L -fuzzy N-convergence structures

Abstract

Based on a completely distributive lattice L, the concept of L-fuzzy N-convergence structures is introduced. It is shown that the category of L-fuzzy topological spaces can be embedded in the category of L-fuzzy N-convergence spaces. It is also proved the category of (topological) pretopological L-fuzzy N-convergence spaces is isomorphic to the category of (topological) L-fuzzy neighborhood spaces and the former is a bireflective subcategory of the category of L-fuzzy N-convergence spaces.

Keywords

category theory

1 Introduction

Since Fischer [5] first introduced convergence structures based on filters in classical sense, the theory developed rapidly [13, 14]. With the development of fuzzy set theory, the classical filter has also been generalized to different kinds of fuzzy filters [2 , 9]. Based on L-filters, many researchers extended convergence structures to fuzzy setting with different definitions and studied their properties [3 , 27–30].

The relations between latticed-valued topological spaces and latticed-valued convergence spaces are investigated extensively. In the framework of L-fuzzy topology, Güloğu et al. [8] introduced the concept of I-fuzzy convergence structures by means of I-filters (I is the real unit interval) and showed that there is one-to-one correspondence between topological I-fuzzy convergence structures and I-fuzzy topologies. Later, Pang and Fang [20] used L-filters to define L-fuzzy Q-convergence structures and proved that the category of topological L-fuzzy Q-convergence spaces is isomorphic to that of L-fuzzy topological spaces. In both of the above convergence spaces, the objects of converging are fuzzy points, which is compatible with the L-fuzzy setting. However, the convergence of an L-filter is not fuzzy, i.e., an L-filter $F$ either converges to a fuzzy point or not. Hence, we point out neither of the above convergence structures may be the best one in L-fuzzy setting.

The aim of this paper is to propose the concept of an L-fuzzy N-convergence structure which assigns to every L-filter a certain degree of converging, other than converging or not. Moreover, the relations betweenL-fuzzy N-convergence spaces and L-fuzzy topological spaces are investigated.

2 Preliminaries

Let (L, ⋁ , ⋀) be a completely distributive lattice. M (L) denotes the set of all non-zero coprimes in L. The smallest element and the largest element in L are denoted by ⊥ and ⊤, respectively. Let a, b be elements in L. We say “a is wedge below b” in symbols a ≺ b if for every subset D ⊆ L, ⋁ D ≥ b implies a ≤ d for some d ∈ D . We denote β (a) = {b | b ≺ a}. Thus a = ⋁ β (a) holds for each a ∈ 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 $\underline{⊥}$ and $\underline{⊤}$ , respectively. L^X is also a completely distributive lattice when it inherits the structure of the lattice L in a natural way, by defining ⋁, ⋀ and ≤ pointwisely. The set of non-zero coprimes in L^X is denoted by $L^{J (L^{X})}$ . It is easy to see that $L^{J (L^{X})}$ is exactly the set of all fuzzy points x_λ (λ ∈ M (L)). 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 operator plays a particular role in the sequel. We list some of its properties.

Lemma 2.1. (Höhle and Šostak [9]) Suppose that (L, ⋁ , ⋀) is a completely distributive lattice and → is the implication operation corresponding to ∧. Then for all a, b, c, d ∈ L, {a_j} _j∈J, {b_j} _j∈J ⊆L, the following conditions hold:

⊤ → a = a.

a ≤ bif and only ifa→ b = ⊤.

(a → b) → b ≥ a.

(a → c) → (b → c) ≥ b → a.

a → ⋀ _j∈Ja_j = ⋀ _j∈J (a → a_j), hence a → b ≤ a → cwheneverb ≤ c.

⋁_j∈Ja_j → b = ⋀ _j∈J (a_j → b) , hence a → c ≥ b → cwhenevera ≤ b.

Lemma 2.2. (Fang [3]) Let $S (-, -)$ be the fuzzy inclusion order of L-subsets, i.e., for any C, D ∈ L^X, $S (C, D) = ⋀_{x \in X} (C (x) \to D (x)) .$ Then for all A, B ∈ L^X, {A_i} _i∈I ⊆ L^X, the following statements hold:

$A \leq B \Leftrightarrow S (A, B) = ⊤$ .

$S (⋁_{i \in I} A_{i}, A) = ⋀_{i \in I} S (A_{i}, A)$ .

$S (A, ⋀_{i \in I} A_{i}) = ⋀_{i \in I} S (A, A_{i})$ .

Definition 2.3. (Höhle and Šostak [9]) A mapping $F : L^{X} \to L$ is called an L-filter on X if it satisfies:

$F (\underline{⊥}) = ⊥, F (\underline{⊤}) = ⊤$ ;

$A \leq B \Rightarrow F (A) \leq F (B)$ ;

$F (A \land B) \geq F (A) \land F (B)$ .

The family of all L-filters on X is denoted by

F_{L} (X) .

Example 2.4. For each x_λ ∈ $L^{J (L^{X})}$ , we define $[x_{λ}]$ as follows: $\forall A \in L^{X}, [x_{λ}] (A) = {\begin{matrix} ⊤, & x_{λ} \leq A, \\ ⊥, & x_{λ} ≨ A . \end{matrix}$ Then $[x_{λ}]$ is an L-filter, called a fuzzy point L-filter.

Let f : X → Y be an ordinary mapping. Define f^→ : L^X → L^Y and f^← : L^Y → L^X by f^→ (A) (y) = ⋁ _f(x)=yA (x) for A ∈ L^X and y ∈ Y, f^← (B) = B ∘ f for B ∈ L^Y, respectively.

Definition 2.5. (Höhle and Šostak [9]) Let $F \in F_{L} (X)$ and f : X → Y be a mapping. Then $f^{\Rightarrow} (F) : L^{Y} \to L, A \to F (f^{\leftarrow} (A))$ is also an L-filter and is called the image of $F$ under f.

In fact, every L-filter is an L-subsets, so we can define fuzzy inclusion order in $F_{L} (X)$ in the following way: $\forall F, G \in F_{L} (X),$ $S_{F} (F, G) = ⋀_{A \in L^{X}} (F (A) \to G (A)) .$ Then the following lemma holds.

