Extensional L -fuzzy Q-convergence structures

Abstract

In this paper, the concept of extensional L-fuzzy Q-convergence structures is proposed. It is shown that there is an adjunction between the category of extensional L-fuzzy topological spaces and the category of extensional L-fuzzy topological spaces. In particular, the category of extensional L-fuzzy topological spaces can be embedded in the category of extensional L-fuzzy Q-convergence spaces as a reflective subcategory. Also, the notion of topological extensional L-fuzzy Q-convergence spaces is introduced and the resulting category is shown to be isomorphic to the category of extensional L-fuzzy topological spaces.

Keywords

reflective subcategory

1 Introduction

In general topology, axiomatic convergence structures not only have nice categorical properties, but also have compatible categorical relations with topological structures [27]. In the fuzzy case, convergence structures are generalized to different kinds of fuzzy convergence structures. In the framework of L-topology, Lowen [16] and Min [21] proposed two kinds of fuzzy convergence structures by means of prefilters. Using stratified L-filters, Jäger [6] introduced stratified L-fuzzy convergence spaces (which are called stratified L-generalized convergence spaces in [7]). There are some works to study the properties of this kind of fuzzy convergence structures [2, 3, 8 –14, 17 –20, 31]. In the situation of L-fuzzifying topology, Xu [30] introduced fuzzifying convergence structures by means of classical filters. Later, Yao [32] introduced L-fuzzifying convergence structures using L-filters of ordinary subsets. Afterwards, Wu and Fang [29] wu introduced L-ordered fuzzifying convergence structures and studied its relations with L-fuzzifying convergence structures. In (L, M)-fuzzy setting, Güloğu et al. [4] introduced the concept of I-fuzzy convergence structures by means of I-filters. In [22, 23], Pang and Fang used L-filters to define L-fuzzy Q-convergence structures. Later, Pang proposed the concepts of (L, M)-fuzzy convergence spaces [24, 25], enriched (L, M)-fuzzy convergence spaces [26] and investigated their relations with (L, M)-fuzzy topological spaces, respectively.

Based on the implication operation on the lattice L, the concept of extensional L-fuzzy topological spaces was introduced and the resulting category was shown to be a coreflective subcategory of the category of L-fuzzy topological spaces [5]. Actually, extensional L-fuzzy topological structure is a special kind of L-fuzzy topological structures, this motivates us to consider a new kind of fuzzy convergence structures, which posses compatible categorical relations with extensional L-fuzzy topologies.

In this paper, we will introduce a new kind of fuzzy convergence structures based on L-fuzzy Q-convergence structures [22] and investigate its relations with extensional L-fuzzy topologies in a categorical sense.

2 Preliminaries

Throughout this paper, L denotes a completely distributive lattice and ^′ is an order-reversing involution on L. The smallest element and the largest element in L are denoted by ⊥_L and ⊤_L, respectively. For a, b ∈ L, we say that a is wedge below b in L, in symbols a ≺ b, if for every subset D ⊆ L, ⋁D ≥ b implies d ≥ a for some d ∈ D. An element a in L is called coprime if a ≤ b ∨ c implies a ≤ b or a ≤ c. The set of nonzero coprime elements in L is denoted by J (L). A complete lattice L is completely distributive iff b = ⋁ {a ∈ J (L) ∣ a ≺ b} for each b ∈ L.

For a nonempty set X, L^X denotes the set of all L-subsets on X. The smallest element and the largest element in L^X are denoted by $⊥_{L}^{X}$ and $⊤_{L}^{X}$ , respectively. The set of nonzero coprime elements in L^X is denoted by $J (L^{X})$ . It is easy to see that $J (L^{X})$ = {x_λ ∣ x ∈ X, λ ∈ J (L)}. We say that a fuzzy point x_λ ∈ $J (L^{X})$ quasi-coincides with A, denoted by $x_{λ} \hat{q} A$ , if λ≰A′ (x). Let φ : X ⟶ Y be a mapping. Define φ^→ : L^X ⟶ L^Y and φ^← : L^Y ⟶ L^X by φ^→ (A) (y) = ⋁ _φ(x)=yA (x) for A ∈ L^X and y ∈ Y, and φ^← (B) = B ∘ φ for B ∈ L^Y, respectively. We define a residual implication operation → : L × L ⟶ L as the right adjoint for the meet operation ∧ by $a \to b = ⋁ {c \in L ∣ a \land c \leq b} .$ This operation plays a particular role in the sequel.

Definition 2.1. [5] Define $𝔼 (-, -) : L^{X} \times L^{X} \to L$ as follows: for each A, B ∈ L^X, $𝔼 (A, B) = ⋀_{x \in X} ((A (x) \to B (x)) \land (B (x) \to A (x))) .$ Then $𝔼 (-, -)$ is called the fuzzy equality of L-subsets.

