Distance functions,upper approximation operators and Alexandrov fuzzy topologies

Abstract

In this paper, we introduce the notions of join preserving maps using distance spaces instead of fuzzy partially ordered sets on complete co-residuated lattices. We investigate the properties of Alexandrov fuzzy topologies, distance functions, join preserving maps and upper approximation operators. Furthermore, we study their relations and examples. We prove that there exist isomorphic categories and Galois correspondences between their categories.

Keywords

Complete co-residuated lattices distance functions join preserving maps upper approximation operators Alexandrov fuzzy topologies

1 Introduction

Ward et al. [34] introduced a complete residuated lattice which is an important mathematical tool as algebraic structures for many valued logics [4 , 28]. Bělohlávek [3, 4] investigated information systems and decision rules over complete residuated lattices. Höhle [11, 12] introduced L-fuzzy topologies with algebraic structure L (cqm, quantales, MV-algebra). Zheng et al. [43] introduced a complete co-residuated lattice as the generalization of t-conorm.

Pawlak [20, 21] introduced the rough set theory as a formal tool to deal with imprecision and uncertainty in the data analysis. For an extension of Pawlak’s rough sets, many researchers [2–4 , 28–31] developed fuzzy rough sets, L-lower and L-upper approximation operators in complete residuated lattices. Junsheng et al. [13] investigated (⊙ , &)-generalized fuzzy rough set on (L, ⊙ , &) where (L, &) is a complete residuated lattice [34] and (L, ⊙) is complete co-residuated lattice in a sense [43].

Zhang et al. [37 , 42] introduced the notion of fuzzy complete lattices using fuzzy partially order based on frames and residuated lattices as generalizations of usual complete lattices. Kim et al. [14 –16] studied the properties of fuzzy join and meet completeness, L-fuzzy upper and lower approximation spaces and Alexandrov L-topologies with fuzzy partially ordered spaces in complete residuated lattices. Ko et al. [17] introduced the concepts of fuzzy join and meet complete lattices using distance spaces instead of fuzzy partially ordered spaces in complete co-residuated lattices.

The goal of this paper is to study the distance functions instead of fuzzy partially ordered sets. We define Alexandrov fuzzy topologies, join preserving maps and upper approximation operators as the sense of a distance function.

This paper is organized as follows. In Section 2, we recall the definitions of complete co-residuated lattices, distance spaces and fuzzy complete. Moreover, we give their examples and properties. In Section 3, we investigate the properties of Alexandrov fuzzy topologies, distance functions, join preserving maps and upper approximation operators. Furthermore their relations and examples are studied. In Theorems 3.18–21, we show that there is a Galois correspondence between the category AFT of Alexandrov fuzzy topologies and the category RE of fuzzy relations. There is a Galois correspondence between the category AFT and the category JP of join preserving maps. Moreover, there is a Galois correspondence between the category AFT and the category AT of Alexandrov topologies. The category RE and the category JP are isomorphic. Finally, the category DIS of distance spaces and the category UP of upper approximation operators are isomorphic.

2 Preliminaries

Definition 2.1. [13, 43] An algebra (L, ∧ , ∨ , ⊕ , ⊥ , ⊤) is called a complete co-residuated lattice if it satisfies the following conditions:

(C1) L = (L, ∧ , ∨ , ⊥ , ⊤) is a complete lattice where ⊥ is the bottom element and ⊤ is the top element.

(C2) a = a⊕ ⊥, a ⊕ b = b ⊕ a and a ⊕ (b ⊕ c) = (a ⊕ b) ⊕ c for all a, b, c ∈ L.

(C3) (⋀ _i∈Γa_i) ⊕ b = ⋀ _i∈Γ (a_i ⊕ b) .

Let (L, ∧ , ∨ , ⊕ , ⊥ , ⊤) be a complete co-residuated lattice. For each x, y ∈ L, we define $x ⊖ y = ⋀ {z \in L ∣ y \oplus z \geq x} .$ Then (x ⊕ y) ≥ z iff x ≥ (z ⊖ y) .

For α ∈ L, A ∈ L^X, we denote (A ⊖ α) , (α ⊕ A) , α_X ∈ L^X as (A ⊖ α) (x) = A (x) ⊖ α, (α ⊕ A) (x) = α ⊕ A (x) , α_X (x) = α.

Put n (x) = ⊤ ⊖ x. The condition n (n (x)) = x for each x ∈ L is called a double negative law. For each x, y ∈ X, $⊥_{x} (y) = {\begin{matrix} ⊥, & if y = x, \\ ⊤, & if y \neq x, \end{matrix} n (⊥_{x}) (y) = {\begin{matrix} ⊤, & if y = x, \\ ⊥, & if y \neq x . \end{matrix}$

Remark 2.2. [17] (1) An infinitely distributive lattice (L, ∧ , ∨ , ⊕ = ∨ , 0, 1) is a complete co-residuated lattice. In particular, the unit interval ([0, 1] , ∨ , ∧ , ⊕ = ∨ , 0, 1) is a complete co-residuated lattice where $\begin{matrix} x ⊖ y = ⋀ {z \in L ∣ y \lor z \geq x} \\ = {\begin{matrix} 0, & if y \geq x, \\ x, & if y ≱ x . \end{matrix} \end{matrix}$ Put n (x) =1 ⊖ x = 1 for x ≠ 1 and n (1) =0. Then n (n (x)) =0 for x ≠ 1 and n (n (1)) =1. Hence n does not satisfy a double negative law.

(2) A t-conorm ([0, 1] , ≤ , ⊕) with (⋀ _i∈Γa_i) ⊕ b = ⋀ _i∈Γ (a_i ⊕ b) for a_i, b ∈ [0, 1] is a complete co-residuated lattice [3 , 28].

(3) ([1, ∞] , ∧ , ∨ , ⊕ = · , 1, ∞) is a complete co-residuated lattice where $\begin{matrix} x ⊖ y = ⋀ {z \in [1, \infty] ∣ yz \geq x} \\ = {\begin{matrix} 1, & if y \geq x, \\ \frac{x}{y}, & if y ≱ x . \end{matrix} \end{matrix}$ $\begin{matrix} \infty \cdot a = a \cdot \infty = \infty, \forall a \in [1, \infty], \infty ⊖ \infty = 1 . \end{matrix}$ Put n (x) =∞ ⊖ x = ∞ for x≠ ∞ and n (∞) =1. Then n (n (x)) =1 for x≠ ∞ and n (n (∞)) = ∞. Hence n does not satisfy a double negative law.

(4) ([0, ∞] , ∧ , ∨ , ⊕ = + , 0, ∞) is a complete co-residuated lattice where $\begin{matrix} y ⊖ x = ⋀ {z \in [0, \infty] ∣ x + z \geq y} \\ = ⋀ {z \in [0, \infty] ∣ z \geq - x + y} = (y - x) \lor 0, \\ \infty + a = a + \infty = \infty, \forall a \in [0, \infty], \infty ⊖ \infty = 0 . \end{matrix}$ Put n (x) =∞ ⊖ x = ∞ for x≠ ∞ and n (∞) =0. Then n (n (x)) =0 for x≠ ∞ and n (n (∞)) = ∞. Hence n does not satisfy a double negative law.

(5) ([0, 1] , ∧ , ∨ , ⊕ , 0, 1) is a complete co-residuated lattice where $\begin{matrix} x \oplus y = (x^{p} + y^{p})^{\frac{1}{p}} \land 1, 1 \leq p < \infty, \\ x ⊖ y = ⋀ {z \in [0, 1] ∣ (z^{p} + y^{p})^{\frac{1}{p}} \geq x} \\ = ⋀ {z \in [0, 1] ∣ z \geq (x^{p} - y^{p})^{\frac{1}{p}}} = (x^{p} - y^{p})^{\frac{1}{p}} \lor 0, \end{matrix}$ Put $n (x) = 1 ⊖ x = (1 - x^{p})^{\frac{1}{p}}$ for 1≤ p < ∞. Then n (n (x)) = x for x ∈ [0, 1]. Hence n satisfies a double negative law.

(6) Let P (X) be the collection of all subsets of X. Then (P (X) , ∩ , ∪ , ⊕ = ∪ , ∅ , X) is a complete co-residuated lattice where $\begin{matrix} A ⊖ B = ⋀ {C \in P (X) ∣ B \cup C \supset A} \\ = A \cap B^{c} = A - B . \end{matrix}$ Put n (A) = X ⊖ A = A^c for each A ⊂ X. Then n (n (A)) = A. Hence n satisfies a double negative law.

Lemma 2.3. [17] Let (L, ∧ , ∨ , ⊕ , ⊖ , ⊥ , ⊤) be a complete co-residuated lattice. For each x, y, z, x_i, y_i ∈ L, we have the following properties.

(1) If y ≤ z, then x ⊕ y ≤ x ⊕ z, y ⊖ x ≤ z ⊖ x and x ⊖ z ≤ x ⊖ y.

(2) (⋁ _i∈Γx_i) ⊖ y = ⋁ _i∈Γ (x_i ⊖ y) and x ⊖ (⋀ _i∈Γy_i) = ⋁ _i∈Γ (x ⊖ y_i) .

(3) (⋀ _i∈Γx_i) ⊖ y ≤ ⋀ _i∈Γ (x_i ⊖ y)

(4) x ⊖ (⋁ _i∈Γy_i) ≤ ⋀ _i∈Γ (x ⊖ y_i) .

(5) x⊖ x = ⊥, x ⊖ ⊥ = x and ⊥⊖ x = ⊥. Moreover, x⊖ y = ⊥ iff x ≤ y.

(6) y ⊕ (x ⊖ y) ≥ x, y ≥ x ⊖ (x ⊖ y) and (x ⊖ y) ⊕ (y ⊖ z) ≥ x ⊖ z.

(7) x ⊖ (y ⊕ z) = (x ⊖ y) ⊖ z = (x ⊖ z) ⊖ y .

(8) x ⊖ y ≥ (x ⊕ z) ⊖ (y ⊕ z), x ⊖ y ≥ (x ⊖ z) ⊖ (y ⊖ z), y ⊖ x ≥ (z ⊖ x) ⊖ (z ⊖ y) and (x ⊕ y) ⊖ (z ⊕ w) ≤ (x ⊖ z) ⊕ (y ⊖ w).

(9) x⊕ y = ⊥ iff x =⊥ and y =⊥.

(10) (x ⊕ y) ⊖ z ≤ x ⊕ (y ⊖ z) and (x ⊖ y) ⊕ z ≥ x ⊖ (y ⊖ z).

