Alexandrov L -topologies and Alexandrov L -convergence structures 1

Abstract

Keywords

1. Introduction

Pawlak [20] introduced the rough set theory as a formal tool to deal with imprecision and uncertainty in the data analysis. For an extension of classical rough sets, many researchers [1–3 , 33] developed L-lower and L-upper approximation operators in complete residuated lattices [28]. By using this concepts, information systems and decision rules were investigated in complete residuated lattices [1 , 33].

An interesting and natural research topic in rough set theory is the study of rough set theory and topological structures. Lai [14] and Ma [15] investigated the Alexandrov L-topology and lattice structures of L-fuzzy rough sets determined by lower and upper sets. Kim [11, 12] studied the relations between L-fuzzy upper and lower approximation spaces and Alexandrov L-topologies in complete residuated lattices. Moreover, categories of fuzzy preorders, approximating operators and Alexandrov topologies are isomorphic [11].

Jäger [5–7 , 19] developed stratified L-convergence structures based on the concepts of L-filters where L is a complete Heyting algebra. Yao [30] extended stratified L-convergence structures to complete residuated lattices and investigated between stratified L-convergence structures and L-fuzzy topological spaces.

Zhang [34] defined a strong L-topology on the concepts of fuzzy complete lattices. As an extension of Yao [30], Fang [2, 3] introduced L-ordered convergence structures on L-ordered filters and investigated between L-ordered convergence structures and strong L-topological spaces. Many researchers [8 , 21–23] developed the properties of L-convergence structures.

Kim [12, 13] introduced the notion of Alexandrov L-(neighborhood) filters as an extension of Fang’s L-ordered filters [3] and defined Alexandrov L-convergence structures on Alexandrov L-(neighborhood) filters. Moreover, We investigate the categorical relations among Alexandrov L-neighborhood filters, L-fuzzy preorders and Alexandrov L-convergence structures.

In this paper, we introduce the concepts of Alexandrov L-neighborhood filters, Alexandrov L-topologies and Alexandrov L-convergence structures in complete residuated lattices. In section 3, we investigate the Galois correspondence between Alexandrov L-neighborhood filters and Alexandrov L-topologies. In section 4, we investigate the relations among Alexandrov L-neighborhood filters, Alexandrov L-topologies and Alexandrov L-convergence structures. There exists the Galois correspondence between Alexandrov L-convergence structures and Alexandrov L-topologies. Moreover, we investigate their topological properties and give their examples.

2. Preliminaries

Definition 2.1. [1 , 28] An algebra (L, ∧, ∨, ⊙, →, ⊥, ⊤) is called a complete residuated lattice if it satisfies the following conditions:

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

(L, ⊙, ⊤) is a commutative monoid;

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

In this paper, we always assume that (L, ≤, ⊙, →, ^∗) is complete residuated lattice with x^* = x→ ⊥ and (x^*) ^* = x for each x ∈ L. For α ∈ L, A ∈ L^X, we denote (α → A), (α ⊙ A), α_X ∈ L^X as (α → A) (x) = α → A (x), (α ⊙ A) (x) = α ⊙ A (x), α_X (x) = α.

Lemma 2.2. [1 , 28] For each x, y, z, x_i, y_i, w ∈ L, we have the following properties.

⊤ → x = x, ⊥ ⊙ x = ⊥,

If y ≤ z, then x ⊙ y ≤ x ⊙ z, x → y ≤ x → z and z → x ≤ y → x,

x ≤ y iff x→ y = ⊤.

x → (⋀ _iy_i) = ⋀ _i (x → y_i),

(⋁ _ix_i) → y = ⋀ _i (x_i → y),

x ⊙ (⋁ _iy_i) = ⋁ _i (x ⊙ y_i),

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

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

(x → y) ⊙ (y → z) ≤ x → z,

⋁_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),

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

(x ⊙ y^*) ^* = x → y and x → y = y^* → x^*.

Definition 2.3. [1, 14] Let X be a set. A function e_X : X × X → L is called:

reflexive if e_X (x, x) =⊤ for all x ∈ X,

transitive if e_X (x, y) ⊙ e_X (y, z) ≤ e_X (x, z), for all x, y, z ∈ X,

if e_X (x, y) = e_X (y, x) =⊤, then x = y.

If e satisfies (E1) and (E2), (X, e_X) is an L-fuzzy preordered set. If e_X satisfies (E1), (E2) and (E3), (X, e_X) is an L-fuzzy partially ordered set (for short, fuzzy poset).

Example 2.4. (1) We define a function e_L : L × L → L as e_L (x, y) = x → y. Then (L, e_L) is a fuzzy poset.

(2) We define a function e_{L
^X} : L^X × L^X → L as e_{L
^X} (A, B) = ⋀ _x∈X (A (x) → B (x)). Then (L^X, e_{L
^X}) is a fuzzy poset from Lemma 2.2 (8).

Definition 2.5. [29 , 35] Let (X, e_X) be a fuzzy poset and A ∈ L^X.

A point x₀ is called a join of A, denoted by x₀ = ⊔ A, if it satisfies

A (x) ≤ e_X (x, x₀),

⋀_x∈X (A (x) → e_X (x, y)) ≤ e_X (x₀, y).

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

A (x) ≤ e_X (x₁, x),

⋀_x∈X (A (x) → e_X (y, x)) ≤ e_X (y, x₁).

Remark 2.6. Let (X, e_X) be a fuzzy poset and Φ ∈ L^X.

If x₀ is a join of Φ, then it is unique because e_X (x₀, y) = e_X (y₀, y) for all y ∈ X, put y = x₀ or y = y₀, then e_X (x₀, y) = e_X (y₀, y) =⊤ implies x₀ = y₀. Similarly, if a meet of Φ exist, then it is unique.

A point x₀ is a join of Φ iff ⋀_x∈X (Φ (x) → e_X (x, y)) = e_X (x₀, y).

A point x₁ is a meet of Φ iff ⋀_x∈X (Φ (x) → e_X (y, x)) = e_X (y, x₁).

Remark 2.7. Let (L^X, e_{L
^X}) be a fuzzy poset and Φ ∈ L^{L
^X}.

⊔Φ = ⋁ _A∈L^X (Φ (A) ⊙ A) from: $\begin{matrix} e_{L^{X}} (⊔ Φ, B) = ⋀_{A \in L^{X}} (Φ (A) \to e_{L^{X}} (A, B)) \\ = e_{L^{X}} (⋁_{A \in L^{X}} (Φ (A) ⊙ A), B) . \end{matrix}$

⊓Φ = ⋀ _A∈L^X (Φ (A) → A) from: $\begin{matrix} e_{L^{X}} (B, ⊔ Φ) = ⋀_{A \in L^{X}} (Φ (A) \to e_{L^{X}} (B, A) \\ = ⋀_{A \in L^{X}} e_{L^{X}} (B, (Φ (A) \to A)) \\ = e_{L^{X}} (B, ⋀_{A \in L^{X}} (Φ (A) \to A)) . \end{matrix}$

Definition 2.8. [11 , 17] A subset τ ⊂ L^X is called an Alexandrov L-topology on X iff it satisfies the following conditions:

α_X ∈ τ.

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

If A ∈ τ and α ∈ L, then α ⊙ A, α → A ∈ τ.

The pair (X, τ) is called an Alexandrov L-topological space.

Let (X, τ₁) and (Y, τ₂) be two Alexandrov L-topological spaces. A mapping φ : X → Y is called continuous if for each B ∈ τ_Y, φ^← (B) ∈ τ_X.

Definition 2.9. [13] Let (L^X, e_{L
^X}) and (L, e_L) be fuzzy posets. A map $F : (L^{X}, e_{L^{X}}) \to (L, e_{L})$ is called an Alexandrov L-filter on X iff $F (⊓ Φ) = ⊓ F^{\to} (Φ)$ for all Φ ∈ L^{L
^X} where $F^{\to} (Φ) (y) = ⋁_{F (A) = y} Φ (A)$ . Let AF (X) denote the set of all Alexandrov L-filters on X.

Theorem 2.10. [13] A map $F : (L^{X}, e_{L^{X}}) \to (L, e_{L})$ is an Alexandrov L-filter on X iff it satisfies the following conditions:

$F (⋀_{i \in Γ} A_{i}) = ⋀_{i \in Γ} F (A_{i})$ for all A_i ∈ L^X.

$F (α \to A) = α \to F (A)$ for all A ∈ L^X and α ∈ L.

3. Alexandrov L-topologies and Alexandrov L-neighborhood spaces

Definition 3.1. [13] A family $N = {N^{x} ∣ x \in X}$ is called an Alexandrov L-neighborhood system on X if, for x ∈ X, a map $N^{x} : L^{X} \to L$ satisfies:

$N^{x}$ is an Alexandrov L-filter on X.

$N^{x} (A) \leq A (x)$ for all A ∈ L^X.

The pair $(X, N)$ is called an Alexandrov L-neighborhood space.

An Alexandrov L-neighborhood system on X is topological if (TN) $N^{x} (A) \leq N^{x} (N^{-} (A))$ for all A ∈ L^X such that $N^{-} (A) (y) = N^{y} (A)$ for all y ∈ X.