Lemma 2.6.Let f : X → Y be an ordinary mapping. Then for all $F, G \in F_{L} (X),$ $S_{F} (F, G) \leq S_{F} (f^{\Rightarrow} (F), f^{\Rightarrow} (G)) .$

Definition 2.7. (Kubiak [12] and Šostak [16]) An L-fuzzy topology on X is a mapping τ : L^X → L which satisfies:

$τ (\underline{⊥}) = τ (\underline{⊤}) = ⊤$ ;

τ (A ∧ B) ≥ τ (A) ∧ τ (B);

τ (⋁ _j∈JA_j) ≥ ⋀ _j∈Jτ (A_j).

For an L-fuzzy topology τ on X, the pair (X, τ) is called an L-fuzzy topological space. Let (X, τ) and (Y, δ) be two L-fuzzy topological spaces and f : X → Y an ordinary mapping. If for any A ∈ L^Y, τ (f^← (A)) ≥ δ (A) , we say that f : (X, τ) → (Y, δ) is continuous. The category of L-fuzzy topological spaces with continuous mappings as morphisms will be denoted by L-FTop.

Definition 2.8. (Shi [25]) An L-fuzzy neighborhood system on X is defined to be a set $N = {N^{x λ} ∣ x_{λ} \in J (L^{X})}$ of mappings $N^{x λ}$ : L^X → L satisfying the following conditions:

$N^{x λ} (\underline{⊤}) = ⊤, N^{x λ} (\underline{⊥}) = ⊥$ ;

$N^{x λ}$ (A) =⊥ for any x_λnotleqslantA;

$N^{x λ}$ (A ∧ B) = $N^{x λ}$ (A) ∧ $N^{x λ}$ (B);

$N^{x λ} (A) = ⋁_{x_{λ} \leq B \leq A} ⋀_{y_{μ} ≺ B} N^{y_{μ}} (B)$ .

The pair $(X, N)$ is called an L-fuzzy neighborhood space. A continuous mapping between two L-fuzzy neighborhood spaces $(X, N_{X})$ and $(Y, N_{Y})$ is a mapping f : X → Y such that $\forall \in x_{λ}, \forall A \in L^{Y}, {N_{X}^{x λ}}_{X} (f^{\leftarrow} (A)) \geq N_{Y}^{f (x)_{λ}} (A)$ .

In this paper, we call $N = {N^{x λ} ∣ x_{λ} \in J (L^{X})}$ satisfying (LFN1)–(LFN3) an L-fuzzy neighborhood system and $N = {N^{x λ} ∣ x_{λ} \in J (L^{X})}$ satisfying (LFN1)–(LFN4) a topological L-fuzzy neighborhood system. The category of L-fuzzy neighborhood spaces with continuous mappings as morphisms is denoted by L-FPN, and L-FTN denote the full subcategory of L-FPN consisting of topological L-fuzzy neighborhood spaces.

In [25], it is shown that $τ^{N} (A) = ⋀_{x λ ≺ A} N^{x λ} (A)$ is an L-fuzzy topology where $N = {N^{x λ} ∣ x_{λ} \in J (L^{X})}$ is an L-fuzzy neighborhood system on X, and $N_{τ} = {{N^{x λ}}_{τ} ∣ x_{λ} \in J (L^{X})}$ is an L-fuzzy neighborhood system if τ is an L-fuzzy topology, where $N^{x λ}$ τ : L^X → L is defined by $N^{x λ}$ τ (A) = ⋁ _{$x_{λ}$ ≤B≤A}τ (B). Moreover, τ (A) = ⋀ _{$x_{λ}$ ≺A} $N^{x λ}$ τ (A).

Theorem 2.9. (Shi [25]) L-FTop is isomorphic to L-FTN.

The objects of a category A is denoted by ∣A∣. For notions related to category theory we refer to [1].

3 L-fuzzy N-convergence structures

In this section, the concept of an L-fuzzy convergence structure is introduced and some properties are discussed.

Definition 3.1. An L-fuzzy N-convergence structure on X is a mapping lim : $F_{L} (X)$ → L $L^{J (L^{X})}$ which satisfies:

lim( $[x_{λ}]$ ) ( $x_{λ}$ ) =⊤;

$F \leq G \Rightarrow lim (F) \leq lim (G)$ .

The pair (X, lim) is called an L-fuzzy N-convergence space.

A mapping $f : (X, lim_{X}) \to (Y, lim_{Y})$ between L-fuzzy N-convergence spaces is called continuous provided that for all $F \in F_{L} (X)$ , $x_{λ}$ ∈ $L^{J (L^{X})}$ , $lim_{X} (F) (x λ) \leq lim_{Y} (f^{\Rightarrow} (F)) (f (x)_{λ})$ . It is easy to check that L-fuzzy convergence spaces and their continuous mappings form a category, denoted by L-FNC.

Definition 3.2. Let $lim_{lfnc} (X)$ denote the fiber ${(X, lim) : (X, lim) \in ∣ L - FNC ∣}$ of X. For all pairs $(X, lim_{1})$ , $(X, lim_{2}) \in lim_{lfnc} (X)$ , we say that $(X, lim_{2})$ is finer than $(X, lim_{1})$ , or $(X, lim_{1})$ is coarser than $(X, lim_{2})$ , if the identity mapping ${id}_{X} : (X, lim_{2}) \to (X, lim_{1})$ is continuous, that is, we have $lim_{2} F (x λ) \leq lim_{1} F (x λ)$ for all $x_{λ}$ ∈ $L^{J (L^{X})}$ and $F \in F_{L} (X)$ . We also write $lim_{2} \leq lim_{1}$ , or $(X, lim_{2}) \leq (X, lim_{1})$ in this case.

Example 3.3. Let X be a nonempty set.

We define $lim_{l} F (x λ) = ⊤$ for all $x_{λ}$ ∈ $L^{J (L^{X})}$ and $F \in F_{L} (X)$ . Then $lim_{l}$ is the coarsest L-fuzzy N-convergence structure on X, called the indiscrete N-convergence structure on X.

We define the discrete L-fuzzy N-convergence structure on X by $lim_{s} F (x λ) = ⊤$ if $F \geq [x_{λ}],$ and =⊥ otherwise. It is the finest L-fuzzy N-convergence structure on X.

Theorem 3.4. $(lim_{lfnc} (X), \leq)$ is a complete lattice.

Proof. Firstly, $lim_{l}$ defined in the above Example is the maximal element.