Definition 2.2. [5] A mapping $F : L^{X} ⟶ L$ is called an L-filter on X if it satisfies: (LF1) $F (⊥_{L}^{X}) = ⊥_{L}, F (⊥_{L}^{X}) = ⊤_{L}$ ;

(LF2) $F (A \land B) = F (A) \land F (B)$ .

The family of all L-filters on X is denoted by $F_{L} (X) .$

Let $F \in F_{L} (X)$ and φ : X ⟶ Y be a mapping. Then $φ^{\Rightarrow} (F) : L^{Y} ⟶ L, A ⟼ F (φ^{\leftarrow} (A))$ is an L-filter on Y and is called the image of $F$ under φ.

Definition 2.3. [15, 28] An L-fuzzy topology on X is a mapping τ : L^X→L which satisfies:

(LFT1) $τ (⊥_{L}^{X}) = τ (⊤_{L}^{X}) = ⊤_{L}$ ;

(LFT2) τ (A ∧ B) ≥ τ (A) ∧ τ (B);

(LFT3) τ (⋁ _j∈JA_j) ≥ ⋀ _j∈Jτ (A_j).

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

A continuous mapping between L-fuzzy topological spaces (X, τ_X) and (Y, τ_Y) is a mapping φ : X→Y such that τ_X (φ^← (A)) ≥ τ_Y (A) for each A ∈ L^Y. The category of L-fuzzy topological spaces with continuous mappings as morphisms will be denoted by L-FTop.

Definition 2.4. [5] An L-fuzzy topology τ on X is called extensional if it satisfies:

(ELFT) ∀A, B ∈ L^X, $τ (A) \land 𝔼 (A, B) \leq τ (B)$ .

The pair (X, τ) is called an extensional L-fuzzy topological space. The full subcategory of L-FTop, consisting of extensional L-fuzzy topological spaces, is denoted by EL-FTop.

Definition 2.5. [1] An L-fuzzy quasi-coincident neighborhood system on X is defined to be a set $Q = {Q x_{λ} ∣ x_{λ} \in J (L^{X})}$ of mappings $Q x_{λ}$ : L^X→L satisfying the following conditions:

(LFQ1) $Q x_{λ} (⊤_{L}^{X}) = ⊤_{L}, Q x_{λ} (⊥_{L}^{X}) = ⊥_{L}$ ;

(LFQ2) $Q x_{λ} (A) \neq ⊥_{L} \Rightarrow x_{λ} \hat{q} A$ ;

(LFQ3) $Q x_{λ}$ (A ∧ B) = $Q x_{λ}$ (A) ∧ $Q x_{λ}$ (B).

(LFQ4) $Q x_{λ} (A) = ⋁_{x_{λ} \hat{q} B \leq A} ⋀_{y_{μ} \hat{q} B} Q_{y_{μ}} (B)$ .

The pair $(X, Q)$ is called an L-fuzzy quasi-coincident neighborhood space.

A continuous mapping between two L-fuzzy quasi-coincident neighborhood spaces $(X, Q^{X})$ and $(Y, Q^{Y})$ is a mapping φ : X→Y such that $Q_{x_{λ}}^{X} (φ^{\leftarrow} (A)) \geq Q_{φ (x)_{λ}}^{Y} (A)$ for each $x_{λ}$ ∈ $J (L^{X})$ and A ∈ L^Y.

Proposition 2.6. [1] (1) Let $Q = {Q x_{λ} ∣ x_{λ} \in J (L^{x})}$ be an L-fuzzy quasi-coincident neighborhood system on X and define $τ^{Q} : L^{X} \to L$ by $τ^{Q} (A) = ⋀_{x_{λ} \hat{q} A} Q x_{λ} (A)$ for each A ∈ L^X. Then $τ^{Q}$ is an L-fuzzy topology on X.

(2) Let τ be an L-fuzzy topology on X and define $Q x_{λ}$ ^τ : L^X→L by ${Q x_{λ}}^{τ} (A) = ⋁_{x_{λ} \hat{q} B \leq A} τ (B)$ for each A ∈ L^X. Then $Q^{τ} = {{Q x_{λ}}^{τ} ∣ x_{λ} \in J (L^{x})}$ is an L-fuzzy quasi-coincident neighborhood system on X. Moreover, $τ (A) = ⋀_{x_{λ} \hat{q} A} {Q x_{λ}}^{τ} (A)$ .

3 Extensional L-fuzzy Q-convergence structures

In this section, we will introduce the concept of extensional L-fuzzy Q-convergence structures and we will study its properties. Now we first recall the notion of L-fuzzy Q-convergence structures.