Let $(X, N_{X})$ and $(Y, N_{Y})$ be Alexandrov L-neighborhood spaces. $ψ : (X, N_{X}) \to (Y, N_{Y})$ is called an N-continuous map if, for each B ∈ L^Y and x ∈ X, $N_{Y}^{ψ (x)} (B) \leq N_{X}^{x} (ψ^{\leftarrow} (B)) .$

Example 3.2. Let ([0, 1], ⊙, →, ^*, 0, 1) be a complete residuated lattice (ref.[1 , 25]) as $\begin{matrix} x ⊙ y = \sqrt{max {0, x^{2} + y^{2} - 1}}, \\ x \to y = min {1, \sqrt{1 - x^{2} + y^{2}}}, \\ x^{*} = \sqrt{1 - x^{2}} . \end{matrix}$

Let X = {x, y, z} be a set and [0, 1]-fuzzy relation $e_{X}^{i} \in [0, 1]^{X \times X}$ as $e_{X}^{1} = (\begin{matrix} 0.8 & 0.4 & 0.5 \\ 1 & 1 & 0.7 \\ 0.3 & 1 & 0.1 \end{matrix}) e_{X}^{2} = (\begin{matrix} 1 & 0.5 & 0.6 \\ 0.8 & 1 & 0.5 \\ 1 & 0.6 & 1 \end{matrix})$

For each i = 1, 2, define a map $N_{e_{X}^{i}}^{y} : L^{X} \to L$ as

N_{e_{X}^{i}}^{y} (A) = ⋀_{x \in X} (e_{X}^{i} (y, x) \to A (x)) .

(1) By (F1) and (F2) in Theorem 2.10, for each y ∈ X, $N_{e_{X}^{1}}^{y}$ is an Alexandrov L-filter such that

\begin{matrix} N_{e_{X}^{1}}^{x} (0_{X}) = (0.8 \to 0) \land (0.4 \to 0) \land (0.5 \to 0) = 0.6, \\ N_{e_{X}^{1}}^{y} (0_{X}) = 0, N_{e_{X}^{1}}^{z} (0_{X}) = 0 . \end{matrix}

Since $N_{e_{X}^{1}}^{y} (0_{X}) ≰ 0_{X} (y) = 0$ , $N_{e_{X}^{1}} = {N_{e_{X}^{1}}^{x} ∣ x \in X}$ is not an Alexandrov L-neighborhood system.

(2) Let $e_{X}^{2} \in L^{X \times X}$ be a fuzzy poset. For each y ∈ X, $N_{e_{X}^{2}}^{y} (A) = ⋀_{x \in X} (e_{X}^{i} (y, x) \to A (x)) \leq e_{X}^{i} (y, y)$ →A (y) = A (y). Moreover, since $⋁_{x \in X} (e_{X}^{2} (y, x) ⊙ e_{X}^{2} (x, z)) = e_{X}^{2} (y, z)$ , we have

\begin{matrix} N_{e_{X}^{2}}^{y} (N_{e_{X}^{2}}^{-} (A)) = ⋀_{x \in X} (e_{X}^{2} (y, x) \to N_{e_{X}^{2}}^{x} (A)) \\ = ⋀_{x \in X} (e_{X}^{2} (y, x) \to ⋀_{z \in X} (e_{X}^{2} (x, z) \to A (z))) \\ = ⋀_{x, z \in X} (e_{X}^{2} (y, x) ⊙ e_{X}^{2} (x, z) \to A (z))) \\ = ⋀_{z \in X} (⋁_{x \in X} e_{X}^{2} (y, x) ⊙ e_{X}^{2} (x, z) \to A (z))) \\ = ⋀_{z \in X} (e_{X}^{2} (y, z) \to A (z))) = N_{e_{X}^{2}}^{y} (A) . \end{matrix}

Hence ${N_{e_{X}^{2}}^{x} ∣ x \in X}$ is a topological Alexandrov L-neighborhood system.

Theorem 3.3. Let $(X, N)$ be an Alexandrov L-neighborhood space and define $τ_{N} = {A \in L^{X} ∣ A = N^{-} (A)}$ . Then $τ_{N}$ is an Alexandrov L-topology. If $N^{x}$ is topological for each x ∈ X, $τ_{N} = {N^{-} (A) ∣ A \in L^{X}}$ .

Proof. If A ≤ B, then N^x (A) = N^x (A) ∧ N^x (B) ≤ N^x (B).

(AT1) Since $N^{-} (α \to A) = α \to N^{-} (A)$ , put α =⊥. Then $N^{-} (⊤_{X}) = ⊤$ . Since A ≤ α → α ⊙ A, then $N^{-} (A) \leq N^{-} (α \to α ⊙ A) = α \to N^{-} (α ⊙ A)$ . Thus $α ⊙ N^{-} (A) \leq N^{-} (α ⊙ A)$ . Put A =⊤. Then $α ⊙ N^{-} (⊤) \leq N^{-} (α ⊙ ⊤) \leq α_{X}$ . Hence $N^{-} (α_{X}) = α_{X}$ ;i.e. $α_{X} \in τ_{N}$ .

(AT2) If $A_{i} = N^{-} (A_{i})$ for all i ∈ Γ, $\begin{matrix} ⋁_{i \in Γ} A_{i} = ⋁_{i \in Γ} N^{-} (A_{i}) \leq N^{-} (⋁_{i \in Γ} A_{i}) \leq ⋁_{i \in Γ} A_{i}, \\ ⋀_{i \in Γ} A_{i} = ⋀_{i \in Γ} N^{-} (A_{i}) = N^{-} (⋀_{i \in Γ} A_{i}) . \end{matrix}$

Then $⋁_{i \in Γ} A_{i}, ⋀_{i \in Γ} A_{i} \in τ_{N}$ .

(AT3) If $A = N^{-} (A)$ , $\begin{matrix} α \to A = α \to N^{-} (A) = N^{-} (α \to A), \\ α ⊙ A = α ⊙ N^{-} (A) \leq N^{-} (α ⊙ A) \leq α ⊙ A . \end{matrix}$

Then $α ⊙ A, α \to A \in τ_{N}$ .

Put $τ = {N^{-} (A) ∣ A \in L^{X}}$ . If $B = N^{-} (A) \in τ$ , then $N^{-} (B) = N^{-} (N^{-} (A)) = N^{-} (A) = B \in τ_{N}$ .

If $A \in τ_{N}$ , $A = N^{-} (A) \in τ$ .

Theorem 3.4. Let τ be an Alexandrov L-topology on X. Define $N_{τ}^{x} : L^{X} \to$ as

N_{τ}^{x} (A) = ⋁_{B \in τ} (e_{L^{X}} (B, A) ⊙ B (x)) .

Then the following properties hold.

$e_{L^{X}} (A, B) \leq ⋀_{x \in X} (N_{τ}^{x} (A) \to N_{τ}^{x} (B))$ , for all A, B ∈ L^X.

For each Φ : L^X → L, we have $N_{τ}^{x} (⋀_{D \in L^{X}} (Φ (D) \to D (x))) = ⋀_{D \in L^{X}} (Φ (D) \to N_{τ}^{x} (D))$ .

$N_{τ} = {N_{τ}^{x} ∣ x \in X}$ is a topological Alexandrov L-neighborhood system on X

$N_{τ}^{x} (α_{X}) = α$ , $N_{τ}^{x} (α \to A) = α \to N_{τ}^{x} (A)$ and $N_{τ} (⋀_{i \in Γ} A_{i}) = ⋀_{i \in Γ} N_{τ}^{x} (A_{i})$ for all A, A_i ∈ L^X.

$τ_{N_{τ}}$ is an Alexandrov L-topology such that $τ = τ_{N_{τ}}$ .

$N_{τ}^{x} (A) = ⋁ {B (x) ∣ B \leq A, B \in τ} .$

Let $(X, N)$ be an Alexandrov L-neighborhood space. Then $N_{τ_{N}} \leq N$ . Moreover, if $(X, N)$ is topological, $N_{τ_{N}} = N$ .

