L -fuzzy pre-proximities and application to L -fuzzy topologies

Abstract

Keywords

Complete residuated lattice Galois correspondence

1 Introduction

Ward et al. [39] introduced a complete residuated lattice which is an algebraic structure for many valued logic. Bělohlávek [3] investigated information systems and decision rules over complete residuated lattices. Höhle [14] introduced L-fuzzy topologies with algebraic structure L (cqm, quantales, MV-algebra). It has developed in many directions [5 , 43].

Recently, Bělohlávek [4] outlined a general theory of fuzzy interior operators and fuzzy interior systems using the structure of the residuated lattice in place of the usual structure of truth value on [0, 1]. Ramadan [29, 31] studied the relationship between L-fuzzy interior systems and L-fuzzy topological spaces from a category viewpoint for a complete residuated lattice L.

Proximity is an important concept in topology and it can be considered either as axiomatizations of geometric notions, close to but quite independent of topology, or as convenient tools for an investigation of topological spaces. Hence proximity has close relations with topology, uniformity and metric. With the development of topology, the theory of proximity makes a massive progress. In the framework of L-topology, many authors generalized the crisp proximity to L-fuzzy setting. Katsaras [18, 19] introduced the concepts of fuzzy topogenous order and fuzzy topogenous structures in completely distributive lattice which are a unified approach to the three spaces: Chang’s fuzzy topologies [6], Katsaras’s fuzzy proximities [16, 17] and Hutton’s fuzzy uniformities [15]. Subsequently, Liu [24], Artico and Moresco [2] extended it into L-fuzzy set theory in view points of Lowen’s fuzzy topology [25]. As an extension of Katsaras’s definition, El-Dardery [8] introduced L-fuzzy topogenous order in view points of Sostak’s fuzzy topology [34 –36], smooth fuzzy topology [28] and Kim’s L-fuzzy proximities [21] on strictly two-sided, commutative quantales. L-fuzzy topogenous structures and L-fuzzy proximities [7 , 42] have been developed in a slightly different sense.

In this paper, we investigate the relations between the L-fuzzy pre-proximities, L-fuzzy interior operators and L-fuzzy topological spaces in complete residuated lattices. In addition, degrees of L-fuzzy continuity, L-fuzzy proximity and L-fuzzy interior mappings are proposed and their connections are studied. Also, we show that there is a Galois correspondence between the category of separated L-fuzzy interior spaces and that of separated L-fuzzy pre-proximity spaces. Finally, we give their examples. In Example 4.4, as an information system as an extension of Pawlak’s rough set [26] and [13 , 41], L-fuzzy pre-proximities, L-fuzzy topologies and L-fuzzy interior operators are introduced. By using these concepts, we can apply to information systems and decision makings.

The content of the paper is organized as follows. In section 2, we recall some fundamental concepts and related definitions of L-fuzzy interior and L-fuzzy topology. In section 3, we investigates the relationships between the L-fuzzy pre-proximities and L-fuzzy interior operators. In section 4, we investigates the relation between the L-fuzzy pre-proximities and L-fuzzy topologies. In section 5, there is a Galois correspondence between the category of L-fuzzy pre-proximity spaces and that of L-fuzzy interior spaces.

2 Preliminaries

Definition 2.1 ([3 , 37]) An algebra (L, ∧ , ∨ , ⊙ , → , ⊥ , ⊤) is called a complete residuated lattice if it satisfies the following conditions

(C1) (L, ≤ , ∨ , ∧ , ⊥ , ⊤) is a complete lattice with the greatest element ⊤ and the least element ⊥;

(C2) (L, ⊙ , ⊤) is a commutative monoid;

(C3) x ⊙ y ≤ z iff x ≤ y → z for x, y, z ∈ L.

In this paper, we assume that (L, ≤ , ⊙ , ^∗) is a complete residuated lattice with an order reversing involution ^∗ which is defined by $\begin{matrix} x \oplus y = (x^{*} ⊙ y^{*})^{*}, x^{*} = x \to ⊥ . \end{matrix}$ For α ∈ L, f ∈ L^X, we denote (α → f) , (α ⊙ f) , α_X ∈ L^X as (α → f) (x) = α → f (x) , (α ⊙ f) (x) = α ⊙ f (x) , α_X (x) = α, $⊤_{x} (y) = {\begin{matrix} ⊤, & if y = x, \\ ⊥, & otherwise, \end{matrix} ⊤_{x}^{*} (y) = {\begin{matrix} ⊥, & if y = x, \\ ⊤, & otherwise . \end{matrix}$ 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 ([3 , 37]).

Lemma 2.2. For each x, y, z, x_i, y_i, w ∈ L, we have the following properties.

(1) ⊤ → x = x, ⊥ ⊙ x = ⊥ ,

(2) If y ≤ z, then x ⊙ y ≤ x ⊙ z, x ⊕ y ≤ x ⊕ z, x → y ≤ x → z and z → x ≤ y → x,

(3) x ≤ y iff x→ y = ⊤,

(4) $(⋀_{i} y_{i})^{*} = ⋁_{i} y_{i}^{*}, (⋁_{i} y_{i})^{*} = ⋀_{i} y_{i}^{*}$ ,

(5) x → (⋀ _iy_i) = ⋀ _i (x → y_i),

(6) (⋁ _ix_i) → y = ⋀ _i (x_i → y),

(7) x ⊙ (⋁ _iy_i) = ⋁ _i (x ⊙ y_i),

(8) (⋀ _ix_i) ⊕ y = ⋀ _i (x_i ⊕ y),

(9) (x ⊙ y) → z = x → (y → z) = y → (x → z),

(10) x ⊙ y = (x → y^∗) ^∗, x ⊕ y = x^∗ → y and x → y = y^* → x^*,

(11) (x → y) ⊙ (z → w) ≤ (x ⊙ z) → (y ⊙ w),

(12) x → y ≤ (x ⊙ z) → (y ⊙ z) and (x → y) ⊙ (y → z) ≤ x → z,

(13) (x → y) ⊙ (z → w) ≤ (x ⊕ z) → (y ⊕ w),

(14) x ⊙ (x → y) ≤ y and y ≤ x → (x ⊙ y) ,

(15) (x ∨ y) ⊙ (z ∨ w) ≤ (x ∨ z) ∨ (y ⊙ w) ≤ (x ⊕ z) ∨ (y ⊙ w) ,

(16) ⋁_i∈Γx_i → ⋁ _i∈Γy_i ≥ ⋀ _i∈Γ (x_i → y_i) and ⋀_i∈Γx_i → ⋀ _i∈Γy_i ≥ ⋀ _i∈Γ (x_i → y_i),

(17) (x ⊙ y) ⊙ (z ⊕ w) ≤ (x ⊙ z) ⊕ (y ⊙ w) ,

(18) z → x ≤ (x → y) → (z → y) and y → z ≤ (x → y) → (x → z).

Definition 2.3 [4] Let X be a set. A mapping R : X × X → L is called an L-fuzzy relation on X, then for all x, y, z ∈ X the relation R is said to be

(1) reflexive if $R (x, x) = ⊤$ ,

(2) symmetric if $R (x, y) = R (y, x)$ ,

(3) transitive if $R (x, y) ⊙ R (y, z) \leq R (x, z)$ .

A L-fuzzy relation on X is called a L-fuzzy pre-order (or L-partial order) if it is reflexive and transitive and called an L-fuzzy equivalence relation if it is reflexive, symmetric and transitive.

Lemma 2.4. [4, 9] For a given set X, define a binary map S : L^X × L^X → L by $\begin{matrix} S (f, g) = ⋀_{x \in X} (f (x) \to g (x)) . \end{matrix}$ Then, for each f, g, h, k ∈ L^X and α ∈ L, the following properties hold.

(1) S is an L-partial order on L^X,

(2) f ≤ g iff S (f, g)≥ ⊤,

(3) If f ≤ g, then S (h, f) ≤ S (h, g) and S (f, h) ≥ S (g, h),

(4) S (f, g) ⊙ S (k, h) ≤ S (f ⊕ k, g ⊕ h) and S (f, g) ⊙ S (k, h) ≤ S (f ⊙ k, g ⊙ h) ,

(5) S (g, h) ≤ S (f, g) → S (f, h) ,

(6) S (f, h) = ⋁ _g∈L^X (S (f, g) ⊙ S (g, h)),

(7) If φ : X → Y is a map, then for f, g ∈ L^X and h, k ∈ L^Y,

$\begin{matrix} S (f, g) \leq S (φ^{\to} (f), φ^{\to} (g)), \\ S (h, k) \leq S (φ^{\to} (h), φ^{\to} (k)), \end{matrix}$ and the equalities hold if φ is bijective.

Definition 2.5 [4, 31] A mapping $I : L^{X} \to L^{X}$ is called an L-fuzzy interior operator on X if $I$ satisfies the following conditions for all f, g ∈ L^X:

(I1) $I (⊤_{X}) = ⊤_{X}$ ,

(I2) $I (f) \leq f$ ,

(I3) If f ≤ g, then $I (f) \leq I (g)$ ,

(I4) $I (f ⊙ g) \geq I (f) ⊙ I (g)$ .

The pair $(X, I)$ is called an L-fuzzy interior space.

An L-fuzzy interior space $(X, I)$ for all α ∈ L and f ∈ L^X is called

(T) topological if $I (I (f)) = I (f)$ ,

(St) stratified if $I (α ⊙ f) \geq α ⊙ I (f)$ ,

(CSt) co-stratified if $I (α \to f) \geq α \to I (f)$ ,

(S) strong if $I (α \to f) = α \to I (f)$ ,

(SE) separated if $I (⊤_{x}) = ⊤_{x}$ for each x ∈ X,

(AL) Alexandrov if $I (⋀_{i \in Γ} f_{i}) = ⋀_{i \in Γ} I (f_{i})$ for each subfamily {f_i : i ∈ Γ} ⊆ L^X.