(11) If L satisfies a double negative law and n (x) =1 ⊖ x, then n (x ⊕ y) = n (x) ⊖ y = n (y) ⊖ x and x ⊖ y = n (y) ⊖ n (x). Moreover, n (⋀ _i∈Γx_i) = ⋁ _i∈Γn (x_i) and n (⋁ _i∈Γx_i) = ⋀ _i∈Γn (x_i).

Definition 2.4. [17] Let (L, ∧ , ∨ , ⊕ , ⊖ , ⊥ , ⊤) be a complete co-residuated lattice. Let X be a set. A function d_X : X × X → L is called a distance function if it satisfies the following conditions:

(M1) d_X (x, x) =⊥ for all x ∈ X,

(M2) d_X (x, y) ⊕ d_X (y, z) ≥ d_X (x, z), for all x, y, z ∈ X,

(M3) If d_X (x, y) = d_X (y, x) =⊥, then x = y.

The pair (X, d_X) is called a distance space.

Remark 2.5. [17] (1) We define a distance function d_X : X × X → [0, ∞]. Then (X, d_X) is called a pseudo-quasi-metric space.

(2) Let (L, ∧ , ∨ , ⊕ , ⊖ , 0, 1) be a complete co-residuated lattice. Define a function d_L : L × L → L as d_L (x, y) = x ⊖ y . By Lemma 2.3 (5) and (6), (L, d_L) is a distance space. Define a function d_{L
^X} : L^X × L^X → L as d_{L
^X} (A, B) = ⋁ _x∈X (A (x) ⊖ B (x)) . Then (L^X, d_{L
^X}) is a distance space.

Definition 2.6. [17] Let (X, d_X) be a distance space and A ∈ L^X.

(1) A point x₀ is called a fuzzy join of A, denoted by x₀ = ⊔ _XA, if it satisfies

(J1) A (x) ≥ d_X (x, x₀),

(J2) ⋁_x∈X (d_X (x, y) ⊖ A (x)) ≥ d_X (x₀, y).

The pair (X, d_X) is called fuzzy join complete if ⊔_XA exists for each A ∈ L^X.

A point x₁ is called a fuzzy meet of A, denoted by x₁ = ⊓ _XA, if it satisfies

(M1) A (x) ≥ d_X (x₁, x),

(M2) ⋁_x∈X (d_X (y, x) ⊖ A (x)) ≥ d_X (y, x₁).

The pair (X, d_X) is called fuzzy meet complete if ⊓_XA exists for each A ∈ L^X.

The pair (X, d_X) is called fuzzy complete if ⊓_XA and ⊔_XA exists for each A ∈ L^X.

Theorem 2.7. [17] Let (X, d_X) be a distance space and Φ ∈ L^X.

(1) A point x₀ is a fuzzy join of Φ iff ⋁_x∈X (d_X (x, y) ⊖ Φ (x)) = d_X (x₀, y).

(2) A point x₁ is a fuzzy meet of Φ iff ⋁_x∈X (d_X (y, x) ⊖ Φ (x)) = d_X (y, x₁).

(3) If ⊔_XΦ is a fuzzy join of Φ ∈ L^X, then it is unique. Moreover, if ⊓_XΦ is a fuzzy meet of Φ ∈ L^X, then it is unique.

Remark 2.8. Let (L^X, d_{L
^X}) be a function space and Φ ∈ L^{L
^X}.

(1) Since ⊔_{L
^X}Φ is a join of Φ iff for all Φ ∈ L^{L
^X}, $\begin{matrix} d_{L^{X}} (⊔_{L^{X}} Φ, B) = ⋁_{A \in L^{X}} (d_{L^{X}} (A, B) ⊖ Φ (A)) \\ = ⋁_{A \in L^{X}} d_{L^{X}} (A ⊖ Φ (A), B) \\ = d_{L^{X}} (⋁_{A \in L^{X}} (A ⊖ Φ (A)), B) \end{matrix}$ by Theorem 2.7(3), then ⊔_{L
^X}Φ = ⋁ _A∈L^X (A ⊖ Φ (A)) .

(2) Since ⊓Φ is a meet of Φ iff for all Φ ∈ L^{L
^X}, $\begin{matrix} d_{L^{X}} (B, ⊓_{L^{X}} Φ) = ⋁_{A \in L^{X}} (d_{L^{X}} (B, A) ⊖ Φ (A)) \\ = ⋁_{A \in L^{X}} d_{L^{X}} (B, Φ (A) \oplus A) \\ = d_{L^{X}} (B, ⋀_{A \in L^{X}} (Φ (A) \oplus A)) \end{matrix}$ By Theorem 2.7(3), ⊓_{L
^X}Φ = ⋀ _A∈L^X (Φ (A) ⊕ A).

3 Distance functions, upper approximation operators and Alexandrov fuzzy topologies

In this section, we assume (L, ∧ , ∨ , ⊕ , ⊖ , ⊥ , ⊤ , n) is a complete coresiduated lattice with a double negation law.

Definition 3.1. Let $J : L^{X} \to L^{Y}$ be a map. Define $J^{*} : L^{L^{X}} \to L^{L^{Y}}$ as $\begin{matrix} J^{*} (Φ) (B) = {\begin{matrix} ⊤, & if J^{\leftarrow} (B) = \emptyset, \\ ⋀_{J (D) = B} Φ (D), & if J^{\leftarrow} (B) \neq \emptyset, \end{matrix} \end{matrix}$ where $J^{\leftarrow} (B) = {D \in L^{X} ∣ J (D) = B} .$

A map $J : L^{X} \to L^{Y}$ is called a join preserving map iff $J (⊔_{L^{X}} Φ) = ⊔_{L^{Y}} J^{*} (Φ)$ for each Φ ∈ L^{L
^X}.

Theorem 3.2. A map $J : L^{X} \to L^{Y}$ is a join preserving map iff it satisfies the following conditions

(J1) $J (A ⊖ α) = J (A) ⊖ α$ , for all A ∈ L^X, and α ∈ L.

(J2) $J (⋁_{i \in I} A_{i}) = ⋁_{i \in I} J (A_{i})$ , for all A_i ∈ L^X.

Proof (⇒) Since $J (⊔_{L^{X}} Φ) = ⊔_{L^{X}} J^{*} (Φ)$ where ⊔_{L
^X}Φ = ⋁ _A∈L^X (A ⊖ Φ (A)) ∈ L^X and $⊔_{L^{Y}} J^{*} (Φ) = ⋁_{C \in L^{Y}} (C ⊖ J^{*} (Φ) (C)) \in L^{Y}$ from Remark 2.8, $J (⋁_{A \in L^{X}} (A ⊖ Φ (A))) = ⋁_{C \in L^{Y}} (C ⊖ J^{*} (Φ) (C))$ .

Define Φ₁ : L^X → L as Φ₁ (A) = α and Φ₁ (B) =⊤, otherwise. Then $(⊔_{L^{X}} Φ_{1}) (x) = ⋁_{D \in L^{X}} (D (x) ⊖ Φ_{1} (D)) = A (x) ⊖ α .$ Since $J^{*} (Φ_{1}) (B) = ⋀_{B = J (A)} Φ_{1} (A)$ and $J (⊔_{L^{X}} Φ_{1}) = ⊔_{L^{Y}} J^{*} (Φ_{1})$ for all Φ₁ ∈ L^{L
^X}, we have $\begin{matrix} ⊔_{L^{Y}} J^{*} (Φ_{1}) (x) = ⋁_{C \in L^{Y}} (C (x) ⊖ J^{*} (Φ_{1}) (C)) \\ = ⋁_{C \in L^{Y}} (C (x) ⊖ ⋀_{C = J (A) \in L^{Y}} Φ_{1} (A)) \\ = J (A) (x) ⊖ Φ_{1} (A) = J (A) (x) ⊖ α \\ = J (⊔_{L^{X}} Φ_{1}) (x) = J (A ⊖ α) (x) . \end{matrix}$ Hence $J (A) ⊖ α = J (A ⊖ α)$ .

Let {A_i ∈ L^X ∣ i ∈ Γ} be given. Define Φ₂ : L^X → L as Φ₂ (A_i) =⊥ for i ∈ Γ and Φ₂ (B) =⊤, otherwise. Then $(⊔_{L^{X}} Φ_{2}) (x) = ⋁_{A \in L^{X}} (A (x) ⊖ Φ_{2} (A)) = ⋁_{i \in Γ} A_{i} (x) .$ Since $J^{*} (Φ_{2}) (B) = ⋀_{B = J (A)} Φ_{2} (A)$ and $J (⊔_{L^{X}} Φ_{2}) = ⊔_{L^{Y}} J^{*} (Φ_{2})$ for Φ₂ ∈ L^{L
^X}, we have $\begin{matrix} J (⊔ Φ_{2}) (y) = J (⋁_{i \in Γ} A_{i}) (y), \\ ⊔_{L^{Y}} J^{*} (Φ_{2}) (y) = ⋁_{B \in L^{Y}} (B (y) ⊖ J^{*} (Φ_{2}) (B)) \\ = ⋁_{B \in L^{Y}} (B (y) ⊖ ⋀_{B = J (A)} Φ_{2} (A)) \\ = ⋁_{A \in L^{X}} (J (A) (y) ⊖ Φ_{2} (A)) \\ = ⋁_{i \in Γ} J (A_{i}) (y) . \end{matrix}$ Hence $J (⋁_{i \in Γ} A_{i}) = ⋁_{i \in Γ} J (A_{i})$ .

(⇐) For all Φ ∈ L^{L
^X}, $\begin{matrix} d_{L^{Y}} (⊔_{L^{Y}} J^{*} (Φ), B) \\ = ⋁_{C \in L^{Y}} (d_{L^{Y}} (C, B) ⊖ J^{*} (Φ) (C)) \\ = ⋁_{C \in L^{Y}} (d_{L^{Y}} (C, B) ⊖ ⋀_{J (A) = C} Φ (A)) \\ (by Lemma 2.3 (2)) \\ = ⋁_{A \in L^{X}} (d_{L^{Y}} (J (A), B) ⊖ Φ (A)) \\ = ⋁_{A \in L^{X}} d_{L^{Y}} (J (A) ⊖ Φ (A)), B) \\ = d_{L^{Y}} (⋁_{A \in L^{X}} (J (A) ⊖ Φ (A)), B) . \end{matrix}$ Hence $J (⊔_{L^{X}} Φ) = ⊔_{L^{Y}} J^{*} (Φ)$ from: $\begin{matrix} ⊔_{L^{Y}} J^{*} (Φ) (y) = ⋁_{A \in L^{X}} (J (A) ⊖ Φ (A)) (y) \\ = J (⋁_{A \in L^{X}} (A ⊖ Φ (A))) (y) (by (J 1) and (J 2)) \\ = J (⊔_{L^{X}} Φ) (y) . \end{matrix}$

Definition 3.3. A join preserving map $J : L^{X} \to L^{X}$ is called an upper approximation operator iff it satisfies the following condition:

(U) $J (A) \geq A$ and $J (J (A)) = J (A)$ for each A ∈ L^X.

