Alexandrov L -preuniform convergence structures

Abstract

In this paper, we introduce the notion of Alexandrov L-preuniform convergence spaces based on the concepts of Alexandrov L-filters on X × X. We investigate the categorical relations among Alexandrov L-preuniform convergence spaces, Alexandrov L-preconvergence spaces and reflexive L-fuzzy relations. Moreover, we investigate their topological properties and give their examples as a viewpoint of the topological structure for fuzzy information and fuzzy rough sets.

Keywords

Complete residuated lattices fuzzy join(meet)-complete lattices

1. Introduction

Höhle [9] introduced an L-filter which is an important mathematical tool as fuzzy topological structures for many valued logics [1–13 , 23–28]. Jäger [10 –13] introduced stratified L-convergence structures based on the concepts of L-filters where L is a complete Heyting algebra. Yao [28] extended stratified L-convergence structures to complete residuated lattices and investigated the relations between stratified L-convergence structures and L-fuzzy topological spaces. Zhang [27 , 32] defined a strong L-topology on the concepts of fuzzy complete lattices. As an extension of Yao [28], Fang [5 –8] introduced L-ordered convergence structures on L-ordered filters and investigated the relations between L-ordered convergence structures and strong L-topological spaces. Many researchers [2–8 , 23–26] developed the properties of L-convergence structures.

Jäger and Burton [13] introduced stratified L-uniform convergence spaces in a complete Heyting algebra. Fang [6 –8] generalized stratified L-semiuniform (L-ordered semiuniform, L-ordered quasiuniform, preuniform) convergence spaces in commutative unital quantales [9].

For an extension of Pawlak’s rough set [22], many researchers [1 , 19] developed L-lower and L-upper approximation operators. By using these concepts, information systems and decision makings were investigated [14 , 30]. Kim et al. [3, 17] introduced the notion of Alexandrov L-(neighborhood) filters defined as a meet preserving map in a Zhang’s fuzzy complete lattice sense [27 , 32]. These filters have stronger conditions than L-filters and are suitable to fuzzy rough sets, Alexandrov L-topologies and Alexandrov L-preconvergence structures.

In this paper, we introduce the notion of Alexandrov L-preuniform convergence spaces based on the concepts of Alexandrov L-filters on X × X. In Theorems 3.3 and 3.4, we show that the set of all Alexandrov L-preuniform convergence structures on X × X is a meet-complete lattice in a Zhang’s fuzzy complete lattice sense [27 , 32] as a generalization of a complete lattice. In Theorems 3.6, 3.14 and 3.17, there exists a Galois correspondence between the category of Alexandrov L-preuniform convergence spaces and that of reflexive L-relations. In Theorems 3.9, 3.11, 3.13 and 3.16, there exists a Galois correspondence between the category of Alexandrov L-preuniform convergence spaces and that of Alexandrov L-convergence spaces. We investigate their topological properties and give their examples as a viewpoint of the topological structure for fuzzy information and fuzzy rough sets in a complete residuated lattice.

2. Preliminaries

Definition 2.1. [1, 9] An algebra (L, ∧, ∨, ⊙, →, ⊥, ⊤) is called a complete residuated lattice if it satisfies the following conditions:

(L1) (L, ≤, ∨, ∧, ⊥, ⊤) is a complete lattice with the greatest element ⊤ and the least element ⊥;

(L2) (L, ⊙, ⊤) is a commutative monoid;

(L3) 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 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 and [x] (A) = A (x).

Lemma 2.2. [1, 9] For each x, y, z, x _i, y _i, w ∈ L, we have the following properties.

(1) ⊤ → x = x, ⊥ ⊙ x = ⊥,

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

(3) x ≤ y iff x→ y = ⊤.

(4) x → (⋀ _i y _i) = ⋀ _i (x → y _i),

(5) (⋁ _i x _i) → y = ⋀ _i (x _i → y),

(6) x ⊙ (⋁ _i y _i) = ⋁ _i (x ⊙ y _i),

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

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

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

(10) ⋁_i∈Γ x _i → ⋁ _i∈Γ y _i ≥ ⋀ _i∈Γ (x _i → y _i) and ⋀_i∈Γ x _i → ⋀ _i∈Γ y _i ≥ ⋀ _i∈Γ (x _i → y _i),

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

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

Definition 2.3. [1, 9] Let X be a set. A function e _X : X × X → L is called: (E1) reflexive if e _X (x, x) =⊤ for all x ∈ X, (E2) transitive if e _X (x, y) ⊙ e _X (y, z) ≤ e _X (x, z), for all x, y, z ∈ X, (E3) anti-symmetric 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.

Definition 2.4. [27 , 32] Let (X, e _X) be an L-fuzzy partially ordered set and A ∈ L ^X. (1) A point x ₀ is called a join of A, denoted by x ₀ = ⊔ A, if it satisfies (J1) A (x) ≤ e _X (x, x ₀), (J2) ⋀_x∈X (A (x) → e _X (x, y)) ≤ e _X (x ₀, y). A point x ₁ is called a meet of A, denoted by x ₁ = ⊓ A, if it satisfies (M1) A (x) ≤ e _X (x ₁, x), (M2) ⋀_x∈X (A (x) → e _X (y, x)) ≤ e _X (y, x ₁).

Remark 2.5. [27 , 32] Let (X, e _X) be an L-fuzzy partially ordered set and Φ ∈ L ^X. (1) A point x ₀ is a join of Φ iff ⋀_x∈X (Φ (x) → e _X (x, y)) = e _X (x ₀, y). (2) A point x ₁ is a meet of Φ iff ⋀_x∈X (Φ (x) → e _X (y, x)) = e _X (y, x ₁).

Definition 2.6. [27 , 32] Let (X, e _X) be an L-fuzzy partially ordered set. The pair (X, e _X) is called a fuzzy join (resp. meet) complete lattice if ⊔Φ (resp. ⊓Φ) exists for each Φ ∈ L ^X. The pair (X, e _X) is called a fuzzy complete lattice if ⊔Φ and ⊓Φ exist for each Φ ∈ L ^X.

Definition 2.7. [17 , 32] Let (X, e _X) and (Y, e _Y) be fuzzy complete lattices and ψ : X → Y a map. (1) ψ is a join preserving map if ψ (⊔ Φ) = ⊔ ψ ^→ (Φ) for all Φ ∈ L ^X, where ψ ^→ (Φ) (y) = ⋁ _ψ(x)=y Φ (x). (2) ψ is a meet preserving map if ψ (⊓ Φ) = ⊓ ψ ^→ (Φ) for all Φ ∈ L ^X. (3) ψ is an order preserving map if e _X (x, y) ≤ e _Y (ψ (x), ψ (y)) for all x, y ∈ X.

Definition 2.8. [3, 17] Let (L ^X, e _{L
^X}) and (L, e _L) be L-fuzzy partially ordered sets. A map $F : (L^{X},$ e _{L
^X}) → (L, e _L) is called an Alexandrov L-filter on X iff $F (⊓ Φ) = ⊓ F^{\to} (Φ)$ for all Φ ∈ L ^{L
^X}. Let AF (X) denote the set of all Alexandrov L-filters on X.

Theorem 2.9. [3, 17] A map $F : L^{X} \to L$ is an Alexandrov L-filter on X iff it satisfies the following conditions:

(F1) $F (⋀_{i \in Γ} A_{i}) = ⋀_{i \in Γ} F (A_{i})$ for all A _i ∈ L ^X.

(F2) $F (α \to A) = α \to F (A)$ for all A ∈ L ^X and α ∈ L.

Definition 2.10. [3, 17] A family $N_{X} = {N^{x} ∣ x \in X}$ is called an Alexandrov L-neighborhood system on X if, for x ∈ X, a map $N^{x} : L^{X} \to L$ satisfies: (N1) $N^{x}$ is an Alexandrov L-filter on X. (N2) $N^{x} (A) \leq A (x)$ for all A ∈ L ^X. The pair $(X, N_{X})$ 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 $N^{-} (A) \in L^{X}$ 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. A map $ψ : (X, N_{X}) \to (Y, N_{Y})$ is called N-continuous if, for each B ∈ L ^Y and x ∈ X, $N_{Y}^{ψ (x)} (B) \leq N_{X}^{x} (ψ^{\leftarrow} (B)) .$

Definition 2.11. [3, 17] A map lim : AF (X) → L ^X is called an Alexandrov L-preconvergence structure on X iff it satisfies the following conditions: (AC1) lim [x] (x) =⊤. (AC2) $e_{AF (X)} (F, G) \leq e_{L^{X}} (lim (F), lim (G))$ for all $F, G \in AF (X)$ . An Alexandrov L-preconvergence structure lim is called Alexandrov L-convergence structure on X if it satisfies: (A) ⋁_y∈X (lim [y] (x) ⊙ lim [z] (y)) = lim [z] (x) for all x, y ∈ X. Let AC (X) denote the set of all Alexandrov L-preconvergence structures on X. A map $ψ : (X, lim_{X}) \to (Y, lim_{Y})$ is called continuous if $lim_{X} (F) (x) \leq lim_{X} ψ^{\Rightarrow} (F) (ψ (x))$ for all $F \in AF (X)$ and x ∈ X where $ψ^{\Rightarrow} (F) (B) = F (ψ^{\leftarrow} (B))$ for all B ∈ L ^Y.

Definition 2.12. [3, 17] An Alexandrov L-preconvergence structure $lim_{X} : AF (X) \to L^{X}$ is called principal if there exists an Alexandrov L-neighborhood system $N = {N^{x} ∣ x \in X}$ such that (P) $lim_{X} (F) (x) = e_{AF (X)} (N^{x}, F)$ for all $F \in AF (X)$ .

3. Alexandrov L-preuniform convergence spaces

Definition 3.1. A map U _X : AF (X × X) → L is called an Alexandrov L-preuniform convergence structure on X × X iff it satisfies the following conditions:

(AU1) U _X ([(x, x)]) =⊤.