(G) generalized if $I (f) (x) \geq ⋀_{x \in X} f (x)$ .

Remark 2.6 An L-fuzzy interior space $(X, I)$ is stratified if and only if $I (α_{X} \to f) \leq α_{X} \to I (f)$ .

Definition 2.7 [14, 32] An L-fuzzy topology on X is a mapping $T : L^{X} \to L$ such that

(T1) $T (⊥_{X}) = T (⊤_{X}) = ⊤$ ,

(T2) $T (f ⊙ g) \geq T (f) ⊙ T (g) \forall f, g \in L^{X},$

(T3) $T (⋁_{i} f_{i}) \geq ⋀_{i} T (f_{i})$ for each subfamily {f_i : i ∈ Γ} ⊆ L^X.

The pair $(X, T)$ is called L-fuzzy topological space.

An L-fuzzy topological space $(X, T)$ for all α ∈ L and f ∈ L^X is

(St) stratified if $T (α ⊙ f) \geq T (f)$ ,

(CSt) co-stratified (enriched) if $T (α \to f) \geq T (f)$ ,

(S) strong if it is both stratified and co-stratified,

(SE) separated if $T (⊤_{x}) = ⊤ \forall x \in X$ ,

(AL) Alexandrov if $T (⋀_{i \in Γ} f_{i}) \geq ⋀_{i \in Γ} T (f_{i})$ for each subfamily {f_i : i ∈ Γ} ⊆ L^X.

A concrete category is a pair $(C, U),$ where $C$ is a category and $U : C \to Set$ is a faithful functor(or a forgetful functor). For each $C$ -object X, U (X) is called the underlying set of X. Thus, every object in a concrete category can be regarded as a structured set. We write $C$ for $(C, U),$ if the concrete functor is obvious. All of the categories considered in this paper are concrete categories. A concrete functor between two concrete categories $(C, U),$ and $(D, V),$ is a functor $G : C \to D$ with U = V ∘ G, which means that G only changes the structures on the underlying sets. Hence, in order to define a concrete functor $G : C \to D,$ we only consider the following two requirements. First, we assign to each $C$ -object X, a $D$ -object G (X) such that V (G (X)) = U (X) . Second, we verify that if a function f : U (X) → U (Y)is a $C$ -morphism X → Y, then it is also a $D$ -morphism G (X) → G (Y).

Theorem 2.8 [1] Suppose that $F : D \to C,$ $G : C \to D$ are concrete functors. Then the following conditions are equivalent:

(1) ${i d_{Y} : F ° G (Y) \to Y ∣ Y \in C}$ is a natural transformation from the functor $F ° G$ to the identity functor on $C$ , and ${i d_{X} : X \to G ° F (X) ∣ X \in D}$ is a natural transformation from the identity functor $D$ to the functor $G ° F$ .

(2) For each $Y \in C, i d_{Y} : F ° G (Y) \to Y$ is a $C$ -morphism, and for each $X \in D, i d_{X} : X \to G ° F (X)$ is a $D$ -morphism.

In this case, (F, G) is called a Galois correspondence between $C$ and $D$ . If (F, G) is a Galois correspondence, then it is easy to check that F is a left adjoint of G, or equivalently that G is a right adjoint of F.

3 The relationships between L-fuzzy pre-proximities and L-fuzzy interior spaces

In this section, we investigate the connections between L-fuzzy pre-proximities and L-fuzzy interior spaces.

Definition 3.1 A L-fuzzy pre-proximity on X is a mapping $δ : L^{X} \times L^{X} \to L$ such that for all $α \in L$ and $f, g, h, f_{1}, h_{1}, f_{2}, h_{2} \in L^{X}$ :

(P1) $δ (⊥_{X}, ⊤_{X}) = δ (⊤_{X}, ⊥_{X}) = ⊥$ ,

(P2) $δ (f, g) \geq \underset{x \in X}{\lor} (f ⊙ g) (x),$

(P3) If $f_{1} \leq f_{2}, h_{1} \leq h_{2}$ , then $δ (f_{1}, h_{1}) \leq δ (f_{2}, h_{2})$ ,

(P4) $δ (f_{1} ⊙ f_{2}, h_{1} \oplus h_{2}) \leq δ (f_{1}, h_{1}) \oplus δ (f_{2}, h_{2})$ . A L-fuzzy pre-proximity δ on X is called (Q) L-fuzzy quasi-proximity δ on X if $δ (f, g) \geq \underset{h}{\land} {δ (f, h) \oplus δ (h^{*}, g)}$ ,

(P) L-fuzzy proximity if $δ^{s} = δ$ , where $δ^{s} (f, g) = δ (g, f)$ ,

(St) stratified if $δ (α ⊙ f, α \to g) \leq δ (f, g),$ and $δ (α \to f, α ⊙ g) \leq δ (f, g)$ ,

(SE) separated if $δ (⊤_{x}, ⊤_{x}^{*}) = δ (⊤_{x}^{*}, ⊤_{x}) = ⊥$ ,

(AL) Alexandrov if $δ (\underset{i \in Γ}{\lor} f_{i}, g) \leq \underset{i \in Γ}{\lor} δ (f_{i}, g), δ (f, \underset{i \in Γ}{\lor} g_{i}) \leq \underset{i \in Γ}{\lor} δ (f, g_{i})$ for each subfamily ${f_{i}, g_{i} : i \in Γ} \subseteq L^{X}$ ,

(G) generalized if $δ (f, g) (x) \leq \underset{x \in X}{\lor} f (x) ⊙ \underset{x \in X}{\lor} g (x) $ .$ From the following theorem, we obtain the L-fuzzy interior operator induced by a L-fuzzy pre-proximity.

Theorem 3.2 Let (X, δ) be an L-fuzzy pre-proximity space. Define two mapping $I_{δ} : L^{X} \to L^{X}$ as follows: $\begin{matrix} I_{δ} (f) = \underset{g \in L^{X}}{\lor} (δ^{*} (g, g^{*}) ⊙ S (g, f) ⊙ g) . \end{matrix}$ Then,

(1) $(X, I_{δ})$ is a stratified L-fuzzy interior space,

(2) If δ is separated, then $I_{δ}$ is separated.

Proof. (1) (I1) Since $δ (⊤_{X}, ⊥_{X}) = ⊥$ , then $\begin{matrix} I_{δ} (⊤_{X}) = \underset{g \in L^{X}}{\lor} (δ^{*} (g, g^{*}) ⊙ S (g, ⊤_{X}) ⊙ g) \\ \geq (δ^{*} (⊤_{X}, ⊥_{X}) ⊙ S (⊤_{X}, ⊤_{X}) ⊙ ⊤_{X}) = ⊤_{X} . \end{matrix}$

(I2) Since $S (g, f) ⊙ g \leq f$ , then $\begin{matrix} I_{δ} (f) \leq \underset{g \in L^{X}}{\lor} (δ^{*} (g, g^{*}) ⊙ f) = f . \end{matrix}$

(I3) For each $f, g \in L^{X}$ , we have $\begin{matrix} S (I_{δ} (f), I_{δ} (h)) \\ = \underset{x \in X}{\land} (\underset{g \in L^{X}}{\lor} (δ^{*} (g, g^{*}) ⊙ S (g, f) ⊙ g (x))) \\ \to (\underset{k \in L^{X}}{\lor} (δ^{*} (k, k^{*}) ⊙ S (k, h) ⊙ k (x)))) \\ (by Lemma 2 .2 (16)) \\ \geq \underset{x \in X}{\land} \underset{g \in L^{X}}{\land} ((δ^{*} (g, g^{*}) ⊙ S (g, f) ⊙ g (x)) \\ \to δ^{*} (g, g^{*}) ⊙ S (g, h) ⊙ g (x)))) \\ \geq \underset{g \in L^{X}}{\land} (S (g, f) \to S (g, h)) = S (f, h) . \end{matrix}$

(I4) From Lemma 2.4, we obtain $\begin{matrix} I_{δ} (f) ⊙ I_{δ} (h) \\ = (\underset{g \in L^{X}}{\lor} (δ^{*} (g, g^{*}) ⊙ S (g, f) ⊙ g)) \\ ⊙ (\underset{k \in L^{X}}{\lor} (δ^{*} (k, k^{*}) ⊙ S (k, h) ⊙ k)) \\ = \underset{g, k \in L^{X}}{\lor} (δ^{*} (g, g^{*}) ⊙ δ^{*} (k, k^{*}) \\ ⊙ S (g, f) ⊙ S (k, h) ⊙ g ⊙ k) \\ \leq \underset{g, k \in L^{X}}{\lor} (δ^{*} (g ⊙ k, g^{*} \oplus k^{*}) \\ ⊙ S (g ⊙ k, f ⊙ h) ⊙ (g ⊙ k)) \leq I_{δ} (f ⊙ h) . \end{matrix}$

(S) Since $S (f, g) \leq S (I_{δ} (f), I_{δ} (g)),$ we have $S (f, α ⊙ f) = α \leq S (I_{δ} (f), I_{δ} (α ⊙ f)) .$ Hence, $α ⊙ I_{δ} (f) \leq I_{δ} (α ⊙ f) .$

(2) By (I2) and $\begin{matrix} I_{δ} (⊤_{x}) & = \underset{g \in L^{X}}{\lor} (δ^{*} (g, g^{*}) ⊙ S (g, ⊤_{x}) ⊙ g) \\ \geq δ^{*} (⊤_{x}, ⊤_{x}^{*}) ⊙ S (⊤_{x}, ⊤_{x}) ⊙ ⊤_{x} = ⊤_{x}, \end{matrix}$ we have $I_{δ} (⊤_{x}) = ⊤_{x} . □$

Example 3.3 Let X be a set and $R \in L^{X \times X}$ be an L-fuzzy pre-order. Define $δ : L^{X} \times L^{X} \to L a s δ (f, g) = \underset{x, y \in X}{\lor} (R (x, y) ⊙ f (x) ⊙ g (y)) .$

(P1) and (P3) are easily proved.