Secondly, let ${lim_{j}}_{j \in J} \subseteq lim_{lfnc} (X) .$ Define ${inf}_{j \in J} {lim}_{j} : F_{L} (X) \to L^{J (L^{X})}$ such that $\forall F \in F_{L} (X), x_{λ} \in L^{J (L^{X})},$ $({inf}_{j \in J} {lim}_{j}) (F) (x_{λ}) = ⋀_{j \in J} {lim}_{j} (F) (x_{λ}) .$ Then inf _j∈J lim _j is an L-fuzzy N-convergence structure on X.

Finally, it’s trivial to check that inf _j∈J lim _j is the maximal lower bound. □

Theorem 3.5. The category L-FNC of all L-fuzzy N-convergence spaces is topological over Set.

Proof. We only prove the existence of initial structure. Let ${(X_{j}, lim_{j})}_{j \in J}$ be a family of L-fuzzy N-convergence spaces and X be a nonempty set. If ${f_{j} : X \to (X_{j}, lim_{j})}_{j \in J}$ is a source, then $lim^{X} : F_{L} (X) \to L^{J (L^{X})}$ defined by ${lim}^{X} (F) (x_{λ}) = ⋀_{j \in J} {lim}_{j} (f_{j}^{\Rightarrow} (F)) (f_{j} (x)_{λ})$ is an L-fuzzy N-convergence structure on X. (LFNC2) is obvious. In order to verify (LFNC1), it suffices to remark that for a mapping f : X → Y and the fuzzy point filter $[x_{λ}]$ , we have for A ∈ L^Y, f^⇒ ( $[x_{λ}]$ ) (A) = $[x_{λ}]$ (f^← (A)) = [f (x) _λ] (A) , i.e., f^⇒ ( $[x_{λ}]$ ) = [f (x) _λ] .

Let further $(Y, lim^{Y}) \in ∣ L$ -FNC∣ and f : Y → X be a mapping. Assume that f_j ∘ f is continuous for every j ∈ J . Then for each $G \in F_{L} (Y),$ we have $\begin{matrix} {lim}^{Y} (G) (y_{μ}) & \leq & ⋀_{j \in J} {lim}_{j} ((f_{j} \circ f)^{\Rightarrow} (G)) (f_{j} \circ f (y)_{μ}) \\ = & ⋀_{j \in J} {lim}_{j} (f_{j}^{\Rightarrow} (f^{\Rightarrow} (G))) (f_{j} (f (y))_{μ}) \\ = & {lim}^{X} (f^{\Rightarrow} (G)) (f (y)_{μ}) . \end{matrix}$ This shows $f : (Y, lim^{Y}) \to (X, lim^{X})$ is continuous. Therefore f is continuous if and only if f_j ∘ f is continuous for each j ∈ J, as desired. □

4 The relations between L-FNC and L-FTop

In this section, the relations between L-fuzzy N-convergence structures and L-fuzzy topologies are discussed in a categorical sense.

For an L-fuzzy N-convergence space (X, lim), define _{$N^{x λ}$ lim} : L^X → L by ∀A ∈ L^X, ${N^{x λ}}_{lim} (A) = ⋀_{F \in F_{L} (X)} (lim (F) (x_{λ}) \to F (A)) .$ Then _{$N^{x λ}$ lim} is an L-filter on X satisfying _{$N^{x λ}$ lim} ≤ [ $x_{λ}$ ]. We call _{$N^{x λ}$ lim} the neighborhood L-filter of $x_{λ}$ .

Theorem 4.1.Let (X, lim) be an L-fuzzy N-convergence space. Define τ_lim : L^X → L as follows: $\forall A \in L^{X}, τ_{lim} (A) = ⋀_{x_{λ} ≺ A} {N^{x λ}}_{lim} (A) .$ Then τ_lim is an L-fuzzy topology on X .

Proof. Check that τ_lim is an L-fuzzy topology on X as follows:

(LFT1) is trivial.

(LFT2) can be proved by the following fact: ∀A, B ∈ L^X, $\begin{matrix} τ_{lim} (A \land B) & = & ⋀_{x_{λ} ≺ A \land B} {N^{x λ}}_{lim} (A \land B) \\ = & ⋀_{x_{λ} ≺ A \land B} ({N^{x λ}}_{lim} (A) \land {N^{x λ}}_{lim} (B)) \\ \geq & ⋀_{x_{λ} ≺ A} {N^{x λ}}_{lim} (A) \land ⋀_{y_{μ} ≺ B} N_{lim}^{y_{μ}} (B) \\ = & τ_{lim} (A) \land τ_{lim} (B) . \end{matrix}$

(LFT3) ∀ {A_j ∣ j ∈ J} ⊆ L^X, $\begin{matrix} τ_{lim} (⋁_{j \in J} A_{j}) & = & ⋀_{x_{λ} ≺ ⋁_{j \in J} A_{j}} {N^{x λ}}_{lim} (⋁_{j \in J} A_{j}) \\ = & ⋀_{j \in J} ⋀_{x_{λ} ≺ A_{j}} {N^{x λ}}_{lim} (⋁_{j \in I} A_{j}) \\ \geq & ⋀_{j \in J} ⋀_{x_{λ} ≺ A_{j}} {N^{x λ}}_{lim} (A_{j}) \\ = & ⋀_{j \in J} τ_{lim} (A_{j}) . \end{matrix}$ This completes the proof. □

Theorem 4.2. If $f : (X, lim_{X}) \to (Y, lim_{Y})$ is continuous, then so is $f : (X, τ_{lim_{X}}) \to (Y, τ_{lim_{Y}})$ .

Proof. Since $f : (X, lim_{X}) \to (Y, lim_{Y})$ is continuous, it follows that $\forall F \in F_{L} (X), \forall x_{λ} \in L^{J (L^{X})},$ ${lim}_{X} (F) (x_{λ}) \leq {lim}_{Y} (f^{\Rightarrow} (F)) (f (x)_{λ}) .$ This implies that ∀B ∈ L^Y, $\begin{matrix} τ_{lim_{X}} (f^{\leftarrow} (B)) \\ = ⋀_{x_{λ} ≺ f^{\leftarrow} (B)} {N^{x λ}}_{lim_{X}} (f^{\leftarrow} (B)) \\ = ⋀_{x_{λ} ≺ f^{\leftarrow} (B)} ⋀_{F \in F_{L} (X)} ({lim}_{X} (F) (x_{λ}) \to F (f^{\leftarrow} (B))) \\ \geq ⋀_{f (x)_{λ} ≺ B} ⋀_{F \in F_{L} (X)} ({lim}_{Y} (f^{\Rightarrow} (F)) (f (x)_{λ}) \\ \to f^{\Rightarrow} (F) (B)) \\ \geq ⋀_{f (x)_{λ} ≺ B} ⋀_{G \in F_{L} (Y)} ({lim}_{Y} (G) (f (x)_{λ}) \to G (B)) \\ = ⋀_{f (x)_{λ} ≺ B} N_{lim_{Y}}^{f (x)_{λ}} (B) \\ \geq ⋀_{y_{μ} ≺ B} N_{lim_{Y}}^{y_{μ}} (B) = τ_{lim_{Y}} (B) . \end{matrix}$ Therefore $f : (X, τ_{lim_{X}}) \to (Y, τ_{lim_{Y}})$ is continuous. □

