Alexandrov L -filters and Alexandrov L -convergence spaces 1

Abstract

In this paper, we introduce the notion of Alexandrov L-(neighborhood) filters and Alexandrov L-convergence structures as a topological viewpoint of fuzzy rough sets. We investigate the categorical relations among Alexandrov L-neighborhood filters, L-fuzzy preorders and Alexandrov L-convergence structures. Moreover, we investigate their topological properties and give examples.

Keywords

Complete residuated lattices fuzzy join(meet)-complete lattices

1 Introduction

Ward et al. [22] introduced a complete residuated lattice which is an algebraic structure for many valued logic. It is an important mathematical tool as algebraic structures for many valued logics [1 , 10–13].

Pawlak [16, 17] 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 [10 , 18–21] developed L-lower and L-upper approximation operators in complete residuated lattices. By using this concepts, information systems and decision rules were investigated in complete residuated lattices [1].

An interesting and natural research topic in rough set theory is the study of rough set theory and topological structures. Lai [13] and Ma [14] investigated the Alexandrov L-topology and lattice structures of L-fuzzy rough sets determined by lower and upper sets. Kim [10, 11] 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 [7–9 , 15] developed stratified L-convergence structures based on the concepts of L-filters where L is a complete Heyting algebra. Yao [24] extended stratified L-convergence structures to complete residuated lattices and investigated between stratified L-convergence structures and L-fuzzy topological spaces.

Zhang [23] defined a strong L-topology on the concepts of fuzzy complete lattices. As an extension of Yao [24], Fang [2, 3] introduced L-ordered convergence structures on L-ordered filters and investigated between L-ordered convergence structures and strong L-topological spaces.

In this paper, we introduce the notion of Alexandrov L-(neighborhood) filters as an extension of Fang’s L-ordered filters [3]. It is motivated by the topological structure of fuzzy information and fuzzy rough sets in a complete residuated lattice. In section 2, some basic concepts and notions used in this paper are listed. In section 3, we define Alexandrov L-(neighborhood) filters. We investigate the categorical relations between topologicla Alexandrov L-neighborhood filters and L-fuzzy preorders. In section 3, we define Alexandrov L-convergence structures. We investigate the categorical relations among Alexandrov L-neighborhood filters, L-fuzzy preorders and Alexandrov L-convergence structures. Moreover, we investigate their topological properties and give examples.

2 Preliminaries

Definition 2.1. [1 , 22] 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 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 , 22] 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 → (⋀ _iy_i) = ⋀ _i (x → y_i),

(5) (⋁_i x_i) → y = ⋀ _i (x_i → y),

(6) x ⊙ (⋁_iy_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,

(10) ⋁_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),

(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, 13] 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) 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.

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 an L-fuzzy partially ordered set.

(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 an L-fuzzy partially ordered set from Lemma 2.2 (9).

Definition 2.5. [23 , 26] Let (X, e_X) be an L-fuzzy partially ordered set and A ∈ L^X.

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

(J1) A (x) ≤ e_X (x, x₀),

(J2) ⋀_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

(M1) A (x) ≤ e_X (x₁, x),

(M2) ⋀_x∈X (A (x) → e_X (y, x)) ≤ e_X (y, x₁).

Remark 2.6. Let (X, e_X) be an L-fuzzy partially ordered set and Φ ∈ L^X.

(1) 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.

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

(3) 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, e_L) be an L-fuzzy partially ordered set and A ∈ L^L.

(1) Since x₀ is a join of A iff ⋀_x∈L (A (x) → e_L (x, y)) = ⋁ _x∈L (x ⊙ A (x)) → y = e_L (x₀, y), x₀ = ⊔ A = ⋁ _x∈L (x ⊙ A (x)) .

(2) Since x₀ is a join of A iff ⋀_x∈L (A (x) → e_L (x, y) = y → ⋀ _x∈L (A (x) → x) = y → ⊓ A, ⊓A = ⋀ _x∈L (A (x) → x) .

Remark 2.8. Let (L^X, e_{L
^X}) be an L-fuzzy partially ordered set and Φ ∈ L^{L
^X}.

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

(2) ⊓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 2.9. [23 , 26] 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 2.10. [23 , 26] 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)=yΦ (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.

3 Alexandrov L-filters and Alexandrov L-neighborhood spaces

We introduce the new notion of Alexandrov L-filters as an extension of Fang’s L-ordered filters [3]. We investigate the relations between topological Alexandrov L-neighborhood filters and L-fuzzy preorders.

Definition 3.1. Let (L^X, e_{L
^X}) and (L, e_L) be L-fuzzy partially ordered sets. 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}. Let AF (X) denote the set of all Alexandrov L-filters on X.

Theorem 3.2. 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:

(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.

Proof (⇒) For all Φ ∈ L^{L
^X}, we have $⊓ F^{\to} (Φ) = ⋀_{D \in L^{X}} (Φ (D) \to F (D))$ from: $\begin{matrix} e_{L} (b, ⊓ F^{\to} (Φ)) = ⋀_{c \in L} (F^{\to} (Φ) (c) \to e_{L} (b, c)) \\ = ⋀_{c \in L} (⋁_{c = F (D)} Φ (D) \to e_{L} (b, F (D))) \\ = e_{L} (b, ⋀_{D \in L^{X}} (Φ (D) \to F (D))) . \end{matrix}$

(F1) Define Φ : L^X → L as Φ (A_i) =⊤ for i ∈ Γ and Φ (B) =⊥, otherwise. Then $(⊓ Φ) (x) = ⋀_{A \in L^{X}} (Φ (A) \to A (x)) = ⋀_{i \in Γ} A_{i} (x) .$ So, ⊓Φ = ⋀ _i∈ΓA_i. $\begin{matrix} ⊓ F^{\to} (Φ) = ⋀_{D \in L^{X}} (Φ (D) \to F (D)) \\ = ⋀_{i \in Γ} F (A_{i}) = F (⊓ Φ) = F (⋀_{i \in Γ} A_{i}) . \end{matrix}$

(F2) Define Φ : L^X → L as Φ (A) = α for A ∈ L^X and Φ (B) =⊥, otherwise. Then $⊓ Φ = ⋀_{C \in L^{X}} (Φ (C) \to C) = α \to A,$ $\begin{matrix} ⊓ F^{\to} (Φ) = ⋀_{D \in L^{X}} (Φ (D) \to F (D)) \\ = α \to F (A) = F (⊓ Φ) = F (α \to A) . \end{matrix}$

(⇐) For all Φ ∈ L^{L
^X}, $F (⊓ Φ) = ⊓ F^{\to} (Φ)$ from: $\begin{matrix} e_{L} (b, ⊓ F^{\to} (Φ)) = ⋀_{c \in L} (F^{\to} (Φ) (c) \to e_{L} (b, c)) \\ = ⋀_{c \in L} (⋁_{c = F (D)} Φ (D) \to e_{L} (b, F (D))) \\ = e_{L} (b, ⋀_{D \in L^{X}} (Φ (D) \to F (D))) \\ = e_{L} (b, F (⋀_{D \in L^{X}} (Φ (D) \to D))) (by (F 1), (F 2)) \\ = e_{L} (b, F (⊓ Φ)) . \end{matrix}$

Theorem 3.3. A map $F : (L^{X}, e_{L^{X}}) \to (L, e_{L})$ is an Alexandrov L-filter on X iff there exists B ∈ L^X such that $F (A) = ⋀_{x \in X} (B (x) \to A (x))$ for all A ∈ L^X.

Proof (⇒) For all A ∈ L^X, since $A = ⋀_{x \in X} (A^{*} (x) \to ⊤_{x}^{*})$ , by Theorem 3.2 (F1) and (F2), we have $F (A) = F (⋀_{x \in X} (A^{*} (x) \to ⊤_{x}^{*}) = ⋀_{x \in X} (A^{*} (x) \to F (⊤_{x}^{*})) = ⋀_{x \in X} (F^{*} (⊤_{x}^{*}) \to A (x))$ . Put $B (x) = F^{*} (⊤_{x}^{*})$ . Then $F (A) = ⋀_{x \in X} (B (x) \to A (x))$ .

(⇐) From Theorem 3.2, it is easily proved.