(P2) For all $f, g \in L^{X}$ , $\begin{matrix} δ (f, g) & = \underset{x, y \in X}{\lor} (R (x, y) ⊙ f (x) ⊙ g (y)) \\ \geq \underset{x \in X}{\lor} (R (x, x) ⊙ f (x) ⊙ g (x)) \\ = \underset{x \in X}{\lor} (f (x) ⊙ g (x)) . \end{matrix}$

(P4) For all $f_{1}, f_{2}, h_{1}, h_{2} \in L^{X}$ , by Lemma 2.2 (2) and (17), $\begin{matrix} δ (f_{1}, h_{1}) \oplus δ (f_{2}, h_{2}) \\ = \underset{x, y \in X}{\lor} (R (x, y) ⊙ f_{1} (x) ⊙ h_{1} (y)) \\ \oplus \underset{z, w \in X}{\lor} (R (z, w) ⊙ f_{2} (z) ⊙ h_{2} (w)) \\ \geq \underset{x, y, z, w \in X}{\lor} (R (x, y) ⊙ R (z, w) \\ ⊙ f_{1} (x) ⊙ f_{2} (z)) ⊙ (h_{1} (y) \oplus h_{2} (w)) \\ \geq \underset{x, y, w \in X}{\lor} (R (x, y) ⊙ R (y, w) \\ ⊙ f_{1} (x) ⊙ f_{2} (x)) ⊙ (h_{1} (w) \oplus h_{2} (w)) \\ = \underset{x, w \in X}{\lor} (\underset{y \in X}{\lor} (R (x, y) ⊙ R (y, w)) \\ ⊙ f_{1} (x) ⊙ f_{2} (x)) ⊙ (h_{1} (w) \oplus h_{2} (w)) \\ = \underset{x, w \in X}{\lor} (R (x, w) ⊙ f_{1} (x) ⊙ f_{2} (x)) \\ ⊙ (h_{1} (w) \oplus h_{2} (w)) = δ (f_{1} ⊙ f_{2}, h_{1} \oplus h_{2}) . \end{matrix}$ Hence, δ is a L-fuzzy pre-proximity on X.

Since $\begin{matrix} δ (α ⊙ f, α \to g) \\ = \underset{x, y \in X}{\lor} (R (x, y) ⊙ (α ⊙ f) (x) ⊙ (α \to g) (y)) \\ \leq \underset{x, y \in X}{\lor} (R (x, y) ⊙ f (x) ⊙ g (y)) = δ (f, g), \end{matrix}$ δ is stratified. Moreover, δ is Alexandrov and generalized.

(1) Let $R = ⊤_{X \times X}$ be given. Then, $δ_{1} (f, g) = \underset{x, y \in X}{\lor} (f (x) ⊙ g (y))$ .

Hence, δ₁ is a L-fuzzy pre-proximity on X. Moreover, δ₁ is stratified, Alexandrov and generalized. Since $δ_{1} (⊤_{x}, ⊤_{x}^{*}) = ⊤$ , δ₁ is not separated.

By Theorem 3.2, we obtain a stratified L-fuzzy interior operator $I_{δ_{1}} : L^{X} \to L^{X}$ as follows: $\begin{matrix} I_{δ_{1}} (f) & = ⋁_{g \in L^{X}} ((⋀_{x, y \in X} (g (x) \to g (y))) \\ ⊙ S (g, f) ⊙ g) \leq f . \end{matrix}$

(2) Let R = ▵ _X×X be given where $▵_{X \times X} (x, y) = {\begin{matrix} ⊤, & if y = x, \\ ⊥, & otherwise . \end{matrix}$ Then, δ₂ (f, g) = ⋁ _x∈X (f (x) ⊙ g (x)) . Hence, δ₂ is a L-fuzzy pre-proximity on X. Moreover,

(Q) For all f, g ∈ L^X, $\begin{matrix} ⋀_{h \in L^{X}} (δ_{2} (f, h) \oplus δ_{2} (h^{*}, g)) \\ = ⋀_{h \in L^{X}} (⋁_{x \in X} (f (x) ⊙ h (x)) \\ \oplus ⋁_{x \in X} (h^{*} (x) ⊙ g (x))) (Put h = g,) \\ \leq ⋁_{x \in X} (f (x) ⊙ g (x)) \oplus ⋁_{x \in X} (g^{*} (x) ⊙ g (x)) \\ = ⋁_{x \in X} (f (x) ⊙ g (x)) \oplus ⊥ = δ_{2} (f, g) . \end{matrix}$

Hence, δ₂ is an L-fuzzy proximity on X. Since $δ_{2} (⊤_{x}, ⊤_{x}^{*}) = ⊥$ , Moreover, δ₂ is separated, stratified, Alexandrov and generalized. By Theorem 3.2, we obtain a strong, separated, generalized and Alexandrov L-fuzzy interior operator $I_{δ_{2}} : L^{X} \to L^{X}$ as follows: $\begin{matrix} I_{δ_{2}} (f) & = ⋁_{g \in L^{X}} ((⋀_{x \in X} (g (x) \to g (x))) \\ ⊙ S (g, f) ⊙ g) \leq f . \end{matrix}$

Lemma 3.4. Let (X, δ) be a L-fuzzy pre-proximity space. Then, δ (α ⊙ f, g) ≥ α ⊙ δ (f, g) iff δ (α → f, g) ≤ α → δ (f, g).

Proof. Let δ (α ⊙ f, g) ≥ α ⊙ δ (f, g). Then,

α ⊙ δ (α → f, g) ≤ δ (α ⊙ (α → f) , g) ≤ δ (f, g).

Thus, δ (α → f, g) ≤ α → δ (f, g).

Let δ (α → f, g) ≤ α → δ (f, g). Then,

δ (f, g) ≤ δ (α → α ⊙ f, g) ≤ α → δ (α ⊙ f, g).

Thus, α ⊙ δ (f, g) ≤ δ (α ⊙ f, g).□

From the following theorem, we obtain the L-fuzzy pre-proximity induced by an L-fuzzy interior operator.

Theorem 3.5 Let $(X, I)$ be an L-fuzzy interior space. Define a map $δ_{I} : L^{X} \times L^{X} \to L$ by $\begin{matrix} δ_{I} (f, g) = ⋁_{x \in X} (f (x) ⊙ I^{*} (g^{*}) (x)) \forall f, g \in L^{X} . \end{matrix}$ Then, we have the following properties.

(1) $δ_{I}$ is an L-fuzzy pre-proximity,

(2) If $I$ is a stratified then, so is $δ_{I}$ and

δ (α ⊙ f, g) ≥ α ⊙ δ (f, g),

(3) $δ_{I} (f, g) \leq ⋁_{h \in L^{X}} (δ_{I} (f, h) ⊙ δ_{I} (h^{*}, g))$ , the equality holds if $I$ is topological,

(4) If $I$ is topological, then $δ_{I}$ is an L-fuzzy quasi- proximity on X,

(5) $I \leq I_{δ_{I}}$ , the equality holds if $I$ is topological,

(6) If $I$ is separated, then $δ_{I}$ is separated,

(7) If δ is a separated L-fuzzy pre-proximity on X, then $δ_{I_{δ}} \leq δ$ ,

(8) If δ is an L-fuzzy pre-proximity on X, then

$δ_{I_{δ}} (f, g) \leq ⋁_{x \in X} (f (x) ⊙ (g^{*} (x) \to δ (g^{*}, g))),$

(9) If $I$ is generalized (resp. Alexandrov), then $δ_{I}$ is generalized (resp.Alexandrov).

Proof.

(1)(P1) Since $I (⊥_{X}) = ⊥_{X}$ and $I (⊤_{X}) = ⊤_{X}$ , we have $\begin{matrix} δ_{I} (⊤_{X}, ⊥_{X}) = ⋁_{x \in X} (⊤_{X} (x) ⊙ I^{*} (⊥_{X}^{*}) (x)) = ⊥, \\ δ_{I} (⊥_{X}, ⊤_{X}) = ⋁_{x \in X} (⊥_{X} (x) ⊙ I^{*} (⊤_{X}^{*}) (x)) = ⊥ . \end{matrix}$

(P2) Since $I (g) \leq g$ , we have $\begin{matrix} δ_{I} (f, g) & = ⋁_{x \in X} (f (x) ⊙ I^{*} (g^{*}) (x)) \\ \geq ⋁_{x \in X} (f (x) ⊙ g (x)) . \end{matrix}$

(P3) If g ≤ g₁, f ≤ f₁ by (I3), then $I^{*} (g^{*}) \leq I^{*} (g_{1}^{*})$ . Thus, $\begin{matrix} δ_{I} (f, g) = ⋁_{x \in X} (f (x) ⊙ I^{*} (g^{*}) (x)) \\ \leq ⋁_{x \in X} (f_{1} (x) ⊙ I^{*} (g_{1}^{*}) (x)) = δ_{I} (f_{1}, g_{1}) . \end{matrix}$

(T) For f₁, f₂, g₁, g₂ ∈ L^X, $\begin{matrix} δ_{I} (f_{1}, g_{1}) \oplus δ_{I} (f_{2}, g_{2}) \\ = ⋁_{x \in X} (f_{1} (x) ⊙ I^{*} (g_{1}^{*}) (x)) \\ \oplus ⋁_{x \in X} (f_{2} (x) ⊙ I^{*} (g_{2}^{*}) (x)) \\ \geq ⋁_{x \in X} (f_{1} (x) ⊙ f_{2} (x)) ⊙ (I^{*} (g_{1}^{*}) (x) \oplus I^{*} (g_{2}^{*}) (x)) \\ (by Lemma 2.2 (11)) \\ \geq ⋁_{x \in X} (f_{1} (x) ⊙ f_{2} (x) ⊙ I^{*} ((g_{1} \oplus g_{2})^{*}) (x)) \\ = δ_{I} (f_{1} ⊙ f_{2}, g_{1} \oplus g_{2}) . \end{matrix}$ Hence, $δ_{I}$ is an L-fuzzy pre-proximity.