By Theorems 4.1 and 4.2, we obtain a concrete functor $𝕋_{𝔽} : L$ -FNC→L-FTop by

$𝕋_{𝔽} : (X, lim) \mapsto (X, τ_{lim})$ and f ↦ f.

Theorem 4.3.Let (X, τ) be an L-fuzzy topological space and _{$N^{x λ}$
τ} be the L-fuzzy neighborhood system with respect to τ. Then $lim_{τ}$ defined by $\forall F \in F_{L} (X), \forall x_{λ} \in L^{J (L^{X})},$ ${lim}_{τ} (F) (x_{λ}) = S_{F} ({N^{x λ}}_{τ}, F)$ is an L-fuzzy N-convergence structure on X.

Proof. By (LFN2), it follows that ${lim}_{τ} (F) (x_{λ}) = ⋀_{x_{λ} \leq A} ({N^{x λ}}_{τ} (A) \to F (A)) .$ Then we have

(LFNC1) Since _{$N^{x λ}$
τ} ≤ $[x_{λ}]$ , it follows that ${lim}_{τ} ([x_{λ}]) (x_{λ}) = ⋀_{x_{λ} \leq A} ({N^{x λ}}_{τ} (A) \to [x_{λ}] (A)) = ⊤ .$

(LFNC2) Straightforward.

Thus $lim_{τ}$ is an L-fuzzy convergence structure. □

Lemma 4.4. Let (X, τ) be an L-fuzzy topological space. Then $lim_{τ} (F) (x_{λ}) = ⋀_{x_{λ} \leq A} (τ (A) \to F (A)) .$

Proof. On one hand, $\begin{matrix} {lim}_{τ} (F) (x_{λ}) & = & ⋀_{x_{λ} \leq A} ({N^{x λ}}_{τ} (A) \to F (A)) \\ = & ⋀_{x_{λ} \leq A} (⋁_{x_{λ} \leq B \leq A} τ (B) \to F (A)) \\ = & ⋀_{x_{λ} \leq A} ⋀_{x_{λ} \leq B \leq A} (τ (B) \to F (A)) \\ \leq & ⋀_{x_{λ} \leq A} (τ (A) \to F (A)) . \end{matrix}$ On the other hand, $\begin{matrix} {lim}_{τ} (F) (x_{λ}) & = & ⋀_{x_{λ} \leq A} ⋀_{x_{λ} \leq B \leq A} (τ (B) \to F (A)) \\ \geq & ⋀_{x_{λ} \leq A} ⋀_{x_{λ} \leq B \leq A} (τ (B) \to F (B)) \\ \geq & ⋀_{x_{λ} \leq A} ⋀_{x_{λ} \leq C} (τ (C) \to F (C)) \\ = & ⋀_{x_{λ} \leq C} (τ (C) \to F (C)) . \end{matrix}$ Therefore $lim_{τ} (F) (x_{λ}) = ⋀_{x_{λ} \leq A} (τ (A) \to F (A)) .$ □

Theorem 4.5. If f : (X, τ_X) → (Y, τ_Y) is continuous, then so is $f : (X, lim_{τ_{X}}) \to (Y, lim_{τ_{Y}})$ .

Proof. Since f : (X, τ_X) → (Y, τ_Y) is continuous, it follows that $\forall B \in L^{Y}, τ_{X} (f^{\leftarrow} (B)) \geq τ_{Y} (B) .$ This implies that $\forall F \in F_{L} (X), \forall x_{λ} \in L^{J (L^{X})}$ , $\begin{matrix} {lim}_{τ_{Y}} (f^{\Rightarrow} (F)) (f (x)_{λ}) \\ = ⋀_{f (x)_{λ} \leq B} (τ_{Y} (B) \to f^{\Rightarrow} (F) (B)) \\ \geq ⋀_{x_{λ} \leq f^{\leftarrow} (B)} (τ_{X} (f^{\leftarrow} (B)) \to F (f^{\leftarrow} (B))) \\ \geq ⋀_{x_{λ} \leq A} (τ_{X} (A) \to F (A)) \\ = {lim}_{τ_{X}} (F) (x_{λ}) . \end{matrix}$ Therefore $f : (X, lim_{τ_{X}}) \to (Y, lim_{τ_{Y}})$ is continuous. □

By Theorems 4.3 and 4.5, we obtain a concrete functor $𝔽_{𝕋} : L$ -FTop→L-FNC by

$𝔽_{𝕋} : (X, τ) \mapsto (X, lim_{τ})$ and f ↦ f.

Theorem 4.6. If lim is an L-fuzzy N-convergence structure on X and τ is an L-fuzzy topology on X, then $τ_{lim_{τ}} = τ$ .

Proof. We first prove ${N^{x λ}}_{lim_{τ}} = {N^{x λ}}_{τ} .$ ∀A ∈ L^X, we have $\begin{matrix} {N^{x λ}}_{lim_{τ}} (A) \\ = ⋀_{F \in F_{L} (X)} ({lim}_{τ} (F) (x_{λ}) \to F (A)) \\ = ⋀_{F \in F_{L} (X)} ((⋀_{B \in L^{X}} ({N^{x λ}}_{τ} (B) \to F (B))) \\ \to F (A)) \\ \geq ⋀_{F \in F_{L} (X)} (({N^{x λ}}_{τ} (A) \to F (A)) \to F (A)) \\ \geq {N^{x λ}}_{τ} (A), \end{matrix}$ and $\begin{matrix} {N^{x λ}}_{lim_{τ}} (A) & = & ⋀_{F \in F_{L} (X)} ({lim}_{τ} (F) (x_{λ}) \to F (A)) \\ \leq & {lim}_{τ} ({N^{x λ}}_{τ}) (x_{λ}) \to {N^{x λ}}_{τ} (A) \\ = & S_{F} ({N^{x λ}}_{τ}, {N^{x λ}}_{τ}) \to {N^{x λ}}_{τ} (A) \\ = & {N^{x λ}}_{τ} (A) . \end{matrix}$ Then it follows that $τ_{lim_{τ}} (A) = ⋀_{x_{λ} ≺ A} {N^{x λ}}_{lim_{τ}} (A) = ⋀_{x_{λ} ≺ A} {N^{x λ}}_{τ} (A) = τ (A) . □$

By Theorems 4.3–4.5, we obtain

Theorem 4.7. The category L-FTop can be embedded in L-FNC as a subcategory. $𝔽_{𝕋}$ is the embedding functor.

