SOFT L-FUZZY QUASI-UNIFORMITIES INDUCED BY SOFT L-NEIGHBORHOOD SYSTEMS

Molodtsov [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 [1,3,4,9.10,16,17,19,20]. Pawlak’s rough set [14,15] can be viewed as a special case of soft rough sets [4]. Kim [9,10] introduced a fuzzy soft F : A → L as an extension as the soft F : A → P (U) where L is a complete residuated lattice [2,5,6]. He introduced soft L-fuzzy interior and closure operators, quasi-uniformities and soft L-fuzzy topogenous orders in complete residuated lattices. In this paper, we obtain soft L-fuzzy quasi-uniformities induced by soft L-neighborhood systems in complete residuated lattices. Moreover, every N continuous surjective soft maps are uniformly continuous soft maps. We give


Introduction
Molodtsov [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 [1,3,4,9.10,16,17,19,20].Pawlak's rough set [14,15] can be viewed as a special case of soft rough sets [4].
Kim [9,10] 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 [2,5,6].He introduced soft L-fuzzy interior and closure operators, quasi-uniformities and soft L-fuzzy topogenous orders in complete residuated lattices.
In this paper, we obtain soft L-fuzzy quasi-uniformities induced by soft L-neighborhood systems in complete residuated lattices.Moreover, every Ncontinuous surjective soft maps are uniformly continuous soft maps.We give their examples.
(1) 1 Definition 2.3.[9,10] 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 2.4.[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 2.5.[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.
(1) The image of (F, A) ∈ S(X, A) under the mapping f φ is denoted by where otherwise. ( (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 2.6.[9,10] Let f φ : S(X, A) → S(Y, B) be a soft mapping.Then we have the following properties.For (F, A), (F i , A) ∈ S(X, A) and (G, B), The triple (X, A, N ) is called a soft L-neighborhood space.Let (X, A, N ) and (Y, B, M ) be soft L-neighborhood spaces.A mapping The triple (X, A, U ) is called a soft L-fuzzy quasi-uniform space.Let (X, A, U X ) and (Y, B, U Y ) be soft L-fuzzy quasi-uniform spaces and 3. Soft L-Fuzzy Quasi-Uniformities Induced by Soft L-Neighborhood Systems Lemma 3.1.For every (F, A), (G, A) ∈ S(X, A), we define (U F , A), Then we have the following properties. ( ( (4) By Lemma 2.2 (11), Theorem 3.2.Let (X, A, N ) be a soft L-neighborhood space.Define a map U N : S(X × X) → L by For all a ∈ A and for some x ∈ X,