If $I$ is a stratified, by Lemma 2.2(14) and Theorem 3.2(1), we have $α ⊙ I (α \to f) \leq I (α ⊙ (α \to f)) \leq I_{δ} (f),$ $I (α \to f) \leq α \to I (f)$ and $I^{*} (α \to f^{*}) \geq α ⊙ I^{*} (f^{*}) .$ Thus, $\begin{matrix} δ_{I} (f, α ⊙ g) = ⋁_{x \in X} (f (x) ⊙ I^{*} (α \to g^{*}) (x)) \\ \geq ⋁_{x \in X} (f (x) ⊙ α ⊙ I^{*} (g^{*}) (x)) \\ = α ⊙ ⋁_{x \in X} (f (x) ⊙ I^{*} (g^{*}) (x)) = α ⊙ δ_{I} (f, g) . \end{matrix}$ It is easy to see that $δ_{I} (α ⊙ f, g) \geq α ⊙ δ_{I} (f, g) .$ $\begin{matrix} δ_{I} (α ⊙ f, α \to g) = ⋁_{x \in X} (α ⊙ f (x) ⊙ I^{*} (α ⊙ g^{*}) (x)) \\ \leq ⋁_{x \in X} (α ⊙ f (x) ⊙ (α \to I^{*} (g^{*}) (x)) \\ \leq ⋁_{x \in X} (f (x) ⊙ I^{*} (g^{*}) (x)) = δ_{I} (f, g) . \end{matrix}$

(3) For f, g, h ∈ L^X, $\begin{matrix} δ_{I}^{*} (f, h) ⊙ δ_{I}^{*} (h^{*}, g) \\ = (⋁_{x \in X} (f (x) ⊙ I^{*} (h^{*}) (x)))^{*} \\ ⊙ (⋁_{x \in X} (h^{*} (x) ⊙ I^{*} (g^{*}) (x)))^{*} \\ = S (f, I (h^{*})) ⊙ S (h^{*}, I (g^{*})) (Since I (h^{*}) \leq h^{*}) \\ \leq S (f, h^{*}) ⊙ S (h^{*}, I (g^{*})) \leq S (f, I (g^{*})) = δ_{I}^{*} (f, g) . \end{matrix}$ Hence, $δ_{I} (f, g) \leq ⋀_{h \in L^{X}} (δ_{I} (f, h) \oplus δ_{I} (h^{*}, g))$ .

If $I$ is topological, then $\begin{matrix} ⋁_{h \in L^{X}} (δ_{I}^{*} (f, h) ⊙ δ_{I}^{*} (h^{*}, g)) \\ = ⋁_{h \in L^{X}} (S (f, I (h^{*})) ⊙ S (h^{*}, I (g^{*}))) \\ (Put h^{*} = I (g^{*})) \\ \geq S (f, I (I (g^{*}))) ⊙ S (I (g^{*}), I (g^{*})) \\ = S (f, I (g^{*})) = δ_{I}^{*} (f, g) . \end{matrix}$

(4) By (2), it is trivial.

(5) Since $S (g, f) \leq S (I (g), I (f))$ from Lemma 2.4, we have $\begin{matrix} I_{δ_{I}} (f) = ⋁_{g \in L^{X}} (δ_{I}^{*} (g, g^{*}) ⊙ S (g, f) ⊙ g)) \\ = S (g, I (g)) ⊙ S (g, f) ⊙ g \\ \leq S (g, I (g)) ⊙ S (I (g), I (f)) ⊙ g \\ \leq S (g, I (f)) ⊙ g \leq I (f) . \end{matrix}$

If $I$ is topological, then $\begin{matrix} I_{δ_{I}} (f) = ⋁_{g \in L^{X}} (δ_{I}^{*} (g, g^{*}) ⊙ S (g, f) ⊙ g) \\ = ⋁_{g \in L^{X}} (S (g, I (g)) ⊙ S (g, f) ⊙ g) \\ \geq S (I (f), I (I (f))) ⊙ S (I (f), f) ⊙ I (f) = I (f) . \end{matrix}$

(6) Let $I$ be separated. Then,

$δ_{I} (⊤_{z}, ⊤_{z}^{*}) = ⋁_{x \in X} (⊤_{z} (x) ⊙ I^{*} (⊤_{z}) (x)) = ⊥ .$

(7) For f, g ∈ L^X, $\begin{matrix} δ_{I_{δ}} (f, g) = ⋁_{x \in X} (f (x) ⊙ I_{δ}^{*} (g^{*}) (x)) \\ = ⋁_{x \in X} (f (x) ⊙ (⋁_{h \in L^{X}} (δ^{*} (h, h^{*}) ⊙ S (h, g^{*}) \\ ⊙ h (x)))^{*}) \\ \leq ⋁_{x \in X} (f (x) ⊙ (δ^{*} (⊤_{x}, ⊤_{x}^{*}) ⊙ S (⊤_{x}, g^{*}) \\ ⊙ ⊤_{x} (x))^{*}) = ⋁_{x \in X} (f (x) ⊙ g (x)) \leq δ (f, g) . \end{matrix}$

(8) $\begin{matrix} δ_{I_{δ}} (f, g) = ⋁_{x \in X} (f (x) ⊙ I_{δ}^{*} (g^{*}) (x)) \\ = ⋁_{x \in X} (f (x) ⊙ (⋁_{h \in L^{X}} (δ^{*} (h, h^{*}) ⊙ S (h, g^{*}) \\ ⊙ h (x)))^{*}) \\ \leq ⋁_{x \in X} (f (x) ⊙ (S (g^{*}, g^{*}) ⊙ g^{*} (x) \to δ (g^{*}, g))) \\ = ⋁_{x \in X} (f (x) ⊙ (g^{*} (x) \to δ (g^{*}, g))) . \end{matrix}$

(9) It is easily proved from definitions.□

Corollary 3.6 Let $(X, I)$ be an L-fuzzy interior space. Define a map $δ_{I}^{s} : L^{X} \times L^{X} \to L$ by $\begin{matrix} δ_{I}^{s} (f, g) = ⋁_{x \in X} (g (x) ⊙ I^{*} (f^{*}) (x)) \forall f, g \in L^{X} . \end{matrix}$

Then, we have the following properties.

(1) $δ_{I}^{s}$ is an L-fuzzy pre-proximity. If $I$ is a stratified, then $δ_{I}^{s}$ is a stratified,

(2) $δ_{I}^{s} (f, g) \leq ⋁_{h \in L^{X}} (δ_{I}^{s} (f, h) ⊙ δ_{I}^{s} (h^{*}, g))$ , the equality holds if $I$ is topological,

(3) If $I$ is topological, then $δ_{I}^{s}$ is an L-fuzzy quasi- proximity on X,

(4) $I \geq I_{δ_{I}^{s}}$ , the equality holds if $I$ is topological,

(5) If $I$ is separated, then $δ_{I}^{s}$ is separated,

(6) If δ is a separated L-fuzzy pre-proximity on X, then $δ_{I_{δ}}^{s} \leq δ^{s}$ ,

(7) If δ^s is an L-fuzzy pre-proximity on X, then

$δ_{I_{δ}}^{s} (f, g) \leq ⋁_{x \in X} (g (x) ⊙ (f^{*} (x) \to δ^{s} (f^{*}, f))) .$

(8) If $I$ is generalized (resp. Alexandrov), then $δ_{I}^{s}$ is generalized (resp. Alexandrov).

Example 3.7

(1) Define $I_{1} : L^{X} \to L^{X}$ as $I_{1} (f) = ⋀_{x \in X} f (x) .$

(I1), (I2), (I3) and (I4) are easily proved. Hence $I_{1}$ is a topological, strong, generalized and Alexandrov L-fuzzy interior operator on X. Since $I_{1} (⊤_{x}) = ⊥_{X}$ and $δ_{I_{1}} (⊤_{x}, ⊤_{x}^{*}) = ⊤$ , $I_{1}$ and $δ_{I_{1}}$ are not separated. By Theorem 3.5, we obtain stratified, Alexandrov and generalized L-fuzzy preproximities $δ_{I_{1}}, δ_{I_{1}}^{s}$ as $\begin{matrix} δ_{I_{1}} (f, g) & = ⋁_{x \in X} (f (x) ⊙ I_{1}^{*} (g^{*}) (y)) \\ = ⋁_{x, y \in X} (f (x) ⊙ g (y)), \\ δ_{I_{1}}^{s} (f, g) & = ⋁_{x \in X} (g (x) ⊙ I_{1}^{*} (f^{*}) (y)) \\ = ⋁_{x, y \in X} (f (y) ⊙ g (x)) . \end{matrix}$

Since $I_{1}$ is topological, then $I_{δ_{I_{1}}} = I_{1}$ .