Remark 3.4. Let (L^X, e_{L
^X}) and (L, e_L) be L-fuzzy partially ordered sets. From Definition 3.1, we can redefine an L-ordered filter as follows. A map $F : (L^{X}, e_{L^{X}}) \to (L, e_{L})$ is an L-ordered filter on X in a Fang’s sense [3] iff $F (⊥_{X}) = ⊥$ and $F (⊓ Φ) = ⊓ F^{\to} (Φ)$ for a finite family Φ ∈ L^{L
^X}.

Theorem 3.5. If $F : (L^{X}, e_{L^{X}}) \to (L, e_{L})$ is an Alexandrov L-filter on X, then, for all A, B ∈ L^X, $e_{L^{X}} (A, B) \leq e_{L} (F (A), F (B))$ .

Proof. For A ≤ B, $F (A) = F (A \land B) = F (A) \land F (B)$ . So, $F (A) \leq F (B) .$ It follows $α \to F (α ⊙ A) = F (α \to α ⊙ A) \geq F (A)$ . Put α = e_{L
^X} (A, B). Then $e_{L^{X}} (A, B) ⊙ F (A) \leq F (e_{L^{X}} (A,$ $B) ⊙ A) \leq F (B)$ implies $e_{L^{X}} (A, B) \leq e_{L} (F (A),$ $F (B))$ .

Theorem 3.6. Let AF (X) denote the set of all Alexandrov L-filters on X. Define e_AF(X) : AF (X) × AF (X) → L as $e_{AF (X)} (F, G) = ⋀_{A \in L^{X}} (F (A) \to G (A)) .$ Then (AF (X) , e_AF(X)) is a meet-complete lattice.

Proof. We easily prove that (AF (X) , e_AF(X)) is an L-fuzzy partially ordered set. For each Φ : AF (X) → L, put $⊓ Φ (A) = ⋀_{G \in AF (X)} (Φ (G) \to G (A))$ .

(F1) For all A_i ∈ L^X, $\begin{matrix} ⊓ Φ (⋀_{i \in Γ} A_{i}) = ⋀_{G \in AF (X)} (Φ (G) \to G (⋀_{i \in Γ} A_{i})) \\ = ⋀_{G \in AF (X)} (Φ (G) \to ⋀_{i \in Γ} G (A_{i})) \\ = ⋀_{i \in Γ} ⋀_{G \in AF (X)} (Φ (G) \to G (A_{i})) \\ = ⋀_{i \in Γ} ⊓ Φ (A_{i}) . \end{matrix}$

(F2) For all A ∈ L^X and α ∈ L, $\begin{matrix} ⊓ Φ (α \to A) = ⋀_{G \in AF (X)} (Φ (G) \to G (α \to A)) \\ = ⋀_{G \in AF (X)} (Φ (G) \to (α \to G (A))) \\ = α \to ⋀_{G \in AF (X)} (Φ (G) \to G (A)) = α \to ⊓ Φ (A) . \end{matrix}$ Hence ⊓Φ is an Alexandrov L-filter on X. Moreover, ⊓Φ is a meet of Φ from: $\begin{matrix} e_{AF (X)} (F, ⊓ Φ) = ⋀_{G \in AF (X)} (Φ (G) \to e_{AF (X)} (F, G)) \\ = e_{AF (X)} (F, ⋀_{G \in AF (X)} (Φ (G) \to G)) . \end{matrix}$

Theorem 3.7. Let AF (X) denote the set of all Alexandrov L-filters on X. Then (AF (X) , e_AF(X)) is a meet-complete lattice iff (1) $α \to F \in AF (X)$ for each $F \in AF (X)$

(2) $⋀_{i \in Γ} F_{i} \in AF (X)$ for each $F_{i} \in AF (X)$ , i ∈ Γ.

Proof. (⇒) (1) Define Φ : AF (X) → L as $Φ (F) = α$ for $F \in AF (X)$ and $Φ (G) = ⊥$ , otherwise. Then $⊓ Φ = ⋀_{G \in AF (X)} (Φ (G) \to G) = α \to F \in AF (X) .$

(2) Let ${F_{i} \in AF (X) ∣ i \in Γ}$ be given. Define Φ : AF (X) → L as $Φ (F_{i}) = ⊤$ for i ∈ Γ and $Φ (G) = ⊥$ , otherwise. Then $⊓ Φ = ⋀_{H \in AF (X)} (Φ (H) \to H) = ⋀_{i \in Γ} F_{i} \in AF (X) .$ (⇐) For Φ : AF (X) → L, $\begin{matrix} e_{AF (X)} (F, ⊓ Φ) = ⋀_{G \in AF (X)} (Φ (G) \to e_{AF (X)} (F, G)) \\ = e_{AF (X)} (F, ⋀_{G \in AF (X)} (Φ (G) \to G)) \end{matrix}$ Hence (AF (X) , e_AF(X)) is a meet-complete lattice.

Remark 3.8. (1) A map [x] : (L^X, e_{L
^X}) → (L, e_L) defined by [x] (A) = A (x) is an Alexandrov L-filter on X. Moreover, $\begin{matrix} e_{L^{X}} (A, B) \leq (A (x) \to B (x)) = ([x] (A) \to [x] (B) . \end{matrix}$ By Theorem 3.7, a map (α → [x]) : (L^X, e_{L
^X}) → (L, e_L) defined by (α → [x]) (A) = α → A (x) is an Alexandrov L-filter on X. The converse of Theorem 3.5 is not true from: $\begin{matrix} e_{L^{X}} (A, B) \leq (α ⊙ [x]) (A) \to (α ⊙ [x]) (B), \end{matrix}$ but α ⊙ [x] is not an Alexandrov L-filter on X.

(2) A map $F : (L^{X}, e_{L^{X}}) \to (L, e_{L})$ defined by $F (A) = ⋀_{x \in X} A (x)$ is an Alexandrov L-filter on X. A map $α \to F$ is an Alexandrov L-filter on X

(3) Put R ∈ L^X×Y, for each x ∈ X, a map $F^{x} : (L^{Y}, e_{L^{Y}}) \to (L, e_{L})$ defined by $F^{x} (A) = ⋀_{y \in Y} (R (x, y) \to A (y))$ is an Alexandrov L-filter on X from Lemma 2.2 (4,7).

Example 3.9. 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 (ref. [1 , 19]) as x^* = 1 - x, $\begin{matrix} x ⊙ y = max {0, x + y - 1}, \\ x \to y = min {1 - x + y, 1} . \end{matrix}$ Let R ∈ [0, 1] ^X×Y be a fuzzy information as follows: $\begin{matrix} R & e & b & w & c & i \\ h_{1} & 0.7 & 0.6 & 0.5 & 0.9 & 0.2 \\ h_{2} & 0.6 & 0.8 & 0.4 & 0.3 & 0.5 \\ h_{3} & 0.4 & 0.9 & 0.8 & 0.6 & 0.6 \end{matrix}$ A map $F^{x} : (L^{Y}, e_{L^{Y}}) \to (L, e_{L})$ defined by $F^{x} (A) = ⋀_{y \in Y} (R (x, y) \to A (y))$ is an Alexandrov L-filter on X. For A = (0.3, 0.5, 0.6, 0.1, 0.1), $F^{h_{1}} (A) = 0.2, F^{h_{2}} (A) = 0.6, F^{h_{3}} (A) = 0.5 .$

Definition 3.10. A family $N_{X} = {N^{x} ∣ x \in X}$ is called an Alexandrov L-neighborhood system on X if a map $N^{x} : (L^{X}, e_{L^{X}}) \to (L, e_{L})$ 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. $ψ : (X, N_{X}) \to (Y, N_{Y})$ is called an N-continuos map if, for each B ∈ L^Y and x ∈ X, $N_{Y}^{ψ (x)} (B) \leq N_{X}^{x} (ψ^{\Leftarrow} (B)) .$

Theorem 3.11. Let x^* = x→ ⊥ and (x^*) ^* = x for each x ∈ L. (1) For each x ∈ X, a map $N^{x} : L^{X} \to L$ is an Alexandrov L-neighborhood filter on X iff there exists a reflexive relation e_N ∈ L^X×X such that $N^{x} (A) = ⋀_{z \in X} (e_{N} (x, z) \to A (z))$ for all A ∈ L^X.

(2) For each x ∈ X, a map $N^{x} : L^{X} \to L$ is a topological Alexandrov L-neighborhood filter on X iff there exists an L-fuzzy preorder e_N ∈ L^X×X such that $N^{x} (A) = ⋀_{z \in X} (e_{N} (x, z) \to A (z))$ for all A ∈ L^X.