Definition 3.1. [22] For each x_λ ∈ $J (L^{X})$ , we define $\hat{q} (x_{λ}) : L^{X} ⟶ L$ as follows: $\begin{matrix} \forall A \in L^{X}, \hat{q} (x_{λ}) (A) = {\begin{matrix} ⊤_{L}, & x_{λ} \hat{q} A, \\ ⊥_{L}, & x_{λ} \neg \hat{q} A . \end{matrix} \end{matrix}$ Then $\hat{q} (x_{λ})$ is an L-filter. We call it fuzzy point L-filter.

Definition 3.2. [22] An L-fuzzy Q-convergence structure on X is a mapping qc : $F_{L} (X)$ ⟶ L^X which satisfies:

(LFQC1) $x_{λ} \leq qc (\hat{q} (x_{λ}))$ ;

(LFQC2) $F \leq G$ implies $qc (F) \leq qc (G)$ .

For an L-fuzzy Q-convergence structure qc on X, the pair (X, qc) is called an L-fuzzy Q-convergence space.

A mapping φ : (X, qc_X) ⟶ (Y, qc_Y) between L-fuzzy Q-convergence spaces is called continuous provided that for each $F \in F_{L} (x)$ and $x_{λ}$ ∈ $J (L^{X})$ , $x_{λ} \leq {qc}_{X} (F) implies φ (x)_{λ} \leq {qc}_{Y} (φ^{\Rightarrow} (F)) .$ That is, $\forall F \in F_{L} (X), φ^{\to} ({qc}_{X} (F)) \leq {qc}_{Y} (φ^{\Rightarrow} (F)) .$

It is easy to check that all L-fuzzy Q-convergence spaces and their continuous mappings form a category, denoted by L-FQC.

Definition 3.3. [22] Let (X, qc) be an L-fuzzy Q-convergence space and $x_{λ}$ ∈ $J (L^{X})$ . Define $F_{x_{λ}}^{q c}$ : L^X ⟶ L by $\forall A \in L^{X}, F_{x_{λ}}^{q c} (A) = ⋀_{x_{λ} \leq qc (F)} F (A) .$ Then $F_{x_{λ}}^{q c}$ is an L-filter, which is called the quasi-coincident neighborhood filter at $x_{λ}$ .

Definition 3.4. An L-fuzzy Q-convergence structure qc on X is called extensional if it satisfies:

(ELFQC) $⋀_{x_{λ} \hat{q} A} (F_{x_{λ}}^{qc} (A) \land 𝔼 (A, B)) \leq ⋀_{y_{μ} \hat{q} B} F_{y_{μ}}^{qc} (B)$ .

For an extensional L-fuzzy Q-convergence structure qc on X, the pair (X, qc) is called an extensional L-fuzzy Q-convergence space.

Definition 3.5. A mapping fai : (X, qc_X) → (Y, qc_Y) between extensional L-fuzzy Q-convergence spaces is called continuous provided that for each $F$ ∈ $F_{L} (X)$ and $x_{λ}$ ∈ $J (L^{X})$ , $x_{λ} \leq {qc}_{X} (F) implies φ (x)_{λ} \leq {qc}_{Y} (φ^{\Rightarrow} (F)) .$ That is, $\forall F \in F_{L} (X), φ^{\to} ({qc}_{X} (F)) \leq {qc}_{Y} (φ^{\Rightarrow} (F)) .$

It is easy to see that all extensional L-fuzzy Q-convergence spaces as objects and all continuous mappings as morphisms form a full subcategory of L-FQC, which is denoted by EL-FQC.

Sometimes, we need to compare more than one extensional L-fuzzy Q-convergence structure on a common domain X, so we define the following:

Definition 3.6. For extensional L-fuzzy Q-convergence spaces (X, qc₁) and (X, qc₂), we say that (X, qc₂) is finer than (X, qc₁), or (X, qc₁) is coarser than (X, qc₂), denoted by (X, qc₂) ≤ _qc (X, qc₁), if the identity mapping id_X : (X, qc₂) → (X, qc₁) is continuous, that is, we have ${qc}_{2} (F) \leq {qc}_{1} (F)$ for each $F \in F_{L} (X)$ . We also write qc₂ ≤ _qcqc₁ in this case.

For a nonempty set X, denote all the extensional L-fuzzy Q-convergence structures on X by EQC (X). Then we have the following result.

Proposition 3.7. (EQC (X) , ≤ _qc) is a partially ordered set.