(AU2) $e_{AF (X \times X)} (U, V) \leq e_{L} (U_{X} (U), U_{X} (V))$ for all $U, V \in AF (X \times X)$ .

An Alexandrov L-preuniform convergence structure U _X is called an Alexandrov L-quasiuniform convergence structure if U _X ([(x, y)]) ⊙ U _X ([(y, z)]) ≤ U _X ([(x, z)]) for each x, y, z ∈ X.

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

A map ψ : (X, U _X) → (Y, U _Y) is called uniformly continuous if $U_{X} (U) \leq U_{Y} (ψ \times ψ)^{\to} (U)$ for all $U \in AF (X \times X)$ where $(ψ \times ψ)^{\to} (U) (w) = U ((ψ \times ψ)^{\leftarrow} (w))$ for all w ∈ L ^Y×Y.

Lemma 3.2. Let U _X : AF (X × X) → L be a map. The following statements are equivalent.

(1) $e_{AF (X \times X)} (V, W) \leq e_{L} (U_{X} (V), U_{X} (W))$ for all $V, W \in AF (X \times X)$ .

(2) If $V \leq W$ , then $U_{X} (V) \leq U_{X} (W)$ and $U_{X} (α \to V) \leq α \to U_{X} (V)$ for all $V, W \in AF (X \times X)$ and α ∈ L.

Proof. (1) ⇒ (2). If $V \leq W$ , then $⊤ = e_{AF (X \times X)} (V, W) \leq e_{L} (U_{X} (V), U_{X} (W))$ . Hence $U_{X} (V) \leq U_{X} (W)$ . Put $W = α \to V$ . Then $α \leq e_{AF (X \times X)} (α \to V, V) \leq e_{L} (U_{X} (α \to V), U_{X} (V)) .$ Hence $U_{X} (α \to V) \leq α \to U_{X} (V) .$

(2) ⇒ (1). Since $V \leq e_{AF (X \times X)} (V, W) \to W$ ,

$U_{X} (V) \leq U_{X} (e_{AF (X \times X)} (V, W) \to W) \leq e_{AF (X \times X)} (V, W) \to U_{X} (W)$ . Hence $e_{AF (X \times X)} (V, W) \leq e_{L} (U_{X} (V), U_{X} (W)) .$ □

Theorem 3.3. Let AU (X) denote the set of all Alexandrov L-preuniform convergence structures on X × X. Then (1) α → U ∈ AU (X) for each U ∈ AU (X)

(2) ⋀_i∈Γ U _i, ⋁ _i∈Γ U _i ∈ AU (X) for each U _i ∈ AU (X), i ∈ Γ.

Proof. (1) (AU1) (α→ U _X) ([(x, x)]) = α → U ([(x, x)]) = α → ⊤ = ⊤.

(AU2) Since $e_{AF (X \times X)} (V, W) \leq e_{L} (U (V), U (W))$ for all $V, W \in AF (X \times X)$ , $\begin{matrix} e_{L} (α \to U_{X} (U), α \to U_{X} (V)) \\ \geq e_{L} (U_{X} (U), U_{X} (V)) (by Lemma 2.2 (11)) \\ \geq e_{AF (X \times X)} (U, V) . \end{matrix}$

(2) It is easily proved. □

Theorem 3.4. Let AU (X) denote the set of all Alexandrov L-preuniform convergence structures on X × X. Define e _AU(X) : AU (X) × AU (X) → L as

e_{AU (X)} (U_{1}, U_{2}) = ⋀_{U \in AF (X \times X)} (U_{1} (U) \to U_{2} (U)) .

Then (AU (X), e _AU(X)) is a meet-complete lattice.

Proof. We easily prove that (AU (X), e _AU(X)) is an L-fuzzy partially ordered set. For Φ : AU (X) → L, since ⋀_U∈AU(X) (Φ (U) → U) ∈ AU (X) from Theorem 3.3, we have $\begin{matrix} e_{AU (X)} (U_{1}, ⊓ Φ) \\ = ⋀_{U \in AU (X)} (Φ (U) \to e_{AU (X)} (U_{1}, U)) \\ = e_{AU (X)} (U_{1}, ⋀_{U \in AU (X)} (Φ (U) \to U)) . \end{matrix}$ Hence ⊓Φ = ⋀ _U∈AU(X) (Φ (U) → U) ∈ AU (X) . □

Remark 3.5. Let AU (X) denote the set of all Alexandrov L-preuniform convergence structures on X × X. Define Φ : AU (X) → L as Φ (U) = α for U ∈ AU (X) and Φ (U ₁) =⊥, otherwise. Then $⊔ Φ = ⋁_{U \in AU (X)} (Φ (U) ⊙ U) = α ⊙ U \notin AU (X) .$

Hence (AU (X), e _AU(X)) is not a join-complete lattice.

Theorem 3.6. (1) Let e _X be a reflexive L-fuzzy relation (resp. L-fuzzy preorder) on X. Define U _{e
_X} : AF (X × X) → L as

U_{e_{X}} (U) = ⋀_{x, y \in X} (e_{X}^{*} (x, y) \to U (⊤_{(x, y)}^{*})) .

Then U _{e
_X} is an Alexandrov L-preuniform (resp. quasiuniform) convergence on X × X.

(2) Let (X, U) be an Alexandrov L-preuniform (resp. quasiuniform) convergence space. Define e _U : X × X → L as $e_{U} (x, y) = U ([(x, y)]) .$

Then e _U is a reflexive L-fuzzy relation (resp. L-fuzzy preorder).

(3) e _{U
_{e
_X}} = e _X and U _{e
_U} ≥ U.

Proof. (1) (AU1) For x ∈ X, $\begin{matrix} U_{e_{X}} ([(x, x)]) \\ = ⋀_{x, y \in L^{X \times X}} (e^{*} (x, y) \to [(x, x)] (⊤_{(x, y)}^{*})) \\ = e^{*} (x, x) \to [(x, x)] (⊤_{(x, x)}^{*}) = ⊤ . \end{matrix}$ (AU2) $e_{AF (X \times X)} (U, V) = ⋀_{x, y \in X} (U (⊤_{(x, y)}^{*}) \to V (⊤_{(x, y)}^{*}))$ from: $\begin{matrix} e_{AF (X \times X)} (U, V) \leq ⋀_{x, y \in X} (U (⊤_{(x, y)}^{*}) \to V (⊤_{(x, y)}^{*})), \end{matrix}$ for $u = ⋀_{x, y \in X} (u^{*} (x, y) \to ⊤_{(x, y)}^{*})$ , by Theorem 2.9 and Lemma 2.2(11), $\begin{matrix} e_{AF (X \times X)} (U, V) = ⋀_{u \in L^{X \times X}} (U (u) \to V (u)) \\ = ⋀_{u \in L^{X \times X}} (⋀_{x, y \in X} (u^{*} (x, y) \to U (⊤_{(x, y)}^{*})) \\ \to ⋀_{x, y \in X} (u^{*} (x, y) \to V (⊤_{(x, y)}^{*}))) \\ \geq ⋀_{x, y \in X} (U (⊤_{(x, y)}^{*}) \to V (⊤_{(x, y)}^{*})) . \end{matrix}$ $\begin{matrix} e_{L} (U_{e_{X}} (U), U_{e_{X}} (V)) \\ = ⋀_{x, y \in X} (e_{X}^{*} (x, y) \to U (⊤_{(x, y)}^{*})) \\ \to ⋀_{x, y \in X} (e_{X}^{*} (x, y) \to V (⊤_{(x, y)}^{*})) \\ \geq ⋀_{x, y \in X} (U (⊤_{(x, y)}^{*}) \to V (⊤_{(x, y)}^{*})) \\ = e_{AF (X \times X)} (U, V) . \end{matrix}$

If e _X is an L-fuzzy preorder on X, U _{e
_X} is an Alexandrov L-quasiuniform convergence on X × X from U _{e
_X} ([(x, y)]) = e _X (x, y).

(2) It is easily proved from (1).

(3) e _{U
_{e
_X}} (x, y) = U _{e
_X} ([(x, y)]) = e _X (x, y) . Since $⋀_{x, y \in X} (U^{*} (⊤_{(x, y)}^{*}) \to [(x, y)] (u)) = ⋀_{x, y \in X} (u (x, y)^{*} \to U (⊤_{(x, y)}^{*})) = U (⋀_{x, y \in X} (u (x, y)^{*} \to ⊤_{(x, y)}^{*})) = U (u),$ $U = ⋀_{x, y \in X} (U^{*} (⊤_{(x, y)}^{*}) \to [(x, y)])$ . Thus, $\begin{matrix} U_{e_{U}} (U) = ⋀_{x, y \in X} (e_{U}^{*} (x, y) \to U (⊤_{(x, y)}^{*})) \\ = ⋀_{x, y \in X} (U^{*} ([(x, y)]) \to U (⊤_{(x, y)}^{*})) \\ = ⋀_{x, y \in X} (U^{*} (⊤_{(x, y)}^{*}) \to U ([(x, y)])) \\ \geq U (⋀_{x, y \in X} (U^{*} (⊤_{(x, y)}^{*}) \to [(x, y)])) = U (U) \\ (by Lemma 3.2 (2)) . \end{matrix}$ □

Theorem 3.7. Let ${N^{x} ∣ x \in X}$ be an Alexandrov L-neighborhood system on X. Define $lim_{N} : AF (X) \to L^{X}$ as

lim_{N} F (x) = e_{AF (X)} (N^{x}, F) .

Then the following properties hold.

(1) $lim_{N}$ is an Alexandrov L-preconvergence structure with $lim_{N} F (x) = ⋀_{y \in X} (N^{x} (⊤_{y}^{*}) \to F (⊤_{y}^{*})) .$

(2) If ${N^{x} ∣ x \in X}$ is topological, then $lim_{N}$ is an Alexandrov L-convergence structure on X.