Proof. (1) (⇒) For all A ∈ L^X, since $A = ⋀_{z \in X} (A^{*} (z) \to ⊤_{z}^{*})$ , by Theorem 3.2 (F1) and (F2), we have $\begin{matrix} N^{x} (A) = N^{x} (⋀_{z \in X} (A^{*} (z) \to ⊤_{z}^{*}) \\ = ⋀_{z \in X} (A^{*} (z) \to N^{x} (⊤_{z}^{*})) \\ = ⋀_{z \in X} ((N^{x} (⊤_{z}^{*}))^{*} \to A (z) . \end{matrix}$ Put $e_{N} (x, y) = (N^{x} (⊤_{z}^{*}))^{*}$ . Then $e_{N} (x, x) = (N^{x} (⊤_{x}^{*}))^{*} \geq (⊤_{x}^{*})^{*} = ⊤ .$ So, e_N is reflexive. Moreover, $N^{x} (A) = ⋀_{z \in X} (e_{N} (x, z) \to A (z))$ .

(⇐) From Theorem 3.2, it satisfies (F1),(F(2) and $N^{x} (A) \leq e_{N} (x, x) \to A (x) = A (x) .$

(2) (⇒) By (1), $\begin{matrix} N^{x} (N^{-} (A)) = ⋀_{y \in X} (e_{N} (x, y) \to N^{y} (A)) \\ = ⋀_{y \in X} (e_{N} (x, y) \to ⋀_{z \in X} (e_{N} (y, z) \to A (z))) \\ = ⋀_{y, z \in X} (e_{N} (x, y) ⊙ e_{N} (y, z) \to A (z))) \\ \geq ⋀_{z \in X} (e_{N} (x, z) \to A (z)) \end{matrix}$ Put $A = ⊤_{z}^{*}$ . Then ⋁_y∈X (e_N (x, y) ⊙ e_N (y, z) ≤ e_N (x, z);i.e. e_N is transitive. By (1), e_N is an L-fuzzy preorder.

(⇐) For each x ∈ X, $N^{x}$ is topological Alexandrov L-neighborhood filter from (1) and for each A ∈ L^X, $\begin{matrix} N^{x} (N^{-} (A)) = ⋀_{y \in X} (e_{N} (x, y) \to N^{y} (A)) \\ = ⋀_{y \in X} (e_{N} (x, y) \to ⋀_{z \in X} (e_{N} (y, z) \to A (z))) \\ = ⋀_{y, z \in X} (e_{N} (x, y) ⊙ e_{N} (y, z) \to A (z))) \\ = ⋀_{z \in X} (e_{N} (x, z) \to A (z)) = N^{x} (A) . \end{matrix}$

Theorem 3.12. Let (X, e_X) be an L-fuzzy preorder set. Define $N_{e_{X}}^{x} : L^{X} \to$ as $N_{e_{X}}^{x} (A) = ⋀_{y \in X} (e_{X} (x, y) \to A (y)) .$ Then (1) For all x, y ∈ X, $N_{e_{X}}^{x} (⊤_{y}^{*}) = e_{X}^{*} (x, y)$ and $e_{X} (x, y) = ⋀_{A \in L^{X}} (N_{e_{X}}^{x} (A) \to N_{e_{X}}^{y} (A)) = ⋀_{z \in X} (N_{e_{X}}^{x} (⊤_{z}^{*}) \to N_{e_{X}}^{y} (⊤_{z}^{*}))$ .

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

Proof. (1) For all x, y ∈ X, $N_{e_{X}}^{x} (⊤_{y}^{*}) = ⋀_{y \in X} (e_{X} (x, y) \to ⊤_{y}^{*} (y)) = e_{X}^{*} (x, y)$ . For each A ∈ L^X, $\begin{matrix} ⋀_{A \in L^{X}} (N_{e_{X}}^{x} (A) \to N_{e_{X}}^{y} (A)) \\ \leq ⋀_{z \in X} (N_{e_{X}}^{x} (⊤_{z}^{*}) \to N_{e_{X}}^{y} (⊤_{z}^{*})) \\ = ⋀_{z \in X} ((e_{X} (x, z))^{*} \to (e_{X} (y, z))^{*}) \\ = ⋀_{z \in X} (e_{X} (y, z) \to e_{X} (x, z)) \\ = e_{X} (y, y) \to e_{X} (x, y) = e_{X} (x, y), \end{matrix}$ $\begin{matrix} N_{e_{X}}^{x} (A) \to N_{e_{X}}^{x} (A) \\ = ⋀_{z \in X} (e_{X} (x, z) \to A (z)) \\ \to ⋀_{w \in X} (e_{X} (x, w) \to A (w))) \\ \geq ⋀_{z \in X} ((e_{X} (x, z) \to A (z)) \to (e_{X} (y, z) \to A (z))) \\ \geq ⋀_{z \in X} (e_{X} (y, z) \to (e_{X} (x, z)) \geq e_{X} (x, y) . \end{matrix}$

(2) (N1) For each Φ : L^X → L, by Remark 2.6 (3), $\begin{matrix} e_{L} (b, ⊓ (N_{e_{X}}^{x})^{\to} (Φ)) = ⋀_{c \in L} (N_{e_{X}}^{x})^{\to} (c) \to e_{L} (b, c)) \\ = ⋀_{c \in L} (⋁_{c = N_{e_{X}}^{x} (D)} Φ (D) \to e_{L} (b, N_{e_{X}}^{x} (D))) \\ = e_{L} (b, ⋀_{D \in L^{X}} (Φ (D) \to N_{e_{X}}^{x} (D))) \\ = e_{L} (b, N_{e_{X}}^{x} (⋀_{D \in L^{X}} (Φ (D) \to D)))) \\ = e_{L} (b, N_{e_{X}}^{x} (⊓ Φ)) . \end{matrix}$ So, $N_{e_{X}}^{x} (⊓ Φ) = ⊓ (N_{e_{X}}^{x})^{\to} (Φ) .$

(N2) Since $N_{e_{X}}^{x} (A) \leq e_{X} (x, x) \to A (x) = A (x)$ , $N_{e_{X}}^{x} (A) \leq A (x)$ , for all A ∈ L^X.

(TN) For all A ∈ L^X, $\begin{matrix} N_{e_{X}}^{x} (N_{e_{X}}^{-} (A)) = ⋀_{y \in X} (e_{X} (x, y) \to N_{e_{X}}^{y} (A)) \\ = ⋀_{y \in X} (e_{X} (x, y) \to ⋀_{z \in X} (e_{X} (y, z) \to A (z))) \\ = ⋀_{z \in X} (⋁_{y \in X} (e_{X} (x, y) ⊙ e_{X} (y, z)) \to A (z))) \\ = ⋀_{z \in X} (e_{X} (x, z) \to A (z)) = N_{e_{X}}^{x} (A) . \end{matrix}$

Hence $N_{e_{X}}^{x}$ is a topological Alexandrov L-neighborhood filter on X

Theorem 3.13. Let $(X, N_{X})$ be an Alexandrov L-fuzzy neighborhood space. Define $e_{N_{X}} : X \times X \to L$ as $e_{N_{X}} (x, y) = ⋀_{A \in L^{X}} (N_{X}^{x} (A) \to N_{X}^{y} (A)) .$

(1) $e_{N_{X}} (x, y) = ⋀_{z \in X} (N_{X}^{x} (⊤_{z}^{*}) \to N_{X}^{y} (⊤_{z}^{*})) = ⋀_{z \in X} (e_{N} (y, z) \to e_{N} (x, z)) \leq e_{N} (x, y)$ , for all x, y ∈ X and $e_{N}^{*} (x, y) = N_{X}^{x} (⊤_{y}^{*})$ .

(2) $N_{e_{N_{X}}} \geq N_{X}$ .

(3) If $N_{X}$ is topological, then $e_{N_{X}} (x, y) = e_{N} (x, y) = (N_{X}^{x} (⊤_{y}^{*}))^{*}$ and $N_{e_{N_{X}}} = N_{X}$ .

(4) If e_X ∈ L^X×X is reflexive, then $e_{N_{e_{X}}} \leq e_{X}$ .

(5) If e_X ∈ L^X×X is an L-fuzzy preorder, then $e_{N_{e_{X}}} = e_{X}$ .