Example 3.4. Let d ∈ L^X×X be a fuzzy relation. Define $J_{d} : L^{X} \to L^{X}$ as follows $J_{d} (A) (y) = ⋁_{x \in X} (A (x) ⊖ d (x, y)) .$

(1) Since $J_{d} (A ⊖ α) (y) = ⋁_{x \in X} ((A ⊖ α) (x) ⊖ d (x, y)) = ⋁_{x \in X} (A (x) ⊖ d (x, y)) ⊖ α = (J_{d} (A) ⊖ α) (y)$ and $J_{d} (⋁_{i \in Γ} A_{i}) = ⋁_{i \in Γ} J_{d} (A_{i})$ from Lemma 2.3(2,7), by Theorem 3.2, $J_{d}$ is a join preserving map.

(2) If d is a distance function, then $J_{d}$ is an upper approximation operator from the followings: $\begin{matrix} ⋀_{y \in X} (d (x, y) \oplus d (y, z)) = d (x, z), \\ J_{d} (A) (y) = ⋁_{x \in X} (A (x) ⊖ d (x, y)) \\ \geq A (y) ⊖ d (y, y) = A (y), \\ J_{d} (J_{d} (A)) (z) = ⋁_{y \in X} (J_{d} (A) (y) ⊖ d (y, z)) \\ = ⋁_{y \in X} ((⋁_{x \in X} (A (x) ⊖ d (x, y))) ⊖ d (y, z)) \\ (by Lemma 2.3 (7)) \\ = ⋁_{x \in X} (A (x) ⊖ ⋀_{y \in X} (d (x, y) \oplus d (y, z))) \\ = ⋁_{x \in X} (A (x) ⊖ d (x, z)) = J_{d} (A) (z) . \end{matrix}$

(3) Define d (x, y) =⊥ for each x, y ∈ X. Then $J_{d}$ is an upper approximation operator with $J_{d} (A) (y) = ⋁_{x \in X} A (x) .$

(4) Define a distance function d as $d (x, y) = {\begin{matrix} ⊥, & if x = y, \\ ⊤, & if x \neq y . \end{matrix}$ Then $J_{d}$ is an upper approximation operator with $J_{d} (A) (y) = A (y) .$

Theorem 3.5. Let $J : L^{X} \to L^{X}$ be an upper approximation operator. If $J^{- 1}$ is a join preserving map such that $J^{- 1} (n (⊥_{x})) (y) = J (n (⊥_{y})) (x)$ for all x, y ∈ X, then $J^{- 1}$ is an upper approximation operator.

Proof. Since A = ⋁ _x∈X (A (x) ⊖ ⊥ _x) = ⋁ _x∈X (n (⊥ _x) ⊖ n (A) (x)) from Lemma 2.3(11), we have

(U) Since $n (J^{- 1} (n (⊥_{x}))) (x) = n (J (n (⊥_{x})) (x) \leq ⊥_{x} (x) = ⊥$ , we have $\begin{matrix} J^{- 1} (A) (x) = J^{- 1} (⋁_{y \in X} (n (⊥_{y}) ⊖ n (A) (y)) (x) \\ = ⋁_{y \in X} (J^{- 1} (n (⊥_{y})) (x) ⊖ n (A) (y)) \\ = ⋁_{y \in X} (A (y) ⊖ n (J^{- 1} (n (⊥_{y}))) (x)) \\ \geq A (x) ⊖ J^{- 1} (⊥_{x}) (x) = A (x) . \end{matrix}$ For all y, z ∈ X, $\begin{matrix} J^{- 1} (J^{- 1} (n (⊥_{y}))) (z) \\ = J^{- 1} (⋁_{x \in X} (n (⊥_{x}) ⊖ n (J^{- 1} (n (⊥_{y}))) (x))) (z) \\ = ⋁_{x \in X} (J^{- 1} (n (⊥_{x})) (z) ⊖ n (J^{- 1} (n (⊥_{y}))) (x)) \\ = ⋁_{x \in X} (J (n (⊥_{z})) (x) ⊖ n (J (n (⊥_{x}))) (y)) \\ = ⋁_{x \in X} (J (n (⊥_{x})) (y) ⊖ n (J (n (⊥_{z}))) (x)) \\ = J (⋁_{x \in X} (n (⊥_{x}) ⊖ n (J (n (⊥_{z}))) (x)) (y) \\ = J (J (n (⊥_{z}))) (y) \\ = J (n (⊥_{z})) (y) = J^{- 1} (n (⊥_{y})) (z), \end{matrix}$ For all A ∈ L^X, z ∈ X, $\begin{matrix} J^{- 1} (J^{- 1} (A)) (z) \\ = J^{- 1} (J^{- 1} (⋁_{x \in X} (n (⊥_{x}) ⊖ n (A) (x))) (z) \\ = J^{- 1} (⋁_{x \in X} (J^{- 1} (n (⊥_{x})) ⊖ n (A) (x))) (z) \\ = ⋁_{x \in X} (J^{- 1} (J^{- 1} (n (⊥_{x}))) (z) ⊖ n (A) (x)) \\ = ⋁_{x \in X} (J^{- 1} (n (⊥_{x})) (z) ⊖ n (A) (x)) \\ = J^{- 1} (⋁_{x \in X} (n (⊥_{x}) ⊖ n (A) (x))) (z) \\ = J^{- 1} (A) (z) . \end{matrix}$

Lemma 3.6. Let (L, ∧ , ∨ , ⊕ , ⊖ , ⊥ , ⊤) be a complete co-residuated lattice. For each x_i, y_i ∈ L, we have the following properties.

(1) (⋁ _i∈Γx_i) ⊖ (⋁ _i∈Γy_i) ≤ ⋁ _i∈Γ (x_i ⊖ y_i).

(2) (⋀ _i∈Γx_i) ⊖ (⋀ _i∈Γy_i) ≤ ⋁ _i∈Γ (x_i ⊖ y_i).

Proof. (1) Since x_i ≤ (x_i ⊖ y_i) ⊕ y_i ≤ ⋁ _i∈Γ (x_i ⊖ y_i) ⊕ (⋁ _i∈Γy_i), then ⋁_i∈Γx_i ≤ ⋁ _i∈Γ (x_i ⊖ y_i) ⊕ (⋁ _i∈Γy_i). Hence (⋁ _i∈Γx_i) ⊖ (⋁ _i∈Γy_i) ≤ ⋁ _i∈Γ (x_i ⊖ y_i).

(2) Since x_i ≤ (x_i ⊖ y_i) ⊕ y_i, then ⋀_i∈Γx_i ≤ (x_i ⊖ y_i) ⊕ y_i iff x_i ⊖ y_i ≥ (⋀ _i∈Γx_i) ⊖ y_i. Thus ⋁_i∈Γ (x_i ⊖ y_i) ≥ ⋁ _i∈Γ ((⋀ _i∈Γx_i) ⊖ y_i) = (⋀ _i∈Γx_i) ⊖ (⋀ _i∈Γy_i) .

Theorem 3.7. Let $J, J^{- 1} : L^{X} \to L^{X}$ be join preserving maps such that $J^{- 1} (⊥_{x}) (y) = J (⊥_{y}) (x)$ for all x, y ∈ X. Then we have the following properties.

(1) For all y, z ∈ X, $\begin{matrix} ⋁_{A \in L^{X}} (J (A) (y) ⊖ J (A) (z)) \\ = ⋁_{x \in X} (J (n (⊥_{x})) (y) ⊖ J (n (⊥_{x})) (z)), \end{matrix}$ $\begin{matrix} ⋁_{A \in L^{X}} (J^{- 1} (A) (y) ⊖ J^{- 1} (A) (z)) \\ = ⋁_{x \in X} (J^{- 1} (n (⊥_{x})) (y) ⊖ J^{- 1} (n (⊥_{x})) (z)) \\ = ⋁_{x \in X} (J (n (⊥_{y})) (x) ⊖ J (n (⊥_{z})) (x)) . \end{matrix}$

(2) If $J$ and $J^{- 1}$ are upper approximation operators, then, for all y, z ∈ X, $\begin{matrix} ⋁_{A \in L^{X}} (J (A) (y) ⊖ J (A) (z)) \\ = ⋁_{x \in X} (J (n (⊥_{x})) (y) ⊖ J (n (⊥_{x})) (z)) \\ = ⋁_{x \in X} (n (J (n (⊥_{x}))) (z) ⊖ n (J (n (⊥_{x}))) (y)) \\ = n (J (n (⊥_{y}))) (z), \end{matrix}$ $\begin{matrix} ⋁_{A \in L^{X}} (J^{- 1} (A) (y) ⊖ J^{- 1} (A) (z)) \\ = ⋁_{x \in X} (J^{- 1} (n (⊥_{x})) (y) ⊖ J^{- 1} (n (⊥_{x})) (z)) \\ = ⋁_{x \in X} (n (J^{- 1} (n (⊥_{x}))) (z) ⊖ n (J^{- 1} (n (⊥_{x}))) (y)) \\ = ⋁_{x \in X} (n (J (n (⊥_{z}))) (x) ⊖ n (J (n (⊥_{y}))) (x)) \\ = n (J^{- 1} (n (⊥_{y}))) (z) = n (J (n (⊥_{z}))) (y) . \end{matrix}$

Proof. (1) Since A = ⋁ _x∈X (A (x) ⊖ ⊥ _x) = ⋁ _x∈X (n (⊥ _x) ⊖ n (A) (x)) from Lemma 2.3(11), we have $\begin{matrix} J (A) (y) ⊖ J (A) (z) \\ = J (⋁_{x \in X} (n (⊥_{x}) ⊖ n (A) (x))) (y) \\ ⊖ J (⋁_{x \in X} (n (⊥_{x}) ⊖ n (A) (x))) (z) \\ = ⋁_{x \in X} (J (n (⊥_{x})) (y) ⊖ n (A) (x))) \\ ⊖ (⋁_{x \in X} (J (n (⊥_{x})) (z) ⊖ n (A) (x))) \\ (Lemma 3.6 (1)) \\ \leq ⋁_{x \in X} ((J (n (⊥_{x})) (y) ⊖ n (A) (x)) \\ ⊖ (J (n (⊥_{x})) (z) ⊖ n (A) (x))) \\ \leq ⋁_{x \in X} (J (n (⊥_{x})) (y) ⊖ J (n (⊥_{x})) (z)) \\ (Lemma 2.3 (8)) \end{matrix}$ Other case is similarly proved.