Proof. (1) $lim_{N} [x] (x) = ⊤$ and $\begin{matrix} e_{L^{X}} (lim_{N} (F), lim_{N} (G)) \\ = ⋀_{x \in X} (e_{AF (X)} (N^{x}, F) \to e_{AF (X)} (N^{x}, G)) \\ \geq e_{AF (X)} (F, G) . \end{matrix}$

For $A = ⋀_{y \in X} (A^{*} (y) \to ⊤_{y}^{*})$ , $\begin{matrix} lim_{N} F (x) = ⋀_{A \in L^{X}} (N^{x} (A) \to F (A)) \\ = ⋀_{A \in L^{X}} (⋀_{y \in X} (A^{*} (y) \to N^{x} (⊤_{y}^{*})) \\ \to ⋀_{y \in X} (A^{*} (y) \to F (⊤_{y}^{*}))) \\ \geq ⋀_{y \in X} (N^{x} (⊤_{y}^{*}) \to F (⊤_{y}^{*})) . \\ lim_{N} F (x) \leq ⋀_{y \in X} (N^{x} (⊤_{y}^{*}) \to F (⊤_{y}^{*})) . \end{matrix}$

(2) Since $N^{-} (⊤_{y}^{*}) = ⋀_{z \in X} ((N^{z} (⊤_{y}^{*}))^{*} \to ⊤_{z}^{*})$ , $\begin{matrix} N^{x} (N^{-} (⊤_{y}^{*})) = N^{x} (⋀_{z \in X} ((N^{z} (⊤_{y}^{*}))^{*} \to ⊤_{z}^{*})) \\ = ⋀_{z \in X} ((N^{z} (⊤_{y}^{*}))^{*} \to N^{x} (⊤_{z}^{*})) = N^{x} (⊤_{y}^{*}) \\ iff ⋁_{z \in X} ((N^{z} (⊤_{y}^{*}))^{*} ⊙ (N^{x} (⊤_{z}^{*}))^{*}) = (N^{x} (⊤_{y}^{*}))^{*} . \end{matrix}$ Since $lim_{N} [y] (x) = ⋀_{y \in X} (N^{x} (⊤_{y}^{*}) \to [y] (⊤_{y}^{*})) = (N^{x} (⊤_{y}^{*}))^{*},$ $\begin{matrix} ⋁_{z \in X} ((N^{z} (⊤_{y}^{*}))^{*} ⊙ (N^{x} (⊤_{z}^{*}))^{*}) \\ = ⋁_{z \in X} (lim_{N} [y] (z) ⊙ lim_{N} [z] (x)) = lim_{N} [y] (x) . \end{matrix}$ □

Lemma 3.8. (1) For $F \in AF (X)$ and x ∈ X, there exists $U_{F}^{x} \in AF (X \times X)$ with $U_{F}^{x} (u) = F (u (x, -)) .$

(2) For each x, y ∈ X, $U_{[y]}^{x} = [(x, y)] .$

(3) Let ψ : X → Y be a map. For $F \in AF (X)$ and x ∈ X, $(ψ \times ψ)^{\Rightarrow} (U_{F}^{x}) = U_{ψ^{\Rightarrow} (F)}^{ψ (x)} .$

(4) For $U \in AF (X \times X)$ , $(ψ \times ψ)^{\Rightarrow} (U) \in AF (Y \times Y)$ . Moreover, $e_{AF (X \times X)} (U, V) \leq e_{AF (X \times X)} ((ψ \times ψ)^{\Rightarrow} (U), (ψ \times ψ)^{\Rightarrow} (V)) .$

(5) $e_{AF (X \times X)} (U_{F}^{x}, U) = (⋀_{y \in X} (F (⊤^{*} (y)) \to U (⊤_{(x, y)}^{*})) \land (⋀_{w \neq x, y \in X} U (⊤_{(w, y)}^{*}))$ .

(6) $e_{AF (X \times X)} (U_{F}^{x}, [(x, y)]) = F^{*} (⊤_{y}^{*})$ . If x ≠ z, $e_{AF (X \times X)} (U_{F}^{x}, [(z, y)]) = ⊥$ .

Proof. (1) Since $F \in AF (X)$ , $\begin{matrix} U_{F}^{x} (α \to u) = F (α \to u (x, -)) \\ = α \to F (u (x, -)) = α \to U_{F}^{x} (u), \\ U_{F}^{x} (⋀_{i \in I} u_{i}) = F (⋀_{i \in I} u_{i} (x, -)) \\ = ⋀_{i \in I} F (u_{i} (x, -)) = ⋀_{i \in I} U_{F}^{x} (u_{i}) . \end{matrix}$

(2) $U_{[y]}^{x} (u) = [y] (u (x, -)) = u (x, y) = [(x, y)] (u) .$

(3) Since (ψ × ψ) ^← (w) (x, y) = ψ ^← (w (ψ (x), -)) (y) = w (ψ (x), ψ (y)), $\begin{matrix} (ψ \times ψ)^{\Rightarrow} (U_{F}^{x}) (w) = U_{F}^{x} ((ψ \times ψ)^{\leftarrow} (w)) \\ = F ((ψ \times ψ)^{\leftarrow} (w) (x, -)), \\ U_{ψ^{\Rightarrow} (F)}^{ψ (x)} (w) = ψ^{\Rightarrow} (F) (w (ψ (x), -)) \\ = F (ψ^{\leftarrow} (w (ψ (x), -))) . \end{matrix}$

(4) Since $(ψ \times ψ)^{\Rightarrow} (U) (α \to v) = U ((ψ \times ψ)^{\leftarrow} (α \to v)) = α \to U ((ψ \times ψ)^{\leftarrow} (v))$ and $(ψ \times ψ)^{\Rightarrow} (U) (⋀_{i \in Γ} v_{i}) = ⋀_{i \in Γ} (ψ \times ψ)^{\Rightarrow} (U) (v_{i})$ , (ψ × ψ) ^⇒ $(U) \in AF (Y \times Y)$ . Moreover, $\begin{matrix} e_{AF (X \times X)} ((ψ \times ψ)^{\Rightarrow} (U), (ψ \times ψ)^{\Rightarrow} (V)) \\ = ⋀_{v \in L^{Y \times Y}} ((ψ \times ψ)^{\Rightarrow} (U) (v) \to (ψ \times ψ)^{\Rightarrow} (V) (v)) \\ = ⋀_{v \in L^{Y \times Y}} (U ((ψ \times ψ)^{\leftarrow} (v)) \to V ((ψ \times ψ)^{\leftarrow} (v))) \\ \geq e_{AF (X \times X)} (U, V) . \end{matrix}$

(5) For $U \in AF (X \times X)$ , $\begin{matrix} e_{AF (X \times X)} (U_{F}^{x}, U) \\ = ⋀_{u \in L^{X \times X}} (U_{F}^{x} (u) \to U (u)) \\ = ⋀_{u \in L^{X \times X}} (⋀_{w, y \in X} (u^{*} (w, y) \to U_{F}^{x} (⊤_{(w, y)}^{*})) \\ \to ⋀_{w, y \in X} (u^{*} (w, y) \to U (⊤_{(w, y)}^{*})) \\ \geq ⋀_{w, y \in X} (U_{F}^{x} (⊤_{(w, y)}^{*}) \to U (⊤_{(w, y)}^{*})), \\ e_{AF (X \times X)} (U_{F}^{x}, U) \\ \leq ⋀_{w, y \in X} (U_{F}^{x} (⊤_{(w, y)}^{*}) \to U (⊤_{(w, y)}^{*})) . \end{matrix}$

Hence $e_{AF (X \times X)} (U_{F}^{x}, U) = ⋀_{w, y \in X} (U_{F}^{x} (⊤_{(w, y)}^{*}) \to U (⊤_{(w, y)}^{*})) .$

Since $u (x, -) = ⋀_{z \in X} (u^{*} (x, z) \to ⊤_{z}^{*})$ , put $u = ⊤_{(w, y)}^{*}$ , $⊤_{(x, y)}^{*} (x, -) = ⋀_{z \in X} (⊤_{(x, y)} (x, z) \to ⊤_{z}^{*}) = ⊤_{y}^{*}$ and $F (⊤_{(w, y)}^{*} (x, -)) = F (⊤_{X}) = ⊤$ for w ≠ x.

$\begin{matrix} e_{AF (X \times X)} (U_{F}^{x}, U) \\ = ⋀_{w, y \in X} (U_{F}^{x} (⊤_{(w, y)}^{*}) \to U (⊤_{(w, y)}^{*})) \\ = ⋀_{w, y \in X} (F (⊤_{(w, y)}^{*} (x, -)) \to U (⊤_{(w, y)}^{*})) \\ = ⋀_{w, y \in X} (F (⋀_{z \in X} (⊤_{(w, y)} (x, z) \to ⊤_{z}^{*})) \\ \to U (⊤_{(w, y)}^{*})) \\ = (⋀_{y \in X} (F (⊤_{y}^{*}) \to U (⊤_{(x, y)}^{*})) \\ \land (⋀_{w \neq x, y \in X} U (⊤_{(w, y)}^{*})) . \end{matrix}$

(6) $e_{AF (X \times X)} (U_{F}^{x}, [(x, y)]) = (⋀_{y \in X} (F (⊤_{y}^{*}) \to [(x, y)] (⊤_{(x, y)}^{*})) \land (⋀_{w \neq x, y \in X} [(x, y)] (⊤_{(w, y)}^{*})) = F^{*} (⊤_{y}^{*})$ . Since $⋀_{w \neq x, y \in X} [(z, y)] (⊤_{(w, y)}^{*}) = [(z, y)] (⊤_{(z, y)}^{*} = ⊥$ for x ≠ z, $e_{AF (X \times X)} (U_{F}^{x}, [(z, y)])$ =⊥. □

Theorem 3.9. Let (X, U) be an Alexandrov L-preuniform convergence space. Define $lim_{U} : AF (X) \to L^{X}$ as

lim_{U} (F) (x) = U (U_{F}^{x})

Then the following properties hold.