Proof. (1) For $A = ⋀_{z \in X} (A^{*} (z) \to ⊤_{z}^{*})$ , we have $\begin{matrix} e_{N_{X}} (x, y) = ⋀_{A \in L^{X}} (N_{X}^{x} (A) \to N_{X}^{y} (A)) \\ = ⋀_{A \in L^{X}} (N_{X}^{x} (⋀_{z \in X} (A^{*} (z) \to ⊤_{z}^{*})) \\ \to N_{X}^{y} (⋀_{w \in X} (A^{*} (w) \to ⊤_{w}^{*}))) \\ = ⋀_{A \in L^{X}} ((⋀_{z \in X} (A^{*} (z) \to N_{X}^{x} (⊤_{z}^{*}))) \\ \to (⋀_{w \in X} (A^{*} (w) \to N_{X}^{y} (⊤_{w}^{*}))) \\ \geq ⋀_{A \in L^{X}} ((⋀_{z \in X} (A^{*} (z) \to N_{X}^{x} (⊤_{z}^{*}))) \\ \to (A^{*} (z) \to N_{X}^{y} (⊤_{z}^{*}))) \\ \geq ⋀_{z \in X} (N_{X}^{x} (⊤_{z}^{*}) \to N_{X}^{y} (⊤_{z}^{*})) . \end{matrix}$ Moreover, put $A = ⊤_{z}^{*}$ . Then $e_{N_{X}} (x, y) \leq ⋀_{z \in X} (N_{X}^{x} (⊤_{z}^{*}) \to N_{X}^{y} (⊤_{z}^{*})) .$ Since $e_{N}^{*} (x, y) = N_{X}^{x} (⊤_{y}^{*})$ from Theorem 3.11, we have $e_{N_{X}} (x, y) =$ $⋀_{z \in X} (N_{X}^{x} (⊤_{z}^{*}) \to N_{X}^{y} (⊤_{z}^{*})) = ⋀_{z \in X} (e_{N} (y, z) \to e_{N}$ (x, z)) ≤ e_N (x, y) .

(2) By (1), $N_{e_{N_{X}}}^{x} (A) = ⋀_{y \in X} (e_{N_{X}} (x, y) \to A (y)) \geq ⋀_{y \in X} (e_{N} (x, y) \to A (y)) = N_{X}^{x} (A)$

(3) Let $N_{X}$ be topological. By Theorem 3.11, e_N is an L-fuzzy preorder. By (1), $e_{N_{X}} (x, y) = ⋀_{z \in X} (e_{N} (y, z) \to e_{N} (x, z)) \geq e_{N} (x, y)$ . Hence $e_{N_{X}} = e_{N}$ and $N_{e_{N_{X}}} = N_{X}$ .

(4) Let e_X ∈ L^X×X be reflexive. Then $N_{e_{X}}$ is an Alexandrov L-neighborhood structure. By (1), $e_{N_{e_{X}}} (x, y) = ⋀_{z \in X} (e_{X} (y, z) \to e_{X} (x, z)) \leq e_{X}$ (x, y).

(5) It follows from (4) and $e_{N_{e_{X}}} (x, y) = ⋀_{z \in X} (e_{X} (y, z) \to e_{X} (x, z)) \geq e_{X} (x, y)$ .

Remark 3.14. (1) A family $N_{X} = {N^{x} = [x] ∣ x \in X}$ is a topological Alexandrov L-neighborhood system on X because [x] ([-] A) = [-] A (x) = [x] (A). By Theorem 3.13, $\begin{matrix} e_{N_{X}} (x, y) = ⋀_{A \in L^{X}} ([x] (A) \to [y] (A)) \\ = ⋀_{z \in X} ([x] (⊤_{z}^{*}) \to [y] (⊤_{z}^{*})) = [x] (⊤_{y}), \\ N_{e_{N_{X}}}^{x} (A) = ⋀_{z \in X} (e_{X} (x, z) \to A (z)) = [x] (A) . \end{matrix}$

(2) A family $N_{X} = {N^{x} = ⋀_{z \in X} A (z) ∣ x \in X}$ is a topological Alexandrov L-neighborhood system on X because $N^{x} (N^{-} (A)) = ⋀_{z \in X} N^{-} (A) (z) = ⋀_{z \in X} N^{z} (A) = ⋀_{z \in X} ⋀_{w \in X} A (w) = N^{x} (A)$ . $\begin{matrix} e_{N_{X}} (x, y) = ⋀_{A \in L^{X}} (⋀_{z \in X} A (z) \to ⋀_{z \in X} A (z)) = ⊤, \\ N_{e_{N_{X}}}^{x} (A) = ⋀_{z \in X} (⊤ \to A (z)) = ⋀_{z \in X} A (z) . \end{matrix}$ (3) Let e_X ∈ L^X×X be reflexive. A family $N_{X} = {N^{x} (A) = ⋀_{z \in X} (e_{X} (x, z) \to A (z)) ∣ x \in X}$ is an Alexandrov L-neighborhood system on X. $\begin{matrix} e_{N_{X}} (x, y) = ⋀_{z \in X} (N^{x} (⊤_{z}^{*}) \to N^{y} (⊤_{z}^{*})) \\ = ⋀_{z \in X} (e_{X} (x, z)^{*} \to e_{X} (y, z)^{*}) \\ \leq e_{X} (x, y)^{*} \to e_{X} (y, y)^{*}) = e_{X} (x, y), \\ N_{e_{N_{X}}}^{x} (A) = ⋀_{z \in X} (e_{N_{X}} (x, z) \to A (z)) \\ \geq ⋀_{z \in X} (e_{X} (x, z) \to A (z)) = N_{e_{X}}^{x} (A) . \end{matrix}$

Example 3.15. Let ([0, 1] , ⊙ , → , ^*, 0, 1) be a complete residuated lattice as in Example 3.9. Let X = {x, y, z} be a set and a reflexive L-fuzzy relation e_X ∈ [0, 1] ^X×X as $e_{X} = (\begin{matrix} 1 & 0.9 & 0.4 \\ 0.6 & 1 & 0.8 \\ 0.3 & 0.9 & 1 \end{matrix}) e_{X} \circ e_{X} = (\begin{matrix} 1 & 0.9 & 0.7 \\ 0.6 & 1 & 0.8 \\ 0.5 & 0.9 & 1 \end{matrix})$ where e_X ∘ e_X (x, z) = ⋁ _y∈X (e_X (x, y) ⊙ e_X (y, z)). Then e_X is not an L-fuzzy preorder. Define $N_{e_{X}}^{x} : L^{X} \to L$ as $N_{e_{X}}^{x} (A) = ⋀_{y \in X} (e_{X} (x, y) \to A (y)) .$ By Theorem 3.10(1), $N_{e_{X}}^{x}$ is an Alexandrov L-neighborhood filter for each x ∈ X, but it is not topological because, for A = (0.7, 1, 0.1), $\begin{matrix} N_{e_{X}}^{-} (A) & = (0.7, 0.2, 0.1) ≰ N_{e_{X}}^{-} (N_{e_{X}}^{-} (A)) \\ = (0.6, 0.2, 0.1) . \end{matrix}$ Since e_X is not an L-fuzzy preorder, $e_{N_{e_{X}}} < e_{X}$ as follows: $\begin{matrix} e_{N_{e_{X}}} (x, y) = ⋀_{z \in X} (N_{e_{X}}^{x} (⊤_{z}^{*}) \to N_{e_{X}}^{y} (⊤_{z}^{*})) \\ = ⋀_{z \in X} (e_{X} (x, z)^{*} \to e_{X} (y, z)^{*}) \\ = ⋀_{z \in X} (e_{X} (y, z) \to e_{X} (x, z)) \end{matrix}$ $e_{N_{e_{X}}} = (\begin{matrix} 1 & 0.6 & 0.4 \\ 0.6 & 1 & 0.8 \\ 0.3 & 0.7 & 1 \end{matrix})$ Since $N_{e_{X}}^{x}$ is not topological, for B = (1, 0.5, 1), $N_{e_{X}}^{-} (B) = (0.6, 0.5, 1) < N_{e_{N_{e_{X}}}}^{-} (B) = (0.9, 0.5, 1) .$

Theorem 3.16. (1) Let (X, e_X) and (Y, e_Y) be L-fuzzy preordered sets. If ψ : (X, e_X) → (Y, e_Y) is an order preserving map, then $ψ : (X, N_{e_{X}}) \to (Y, N_{e_{Y}})$ is an N-continuous map.

