X, A) and S(Y, B) be the families of all fuzzy soft sets over X and Y , respectively.The mapping f φ : S(X, A) → S(Y, B) is a soft mapping where f : X → Y and φ : A → B are mappings.
(2) The inverse image of (G, B) ∈ S(Y, B) under the mapping f φ is denoted by (3) The soft mapping f φ : S(X, A) → S(Y, B) is called injective (resp.surjective, bijective) if f and φ are both injective (resp.surjective, bijective).Lemma 6. (see [9,10]) Let f φ : S(X, A) → S(Y, B) be a soft mapping.Then we have the following properties.For (F, A), Definition 7. (see [9,10]) A mapping U : S(X × X, A) → L is called a soft L-fuzzy quasi-uniformity on X iff it satisfies the properties.
). Lemma 8. (see [10]) Let (X, A, U ) be a soft L-fuzzy quasi uniform space.For each (U, A) ∈ S(X × X, A) and (F, A) ∈ S(X, A), we define , for all x ∈ X, a ∈ A, Then we have the following properties. ( 3. Initial Soft L-Fuzzy Quasi-Uniform Spaces Lemma 9. Let (X, A, U ) be a soft L-fuzzy quasi-uniform space.We define a function U s : S(X × X, A) → L by Proof.We easily proved from (V, A) where the is taken over every finite index K = {k 1 , ..., k n } ⊂ Γ. Then: (1) The structure U is the coarsest soft L-fuzzy (resp.quasi-)uniformity on X for which each (f k ) φ k is an uniformly continuous soft map.
(2) A map where the is taken over every finite index K = {k 1 , ..., k n } ⊂ Γ, for all r > s. ( Proof. (1) First, we will show that U is a soft L-fuzzy quasi-uniformity on X.
By Lemma 2(5) and the definition of U ((U, A)), there exists a finite index set Also, by definition of U ((W, A)), there exists a finite index set Since and (SU4) Let U ((U, A)) = 0. Then there exists a finite index (SU5) Suppose there exists (U, A) ∈ S(X × X, A) such that By the definition of U ((U, A)), there exists a finite index , By Lemma 2(5), for each Then we have (W, A) • (W, A) ≤ (U, A) and , for all (U, A) ∈ S(X × X, A).
(U) Let {(X k , U k ) | k ∈ Γ} be a family of soft L-fuzzy uniform spaces.Suppose that there exists (U, A) ∈ S(X × X, A) such that By the definition of U , there exists a finite index For each k i ∈ K, since (X k i , V k i ) is a soft L-fuzzy uniform space, by (U), For each On the other hand, we have Hence U, A)) for all k ∈ Γ, then it is proved that U ′ ≥ U from the following: (2) Necessity of the composition condition is clear since the composition of an uniformly continuous soft maps is an uniformly continuous soft map.
Conversely, suppose that f φ : (Z, C, W) → (X, A, U ) is not an uniformly continuous soft map.There exists (U, A) ∈ S(X × X, A) such that By the definition of U , there exists a finite index set On the other hand, for each is an uniformly continuous soft map, we have It follows that It is a contradiction.
(3) Since we have Conversely, for s < r, suppose that On the other hand, since U ((U, A)) ≥ r, for r > s, there exists a finite index It is a contradiction.
(4) From the definition of U ((U, A)), Then there exists a finite subsets It is a contradiction.Hence the result follows.
From Theorem 10, we define the following definition.Definition 11.Let U i be soft L-fuzzy quasi-uniformities on X i for i ∈ Γ.Let X be a set and, for each i ∈ Γ, (f i ) φ i : (X, A) → (X i , B i ) a soft map.The initial soft L-fuzzy quasi-uniformity on X induced by {(f i ) φ i | i ∈ Γ} is the coarsest soft L-fuzzy quasi-uniformity on X for which (f i ) φ i is an uniformly continuous soft map.
The product soft L-fuzzy quasi-uniformity on X = Π i∈Γ X i is the initial soft L-fuzzy quasi-uniformity induced by {(π i ) The pair (Y, B, U Y ) is a subspace of (X, A, U ) where U Y is the initial soft L-fuzzy quasi-uniformity induced by an inclusion soft map i φ : (Y, B) → (X, A).
From Theorem 10, we obtain the following corollary.Corollary 12. Let U i be soft L-fuzzy quasi-uniformity on X for i ∈ Γ.Let X be a set and, for each i ∈ Γ, (id i ) id : (X, A) → (X, A, U i ) an identity soft map.Define the map U : S(X × X, A) → L on X by where the is taken over every finite index K = {k 1 , ..., k n } ⊂ Γ.Then U is the coarsest soft L-fuzzy quasi-uniformity on X which U is finer than U i for each i ∈ Γ.Then ([0, 1], ⊙, →, * 0, 1) is a complete residuated lattice (ref.[2,6]).