5 Pretopological and topological L-fuzzy N-convergence structures

In this section, the concepts of pretopological and topological L-fuzzy N-convergence structures are proposed. The results show that the category of (topological) pretopological L-fuzzy N-convergence spaces is isomorphic to the category of (topological) L-fuzzy neighborhood spaces. It is also proved the category of pretopological L-fuzzy N-convergence spaces is a bireflective subcategory of the category of L-fuzzy N-convergence spaces.

Definition 5.1. A pretopological L-fuzzy N-convergence structure on X is a mapping satisfying (LFNC1), (LFNC2) and

$(LFPNC) lim (F) (x_{λ}) = S_{F} ({N^{x λ}}_{lim}, F) .$

The pair (X, lim) is called a pretopological L-fuzzy N-convergence space, and it will be called topological if it satisfies moreover:

$(LFTNC) {N^{x λ}}_{lim} (A) = ⋁_{x_{λ} \leq B \leq A} ⋀_{y_{μ} ≺ B} N_{lim}^{y_{μ}} (B) .$

Let L-FPNC denote the full subcategory of L-FNC formed by all pretopological L-fuzzy N-convergence spaces, and L-FTNC the full subcategory of L-FPNC formed by all topological L-fuzzy N-convergence spaces.

Next we shall discuss the relations between pretopological L-fuzzy N-convergence structures and L-fuzzy neighborhood systems.

Theorem 5.2.Let (X, lim) be a pretopological L-fuzzy N-convergence space. Then _{$N^{x λ}$ lim} defined by ${N^{x λ}}_{lim} (A) = ⋀_{F \in F_{L} (X)} (lim (F) (x_{λ}) \to F (A))$ is an L-fuzzy neighborhood system. Moreover, if (X, lim) is topological, then so is _{$N^{x λ}$ lim}.

Proof. (LFN1) is straightforward.

(LFN2) For ∀mhdnotleqslantA, $[x_{λ}]$ (A) =⊥, ${N^{x λ}}_{lim} (A) \leq lim ([x_{λ}]) (x_{λ}) \to [x_{λ}] (A) = ⊤ \to ⊥ = ⊥ .$

(LFN3) For each A, B ∈ L^X, $\begin{matrix} {N^{x λ}}_{lim} (A \land B) \\ = ⋀_{F \in F_{L} (X)} (lim (F) (x_{λ}) \to F (A \land B)) \\ = ⋀_{F \in F_{L} (X)} ((lim (F) (x_{λ}) \to F (A)) \\ \land (lim (F) (x_{λ}) \to F (B))) \\ = ⋀_{F \in F_{L} (X)} (lim (F) (x_{λ}) \to G (A)) \\ \land ⋀_{G \in F_{L} (X)} (lim (G) (x_{λ}) \to G (B)) \\ = {N^{x λ}}_{lim} (A) \land {N^{x λ}}_{lim} (B) . \end{matrix}$ Hence _{$N^{x λ}$ lim} is an L-fuzzy neighborhood system. Moreover, that _{$N^{x λ}$ lim} satisfies (LFN4) can be easily obtained from (LFTNC). □

Theorem 5.3. If $f : (X, lim_{X}) \to (Y, lim_{Y})$ is continuous with respect to (topological) pretopological L-fuzzy N-convergence structures $lim_{X}$ and $lim_{Y}$ , then $f : (X, N_{lim_{X}}) \to (Y, N_{lim_{Y}})$ is continuous with respect to (topological) L-fuzzy neighborhood systems $N_{lim_{X}}$ and $N_{lim_{Y}}$ .

Proof. Since $f : (X, lim_{X}) \to (Y, lim_{Y})$ is continuous, it follows that $\forall F \in F_{L} (X), \forall x_{λ} \in L^{J (L^{X})},$ ${lim}_{X} (F) (x_{λ}) \leq {lim}_{Y} (f^{\Rightarrow} (F)) (f (x)_{λ}) .$ This implies that ∀B ∈ L^Y, $\begin{matrix} {N^{x λ}}_{{lim}_{X}} (f^{\leftarrow} (B)) \\ = ⋀_{F \in F_{L} (X)} ({lim}_{X} (F) (x_{λ}) \to F (f^{\leftarrow} (B))) \\ \geq ⋀_{F \in F_{L} (X)} ({lim}_{Y} (f^{\Rightarrow} (F)) (f (x)_{λ}) \to f^{\Rightarrow} (F) (B)) \\ \geq ⋀_{G \in F_{L} (X)} ({lim}_{Y} (G) (f (x)_{λ}) \to G (B)) = N_{{lim}_{Y}}^{f (x)_{λ}} (B) . \end{matrix}$ Therefore $f : (X, N_{lim_{X}}) \to (Y, N_{lim_{Y}})$ is continuous. □

Theorem 5.4.Let $(X, N)$ be an L-fuzzy neighborhood system. Then $lim_{N}$ defined by ${lim}_{N} (F) (x_{λ}) = S_{F} (N^{x λ}, F)$ is a pretopological L-fuzzy N-convergence structure. Moreover, if $(X, N)$ is topological, then so is ${lim}_{N}$ .

Proof. (LFNC1) By (LFN2), we have $\begin{matrix} {lim}_{N} (F) (x_{λ}) & = & ⋀_{A \in L^{X}} (N^{x λ} (A) \to F (A)) \\ = & ⋀_{x_{λ} \leq A} (N^{x λ} (A) \to F (A)) . \end{matrix}$ This means that $lim_{N} ([x_{λ}]) (x_{λ}) = ⋀_{x_{λ} \leq A} (N^{x λ} (A) \to [x_{λ}] (A)) = ⊤ .$

(LFNC2) Straightforward.