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

(3) The converses of (1) and (2) are true.

Proof. (1) For each B ∈ L^Y, $\begin{matrix} N_{e_{Y}}^{ψ (x)} (B) = ⋀_{y \in Y} (e_{Y} (ψ (x), y) \to B (y)) \\ \leq ⋀_{y \in Y} (e_{Y} (ψ (x), ψ (z)) \to B (ψ (z))) \\ \leq ⋀_{z \in X} (e_{X} (x, z) \to ψ^{\Leftarrow} (B) (z)) = N_{e_{X}}^{x} (ψ^{\Leftarrow} (B)) . \end{matrix}$

(2) From Theorem 3.13(3), $\begin{matrix} e_{N_{Y}} (ψ (x), ψ (y)) = (N_{Y}^{ψ (x)} (⊤_{ψ (y)}^{*}))^{*} \\ \geq (N_{X}^{x} (ψ^{\Leftarrow} (⊤_{ψ (y)}^{*})))^{*} \geq (N_{X}^{x} (⊤_{y}^{*}))^{*} = e_{N_{X}} (x, y) . \end{matrix}$

(3) From Theorem 3.13 (3,4), since $e_{N_{e_{X}}} = e_{X}$ and $N_{e_{N_{X}}} = N_{X}$ , the converses of (1) and (2) are true.

Let POS be denote the category of L-fuzzy preordered sets and order preserving maps for morphisms. Let TANS be denote the category of topological Alexandrov L-neighborhood spaces and N-continuous maps for morphisms.

Theorem 3.17. Two categories POS and TANS are isomorphic.

Proof. Define F:POS→ TANS as $F (X, e_{X}) = (X, N_{e_{X}})$ . Hence F is a functor from Theorem 3.16. Define a functor G : TANS → POS as $G (X, N_{X}) = (X, e_{N_{X}})$ . Hence G is a functor.

From Theorem 3.13, since G (F (X, e_X)) = G (X, $N_{e_{X}}) = (X, e_{N_{e_{X}}}) = (X, e_{X})$ and $F (G (X, N_{X})) = F (X, e_{N_{X}}) = (X, N_{e_{N_{X}}}) = (X, N_{X})$ . Thus, POS and TANS are isomorphic.

4 Alexandrov L-convergence spaces

In this section, we introduce the new notion of Alexandrov L-convergence structure on Alexandrov L-filters. We investigate the categorical relations among Alexandrov L-neighborhood filters, L-fuzzy preorders and Alexandrov L-convergence structures.

Definition 4.1. A map lim : AF (X) → L^X is called an Alexandrov L-convergence 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 $F, G \in AF (X)$ .

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

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

(1) $e_{AF (X)} (F, G) \leq e_{L^{X}} (lim (F), lim (G))$ for all $F, G \in AF (X)$ .

(2) If $F \leq G$ , then $lim (F) \leq lim (G)$ and $lim (α \to F) \leq α \to lim (F)$ for all $F, G \in AF (X)$ and α ∈ L.

Proof. (1) ⇒ (2). If $F \leq G$ , then $⊤ = e_{AF (X)} (F,$ $G) \leq e_{L^{X}} (lim (F), lim (G))$ . Hence $lim (F) \leq lim (G)$ . Put $G = α \to F$ . Then $α \leq e_{AF (X)} (α \to F, F) \leq e_{L^{X}} (lim (α \to F), lim (F)) .$ Hence $lim (α \to F) \leq α \to lim (F) .$

(2) ⇒ (1). Since $F \leq e_{AF (X)} (F, G) \to G$ ,

$lim (F) \leq lim (e_{AF (X)} (F, G) \to G) \leq e_{AF (X)} (F, G)$ $\to lim (G)$ . Hence $e_{AF (X)} (F, G) \leq e_{L^{X}} (lim (F),$ $lim (G)) .$

Theorem 4.3. Let AC (X) denote the set of all Alexandrov L-convergence structures on X. Then (1) α → lim ∈ AC (X) for each lim ∈ AC (X)

(2) $⋀_{i \in Γ} lim_{i}, ⋁_{i \in Γ} lim_{i} \in AC (X)$ for each $lim_{i} \in AC (X)$ , i ∈ Γ.

Proof. (1) (AC1) (α→ lim) [x] (x) = α → lim [x] (x) = α → ⊤ = ⊤.

(AC2) $e_{AF (X)} (F, G) \leq e_{L^{X}} (lim (F), lim (G))$ for all $F, G \in AF (X)$ .

$\begin{matrix} e_{L^{X}} (α \to lim (F), α \to lim (G)) \\ \geq e_{L^{X}} (lim (F), lim (G)) \geq e_{AF (X)} (F, G) . \end{matrix}$

(2) (AC1) $⋀_{i \in Γ} lim_{i} [x] (x) = ⊤$ .

(AC2) $\begin{matrix} e_{L^{X}} (⋀_{i \in Γ} lim_{i} (F), ⋀_{i \in Γ} lim_{i} (G)) \\ \geq ⋀_{i \in Γ} e_{L^{X}} (lim_{i} (F), lim_{i} (G)) \geq e_{AF (X)} (F, G) . \end{matrix}$ Hence $⋀_{i \in Γ} lim_{i} \in AC (X)$ . Similarly, $⋁_{i \in Γ} lim_{i} \in AC (X)$ .

Theorem 4.4. Let AC (X) denote the set of all Alexandrov L-convergence structures on X. Define e_AC(X) : AC (X) × AC (X) → L as $\begin{matrix} e_{AC (X)} (lim_{1}, lim_{2}) \\ = ⋀_{x \in X} ⋀_{F \in AF (X)} (lim_{1} (F) (x) \to lim_{2} (F) (x)) . \end{matrix}$ Then (AC (X) , e_AC(X)) is a meet-complete lattice.

Proof. We easily prove that (AC (X) , e_AC(X)) is an L-fuzzy partially ordered set. For Φ : AC (X) → L, since ⋀_lim∈AC(X) (Φ (lim) → lim) ∈ AC (X) from Theorem 4.3, we have $\begin{matrix} e_{AC (X)} (lim_{1}, ⊓ Φ) \\ = ⋀_{lim \in AC (X)} (Φ (lim) \to e_{AC (X)} (lim_{1}, lim)) \\ = e_{AC (X)} (lim_{1}, ⋀_{lim \in AC (X)} (Φ (lim) \to lim)) . \end{matrix}$ Hence ⊓Φ ∈ AC (X) .

Remark 4.5. Let AC (X) denote the set of all Alexandrov L-convergence structures on X. Define Φ : AC (X) → L as Φ (lim) = α for lim ∈ AC (X) and $Φ (lim_{1}) = ⊥$ , otherwise. Then $⊔ Φ = ⋁_{lim \in AC (X)} (Φ (lim) ⊙ lim) = α ⊙ lim \notin AC (X) .$ Hence (AC (X) , e_AC(X)) is not a join-complete lattice.

Theorem 4.6. 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 (1) $lim_{N}$ is an Alexandrov L-convergence structure on X.

(2) Define Φ^x : AF (X) → L as $Φ^{x} (F) = lim_{N} F (x) .$ Then ${⊓ Φ^{x} (F) ∣ x \in X}$ is the system of Alexandrov L-neighborhood filters such that $N^{x} = ⊓ Φ^{x} (F)$ .