(1) $lim_{U}$ is an Alexandrov L-preconvergence structure.

(2) $lim_{U} ([y]) (x) = U (U_{[y]}^{x}) = U ([(x, y)]) = e_{U} (x, y) .$

(3) If (X, U) is an Alexandrov L-quasiuniform convergence space, then $lim_{U}$ is an Alexandrov L-convergence structure.

(4) If e _X is an L-fuzzy preorder on X, then $lim_{U_{e_{X}}}$ is an Alexandrov L-convergence structure such that $\begin{matrix} lim_{U_{e_{X}}} (F) (x) = U_{e_{X}} (U_{F}^{x}) \\ = ⋀_{y \in X} (e_{X}^{*} (x, y) \to F (⊤_{y}^{*})) \\ = lim_{N_{e_{X}}} (F) (x), \end{matrix}$ where $N_{e_{X}}^{x} (A) = ⋀_{y \in X} (e_{X} (x, y) \to A (y))$ .

Proof. (1) Since $U_{[x]}^{x} = [(x, x)]$ , $lim_{U} ([x]) (x) = U (U_{[x]}^{x}) = U ([(x, x)]) = ⊤ .$

For each $F, G \in AF (X)$ , $\begin{matrix} e_{L^{X}} (lim_{U} (F), lim_{U} (G)) \\ = ⋀_{x \in X} (lim_{U} (F) (x) \to lim_{U} (G) (x)) \\ = ⋀_{x \in X} (U (U_{F}^{x}) \to U (U_{G}^{x})) \\ \geq ⋀_{x \in X} e_{L^{X \times X}} (U_{F}^{x}, U_{G}^{x}) \\ = ⋀_{x \in X} ⋀_{u \in L^{X \times X}} (U_{F}^{x} (u) \to U_{G}^{x} (u)) \\ = ⋀_{x \in X} ⋀_{u \in L^{X \times X}} (F (u (x, -)) \to G (u (x, -))) \\ = ⋀_{x \in X} ⋀_{A \in L^{X}} (F (A) \to G (A)) \\ = e_{AF (X)} (F, G) . \end{matrix}$

(2) It follows from Lemma 3.8(2) and Theorem 3.6(2).

(3) For each x, z ∈ X, $\begin{matrix} ⋁_{y \in X} (U ([(x, y)]) ⊙ U ([(y, z)])) \\ = ⋁_{y \in X} (lim_{U} [y] (x) ⊙ lim_{U} [z] (y)) = lim_{U} [z] (x) . \end{matrix}$

(4) Since $u (x, -) = ⋀_{z \in X} (u^{*} (x, z) \to ⊤_{z}^{*})$ , put $u = ⊤_{(w, y)}^{*}$ , $⊤_{(x, y)}^{*} (x, -) = ⋀_{z \in X} (⊤_{(x, y)} (x, z) \to ⊤_{z}^{*}) = ⊤_{y}^{*}$ and $F (⊤_{(w, y)}^{*} (x, -)) = F (⊤_{X}) = ⊤$ for w ≠ x. Thus, $\begin{matrix} lim_{U_{e_{X}}} (F) (x) = ⋀_{w, y \in X} (e_{X}^{*} (w, y) \to U_{F}^{x} (⊤_{(w, y)}^{*})) \\ = ⋀_{w, y \in X} (e_{X}^{*} (w, y) \to F (⊤_{(w, y)}^{*} (x, -))) \\ = ⋀_{w, y \in X} (e_{X}^{*} (w, y) \to F (⋀_{z \in X} (⊤_{(w, y)} (x, z) \to ⊤_{z}^{*})) \\ = ⋀_{y \in X} (e_{X}^{*} (x, y) \to F (⋀_{z \in X} (⊤_{(x, y)} (x, z) \to ⊤_{z}^{*})) \\ = ⋀_{y \in X} (e_{X}^{*} (x, y) \to F (⊤_{y}^{*})) \\ = ⋀_{y \in X} (N_{e_{X}}^{x} (⊤_{y}^{*}) \to F (⊤_{y}^{*})) \\ = lim_{N_{e_{X}}} (F) (x) . □ \end{matrix}$ Example 3.10. Let ([0, 1], ⊙, →, ^*, 0, 1) be a complete residuated lattice (ref.[1, 9]) as $\begin{matrix} x ⊙ y & = & max {0, x + y - 1}, \\ x \to y & = & min {1 - x + y, 1} \\ x^{*} & = & 1 - x . \end{matrix}$

Let X = {x, y, z} and U _{e
_X} : AF (X × X) → [0, 1] with $U_{e_{X}} (U) = ⋀_{x, y \in X} (e_{X}^{*} (x, y) \to U (1_{(x, y)}^{*})$ for $e_{X} \in {e_{X}^{1}, e_{X}^{2}}$ with [0, 1]-fuzzy relations $e_{X}^{1}, e_{X}^{2} \in [0, 1]^{X \times X}$ as $e_{X}^{1} = (\begin{matrix} 1 & 0.8 & 0.5 \\ 0.6 & 1 & 0.7 \\ 0.9 & 0.3 & 1 \end{matrix}) e_{X}^{2} = (\begin{matrix} 0.5 & 0.7 & 0.3 \\ 0.4 & 0.6 & 0.3 \\ 0.2 & 0.8 & 0.8 \end{matrix})$ (1) Since $U_{e_{X}^{2}} ([(x, x)]) = ⋀_{x, y \in X} ((e_{X}^{2})^{*} (x, y) \to [(x, x)] (1_{(x, y)}^{*}) = ⋀_{x \in X} e_{X}^{2} (x, x) = 0.5,$ $U_{e_{X}^{2}}$ is not an Alexandrov [0, 1]-preuniform convergence structure. Since $e_{X}^{1}$ is a reflexive [0, 1]-fuzzy relation, by Theorem 3.6, $U_{e_{X}^{1}}$ is an Alexandrov [0, 1]-preuniform convergence structure such that $\begin{matrix} U_{e_{X}^{1}} (U) = 1 \land (0.8 + U (1_{(x, y)}^{*})) \land (0.5 + U (1_{(x, z)}^{*})) \\ \land (0.6 + U (1_{(y, x)}^{*})) \land (0.7 + U (1_{(y, z)}^{*})) \\ \land (0.9 + U (1_{(z, x)}^{*})) \land (0.3 + U (1_{(z, y)}^{*})) . \end{matrix}$

But $U_{e_{X}^{1}}$ is not an Alexandrov [0, 1]-quasiuniform convergence structure from: $\begin{matrix} 0.7 & = & U_{e_{X}^{1}} ([(z, x)]) ⊙ U_{e_{X}^{1}} ([(x, y)]) \\ ≰ U_{e_{X}^{1}} ([(z, y)]) = 0.3 . \end{matrix}$

$N_{e_{X}^{1}}^{x}$ is an Alexandrov [0, 1]-neighborhood filter with $\begin{matrix} (\begin{matrix} N_{e_{X}^{1}}^{x} (A) \\ N_{e_{X}^{1}}^{y} (A) \\ N_{e_{X}^{1}}^{z} (A) \end{matrix}) \\ = (\begin{matrix} A (x) \land (0.2 + A (y)) \land (0.5 + A (z)) \\ (0.4 + A (x)) \land A (y) \land (0.3 + A (z)) \\ (0.1 + A (x)) \land (0.7 + A (y)) \land A (z) \end{matrix}) \end{matrix}$

Then $N_{e_{X}^{1}}^{x} (1_{y}^{*}) = (e_{X}^{1})^{*} (x, y) = e_{U_{e_{X}^{1}}}^{*} (x, y) .$

From Theorem 3.9, we obtain an Alexandrov [0, 1]-preconvergence structure as $lim_{U_{e_{X}^{1}}} (F) (-) = U_{e_{X}^{1}} (U_{F}^{-}) = lim_{N_{e_{X}^{1}}} (F) (-)$ with $\begin{matrix} lim_{N_{e_{X}^{1}}} (F) (x) = (0.8 + F (1_{y}^{*})) \land (0.5 + F (1_{z}^{*})) \land 1, \\ lim_{N_{e_{X}^{1}}} (F) (y) = (0.6 + F (1_{x}^{*})) \land (0.7 + F (1_{z}^{*})) \land 1, \\ lim_{N_{e_{X}^{1}}} (F) (z) = (0.9 + F (1_{x}^{*})) \land (0.3 + F (1_{y}^{*})) \land 1 . \end{matrix}$

But $lim_{U}$ is not an Alexandrov [0, 1]-convergence structure because $0.7 = lim_{U_{e_{X}^{1}}} ([x]) (z) ⊙ lim_{U_{e_{X}^{1}}} ([y]) (x) ≰ lim_{U_{e_{X}^{1}}} ([y]) (z) = 0.3 .$

(2) In (1), if e _X = ▵ _X×X with $▵_{X \times X} (x, y) = {\begin{matrix} 1, & if y = x, \\ 0, & otherwise, \end{matrix}$ $\begin{matrix} U_{▵_{X \times X}} (U) = ⋀_{x \neq y \in X} U (1_{(x, y)}^{*}) . \end{matrix}$

Since ▵_X×X is a [0, 1]-fuzzy preorder, U _{▵_X×X} is an Alexandrov [0, 1]-quasiuniform convergence structure on X × X. Moreover, $\begin{matrix} e_{U_{▵_{X \times X}}} (x, y) = U_{▵_{X \times X}} ([(x, y)]) = ▵_{X \times X} (x, y) . \\ N_{▵_{X \times X}}^{-} (A) = ⋀_{y \in X} (▵_{X \times X} (-, y) \to A (y)) = A (-), \\ lim_{U_{▵_{X \times X}}} (F) (x) = ⋀_{x \neq y \in X} F (1_{y}^{*}) . \end{matrix}$ Since $lim_{U_{▵_{X \times X}}} ([y]) (x) = ▵_{X \times X} (x, y)$ , by Theorem 3.9(4), $lim_{U_{▵_{X \times X}}}$ is an Alexandrov L-convergence structure.