Proof. It is obvious and the proof is omitted. □

Proposition 3.8.Let {qc_i ∣ i = 1, 2, ⋯ , n} be a family of extensional L-fuzzy Q-convergence structures on X, where n is a nonnegative natural number. Define $\lor_{i = 1}^{i = n} {qc}_{i} : F_{L} (X) \to L^{X}$ by $\forall F \in F_{L} (X), (\lor_{i = 1}^{i = n} {qc}_{i}) (F) = \lor_{i = 1}^{i = n} {qc}_{i} (F) .$ Then $\lor_{i = 1}^{i = n} {qc}_{i}$ is an extensional L-fuzzy Q-convergence structure on X.

Proof. It suffices to verify that $\lor_{i = 1}^{i = n} {qc}_{i}$ satisfies (LFQC1), (LFQC2) and (ELFQC). (LFQC1) and (LFQC2) are obvious.

(ELFQC) Take any A ∈ L^X. Then $\begin{matrix} F_{x_{λ}}^{\lor_{i = 1}^{i = n} {qc}_{i}} (A) & = & ⋀_{x_{λ} \leq (\lor_{i = 1}^{i = n} {qc}_{i}) (F)} F (A) \\ = & ⋀_{\exists i \in {1, \dots, n}, x_{λ} \leq {qc}_{i} (F)} F (A) \\ = & \land_{i = 1}^{i = n} ⋀_{x_{λ} \leq {qc}_{i} (F)} F (A) \\ = & \land_{i = 1}^{i = n} F_{x_{λ}}^{{qc}_{i}} (A) . \end{matrix}$ This shows $F_{x_{λ}}^{\lor_{i = 1}^{i = n} {qc}_{i}} = \land_{i = 1}^{i = n} F_{x_{λ}}^{{qc}_{i}}$ . Then for each A, B ∈ L^X, it follows that $\begin{matrix} ⋀_{x_{λ} \hat{q} A} (F_{x_{λ}}^{\lor_{i = 1}^{i = n} {qc}_{i}} (A) \land 𝔼 (A, B)) \\ = ⋀_{x_{λ} \hat{q} A} (\land_{i = 1}^{i = n} F_{x_{λ}}^{{qc}_{i}} (A) \land 𝔼 (A, B)) \\ = ⋀_{x_{λ} \hat{q} A} (\land_{i = 1}^{i = n} (F_{x_{λ}}^{{qc}_{i}} (A) \land 𝔼 (A, B))) \\ = \land_{i = 1}^{i = n} (⋀_{x_{λ} \hat{q} A} (F_{x_{λ}}^{{qc}_{i}} (A) \land 𝔼 (A, B))) \\ \leq \land_{i = 1}^{i = n} ⋀_{y_{μ} \hat{q} B} F_{y_{μ}}^{{qc}_{i}} (B) \\ = ⋀_{y_{μ} \hat{q} B} \land_{i = 1}^{i = n} F_{y_{μ}}^{{qc}_{i}} (B) \\ = ⋀_{y_{μ} \hat{q} B} F_{y_{μ}}^{\lor_{i = 1}^{i = n} {qc}_{i}} (B), \end{matrix}$ as desired. □

Corollary 3.9. (EQC (X) , ≤ _qc)) is a union semilattice.

2 Relations between EL-FQC and EL-FTop

As we all know, there are close relations between convergence spaces and topological spaces from a categorical aspect. In this section, we will generalize the relations to extensional L-fuzzy Q-convergence spaces and extensional L-fuzzy topological spaces.

Proposition 4.1.Let (X, qc) be an extensional L-fuzzy Q-convergence space and define τ^qc : L^X ⟶ L by $\forall A \in L^{X}, τ^{qc} (A) = ⋀_{x_{λ} \hat{q} A} F_{x_{λ}}^{q c} (A) .$ Then τ^qc is an extensional L-fuzzy topology on X.

Proof. It suffices to verify that τ^qc satisfies (LFT1)–(LFT3) and (ELFT). In fact,

(LFT1) By the definition of τ^qc, we have

$τ^{qc} (⊥_{L}^{X}) = ⋀_{x_{λ} \hat{q} ⊥_{L}^{X}} F_{x_{λ}}^{q c} (⊥_{L}^{X}) = ⋀ \emptyset = ⊤_{L}$ ,

$τ^{qc} (⊤_{L}^{X}) = ⋀_{x_{λ} \hat{q} ⊤_{L}^{X}} F_{x_{λ}}^{q c} (⊤_{L}^{X}) = ⋀ ⊤_{L} = ⊤_{L}$ .