(LFPNC) By the definition of $lim_{N}$ , it suffices to prove that ${N^{x λ}}_{lim_{N}} = N^{x λ}$ . On one hand, ∀A ∈ L^X, $\begin{matrix} {N^{x λ}}_{lim_{N}} (A) & = & ⋀_{F \in F_{L} (X)} ({lim}_{N} (F) (x_{λ}) \to F (A)) \\ \leq & {lim}_{N} (N^{x λ}) (x_{λ}) \to N^{x λ} (A) \\ = & S_{F} (N^{x λ}, N^{x λ}) \to N^{x λ} (A) = N^{x λ} (A) . \end{matrix}$ On the other hand, $\begin{matrix} {N^{x λ}}_{lim_{N}} (A) \\ = ⋀_{F \in F_{L} (X)} ({lim}_{N} (F) (x_{λ}) \to F (A)) \\ = ⋀_{F \in F_{L} (X)} (S_{F} (N^{x λ}, F) \to F (A)) \\ = ⋀_{F \in F_{L} (X)} (⋀_{B \in L^{X}} (N^{x λ} (B) \to F (B)) \to F (A)) \\ \geq ⋀_{F \in F_{L} (X)} ((N^{x λ} (A) \to F (A)) \to F (A)) \\ \geq N^{x λ} (A) . \end{matrix}$ Therefore, we have $lim_{N} (F) (x_{λ}) = S_{F} (N^{x λ}, F) = S_{F} ({N^{x λ}}_{{lim}_{N}}, F)$ , as desired. If $(X, N)$ is topological, then (LFNTC) can be obtained from $N_{lim_{N}} = N .$ □

Corollary 5.5. If $(X, N)$ is an L-fuzzy neighborhood space, then $N_{lim_{N}} = N .$

Theorem 5.6. If $f : (X, N_{X}) \to (Y, N_{Y})$ is continuous with respect to (topological) L-fuzzy neighborhood systems $N_{X}$ and $N_{Y}$ , then $f : (X, lim_{N_{X}}) \to (Y, lim_{N_{Y}})$ is continuous with respect to (topological) pretopological L-fuzzy N-convergence structures $lim_{N_{X}}$ and $lim_{N_{Y}}$ .

Proof. Since $f : (X, N_{X}) \to (Y, N_{Y})$ is continuous, it follows that $\forall B \in L^{Y}, N_{X}^{x λ} (f^{\leftarrow} (B)) \geq N_{Y}^{f (x)_{λ}} (B) .$ This implies that $\forall F \in F_{L} (X), \forall x_{λ} \in L^{J (L^{X})}$ , $\begin{matrix} {lim}_{N_{Y}} (f^{\Rightarrow} (F)) (f (x)_{λ}) \\ = ⋀_{B \in L^{Y}} (N_{Y}^{f (x)_{λ}} (B) \to f^{\Rightarrow} (F) (B)) \\ \geq ⋀_{B \in L^{Y}} ({N^{x λ}}_{X} (f^{\leftarrow} (B)) \to F (f^{\leftarrow} (B)) \\ \geq ⋀_{A \in L^{X}} ({N^{x λ}}_{X} (A) \to F (A)) \\ = {lim}_{N_{X}} (F) (x_{λ}) . \end{matrix}$ Therefore $f : (X, lim_{N_{X}}) \to (Y, lim_{N_{Y}})$ is continuous. □

Theorem 5.7. If (X, lim) is a pretopological L-fuzzy N-convergence structure, then $lim_{N_{lim}} = lim$ .

Proof. By the definition of $lim_{N}$ and (LFPNC), we obtain $\lim_{n_{\lim}} (F) (x_{λ}) = S_{F} (N_{\lim}^{x_{λ}}, F) = \lim (F) (x_{λ}) . □$

The following results follow from Theorem 2.9 and Theorems 5.2–5.7.

Theorem 5.8. L-FPNC is isomorphic to L-FPN.

Theorem 5.9. L-FTNC, L-FTN and L-FTop are all isomorphic.

Next we discuss the relations between L-FPNC and L-FNC.

Let X be a nonempety set. We write $lim_{lfpnc} (X)$ for the set of all pretopological L-fuzzy N-convergence structures on X and still write ≤ for the restriction of the order on $lim_{lfnc} (X)$ to $lim_{lfpnc} (X) .$ Then the following theorem holds.

Theorem 5.10. $(lim_{lfpnc} (X), \leq)$ is a complete lattice.

Proof. It is similar to the proof of Theorem 3.4. We need only verify for ${lim_{j}}_{j \in J} \subseteq lim_{lfpnc} (X),$ $inf_{j \in J} lim_{j} : F_{L} (X) \to L^{J (L^{X})}$ defined by $\forall F \in F_{L} (X), x_{λ} \in L^{J (L^{X})},$ $({inf}_{j \in J} {lim}_{j}) (F) (x_{λ}) = ⋀_{j \in J} {lim}_{j} (F) (x_{λ})$ is a pretopological L-fuzzy N-convergence structure on X.

(LFNC1) and (LFNC2) are true trivially.

In order to prove (LFPNC), we first check

${(LFNC) {N^{x λ}}_{inf_{j \in J} lim_{j}} \geq ⋁_{j \in J} {N^{x λ}}_{lim_{j}} .$