(3) If e _X = 1_X×X, then $U_{1_{X \times X}} (U) = 1 .$ Then U _{1_X×X} is an Alexandrov L-quasiuniform convergence structure on X × X. Moreover, $\begin{matrix} e_{U_{1_{X \times X}}} (x, y) = U_{1_{X \times X}} ([(x, y)]) = 1, \\ N_{1_{X \times X}}^{-} (A) = ⋀_{y \in X} A (y), \\ lim_{U_{1_{X \times X}}} (F) (x) = ⋀_{y \in X} (1_{X \times X}^{*} (x, y) \to F (1_{y}^{*})) = 1, \end{matrix}$

Since $lim_{U_{1_{X \times X}}} ([y]) (x) = 1_{X \times X} (x, y)$ , by Theorem 3.9(4), $lim_{U_{1_{X \times X}}}$ is an Alexandrov L-convergence structure.

(4) Define e _A : X × X → [0, 1] as e _A (x, y) = A (x) → A (y). Then e _A is an [0, 1]-fuzzy preorder. By Theorem 3.6, we obtain an Alexandrov L-quasiuniform convergence U _{e
_A} : AF (X × X) → [0, 1] as $\begin{matrix} U_{e_{A}} (U) = ⋀_{x, y \in X} (A (x) ⊙ A^{*} (y) \to U (1_{(x, y)}^{*})) \\ U_{e_{A}} ([(x, y)]) = A (x) \to A (y) = e_{U_{e_{A}}} (x, y) \\ = lim_{U_{e_{A}}} ([y]) (x) . \end{matrix}$

Theorem 3.11. Let (X, lim) be an Alexandrov L-pre-convergence space. Define U _lim : AF (X × X) → L as

U_{lim} (U) = ⋁_{x \in X, F \in AF (X)} lim (F) (x) ⊙ e_{AF (X \times X)} (U_{F}^{x}, U)

Then the following properties hold.

(1) U _lim is an Alexandrov L-preuniform convergence structure. Moreover, $lim_{U_{lim}} \geq lim$ and U _lim ([(x, y)]) ≥ lim([y]) (x) .

(2) If (X, U) be an Alexandrov L-preuniform convergence space, then $U_{lim_{U}} \leq U$ .

(3) If (X, lim) is a principle Alexandrov L-preconvergence space, then there exists an Alexandrov L-neighborhood system ${N^{x} ∣ x \in X}$ such that $\begin{matrix} U_{lim} (U) & = ⋁_{x \in X} (⋀_{y \in X} (N^{x} (⊤_{y}^{*}) \to U (⊤_{(x, y)}^{*})) \\ \land ⋀_{w \neq x, y \in X} (U (⊤_{(w, y)}^{*}))) . \end{matrix}$ Moreover, $lim_{U_{lim}} = lim$ and U _lim ([(x, y)]) = lim([y]) (x) .

(4) If (X, lim) is a principle Alexandrov L-convergence space, then U _lim is an Alexandrov L-quasiuniform convergence structure.

Proof. (1) Since $U_{[x]}^{x} = [(x, x)]$ from Lemma 3.8(2), $\begin{matrix} U_{lim} ([(x, x)]) \\ \geq lim ([x]) (x) ⊙ e_{AF (X \times X)} (U_{[x]}^{x}, U_{[x]}^{x}) = ⊤ . \end{matrix}$

$\begin{matrix} U_{lim} (U) \to U_{lim} (V) \\ = ⋁_{x \in X, F \in AF (X)} lim (F) (x) ⊙ e_{AF (X \times X)} (U_{F}^{x}, U) \\ \to ⋁_{x \in X, G \in AF (X)} lim (G) (x) ⊙ e_{AF (X \times X)} (U_{G}^{x}, V) \\ \geq ⋀_{x \in X, F \in AF (X)} (lim (F) (x) ⊙ e_{AF (X \times X)} (U_{F}^{x}, U) \\ \to lim (F) (x) ⊙ e_{AF (X \times X)} \\ U_{F}^{x}, V) (by Lemma 2.2 (10)) \\ \geq ⋀_{x \in X, F \in AF (X)} (e_{AF (X \times X)} (U_{F}^{x}, U) \\ \to e_{AF (X \times X)} (U_{F}^{x}, V) \\ \geq e_{AF (X \times X)} (U, V) (by Lemma 2.2 (8)) . \end{matrix}$

Moreover, $\begin{matrix} lim_{U_{lim}} (F) (x) = U_{lim} (U_{F}^{x}) \\ = ⋁_{z \in X, G \in AF (X)} lim (G) (z) ⊙ e_{AF (X \times X)} (U_{G}^{z}, U_{F}^{x}) \\ \geq lim (F) (x) ⊙ e_{AF (X \times X)} (U_{F}^{x}, U_{F}^{x}) \\ = lim (F) (x) . \end{matrix}$

By Lemma 3.8 (6), since $e_{AF (X \times X)} (U_{F}^{x}, [(x, y)]) = F^{*} (⊤_{y}^{*})$ and $e_{AF (X \times X)} (U_{F}^{z}, [(x, y)]) = 0$ , $\begin{matrix} U_{lim} ([(x, y)]) = ⋁_{z \in X, F \in AF (X)} lim (F) (z) \\ ⊙ e_{AF (X \times X)} (U_{F}^{z}, [(x, y)]) \\ = ⋁_{F \in AF (X)} (lim (F) (x) ⊙ F^{*} (⊤_{y}^{*})) \\ \geq lim ([y]) (x) ⊙ [y]^{*} (⊤_{y}^{*}) = lim ([y]) (x) . \end{matrix}$

(2) For each $U \in AF (X \times X)$ , $\begin{matrix} U_{lim_{U}} (U) \\ = ⋁_{x \in X, F \in AF (X)} lim_{U} (F) (x) ⊙ e_{AF (X \times X)} (U_{F}^{x}, U) \\ \leq ⋁_{x \in X, F \in AF (X)} U (U_{F}^{x}) ⊙ e_{L} (U (U_{F}^{x}), U (U)) \\ \leq U (U) . \end{matrix}$

(3) Since (X, lim) is a principle Alexandrov L-preconvergence space, there exists an Alexandrov L-neighborhood system ${N^{x} ∣ x \in X}$ such that $lim (F) (x) = e_{AF (X)} (N^{x}, F) = ⋀_{y \in X} (N^{x} (⊤_{y}^{*}) \to F (⊤_{y}^{*})) .$ By Lemma 3.8(5), $\begin{matrix} U_{lim} (U) \\ = ⋁_{x \in X, F \in AF (X)} lim (F) (x) ⊙ e_{AF (X \times X)} (U_{F}^{x}, U) \\ = ⋁_{x \in X, F \in AF (X)} ⋀_{y \in X} (N^{x} (⊤_{y}^{*}) \to F (⊤_{y}^{*})) \\ ⊙ (⋀_{y \in X} (F (⊤_{y}^{*}) \to U (⊤_{(x, y)}^{*})) \\ \land ⋀_{w \neq x, y \in X} (U (⊤_{(w, y)}^{*}))) \\ \leq ⋁_{x \in X, F \in AF (X)} (⋀_{y \in X} (N^{x} (⊤_{y}^{*}) \to F (⊤_{y}^{*})) \\ ⊙ ⋀_{y \in X} (F (⊤_{y}^{*}) \to U (⊤_{(x, y)}^{*}))) \\ \land ⋀_{w \neq x, y \in X} (U (⊤_{(w, y)}^{*})) \\ \leq ⋁_{x \in X} ⋀_{y \in X} (N^{x} (⊤_{y}^{*}) \to U (⊤_{(x, y)}^{*})) \\ \land ⋀_{w \neq x, y \in X} (U (⊤_{(w, y)}^{*})) \end{matrix}$ $\begin{matrix} U_{lim} (U) \geq ⋁_{x \in X} ⋀_{y \in X} (N^{x} (⊤_{y}^{*}) \to N^{x} (⊤_{y}^{*})) \\ ⊙ ⋀_{y \in X} (N^{x} (⊤_{y}^{*}) \to U (⊤_{(x, y)}^{*})) \\ \land ⋀_{w \neq x, y \in X} (U (⊤_{(w, y)}^{*})) \\ \geq ⋁_{x \in X} ⋀_{y \in X} (N^{x} (⊤_{y}^{*}) \to U (⊤_{(x, y)}^{*})) \\ \land ⋀_{w \neq x, y \in X} (U (⊤_{(w, y)}^{*})) . \end{matrix}$

$\begin{matrix} U_{lim} (U) & = ⋁_{x \in X} (⋀_{y \in X} (N^{x} (⊤_{y}^{*}) \to U (⊤_{(x, y)}^{*})) \\ \land ⋀_{w \neq x, y \in X} (U (⊤_{(w, y)}^{*}))) . \end{matrix}$

Since $U_{F}^{x} (⊤_{(x, y)}^{*}) = F (⊤_{y})$ , if x ≠ w, $U_{F}^{x} (⊤_{(w, y)}^{*}) = F (⊤_{X}) = ⊤$ from Lemma 3.8(5), $\begin{matrix} lim_{U_{lim}} F (x) = U_{lim} (U_{F}^{x}) \\ = ⋁_{z \in X} (⋀_{y \in X} (N^{z} (⊤_{y}^{*}) \to U_{F}^{x} (⊤_{(z, y)}^{*})) \\ \land ⋀_{w \neq z, y \in X} (U_{F}^{x} (⊤_{(w, y)}^{*}))) \\ = ⋀_{y \in X} (N^{x} (⊤_{y}^{*}) \to F (⊤_{y}^{*})) \\ = lim F (x) . \end{matrix}$

Moreover, $lim_{U_{lim}} = lim$ .