(LFT2) Take any A, B ∈ L^X. Then $\begin{matrix} τ^{qc} (A) \land τ^{qc} (B) \\ = ⋀_{x_{λ} \hat{q} A} F_{x_{λ}}^{q c} (A) \land ⋀_{y_{μ} \hat{q} B} F_{y_{μ}}^{qc} (B) \\ = ⋀_{x_{λ} \hat{q} A} ⋀_{y_{μ} \hat{q} B} (F_{x_{λ}}^{q c} (A) \land F_{y_{μ}}^{qc} (B)) \\ \leq ⋀_{x_{λ} \hat{q} (A \land B)} ⋀_{y_{μ} \hat{q} (A \land B)} (F_{x_{λ}}^{q c} (A) \land F_{y_{μ}}^{qc} (B)) \\ \leq ⋀_{x_{λ} \hat{q} (A \land B)} (F_{x_{λ}}^{q c} (A) \land F_{x_{λ}}^{q c} (B)) \\ = ⋀_{x_{λ} \hat{q} (A \land B)} F_{x_{λ}}^{q c} (A \land B) \\ = τ^{qc} (A \land B) . \end{matrix}$

(LFT3) For {A_j ∣ j ∈ J} ⊆ L^X, it follows that $\begin{matrix} τ^{qc} (⋁_{j \in J} A_{j}) & = & ⋀_{x_{λ} \hat{q} ⋁_{j \in J} A_{j}} F_{x_{λ}}^{q c} (⋁_{j \in J} A_{j}) \\ = & ⋀_{j \in J} ⋀_{x_{λ} \hat{q} A_{j}} F_{x_{λ}}^{q c} (⋁_{j \in J} A_{j}) \\ \geq & ⋀_{j \in J} ⋀_{x_{λ} \hat{q} A_{j}} F_{x_{λ}}^{q c} (A_{j}) \\ = & ⋀_{j \in J} τ^{qc} (A_{j}) . \end{matrix}$

(ELFT) Take A, B ∈ L^X. By (ELFQC), we have $\begin{matrix} τ^{qc} (A) \land 𝔼 (A, B) & = & ⋀_{x_{λ} \hat{q} A} (F_{x_{λ}}^{qc} (A) \land 𝔼 (A, B)) \\ \leq & ⋀_{y_{μ} \hat{q} B} F_{y_{μ}}^{qc} (B) \\ = & τ^{qc} (B) . \end{matrix}$ This shows that τ^qc is an extensional L-fuzzy topology on X. □

Proposition 4.2. If φ : (X, qc_X) ⟶ (Y, qc_Y) is continuous, then so is φ : (X, τ^{qc
_X}) ⟶ (Y, τ^{qc
_Y}).

Proof. Since φ : (X, qc_X) ⟶ (Y, qc_Y) is continuous, it follows that for each $F$ ∈ $F_{L} (X)$ , $φ^{\to} ({qc}_{X} (F)) \leq {qc}_{Y} (φ^{\Rightarrow} (F)) .$ Then for each B ∈ L^Y, we have $\begin{matrix} τ^{{qc}_{X}} (φ^{\leftarrow} (B)) & = & ⋀_{x_{λ} \hat{q} φ^{\leftarrow} (B)} F_{x_{λ}}^{{qc}_{X}} (φ^{\leftarrow} (B)) \\ = & ⋀_{x_{λ} \hat{q} φ^{\leftarrow} (B)} φ^{\Rightarrow} (F_{x_{λ}}^{{qc}_{X}}) (B) \\ = & ⋀_{φ (x)_{λ} \hat{q} B} ⋀_{x_{λ} \leq {qc}_{X} (F)} φ^{\Rightarrow} (F) (B) \\ \geq & ⋀_{φ (x)_{λ} \hat{q} B} ⋀_{φ (x)_{λ} \leq {qc}_{Y} (φ^{\Rightarrow} (F))} φ^{\Rightarrow} (F) (B) \\ \geq & ⋀_{y_{μ} \hat{q} B} ⋀_{y_{μ} \leq {qc}_{Y} (G)} G (B) \\ = & ⋀_{y_{μ} \hat{q} B} F_{y_{μ}}^{{qc}_{Y}} (B) \\ = & τ^{{qc}_{Y}} (B) . \end{matrix}$ Thus, φ : (X, τ^{qc
_X}) ⟶ (Y, τ^{qc
_Y}) is continuous. □

By Propositions 4.1 and 4.2, we obtain a functor $𝕋_{ℂ} : EL$ -FQC→EL-FTop by

$𝕋_{ℂ} : (X, qc) ⟼ (X, τ^{qc})$ and φ ⟼ φ.

