On L-fuzzy closure operators and L-fuzzy pre-proximities

The aim of this paper is to investigate the relations among the L-fuzzy pre-proximities, L-fuzzy closure operators and L-fuzzy co-topologies in complete residuated lattices. We show that there is a Galois correspondence between the category of separated L-fuzzy closure spaces and that of separated L-fuzzy pre-proximity spaces and we give their examples.

In this paper, we assume that (L, ≤, ⊙, * ) is a complete residuated lattice with an order reversing involution * which is defined by For α ∈ L and f ∈ L X , we denote Some basic properties of the binary operation ⊙ and residuated operation → are collected in the following lemma, and they can be found in many works, for instance [8][9][10][11]38].

Lemma 2
For each x, y, z, x i , y i , w ∈ L , we have the following properties.

Remark 5
Definition 6 [16,17,39] A mapping F : L X → L is called L-fuzzy co-topology on X if it satisfies the following conditions: The pair (X, F) is called L-fuzzy co-topological space. An L-fuzzy co-topological space is said to be

Definition 7
Let (X, F X ) and (Y , F Y ) be L-fuzzy co-topological spaces and ϕ : (X, F X ) → (Y , F Y ) be a mapping. Then, D F (φ) defined by is called the degree of LF-continuous for ϕ .
which is exactly the definition of LF-continuous mappings between L-fuzzy co-topological spaces.
Definition 8 [8,36] Let X be a set. A map R : X × X → L is called an L-partial order if it satisfies the following conditions The relationships between L-fuzzy pre-proximities and topological structures Definition 9 A mapping δ : L X × L X → L is called an L-fuzzy pre-proximity on X if it satisfies the following axioms.
. The pair (X, δ) is called L-fuzzy preproximity space. An L-fuzzy pre-proximity is called an (L, ⊙, ⊕)-fuzzy pre-proximity if (P4) For every f 1 , f 2 , h 1 , h 2 ∈ L X we have An L-fuzzy pre-proximity is called an L-fuzzy quasi-proximity on X if it satisfies (P4) and

Definition 10
Let (X, δ X ) and (Y , δ Y ) be L-fuzzy pre-proximity spaces and ϕ : , which is exactly the definition of LF-proximity mappings between L-fuzzy pre-proximity spaces.

Lemma 11
Let (X, δ) be an L-fuzzy pre-proximity space. Then, From the following theorem, we obtain the L-fuzzy closure operator induced by an L-fuzzy pre-proximity.

Theorem 12
Let δ be an L-fuzzy pre-proximity on X. Define C δ : L X → L X as follows: Then, . From Lemma 2, we obtain Hence, C δ is an L-fuzzy closure operator on X. (2) (3) By (C2) and

Example 13
Let X be a set and R ∈ L X×X be an L-fuzzy pre-order. Define δ : L X × L X → L as (P1) and (P3) are easily proved.
Hence, δ is an L-fuzzy pre-proximity on X. Since δ is stratified. Moreover, δ is Alexandrov and generalized. By Theorem 12, we obtain a stratified L-fuzzy closure operator C δ : L X → L X as Hence, δ 1 is an L-fuzzy pre-proximity on X. Moreover, δ 1 is stratified, Alexandrov and generalized. Since By Theorem 12, we obtain a stratified L-fuzzy closure operator C δ 1 : L X → L X as . .

Theorem 14
Let (X, C) be an L-fuzzy closure space. Define a mapping δ C : L X × L X → L by Then, we have the following properties.
(1) δ C is an L-fuzzy pre-proximity, g)),the equality holds if C is topological, (4) If C is topological, then δ C is an L-fuzzy quasi-proximity on X, If C is generalized (resp. Alexandrov), then δ C is generalized (resp. Alexandrov).
(5) From Lemma 2, we have, . If C is topological, then (8) It is easily proved from definitions.

Corollary 15
Let (X, C) be an L-fuzzy closure space. Define a mapping δ s C : L X × L X → L by Then, we have the following properties.
(1) δ s C is an L-fuzzy pre-proximity, g)) , the equality holds if C is topological, (4) If C is topological, then δ s C is a L-fuzzy quasi-proximity on X,

The relationships between L-fuzzy pre-proximities and L-fuzzy co-topologies
Theorem 16 Let δ be an Alexandrov L-fuzzy pre-proximity on X. Define a mapping F δ : L X → L by F δ (f ) = δ * (f * , f ) . Then, . (1) F δ is an L-fuzzy co-topology on X, (2) If δ is stratified, then F δ is strong, (1) F C δ is an L-fuzzy co-topology on X with F C δ ≥ F δ , (2) If C is Alexandrov (resp. strong, separated), then F δ C is Alexandrov (resp. strong, separated).
(T3) By Lemma 2(16), we have Hence, F C δ is an L-fuzzy co-topology on X. Moreover, Hence, F C δ is an Alexandrov L-fuzzy co-topology on X. By Lemma 2(14)(18), we have Other cases are easily proved.

Theorem 18
Let (X, δ) be an L-fuzzy pre-proximity space. Then, the mapping ) is an L-fuzzy co-topology on X. Moreover, if δ is Alexandrov and Moreover, if δ is Alexandrov, then Hence, F (1) δ is Alexandrov L-fuzzy co-topology on X.