By Lemma 3.8 (6), since $e_{AF (X \times X)} (U_{F}^{x}, [(x, y)]) = F^{*} (⊤_{y}^{*})$ , $\begin{matrix} U_{lim} ([(x, y)]) = ⋁_{F \in AF (X)} lim (F) (x) \\ ⊙ e_{AF (X \times X)} (U_{F}^{x}, [(x, y)]) \\ = ⋁_{F \in AF (X)} (⋀_{y \in X} (N^{x} (⊤_{y}^{*}) \to F (⊤_{y}^{*})) ⊙ F^{*} (⊤_{y}^{*})) \\ \leq ⋁_{F \in AF (X)} (⋀_{y \in X} (F^{*} (⊤_{y}^{*}) \\ \to (N^{x})^{*} (⊤_{y}^{*})) ⊙ F^{*} (⊤_{y}^{*})) \\ \leq (N^{x})^{*} (⊤_{y}^{*}) = lim ([y]) (x) . \end{matrix}$

By (1), U _lim ([(x, y)]) = lim([y]) (x) .

(4) It follows from U _lim ([(x, y)]) = lim([y]) (x) . □

Example 3.12. (1) Let $N = {[x] ∣ x \in X}$ be an Alexandrov L-neighborhood system on X. Define $lim_{X} : AF (X) \to L^{X}$ as $\begin{matrix} lim_{X} F (x) = ⋀_{A \in L^{X}} ([x] (A) \to F (A)) \\ = ⋀_{y \in X} ([x] (⊤_{y}^{*}) \to F (⊤_{y}^{*})) = ⋀_{x \neq y \in X} F^{x} (⊤_{y}^{*}) . \end{matrix}$ Hence $lim_{X} ([y]) (x) = ⊤_{y} (x)$ . Since [x] is a topological L-neighborhood filter on X and $⋁_{y \in X} (lim_{X} ([y]) (x) ⊙ lim_{X} ([z]) (y)) = lim_{X} ([z]) (x)$ , $lim_{X}$ is a principle Alexandrov L-convergence structure. We obtain an Alexandrov L-quasiuniform convergence structure $U_{lim_{X}} : AF (X \times X) \to L$ as $\begin{matrix} U_{lim_{X}} (U) = ⋁_{x \in X} (⋀_{y \in X} ([x] (⊤_{y}^{*}) \to U (⊤_{(x, y)}^{*})) \\ \land ⋀_{w \neq x, y \in X} (U (⊤_{(w, y)}^{*}))) \\ = ⋁_{x \in X} (⋀_{x \neq y \in X} U (⊤_{(x, y)}^{*})) \\ \land ⋀_{w \neq x, y \in X} (U (⊤_{(w, y)}^{*}))) . \end{matrix}$ Then $U_{lim_{X}} ([(x, y)]) = lim ([y]) (x) = ⊤_{y} (x) .$ Since $U_{F}^{x} (⊤_{(w, y)}^{*}) = F (⊤_{(x, y)}^{*} (w, -)) = ⊤$ for w ≠ x and $U_{F}^{x} (⊤_{(x, y)}^{*}) = F (⊤_{(x, y)}^{*} (x, -)) = F (⊤_{y}^{*})$ , $\begin{matrix} lim_{U_{lim_{X}}} F (x) = U_{lim_{X}} (U_{F}^{x}) \\ = (⋀_{x \neq y \in X} U_{F}^{x} (⊤_{(x, y)}^{*})) \\ \land (⋀_{w \neq x, y \in X} (U_{F}^{x} (⊤_{(w, y)}^{*}))) \\ = ⋀_{x \neq y \in X} F^{x} (⊤_{y}^{*})) = lim_{X} F (x) . \end{matrix}$ (2) Let $N = {⋀_{z \in X} [z] ∣ z \in X}$ be an Alexandrov L-neighborhood system on X. Define $lim_{X} : AF (X) \to L^{X}$ as $\begin{matrix} lim_{X} F (x) = ⋀_{A \in L^{X}} (⋀_{z \in X} A (z) \to F (A)) \\ = ⋀_{y \in X} (⋀_{z \in X} [z] (⊤_{y}^{*}) \to F (⊤_{y}^{*})) = ⊤ . \end{matrix}$ Moreover, $lim_{X} ([y]) (x) = (⋀_{z \in X} [z]) (⊤_{y}^{*}) \to [y] (⊤_{y}^{*}) = ⊥ \to 0 = ⊤ .$ Hence $lim_{X}$ is a principle Alexandrov L-convergence structure. We obtain an Alexandrov L-quasiuniform convergence structure $U_{lim_{X}} : AF (X \times X) \to L$ as $\begin{matrix} U_{lim_{X}} (U) = ⋁_{x \in X} ⋀_{y \in X} (⋀_{z \in X} [z] (⊤_{y}^{*}) \to U (⊤_{(x, y)}^{*})) \\ \land ⋀_{w \neq x, y \in X} (U (⊤_{(w, y)}^{*})) \\ = ⋁_{x \in X} (⋀_{w \neq x, y \in X} U (⊤_{(w, y)}^{*})), \\ lim_{U_{lim_{X}}} F (x) = ⋁_{x \in X} (⋀_{w \neq x, y \in X} U_{F}^{x} (⊤_{(w, y)}^{*})) = ⊤ . \end{matrix}$

Theorem 3.13. (1) Let (X, U _X) and (Y, U _Y) be Alexandrov L-preuniform convergence spaces. If ψ : (X, U _X) → (Y, U _Y) is a uniformly continuous map, then $ψ : (X, lim_{U_{X}}) \to (Y, lim_{U_{Y}})$ is a continuous map.

(2) Let $(X, lim_{X})$ and $(Y, lim_{Y})$ be Alexandrov L-preconvergence spaces. If $ψ : (X, lim_{X}) \to (Y, lim_{Y})$ is a continuous map, then $ψ : (X, U_{lim_{X}}) \to (Y, U_{lim_{Y}})$ is a uniformly continuous map.

Proof. (1) Since $U_{X} (U) \leq U_{Y} (U) ((ψ \times ψ)^{\Rightarrow} (U))$ for each $U \in AF (X \times X)$ , $\begin{matrix} lim_{U_{X}} (F) (x) = U_{X} (U_{F}^{x}) \\ \leq U_{Y} (U) ((ψ \times ψ)^{\Rightarrow} (U_{F}^{x})) = U_{Y} (U_{ψ^{\Rightarrow} (F)}^{ψ (x)}) \\ = lim_{U_{Y}} (ψ^{\Rightarrow} (F)) (ψ (x)) . \end{matrix}$

(2) For each $U \in AF (X \times X)$ ,

$\begin{matrix} U_{lim_{X}} (U) \\ = ⋁_{x \in X, F \in AF (X)} lim (F) (x) ⊙ e_{AF (X \times X)} (U_{F}^{x}, U) \\ \leq ⋁_{x \in X, F \in AF (X)} lim (ψ^{\Rightarrow} (F)) (ψ (x)) \\ ⊙ e_{AF (Y \times Y)} (U_{ψ^{\Rightarrow} (F)}^{ψ (x)}, (ψ \times ψ)^{\Rightarrow} (U)) \\ \leq U_{lim_{Y}} ((ψ \times ψ)^{\Rightarrow} (U)) . □ \end{matrix}$

Theorem 3.14. (1) Let (X, U _X) and (Y, U _Y) be an Alexandrov L-preuniform convergence spaces. If ψ : (X, U _X) → (Y, U _Y) is a uniform continuous map, then ψ : (X, e _{U
_X}) → (Y, e _{U
_Y}) is an order preserving map.

(2) Let (X, e _X) and (Y, e _Y) be L-fuzzy preorder spaces. If ψ : (X, e _X) → (Y, e _Y) is an order preserving map, then ψ : (X, U _{e
_X}) → (Y, U _{e
_Y}) is a uniform continuous map.

Proof. (1) Since (ψ × ψ) ^⇒ ([(x, y)]) (u) = [(x, y)] ((ψ × ψ) ^← (u)) = (ψ × ψ) ^← (u) (x, y) = u (ψ (x), ψ (y)) = [(ψ (x), ψ (y))] (u), $\begin{matrix} e_{U_{Y}} (ψ (x), ψ (y)) = U_{Y} ([(ψ (x), ψ (y))]) \\ = U_{Y} ((ψ \times ψ)^{\Rightarrow} ([(x, y)])) \\ \geq U_{X} ([(x, y)]) = e_{U_{X}} (x, y) . \end{matrix}$

(2) Since $(ψ \times ψ)^{\Rightarrow} (U_{X}) (⊤_{ψ (x), ψ (y)}^{*}) = U_{X} ((ψ \times ψ)^{\leftarrow} (⊤_{ψ (x), ψ (y)}^{*})) \geq U_{X} (⊤_{(x, y)}^{*})$ , $\begin{matrix} U_{e_{X}} (U) = ⋀_{x, y \in X} (e_{X}^{*} (x, y) \to U_{X} (⊤_{(x, y)}^{*})) \\ \leq ⋀_{x, y \in X} (e_{X}^{*} (ψ (x), ψ (y)) \\ \to (ψ \times ψ)^{\Rightarrow} (U_{X}) (⊤_{ψ (x), ψ (y)}^{*})) \\ = U_{e_{Y}} ((ψ \times ψ)^{\Rightarrow} (U_{X})) . □ \end{matrix}$

Definition 3.15. [1, 9] 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 AUC be denote the category of Alexandrov L-preuniform convergence spaces and uniform continuous mappings for morphisms.

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

Theorem 3.16. (1) F : AC → AUC defined as $F (X, lim_{X}) = (X, U_{lim_{X}})$ is a functor.

(2) G : AUC → AC defined as $G (X, U_{X}) = (X, lim_{U_{X}})$ is a functor.

(3) The pair (F, G) is a Galois correspondence between AUC and AC.

Proof. (1) and (2) are follows from Theorems 3.9, 3.11 and 3.13.