(2) For all y, z ∈ X, $\begin{matrix} J (n (⊥_{y})) (z) = J (J (n (⊥_{y}))) (z) \\ = J (⋁_{x \in X} (n (⊥_{x}) ⊖ n (J (n (⊥_{y}))) (x))) (z) \\ = ⋁_{x \in X} (J (n (⊥_{x})) (z) ⊖ n (J (n (⊥_{y}))) (x)) . \end{matrix}$ Since $J (n (⊥_{y})) (z) \oplus n (J (n (⊥_{y}))) (x) \geq J (n (⊥_{x})) (z)$ , $\begin{matrix} ⋁_{y \in X} (n (J (n (⊥_{y}))) (z) ⊖ n (J (n (⊥_{y}))) (x)) \\ \leq n (J (n (⊥_{x}))) (z) . \end{matrix}$ Moreover, $\begin{matrix} ⋁_{y \in X} (n (J (n (⊥_{y}))) (z) ⊖ n (J (n (⊥_{y}))) (x)) \\ \geq n (J (n (⊥_{x}))) (z) ⊖ n (J (n (⊥_{x}))) (x) \\ = n (J (n (⊥_{x}))) (z), \end{matrix}$ $\begin{matrix} ⋁_{A \in L^{X}} (J (A) (y) ⊖ J (A) (z)) \\ = ⋁_{x \in X} (J (n (⊥_{x})) (y) ⊖ J (n (⊥_{x})) (z)) \\ = ⋁_{x \in X} (n (J (n (⊥_{x}))) (z) ⊖ n (J (n (⊥_{x}))) (y)) \\ = n (J (n (⊥_{y}))) (z) . \end{matrix}$ Other case is similarly proved.

Example 3.8. Let d ∈ L^X×X be a fuzzy relation. We obtain join preserving maps $J_{d}, J_{d}^{- 1} : L^{X} \to L^{X}$ as follows $\begin{matrix} J_{d} (A) (y) = ⋁_{x \in X} (A (x) ⊖ d (x, y)), \\ J_{d}^{- 1} (A) (y) = J_{d^{- 1}} (A) (y) = ⋁_{x \in X} (A (x) ⊖ d^{- 1} (x, y)) \\ = ⋁_{x \in X} (A (x) ⊖ d (y, x)) . \end{matrix}$

(1) Since $J_{d} (n (⊥_{x})) (y) = n (d (x, y))$ and

$J_{d^{- 1}} (n (⊥_{x})) (y) = n (d (y, x))$ , $\begin{matrix} ⋁_{A \in L^{X}} (J_{d} (A) (y) ⊖ J_{d} (A) (z)) \\ = ⋁_{x \in X} (J_{d} (n (⊥_{x})) (y) ⊖ J_{d} (n (⊥_{x})) (z)) \\ = ⋁_{x \in X} (n (d (x, y)) ⊖ n (d (x, z))) \\ = ⋁_{x \in X} (d (x, z) ⊖ d (x, y)), \\ ⋁_{A \in L^{X}} (J_{d}^{- 1} (A) (x) ⊖ J_{d}^{- 1} (A) (z)) \\ = ⋁_{x \in X} (J_{d}^{- 1} (n (⊥_{x})) (y) ⊖ J_{d}^{- 1} (n (⊥_{x})) (z)) \\ = ⋁_{x \in X} (n (d (y, x)) ⊖ n (d (z, x))) \\ = ⋁_{x \in X} (d (z, x) ⊖ d (y, x)) . \end{matrix}$

(2) If d is a distance function, then d^-1 is a distance function. Moreover, $J_{d}$ and $J_{d}^{- 1}$ are upper approximation operator such that $\begin{matrix} ⋁_{A \in L^{X}} (J_{d} (A) (y) ⊖ J_{d} (A) (z)) \\ = ⋁_{x \in X} (J_{d} (n (⊥_{x})) (y) ⊖ J_{d} (n (⊥_{x})) (z)) \\ = ⋁_{x \in X} (n (J_{d} (n (⊥_{x}))) (z) ⊖ n (J_{d} (n (⊥_{x}))) (y)) \\ = ⋁_{x \in X} (n (J_{d} (n (⊥_{x}))) (z) ⊖ n (J_{d} (n (⊥_{x}))) (y)) \\ = ⋁_{x \in X} (d (x, z) ⊖ d (x, y)) = d (y, z) \\ = n (J_{d} (n (⊥_{y}))) (z), \end{matrix}$ $\begin{matrix} ⋁_{A \in L^{X}} (J_{d}^{- 1} (A) (y) ⊖ J_{d}^{- 1} (A) (z)) \\ = ⋁_{x \in X} (J_{d}^{- 1} (n (⊥_{x})) (y) ⊖ J_{d}^{- 1} (n (⊥_{x})) (z)) \\ = ⋁_{y \in X} (n (d (y, x)) ⊖ n (d (z, x))) \\ = ⋁_{y \in X} (d (z, x) ⊖ d (y, x)) \\ = d (z, y) = n (J_{d} (n (⊥_{z}))) (y) . \end{matrix}$

Definition 3.9. (1) A map T : L^X → L is called an Alexandrov fuzzy topology on X iff it satisfies the following conditions:

(T1) T (α_X) =⊥, for each α ∈ L,

(T2) T (⋀ _i∈ΓA_i) ≤ ⋁ _i∈ΓT (A_i) and T (⋁ _i∈ΓA_i) ≤ ⋁ _i∈ΓT (A_i),

(T3) T (α ⊕ A) ≤ T (A),

(T4) T (A ⊖ α) ≤ T (A).

(2) A subset τ ⊂ L^X is called an Alexandrov topology on X iff it satisfies the following conditions:

(A1) α_X ∈ τ.

(A2) If A_i ∈ τ for all i ∈ I, then ⋁_i∈IA_i, ⋀ _i∈IA_i ∈ τ.

(A3) If A ∈ τ and α ∈ L, then A ⊖ α, A ⊕ α ∈ τ.

Theorem 3.10. Let T : L^X → L be an Alexandrov fuzzy topology.

(1) Define Tⁿ (A) = T (n (A)). Then Tⁿ is an Alexandrov fuzzy topology.

(2) Define τ_T = {A ∈ L^X ∣ T (A) = ⊥}. Then τ_T is an Alexandrov topology.

Proof. (1) (T1) Tⁿ (α_X) = T (n (α_X)) = ⊥ .

(T2) For each A_i ∈ L^X, i ∈ I, by Lemma 2.3(11), $\begin{matrix} T^{n} (⋁_{i \in Γ} A_{i}) = T (n (⋁_{i \in Γ} A_{i})) = T (⋀_{i \in Γ} n (A_{i})) \\ \leq ⋁_{i \in Γ} T (n (A_{i})) = ⋁_{i \in Γ} T^{n} (A_{i}), \\ T^{n} (⋀_{i \in Γ} A_{i}) = T (⋁_{i \in Γ} n (A_{i})) \\ \leq ⋁_{i \in Γ} T (n (A_{i})) = ⋁_{i \in Γ} T^{n} (A_{i}) . \end{matrix}$

(T3) For each A ∈ L^X and α ∈ L, by Lemma 2.3(11), $\begin{matrix} T^{n} (α \oplus A) & = T (n (α \oplus A)) = T (n (A) ⊖ α) \\ \leq T (n (A)) = T^{n} (A) . \end{matrix}$

(T4) For each A ∈ L^X and α ∈ L, by Lemma 2.3(11), $\begin{matrix} T^{n} (A ⊖ α) & = T (n (A ⊖ α)) = T (n (A) \oplus α) \\ \leq T (n (A)) = T^{n} (A) . \end{matrix}$

(2) (A1) By (T1), α_X ∈ τ_T.

(A2) If A_i ∈ τ_T for all i ∈ I, by (T2), then ⋁_i∈IA_i, ⋀ _i∈IA_i ∈ τ_T.

(A3) If A ∈ τ_T and α ∈ L, by (T3) and (T4), then A ⊖ α, A ⊕ α ∈ τ_T.

Theorem 3.11. Let $J : L^{X} \to L^{X}$ be a join preserving map. Define $T_{J} : L^{X} \to L$ as $T_{J} (A) = ⋁_{x \in X} (J (A) (x) ⊖ A (x)) = d_{L^{X}} (J (A), A) .$ Then we have the following properties.

(1) $T_{J}$ is an Alexandrov fuzzy topology on X.

(2) $T_{J} (A) = ⋁_{x, y \in X} ((A (x) ⊖ A (y)) ⊖ n (J (n (⊥_{x})) (y)))$ such that $T_{J} (A) \leq ⋁_{x \neq y \in X} J (n (⊥_{x})) (y)$ .

(3) If $J$ is an upper approximation operator, then $T_{J} (J (A)) = ⊥$ for each A ∈ L^X.

(4) If $J^{- 1}$ is a join preserving map such that $J^{- 1} (n (⊥_{x})) (y) = J (n (⊥_{y})) (x)$ for all x, y ∈ X. Define $T_{J}^{n} (A) = T_{J} (n (A))$ . Then $T_{J}^{n} = T_{J^{- 1}}$ is an Alexandrov fuzzy topology.