(2) Define $I_{2} : L^{X} \to L^{X}$ as $I_{2} (f) = f .$

(I1), (I2), (I3) and (I4) are easily proved. Hence, $I_{2}$ is a topological, strong, generalized and Alexandrov L-interior operator on X. Since $I_{2} (⊤_{x}) = ⊤_{x}$ and $δ_{I_{2}} (⊤_{x}, ⊤_{x}^{*}) = ⊥$ , $I_{2}$ and $δ_{I_{2}}$ are separated. By Theorem 3.5, we obtain stratified, Alexandrov and generalized L-fuzzy preproximities $δ_{I_{2}}, δ_{I_{2}}^{s}$ as $\begin{matrix} δ_{I_{1}} (f, g) & = ⋁_{x \in X} (f (x) ⊙ I_{1}^{*} (g^{*}) (x)) \\ = ⋁_{x \in X} (f (x) ⊙ g (x)), \\ δ_{I_{1}}^{s} (f, g) & = ⋁_{x \in X} (g (x) ⊙ I_{1}^{*} (f^{*}) (x)) \\ = ⋁_{x \in X} (g (x) ⊙ f (x)) . \end{matrix}$ Since $I_{2}$ is topological, then $I_{δ_{I_{2}}} = I_{2}$ .

4 The relationships between L-fuzzy pre-proximities and L-fuzzy topologies

In this section, we investigate the connections between L-fuzzy pre-proximities and L-fuzzy topologies.

Theorem 4.1 If δ is Alexandrov L-fuzzy pre-proximity on X. Define a mapping $T_{δ} : L^{X} \to L$ by: $T_{δ} (f) = δ^{*} (f, f^{*})$ . Then,

(1) $T_{δ}$ is an L-fuzzy topology on X,

(2) If δ is stratified, then $T_{δ}$ is strong,

(3) If δ is separated, then so is $T_{δ}$ .

Proof. (1)(T1) $T_{δ} (⊥_{X}) = δ^{*} (⊥_{X}, ⊤_{X}) = ⊤,$

$T_{δ} (⊤_{X}) = δ^{*} (⊤_{X}, ⊥_{X}) = ⊤ .$

(T2) $\begin{matrix} T_{δ} (f ⊙ g) = δ^{*} (f ⊙ g, f^{*} \oplus g^{*}) \\ \geq δ^{*} (f, f^{*}) ⊙ δ^{*} (g, g^{*}) = T_{δ} (f) ⊙ T_{δ} (g) . \end{matrix}$

(T3) $\begin{matrix} T_{δ} (⋁_{i \in Γ} f_{i}) = δ^{*} (⋁_{i \in Γ} f_{i}, ⋀_{i \in Γ} f_{i}^{*}) \\ \geq ⋀_{i \in Γ} δ^{*} (f_{i}, f_{i}^{*}) = ⋀_{i \in Γ} T_{δ} (f_{i}) . \end{matrix}$

(2) $\begin{matrix} T_{δ} (α ⊙ f) & = δ^{*} (α ⊙ f, α \to f^{*}) \\ \geq δ^{*} (f, f^{*}) = T_{δ} (f), \\ T_{δ} (α \to f) & = δ^{*} (α \to f, α ⊙ f^{*}) \\ \geq δ^{*} (f, f^{*}) = T_{δ} (f) . \end{matrix}$

(3) It is easy.□

Theorem 4.2 Let $(X, I)$ be an L-fuzzy interior space. Define a mapping $T_{I_{δ}} : L^{X} \to L$ by: $T_{I_{δ}} (f) = S (f, I_{δ} (f)) .$ Then,

(1) $T_{I_{δ}}$ is an L-fuzzy topology on X with $T_{I_{δ}} \geq T_{δ}$ ,

(2) If $I$ is Alexandrov (resp. strong, separated), then $T_{δ_{I}}$ is Alexandrov (resp. strong, separated).

Proof. (1) (T1) $T_{I_{δ}} (⊤_{X}) = S (⊤_{X}, I_{δ} (⊤_{X})) = ⊤,$

$T_{I_{δ}} (⊥_{X}) = S (⊥_{X} \to I_{δ} (⊥_{X})) = ⊤ .$

(T2) From Lemma 2.4, we have $\begin{matrix} T_{I_{δ}} (f ⊙ g) = S (f ⊙ g, I_{δ} (f ⊙ g)) \\ \geq S (f ⊙ g, I_{δ} (f) ⊙ I_{δ} (g)) \\ \geq S (f, I_{δ} (f)) ⊙ S (g, I_{δ} (g)) \\ = T_{I_{δ}} (f) ⊙ T_{I_{δ}} (g) . \end{matrix}$

(T3) $\begin{matrix} T_{I_{δ}} (⋁_{i} f_{i}) = S (⋁_{i} f_{i}, I_{δ} (⋁_{i} f_{i}) \\ = ⋀_{x \in X} (⋁_{i} f_{i} (x) \to I_{δ} (⋁_{i} f_{i}) (x)) \\ \geq ⋀_{x \in X} (⋁_{i} f_{i} (x) \to ⋁_{i} I_{δ} (f_{i}) (x)) \\ (by Lemma 2.2 (16)) \\ \geq ⋀_{i} ⋀_{x \in X} (f_{i} (x) \to I_{δ} (f_{i}) (x))) = ⋀_{i} T_{I_{δ}} (f_{i}) . \end{matrix}$ $\begin{matrix} T_{I_{δ}} (f) = S (f, I_{δ} (f)) \\ = (⋁_{x \in X} (f (x) ⊙ I_{δ}^{*} (f) (x)))^{*} \\ \geq (⋁_{x \in X} (f (x) ⊙ f^{*} (x)))^{*} \\ \geq δ^{*} (f, f^{*}) = T_{δ} (f) . \end{matrix}$

(2) By Lemma 2.2 (16), we have $\begin{matrix} T_{I_{δ}} (⋀_{i \in Γ} f_{i}) = S (⋀_{i \in Γ} f_{i}, I_{δ} (⋀_{i \in Γ} f_{i})) \\ = ⋀_{x \in X} (⋀_{i \in Γ} (f_{i} (x) \to I_{δ} (⋀_{i \in Γ} f_{i}) (x)) \\ = ⋀_{x \in X} (⋀_{i \in Γ} (f_{i} (x) \to ⋀_{i \in Γ} I_{δ} (f_{i}) (x)) \\ \geq ⋀_{i \in Γ} (⋀_{x \in X} (f_{i} (x) \to I_{δ} (f_{i}) (x)) = ⋀_{i \in Γ} T_{I_{δ}} (f_{i}) . \end{matrix}$ Hence, $T_{I_{δ}}$ is Alexandrov L-fuzzy topology on X.

By Lemma 2.2 (14), (18), we have $\begin{matrix} T_{I_{δ}} (α ⊙ f) \\ = ⋀_{x \in X} ((α ⊙ f (x)) \to (I_{δ} (α ⊙ f) (x))) \\ \geq ⋀_{x \in X} ((α ⊙ f (x)) \to (α ⊙ I_{δ} (f) (x))) \\ \geq ⋀_{x \in X} (f (x) \to I_{δ} (f) (x)) = T_{I_{δ}} (f) . \\ T_{I_{δ}} (α \to f) \\ = ⋀_{x \in X} ((α \to f (x)) \to I_{δ} (α \to f) (x)) \\ = ⋀_{x \in X} ((α \to f (x)) \to (α \to I_{δ} (f) (x))) \\ \geq ⋀_{x \in X} (f (x) \to I_{δ} (f) (x)) = T_{I_{δ}} (f) . \end{matrix}$ Other cases are easily proved.□

Theorem 4.3 Let (X, δ) be an L-fuzzy pre-proximity space. Define a mapping $T_{δ}^{(1)} : L^{X} \to L$ by: $T_{δ}^{(1)} (f) = ⋀_{x \in X} (f (x) \to δ^{*} (⊤_{x}, f^{*})) .$ Then,

(1) $T_{δ}^{(1)}$ is an L-fuzzy topology on X,

(2) If δ is Alexandrov and δ (α ⊙ f, g) ≥ α ⊙ δ (f, g), then so is $T^{(1)}$ and $T_{δ}^{(1)} \geq T_{δ}$ ,

(3) If δ is separated, then $T_{δ}^{(1)}$ is separated.

Proof. (1) (T1) $T_{δ}^{(1)} (⊥_{X}) = ⊤,$ $\begin{matrix} T_{δ}^{(1)} (⊤_{X}) & = ⋀_{x \in X} (⊤_{X} (x) \to δ^{*} (⊤_{x}, ⊤_{X}^{*})) \\ = (⋁_{x \in X} δ (⊤_{x}, ⊤_{X}^{*}))^{*} = ⊤ . \end{matrix}$

(T2) $\begin{matrix} T_{δ}^{(1)} (f ⊙ g) = ⋀_{x \in X} ((f ⊙ g) (x) \to δ^{*} (⊤_{x}, f^{*} \oplus g^{*})) \\ \geq ⋀_{x \in X} ((f ⊙ g) (x) \to δ^{*} (⊤_{x}, f^{*}) ⊙ δ^{*} (⊤_{x}, g^{*})) \\ \geq ⋀_{x \in X} ((f (x) \to δ^{*} (⊤_{x}, f^{*})) ⊙ (g (x) \to δ^{*} (⊤_{x}, g^{*}))) \\ \geq T_{δ}^{(1)} (f) ⊙ T_{δ}^{(1)} (g) . \end{matrix}$

(T3) $\begin{matrix} T_{δ}^{(1)} (⋁_{i \in Γ} f_{i}) \\ = ⋀_{x \in X} (⋁_{i \in Γ} f_{i} (x) \to δ^{*} (⊤_{x}, ⋀_{i \in Γ} f_{i}^{*})) \\ \geq ⋀_{x \in X} (⋁_{i \in Γ} f_{i} (x) \to ⋁_{i \in Γ} δ^{*} (⊤_{x}, f_{i}^{*})) \\ \geq ⋀_{i \in Γ} ⋀_{x \in X} (f_{i} (x) \to δ^{*} (⊤_{x}, f_{i}^{*})) \\ = ⋀_{i \in Γ} T_{δ}^{(1)} (f_{i}) . \end{matrix}$

(2) If δ is Alexandrov, then $\begin{matrix} T_{δ}^{(1)} (⋀_{i \in Γ} f_{i}) \\ = ⋀_{x \in X} (⋀_{i \in Γ} f_{i} (x) \to δ^{*} (⊤_{x}, ⋁_{i \in Γ} f_{i}^{*})) \\ = ⋀_{x \in X} (⋀_{i \in Γ} f_{i} (x) \to ⋀_{i \in Γ} δ^{*} (⊤_{x}, f_{i}^{*})) \\ \geq ⋀_{i \in Γ} ⋀_{x \in X} (f_{i} (x) \to δ^{*} (⊤_{x}, f_{i}^{*})) \\ = ⋀_{i \in Γ} T_{δ}^{(1)} (f_{i}) . \end{matrix}$

Hence, $T_{δ}^{(1)}$ is Alexandrov L-fuzzy topology on X.