Proof. (1) For each A, B ∈ L^X, by Lemma 2.2, $\begin{matrix} ⋀_{x \in X} (N_{τ}^{x} (A) \to N_{τ}^{x} (B)) \\ = ⋀_{x \in X} (⋁_{C \in τ^{c}} (e_{L^{X}} (C, A) ⊙ C (x)) \\ \to ⋁_{D \in τ^{c}} (e_{L^{X}} (D, B) ⊙ D (x))) \\ \geq ⋀_{x \in X} (⋀_{C \in τ^{c}} (e_{L^{X}} (C, A) ⊙ C (x) \\ \to (e_{L^{X}} (C, B) ⊙ C (x)))) \\ \geq ⋀_{C \in τ^{c}} (e_{L^{X}} (C, A) \to e_{L^{X}} (C, B)) \\ \geq e_{L^{X}} (A, B) . \end{matrix}$

(2) For each Φ : L^X → L, we will show that $N_{τ}^{x} (⋀_{D \in L^{X}} (Φ (D) \to D (x))) = ⋀_{D \in L^{X}} (Φ (D) \to N_{τ}^{x} (D))$ . By (AT2) and (AT3), $N_{τ}^{-} (D) \in τ$ . Since $⋀_{D \in L^{X}} (Φ (D) \to D) \geq ⋀_{D \in L^{X}} (Φ (D) \to N_{τ}^{-} (D))$ and $⋀_{D \in L^{X}} (Φ (D) \to N_{τ}^{-} (D)) \in τ$ , $\begin{matrix} N_{τ}^{x} (⋀_{D \in L^{X}} (Φ (D) \to D)) \\ = ⋁_{C \in τ} e_{L^{X}} (C, ⋀_{D \in L^{X}} (Φ (D) \to D)) ⊙ C (x)) \\ \geq e_{L^{X}} (⋀_{D \in L^{X}} (Φ (D) \to N_{τ}^{-} (D)), ⋀_{D \in L^{X}} (Φ (D) \to D)) \\ ⊙ ⋀_{D \in L^{X}} (Φ (D) \to N_{τ}^{x} (D))) \\ \geq ⋀_{D \in L^{X}} (Φ (D) \to N_{τ}^{x} (D)), \end{matrix}$ $\begin{matrix} N_{τ}^{x} (⋀_{D \in L^{X}} (Φ (D) \to D)) \\ = ⋁_{C \in τ} e_{L^{X}} (C, ⋀_{D \in L^{X}} (Φ (D) \to D)) ⊙ C (x)) \\ = ⋁_{C \in τ} (⋀_{D \in L^{X}} (Φ (D) \to e_{L^{X}} (C, D)) ⊙ C (x))) \\ ((a \to b) ⊙ c \leq a \to b ⊙ c) \\ \leq ⋀_{D \in L^{X}} (Φ (D) \to ⋁_{C \in τ} e_{L^{X}} (C, D) ⊙ C (x))) \\ = ⋀_{D \in L^{X}} (Φ (D) \to N_{τ}^{x} (D)) . \end{matrix}$

(3) (N1) For each Φ : L^X → L, by (2), $\begin{matrix} e_{L} (b, ⊓ (N_{τ}^{x})^{\to} (Φ)) \\ = ⋀_{c \in L} (N_{τ}^{x})^{\to} (c) \to e_{L} (b, c)) \\ = ⋀_{c \in L} (⋁_{c = N_{τ}^{x} (D)} Φ (D) \to e_{L} (b, N_{τ}^{x} (D))) \\ = e_{L} (b, ⋀_{D \in L^{X}} (Φ (D) \to N_{τ}^{x} (D))) \\ = e_{L} (b, N_{τ}^{x} (⋀_{D \in L^{X}} (Φ (D) \to D)))) \\ = e_{L} (b, N_{τ}^{x} (⊓ Φ)) . \end{matrix}$

So, $N_{τ}^{x} (⊓ Φ) = ⊓ (N_{τ}^{x})^{\to} (Φ) .$ Hence $N^{x}$ is an Alexandrov L-filter on X.

(N2) Since e_{L
^X} (B, A) ⊙ B (x) ≤ (B (x) → A (x)) ⊙ B (x) ≤ A (x), $N_{τ}^{x} (A) \leq A (x)$ , for all A ∈ L^X.

(TN) Since $N_{τ}^{-} (A) = ⋁_{B \in τ} (e_{L^{X}} (B, A) ⊙ B) \in τ$ , $\begin{matrix} N_{τ}^{x} (N_{τ}^{-} (A)) = ⋁_{B \in τ} (e_{L^{X}} (B, N_{τ}^{-} (A)) ⊙ B (x)) \\ \geq e_{L^{X}} (N_{τ}^{-} (A), N_{τ}^{-} (A)) ⊙ N_{τ}^{-} (A) (x) = N_{τ}^{-} (A) (x) \end{matrix}$

(4) It follows from (3) and Theorem 2.10.

(5) Let A ∈ τ. Then $N_{τ}^{x} (A) = ⋁_{B \in τ} (e_{L^{X}} (B, A) ⊙ B (x)) \geq e_{L^{X}} (A, A) ⊙ A (x) = A (x)$ . So $A \in τ_{N_{τ}}$ .

Let $A \in τ_{N_{τ}}$ . Then $A = N_{τ}^{-} (A) \in τ$ .

(6) Put $N^{x} (A) = ⋁ {B (x) ∣ B \leq A, B \in τ} .$ For B ≤ A, B ∈ τ, ⋁_B∈τ (e_{L
^X} (B, A) ⊙ B (x)) ≥ B (x). Hence $N^{x} (A) \leq N_{τ}^{x} (A)$ .

Since ⋁_B∈τ (e_{L
^X} (B, A) ⊙ B (x)) ≤ A (x) and ⋁_B∈τ (e_{L
^X} (B, A) ⊙ B (x)) ∈ τ. Hence $N^{x} (A) \geq N_{τ}^{x} (A)$ .

(7) By (6), $N_{τ_{N}}^{x} (A) = ⋁ {B (x) ∣ B \leq A, N (B) = B \in τ_{N}} \leq N^{x} (A) .$ If $N^{x} (N^{-} (A)) = N^{x} (A)$ , then $N^{-} (A) \in τ_{N}$ . Thus $N_{τ_{N}}^{x} = N^{x} .$

Example 3.5. (1) Let τ = L^X be an Alexandrov L-topology on X. Then $\begin{matrix} N_{τ}^{x} (A) & = ⋁ {B (x) ∣ B \leq A, B \in τ = L^{X}} \\ = A (x) = [x] (A) . \end{matrix}$

Since $N_{τ}^{x} (N_{τ}^{-} (A)) = N_{τ}^{x} (A)$ , $N_{τ} = {N_{τ}^{x} = [x] ∣ x \in X}$ is a topological Alexandrov L-neighborhood system on X. By Theorem 3.4(5), $τ_{N_{τ}} = {A \in L^{X} ∣ N_{τ}^{x} (A) = [x] (A) = A (x), \forall x \in X} = L^{X}$ is an Alexandrov L-topology on X.

(2) Let τ = {α_X ∣ α ∈ L} be an Alexandrov L-topology on X. Then $\begin{matrix} N_{τ}^{x} (A) & = ⋁ {B (x) ∣ B \leq A, B \in τ} \\ = ⋀_{x \in X} A (x) = ⋀_{x \in X} [x] (A) . \end{matrix}$

Since $N_{τ}^{x} (N_{τ}^{-} (A)) = ⋀_{x \in X} N_{τ}^{x} (A) = ⋀_{x \in X} A (x) =$ $N_{τ}^{x} (A)$ , $N_{τ} = {N_{τ}^{x} = ⋀_{x \in X} [x] ∣ x \in X}$ is a topological Alexandrov L-neighborhood system on X. Moreover, $τ_{N_{τ}} = {A \in L^{X} ∣ N^{x} (A) = ⋀_{z \in X} A (z) = A (x), x \in X} = {α_{X} ∣ α \in L}$ is an Alexandrov L-topology on X.

(3) Let e_X ∈ L^X×X be reflexive. A family $N = {N^{x} (A) = ⋀_{z \in X} (e_{X} (x, z) \to A (z)) ∣ x \in X}$ is an Alexandrov L-neighborhood system on X. $τ_{N_{X}} = {A \in L^{X} ∣ N^{x} (A) = A (x), x \in X}$ is an Alexandrov L-topology on X.

(4) Let e_X ∈ L^X×X be an L-fuzzy preorder. Since ⋁_x∈X (e_X (y, x) ⊙ e_X (x, z) = e_X (y, z), by (3) and a similar method in Example 3.2(2), we have $N^{y} (N^{-} (A)) (y) = N^{y} (A) .$ Hence ${N^{x} ∣ x \in X}$ is a topological Alexandrov L-neighborhood system. From Theorem 3.3, we can obtain Alexandrov L-topology $τ_{N}$ as $τ_{N} = {⋀_{y \in Y} (e_{X} (-, y) \to A (y)) ∣ A \in L^{X}} .$

Theorem 3.6. (1) Let (X, τ_X) and (Y, τ_Y) be Alexandrov L-topological spaces. If ψ : (X, τ_X) → (Y, τ_Y) is a continuous map, then $ψ : (X, N_{τ_{X}}) \to (Y, N_{τ_{Y}})$ is an N-continuous map.

(2) Let $(X, N_{X})$ and $(Y, N_{Y})$ be Alexandrov L-neighborhood spaces. If $ψ : (X, N_{X}) \to (Y, N_{Y})$ is an N-continuous map, then $ψ : (X, τ_{N_{X}}) \to (Y, τ_{N_{X}})$ is a continuous map.