(5) If $J$ is an upper approximation operator, then $J^{- 1}$ is an upper approximation operator such that $T_{J} (n (J^{- 1} (A)) = T_{J^{- 1}} (J^{- 1} (A)) = ⊥ .$

Proof. (1) (T1) For α_X = ⋁ _y∈X (α (y) ⊖ ⊥ _y) = ⋁ _y∈X (n (⊥ _y) ⊖ n (α (y))), $\begin{matrix} J (α_{X}) (x) = J (⋁_{y \in X} (n (⊥_{y}) ⊖ n (α (y)))) (x) \\ = ⋁_{y \in X} (J (n (⊥_{y}) (x) ⊖ n (α (y)))) \\ = ⋁_{y \in X} (α ⊖ n (J (n (⊥_{y}))) (x)) \leq α . \end{matrix}$ Thus, $J (α_{X}) \leq α_{X}$ . By Lemma 2.3(5), $T_{J} (α_{X}) = ⊥ .$

(T2) $\begin{matrix} T_{J} (⋁_{i \in Γ} A_{i}) \\ = ⋁_{x} (J (⋁_{i \in Γ} A_{i}) (x) ⊖ ⋁_{i \in Γ} A_{i} (x)) \\ = ⋁_{x} (⋁_{i \in Γ} J (A_{i}) (x) ⊖ ⋁_{i \in Γ} A_{i} (x)) \\ \leq ⋁_{x} (⋁_{i \in Γ} (J (A_{i}) (x) ⊖ A_{i} (x))) \\ (by Lemma 3.6 (1)) \\ = ⋁_{i \in Γ} T_{J} (A_{i}) . \end{matrix}$ Since $J (⋀_{i \in Γ} A_{i}) \leq ⋀_{i \in Γ} J (A_{i})$ , we have $\begin{matrix} T_{J} (⋀_{i \in Γ} A_{i}) \\ = ⋁_{x} (J (⋀_{i \in Γ} A_{i}) (x) ⊖ ⋀_{i \in Γ} A_{i} (x)) \\ \leq ⋁_{x} (⋀_{i \in Γ} J (A_{i}) (x) ⊖ ⋀_{i \in Γ} A_{i} (x)) \\ \leq ⋁_{x} (⋁_{i \in Γ} (J (A_{i}) (x) ⊖ A_{i} (x))) \\ (by Lemma 3.6 (2)) \\ = ⋁_{i \in Γ} T_{J} (A_{i}) . \end{matrix}$

(T3) Since A ≥ (α ⊕ A) ⊖ α, $J (A) \geq J ((α \oplus A) ⊖ α) = J (α \oplus A) ⊖ α$ . Then $J (α \oplus A) \leq J (A) \oplus α$ . Thus, $\begin{matrix} T_{J} (α \oplus A) \\ = ⋁_{x} (J (α \oplus A) (x) ⊖ (α \oplus A (x))) \\ \leq ⋁_{x} ((α \oplus J (A) (x)) ⊖ (α \oplus A (x))) \\ (by Lemma 2.3 (8)) \\ \leq ⋁_{x} (J (A) (x) ⊖ A (x)) = T_{J} (A) . \end{matrix}$ $\begin{matrix} T_{J} (A ⊖ α) \\ = ⋁_{x} (J (A ⊖ α) (x) ⊖ (A (x) ⊖ α)) \\ = ⋁_{x} ((J (A) (x) ⊖ α) ⊖ (A (x) ⊖ α)) \\ (by Lemma 2.3 (8)) \\ \leq ⋁_{x} (J (A) (x) ⊖ A (x)) = T_{J} (A) . \end{matrix}$

(2) For A = ⋁ _x∈X (A (x) ⊖ ⊥ _x) = ⋁ _x∈X (n (⊥ _x) ⊖ n (A (x))), we have $\begin{matrix} T_{J} (A) = ⋁_{y \in X} (J (A) (y) ⊖ A (y)) \\ = ⋁_{y \in X} (J (⋁_{x \in X} (n (⊥_{x}) ⊖ n (A (x)))) (y) ⊖ A (y)) \\ = ⋁_{y \in X} (⋁_{x \in X} (J (n (⊥_{x})) (y) ⊖ n (A (x))) ⊖ A (y)) \\ = ⋁_{x, y \in X} (J (n (⊥_{x})) (y) ⊖ (n (A (x)) \oplus A (y))) \\ (by Lemma 2.3 (11)) \\ = ⋁_{x, y \in X} ((A (x) ⊖ A (y)) ⊖ n (J (n (⊥_{x})) (y))) . \end{matrix}$

$\begin{matrix} T_{J} (A) \\ = ⋁_{y \in X} (⋁_{x \in X} (J (n (⊥_{x})) (y) ⊖ n (A (x))) ⊖ A (y)) \\ = ⋁_{y \in X} (⋁_{x \in X} (J (n (⊥_{x})) (y) ⊖ n (A (x))) \\ ⊖ ⋁_{x \in X} (n (⊥_{x}) (y) ⊖ n (A (x)))) \\ \leq ⋁_{x, y \in X} ((J (n (⊥_{x})) (y) ⊖ n (A (x))) \\ ⊖ (n (⊥_{x}) (y) ⊖ n (A (x)))) (by Lemma 2.3 (8)) \\ \leq ⋁_{x, y \in X} ((J (n (⊥_{x})) (y) ⊖ n (⊥_{x}) (y)) \\ = ⋁_{x \neq y \in X} J (n (⊥_{x})) (y) . \end{matrix}$

(3) It follows from $T_{J} (J (A)) = ⋁_{x \in X}$ $(J (J (A)) (x) ⊖ J (A) (x)) = d_{L^{X}} (J (A), J (A)) = ⊥ .$

(4) By (2) and Lemma 2.3(11), $\begin{matrix} T_{J^{- 1}} (A) = ⋁_{x, y \in X} ((A (x) ⊖ A (y)) ⊖ n (J^{- 1} (n (⊤_{x}))) (y)) \\ = ⋁_{x, y \in X} ((n (A) (y) ⊖ n (A) (x)) ⊖ n (J (n (⊤_{y}))) (x)) \\ = T_{J} (n (A)) . \end{matrix}$

(5) By (3), $T_{J} (n (J^{- 1} (A)) = T_{J^{- 1}} (J^{- 1} (A)) = ⊥ .$

Example 3.12. Let d ∈ L^X×X be a fuzzy relation. By Example 3.4, $J_{d}$ is a join preserving map with $J_{d} (A) (y) = ⋁_{x \in X} (A (x) ⊖ d (x, y))$ . Define $T_{J_{d}} : L^{X} \to L$ as $T_{J_{d}} (A) = ⋁_{x \in X} (J_{d} (A) (x) ⊖ A (x)) = d_{L^{X}} (J_{d} (A), A) .$ From Theorem 3.11, we obtain the following results.

(1) $T_{J_{d}}$ is an Alexandrov fuzzy topology on X.

(2) Since $n (J_{d} (n (⊥_{x})) (y) = n (⋁_{x \in X} (n (⊥_{x}) (x) ⊖ d (x, y))) = n (⊤_{x} (x) ⊖ d (x, y)) = d (x, y)$ , by Theorem 3.11 (1), we have $T_{J_{d}} (A) = ⋁_{x, y \in X} ((A (x) ⊖ A (y)) ⊖ n (J_{d} (n (⊥_{x})) (y))) = ⋁_{x, y \in X} ((A (x) ⊖ A (y)) ⊖ d (x, y))$ such that $T_{J_{d}} (A) \leq ⋁_{x \neq y \in X} n (d (x, y))$ .

(3) If d is a distance function, then $J_{d}$ is an upper approximation operator such that $T_{J_{d}} (J_{d} (n (⊥_{x}))) = T_{J_{d}} (nd (x, -)) = ⊥$ for all x ∈ X.

(4) If $d^{- 1} (x, y) = d (y, x) = n (J (n (⊥_{y}))) (x)$ for all x, y ∈ X, then $J_{d^{- 1}} = J_{d}^{- 1}$ is a join preserving map. Define $T_{J_{d}}^{n} (A) = T_{J_{d}} (n (A))$ . Then $T_{J_{d}}^{n} = T_{J_{d^{- 1}}}$ is an Alexandrov fuzzy topology.

(5) If d is a distance function, then d^-1 is a distance function. Since $J_{d}^{- 1} (n (⊥_{x})) = n (d^{- 1} (x, -)) = n (d (-, x))$ and $J_{d} (n (⊥_{x})) = n (d (x, -))$ , then $J_{d}$ and $J_{d^{- 1}}$ are upper approximation operators such that $T_{J_{d}} (n (d (x, -))) = T_{J_{d^{- 1}}} (d (x, -)) = ⊥ .$

(6) Define d (x, y) =⊥ for each x, y ∈ X. Then $J_{d}$ is an upper approximation operator with $J_{d} (A) (y) = ⋁_{x \in X} (A (x)) .$ By (2), $T_{J_{d}} (A) = ⋁_{x, y \in X} (A (x) ⊖ A (y)) .$

(7) Define a distance function d as $d (x, y) = {\begin{matrix} ⊥, & if x = y, \\ ⊤, & if x \neq y . \end{matrix}$ Then $J_{d}$ is an upper approximation operator with $J_{d} (A) (y) = A (y) .$ $T_{J_{d}} (A) = ⋁_{x \in X} (J_{d} (A) (x) ⊖ A (x)) = ⋁_{x \in X} (A (x) ⊖ A (x)) = ⊥ .$

Theorem 3.13. Let $J, J^{- 1} : L^{X} \to L^{X}$ be join preserving maps. Then we have the following properties.

(1) $T_{J} (A) \geq T_{J^{- 1}} (α ⊖ A)$ and $T_{J^{- 1}} (A) \geq T_{J} (α ⊖ A) .$

(2) $T_{J} (A) \geq T_{J^{- 1}} (α ⊖ A) \geq T_{J^{- 1}} (β ⊖ (α ⊖ A))$ and $T_{J^{- 1}} (A) = T_{J} (n (A)) .$

Proof. (1) Since A (x) ⊖ A (y) ≥ (α ⊖ A (y)) ⊖ (α ⊖ A (x)) from Lemma 2.3(8), $\begin{matrix} T_{J} (A) = ⋁_{x, y \in X} ((A (x) ⊖ A (y)) ⊖ n (J (n (⊥_{x})) (y))) \\ \geq ⋁_{x, y \in X} (((α ⊖ A (y)) ⊖ (α ⊖ A (x))) \\ ⊖ n (J^{- 1} (n (⊥_{y})) (x))) = T_{J^{- 1}} (α ⊖ A) . \end{matrix}$

(2) By (1), $T_{J} (A) \geq (T_{J^{- 1}} (α ⊖ A) \geq T_{J} (β ⊖ (α ⊖ A))$ . Put α = β =⊥. Then $T_{J} (A) \geq T_{J^{- 1}} (n (A)) \geq T_{J} (n (n (A))) = T_{J} (A) .$

Theorem 3.14. Let $J, J^{- 1} : L^{X} \to L^{X}$ be join preserving maps.