Take any A ∈ L^X. Then $\begin{matrix} N_{\inf_{j \in J}}^{x λ} {lim}_{j} (A) \\ = ⋀_{G \in F_{L} (X)} (({inf}_{j \in J} {lim}_{j}) (G) (x_{λ}) \to G (A)) \\ = ⋀_{G \in F_{L} (X)} (⋀_{j \in J} {lim}_{j} (G) (x_{λ}) \to G (A)) \\ \geq ⋀_{G \in F_{L} (X)} ⋁_{j \in J} ({lim}_{j} (G) (x_{λ}) \to G (A)) \\ \geq ⋁_{j \in J} ⋀_{G \in F_{L} (X)} ({lim}_{j} (G) (x_{λ}) \to G (A)) \\ = ⋁_{j \in J} N_{\inf_{j \in J}}^{x λ} {lim}_{j} (A) . \end{matrix}$ By (LFNC) we have $\begin{matrix} S_{F} (N_{\inf_{j \in J}}^{x λ} {lim}_{j}, F) \\ = ⋀_{A \in L^{X}} (N_{\inf_{j \in J}}^{x λ} {lim}_{j} (A) \to F (A)) \\ \leq ⋀_{A \in L^{X}} (⋁_{j \in J} N_{\inf_{j \in J}}^{x λ} {lim}_{j} (A) \to F (A)) \\ = ⋀_{A \in L^{X}} ⋀_{j \in J} (N_{\inf_{j \in J}}^{x λ} lim j (A) \to F (A)) \\ = ⋀_{j \in J} ⋀_{A \in L^{X}} (N_{\inf_{j \in J}}^{x λ} lim j (A) \to F (A)) \\ = ⋀_{j \in J} S_{F} (N_{\inf_{j \in J}}^{x λ} lim j, F) = ⋀_{j \in J} {lim}_{j} (F) (x_{λ}) \\ = ({inf}_{j \in J} {lim}_{j}) (F) (x_{λ}), \end{matrix}$ and $\begin{matrix} S_{F} (N_{\inf_{j \in J}}^{x λ} inf j \in J {lim}_{j}, F) \\ = ⋀_{A \in L^{X}} (N_{\inf_{j \in J}}^{x λ} inf j \in J {lim}_{j} (A) \to F (A)) \\ \geq ⋀_{A \in L^{X}} (({inf}_{j \in J} {lim}_{j}) (F) (x_{λ}) \to F (A)) \\ \to F (A) \\ \geq ({inf}_{j \in J} {lim}_{j}) (F) (x_{λ}) . \end{matrix}$ Hence ${inf}_{j \in J} lim_{j}$ is a pretopological L-fuzzy N-convergence structure, as desired. □

Lemma 5.11.Let $(Y, lim_{Y})$ be a pretopological L-fuzzy N-convergence space and f a mapping from X to Y. Define $lim_{X} : F_{L} (X) \to L^{J (L^{X})}$ such that $\forall F \in F_{L} (X), \forall x_{λ} \in J (L^{X}),$ ${lim}_{X} (F) (x_{λ}) = {lim}_{Y} (f^{\Rightarrow} (F)) (f (x)_{λ}) .$ Then lim _X is a pretopological L-fuzzy N-convergence structure on X.

Proof. (LFNC1) and (LFNC2) hold obviously.

(LFPNC) First we prove that $f^{\Rightarrow} (N_{\inf_{j \in J}}^{x λ} lim_{X}) \geq N_{lim_{Y}}^{f (x)_{λ}} .$

For each B ∈ L^Y, we have $\begin{matrix} f^{\Rightarrow} (N_{\inf_{j \in J}}^{x λ} {lim}_{X}) (B) \\ = N_{\inf_{j \in J}}^{x λ} lim X (f^{\leftarrow} (B)) \\ = ⋀_{F \in F_{L} (X)} ({lim}_{X} (F) (x_{λ}) \to F (f^{\leftarrow} (B))) \\ = ⋀_{F \in F_{L} (X)} ({lim}_{Y} (f^{\Rightarrow} (F)) (f (x)_{λ}) \to f^{\Rightarrow} (F) (B)) \\ \geq ⋀_{G \in F_{L} (Y)} ({lim}_{Y} (G) (f (x)_{λ}) \to G (B)) \\ = N_{{lim}_{Y}}^{f (x)_{λ}} (B) . \end{matrix}$ Then it follows that $\begin{matrix} {lim}_{X} (F) (x_{λ}) \\ = {lim}_{Y} (f^{\Rightarrow} (F)) (f (x)_{λ}) \\ = S_{F} (N_{{lim}_{Y}}^{f (x)_{λ}}, f^{\Rightarrow} (F)) \\ \geq S_{F} (f^{\Rightarrow} (N_{\inf_{j \in J}}^{x λ} lim X), f^{\Rightarrow} (F)) \\ \geq S_{F} (N_{\inf_{j \in J}}^{x λ} {lim}_{X}, F) \\ = ⋀_{A \in L^{X}} (N_{\inf_{j \in J}}^{x λ} {lim}_{X} (A) \to F (A)) \\ \geq ⋀_{A \in L^{X}} (({lim}_{X} (F) (x_{λ}) \to F (A)) \to F (A)) \\ \geq {lim}_{X} (F) (x_{λ}) . \end{matrix}$ Therefore ${lim}_{X} (F) (x_{λ}) = S_{F} (N_{\inf_{j \in J}}^{x λ} {lim}_{X}, F)$ , as desired. □

Theorem 5.12. The category L-FPNC of all pretopological L-fuzzy N-convergence spaces is a bireflective full subcategory in L-FNC.

Proof. Let $(X, lim^{¯}) \in$ ∣L-FNC∣ and $E_{lim^{¯}} = {lim ∣ (X, lim) \in ∣ L$ - $FPNC ∣, lim^{¯} \leq lim} .$ Define ${lim^{¯}}^{*} : F_{L} (X) \to L^{J (L^{X})}$ as follows: $\forall F \in F_{L} (X), x_{λ} \in L^{J (L^{X})},$ ${lim^{¯}}^{*} (F) (x_{λ}) = ⋀_{lim \in E_{lim^{¯}}} lim (F) (x_{λ}) .$ By Theorem 5.10, we know ${lim^{¯}}^{*} \in E_{lim^{¯}}$ . Further, we claim that ${id}_{X} : (X, lim^{¯}) \to (X, {lim^{¯}}^{*})$ is the L-FPNC-bireflector.

For this it suffices to prove:

${id}_{X} : (X, lim^{¯}) \to (X, {lim^{¯}}^{*})$ is continuous.

For each pretopological L-fuzzy N-convergence space (Y, lim _Y), and each mapping f : X → Y, the continuity of $f : (X, lim^{¯}) \to (Y, {lim}_{Y})$ implies the continuity of $f : (X, {lim^{¯}}^{*}) \to (Y, {lim}_{Y})$ .

(1) Since ${lim^{¯}}^{*} \in E_{lim^{¯}},$ we have $lim^{¯} (F) (x_{λ}) \leq {lim^{¯}}^{*} (F) (x_{λ})$ for each $F \in F_{L} (X)$ and $x_{λ}$ ∈ $L^{J (L^{X})}$ . This implies that ${id}_{X} : (X, lim^{¯}) \to (X, {lim^{¯}}^{*})$ is continuous.