(3) By Theorem 3.11(2), if (X, U _X) is an Alexandrov L-preuniform convergence space, then $F (G (X, U_{X})) = (X, U_{lim_{U_{X}}}) \leq (X, U_{X})$ . Hence, the identity map ${id}_{X} : (X, U_{lim_{U_{X}}}) = F (G (X, U_{X})) \to (X, U_{X})$ is a uniformly continuous map. Moreover, if $(X, lim_{X})$ is an Alexandrov L-preconvergence space, by Theorem 3.11(1), $G (F (X, lim_{X})) = lim_{U_{lim_{X}}} \geq lim_{X}$ . Hence the identity map ${id}_{X} : (X, lim_{X}) \to G (F (X, lim_{X})) = (X, lim_{U_{lim_{X}}})$ is a continuous map. Therefore (F, G) is a Galois correspondence.

Let RFR be denote the category of reflexive L-fuzzy relations and order preserving mappings for morphisms. □

Theorem 3.17. (1) P : RFR → AUC defined as P (X, e _X) = (X, U _{e
_X}) is a functor.

(2) Q : AUC → RFR defined as Q (X, U _X) = (X, e _{U
_X}) is a functor.

(3) The pair (Q, P) is a Galois correspondence between AUC and RFR.

Proof. (1) and (2) are follows from Theorems 3.6 and 3.14.

(3) By Theorem 3.6(3), if (X, U _X) is an Alexandrov L-preuniform convergence space, then P (Q (X, U _X)) = (X, U _{e
_{U
_X}}) ≥ (X, U _X). Hence, the identity map id _X : (X, U _X) → (X, U _{e
_{U
_X}}) = P (Q (X, U _X)) is a uniformly continuous map. Moreover, if (X, e _X) is a reflexive L-fuzzy relation, by Theorem 3.6(3), Q (P (X, e _X)) = (X, e _{U
_{e
_X}}) = (X, e _X). Hence the identity map id _X : Q (P (X, e _X)) = (X, e _{U
_{e
_X}}) → (X, e _X) is an oder preserving map. Therefore (Q, P) is a Galois correspondence. □