(1) For all x, y ∈ X, $\begin{matrix} n (J (n (⊥_{x})) (y) \geq ⋁_{A \in L^{X}} ((A (x) ⊖ A (y)) ⊖ T_{J} (A)) \\ = ⋁_{A \in L^{X}} ((A (y) ⊖ A (x)) ⊖ T_{J^{- 1}} (A)) . \end{matrix}$

(2) If $J$ is an upper approximation operator, then, for all x, y ∈ L^X, $\begin{matrix} n (J (n (⊥_{x})) (y) = ⋁_{A \in L^{X}} ((A (x) ⊖ A (y)) ⊖ T_{J} (A)) \\ = ⋁_{z \in X} (n (J (n (⊥_{z}))) (y) ⊖ n (J (n (⊥_{z}))) (x)) \\ = ⋁_{z \in X} (J (n (⊥_{z}))) (x) ⊖ J (n (⊥_{z}))) (y)) \\ = ⋁_{A \in L^{X}} ((A (y) ⊖ A (x)) ⊖ T_{J^{- 1}} (A)) \\ = ⋁_{z \in X} (n (J^{- 1} (n (⊥_{z}))) (x) ⊖ n (J^{- 1} (n (⊥_{z}))) (y)) \end{matrix}$

(3) Define $d_{J} (x, y) = n (J (n (⊥_{x})) (y)$ for each x, y ∈ X. Then $J_{d_{J}} = J$ and $d_{J_{d}} = d$ . If $J$ is an upper approximation operator, then $d_{J}$ is a distance function.

Proof. (1) Since a ⊕ (b ⊖ a) ≥ b iff a ≥ b ⊖ (b ⊖ a), we have $\begin{matrix} n (J (n (⊥_{x})) (y) \\ \geq ⋁_{A \in L^{X}} ((A (x) ⊖ A (y)) ⊖ ((A (x) ⊖ A (y)) \\ ⊖ n (J (n (⊥_{x})) (y)))) \\ \geq ⋁_{A \in L^{X}} ((A (x) ⊖ A (y)) ⊖ ⋁_{x, y \in X} ((A (x) ⊖ A (y)) \\ ⊖ n (J (n (⊥_{x})) (y)))) (by Theorem 3.11 (2)) \\ = ⋁_{A \in L^{X}} ((A (x) ⊖ A (y)) ⊖ T_{J} (A)) \\ = ⋁_{A \in L^{X}} ((n (A) (y) ⊖ n (A) (x)) ⊖ T_{J^{- 1}} (n (A))) \end{matrix}$

(2) Since $T_{J} (J (n (⊥_{x})) = T_{J^{- 1}} (J^{- 1} (n (⊥_{x}))) = T_{J} (n (J^{- 1} (n (⊥_{x})))) = T_{J^{- 1}} (n (J (n (⊥_{x})))) = ⊥$ and $n (J (n (⊥_{x}))) (x) \leq ⊥_{x} (x) = ⊥$ , $\begin{matrix} n (J (n (⊥_{x})) (y) \geq ⋁_{A \in L^{X}} ((A (x) ⊖ A (y)) ⊖ T_{J} (A)) \\ \geq ⋁_{z \in X} ((J (n (⊥_{z})) (x) ⊖ J (n (⊥_{z})) (y)) ⊖ T_{J} (J (⊥_{z}))) \\ = ⋁_{z \in X} (n (J (n (⊥_{z}))) (y) ⊖ n (J (n (⊥_{z}))) (x)) \\ \geq n (J (n (⊥_{x}))) (y) ⊖ n (J (n (⊥_{x}))) (x) \\ = n (J (n (⊥_{x}))) (y), \end{matrix}$ $\begin{matrix} n (J (n (⊥_{x})) (y) \\ \geq ⋁_{A \in L^{X}} ((A (y) ⊖ A (x)) ⊖ T_{J^{- 1}} (A)) \\ \geq ⋁_{z \in X} ((J^{- 1} (n (⊥_{z})) (y) ⊖ J^{- 1} (n (⊥_{z})) (x)) \\ ⊖ T_{J^{- 1}} (J^{- 1} (⊥_{z}))) \\ = ⋁_{z \in X} (n (J^{- 1} (n (⊥_{z}))) (x) ⊖ n (J^{- 1} (n (⊥_{z}))) (y)) \\ \geq n (J^{- 1} (n (⊥_{y}))) (x) ⊖ n (J^{- 1} (n (⊥_{y}))) (y) \\ = n (J (n (⊥_{x}))) (y) . \end{matrix}$

(3) For each A ∈ L^X, y ∈ X, $\begin{matrix} J_{d_{J}} (A) (y) = ⋁_{x \in X} (A (x) ⊖ d_{J} (x, y)) \\ = ⋁_{x \in X} (A (x) ⊖ n (J (n (⊥_{x}))) (y)) \\ = ⋁_{x \in X} (J (n (⊥_{x})) (y) ⊖ n (A (x))) \\ = J (⋁_{x \in X} (n (⊥_{x}) ⊖ n (A (x)))) (y) \\ = J (A) (y) . \end{matrix}$

$\begin{matrix} d_{J_{d}} (x, z) = n (J_{d} (n (⊥_{x})) (z) \\ = n (⋁_{y \in X} (n (⊥_{y}) (z) ⊖ d (x, y))) = d (x, z) . \end{matrix}$

(D1) $d_{J} (x, x) = n (J (n (⊥_{x})) (x) \leq ⊥_{x} (x) = ⊥$ .

(D2) Since

$J (n (⊥_{x})) = ⋁_{y \in X} (n (⊥_{y}) ⊖ n (J (n (⊥_{x}))) (y))$ , $\begin{matrix} ⋁_{y \in X} (J (n (⊥_{y})) (z) ⊖ n (J (n (⊥_{x}))) (y)) \\ = J (⋁_{y \in X} (n (⊥_{y}) ⊖ n (J (n (⊥_{x}))) (y))) (z) \\ = J (J (n (⊥_{x}))) (z) = J (n (⊥_{x})) (z), \end{matrix}$ $\begin{matrix} d_{J} (x, z) = n (J (n (⊥_{x})) (z) \\ = n (⋁_{y \in X} (J (n (⊥_{y})) (z) ⊖ n (J (n (⊥_{x}))) (y))) \\ = ⋀_{y \in X} (n (J (n (⊥_{y}))) (z) \oplus n (J (n (⊥_{x}))) (y))) \\ = ⋀_{y \in X} (d_{J} (y, z) \oplus d_{J} (x, y)) . \end{matrix}$

Example 3.15. Let d ∈ L^X×X be a fuzzy relation and $J_{d}, J_{d^{- 1}}$ be join preserving maps in Example 3.4.

(1) For all x, y ∈ X, $\begin{matrix} d (x, y) = n (J_{d} (n (⊥_{x})) (y) \\ \geq ⋁_{A \in L^{X}} ((A (x) ⊖ A (y)) ⊖ T_{J_{d}} (A)) \\ = ⋁_{A \in L^{X}} ((A (y) ⊖ A (x)) ⊖ T_{J_{d^{- 1}}} (A)) . \end{matrix}$

(2) If d is a distance function, then for all x, y ∈ X, $\begin{matrix} d (x, y) = n (J_{d} (n (⊥_{x})) (y) \\ = ⋁_{A \in L^{X}} ((A (x) ⊖ A (y)) ⊖ T_{J_{d}} (A)) \\ = ⋁_{z \in X} (d (z, y) ⊖ d (z, x)) \end{matrix}$ $\begin{matrix} = ⋁_{A \in L^{X}} ((A (y) ⊖ A (x)) ⊖ T_{J_{d^{- 1}}} (A)) \\ = ⋁_{z \in X} (d (x, z) ⊖ d (y, z)) . \end{matrix}$

Theorem 3.16. Let T be an Alexandrov fuzzy topology on X. For all x, y ∈ X, define a fuzzy relation d_T (x, y) = ⋁ _A∈L^X ((A (x) ⊖ A (y)) ⊖ T (A)).

(1) Define T_{d
_T} : L^X → L as $T_{d_{T}} (A) = ⋁_{x, y \in X} ((A (x) ⊖ A (y)) ⊖ d_{T} (x, y)) .$ Then T_{d
_T} is an Alexandrov fuzzy topology with T_{d
_T} ≤ T and d_{T
_{d
_X}} ≤ d_X for each d_X ∈ L^X×X.

(2) If d is a distance function, then d_{T
_{d
_X}} = d_X.

(3) If $J_{T}$ is a join preserving map defined as $J_{T} (A) (y) = ⋁_{x \in X} (A (x) ⊖ d_{T} (x, y))$ , then $T_{J_{T}} \leq T$ .

(4) If $J$ is a join preserving map, then $J_{T_{J}} \leq J$ . Moreover, if $J$ is an upper approximation operator, then $J_{T_{J}} = J$ .

Proof. (1) Put $J_{d_{T}} (A) (y) = ⋁_{x \in X} (A (x) ⊖ d_{T} (x, y))$ . By Example 3.4(1), $J_{d_{T}}$ is a join preserving map. Thus, by Lemma 2.3(2,7), $\begin{matrix} T_{d_{T}} (A) = ⋁_{x, y \in L^{X}} ((A (x) ⊖ A (y)) ⊖ d_{T} (x, y)) \\ = ⋁_{x, y \in L^{X}} ((A (x) ⊖ d_{T} (x, y)) ⊖ A (y)) \\ = ⋁_{y \in L^{X}} (⋁_{x \in L^{X}} (A (x) ⊖ d_{T} (x, y)) ⊖ A (y)) \\ = ⋁_{y \in L^{X}} (J_{d_{T}} (A) (y) ⊖ A (y)) = T_{J_{d_{T}}} (A) . \end{matrix}$ By Theorem 3.11, T_{d
_T} is an Alexandrov fuzzy topology. Since a ⊕ (b ⊖ a) ≥ b iff a ≥ b ⊖ (b ⊖ a), by Example 3.12(2), $\begin{matrix} T_{d_{T}} (A) = ⋁_{x, y \in X} ((A (x) ⊖ A (y)) ⊖ d_{T} (x, y)) \\ = ⋁_{x, y \in X} ((A (x) ⊖ A (y)) ⊖ ⋁_{B \in L^{X}} ((B (x) ⊖ B (y)) \\ ⊖ T (B))) \\ \leq ⋁_{x, y \in X} ((A (x) ⊖ A (y)) ⊖ ((A (x) ⊖ A (y)) ⊖ T (A))) \\ \leq T (A), \end{matrix}$

$\begin{matrix} d_{T_{d_{X}}} (x, y) = ⋁_{A \in L^{X}} ((A (x) ⊖ A (y)) ⊖ T_{d_{X}} (A)) \\ = ⋁_{A \in L^{X}} ((A (x) ⊖ A (y)) ⊖ ⋁_{A \in L^{X}} ((A (x) ⊖ A (y)) \\ ⊖ d_{X} (x, y))) (by Lemma 3.6 (1)) \\ \leq ⋁_{A \in L^{X}} ((A (x) ⊖ A (y)) ⊖ ((A (x) ⊖ A (y)) \\ ⊖ d_{X} (x, y))) \leq d_{X} (x, y) . \end{matrix}$

(2) It follows from (1) and $\begin{matrix} d_{T_{d_{X}}} (x, y) = ⋁_{A \in L^{X}} ((A (x) ⊖ A (y)) ⊖ T_{d_{X}} (A)) \\ \geq ((nd (x, -) (x) ⊖ nd (x, -) (y)) ⊖ T_{d_{X}} (nd (x, -))) \\ = d_{X} (x, y) . \end{matrix}$

(3) In Proof (1), $J_{d_{T}} = J_{T}$ and $T_{J_{T}} = T_{d_{T}} \leq T$ .