Proof. (1) We show that for each B ∈ L^Y, $N_{τ_{Y}}^{ψ (x)} (B) \leq N_{τ_{X}}^{x} (ψ^{\leftarrow} (B)) .$ $\begin{matrix} N_{τ_{Y}}^{ψ (x)} (B) = ⋁_{C \in τ_{Y}} (e_{L^{Y}} (C, B) ⊙ C (ψ (x))) \\ \leq ⋁_{C \in τ_{Y}} ((⋀_{x \in X} (C (ψ (x)) \to B (ψ (x))) ⊙ C (ψ (x))) \\ \leq ⋁_{ψ^{\leftarrow} (C) \in τ_{X}} (e_{L^{X}} (ψ^{\leftarrow} (C), ψ^{\leftarrow} (B)) ⊙ ψ^{\leftarrow} (C) (x)) \\ \leq ⋁_{D \in τ_{X}} (e_{L^{X}} (D, ψ^{\leftarrow} (B)) ⊙ D (x)) \\ = N_{τ_{X}}^{x} (ψ^{\leftarrow} (B)) . \end{matrix}$

(2) For each $B \in τ_{N_{Y}}$ and y ∈ Y, $B (y) = N_{Y}^{y} (B)$ . Then $B (ψ (x)) = N_{Y}^{ψ (x)} (B)$ . Since $N_{Y}^{ψ (x)} (B) \leq N_{X}^{x} (ψ^{\leftarrow} (B))$ , $B (ψ (x)) = N_{Y}^{ψ (x)} (B) \leq N_{X}^{x} (ψ^{\leftarrow} (B)) .$ Thus, $ψ^{\leftarrow} (B) \in τ_{N_{Y}}$ .

Definition 3.7. [4, 25] Suppose that $F : D \to C,$ $G : C \to D$ are concrete functors. The pair (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 ATS be denote the category of Alexandrov L-topological spaces and continuous mappings for morphisms. Let ANS be denote the category of Alexandrov L-neighborhood spaces and N-continuous mappings for morphisms.

Theorem 3.8. (1) F : ANS→ ATS defined as F $(X, N_{X}) = (X, τ_{N_{X}})$ is a functor.

(2) G : ATS→ ANS defined as G $(X, τ_{X}) = (X, N_{τ_{X}})$ is a functor.

(3) The pair (F, G) is a Galois correspondence between ANS and ATS.

Proof. By Theorem 3.4(7), if $(X, N_{X})$ is an Alexandrov L-neighborhood space, then G(F $(X, N_{X}) = (X, N_{τ_{N_{X}}})$ . Since $N_{τ_{N_{X}}} \leq N_{X}$ , the identity map ${id}_{X} : (X, N_{X})$ → $(X, N_{τ_{N_{X}}})$ =G(F $(X, N_{X}))$ is an N-continuous map. Moreover, if (X, τ_X) is an Alexandrov L-topological space, by Theorem 3.4(5), F(G $(X, τ_{X})) = τ_{N_{τ_{X}}} = τ_{X}$ . Hence the identity map id_X : G(F $(X, τ_{X})) = τ_{N_{τ_{X}}} \to (X, τ_{X})$ is a continuous map. Therefore (F,G) is a Galois correspondence.

Theorem 3.9. Let ψ : X → Y be a map. Let $F : L^{X} \to L$ is an Alexandrov L-filter on X and define $ψ^{\Rightarrow} (F L$ as

φ \Rightarrow (F) (B) = F (φ \leftarrow (B)) .

Then ψ^⇒(F) is an Alexandrov L-filter on Y. Moreover, if N^x : L^X → L is an Alexandrov L-neighborhood filter on X, then ψ^⇒(N^x)ψ(x) : L^Y →L is an Alexandrov L-neighborhood filter on Y.

Proof. For each B_i ∈ L^Y,

$\begin{array}{l} ψ^{\Rightarrow} (F) (\land_{i \in Γ} B_{i}) = F (ψ^{\leftarrow} (\land_{i \in Γ} B_{i})) \\ = F (\land_{i \in Γ} ψ^{\leftarrow} (B_{i})) = \land_{i \in Γ} F (ψ^{\leftarrow} (B_{i})) \\ = \land_{i \in Γ} ψ^{\Rightarrow} (F) (B_{i}) . \end{array}$

For each B ∈ L^Y and α ∈ L,

$\begin{array}{l} ψ^{\Rightarrow} (F) (α \to B) = F (ψ^{\leftarrow} (α \to B)) \\ = F (α \to ψ^{\leftarrow} (B)) = α \to F (ψ^{\leftarrow} (B)) \\ = α \to ψ^{\Rightarrow} (F) (B) . \end{array}$

By Theorem 2.10, ψ^⇒(F) is an Alexandrov L-filter on Y. Let N^x be an Alexandrov L-neighborhood filter on X. Then ψ^⇒(N^x)ψ(x)(B) = N^x(ψ^←(B)) ≤ψ^←(B)(x) = B(ψ(x)). Thus the result hold.

4. Alexandrov L-convergence spaces

Definition 4.1. [13] A map lim : AF(X) → LX is called an Alexandrov L-convergence structure on X iff it satisfies the following conditions:

(AC1) lim[x](x) = T.

(AC2) e_AF(X)(F, G) ≤ e_LX(lim(F), lim(G)) for all F, G ∈ AF(X).

Let AC(X) denote the set of all Alexandrov L-convergence structures on X.

A map ψ : (X, lim_X) → (Y, lim_Y) is called continuous if limX(F)(x) ≤ limX ψ^⇒(F)(ψ(x)) for all F ∈ AF(X) and x ∈ X

Lemma 4.2. Let lim : AF(X) → L^X be a map. The following statements are equivalent.

(1) $e_{A F (X)} (F, G) \leq e_{L^{X}} (\lim (F), \lim (G)) f o r a l l F, G \in A F (X) .$

(2) $I f F \leq G, t h e n \lim (F) \leq \lim (G) a n d \lim (α \to F) \leq α \to \lim (F) f o r a l l F, G \in A F (X) a n d α \in L .$

Proof. (1) ⇒ (2). If F ≤ G, then $\begin{array}{l} T = e_{A F (X)} (F, G) \leq e_{L^{X}} (\lim (F), l i m (G)) . Hence l i m (F) \leq \lim (G) . put G = α \to F . Then α \leq e_{A F (X)} (α \to F) \leq e_{L^{X}} (\lim (α \to F), \lim (F)) . \\ Hence l i m (α \to F) \leq α \to \lim (F) . \end{array}$

$\begin{array}{l} (2) \Rightarrow (1) . since F \leq e_{A F (X)} (F, G) \to G, \lim (F) \leq \lim (e_{A F (X)} (F, G) \to G) \leq \\ e_{A F (X)} (F, G) \to \lim (G) . Hence e_{A F (X)} (F, G) \leq e_{L^{X}} (\lim (F), \lim (G)) . \end{array}$

Theorem 4.3. Let lim be an Alexandrov convergence space on X. Then the following properties hold.

$(1) F o r Φ : A F (X) \to L^{X}, \lim (⊓ Φ) \leq ⊓ \lim^{\to} (Φ) .$

$\begin{array}{l} (2) F o r Φ_{\lim}^{x} : A F (X) \to L a s \\ Φ_{\lim}^{x} (F) = \lim F (x) . \end{array}$

Define ${N_{\lim}^{x} = ⊓ Φ_{\lim}^{x} | x \in X}$ Then N^x _lim is an Alexandrov L-neighborhood filter on X such that N^x _lim(A) = ∧F∈AF(X)(lim F(x) → F(A)).

Proof. (1) For Φ : AF(X) → LX,

$\begin{array}{l} e_{L^{X}} (B, ⊓ \lim^{\to} (Φ)) \\ = \land_{C \in L^{X}} \lim^{\to} (Φ) (C) \to e_{L^{X}} (B, C)) \\ = \land_{C \in L^{X}} (\lor_{C} = \lim (F) Φ (F) \to e_{L^{X}} (B, \lim (F))) \\ = e_{L^{X}} (B, \land_{F \in A F (X)} (Φ (F) \to \lim (F))) \\ (by Lemma 4 .2(2)) \\ \geq e_{L^{X}} (B, \lim (\land_{F \in A F (X)} (Φ (F) \to F))) \\ = e_{L^{X}} (B, \lim (⊓ Φ)) . \end{array}$

Hence lim(⊓Φ) ≤ lim^→(⊓Φ).

(2) For Φ^x _lim : AF(X) → L,

$\begin{array}{l} e_{A F (X)} (G, ⊓ Φ_{\lim}^{x}) \\ \land_{F \in A F (X)} (Φ_{\lim}^{x} (F) \to e_{A F (X)} (G, F)) \\ = \land_{F \in A F (X)} (limF (x) \to e_{A F (X)} (G, F)) \\ = e_{A F (X)} (G, \land_{F \in A F (X)} (\lim F (x) \to F)) \end{array}$

Hence $\begin{array}{l} ⊓ Φ_{\lim}^{x} (A) = \land_{F \in A F (X)} (\lim F (x) \to F (A)) . \\ Moreover, ⊓ Φ_{\lim}^{x} (α \to A) = α \to ⊓ Φ_{\lim}^{x} (A) \\ and ⊓ Φ_{\lim}^{x} (\land_{i \in I} A_{i}) = \land_{i \in I} ⊓ Φ_{\lim}^{x} (A_{i}) . Put N_{\lim}^{x} \\ = ⊓ Φ_{\lim}^{x} . For Φ \in L^{L^{X}}, \end{array}$

$\begin{array}{l} e_{L} (b, ⊓ {(N_{\lim}^{x})}^{\to} (Φ)) \\ = \land_{c \in L} ({(N_{\lim}^{x})}^{\to} (c) \to e_{L} (b, c)) \\ = \land_{c \in L} (\lor_{c} = N_{\lim}^{x} (D) Φ (D) \to e_{L} (b, N_{\lim}^{x} (D))) \\ = e_{L} (b, N_{\lim}^{x} (\land_{D \in L^{X}} (Φ (D) \to D))) \\ = e_{L} (b, N_{\lim}^{x} (⊓ Φ)) . \end{array}$

Hence $\begin{array}{l} N_{\lim}^{x} (⊓ Φ) = ⊓ (N_{\lim}^{x}) \to (Φ) . Moreover, ⊓ \\ Φ_{\lim}^{x} (A) = \land_{F \in A F (X)} (\lim F (x) \to F (A)) \leq \lim [x] (x) \\ \to [x] (A) = A (x) . Thus N_{\lim}^{x} \end{array}$ is an Alexandrov L-neighborhood filter on X.

