L-FUZZY UNIFORMITIES INDUCED BY L-NEIGHBORHOOD SYSTEMS AND L-FUZZY TOPOLOGIES

In this paper, we obtain L-fuzzy uniformities induced by Lneighborhood systems and L-fuzzy topologies in complete residuated lattices. Moreover, every N -continuous maps are fuzzy uniformly continuous and every fuzzy continuous maps are fuzzy uniformly continuous. AMS Subject Classification: 03E72, 06A15, 06F07, 54A05

Using the Lowen neighborhood system [10], Katsaras proved that every linear fuzzy neighborhood space is uniformizable in the sense of Lowen uniformity.
Received: September 18, 2014 c 2015 Academic Publications, Ltd. url: www.acadpubl.eu§ Correspondence author Kim [8] introduced the notion of fuzzy uniformities as an extension of Lowen in a stsc-quantale lattice L. Hájek [4] introduced a complete residuated lattice which is an algebraic structure for many valued logic.Bělohlávek [2] investigated information systems and decision rules in complete residuated lattices.
In this paper, we obtain L-fuzzy uniformities induced by L-neighborhood systems and L-fuzzy topologies in complete residuated lattices.Moreover, every N -continuous maps are fuzzy uniformly continuous and every fuzzy continuous maps are fuzzy uniformly continuous.
Lemma 2.2.[2,4] For each x, y, z, w, x i , y i ∈ L, the following properties hold: The pair (X, T ) is called an L-fuzzy topological space.Let (X, T 1 ) and (Y, T 2 ) be two L-fuzzy topological spaces.A mapping φ : Definition 2.4.[8] A map U : L X×X → L is called an L-fuzzy uniformity on X iff the following conditions hold: An L-fuzzy uniformity U on X is said to be stratified if The pair (X, U ) is called an L-fuzzy uniform space.
Let (X, U ) and (Y, V) be L-fuzzy uniform spaces, and φ : X → Y ba a mapping.Then φ is said to be fuzzy uniformly continuous if Remark 2.5.Let (X, U ) be an L-fuzzy uniform space.
(1) By (U1) and (U2), we have The previous axiom can be reformulated in the following way The pair (X, N ) is called an L-neighborhood space.Let (X, N ) and (Y, M ) be two L-neighborhood spaces.A mapping φ : 3. L-Fuzzy Uniformities Induced by L-Neighborhood Systems and L-Fuzzy Topologies Definition 3.1.For λ ∈ L X , we define u λ ∈ L X×X associated with λ by Thus, Finally, let u ∈ L X×X , λ ∈ L X and α ∈ L, we have Proof.We want to show that for all v ∈ L Y ×Y , we have Theorem 3.4.Let (X, T ) be an L-fuzzy topological space.Define a map Then (1) (X, U T ) is an L-fuzzy uniform space, (2) If (X, T ) is enriched, then U T (α ⊙ u) ≥ U T (u) for all α ∈ L and u ∈ L X×X .