Proof. (1) (AC1) $lim_{N} [x] (x) = e_{AF (X)} (N^{x}, [x]) = ⊤ .$

(AC2) For all $F, G \in AF (X)$ , $\begin{matrix} e_{L^{X}} (lim_{N} (F), lim_{N} (G)) \\ = ⋀_{x \in X} (lim_{N} (F) (x) \to lim_{N} (G) (x)) \\ = ⋀_{x \in X} (e_{AF (X)} (N^{x}, F) \to e_{AF (X)} (N^{x}, G)) \\ \geq e_{AF (X)} (F, G) . \end{matrix}$

(2) For Φ^x : AF (X) → L as $Φ^{x} (F) = lim_{N} F (x)$ , $\begin{matrix} ⋀_{F \in AF (X)} (Φ^{x} (F) \to e_{AF (X)} (G, F)) \\ = ⋀_{F \in AF (X)} (lim_{N} (F) (x) \to e_{AF (X)} (G, F)) \\ = ⋀_{x \in X} (e_{AF (X)} (N^{x}, F) \to e_{AF (X)} (G, F)) \\ \geq e_{AF (X)} (G, N^{x}) . \end{matrix}$

Since $Φ^{x} (N^{x}) = lim_{N} (N^{x}) (x) = e_{AF (X)} (N^{x},$ $N^{x}) = ⊤$ , $\begin{matrix} ⋀_{F \in AF (X)} (Φ^{x} (F) \to e_{AF (X)} (G, F)) \\ \leq lim_{N} (N^{x}) (x) \to e_{AF (X)} (G, N^{x}) \\ = ⊤ \to e_{AF (X)} (G, N^{x}) = e_{AF (X)} (G, N^{x}) . \end{matrix}$ We have $⊓ Φ^{x} (F) = N^{x}$ from: $\begin{matrix} e_{AF (X)} (G, ⊓ Φ^{x}) \\ = ⋀_{F \in AF (X)} (Φ^{x} (F) \to e_{AF (X)} (G, F)) \\ = e_{AF (X)} (G, N^{x}) . \end{matrix}$

Theorem 4.7. Let $(X, lim_{X})$ be an Alexandrov L-convergence space. Define $N_{lim_{X}}^{x} : L^{X} \to L$ as $N_{lim_{X}}^{x} (A) = ⋀_{F \in AF (X)} (lim_{X} (F) (x) \to F (A)) .$

Then the following properties hold.

(1) $N_{lim}^{x}$ is an Alexandrov L-neghborhood filter on X.

(2) $N_{lim_{N}}^{x} = N^{x}$ .

(3) $lim_{N_{lim}} \geq lim$ .

Proof. (1) By Theorem 3.2, $N_{lim}^{x}$ is an Alexandrov L-neghborhood filter on X from: $\begin{matrix} N_{lim_{X}}^{x} (α \to A) \\ = α \to ⋀_{F \in AF (X)} (lim_{X} (F) (x) \to F (A)), \\ N_{lim_{X}}^{x} (⋀_{i \in I} A_{i}) = ⋀_{i \in I} N_{lim_{X}}^{x} (A_{i}), \\ N_{lim_{X}}^{x} (A) \leq lim_{X} ([x]) (x) \to [x] (A) = A (x) . \end{matrix}$

(2) For each A ∈ L^X, $N_{lim_{N}}^{x} = N^{x}$ from: $\begin{matrix} N_{lim_{N}}^{x} (A) = ⋀_{F \in AF (X)} (lim_{N} (x) \to F (A)) \\ = ⋀_{F \in AF (X)} (⋀_{B \in L^{X}} (N^{x} (B) \to F (B)) \to F (A)) \\ \geq ⋀_{F \in AF (X)} ((N^{x} (A) \to F (A)) \to F (A)) \\ \geq N^{x} (A), \\ N_{lim_{N}}^{x} (A) = ⋀_{F \in AF (X)} (lim_{N} (x) \to F (A)) \\ \leq ⋀_{B \in L^{X}} (N^{x} (B) \to N^{x} (B)) \to N^{x} (A)) . \end{matrix}$

(3) For each $F \in AF (X), x \in X$ , $\begin{matrix} lim_{N_{lim}} (F) (x) = e_{AF (X)} (N_{lim}^{x}, F) \\ = ⋀_{A \in L^{X}} ((⋀_{G \in AF (X)} (lim (G) (x) \to G (A)) \to F (A)) \\ \geq (lim (F) (x) \to F (A)) \to F (A) \\ \geq lim (F (x) . \end{matrix}$

Definition 4.8. An Alexandrov L-convergence structure $lim_{X} : AF (X) \to L^{X}$ is called principal if

(P) $lim_{X} (F) (x) = e_{AF (X)} (N_{lim_{X}}^{x}, F)$ for all $F \in AF (X)$ .

If $N_{lim_{X}}^{x}$ is topological, then $lim_{X}$ is called topological.

Remark 4.9. Let ${N^{x} ∣ x \in X}$ be an Alexandrov L-neighborhood system on X. By Theorem 4.6, $lim_{N}$ is a pricipal Alexandrov L-convergence structure on X. If $N^{x}$ is topological, then $lim_{N}$ is topological.

Theorem 4.10. Let $(X, lim_{X})$ be an Alexandrov L-convergence space. Define $e_{lim_{X}} (x, y) = ⋀_{F \in AF (X)} (lim_{X} (F) (y) \to lim_{X} (F) (x)) .$ Then (1) $e_{lim_{X}}$ is an L-fuzzy preorder on X.

(2) If $lim_{X}$ is principal, then $e_{N_{lim_{X}}} = e_{lim_{X}}$ .

(3) If $lim_{X}$ is topological principal, then, for all x, y ∈ X, $e_{N_{lim_{X}}} (x, y) = e_{lim_{X}} (x, y) = (N_{lim_{X}}^{x} (⊤_{y}^{*}))^{*} .$

(4) If $N_{X}$ is a topological Alexandrov L-neighborhood filter on X, then $e_{N_{X}} (x, y) = e_{lim_{N_{X}}} (x, y) = (N^{x} (⊤_{y}^{*}))^{*}$ for all x, y ∈ X.

(5) If e_X is an L-fuzzy preorder on X and define $lim_{e_{X}} (F) = e_{AF (X)} (N_{e_{X}}^{x}, F)$ then $(X, lim_{e_{X}})$ is a topological principal Alexandrov L-convergence space such that $e_{X} = e_{N_{e_{X}}} = e_{N_{lim e_{X}}} = e_{lim_{e_{X}}} .$

(6) If $lim_{X}$ is principal, then $lim_{e_{lim_{X}}} \leq lim_{X}$ . Moreover, if $lim_{X}$ is topological principal, then $lim_{e_{lim_{X}}} = lim_{X}$ .

Proof. (1) (E1) $e_{lim_{X}} (x, x) = ⊤ .$ (E2) $\begin{matrix} e_{lim_{X}} (y, x) ⊙ e_{lim_{X}} (z, y) \\ = ⋀_{F \in AF (X)} (lim_{X} (F) (x) \to lim_{X} (F) (y)) \\ ⊙ ⋀_{F \in AF (X)} (lim_{X} (F) (y) \to lim_{X} (F) (z)) \\ = ⋀_{F \in AF (X)} ((lim_{X} (F) (x) \to lim_{X} (F) (y)) \\ ⊙ (lim_{X} (F) (y) \to lim_{X} (F) (z))) \\ \leq ⋀_{F \in AF (X)} (lim_{X} (F) (x) \to lim_{X} (F) (z)) \\ = e_{lim_{X}} (z, x) . \end{matrix}$ Hence $e_{lim_{X}}$ is an L-fuzzy preorder on X.

(2) $\begin{matrix} e_{lim_{X}} (x, y) = ⋀_{F \in AF (X)} (lim_{X} (F) (y) \to lim_{X} (F) (x)) \\ = ⋀_{F \in AF (X)} (e_{AF (X)} (N_{lim_{X}}^{y}, F) \to e_{AF (X)} (N_{lim_{X}}^{x}, F)) \\ \geq e_{AF (X)} (N_{lim_{X}}^{x}, N_{lim_{X}}^{y}) = e_{N_{lim_{X}}} (x, y) . \end{matrix}$ $\begin{matrix} e_{lim_{N_{X}}} (x, y) \\ = ⋀_{F \in AF (X)} (e_{AF (X)} (N_{lim_{X}}^{y}, F) \to e_{AF (X)} (N_{lim_{X}}^{x}, F)) \\ \leq e_{AF (X)} (N_{lim_{X}}^{y}, N_{lim_{X}}^{y}) \to e_{AF (X)} (N_{lim_{X}}^{x}, N_{lim_{X}}^{y}) \\ = e_{AF (X)} (N_{lim_{X}}^{x}, N_{lim_{X}}^{y}) = e_{N_{lim_{X}}} (x, y) . \end{matrix}$

(3) and (4) follow from Theorem 3.13(3).