Theorem 4.4. Let {N^x | x ∈ X} be an Alexandrov L-neighborhood system on X. Define lim _N : AF(X) → L^X as

$\lim_{N} F (x) = e_{A F (X)} (N^{x}, F) .$

Then the following properties hold.

(1) lim_N is an Alexandrov convergence structure on X.

(2) Define $\begin{array}{l} Φ^{x} : A F (X) \to L a s \\ Φ^{x} (F) = \lim_{N} F (x) . \end{array}$

Then $\begin{array}{l} {N^{x} = ⊓ Φ^{x} | x \in X} i s a n A l e x a n d r o v L - n e i g h b o r h o o d s y s t e m . \\ (3) l i m_{N} F (x) = \land_{y \in X} (N^{x} (T_{y}^{*}) \to (T_{y}^{*})) . \end{array}$

$(3) For A = \land_{y \in Y} (A^{*} (y) \to T_{y}^{*}),$

Proof. (1) and (2) are follows from Theorem 4.6 in [13].

(3) For $\begin{array}{l} A = \land_{y \in Y} (A^{*} (y) \to T_{y}^{*}), \\ \lim_{N} F (x) = \land_{A \in L^{X}} (N^{x} (\land_{y \in Y} (A^{*} (y) \to T_{y}^{*}))) \\ \to F ((\land_{y \in Y} (A^{*} (y) \to T_{y}^{*}))) \\ \geq \land_{A \in L^{X}} \land_{y \in Y} (A^{*} (y) \to N^{x} (T_{y}^{*})) \\ \to \land_{y \in Y} (A^{*} (y) \to F (T_{y}^{*})) \\ \geq \land_{y \in Y} (N^{x} (T_{y}^{*}) \to F (T_{y}^{*})) . \end{array}$

Moreover,

$\begin{array}{l} \lim_{N} F (x) = e_{A F (X)} (N^{x}, F) \\ \leq \land_{y \in X} (N^{x} (T_{y}^{*}) \to F (T_{y}^{*})) . \end{array}$

Hence $\lim_{N} F (x) = \land_{y \in X} (N^{x} (T_{y}^{*}) \to F (T_{y}^{*})) .$

Definition 4.5. An Alexandrov L-convergence structure lim_X : AF(X) → L^X is called principal if it satisfies:

(P) lim_X(F)(x) = e_LX(N^x_limX,F) for all F, G ∈ AF(X).

Theorem 4.6. Let τ be an Alexandrov L-topology on X. Define lim_τ : AF(X) → L^X as

$\lim_{τ} F (x) = e_{A F (X)} (N_{τ}^{x}, F) .$

Then the following properties hold.

(1) lim_τ is an Alexandrov convergence structure on X.

(2) $\lim_{τ} F (x) = \land_{y \in X} (N_{τ}^{x} (T_{y}^{*}) \to F (T_{y}^{*})) .$

(3) Define $\begin{array}{l} Φ_{\lim_{τ}}^{x} : A F (X) \to L a s \\ Φ_{\lim_{τ}}^{x} (F) = \lim_{τ} F (x) . \end{array}$

Then ${⊓ Φ_{\lim_{τ}}^{x} | x \in X}$ is the system of Alexandrov L-neighborhood filters such that $N_{τ}^{x} = ⊓ Φ_{\lim_{τ}}^{x} .$

(4) $N_{\lim_{τ}}^{x} (A) = ⊓ Φ_{\lim_{τ}}^{x} (A) = \land_{F \in A F (X)} (\lim_{τ} F (x) \to F (A)) = N_{τ}^{x} (A) f o r a l l A \in L^{X} .$

(5) (X, lim_τ) is a principal Alexandrov L-convergence space.

Proof. (1) (AC1) Since $N_{τ}^{x} (A) \leq A (x) = [x] (A) for A \in L^{X}, \lim_{τ} [x] (x) = e_{A F (X)} (N_{τ}^{x}, [x]) = T .$

(AC2) $e_{A F (X)} (F, G) \leq e_{L^{X}} (\lim_{τ} (F), \lim_{τ} (G)) for all F, G \in A F (X) .$

$\begin{array}{l} e_{L^{X}} (\lim (F), \lim (G)) \\ = \land_{x \in X} (\lim (F) (x) \to \lim (G) (x)) \\ = \land_{x \in X} (e_{A F (X)} (N_{τ}^{x}, F) \to e_{A F (X)} (N_{τ}^{x}, G)) \\ \geq e_{A F (X)} (F, G) (by Lemma 2 .2 (11)) . \end{array}$

(2) It follows from Theorem 4.4(3).

(3) For G ∈ AF(X),

$\begin{array}{l} \land_{F \in A F (X)} (Φ_{\lim τ}^{x} (F) \to e_{A F (X)} (G, F)) \\ = \land_{F \in A F (X)} (\lim τ (F) (x) \to e_{A F (X)} (G, F)) \\ = \land_{x \in X} (e_{A F (X)} (N_{τ}^{x}, F) \to e_{A F (X)} (G, F)) \\ \geq e_{A F (X)} (G, N_{τ}^{x}) . \end{array}$

Since $Φ_{\lim_{τ}}^{x} (N_{τ}^{x}) = \lim (N_{τ}^{x}) (x) = e_{A F (X)} (N_{τ}^{x}, N_{τ}^{x}) = T, f o r G \in A F (X),$

$\begin{array}{l} \land_{F \in A F (X)} (Φ_{\lim_{τ}}^{x} (F) \to e_{A F (X)} (G, F)) \\ \leq \lim_{τ} (N_{τ}^{x} (x) \to e_{A F (X)} (G, N_{τ}^{x})) \\ = T \to e_{A F (X)} (G, N_{τ}^{x}) = e_{A F (X)} (G, N_{τ}^{x}) . \end{array}$

Thus, $⊓ Φ_{\lim_{τ}}^{x} = N_{τ}^{x} from :$

$\begin{array}{l} e_{A F (X)} (G, ⊓ Φ_{\lim_{τ}}^{x}) \\ = \land_{F \in A F (X)} (Φ_{\lim_{τ}}^{x} (F) \to e_{A F (X)} (G, F)) \\ = e_{A F (X)} (G, N_{τ}^{x}) . \end{array}$

(4) By (3),

$\begin{array}{l} e_{A F (X)} (G, ⊓ Φ_{\lim_{τ}}^{x}) \\ = e_{A F (X)} (G, \land_{F \in A F (X)} (Φ_{\lim_{τ}}^{x} (F) \to F)) \\ = e_{A F (X)} (G, N_{τ}^{x}) . \end{array}$

Hence $N_{\lim_{τ}}^{x} (A) = ⊓ Φ_{\lim_{τ}}^{x} (A) = \land_{F \in A F (X)} (\lim_{τ} F (x) \to F (A)) = N_{τ}^{x} (A) for all A \in L^{X} .$

(5) lim_τ is a principal Alexandrov L-convergence structure from:

$\begin{array}{l} \lim_{τ} (F) (x) = e_{L^{X}} (N_{\lim_{τ}}^{x}, F) \\ = e_{L^{X}} (N_{τ}^{x}, F) . \end{array}$

Theorem 4.7. Let (X, τX) be an Alexandrov L-topological space. Define

$e_{τ X} (x, y) = \underset{A \in τ X}{\land} (A (x) \to A (y)) .$

Then the following properties hold.

(1) e_τX is an L-fuzzy preorder on X.

(2) If $N_{e_{τ X}}^{x} (A) = \land_{y \in X} (e_{τ X} (x, y) \to A (y)) f o r e a c h x \in X, t h e n {N_{e_{τ X}}^{x} | x \in X} i s a t o p o l o g i c a l A l e x a n d r o v L - n e i g h b o r h o o d s y s t e m .$

(3) $τ_{X} = {N_{e_{τ X}}^{-} (A) | A \in L^{X}} .$

(4) $N_{e_{τ X}}^{x} = N_{_{τ X}}^{x}$

(5) Define

$\lim_{e_{τ_{X}}} (F) = e_{A F (X)} (N_{e_{τ_{X}}}^{x}, F)$

then (X, lim_eτX) is a principal Alexandrov L-convergence space.

Proof. (1) and (2) are easily proved from Example 2.4, Theorem 2.10 and Example 3.2(2).