Lemma 4.3.Let (X, τ) be an extensional L-fuzzy topological space. Then for each $x_{λ}$ ∈ $J (L^{X})$ and A, B ∈ L^X, $⋀_{x_{λ} \hat{q} A} (Q_{x_{λ}}^{τ} (A) \land 𝔼 (A, B)) \leq ⋀_{y_{μ} \hat{q} B} Q_{y_{μ}}^{τ} (B) .$

Proof. By Proposition 2.6, we have $\begin{matrix} ⋀_{x_{λ} \hat{q} A} (Q_{x_{λ}}^{τ} (A) \land 𝔼 (A, B)) \\ = & ⋀_{x_{λ} \hat{q} A} Q_{x_{λ}}^{τ} (A) \land 𝔼 (A, B) \\ = & τ (A) \land 𝔼 (A, B) \\ \leq & τ (B) \\ = & ⋀_{y_{μ} \hat{q} B} Q_{y_{μ}}^{τ} (B) . \end{matrix}$ □

Lemma 4.4.Let (X, τ) be an L-fuzzy topological space and define qc^τ : $F_{L} (X)$ ⟶ L^X by $\forall F \in F_{L} (X), {qc}^{τ} (F) = ⋁_{Q_{y_{μ}}^{τ} \leq F} y_{μ} .$ Then for each $x_{λ}$ ∈ $J (L^{X})$ and $F$ ∈ $F_{L} (X)$ , $x_{λ} \leq {qc}^{τ} (F) if and only if Q_{x_{λ}}^{τ} \leq F .$

Proof. The sufficiency is obvious. It suffices to show the necessity. Suppose that $x_{λ}$ ≤ qc^τ ( $F$ ). Take any A ∈ L^X and α ∈ J (L) such that $α ≺ Q_{x_{λ}}^{τ} (A) = ⋁_{x_{λ} \hat{q} B \leq A} ⋀_{y_{μ} \hat{q} B} Q_{y_{μ}}^{τ} (B) .$ Then there exists B_α ∈ L^X such that $x_{λ} \hat{q} B_{α} \leq A$ and for each $y_{μ} \hat{q} B_{α}$ , $α \leq Q_{y_{μ}}^{τ} (B_{α})$ . Then we have ${qc}^{τ} (F) \geq x_{λ} ≰ B_{α}^{'}$ . By the definition of qc^τ, there exists z_ν ∈ $J (L^{X})$ such that $Q_{z_{ν}}^{τ} \leq F$ and $z_{ν} ≰ B_{α}^{'}$ . Thus, $α \leq Q_{z_{ν}}^{τ} (B_{α}) \leq F (B_{α}) \leq F (A) .$ By the arbitrariness of α, we get $F (A) \geq ⋁ {α \in J (L) ∣ α ≺ Q_{x_{λ}}^{τ} (A)} = Q_{x_{λ}}^{τ} (A),$ as desired. □

Proposition 4.5. Let (X, τ) be an extensional L-fuzzy topological space. Then qc^τ is an extensional L-fuzzy Q-convergence structure on X.

Proof. It is enough to show that qc^τ satisfies (LFQC1), (LFQC2) and (ELFQC).

(LFQC1) Take any $x_{λ}$ ∈ $J (L^{X})$ . Then $Q_{x_{λ}}^{τ} \in F_{L} (X)$ and $Q_{x_{λ}}^{τ} \leq \hat{q} (x_{λ})$ . Further, it follows that ${qc}^{τ} (\hat{q} (x_{λ})) = ⋁_{Q_{y_{μ}}^{τ} \leq \hat{q} (x_{λ})} y_{μ} \geq x_{λ} .$

(LFQC2) Obviously.

(ELFQC) By Lemma 4.4, it follows that $F_{x_{λ}}^{{qc}^{τ}} = ⋀_{x_{λ} \leq {qc}^{τ} (F)} F = ⋀_{Q_{x_{λ}}^{τ} \leq F} F = Q_{x_{λ}}^{τ} .$ Then by Lemma 4.3, we know that $F_{x_{λ}}^{{qc}^{τ}}$ satisfies (ELFQC). □

Proposition 4.6.If φ : (X, τ_X) ⟶ (Y, τ_Y) is continuous, then so is φ : (X, qc^{τ
_X}) ⟶ (Y, qc^{τ
_Y}).

Proof. Since φ : (X, τ_X) ⟶ (Y, τ_Y) is continuous, it follows that $φ : (X, Q^{τ_{X}}) ⟶ (Y, Q^{τ_{Y}})$ is continuous, that is, $\forall x_{λ} \in J (L^{X}), φ^{\Rightarrow} (Q_{x_{λ}}^{τ_{X}}) \geq Q_{φ (x)_{λ}}^{τ_{Y}} .$ Then for each $F$ ∈ $F_{L} (X)$ , by Lemma 4.4, we have $\begin{matrix} x_{λ} \leq {qc}^{τ_{X}} (F) & \Leftrightarrow & Q_{x_{λ}}^{τ_{X}} \leq F \\ \Rightarrow & φ^{\Rightarrow} (Q_{x_{λ}}^{τ_{X}}) \leq φ^{\Rightarrow} (F) \\ \Rightarrow & Q_{φ (x)_{λ}}^{τ_{Y}} \leq φ^{\Rightarrow} (F) \\ \Leftrightarrow & φ (x)_{λ} \leq {qc}^{τ_{Y}} (φ^{\Rightarrow} (F)) \\ \Leftrightarrow & x_{λ} \leq φ^{\leftarrow} ({qc}^{τ_{Y}} (φ^{\Rightarrow} (F))) . \end{matrix}$ By the arbitrariness of $x_{λ}$ , we obtain that qc^{τ
_X} ( $F$ ) ≤ φ^← (qc^{τ
_Y} (φ^⇒ ( $F$ ))). Thus we have φ^→ (qc^{τ
_X} ( $F$ )) ≤ qc^{τ
_Y} (φ^⇒ ( $F$ )). This shows φ : (X, qc^{τ
_X}) ⟶ (Y, qc^{τ
_Y}) is continuous. □