Example 3.18. Let X = {h _i ∣ i = {1, …, 3}} with h _i = house and Y = {e, b, w, c, i} with e = expensive,b = beautiful, w = wooden, c = creative, i = in the green surroundings. Let ([0, 1], ⊙, →, ^*, 0, 1) be a complete residuated lattice as in Example 3.10. Let R ∈ [0, 1] ^X×Y be a fuzzy information as follows: $\begin{matrix} R & e & b & w & c & i \\ h_{1} & 0.7 & 0.6 & 0.5 & 0.9 & 0.2 \\ h_{2} & 0.6 & 0.8 & 0.4 & 0.3 & 0.5 \\ h_{3} & 0.4 & 0.9 & 0.8 & 0.6 & 0.6 \end{matrix}$ Define L-fuzzy preordered relations $e_{X}^{Y}, e_{X}^{{b, w}} \in [0, 1]^{X \times X}$ by $\begin{matrix} e_{X}^{Y} (h_{i}, h_{j}) = ⋀_{y \in Y} (R (h_{i}, y) \to R (h_{j}, y)), \\ e_{X}^{{b, w}} (h_{i}, h_{j}) = ⋀_{y \in {b, w}} (R (h_{i}, y) \to R (h_{j}, y)) . \end{matrix}$ $e_{X}^{Y} = (\begin{matrix} 1 & 0.4 & 0.7 \\ 0.7 & 1 & 0.8 \\ 0.6 & 0.6 & 1 \end{matrix}) e_{X}^{{b, w}} = (\begin{matrix} 1 & 0.9 & 1 \\ 0.8 & 1 & 1 \\ 0.7 & 0.6 & 1 \end{matrix})$ (1) We obtain a topological Alexandrov L-neighborhood system ${N_{Y}^{h_{1}} ∣ h_{1} \in X}$ and by Theorem 3.7, a principal Alexandrov L-convergence structure $lim_{N_{Y}}$ as follows: $\begin{matrix} N_{Y}^{h_{-}} (A) = ⋀_{y \in X} (e_{X}^{Y} (h_{-}, h_{2}) \to A (h_{2})), \\ lim_{N_{Y}} (F) (h_{i}) = ⋀_{j \in {1, 2, 3}} (e_{X}^{*} (h_{i}, h_{j}) \to F (1_{h_{j}}^{*})), \\ N_{Y}^{h_{i}} (1_{h_{j}}^{*}) = (e_{X}^{Y})^{*} (h_{i}, h_{j}), \end{matrix}$ By Theorem 3.11(4), $U_{lim_{N_{Y}}}$ is Alexandrov L-quasiuniform convergence structure on X × X such that $\begin{matrix} U_{lim_{N_{Y}}} (U) \\ = ⋁_{i \in {1, 2, 3}} ⋀_{j \in {1, 2, 3}} (N_{Y}^{h_{i}} (1_{h_{j}}^{*}) \to U (1_{(h_{i}, h_{j})}^{*})) \\ = ⋁_{i \in {1, 2, 3}} (⋀_{j \in {1, 2, 3}} ((e_{X}^{Y})^{*} (h_{i}, h_{j}) \\ \to U (1_{(h_{i}, h_{j})}^{*})) \land ⋀_{i \neq k, j \in {1, 2, 3}} U (1_{(h_{k}, h_{j})}^{*})) . \end{matrix}$ Since $e_{X}^{Y}$ is an L-fuzzy preorder, by Theorem 3.6, $U_{e_{X}^{Y}}$ is an Alexandrov L-quasiuniform convergence structure on X × X such that $\begin{matrix} U_{e_{X}^{Y}} (U) = ⋀_{i, j \in {1, 2, 3}} ((e_{X}^{Y})^{*} (h_{i}, h_{j}) \to U (1_{(h_{i}, h_{j})}^{*})), \\ e_{U_{e_{X}^{Y}}} (x, y) = U_{e_{X}^{Y}} ([(x, y)]) = e_{X}^{Y} (x, y) . \end{matrix}$ $\begin{matrix} lim_{U_{e_{X}^{Y}}} (F) (h_{i}) = ⋀_{i, j \in {1, 2, 3}} (e_{X}^{*} (h_{i}, h_{j}) \to \\ U_{F}^{h_{i}} (1_{(h_{i}, h_{j})}^{*})) \\ = ⋀_{j \in {1, 2, 3}} (e_{X}^{*} (h_{i}, h_{j}) \to F (1_{h_{j}}^{*})) = lim_{N_{Y}} (F) (h_{i}), \\ lim_{U_{e_{X}^{Y}}} (N_{Y}^{h_{k}}) (h_{i}) \\ = ⋀_{i, j \in {1, 2, 3}} (e_{X}^{*} (h_{i}, h_{j}) \to U_{N_{Y}^{h_{k}}}^{h_{i}} (1_{(h_{i}, h_{j})}^{*})) \\ = ⋀_{j \in {1, 2, 3}} (e_{X}^{*} (h_{i}, h_{j}) \to N_{Y}^{h_{k}} (1_{h_{j}}^{*})) \\ = ⋀_{j \in {1, 2, 3}} ((e_{X}^{Y})^{*} (h_{i}, h_{j}) \to (e_{X}^{Y})^{*} (h_{k}, h_{j})) \\ = e_{X}^{Y} (h_{i}, h_{k}) . \end{matrix}$ (1-A) For B ∈ [0, 1] ^X with B = (0.4, 0.7, 0.1), put $u_{B} (h_{i}, h_{j}) = B (h_{i}) \to B (h_{j}) = V^{*} (1_{(h_{i}, h_{j})}^{*})$ . Then $\begin{matrix} V (u) = ⋀_{i, j \in {1, 2, 3}} (V^{*} (1_{(h_{i}, h_{j})}^{*}) \to u (h_{i}, h_{j})), \\ N_{B}^{h_{-}} (A) = ⋀_{y \in X} (u_{B} (h_{-}, h_{2}) \to A (h_{2})), \\ N_{B}^{h_{i}} (1_{h_{j}}^{*}) = ⋀_{k = 1}^{3} (u_{B} (h_{i}, h_{k}) \to 1_{h_{j}}^{*} (h_{k})) \\ = u_{B}^{*} (h_{i}, h_{j}) . \end{matrix}$ Then $V \in AF (X \times X), N_{B}^{h_{-}} \in AF (X)$ . By Theorems 3.6 and 3.11, $\begin{matrix} U_{e_{X}^{Y}} (V) = ⋀_{j \in {1, 2, 3}} ((e_{X}^{Y})^{*} (h_{i}, h_{j}) \\ \to V (1_{(h_{i}, h_{j})}^{*})) = 0.4, \\ U_{lim_{N_{Y}}} (V) = ⋁_{i \in {1, 2, 3}} (⋀_{j \in {1, 2, 3}} (e_{X}^{*} (h_{i}, h_{j}) \\ \to V (1_{(h_{i}, h_{j})}^{*})) \land ⋀_{i \neq k, j \in {1, 2, 3}} V (1_{(h_{k}, h_{j})}^{*})) = 0, \\ lim_{U_{e_{X}^{Y}}} (N_{B}^{h_{1}}) = (0.4, 0.7, 0.6), \\ lim_{U_{e_{X}^{Y}}} (N_{B}^{h_{2}}) = (0.4, 1, 0.6), \\ lim_{U_{e_{X}^{Y}}} (N_{B}^{h_{3}}) = (0.4, 0.7, 0.6) . \end{matrix}$ (1-B) Let $W (u) = ⋀_{i, j \in {1, 2, 3}} (e_{X}^{{b, w}} (h_{i}, h_{j}) \to u (h_{i}, h_{j})) .$ Then $W \in AF (X \times X)$ . $\begin{matrix} U_{e_{X}^{Y}} (W) = ⋀_{j \in {1, 2, 3}} ((e_{X}^{Y})^{*} (h_{i}, h_{j}) \to W (1_{(h_{i}, h_{j})}^{*})) \\ = ⋀_{j \in {1, 2, 3}} ((e_{X}^{Y})^{*} (h_{i}, h_{j}) \to (e_{X}^{{b, w}})^{*} (h_{i}, h_{j})) = 0.5, \\ U_{lim_{N_{Y}}} (W) = ⋁_{i \in {1, 2, 3}} ⋀_{j \in {1, 2, 3}} (e_{X}^{*} (h_{i}, h_{j}) \\ \to (e_{X}^{{b, w}})^{*} (1_{(h_{i}, h_{j})}^{*})) \\ \land ⋀_{i \neq k, j \in {1, 2, 3}} (e_{X}^{{b, w}})^{*} (1_{(h_{k}, h_{j})}^{*})) = 0, \\ N_{{b, w}}^{h_{i}} (1_{h_{j}}^{*}) = (e_{X}^{{b, w}})^{*} (h_{i}, h_{j}) . \end{matrix}$ $\begin{matrix} lim_{U_{e_{X}^{Y}}} (N_{{b, w}}^{h_{1}}) = (0.5, 0.7, 0.6), \\ lim_{U_{e_{X}^{Y}}} (N_{{b, w}}^{h_{2}}) = (0.4, 0.8, 0.6), \\ lim_{U_{e_{X}^{Y}}} (N_{{b, w}}^{h_{3}}) = (0.7, 0.8, 0.9) . \end{matrix}$ (1-C) Let $U (u) = ⋀_{i, j \in {1, 2, 3}} u (h_{i}, h_{j})$ and $U (1_{(h_{i}, h_{j})}^{*}) = 0$ . Then $U \in AF (X \times X)$ . By Theorems 3.6 and 3.11, $\begin{matrix} U_{e_{X}^{Y}} (U) = ⋀_{i, j \in {1, 2, 3}} ((e_{X}^{Y})^{*} (h_{i}, h_{j}) \to U (1_{(h_{i}, h_{j})}^{*})) \\ = ⋀_{i, j \in {1, 2, 3}} e_{X}^{Y} (h_{i}, h_{j}) = 0.4, \\ U_{lim_{N_{Y}}} (U) = ⋁_{i \in {1, 2, 3}} (⋀_{j \in {1, 2, 3}} e_{X} (h_{i}, h_{j})) \\ \land ⋀_{i \neq k, j \in {1, 2, 3}} (U (1_{(h_{k}, h_{j})}^{*})) = 0 . \end{matrix}$ (1-D) For $W \in AF (X \times X)$ such that $W (u) = ⋀_{i, j \in {1, 2, 3}} (u^{*} (h_{i}, h_{j}) \to W (1_{(h_{i}, h_{j})}^{*}))$ with $W (1_{(h_{i}, h_{j})}^{*}) = (\begin{matrix} 0.2 & 0.3 & 0.7 \\ 0.5 & 0.1 & 0.6 \\ 0.4 & 0.2 & 0.7 \end{matrix})$ $\begin{matrix} U_{e_{X}^{Y}} (W) = ⋀_{i, j \in {1, 2, 3}} ((e_{X}^{Y})^{*} (h_{i}, h_{j}) \\ \to W (1_{(h_{i}, h_{j})}^{*})) = 0.7 \\ U_{lim_{N^{Y}}} (W) = ⋁_{i \in {1, 2, 3}} (⋀_{j \in {1, 2, 3}} (e_{X}^{Y})^{*} (h_{i}, h_{j}) \\ \land ⋀_{i \neq k, j \in {1, 2, 3}} (W (1_{(h_{k}, h_{j})}^{*})) \\ = 0.1 \lor 0.2 \lor 0.1 = 0.2 . \end{matrix}$ (2) By a similar way in (1), ${N_{{b, w}}^{h_{1}} ∣ h_{1} \in X}$ is a topological Alexandrov L-neighborhood system as $\begin{matrix} N_{{b, w}}^{h_{1}} (A) = A (h_{1}) \land (0.1 + A (h_{2})) \land A (h_{3}) \\ N_{{b, w}}^{h_{2}} (A) = (0.2 + A (h_{1})) \land A (h_{2}) \land A (h_{3}) \\ N_{{b, w}}^{h_{3}} (A) = (0.3 + A (h_{1})) \land (0.4 + A (h_{2})) \land A (h_{3}) \\ N_{{b, w}}^{h_{i}} (1_{h_{j}}^{*}) = (e_{X}^{{b, w}})^{*} (h_{i}, h_{j}) . \end{matrix}$ By Theorem 3.7, we obtain a principal Alexandrov L-convergence structure $\begin{matrix} lim_{N_{{b, w}}} (F) (h_{1}) = 1 \land (0.1 + F (1_{h_{2}}^{*})) \\ lim_{N_{{b, w}}} (F) (h_{2}) = 1 \land (0.2 + F (1_{h_{1}}^{*})) \\ lim_{N_{{b, w}}} (F) (h_{3}) = 1 \land (0.3 + F (1_{h_{1}}^{*})) \\ \land (0.4 + F (1_{h_{2}}^{*})) \end{matrix}$ By Theorem 3.11(4), $U_{lim_{N_{{b, w}}}}$ is Alexandrov L-quasiuniform convergence structure on X × X such that $\begin{matrix} U_{lim_{N_{{b, w}}}} (U) \\ = ⋁_{i \in {1, 2, 3}} ⋀_{j \in {1, 2, 3}} (N_{{b, w}}^{h_{i}} (1_{h_{j}}^{*}) \to U (1_{(h_{i}, h_{j})}^{*})) \\ = ⋁_{i \in {1, 2, 3}} ⋀_{j \in {1, 2, 3}} ((e_{X}^{{b, w}})^{*} (h_{i}, h_{j}) \\ \to U (1_{(h_{i}, h_{j})}^{*})) . \end{matrix}$ Since $e_{X}^{{b, w}}$ is an L-fuzzy preorder, by Theorem 3.6, $U_{e_{X}^{{b, w}}}$ is an Alexandrov L-quasiuniform convergence structure on X × X such that $\begin{matrix} U_{e_{X}^{{b, w}}} (U) = 1 \land (0.1 + U (1_{(h_{1}, h_{2})}^{*})) \\ \land (0.2 + U (1_{(h_{2}, h_{1})}^{*})) \\ \land (0.3 + U (1_{(h_{3}, h_{1})}^{*})) \land (0.4 + U (1_{(h_{3}, h_{2})}^{*})), \\ e_{U_{e_{X}^{{b, w}}}} (x, y) = U_{e_{X}^{{b, w}}} ([(x, y)]) = e_{X}^{{b, w}} (x, y) . \end{matrix}$ $\begin{matrix} lim_{U_{e_{X}^{{b, w}}}} (F) (h_{i}) = lim_{N_{{b, w}}} (F) (h_{i}), \\ lim_{U_{e_{X}^{{b, w}}}} (N_{{b, w}}^{h_{k}}) (h_{i}) \\ = ⋀_{j \in {1, 2, 3}} (e_{X}^{*} (h_{i}, h_{j}) \to N_{{b, w}}^{h_{k}} (1_{h_{j}}^{*})) \\ = ⋀_{j \in {1, 2, 3}} ((e_{X}^{{b, w}})^{*} (h_{i}, h_{j}) \to (e_{X}^{{b, w}})^{*} (h_{k}, h_{j})) \\ = e_{X}^{{b, w}} (h_{i}, h_{k}) . \end{matrix}$ (2-A) By (1-A), $\begin{matrix} U_{e_{X}^{{b, w}}} (V) = 0.6 . \\ U_{lim_{N_{{b, w}}}} (V) = ⋁_{i \in {1, 2, 3}} ⋀_{j \in {1, 2, 3}} ((e_{X}^{{b, w}})^{*} (h_{i}, h_{j}) \\ \to V (1_{(h_{i}, h_{j})}^{*})) \land ⋀_{i \neq k, j \in {1, 2, 3}} (V (1_{(h_{k}, h_{j})}^{*})) = 0 \\ lim_{U_{e_{X}^{{b, w}}}} (N_{B}^{h_{1}}) = (0.9, 0.8, 0.6), \\ lim_{U_{e_{X}^{{b, w}}}} (N_{B}^{h_{2}}) = (0.9, 1, 0.6), \\ lim_{U_{e_{X}^{{b, w}}}} (N_{B}^{h_{3}}) = (0.9, 0.8, 0.6) . \end{matrix}$ (2-B) For $U \in AF (X \times X)$ in (1-C), $\begin{matrix} U_{e_{X}^{{b, w}}} (U) = 0.6, \\ U_{lim_{N^{{b, w}}}} (U) = ⋁_{i \in {1, 2, 3}} (⋀_{j \in {1, 2, 3}} e_{X}^{{b, w}} (h_{i}, h_{j}) \\ \land ⋀_{i \neq k, j \in {1, 2, 3}} (U (1_{(h_{k}, h_{j})}^{*})) = 0 . \end{matrix}$ (2-C) For $W \in AF (X \times X)$ in (1-D), $\begin{matrix} U_{e_{X}^{{b, w}}} (W) = 0.8, U_{lim_{N^{{b, w}}}} (W) = 0.2 . \end{matrix}$

4. Conclusion

In this paper, we introduce the notion of Alexandrov L-preuniform convergence spaces in a complete residuated lattice. We investigate the relations among Alexandrov L-(neighborhood) filters, L-fuzzy preorders and Alexandrov L-(preuniform)convergence structures. There exists a Galois correspondence between the category of Alexandrov L-preuniform convergence spaces and that of Alexandrov L-convergence spaces (resp. reflexive L-relations). Finally, we give the examples of Alexandrov L-preuniform convergence spaces and Alexandrov L-convergence for a fuzzy information.

In the future, by using the concepts of Alexandrov L-preuniform convergence spaces and Alexandrov L-convergence spaces, information systems and decision rules with a view point of applications to mult-attribute decision-making [14 , 30] are investigated in residuated lattices.

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