(3) Put $\begin{array}{l} τ N_{e τ_{x}} = {N_{e_{τ X}}^{-} (A) | A \in L^{X}} . Let B \in τ_{X} . Then \land_{y \in X} \\ (e_{τ X} (x, y) \to B (y)) \leq e_{τ X} (x, x) \to B (x) = B (x) . Since \\ e_{τ X} (x, y) \to B (y) \geq (B (x) \to B (y) \geq B (x)) . Hence \\ B (y) \geq (B (x) \to B (y)) \to B (y) \geq B (x) . Hence B = N_{e_{τ X}}^{-} (B) \in τ N_{e τ_{X}} . \end{array}$

Let $\begin{array}{l} N_{e_{τ X}}^{-} (A) = \land_{y \in X} (e_{τ X} (-, y) \to A (y)) = \\ \land_{y \in X} (A^{*} (y) \to e_{τ X}^{*} (-, y)) \in τ_{N_{e τ X}} . since e_{τ X}^{*} (-, y) \\ = \lor_{y \in X} {(B (-) \to B^{*} (y))}^{*} = \lor_{B \in τ_{X}} (B^{*} (y) ⊙ B (-)) \in τ_{X}, N_{e_{τ X}}^{-} (A) \in τ_{X} . \end{array}$

(4) Since $N_{e_{τ X}}^{-} (A) \leq A and N_{e_{τ X}}^{-} (A) \in τ_{X}, N_{e_{τ X}}^{x} \leq N_{_{τ X}}^{x} .$

Since $N_{e_{τ X}}^{-} (A) = \lor {B_{i} | B_{i} \leq A, B_{i} \in τ_{X}} and B_{i} = N_{e_{τ X}}^{-} (B_{i}), \lor B_{i} = N_{e_{τ X}}^{-} (\lor B_{i}) \leq N_{e_{τ X}}^{-} (A) .$

(5) lim_eτX is a principal Alexandrov L-convergence structure from:

$\begin{array}{l} N_{\lim_{e τ_{X}}}^{x} (A) \\ = \land_{F \in A F (X)} (\lim_{e τ X} (F) (x) \to F (A)) \\ = \land_{F \in A F (X)} (e_{A F (X)} (N_{e τ X}^{x}, F) \to F (A)) \\ = N_{e τ X}^{x} (A) .. \end{array}$

For all $\begin{array}{l} F \in A F (X), \lim_{e τ X} (F) (x) = \\ e_{L^{X}} (N_{\lim_{e τ X}}^{x}, F) = e_{L^{X}} (N_{e_{τ X}}^{x}, F) . \end{array}$

Theorem 4.8. Let lim_X be an Alexandrov convergence structure on X. Define Φ^x_limX :AF(X) → L as

$Φ_{\lim_{X}}^{x} (F) = \lim_{X} (F) (x) .$

Then the following properties hold.

(1) Define τ_limX ⊂ L^X as

$τ_{\lim_{X}} = {A \in L^{X} | \begin{array}{l} ⊓ \end{array} Φ_{\lim_{X}}^{x} (A) = A (x), \forall x \in X} .$

Then τ_limX is an Alexandrov L-topology on X.

(2) lim_τlimX ≥ lim _X .

(3) Let τ be an Alexandrov L-topology on X. Then τ_limτ = τ.

Proof. (1) For G ∈ AF(X) and x ∈ X,

$\begin{array}{l} e_{A F (X)} (G ⊓ Φ_{\lim_{X}}^{x}) \\ = \land_{F \in A F (X)} (\lim_{X} (F) (x) \to e_{A F (X)} (G, F)) \\ = e_{A F (X)} (G, \land_{F \in A F (X)} (\lim_{X} (F) (x) \to F)) . \end{array}$

Hence $\begin{array}{l} ⊓ Φ_{\lim_{X}}^{x} (A) = \land_{F \in A F (X)} (\lim_{X} (F) (x) \to F (A)) . \\ Moreover, ⊓ Φ_{\lim_{X}}^{x} (A) \leq \lim_{X} ([x]) (x) \to [x] (A) = A (x) \end{array}$

(AT1) Since $\begin{array}{l} ⊓ Φ_{\lim_{X}}^{-} (α \to α ⊙ A) = α \to ⊓ Φ_{\lim_{X}}^{-} (A), put α = ⊥ . \\ Then ⊓ Φ_{\lim_{X}}^{-} (T_{X}) = T . Since ⊓ Φ_{\lim_{X}}^{-} (A) \leq ⊓ Φ_{\lim_{X}}^{-} \\ (α \to α ⊙ A) = α \to ⊓ Φ_{\lim_{X}}^{-} (α ⊙ A), then α ⊙ ⊓ Φ_{\lim_{X}}^{-} (A) \\ \leq ⊓ Φ_{\lim_{X}}^{-} (α ⊙ A) . Put A = T_{X} . Then α ⊙ ⊓ Φ_{\lim_{X}}^{-} (T_{X}) \leq \\ ⊓ Φ_{\lim_{X}}^{-} (α ⊙ T_{X}) \leq α_{X} . Hence ⊓ Φ_{\lim_{X}}^{-} (α_{X}) = \\ α_{X}; i . e . α_{X} \in τ_{\lim_{X}} . \end{array}$

(AT2) If $\begin{array}{l} A_{i} = ⊓ Φ_{\lim_{X}}^{-} (A_{i}) for all i \in Γ, \\ \lor_{i \in Γ} A_{i} = \lor_{i \in Γ} ⊓ Φ_{\lim X}^{-} (A_{i}) \\ \leq ⊓ Φ_{\lim X}^{-} (\lor_{i \in Γ} A_{i}) \leq \lor_{i \in Γ} A_{i}, \\ \land_{i \in Γ} A_{i} = \land_{i \in Γ} ⊓ Φ_{\lim_{X}}^{-} (A_{i}) = ⊓ Φ_{\lim_{X}}^{-} (\land_{i \in Γ} A_{i}) . \end{array}$

Then $\lor_{i \in Γ} A_{i}, \land_{i \in Γ} A_{i} \in τ_{\lim X} .$

(AT3) If $A = ⊓ Φ_{\lim_{X}}^{-} (A),$

$\begin{array}{l} α \to A = α \to ⊓ Φ_{\lim_{X}}^{-} (A) = ⊓ Φ_{\lim_{X}}^{-} (α \to A), \\ α ⊙ A = α ⊙ ⊓ Φ_{\lim_{X}}^{-} (A) = ⊓ Φ_{\lim_{X}}^{-} \leq α ⊙ A . \end{array}$

Then α ⊙ A, α → A ∈ τlim _X.

(2) For A ∈ L ^X ,

$\begin{array}{l} N_{τ_{\lim X}}^{-} (A) = \lor {B_{i} | ⊓ Φ_{\lim_{X}}^{-} (B_{i}) = B_{i} \leq A} \\ = \lor {B_{i} | N_{\lim_{X}}^{} (B_{i}) = B_{i} \leq A} \\ \leq N_{\lim_{X}}^{-} (A) \end{array}$

because $\lor B_{i} = N_{\lim_{X}}^{-} (\lor B_{i}) \leq N_{\lim_{X}}^{-} (A) .$

For F ∈ AF(X) and x ∈ X,

$\begin{array}{l} \lim_{τ \lim} (F) (x) = e_{A F (X)} (N_{τ \lim}^{x}, F) \geq e_{A F (X)} (N_{\lim}^{x}, F) \\ = \lor_{A \in L^{x}} ((\land_{G \in A F (X)} (\lim (G (x) \to G (A)) \to F (A)) \\ \geq (\lim (F) (x) \to F (A)) \to F (A) \\ \geq \lim (F) (x) . \end{array}$

(3) By Theorem 4.6(4), $τ_{\lim_{τ}} = {A \in L^{X} | N_{\lim_{τ}}^{-} (A) = N_{τ}^{-} (A) = A} = τ_{N_{τ}} = τ .$

Example 4.9. (1) A map [x] : L^X → L defined by [x](A) = A(x) is an Alexandrov L-neighborhood filter on X. From Theorem 4.4, Alexandrov L-convergence lim : AF(X) → L^X is obtained as

$\begin{array}{l} \lim F (x) = e_{A F (X)} ([x], F) = \land_{A \in L^{X}} ([x] (A) \to F (A)) \\ = \land_{y \in X} ([x] (T_{y}^{*}) \to F (T_{y}^{*})) = \land_{x \neq y \in X} F (T_{y}^{*}) . \end{array}$

Moreover, since $\land_{F \in A F (X)} (\lim F (x) \to F (A)) \geq ([x] (A) \to F (A)) \to F (A) \geq [x] (A) and F = [x], \land_{F \in A F (X)} (\lim F (x) \to F (A)) \leq \lim \cup x] (x) \to [x] (A) = [x] (A),$

$\begin{array}{l} N_{\lim}^{x} (A) = ⊓ Φ_{\lim}^{x} (A) \\ = \land_{F \in A F (X)} (\lim F (x) \to F (A)) = [x] (A), \\ τ_{\lim} = {A \in L^{X} | A (x) = N_{\lim}^{x} (A)} = L^{X} \\ N_{τ \lim}^{x} (A) = \lor {B (x) | B \leq A, B \in τ_{\lim} = L^{X}} = [x] (A) \\ \lim_{τ \lim} F (x) = e_{A F (X)} ([x], F) = \lim F (x) . \end{array}$