(4) For each A ∈ L^X, y ∈ X, $\begin{matrix} J_{T_{J}} (A) (y) \\ = ⋁_{x \in X} (A (x) ⊖ d_{T_{J}} (x, y)) \\ = ⋁_{x \in X} (A (x) ⊖ ⋁_{A \in L^{X}} ((A (x) ⊖ A (y)) ⊖ T_{J} (A))) \\ \geq ⋁_{x \in X} (A (x) ⊖ n (J (n (⊥_{X}))) (y)) \\ (by Theorem 3.14 (1)) \\ = ⋁_{x \in X} (J (n (⊥_{X})) (y) ⊖ n (A) (x)) \\ = J (⋁_{x \in X} (n (⊥_{X})) (y) ⊖ n (A) (x)) \\ = J (A) (y), \end{matrix}$ For each A ∈ L^X, y ∈ X, $\begin{matrix} J_{T_{J}} (A) (y) \\ = ⋁_{x \in X} (A (x) ⊖ d_{T_{J}} (x, y)) \\ = ⋁_{x \in X} (A (x) ⊖ ⋁_{A \in L^{X}} ((A (x) ⊖ A (y)) ⊖ T_{J} (A))) \\ = ⋁_{x \in X} (A (x) ⊖ n (J (n (⊥_{X}))) (y)) \\ (by Theorem 3.14 (2)) \\ = J (⋁_{x \in X} (n (⊥_{X})) (y) ⊖ n (A) (x)) \\ = J (A) (y) . \end{matrix}$

Definition 3.17. [1, 9] Let $F : D \to C,$ $G : C \to D$ be functors.

(1) $C$ and $D$ are said to be isomorphic if $G \circ F = {id}_{D}$ and $F \circ G = {id}_{C}$ .

(2) (F, G) is called a Galois correspondence between $C$ and $D$ if for each $Y \in C,$ id_Y : F ∘ G (Y) → Y is a $C$ -morphism, and for each $X \in D$ , id_X : X → G ∘ F (X) is a $D$ -morphism.

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 .

Let AFT be a category with objects (X, T_X) and (Y, T_Y) where T_X and T_Y are Alexandrov fuzzy topologies with an A-morphism f : (X, T_X) → (Y, T_Y) such that T_X (f^← (B)) ≤ T_Y (B) for all B ∈ L^Y.

Let RE (resp. DIS) be a category with objects (X, d_X) and (Y, d_Y) where d_X and d_Y are fuzzy relations (resp. distance functions) with a D-morphism f : (X, d_X) → (Y, d_Y) such that d_Y (f (x) , f (y)) ≤ d_X (x, y) for all x, y ∈ X.

Let JP (resp. UA) be a category with objects $(X, J_{X})$ and $(Y, J_{Y})$ where $J_{X}$ and $J_{X}$ are join preserving maps (resp. upper approximation operators) with a J-morphism $f : (X, J_{X}) \to (Y, J_{Y})$ such that $f^{\leftarrow} (J_{Y} (B)) \geq J_{X} (f^{\leftarrow} (B))$ for all B ∈ L^Y.

Theorem 3.18. Θ : RE → AFT is a left adjoint of Φ : AFT → RE,i.e., (Θ, Φ) is a Galois correspondence.

Proof. Define Θ : RE → AFT as Θ (X, d_X) = (X, T_{d
_X}) where T_{d
_X} is an Alexandrov fuzzy topology in Example 3.12(2) defined as $T_{d_{X}} (A) = ⋁_{x, y \in X} ((A (x) ⊖ A (y)) ⊖ d_{X} (x, y)) .$ Let d_Y (f (x) , f (z)) ≤ d_X (x, z). Then f : (X, T_{d
_X}) → (Y, T_{d
_Y}) is an A-morphism from: $\begin{matrix} T_{d_{X}} (f^{\leftarrow} (B)) \\ = ⋁_{x, z \in X} ((f^{\leftarrow} (B) (x) ⊖ f^{\leftarrow} (B) (z)) ⊖ d_{X} (x, z)) \\ \leq ⋁_{x, z \in X} ((B (f (x)) ⊖ B (f (z))) ⊖ d_{Y} (f (x), f (z))) \\ \leq ⋁_{y, w \in X} ((B (y) ⊖ B (w)) ⊖ d_{Y} (y, w)) \\ = T_{d_{Y}} (B) . \end{matrix}$ Hence Θ is a functor.

Define a functor Φ : AFT → RE as Φ (X, T_X) = (X, d_{T
_X}) where d_{T
_X} (x, y) = ⋁ _A∈L^X ((A (x) ⊖ A (y)) ⊖ T_X (A)).

Let T_X (f^← (B)) ≤ T_Y (B). Then $\begin{matrix} d_{T_{Y}} (f (x), f (z)) \\ = ⋁_{B \in L^{Y}} ((B (f (x)) ⊖ B (f (y))) ⊖ T_{Y} (B)) \\ \leq ⋁_{B \in L^{Y}} ((f^{\leftarrow} (B) (x) ⊖ f^{\leftarrow} (B) (y)) ⊖ T_{X} (f^{\leftarrow} (B))) \\ \leq ⋁_{A \in L^{X}} ((A (x) ⊖ A (y)) ⊖ T_{X} (A)) \\ = d_{T_{X}} (x, z) . \end{matrix}$

Hence Φ is a functor. By Theorem 3.16(1), Θ (Φ (X, T_X)) = Θ (X, d_{T
_X}) = (X, T_{d
_{T
_X}}) and id_X : Θ (Φ (X, T_X)) = (X, T_{d
_{T
_X}}) → (X, T_X) is an A-morphism because T_{d
_{T
_X}} ≤ T_X. Moreover, by Theorem 3.16(1), Φ (Θ (X, d_X)) = Ψ (X, T_{d
_X}) = (X, d_{T
_{d
_X}}). id_X : (X, d_X) → Φ (Θ (X, d_X)) = (X, d_{T
_{d
_X}}) is a D-morphism because d_{T
_{d
_X}} ≤ d_X.

Theorem 3.19 Δ : JP → AFT is a left adjoint of Ψ : AFT → JP,i.e., (Δ, Ψ) is a Galois correspondence.

Proof. Define Δ : JP → AFT as $Δ (X, J_{X}) = (X, T_{J_{X}})$ where $T_{J_{X}}$ is an Alexandrov fuzzy topology in Theorem 3.11 defined as $T_{J_{X}} (A) = ⋁_{x \in X} (J_{X} (A) (x) ⊖ A (x)) .$ Let $f^{\leftarrow} (J_{Y} (B)) \geq J_{X} (f^{\leftarrow} (B))$ for all B ∈ L^Y. Then $f : (X, T_{J_{X}}) \to (Y, T_{J_{Y}})$ is an A-morphism from: $\begin{matrix} T_{J_{X}} (f^{\leftarrow} (B)) = ⋁_{x \in X} (J_{X} (f^{\leftarrow} (B)) (x) ⊖ f^{\leftarrow} (B) (x)) \\ \leq ⋁_{x \in X} (f^{\leftarrow} (J_{Y} (B)) (x) ⊖ f^{\leftarrow} (B) (x)) \\ \leq ⋁_{x \in X} (J_{Y} (B) (f (x)) ⊖ B (f (x))) \\ \leq ⋁_{y \in Y} (J_{Y} (B)) (y) ⊖ B (y)) \\ = T_{J_{Y}} (B) . \end{matrix}$ Hence Δ is a functor.

Define a functor Ψ : AFT → JP as $Ψ (X, T_{X}) = (X, J_{T_{X}})$ where $J_{T_{X}} (A) (y) = ⋁_{A \in L^{X}} (A (x) ⊖ d_{T_{X}} (x, y))$ .

Let T_X (f^← (B)) ≤ T_Y (B). By Theorem 3.18, d_{T
_Y} (f (x) , f (z)) ≤ d_{T
_X} (x, z) for each x, z ∈ X. Then $\begin{matrix} f^{\leftarrow} (J_{T_{Y}} (B)) (x) = J_{T_{Y}} (B) (f (x)) \\ = ⋁_{y \in Y} (B (y) ⊖ d_{T_{Y}} (y, f (x))) \\ \geq ⋁_{z \in X} (B (f (z)) ⊖ d_{T_{Y}} (f (z), f (x))) \\ \geq ⋁_{z \in X} (f^{\leftarrow} (B) (z) ⊖ d_{T_{X}} (z, x)) \\ = J_{T_{X}} (f^{\leftarrow} (B)) (x) . \end{matrix}$

Hence Ψ is a functor. By Theorem 3.16(3), $Δ (Ψ (X, T_{X})) = Δ (X, J_{T_{X}}) = (X, T_{J_{T_{X}}})$ and ${id}_{X} : Δ (Ψ (X, T_{X})) = (X, T_{J_{T_{X}}}) \to (X, T_{X})$ is an A-morphism because $T_{J_{T_{X}}} \leq T_{X}$ . Moreover, by Theorem 3.16(4), $Ψ (Δ (X, J_{X})) = Ψ (X, T_{J_{X}}) = (X, J_{T_{J_{X}}})$ . ${id}_{X} : (X, J_{X}) \to Ψ (Δ (X, J_{X})) = (X, J_{T_{J_{X}}})$ is a J-morphism because $J_{T_{J_{X}}} \geq J_{X}$ .

Theorem 3.20. (1) DIS and UP are isomorphic.

(2) RE and JP are isomorphic.

Proof. Define Λ : DIS → UP as $Λ (X, d_{X}) = (X, J_{d_{X}})$ where $J_{d_{X}}$ is an upper approximation operator from Example 3.4(2).