By Propositions 4.5 and 4.6, we obtain a functor $ℂ_{𝕋} : EL$ -FTop→EL-FQC by $ℂ_{𝕋} : (X, τ) ⟼ (X, {qc}^{τ}) and φ ⟼ φ .$

Theorem 4.7. $(𝕋_{ℂ}, ℂ_{𝕋})$ is an adjunction between EL-FQC and EL-FTop. Moreover, $𝕋_{ℂ}$ is a left inverse of $ℂ_{𝕋}$ .

Proof. Suppose that $𝕀_{EL - FQC}$ is the identity functor on EL-FQC and $𝕀_{EL - FTop}$ is the identity functor on EL-FTop. It suffices to verify that $ℂ_{𝕋} \circ 𝕋_{ℂ} \geq 𝕀_{EL - FQC}$ and $𝕋_{ℂ} \circ ℂ_{𝕋} = 𝕀_{EL - FTop}$ . That is, for an extensional L-fuzzy Q-convergence structure qc on X and an extensional L-fuzzy topology τ on X, we have

(1) qc^{τ
^qc} ≥ _qcqc;

(2) τ^{qc
^τ} = τ.

For (1), take any $x_{λ}$ ∈ $J (L^{X})$ and A ∈ L^X. It follows that $Q_{x_{λ}}^{τ^{qc}} (A) = ⋁_{x_{λ} \hat{q} B \leq A} ⋀_{y_{μ} \hat{q} B} F_{y_{μ}}^{qc} (B) \leq F_{x_{λ}}^{qc} (A) .$ Then $Q_{x_{λ}}^{τ^{qc}} \leq F_{x_{λ}}^{qc}$ . Further, take any $F$ ∈ $F_{L} (X)$ . Then $\begin{matrix} x_{λ} \leq qc (F) & \Rightarrow & F_{x_{λ}}^{qc} \leq F \\ \Rightarrow & Q_{x_{λ}}^{τ^{qc}} \leq F \\ \Leftrightarrow & x_{λ} \leq {qc}^{τ^{qc}} (F) . (by Lemma 44) \end{matrix}$ By the arbitrariness of $x_{λ}$ , we obtain qc ≤ _qcqc^{τ
^qc}.

For (2), by Propositions 2.6 and 4.1, it suffices to show that $Q_{x_{λ}}^{τ} = F_{x_{λ}}^{{qc}^{τ}}$ for each $x_{λ}$ ∈ $J (L^{X})$ . By Lemma 4.4, for each A ∈ L^X, we have $F_{x_{λ}}^{{qc}^{τ}} (A) = ⋀_{x_{λ} \leq {qc}^{τ} (F)} F (A) = ⋀_{Q_{x_{λ}}^{τ} \leq F} F (A) = Q_{x_{λ}}^{τ} (A),$ as desired. □

Corollary 4.8. The category EL-FTop can be embedded in the category EL-FQC as a reflective subcategory.

In general topology, we all know that the category of topological convergence spaces, which is a special kind of convergence spaces, is isomorphic to the category of topological spaces. In the framework of extensional L-fuzzy topology, we have the similar conclusion.

Definition 4.9. An extensional L-fuzzy Q-convergence structure qc on X is called topological if it satisfies

(LFQC3) $x_{λ} \leq qc (F_{x_{λ}}^{qc})$ ;

(LFQC4) $F_{x_{λ}}^{qc} (A) = ⋁_{x_{λ} \hat{q} B \leq A} ⋀_{y_{μ} \hat{q} B} F_{y_{μ}}^{qc} (B)$ .

For a topological extensional L-fuzzy Q-convergence structure qc on X, the pair (X, qc) is called a topological extensional L-fuzzy Q-convergence space.

The full subcategory of EL-FQC, consisting of topological extensional L-fuzzy Q-convergence spaces, is denoted by EL-FTQC.