If δ (α ⊙ f, g) ≥ α ⊙ δ (f, g), we have $\begin{matrix} T_{δ}^{(1)} (f) & = ⋀_{x \in X} (f (x) \to δ^{*} (⊤_{x}, f^{*})) \\ = (⋁_{x \in X} (f (x) ⊙ δ (⊤_{x}, f^{*})))^{*} \\ \geq (δ (⋁_{x \in X} (f (x) ⊙ ⊤_{x}), f^{*}))^{*} \\ = δ^{*} (f, f^{*}) = T_{δ} (f) . \end{matrix}$

(3) It is easily proved. □

Example 4.4 Let X = {h_i ∣ i = {1, . . . , 3}} with h_i=house and Y = {e, b, w, c, i} with e=expensive, b= beautiful, w=wooden, c= creative, i=in the green surroundings. Let ([0, 1] , ⊙ , → , ^*, 0, 1) be a complete residuated lattice as $\begin{matrix} x ⊙ y = max {0, x + y - 1}, \\ x \to y = min {1 - x + y, 1}, x^{*} = 1 - x . \end{matrix}$ Let I ∈ [0, 1] ^X×Y be a fuzzy information as follows:

$\begin{matrix} I & e & b & w & c & i \\ h_{1} & 0.7 & 0.6 & 0.5 & 0.9 & 0.2 \\ h_{2} & 0.6 & 0.8 & 0.4 & 0.3 & 0.5 \\ h_{3} & 0.4 & 0.9 & 0.8 & 0.6 & 0.6 \end{matrix}$

Define [0, 1]-fuzzy pre-orders $R_{Y}^{X}$ , $R_{X}^{{b, w}}$ ∈ [0, 1]^X×X by $\begin{matrix} R_{X}^{Y} (h_{i}, h_{j}) & = ⋀_{y \in Y} (I (h_{i}, y) \to I (h_{j}, y)), \\ R_{X}^{{b, w}} (h_{i}, h_{j}) & = ⋀_{y \in {b, w}} (R (h_{i}, y) \to R (h_{j}, y)) . \end{matrix}$ $R_{X}^{Y} = (\begin{matrix} 1 & 0.4 & 0.7 \\ 0.7 & 1 & 0.8 \\ 0.6 & 0.6 & 1 \end{matrix}), R_{X}^{{b, w}} = (\begin{matrix} 1 & 0.9 & 1 \\ 0.8 & 1 & 1 \\ 0.7 & 0.6 & 1 \end{matrix}) .$

(1) For each $R \in {R_{X}^{Y}, R_{X}^{{b, w}}}$ , by Example 3.3, we obtain a stratified, Alexandrov and generalized [0, 1]-fuzzy pre-proximity δ_R : [0, 1] ^X × [0, 1] ^X → [0, 1] as $\begin{matrix} δ_{R} (f, g) = ⋁_{h_{i}, h_{j} \in X} (R_{X}^{Y} (h_{i}, h_{j}) ⊙ f (h_{i}) ⊙ g (h_{j})) . \end{matrix}$ By Theorem 3.2, we obtain a stratified [0, 1]-fuzzy interior operator $I_{δ_{R}} : [0, 1]^{X} \to [0, 1]^{X}$ as $\begin{matrix} I_{δ_{R}} (f) (h_{i}) \\ = ⋁_{g \in L^{X}} (S (g, f) ⊙ g (h_{i}) ⊙ δ_{R}^{*} (g, g^{*})) \\ = ⋀_{g \in L^{X}} (S (f, g^{*}) ⊙ g (h_{i}) \\ ⊙ ⋀_{h_{j}, h_{k} \in X} (R_{X}^{Y} (h_{j}, h_{k}) ⊙ g (h_{j}) \to g (h_{k}))) . \end{matrix}$ By Theorem 4.1, we obtain a strong [0, 1]-fuzzy topology $T_{δ_{R}} : [0, 1]^{X} \to [0, 1]$ as

$\begin{matrix} T_{δ_{R}} (f) = δ_{R}^{*} (f, f^{*}) \\ = (⋁_{h_{i}, h_{j} \in X} (R_{X}^{Y} (h_{i}, h_{j}) ⊙ f (h_{i}) ⊙ f^{*} (h_{j})))^{*} \\ = ⋀_{h_{i}, h_{j} \in X} (R_{X}^{Y} (h_{i}, h_{j}) ⊙ f (h_{i}) \to f (h_{j})) . \end{matrix}$

Since, $\begin{matrix} δ_{R} (⊤_{h_{i}}, f) \\ = ⋁_{h_{i}, h_{j} \in X} (R_{X}^{Y} (h_{i}, h_{j}) ⊙ ⊤_{h_{i}} (h_{i}) ⊙ f (h_{j})) \\ = ⋁_{h_{i} \in X} (R (h_{i}, h_{j}) ⊙ f (h_{j})), \end{matrix}$ by Theorem 4.3, we obtain [0, 1]-fuzzy topology $T_{δ_{R}}^{(1)} : [0, 1]^{X} \to [0, 1]$ as $\begin{matrix} T_{δ_{R}}^{(1)} (f) = ⋀_{h_{j} \in X} (f (h_{j}) \to δ_{R}^{*} (⊤_{h_{j}}, f^{*})) \\ = ⋀_{h_{j} \in X} (f (h_{j}) \to (⋁_{h_{i} \in X} (R (h_{i}, h_{j}) ⊙ f^{*} (h_{j}))^{*}) \\ = ⋀_{h_{i}, h_{j} \in X} ((R (h_{i}, h_{j}) ⊙ f (h_{i})) \to f (h_{j})) . \end{matrix}$

(2) For each $R \in {R_{X}^{Y}, R_{X}^{{b, w}}}$ , we obtain a strong, generalized, topological and Alexandrov [0, 1]-fuzzy interior operator $I_{R} : [0, 1]^{X} \to [0, 1]^{X}$ as $\begin{matrix} I_{R} (f) (h_{i}) = ⋀_{h_{j} \in X} (R (h_{i}, h_{j}) \to f (h_{j})) . \end{matrix}$

By Theorem 3.5, we obtain a generalized, topological and Alexandrov [0, 1]-fuzzy quasi-proximity $δ_{I_{R}}$ as $\begin{matrix} δ_{I_{R}} (f, g) = ⋁_{h_{i} \in X} (f (h_{i}) ⊙ I_{R}^{*} (g^{*}) (h_{i})) \\ = ⋁_{h_{i} \in X} (f (h_{i}) ⊙ ⋁_{h_{j} \in X} (R (h_{i}, h_{j}) ⊙ g (h_{j}))) \\ = ⋁_{h_{i}, h_{j} \in X} (R (h_{i}, h_{j}) ⊙ f (h_{i}) ⊙ g (h_{j})) . \end{matrix}$ By Theorem 4.3, we obtain [0, 1]-fuzzy topologies $T_{δ_{I_{R}}}$ and $T_{δ_{I_{R}}}^{(1)}$ as follows:

$\begin{matrix} T_{δ_{I_{R}}} (f) = δ_{I_{R}}^{*} (f, f^{*}) \\ = (⋁_{h_{i}, h_{j} \in X} (R (h_{i}, h_{j}) ⊙ f (h_{i}) ⊙ f^{*} (h_{j})))^{*} \\ = ⋀_{h_{i}, h_{j} \in X} (R_{X}^{Y} (h_{i}, h_{j}) ⊙ f (h_{i}) \to f (h_{j})) . \end{matrix}$

Since $δ_{I_{R}} (⊤_{h_{j}}, f) = ⋁_{h_{i}, h_{j} \in X} (R (h_{i}, h_{j}) ⊙ f (h_{i}) ⊙ ⊤_{h_{j}} (h_{j})) = ⋁_{h_{i} \in X} (R (h_{i}, h_{j}) ⊙ f (h_{i}))$ , then

$\begin{matrix} T_{δ_{I_{R}}}^{(1)} (f) = ⋀_{h_{j} \in X} (f (h_{j}) \to δ_{R}^{*} (⊤_{h_{j}}, f^{*}))) \\ = ⋀_{h_{j} \in X} (f (h_{j}) \to (⋁_{h_{i} \in X} (R (h_{i}, h_{j}) ⊙ f^{*} (h_{i}))^{*}) \\ = ⋀_{h_{i}, h_{j} \in X} ((R (h_{i}, h_{j}) ⊙ f (h_{i})) \to f (h_{j})) . \end{matrix}$

5 Galois correspondences

We devote this section to a Galois correspondence between the category of separated L-fuzzy interior spaces and that of separated L-fuzzy pre-proximity spaces. Also, we mainly define degrees of LF-continuity [40], LF-proximity and LF-interior of mappings to equip each mapping between L-fuzzy topological, L-fuzzy proximity and L-fuzzy interior spaces with some degree to be an LF-continuous, LF-proximity and an LF-interior mapping, respectively. Then, we will study their connections in a degree sense. Moreover, φ always denotes a mapping from X to Y in the following sections.