(5) $lim_{e_{X}}$ is a principal Alexandrov L-convergence structure from: $\begin{matrix} N_{lim_{e_{X}}}^{x} (A) = ⋀_{F \in AF (X)} (lim_{e_{X}} (F) (x) \to F (A)) \\ = ⋀_{F \in AF (X)} (e_{AF (X)} (N_{e_{X}}^{x}, F) \to F (A)) = N_{e_{X}}^{x} (A) . \end{matrix}$ By Theorem 3.13(5), $e_{X} = e_{N_{e_{X}}}$ .

(6) For each x, y ∈ X, $\begin{matrix} e_{lim_{X}} (x, y) = e_{N_{lim_{X}}} (x, y) \\ = ⋀_{z \in X} ((N_{lim_{X}}^{x} (⊤_{z}^{*})) \to (N_{lim_{X}}^{y} (⊤_{z}^{*}))) \\ = ⋀_{z \in X} (((N_{lim_{X}}^{y} (⊤_{z}^{*})))^{*} \to (N_{lim_{X}}^{x} (⊤_{z}^{*})^{*})) \\ \leq (N_{lim_{X}}^{y} (⊤_{y}^{*}))^{*} \to (N_{lim_{X}}^{x} (⊤_{y}^{*}))^{*} \\ ((N_{lim_{X}}^{y} (⊤_{y}^{*}))^{*} \geq (⊤_{y}^{*})^{*} (y) = ⊤) \\ = (N_{lim_{X}}^{x} (⊤_{y}^{*}))^{*} . \end{matrix}$ By (5), $\begin{matrix} lim_{e_{lim_{X}}} (F) (x) = e_{AF (X)} (N_{e_{lim_{X}}}^{x}, F) \\ = e_{AF (X)} (N_{e_{N_{lim_{X}}}}^{x}, F) \\ = ⋀_{A \in L^{X}} ((⋀_{y \in X} (e_{N_{lim_{X}}} (x, y) \to A (y)) \to F (A)) \\ \leq ⋀_{A \in L^{X}} ((⋀_{y \in X} ((N_{lim_{X}}^{x} (⊤_{y}^{*}))^{*} \to A (y)) \to F (A)) \\ = ⋀_{A \in L^{X}} ((⋀_{y \in X} (A^{*} (y) \to N_{lim_{X}}^{x} (⊤_{y}^{*})) \to F (A)) \\ = ⋀_{A \in L^{X}} (N_{lim_{X}}^{x} ((⋀_{y \in X} (A^{*} (y) \to ⊤_{y}^{*}))) \to F (A)) \\ = ⋀_{A \in L^{X}} (N_{lim_{X}}^{x} (A) \to F (A)) = lim (F) (x) . \end{matrix}$ Let $lim_{X}$ be topological principal. By (2), $e_{N_{lim_{X}}} = lim_{X}$ and $lim_{e_{lim_{X}}} = lim_{X}$ .

Remark 4.11. (1) Define $lim_{X} : AF (X) \to L^{X}$ as $lim_{X} F (x) = e_{AF (X)} ([x], F) = ⋀_{A \in L^{X}} ([x] (A) \to F (A)) .$ Then $lim_{X}$ is a principle Alexandrov L-convergence structure from $\begin{matrix} N_{lim_{X}}^{x} (A) = ⋀_{F \in AF (X)} (lim_{X} (F) (x) \to F (A)) = [x] (A) . \end{matrix}$ Moreover, $e_{lim_{X}} (x, y) = e_{N_{lim_{X}}} (x, y) = ⋀_{A \in L^{X}} ([y]$ (A) → [x] (A)) = ▵ _X×X (x, y) . $▵_{X \times X} (x, y) = {\begin{matrix} ⊤, & if y = x, \\ ⊥, & otherwise, \end{matrix}$ $\begin{matrix} N_{e_{lim_{X}}}^{x} (A) = ⋀_{y \in X} (e_{lim_{X}} (x, y) \to A (y)) = [x] (A), \\ lim_{e_{lim_{X}}} (F) (x) = ⋀_{A \in L^{X}} (N_{e_{lim_{X}}}^{x} (A) \to F (A)) \\ = ⋀_{A \in L^{X}} ([x] (A) \to F (A)) = lim_{X} (F) (x) . \end{matrix}$

(2) Define $lim_{X} : AF (X) \to L^{X}$ as $lim_{X} (F) (x) = ⋀_{A \in L^{X}} (⋀_{z \in X} A (z) \to F (A)) .$ Then $lim_{X}$ is a principle Alexandrov L-convergence structure from $\begin{matrix} N_{lim_{X}}^{x} (A) = ⋀_{F \in AF (X)} (lim_{X} (F) (x) \to F (A)) \\ = ⋀_{z \in X} A (z) . \end{matrix}$ Moreover, $e_{lim_{X}} (x, y) = e_{N_{lim_{X}}} (x, y) = ⋀_{A \in L^{X}} (⋀_{z \in X}$ A (z) → ⋀ _z∈XA (z)) = ⊤ . $\begin{matrix} N_{e_{lim_{X}}}^{x} (A) = ⋀_{y \in X} (e_{lim_{X}} (x, y) \to A (y)) = ⋀_{y \in X} A (y), \\ lim_{e_{lim_{X}}} (F) (x) = ⋀_{A \in L^{X}} (N_{e_{lim_{X}}}^{x} (A) \to F (A)) \\ = ⋀_{A \in L^{X}} (⋀_{y \in X} A (y) \to F (A)) = lim_{X} (F) (x) . \end{matrix}$

(3) Let e_X ∈ L^X×X be reflexive. Define $lim_{X} : AF (X) \to L^{X}$ as $\begin{matrix} lim_{X} (F) (x) \\ = ⋀_{A \in L^{X}} ((⋀_{y \in X} (e_{X} (x, y) \to A (y)) \to F (A)) . \end{matrix}$ Then $lim_{X}$ is a principle Alexandrov L-convergence structure from $\begin{matrix} N_{lim_{X}}^{x} (A) = ⋀_{F \in AF (X)} (lim_{X} (F) (x) \to F (A)) \\ = (⋀_{y \in X} (e_{X} (x, y) \to A (y)) . \end{matrix}$ Moreover, $e_{lim_{X}} (x, y) = e_{N_{lim_{X}}} (x, y) = ⋀_{z \in X} (N_{lim_{X}}^{x}$ $(⊤_{z}^{*}) \to N_{lim_{X}}^{y} (⊤_{z}^{*})) = ⋀_{z \in X} (e_{X}^{*} (x, z) \to e_{X}^{*} (y, z))$ ≤e_X (x, y). If e_X is transitive, $e_{N_{lim_{X}}} = e_{X} (x, y)$ . $\begin{matrix} N_{e_{lim_{X}}}^{x} (A) = ⋀_{y \in X} (e_{lim_{X}} (x, y) \to A (y)) \\ \geq ⋀_{y \in X} (e_{X} (x, y) \to A (y)) = N_{lim_{X}}^{x} (A), \\ lim_{e_{lim_{X}}} (F) (x) = ⋀_{A \in L^{X}} (N_{e_{lim_{X}}}^{x} (A) \to F (A)) \\ \leq ⋀_{A \in L^{X}} (N_{lim_{X}}^{x} (A) \to F (A)) = lim_{X} (F) (x) . \end{matrix}$

If e_X is transitive, $lim_{e_{lim_{X}}} = lim_{X} .$

Theorem 4.12. (1) Let $(X, lim_{X})$ and $(Y, lim_{Y})$ be Alexandrov convergence spaces. If $ψ : (X, lim_{X}) \to (Y, lim_{Y})$ is a continuous map, then $ψ : (X, N_{lim_{X}}) \to (Y, N_{lim_{Y}})$ 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 $ψ : (X, lim_{N_{X}}) \to (Y, lim_{N_{X}})$ is a continuous map.