(2) A map G^x : L^X → L defined by G^x(A) = ∧z∈X[z](A) = ∧z∈X A(z) is an Alexandrov Lneighborhood filter on X. lim : AF(X) → LX as

$\begin{array}{l} \lim F (x) = e_{A F (X)} (G^{x}, F) = \land_{A \in L^{X}} (\land_{z \in X} A (z) \to F (A)) \\ = \land_{y \in X} (\land_{z \in X} Τ_{y}^{*} (z) \to F (T_{y}^{*})) = T . \end{array}$

Since $\begin{array}{l} F (A) = \land_{y \in X} (A^{*} (y) \to F (T_{y}^{*})) = \land_{y \in X} (F^{*} (T_{y}^{*}) \to A (y)) \geq \land_{y \in X} A (y), \land_{F \in A F (X)} F (A) = \land_{y \in X} A (y) . Thus, \\ N_{\lim}^{x} (A) = ⊓ Φ_{\lim}^{x} (A) = \land_{F \in A F (X)} (\lim (F) (x) \to F (A)) \\ = \land_{F \in A F (X)} (T \to F (A)) = \land_{F \in A F (X)} F (A) \\ = G^{x} (A) = \land_{z \in X} A (z), \\ τ_{\lim} = {A \in L^{X} | A (x) = \land_{z \in X} A (z)} = {α \in L} \\ N_{τ \lim}^{x} (A) = \lor {B (x) | B \leq A, B \in τ_{\lim}} = \land_{x \in X} [x] (A) \\ \lim_{τ_{\lim}} F (x) = e_{A F (X)} (\land_{x \in X} [x], F) = \lim F (x) . \end{array}$

(3) Let e_X ∈ L^X×X be an L-fuzzy preorder. For each y ∈ X, maps G^x, F_x : L^X → L defined by G^x(A) = ∧_y∈X(e_X(x, y) → A(y)), F_x(A) = ∧_y∈X(e_X(y, x) → A(y)) are Alexandrov L-neighborhood filters on X.

(3-a) lim₁ : AF(X) → L^X as

$\begin{array}{l} \lim_{1} (F) (x) = e_{A F (X)} (G^{x}, F) \\ = \land_{A \in L^{X}} ((\land_{y \in X} (e_{X} (x, y) \to A (y)) \to F (A)) \\ = \land_{y \in X} (e_{X}^{*} (x, y) \to F (T_{y}^{*})) . \end{array}$

Moreover,

$\begin{array}{l} N_{\lim_{1}}^{x} (A) = ⊓ Φ_{\lim_{1}}^{x} (A) \\ = \land_{F \in A F (X)} (\lim_{1} (F) (x) \to F (A)) \\ = G^{x} (A) = \land_{z \in X} (e_{X} (x, z) \to A (z)), \\ τ_{\lim_{1}} \\ = {A \in L^{X} | A (x) = \land_{z \in X} (e_{X} (x, z) \to A (z))} \\ = {\land_{z \in X} (e_{X} (-, z) \to A (z)) | A \in L^{X}} . \end{array}$

Since $\begin{array}{l} N_{τ_{\lim_{1}}}^{x} (A) = \lor {B (x) | B \leq A, B \in τ_{\lim_{1}}}, \lor B (x) = \\ \lor N_{_{\lim_{1}}}^{x} (B) = N_{_{\lim_{1}}}^{x} (\lor B) \leq N_{_{\lim_{1}}}^{x} (A) . S o, N_{τ_{\lim_{1}}}^{x} (A) \leq N_{_{\lim_{1}}}^{x} (A) . Moreover, s i n c e \\ N_{_{\lim_{1}}}^{x} (N_{_{\lim_{1}}}^{-} (A)) = N_{_{\lim_{1}}}^{x} (A) \leq A, N_{τ_{\lim_{1}}}^{x} (A) = \\ \lor {B (x) | B \leq A, B \in τ_{\lim_{1}}} \geq N_{_{\lim_{1}}}^{x} (A) . Thus, \\ N_{τ_{\lim_{1}}}^{x} (A) = \lor {B (x) | B \leq A, B \in τ_{\lim_{1}}} = N_{_{\lim_{1}}}^{x} (A) \\ {(\lim_{1})}_{τ_{\lim_{1}}} F (x) = e_{A F (X)} (G^{x}, F) = \lim_{1} F (x) . \\ (3 - b) \lim_{2} : A F (X) \to L^{X} as \\ l i m_{2} F (x) = e_{A F (X)} (F_{x}, F) \\ = \land_{A \in L^{X}} (\land_{y \in X} (e_{X} (y, x) \to A (y)) \to F (A)) \\ \land_{y \in X} (e_{X}^{*} (y, x) \to F (T_{y}^{*})) . \end{array}$

Moreover, $\begin{array}{l} N_{\lim_{2}}^{x} (A) = ⊓ Φ_{\lim_{2}}^{x} (A) \\ = \land_{F \in A F (X)} (\lim_{2} (F) (x) \to F (A)) \\ = F_{x} (A) = \land_{z \in X} (e_{X} (z, x) \to A (z)), \\ τ_{\lim_{2}} \\ = {A \in L^{X} | A (x) = \land_{z \in X} (e_{X} (z, x) \to A (z))} \\ = {\land_{z \in X} (e_{X} (z, -) \to A (z)) | A \in L^{X}} \\ N_{τ_{\lim_{2}}}^{x} (A) = \lor {B (x) | B \leq A, B \in τ_{\lim_{2}}} = N_{\lim_{2}}^{x} (A) \\ {(\lim_{2})}_{τ_{\lim_{2}}} F (x) = e_{A F (X)} (F_{x}, F) = \lim_{2} F (x) . \end{array}$

Theorem 4.10. (1) Let (X, lim_X) and (Y, lim_Y) be Alexandrov convergence spaces. If ψ : (X, lim_X) → (Y, lim_Y) is a continuous map, then ψ : (X, lim_τX → (Y, lim_τY) is a continuous map.

(2) Let (X, τX) and (Y, τY) be Alexandrov L-topological spaces. If ψ : (X, τX) → (Y, τY) is a continuous map, then ψ : (X, lim_τX) → (Y, lim_τY) is a continuous map.

Proof. (1) For each B ∈ τ_limY; i.e. $\prod ϕ_{\lim Y}^{y} (B) = B (y), \prod ϕ_{\lim X}^{x} (ψ \leftarrow B))$

\begin{array}{l} = \land_{F \in A F (X)} (ϕ_{\lim_{x}}^{x} (F) \to F (ψ \leftarrow (B))) \\ = \land_{F \in A F (X)} (l i m_{X} (F) (x) \to ψ \Rightarrow (F) (B)) \\ \geq \land_{F \in A F (X)} (l i m_{Y} (ψ \Rightarrow (F)) (ψ (x)) \to ψ \Rightarrow (F) (B)) \\ \geq \land_{G \in A F (Y)} (l i m_{Y} (G) (ψ (x)) \to G (B)) \\ = \land_{G \in A F (Y)} (ϕ_{\lim Y}^{ψ (x)} (G) \to G (B)) \\ = \prod ϕ_{\lim Y}^{ψ (x)} (B) = ψ \leftarrow (B) (x) . \end{array}

Hence ψ←(B) ∈ τlim_X.

(2) For each F ∈ AF(Y),

\begin{array}{l} \lim_{τ Y} (ψ \Rightarrow (F)) (ψ (x)) = e_{A F (Y)} (N_{τ Y}^{ψ (x)}, ψ \Rightarrow (F)) \\ = \land_{B \in L^{Y}} (N_{τ Y}^{ψ (x)} (B)) \to ψ \Rightarrow (F) (B) \\ \geq \land_{B \in L^{Y}} (N_{τ X}^{x} (ψ \leftarrow (B)) \to F (ψ \leftarrow (B))) \\ \geq \land_{A \in L^{X}} (N_{τ X}^{x} (A) \to F (A)) \\ = l i m_{τ X} (F) (x) . \end{array}

Hence lim_τx(F)(x) ≤ lim_τY(ψ⇒F))(ψ(x).

Let ACS be denote the category of Alexandrov L-convergence spaces and continuous mappings for morphisms.

Theorem 4.11. (1) F: ACS→ ATS defined asF(X, lim_X) = (X, τ_limX) is a functor.

(2) G: ATS→ ACS defined as G(X, τx) = (X, lim_τx) is a functor.

(3) The pair (F, G) is a Galois correspondence between ATS and ACS.

Proof. (1) and (2) follow from Theorem 4.10.

(3) By Theorem 4.8(3), if (X, τ_X) is an Alexandrov L-topological space, then F(G(X, τ_X)) = (X, τlim_τX = τ_X). Hence, the identity map id_X : (X, τlimτ_X) = F(G(X, τ_X)) → (X, τ_X) is a continuous map. Moreover, if (X, lim_X) is an Alexandrov L-convergence space, by Theorem 4.8(2), lim_τlimX ≥ lim_X. Hence the identity map id_X : (X, lim_X) → G(F(X, lim_X)) = (X, lim_τlimX) is a continuous map. Therefore (F, G) is a Galois correspondence.