Definition 5.1 Let $(X, I_{X})$ and $(Y, I_{Y})$ be two L-fuzzy interior spaces. Then, $D_{I} (φ)$ is defined by

$D_{I} (φ) = ⋀_{h \in L^{Y}} S (φ^{\to} (I_{Y} (h)), I_{X} (φ^{\to} (h)))$

is called the degree of LF-interior map for φ. If $D_{I} (φ) = ⊤$ , then $φ^{\to} (I_{Y} (h)) \leq I_{X} (φ^{\to} (h))$ for all h ∈ L^Y, which is exactly the definition of L-fuzzy interior mapping between L-fuzzy interior spaces.

Definition 5.2 Let $(X, T_{X})$ and $(Y, T_{Y})$ be two L-fuzzy topological spaces. Then, $D_{T} (φ)$ is defined by

$D_{T} (φ) = ⋀_{h \in L^{Y}} (T_{Y} (h) \to T_{X} (φ^{\to} (h)))$

is called the degree of LF-continuous map for φ. If $D_{T} (φ) = ⊤$ , then $T_{Y} (h) \leq T_{X} (φ^{\to} (h))$ for all h ∈ L^Y, which is exactly the definition of L-fuzzy continuous mapping between L-fuzzy topological spaces.

Definition 5.3 Let (X, δ_X) and (Y, δ_Y) be two L-fuzzy proximity spaces. Then, D_δ (φ) is defined by

$D_{δ} (φ) = ⋀_{k, l \in L^{Y}} (δ_{X} (φ^{\to} (k), φ^{\to} (l)) \to δ_{Y} (k, l))$

is called the degree of LF-proximity map for φ. If D_δ (φ) =⊤, then

δ_X (φ^→ (k) , φ^→ (l)) ≤ δ_Y (k, l) ∀ k, l ∈ L^Y

which is exactly the definition of L-fuzzy proximity mapping between L-fuzzy pre-proximity spaces.

Theorem 5.4 Let (X, δ_X) and (Y, δ_Y) be L-fuzzy pre-proximity spaces. Then

(1) $D_{δ} (φ) \leq D_{I_{δ}} (φ)$ ,

(2) $D_{δ} (φ) = D_{T_{δ}} (φ)$ ,

(3) $D_{δ} (φ) \leq D_{T_{δ}^{(1)}} (φ)$ .

Proof. (1) $\begin{matrix} D_{I_{δ}} (φ) \\ = ⋀_{h \in L^{Y}} S (φ^{\to} (I_{δ_{Y}} (h)), I_{δ_{X}} (φ^{\to} (h))) \\ = ⋀_{h \in L^{Y}} ⋀_{x \in X} (φ^{\to} (I_{δ_{Y}} (h)) (x) \to I_{δ_{X}} (φ^{\to} (h)) (x)) \\ = ⋀_{h \in L^{Y}} ⋀_{x \in X} (I_{δ_{Y}} (h) (φ (x)) \to I_{δ_{X}} (φ^{\to} (h)) (x)) \\ = ⋀_{h \in L^{Y}} ⋀_{x \in X} [(⋁_{g_{1} \in L^{Y}} δ_{Y}^{*} (g_{1}, g_{1}^{*}) ⊙ S (g_{1}, h) \\ ⊙ g_{1} (φ (x)) \to (⋁_{g \in L^{X}} δ_{X}^{*} (g, g^{*}) ⊙ S (g, φ^{\to} (h)) ⊙ g (x))] \\ = ⋀_{h \in L^{Y}} ⋀_{g_{1} \in L^{Y}} ⋀_{x \in X} [δ_{Y}^{*} (g_{1}, g_{1}^{*}) ⊙ S (g_{1}, h) \\ ⊙ φ^{\to} (g_{1}) (x) \to (⋁_{g = φ^{\to} (g_{1}) \in L^{X}} δ_{X}^{*} (φ^{\to} (g_{1}), (φ^{\to} (g_{1}))^{*}) \\ ⊙ S (φ^{\to} (g_{1}), φ^{\to} (h)) ⊙ φ^{\to} (g_{1}) (x))] \\ \geq ⋀_{h \in L^{Y}} ⋀_{g_{1} \in L^{Y}} ⋀_{x \in X} [(δ_{Y}^{*} (φ^{\to} (g_{1}), φ^{\to} (g_{1}^{*})) \\ ⊙ S (φ^{\to} (g_{1}), φ^{\to} (h)) ⊙ φ^{\to} (g_{1}) (x)) \\ \to (δ_{X}^{*} (φ^{\to} (g_{1}), φ^{\to} (g_{1}^{*})) \\ ⊙ S (φ^{\to} (g_{1}), φ^{\to} (h)) ⊙ φ^{\to} (g_{1}) (x))] \\ \geq ⋀_{g_{1} \in L^{Y}} (δ_{Y}^{*} (g_{1}, g_{1}^{*}) \to δ_{X}^{*} (φ^{\to} (g_{1}), φ^{\to} (g_{1}^{*}))) \\ = ⋀_{g_{1} \in L^{Y}} (δ_{X} (φ^{\to} (g_{1}), φ^{\to} (g_{1}^{*})) \to δ_{Y} (g_{1}, g_{1}^{*})) \\ = ⋀_{g_{1}, g_{2} \in L^{Y}} (δ_{X} (φ^{\to} (g_{1}), φ^{\to} (g_{2})) \to δ_{Y} (g_{1}, g_{2})) \\ = D_{δ} (φ) . \end{matrix}$

(2) $\begin{matrix} D_{T_{δ}} (φ) \\ = ⋀_{h \in L^{Y}} (T_{δ_{Y}} (h) \to T_{δ_{X}} (φ^{\to} (h))) \\ = ⋀_{h \in L^{Y}} (δ_{Y}^{*} (h, h^{*}) \to δ_{X}^{*} (φ^{\to} (h), (φ^{\to} (h))^{*})) \\ = ⋀_{h \in L^{Y}} (δ_{Y}^{*} (h, h^{*}) \to δ_{X}^{*} (φ^{\to} (h), φ^{\to} (h^{*}))) \\ = ⋀_{h \in L^{Y}} (δ_{X} (φ^{\to} (h), φ^{\to} (h^{*})) \to δ_{Y} (h, h^{*})) \\ = ⋀_{h, g \in L^{Y}} (δ_{X} (φ^{\to} (h), φ^{\to} (g)) \to δ_{Y} (h, g)) = D_{δ} (φ) . \end{matrix}$

(3) $\begin{matrix} D_{T_{δ}^{(1)}} (φ) \\ = ⋀_{f \in L^{Y}} (T_{δ_{Y}}^{(1)} (f) \to T_{δ_{X}}^{(1)} (φ^{\to} (f))) \\ = ⋀_{f \in L^{Y}} (⋀_{y \in Y} (δ_{Y} (f, ⊤_{y}) \to f (y)) \\ \to ⋀_{x \in X} (δ_{X} (φ^{\to} (f), φ^{\to} (⊤_{φ (x)}) \to φ^{\to} (f) (x))) \\ \geq ⋀_{f \in L^{Y}} (⋀_{x \in X} (δ_{Y} (f, ⊤_{φ (x)}) \to f (φ (x))) \\ \to ⋀_{x \in X} (δ_{X} (φ^{\to} (f), φ^{\to} (⊤_{φ (x)}) \to f (φ (x)))) \\ \geq ⋀_{f \in L^{Y}} ⋀_{x \in X} (δ_{X} (φ^{\to} (f), φ^{\to} (⊤_{φ (x)}) \\ \to δ_{Y} (f, ⊤_{φ (x)})) \\ = ⋀_{f \in L^{Y}} (δ_{X} (φ^{\to} (f), φ^{\to} (⊤_{φ (x)})) \\ \to δ_{Y} (f, ⊤_{φ (x)})) = D_{δ} (φ) . \end{matrix}$ □

Theorem 5.5 Let $(X, I_{X})$ and $(Y, I_{Y})$ be L-fuzzy

interior spaces. Then,

(1) $D_{I} (φ) \leq D_{δ_{I}} (φ)$ ,

(2) $D_{I_{δ}} (φ) \leq D_{T_{I_{δ}}} (φ)$ .

Proof. (1) $\begin{matrix} D_{δ_{I}} (φ) \\ = ⋀_{k, l \in L^{Y}} (δ_{I_{X}} (φ^{\to} (k), φ^{\to} (l)) \to δ_{I_{Y}} (k, l)) \\ = ⋀_{k, l \in L^{Y}} ((⋁_{x \in X} φ^{\to} (k) (x) ⊙ I_{X}^{*} ((φ^{\to} (l))^{*}) (x)) \\ \to (⋁_{y \in Y} k (y) ⊙ I_{Y}^{*} (l^{*}) (y))) \\ = ⋀_{k, l \in L^{Y}} ⋀_{x \in X} ((k (φ (x)) ⊙ I_{X}^{*} (φ^{\to} (l^{*})) (x)) \\ \to (⋁_{x \in X} k (φ (x)) ⊙ I_{Y}^{*} (l^{*}) (φ (x)))) \\ \geq ⋀_{l \in L^{Y}} ⋀_{x \in X} (I_{Y} (l^{*}) (φ (x)) \to I_{X} (φ^{\to} (l^{*})) (x)) \\ \geq ⋀_{l \in L^{Y}} ⋀_{x \in X} (φ^{\to} (I_{Y} (l^{*})) (x) \to I_{X} (φ^{\to} (l^{*})) (x)) \\ = ⋀_{g = l^{*} \in L^{Y}} S (φ^{\to} (I_{Y} (g)), I_{X} (φ^{\to} (g))) = D_{I} (φ) . \end{matrix}$

(2) $\begin{matrix} D_{T_{I_{δ}}} (φ) \\ = ⋀_{h \in L^{Y}} (T_{I_{δ_{Y}}} (h) (x) \to T_{I_{δ_{X}}} (φ^{\to} (h)) (x)) \\ = ⋀_{h \in L^{Y}} (S (h, I_{δ_{Y}} (h)) \to S (φ^{\to} (h), I_{δ_{X}} (φ^{\to} (h)))) \\ \geq ⋀_{h \in L^{Y}} (S (φ^{\to} (h), φ^{\to} (I_{δ_{Y}}) (h)) \\ \to S (φ^{\to} (h), I_{δ_{X}} (φ^{\to} (h)))) \\ \geq ⋀_{h \in L^{Y}} S (φ^{\to} (I_{δ_{Y}}) (h), I_{δ_{X}} (φ^{\to} (h))) = D_{I_{δ}} (φ) . \end{matrix}$ □

The category of separated L-fuzzy pre-proximity spaces with LF-proximity mappings as morphisms is denoted by SPROX.