Proof. (1) For each B ∈ L^Y, $\begin{matrix} N_{lim_{X}}^{x} (ψ^{\Leftarrow} (B)) = ⋀_{F \in AF (X)} (lim (F) (x) \to F (ψ^{\Leftarrow} (B))) \\ = ⋀_{F \in AF (X)} (lim (F) (x) \to ψ^{\Rightarrow} (F) (B)) \\ \geq ⋀_{F \in AF (X)} (lim_{Y} (ψ^{\Rightarrow} (F)) (ψ (x)) \to ψ^{\Rightarrow} (F) (B)) \\ \geq ⋀_{G \in AF (Y)} (lim_{Y} (G) (ψ (x)) \to G (B)) \\ = ⋀_{G \in AF (Y)} (Φ_{Y}^{ψ (x)} (G) \to G (B)) = N_{lim_{Y}}^{ψ (x)} (B) . \end{matrix}$

(2) For each $F \in AF (Y)$ , $\begin{matrix} lim_{N_{Y}} (ψ^{\Rightarrow} (F)) (ψ (x)) = e_{AF (Y)} (N_{Y}^{ψ (x)}, ψ^{\Rightarrow} (F)) \\ = ⋀_{B \in L^{Y}} (N_{Y}^{ψ (x)} (B) \to ψ^{\Rightarrow} (F) (B)) \\ \geq ⋀_{B \in L^{Y}} (N_{X}^{x} (ψ^{\Leftarrow} (B)) \to F (ψ^{\Leftarrow} (B))) \\ \geq ⋀_{A \in L^{X}} (N_{X}^{x} (A) \to F (A)) = lim_{N_{X}} (F) (x) . \end{matrix}$

Theorem 4.13. (1) Let $(X, lim_{X})$ and $(Y, lim_{Y})$ be a topological principal Alexandrov convergence spaces. If $ψ : (X, lim_{X}) \to (Y, lim_{Y})$ is a continuous map, then $ψ : (X, e_{lim_{X}}) \to (Y, e_{lim_{Y}})$ 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 $ψ : (X, lim_{e_{X}}) \to (Y, lim_{e_{X}})$ is a continuous map.

Proof. (1) Since $N_{lim_{X}}$ and $N_{lim_{Y}}$ are topological, by Theorem 3.13(3), $\begin{matrix} e_{lim_{Y}} (ψ (x), ψ (y)) = e_{N_{lim_{Y}}} (ψ (x), ψ (y)) \\ = (N_{lim_{Y}}^{ψ (x)} (⊤_{ψ (y)}^{*}))^{*} \geq (N_{lim_{X}}^{x} (ψ^{\Leftarrow} (⊤_{ψ (y)}^{*})))^{*} \\ \geq (N_{lim_{X}}^{x} (⊤_{y}^{*}))^{*} = e_{N_{lim_{X}}} (x, y) = e_{lim_{X}} (x, y) . \end{matrix}$

(2) From Theorem 3.16, for each B ∈ L^Y, $N_{e_{Y}}^{ψ (x)} (B) \leq N_{e_{X}}^{x} (ψ^{\Leftarrow} (B)) .$ Thus $\begin{matrix} lim_{e_{Y}} (ψ^{\Rightarrow} (F)) (ψ (x)) = e_{AF (Y)} (N_{e_{Y}}^{ψ (x)}, ψ^{\Rightarrow} (F)) \\ = ⋀_{B \in L^{Y}} (N_{e_{Y}}^{ψ (x)} (B) \to ψ^{\Rightarrow} (F) (B)) \\ \geq ⋀_{B \in L^{Y}} (N_{e_{X}}^{x} (ψ^{\Leftarrow} (B)) \to F (ψ^{\Leftarrow} (B))) \\ \geq ⋀_{A \in L^{X}} (N_{e_{X}}^{x} (A) \to F (A)) = lim_{e_{X}} (F) (x) . \end{matrix}$

Definition 4.14. [1] 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 ACS (resp.TPACS) denote the category of (resp. topological principal) Alexandrov L-convergence spaces and continuous mappings for morphisms. Let ANS denote the category of Alexandrov L-neighborhood spaces continuous mappings for morphisms.

Theorem 4.15. (1) F : ACS → ANS defined as $F (X, lim_{X}) = (X, N_{lim_{X}})$ is a functor.

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

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

(4) Two categories TPACS and POS are isomorphic.

Proof. (1) and (2) are follows from Theorems 4.6, 4.7 and 4.12.

(3) By Theorem 4.7(2), if $(X, N_{X})$ is an Alexandrov L-neighborhood space, then $F (G (X, N_{X}) = (X, N_{lim_{N_{X}}} = N_{X})$ . Hence, the identity map ${id}_{X} : (X, N_{lim_{N_{X}}}) = F (G (X, N_{X}) \to (X, N_{X})$ is a continuous map. Moreover, if $(X, lim_{X})$ is an Alexandrov L-convergence space, by Theorem 4.7(3), $G (F (X, lim_{X})) = lim_{N_{lim_{X}}} \geq lim_{X}$ . Hence the identity map ${id}_{X} : (X, lim_{X}) \to G (F (X, lim_{X})) = lim_{N_{lim_{X}}}$ is a continuous map. Therefore (F, G) is a Galois correspondence.

(4) Define H : POS → TPACS as $H (X, e_{X}) = (X, lim_{e_{X}})$ . Define a functor K : TPACS → POS as $K (X, lim_{X}) = (X, e_{lim_{X}})$ . By Theorem 4.13, H and K are functors. Since $K (H (X, e_{X})) = K (X, lim_{e_{X}}) = (X, e_{lim_{e_{X}}}) = (X, e_{X})$ from Theorem 4.10(5) and $H (K (X, lim_{X})) = H (X, e_{lim_{X}}) = (X, lim_{e_{lim_{X}}}) = (X, lim_{X})$ from Theorem 4.10(6). Thus, POS and TPACS are isomorphic.

Example 4.16. Let ([0, 1] , ⊙ , → , ^*, 0, 1) be a complete residuated lattice (ref. [1 , 19]) 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 an L-fuzzy preorder e_X ∈ [0, 1] ^X×X as $e_{X} = (\begin{matrix} 1 & 0.6 & 0.5 \\ 0.8 & 1 & 0.6 \\ 0.8 & 0.7 & 1 \end{matrix})$ Define an Alexandrov L-neighborhood filter $N^{y} : L^{X} \to L$ as $N^{y} (A) = ⋀_{x \in X} (e_{X} (y, x) \to A (x)) .$

(TN) Since e_X ∘ e_X = e_X, we have $\begin{matrix} N^{y} (N^{-} (A)) = ⋀_{x \in X} (e_{X} (y, x) \to N^{x} (A)) \\ = ⋀_{x \in X} (e_{X} (y, x) \to ⋀_{z \in X} (e_{X} (x, z) \to A (z))) \\ = ⋀_{x, z \in X} (e_{X} (y, x) ⊙ e_{X} (x, z) \to A (z))) \\ = ⋀_{z \in X} (⋁_{x \in X} e_{X} (y, x) ⊙ e_{X} (x, z) \to A (z))) \\ = ⋀_{z \in X} (e_{X} (y, z) \to A (z))) = N^{y} (A) . \end{matrix}$ Hence ${N^{x} ∣ x \in X}$ is a topological Alexandrov L-neighborhood system. We obtain: $\begin{matrix} lim_{N} (F) (x) = e_{AF (X)} (N^{x}, F) \\ = ⋀_{A \in L^{X}} (⋀_{y \in X} (e_{X} (x, y) \to A (y)) \to F (A)), \\ N_{lim_{N}}^{x} (A) = ⋀_{F \in AF (X)} (lim_{N} (F) (x) \to F (A)) \\ = ⋀_{F \in AF (X)} (e_{AF (X)} (N^{x}, F) \to F (A)) \\ = N^{x} (A) = ⋀_{y \in X} (e_{X} (x, y) \to A (y)) . \end{matrix}$ Hence $lim_{N}$ is a topological principal Alexandrov convergence structure. By Theorem 4.10(4), $\begin{matrix} e_{lim_{N}} (x, y) = (N_{lim_{N}}^{x} (⊤_{y}^{*}))^{*} = e_{X} (x, y) . \end{matrix}$

5 Conclusion

In this paper, as a viewpoint for fuzzy rough sets, we can consider Alexandrov L-(neighborhood) filters, L-fuzzy preorders and Alexandrov L-convergence structures. In section 2, the category of L-fuzzy preordered sets and that of topological Alexandrov L-neighborhood spaces are isomorphic. In section 3, there exists a Galois correspondence between the category of Alexandrov L-convergence spaces and that of Alexandrov L-neighborhood spaces. Moreover, the category of L-fuzzy preordered sets and that of topological principal Alexandrov L-convergence spaces are isomorphic. Moreover, we investigate the their topological properties and give their examples.

In the future, we study Alexandrov L-uniform convergence spaces and Alexandrov L-fuzzy topological spaces.

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