Theorem 4.12. Let (Y_i, lim_i) be Alexandrov convergence spaces and ψ_i : X → Y_i be a map for each i ∈ Γ. Define lim : AF(X) → LX as

$\lim (F) (x) = \underset{i \in Γ}{\land} \lim_{i} (ψ_{i}^{\Rightarrow} (F) (ψ (x))) .$

Then the following properties hold.

(1) (X, lim) is an Alexandrov convergence space.

(2) Let ψ : (Z, limZ) → (X, lim) be a map. If ψ_i o ψ is continuous, then ψ is a continuous map.

Proof. (1) Since $\begin{array}{l} ψ_{i}^{\Rightarrow} ([x]) (B) = [x] (ψ_{i}^{\leftarrow} (B)) = ψ_{i}^{\leftarrow} (B) (x) = B (ψ_{i} (x)) = [ψ_{i} (x)] (B), \\ \lim ([x]) (x) = \land_{i \in Γ} \lim_{i} (ψ_{i}^{\Rightarrow} ([x]) (ψ_{i} (x)) \\ = \land_{i \in Γ} \lim_{i} ([ψ_{i} (x)]) (ψ (x)) = T . \\ l i m (F) (x) \to \lim (G) (x) \\ \geq \land_{i \in Γ} \lim_{i} (ψ_{i}^{\Rightarrow} (F) (ψ (x)) \to \land_{i \in Γ} \lim_{i} (ψ_{i}^{\Rightarrow} (G) (ψ (x)))) \\ \geq \land_{i \in Γ} e_{A F (Y)} (ψ_{i}^{\Rightarrow} (F), ψ_{i}^{\Rightarrow} (G)) \\ \geq \land_{i \in Γ} e_{A F (X)} (F, G) = e_{A F (X)} (F, G) . \end{array}$

(2) We show that for each F ∈ AF(Z),

$\lim_{Z} (F) (z) \leq \lim (ψ^{\Rightarrow} (F)) (ψ (z)) .$

Since ψ_i o ψ is continuous, lim_z(F)(z) ≤ $\lim_{i} ({(ψ_{i} o ψ)}^{\Rightarrow} (F)) (ψ_{i} o ψ (x)) .$

$\begin{array}{l} \lim_{Z} (F) (z) \leq \land_{i \in Γ} \lim_{i} ({(ψ_{i} o ψ)}^{\Rightarrow} (F)) (ψ_{i} o ψ (z)) \\ = \land_{i \in Γ} \lim_{i} (ψ_{i}^{\Rightarrow} (ψ^{^{\Rightarrow}} (F)) (ψ_{i} (ψ (z)))) \\ = \lim (ψ^{\Rightarrow} (F)) (ψ (z)) . \end{array}$

Example 4.13. Let ([0, 1],⊙, →, *, 0, 1) be a complete residuated lattice (ref.[1, 4, 25]) as

$\begin{array}{l} x ⊙ y = \max {0, x + y - 1}, \\ x \to y = \min {1 - x + y, 1} \\ x^{*} = 1 - x . \end{array}$

Let X = {x, y, z} and B ∈ [0, 1] ^X with B(x) = 0.6, B(y) = 0.8, B(z) = 0.3. Then we obtain an [0, 1]-fuzzy preorder e_X with e_X(x, y) = B(x) → B(y) as

e_{X} = (\begin{matrix} 1 & 1 & 0.7 \\ 0.8 & 1 & 0.5 \\ 1 & 1 & 1 \end{matrix})

Define N^x : L^X → L as

N^{x} (A) = \underset{y \in X}{\land} (e_{X} (x, y) \to A (y))

From Example 3.5(4), {N^x | x ∈ X} is a topological Alexandrov [0, 1]-neighborhood system. From Theorem 3.4, we can obtain Alexandrov [0, 1]-topology τ_N as

τ_{N} = {\underset{y \in Y}{\land} (e_{X} (-, y) \to A ()) | A \in {[0, 1]}^{X}} .

Put $A = T_{z}^{*}, e_{X}^{*} (-, z) \in τ_{N} .$ From Theorem 4.7,

\begin{matrix} e_{τ N} (x, z) = \land_{A \in τ_{N}} (A (x)) \to A (z)) \\ \leq (e_{X}^{*} (-, z) (x) \to e_{X}^{*} (-, z) (z)) = e_{x} (x, z), \end{matrix}

\begin{array}{l} e_{τ N} (x, z) = \land_{A \in τ_{N}} (A (x)) \to A (z)) \\ = \land_{A \in L^{X}} (\land_{y \in Y} (e_{X} (x, y)) \to A (y)) \\ \to \land_{y \in Y} ((e_{X} (z, y)) \to A (y)) \\ \geq \land_{A \in L^{X}} (\land_{y \in Y} (e_{X} (x, y)) \to A (y)) \to (e_{X} (z, y)) \to A (y)) \\ \geq \land_{y \in Y} ((e_{X} (z, y)) \to (e_{X} (x, y)) \geq e_{X} (x, z) . \end{array}

Hence e_τN (x, z) = e_X(x, z).

Since {N_x | x ∈ X} is a topological Alexandrov[0, 1]-neighborhood system, by Theorem 3.4(7),

\begin{array}{l} N_{τ N}^{x} (A) = N^{x} (A) = \land_{y \in X} (e_{X} (x, y) \to A (y)), \\ l i m_{N} (F) (x) = e_{A F} (X) (N^{x}, F) \\ = \land_{A \in {[0, 1]}^{X}} (\land_{y \in X} (e_{X} (x, y) \to A (y)) \to F (A)) \\ \land_{y \in X} (e_{X}^{*} (x, y) \to F (1_{y}^{*})) . \end{array}

\begin{array}{l} N_{\lim N}^{x} (A) \\ = \land_{F \in A F (X)} (l i m_{N} (F) (x) \to F (A)) \\ = \land_{F \in A F (X)} (e_{A F (X)} (N^{x}, F) \to F (A)) \\ = N^{x} (A) = \land_{y \in X} (e_{X} (x, y) \to A (y)) . \end{array}

(2) Let Y = {a, b, c} and ψ : X → Y defined as ψ(x) = ψ(y) = a, ψ(z) = b. Let N^x be an Alexandrov filter on X. From Theorem 3.9,

ψ \Rightarrow (N^{x}) (B) = N^{x} (ψ \leftarrow (B)) .

Then ψ⇒(N^x) is an Alexandrov [0, 1]-filter on Y.

Let e_Y ∈ [0, 1]^Y×Y be an [0, 1]-fuzzy preorder as

e Y = (\begin{matrix} 1 & 0.6 & 0.5 \\ 0.8 & 1 & 0.6 \\ 0.8 & 0.7 & 1 \end{matrix})

Define N^y : [0, 1]^Y → [0, 1] as

N^{y} (B) = \underset{z \in Y}{\land} (e_{Y} (y, z)) \to B (z)) .

From Example 3.5(4), N^y is a topological Alexandrov [0, 1]-neighborhood filter.

\begin{array}{l} \lim_{Y} (G) (x) = e_{A F (Y)} (N^{y}, G) \\ = \land_{B \in L^{Y}}^{(\land_{y \in X} (e_{Y}^{*} (x, y) \to) G (T_{y}^{*})),} \end{array}

\begin{array}{l} l i m_{X} (F) (x) = l i m y (ψ (x)) \\ = e_{A F} (Y) (N^{ψ (x)}), ψ \Rightarrow (F)) \\ = \land_{y \in X} (e_{Y}^{*} (ψ (x), y) \to F (ψ \leftarrow (T_{y}^{*}))) \\ = (0.4 \to F (ψ \leftarrow (T_{b}^{*}))) \land (0.5 \to F (ψ \leftarrow (T_{C}^{*}))), \\ \lim_{X} (F) (y) = l i m_{X} (F) (x), \\ \lim_{X} (F) (z) = l i m_{y} (ψ \Rightarrow F) (F)) (ψ (z)) \\ = \land_{y \in X} (e_{Y}^{*} (b, y) \to F (ψ \leftarrow (T_{y}^{*})) \\ = (0.2 \to F (ψ \leftarrow (T_{a}^{*}))) \land (0.4 \to F (ψ \leftarrow (T_{c}^{*}))) . \end{array}

5. Conclusion

In this paper, we investigate the relations among Alexandrov L-neighborhood filters, Alexandrov L-topologies and Alexandrov L-convergence structures as a viewpoint for fuzzy rough sets. There exists a Galois correspondence between the category of Alexandrov L-neighborhood spaces and that of Alexandrov L-topological spaces. Moreover, there exists a Galois correspondence between the category of Alexandrov L-topological spaces and that of Alexandrov L-convergence spaces. Moreover, we investigate the their topological properties and give their examples.

In the future, we study the relations among Alexandrov L-uniform convergence spaces, Alexandrov L-convergence structures and Alexandrov L-fuzzy topological spaces. Using Alexandrov L-topological structures, we investigate some applications in rough sets or soft sets.

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