The category of separated L-fuzzy interior spaces with LF-interior mappings as morphisms is denoted by SFI.

From Theorems 3.2 and 5.4, we obtain a concrete functor ϒ : SPROX → SFI defined as $ϒ (X, δ) = (X, I_{δ}), ϒ (φ) = φ .$

From Theorems 3.4 and 5.5, we obtain a concrete functor Ω : SFI → SPROX defined as $Ω (X, I) = (X, δ_{I}), Ω (φ) = φ .$

Theorem 5.6 Ω : SFI → SPROX is a left adjoint of ϒ : SPROX → SFI, i.e., (ϒ, Ω) is a Galois correspondence.

Proof. By Theorem 3.5(5), if $I_{X}$ is an separated L-fuzzy interior operator on a set X, then $ϒ (Ω (I_{X})) = I_{δ_{I_{X}}} \leq I_{X}$ . Hence, the identity map ${id}_{X} : (X, I_{X}) \to (X, I_{T_{I_{X}}}) = (X, ϒ (Ω (I_{X})))$ is an LF-interior map.

Moreover, if δ_Y is a separated L-fuzzy pre-proximity on a set Y, by Theorem 3.5(7), $Ω (ϒ (δ_{Y})) = δ_{I_{δ_{Y}}} \leq δ_{Y}$ . Hence, the identity map ${id}_{Y} : (Y, δ_{I_{δ_{Y}}}) \to (Y, δ_{Y})$ is LF-proximity map.

Therefore, (ϒ, Ω) is a Galois correspondence.□

5 Conclusion

In this paper, L-fuzzy pre-proximities and L-fuzzy interior operators in complete residuated lattice are investigated. It is also shown that there is a Galois correspondence between the category of (separated) L-fuzzy interior spaces and that of (separated) L-fuzzy pre-proximity spaces. As extension of Pawlak’s rough set [31], we give Example 4.4 as a viewpoint of the topological structure for fuzzy information and fuzzy rough sets in a complete residuated lattice.

In the future, by using the concepts of L-fuzzy pre-proximity spaces, information systems and decision rules with a view point of applications to multi-attribute decision-making are investigated in residuated lattices [40].

References

Adámek

, Herrlich

, Strecker

G.E.

, Abstract and Concrete Categories, Wiley, New York, 1990.

Artico

and Moresco

, Fuzzy proximities and totally bounded fuzzy uniformities, J Math Anal Appl 99 (1984), 320–337.

Bělohlávek

, Fuzzy Relational Systems, Kluwer Academic Publishers, New York, 2002.

Bělohlávek

and Funiokova

, Fuzzy interior operators, Int J Gen Syst 33(4) (2004), 415–430.

Bělohlávek

and Krupka

, Central points and approximation in residuated lattices,, Int J Approx Reason 66 (2015), 27–38.

Chang

C.L.

, Fuzzy topological spaces,, J Math Anal Appl 24 (1968), 182–190.

Čimoka

and Šostak

A.P.

, L-fuzzy syntopogenous structures, Part I: Fundamentals and application to L-fuzzy topologies, L-fuzzy proximities and L-fuzzy uniformities, }, Fuzzy Sets and Systems 232 (2013), 74–97.

El-Dardery

, Ramadan

A.A.

and Kim

Y.C.

, L-fuzzy topogenous orders and L-fuzzy topologies, , Journal of Intelligent and Fuzzy Systems 24 (2013), 601–609.

Fang

, The relationship between L-ordered convergence structures and strong L-topologies,, Fuzzy Sets Syst 161 (2010), 2923–2944.

10.

Fang

and Yue

, L-fuzzy closure systems, Fuzzy sets Syst 161 (2010), 1242–1252.

11.

Goguen

J.A.

, L-fuzzy sets, J Math Anal Appl (1967), 145–174.

12.

Hájek

, Metamathematices of Fuzzy Logic, Kluwer Academic Publishers, Dordrecht, 1998.

13.

Hao

and Li

, The relationship between L-fuzzy rough set and L-topology, Fuzzy Sets Syst 178 (2011), 74–83.

14.

Höhle

and Rodabaugh

S.E.

, Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, The Handbooks of Fuzzy Sets Series 3, Kluwer Academic Publishers, Boston, 1999.

15.

Hutton

, Uniformities in fuzzy topological spaces, J Math Anal Appl 58 (1977), 74–79.

16.

Katsaras

A.K.

, Fuzzy proximity spaces, J Math Anal Appl 68 (1979), 100–110.

17.

Katsaras

A.K.

and Petalas

C.G.

, A unified theory of fuzzy topologies, fuzzy proximities and fuzzy uniformities, Rev Roum Math Pures Appl 28 (1983), 845–896.

18.

Katsaras

A.K.

, On fuzzy syntopogenous structures, Rev Roum Math Pures Appl 30 (1985), 419–431.

19.

Katsaras

A.K.

, Fuzzy syntopogenous structures compatible with Lowen fuzzy uniformities and Artico-Moresco fuzzy proximities, Fuzzy Sets Syst 36 (1990), 375–393.

20.

Kim

Y.C.

, Categories of fuzzy preorders, approximation operators and Alexandrov topologies, J Int Fuzzy Syst 31 (2016), 1787–1793.

21.

Kim

Y.C.

and Min

K.C.

, L-fuzzy proximities and L-fuzzy topologies, Inf Sci 173 (2005), 93–113.

22.

Kortelainen

, On relationship between modified sets, topological space and rough sets, Fuzzy Sets Syst 61 (1994), 91–95.

23.

Zhang

, Zhan

J.M.

and Xu

, Covering-based generalized IF rough sets with applications to multi-attribute decision-making,, Inf Sci 478 (2019), 275–302.

24.

Liu

W.J.

, Fuzzy proximity spaces redefined, Fuzzy Sets Syst 15 (1985), 241–248.

25.

Lowen

, Fuzzy topological spaces and fuzzy compactness, J Math Anal Appl 56 (1976), 621–633.

26.

Pawlak

, Rough sets, Int J Comput Inf Sci 11 (1982), 341–356.

27.

Radzikowska

A.M.

, Kerre

E.E.

Transactions on Rough Sets II 3135 (2004), 278–296 Fuzzy rough sets based on residuated lattices, LNCS.

28.

Ramadan

A.A.

, Smooth topological spaces, Fuzzy Sets Syst 48 (1992), 371–375.

29.

Ramadan

A.A.

, L-fuzzy interior systems, Comp Math Appl 62 (2011), 4301–4307.

30.

Ramadan

A.A.

, Elkordy

E.H.

and Kim

Y.C.

, Perfect L-fuzzy topogenous spaces, L-fuzzy quasi-proximities and L-fuzzy quasi-uniform spaces, J Int Fuzzy Syst 28 (2015), 2591–2604.

31.

Ramadan

A.A.

, On L-fuzzy interior operators and L-fuzzy quasi-uniform spaces, J Int Fuzzy Syst 30 (2016), 3717–3752.

32.

Rodabaugh

S.E.

and Klement

E.P.

, Topological and Algebraic Structures In Fuzzy Sets, The Handbook of Recent Developments in the Mathematics of Fuzzy Sets, Kluwer Academic Publishers, Boston, Dordrecht, London, 2003.

33.

Shi

F.G.

and Pang

, Categories isomorphic to the category of L-fuzzy closure system spaces, Ir J Fuzzy Syst 10 (2013), 127–146.

34.

Šostak

, On a fuzzy topological structure, Suppl Rend Circ Mat Palermo Ser II 11 (1985), 89–103.

35.

Šostak

, Two decades of fuzzy topology: basic ideas, notions and results, Russ Math Surv 44 (1989), 125–186.

36.

Šostak

, Basic structures of fuzzy topology, J Math Sci 78(6) (1996), 662–701.

37.

Turunen

, Mathematics Behind Fuzzy Logic, A Springer-Verlag Co., Heidelberg, 1999.

38.

Wang

C.Y.

, Topological characterizations of generalized fuzzy rough sets, Fuzzy Sets Syst 312 (2017), 109–125.

39.

Ward

and Dilworth

R.P.

, Residuated lattices, Trans Amer Math Soc 45 (1939), 335–354.

40.

Xiu

and Li

, Degrees of L-Continuity for Mappings between L-Topological Spaces,, Mathematics 7 (2019), 1013.

41.

Yao

Y.Y.

and Lin

T.Y.

, Generalization of Rough Sets using Modal Logics, Int Aut Soft Comp 2(2) (1996), 103–19.

42.

Yue

and Shi

F.G.

, Generalization quasi-proximities, Fuzzy Sets Syst 158 (2007), 386–398.

43.

Zhang

, An enriched category approach to many valued topology, Fuzzy Sets Syst 158 (2007), 349–366.

44.

Zadeh

L.A.

, Fuzzy Sets, Inf Cont 8 (1965), 338–353.