Let d_X (x, z) ≥ d_Y (f (x) , f (z)). Then $f : (X, J_{X}) \to (Y, J_{Y})$ is a J-morphism from: $\begin{matrix} f^{\leftarrow} (J_{d_{Y}} (B)) (x) = J_{d_{Y}} (B) (f (x)) \\ = ⋁_{y \in Y} (B (y) ⊖ d_{Y} (y, f (x))) \\ \geq ⋁_{z \in X} (B (f (z)) ⊖ d_{Y} (f (z), f (x))) \\ \geq ⋁_{z \in X} (f^{\leftarrow} (B) (z) ⊖ d_{X} (z, x)) = J_{d_{X}} (f^{\leftarrow} (B)) (x) . \end{matrix}$ Hence Λ is a functor.

Define Π : UP → DIS as $Π (X, J_{X}) = (X, d_{J_{X}})$ where $d_{J_{X}}$ is a distance function from Theorem 3.14(3).

Let $f^{\leftarrow} (J_{Y} (B)) \geq J_{X} (f^{\leftarrow} (B))$ for all B ∈ L^Y. Put B = n (⊥ _f(x)), since f^← (n (⊥ _f(x))) (y) = n (⊥ _f(x)) (f (y)) ≥ n (⊥ _x) (y), $\begin{matrix} f^{\leftarrow} (J_{Y} (n (⊥_{f (x)}))) (y) \geq J_{X} (f^{\leftarrow} (n (⊥_{f (x)}))) (y) \\ \geq J_{X} (n (⊥_{x})) (y) . \end{matrix}$ So, $J_{Y} (n (⊥_{f (x)})) (f (y)) \geq J_{X} (n (⊥_{x})) (y)$ . Thus $\begin{matrix} d_{J_{X}} (x, y) = n (J_{X} (n (⊥_{x}))) (y) \\ \geq n (J_{Y} (n (⊥_{f (x)}))) (f (y)) = d_{J_{Y}} (f (x), f (y)) . \end{matrix}$ Hence Π is a functor.

We have $Π (Λ (X, d_{X})) = Π (X, J_{d_{X}}) = (X, d_{J_{d_{X}}}) = (X, d_{X})$ from Theorem 3.14(3). We have $Λ (Π (X, J_{X})) = Λ (X, d_{J_{X}}) = (X, J_{d_{J_{X}}}) = (X, J_{X})$ . Hence DIS and UP are isomorphic.

(2) It is similarly proved as (1).

Let AT be a category with objects (X, τ_X) and (Y, τ_Y) where τ_X and τ_Y are Alexandrov topologies with a continuous map f : (X, τ_X) → (Y, τ_Y) such that f^← (B) ∈ τ_X for all B ∈ τ_Y.

Theorem 3.21. Σ : AT → AFT is a left adjoint of ϒ : AFT → AT,i.e., (Σ, ϒ) is a Galois correspondence. Moreover, τ_{T
_{τ
_X}} = τ_X.

Proof. Define Σ : AT → AFT as Σ (X, τ_X) = (X, T_{τ
_X}) where $T_{τ_{X}} (A) = {\begin{matrix} ⊥, & if A \in τ_{X}, \\ ⊤, & if A \notin τ_{X}, \end{matrix}$ Then T_{τ
_X} is an Alexandrov fuzzy topology. Let f : (X, τ_X) → (Y, τ_Y) be a continuous map. Then f : (X, T_{τ
_X}) → (Y, T_{τ
_Y}) is an A-morphism. Hence Σ is a functor.

Define a functor ϒ : AFT → AT as ϒ (X, T_X) = (X, τ_{T
_X}) where τ_{T
_X} = {A ∈ L^X ∣ T_X (A) = ⊥}. Let T_X (f^← (B)) ≤ T_Y (B). For each B ∈ τ_{T
_Y}, f^← (B) ∈ τ_{T
_X}. Hence ϒ is a functor. Since Σ (ϒ (X, T_X)) = Σ (X, τ_{T
_X}) = (X, T_{τ
_{T
_X}}), id_X : Σ (ϒ (X, T_X)) = (X, T_{τ
_{T
_X}}) → (X, T_X) is an A-morphism because T_{τ
_{T
_X}} ≤ T_X. Moreover, Since ϒ (Σ (X, τ_X)) = ϒ (X, T_{τ
_X}) = (X, τ_{T
_{τ
_X}}), id_X : (X, τ_X) → ϒ (Σ (X, τ_X)) = (X, τ_{T
_{τ
_X}}) is a continuous map because τ_{T
_{τ
_X}} = τ_X.

Example 3.22. Let (L = [0, 1] , ∧ , ∨ , ⊕ , ⊖ , 0, 1, n) be a complete residuated lattice with a double law negative which is defined by $x \oplus y = (x + y) \land 1, x ⊖ y = (x - y) \lor 0, n (x) = 1 - x .$

Let X = {x, y, z} be a set and $J_{d} : L^{X} \to L^{X}$ be a join preserving map defined as, for all x, y ∈ X, $J_{d} (A) (y) = ⋁_{x \in X} (A (x) ⊖ d (x, y))$ with $d = (\begin{matrix} 0.3 & 0.6 & 0.5 \\ 0.5 & 0.2 & 0.4 \\ 0.1 & 0.6 & 0.7 \end{matrix})$ Put A = (0.5, 0.7, 0.2) ∈ L^X. Then $J_{d}$ is not an upper approximation operator because $J_{d} (A) = (0.2, 0.5, 0.3) ≱ A$ and $J_{d} (J_{d} (A)) = (0.2, 0.3, 0.1) \neq J_{d} (A) .$ Moreover, $T_{J_{d}} (A) = d_{L^{X}} (J_{d} (A), A) = 0.1$ . Since $J_{d} (n (0_{x})) (y) = n (d (x, y))$ , $\begin{matrix} J_{d} (n (0_{x})) = (0.7, 0.4, 0.5), \\ \neq J_{d} (J (n (0_{x})) = (0.4, 0.2, 0.2) \\ J_{d} (n (0_{y})) = (0.5, 0.8, 0.6), \\ \neq J_{d} (J (n (0_{y})) = (0.5, 0.6, 0.4) \\ J_{d} (n (0_{z})) = (0.9, 0.4, 0.7), \\ \neq J_{d} (J (n (0_{z}))) = (0.6, 0.3, 0.4), \end{matrix}$ $\begin{matrix} T_{J_{d}} (J_{d} (⊥_{x})) = d_{L^{X}} (J_{d} (J_{d} (⊥_{x})), J_{d} (⊥_{x})) = 0, \\ T_{J_{d}} (J_{d} (⊥_{y})) = d_{L^{X}} (J_{d} (J_{d} (⊥_{y})), J_{d} (⊥_{y})) = 0, \\ T_{J_{d}} (J_{d} (⊥_{z})) = d_{L^{X}} (J_{d} (J_{d} (⊥_{z})), J_{d} (⊥_{z})) = 0, \end{matrix}$ $\begin{matrix} 0.6 = d (x, y) = n (J_{d} (n (⊥_{x})) (y) \\ \neq ⋁_{z \in X} (n (J_{d} (n (⊥_{z}))) (y) ⊖ n (J_{d} (n (⊥_{z}))) (x)) \\ = ⋁_{z \in X} (d (z, y) ⊖ d (z, x)) = 0.5 . \end{matrix}$ For each B ∈ L^X, by Example 3.12(2), $T_{J_{d}} (B) \leq ⋁_{x \neq y} n (d (x, y) = 0.9 .$

Example 3.23. Let (L = [0, 1] , ∧ , ∨ , ⊕ , ⊖ , 0, 1, n) be a complete residuated lattice as in Example 3.22. Let d ∈ L^X×X as follows $d = (\begin{matrix} 0 & 0.3 & 0.1 \\ 0.4 & 0 & 0 \\ 0.5 & 0.6 & 0 \end{matrix})$ Since d (x, x) =0 and d (x, z) = ⋀ _y∈X (d (x, y) ⊕ d (y, z)), d is a distance function. By Example 3.4(2), $J_{d} : L^{X} \to L^{X}$ is an upper approximation map defined as, for all x, y ∈ X, $J_{d} (A) (y) = ⋁_{x \in X} (A (x) ⊖ d (x, y)) .$ We obtain an Alexandrov fuzzy topology $T_{d} = T_{J_{d}}$ as $T_{d} (A) = ⋁_{x, y \in X} ((A (x) ⊖ A (y)) ⊖ d (x, y)) .$ Since $J_{d} (n (0_{x})) (y) = n (d (x, y))$ , $\begin{matrix} J_{d} (n (0_{x})) = (1, 0.7, 0.9) = J_{d} (J (n (0_{x})), \\ J_{d} (n (0_{y})) = (0.7, 1, 0.4) = J_{d} (J (n (0_{y})), \\ J_{d} (n (0_{z})) = (0.9, 1, 1) = J_{d} (J (n (0_{z}))), \end{matrix}$ $\begin{matrix} T_{J_{d}} (J_{d} (⊥_{x})) = d_{L^{X}} (J_{d} (J_{d} (⊥_{x})), J_{d} (⊥_{x})) = 0, \\ T_{J_{d}} (J_{d} (⊥_{y})) = d_{L^{X}} (J_{d} (J_{d} (⊥_{y})), J_{d} (⊥_{y})) = 0, \\ T_{J_{d}} (J_{d} (⊥_{z})) = d_{L^{X}} (J_{d} (J_{d} (⊥_{z})), J_{d} (⊥_{z})) = 0, \end{matrix}$ $\begin{matrix} 0.3 = d (x, y) = n (J_{d} (n (⊥_{x})) (y) \\ = ⋁_{z \in X} (n (J_{d} (n (⊥_{z}))) (y) ⊖ n (J_{d} (n (⊥_{z}))) (x)) \\ = ⋁_{z \in X} (d (z, y) ⊖ d (z, x)) . \end{matrix}$ For each B ∈ L^X, by Example 3.12(2), $T_{J_{d}} (B) \leq ⋁_{x \neq y} n (d (x, y) = 0.9 .$

4 Conclusion

In this paper, we are interested the distance functions as a new definition on complete co-residuated lattices. As main results, there is a Galois correspondence between the category of Alexandrov fuzzy topologies and the category of fuzzy relations. There is a Galois correspondence between the category of Alexandrov fuzzy topologies and the category of join preserving maps. Moreover, there is a Galois correspondence between the category of Alexandrov fuzzy topologies and the category of Alexandrov topologies. The category of of fuzzy relations and the category of join preserving maps are isomorphic. Finally, the category of distance spaces and the category of upper approximation operators are isomorphic.

In the future, we plan to investigate fuzzy rough sets, information systems and decision rules by using the concepts of distance spaces in complete co-residuated lattices.

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