By Proposition 4.5, we know that an extensional L-fuzzy topology can induce an extensional L-fuzzy Q-convergence structure. The following result demonstrates that an extensional L-fuzzy Q-convergence structure, which is induced by an L-fuzzy topology, is topological.

Proposition 4.10.Let (X, τ) be an extensional L-fuzzy topological space. Then qc^τ is a topological extensional L-fuzzy Q-convergence structure on X.

Proof. By Proposition 4.5, it suffices to show that qc^τ satisfies (LFQC3) and (LFQC4). Since we have proved that $F_{x_{λ}}^{{qc}^{τ}} = Q_{x_{λ}}^{τ}$ in Theorem 4.7, it follows that

(LFQC3) ${qc}^{τ} (F_{x_{λ}}^{{qc}^{τ}}) = ⋁_{Q_{y_{μ}}^{τ} \leq F_{x_{λ}}^{{qc}^{τ}}} y_{μ} \geq x_{λ}$ .

(LFQC4) By Proposition 2.6, we know that $Q_{x_{λ}}^{τ}$ satisfies (LFQ4), i.e., $Q_{x_{λ}}^{τ}$ satisfies (LFQC4). This means $F_{x_{λ}}^{{qc}^{τ}}$ satisfies (LFQC4). □

By Propositions 4.6 and 4.10, we know that $ℂ_{𝕋} : EL$ -FTop→EL-FTQC is a functor. Obviously, $𝕋_{ℂ} : EL$ -FTQC→EL-FTop is also a functor. Further, they are both isomorphic, which is shown in the next theorem.

Theorem 4.11. EL-FTQC is isomorphic to EL-FTop.

Proof. By Theorem 4.7, it suffices to show that for each topological extensional L-fuzzy Q-convergence space (X, qc), qc^{τ
^qc} = qc.

Take any $x_{λ}$ ∈ $J (L^{X})$ and A ∈ L^X. By (LFQC4), $\begin{matrix} Q_{x_{λ}}^{τ^{qc}} (A) & = & ⋁_{x_{λ} \hat{q} B \leq A} τ^{qc} (B) \\ = & ⋁_{x_{λ} \hat{q} B \leq A} ⋀_{y_{μ} \hat{q} B} F_{y_{μ}}^{qc} (B) \\ = & F_{x_{λ}}^{q c} (A) . \end{matrix}$ Then for each $F$ ∈ $F_{L} (X)$ , by (LFQC3), $\begin{matrix} x_{λ} \leq qc (F) & \Leftrightarrow & F_{x_{λ}}^{q c} \leq F \\ \Leftrightarrow & Q_{x_{λ}}^{τ^{qc}} \leq F \\ \Leftrightarrow & x_{λ} \leq {qc}^{τ^{qc}} (F) . (by Lemma 4.4) \end{matrix}$ By the arbitrariness of $x_{λ}$ , we get qc = qc^{τ
^qc}, as desired. □

3 Conclusion

In [5], the researchers introduced extensional L-fuzzy topological spaces and showed that it was a new concept, which is different from L-fuzzy topological spaces. That is, the axiom (ELFT) is independent from (LFT1)–(LFT3). In this paper, we proved that the category of extensional L-fuzzy topological spaces (fulfilling (LFT1)–(LFT3) and (ELFT)) is isomorphic to the category of topological extensional L-fuzzy Q-convergence spaces (fulfilling (LFQC1)–(LFQC4) and (ELFQC)). Also, in [22], the researchers showed that the category of L-fuzzy topological spaces (fulfilling (LFT1)–(LFT3)) is isomorphic to the category of topological L-fuzzy Q-convergence spaces (fulfilling (LFQC1)–(LFQC4)). This means that (ELFQC) is independent from (LFQC1)–(LFQC4). Certainly, (ELFQC) is independent from (LFQC1) and (LFQC2). From this aspect, we know that extensional L-fuzzy Q-convergence space is an independent concept, which is different from L-fuzzy Q-convergence space.

Besides, there are close relations between extensional L-fuzzy topologies and L-fuzzy topologies from a categorical viewpoint. Inspired by this, we will consider the following problems in the future.

(1) The categorical relations between extensional L-fuzzy Q-convergence spaces and L-fuzzyQ-convergence spaces. (2) Categorical properties of extensional L-fuzzyQ-convergence spaces, such as Cartesian-closedness.

Footnotes

Acknowledgments

The authors would like to express their sincere thanks to the anonymous reviewers and the area editor for their careful reading and constructive comments. This work is supported by the Science and Technology Project of Yulin City (No. 2014CXY-08), the Science and Technology Program of Department of Education in Shaanxi Province (No. 12JK0890), China Postdoctoral Science Foundation (No. 2015M581434) and the National Natural Science Foundation of China (No. 11501435).

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