(2) We need only prove $\forall F \in F_{L} (X), x_{λ} \in L^{J (L^{X})},$ ${lim^{¯}}^{*} (F) (x_{λ}) \leq {lim}_{Y} (f^{\Rightarrow} (F)) (f (x)_{λ}) .$ By Lemma 5.11, lim _X defined by $\forall F \in F_{L} (X), x_{λ} \in L^{J (L^{X})},$ ${lim}_{X} (F) (x_{λ}) = {lim}_{Y} (f^{\Rightarrow} (F)) (f (x)_{λ})$ is a pretopological L-fuzzy convergence structure on X . Further by the continuity of $f : (X, lim^{¯}) \to (Y, {lim}_{Y})$ , we have $lim^{¯} (F) (x_{λ}) \leq {lim}_{Y} (f^{\Rightarrow} (F)) (f (x)_{λ}) = {lim}_{X} (F) (x_{λ}) .$ Hence it follows that $lim^{¯} \leq {lim}_{X}$ and ${lim}_{X} \in E_{lim^{¯}} .$ This shows ${lim^{¯}}^{*} \leq {lim}_{X} .$ Therefore $\begin{matrix} {lim^{¯}}^{*} (F) (x_{λ}) & \leq & {lim}_{X} (F) (x_{λ}) \\ = & {lim}_{Y} (f^{\Rightarrow} (F)) (f (x)_{λ}) . \end{matrix}$ The continuity of $f : (X, {lim^{¯}}^{*}) \to (Y, {lim}_{Y})$ is proved. □

By Corollary 3.5 and Theorem 5.12, we have

Corollary 5.13. The category L-FPNC of all pretopological L-fuzzy N-convergence spaces is topological over Set.

6 Applications of L-fuzzy N-convergence structures

In general topology, continuity of mappings between topological spaces can be characterized by convergence of filters. In other words, continuity of mappings between topological spaces can be characterized by their induced convergence structures. In [18], Liang and Shi endowed the continuity of mappings between L-fuzzy topological spaces with degrees, while the concept of L-fuzzy N-convergence structures is also equipped with degrees. This motivates us to consider characterizing continuous degrees of mappings between L-fuzzy topological spaces by using their induced L-fuzzy N-convergence structures.

Definition 6.1. ([18]) Let f : (X, τ_X) → (Y, τ_Y) be a mapping between L-fuzzy topological spaces. Define the degree Cont (f) to which f is continuous as follows: $Cont (f) = ⋀_{B \in L^{Y}} (τ_{Y} (B) \to τ_{X} (f^{\leftarrow} (B))) .$

Lemma 6.2.Let (X, τ) be an L-fuzzy topological space. Then for each A ∈ L^X, $τ (A) = ⋀_{x_{λ} ≺ A} ⋀_{F \in F_{L} (X)} ({lim}_{τ} (F) (x_{λ}) \to F (A)) .$

Proof. By Theorem 4.6, we know τ (A) = τ_{lim_τ} (A) for all A ∈ L^X. Then by Theorem 4.1 and the definition of $N_{x_{λ}}^{lim}$ , it follows that $\begin{matrix} τ (A) & = & τ_{{lim}_{τ}} (A) \\ = & ⋀_{x_{λ} ≺ A} ⋀_{F \in F_{L} (X)} ({lim}_{τ} (F) (x_{λ}) \to F (A)) . □ \end{matrix}$

Theorem 6.3.Letf : (X, τ_X) → (Y, τ_Y) be a mapping between L-fuzzy topological spaces. Then $\begin{matrix} Cont (f) & = & ⋀_{x_{λ} \in J (L^{X})} ⋀_{F \in F_{L} (X)} ({lim}_{τ_{X}} (F) (x_{λ}) \\ \to {lim}_{τ_{Y}} (f^{\Rightarrow} (F)) (f (x)_{λ})) . \end{matrix}$

Proof. On one hand, by Lemma 4.4, it follows that $\begin{matrix} ⋀_{x_{λ} \in J (L^{X})} ⋀_{F \in F_{L} (X)} ({lim}_{τ_{X}} (F) (x_{λ}) \\ \to {lim}_{τ_{Y}} (f^{\Rightarrow} (F)) (f (x)_{λ})) . \\ = ⋀_{x_{λ} \in J (L^{X})} ⋀_{F \in F_{L} (X)} (⋀_{x_{λ} ≺ A} (τ_{X} (A) \to F (A)) \\ \to ⋀_{f (x)_{λ} ≺ B} (τ_{Y} (B) \to f^{\Rightarrow} (F) (B))) \\ = ⋀_{x_{λ} \in J (L^{X})} ⋀_{F \in F_{L} (X)} ⋀_{x_{λ} ≺ f^{\leftarrow} (B)} (⋀_{x_{λ} ≺ A} (τ_{X} (A) \to \\ F (A)) \to (τ_{Y} (B) \to F (f^{\leftarrow} (B)))) \\ \geq ⋀_{x_{λ} \in J (L^{X})} ⋀_{F \in F_{L} (X)} ⋀_{x_{λ} ≺ f^{\leftarrow} (B)} ((τ_{X} (f^{\leftarrow} (B)) \to \\ F (f^{\leftarrow} (B))) \to (τ_{Y} (B) \to F (f^{\leftarrow} (B)))) \\ \geq ⋀_{x_{λ} \in J (L^{X})} ⋀_{F \in F_{L} (X)} ⋀_{x_{λ} ≺ f^{\leftarrow} (B)} (τ_{Y} (B) \to τ_{X} (f^{\leftarrow} (B)) \\ \geq ⋀_{B \in L^{Y}} (τ_{Y} (B) \to τ_{X} (f^{\leftarrow} (B)) = Cont (f) . \end{matrix}$ On the other hand, by Lemma 6.2, we have $\begin{matrix} Cont (f) \\ = ⋀_{B \in L^{Y}} (τ_{Y} (B) \to τ_{X} (f^{\leftarrow} (B)) \\ = ⋀_{B \in L^{Y}} (⋀_{y_{μ} ≺ B} ⋀_{G \in F_{L} (Y)} ({lim}_{τ_{Y}} (G) (y_{μ}) \to G (B)) \\ \to ⋀_{x_{λ} ≺ f^{\leftarrow} (B)} ⋀_{F \in F_{L} (X)} ({lim}_{τ_{X}} (F) (x_{λ}) \to F (f^{\leftarrow} (B))) \\ \geq ⋀_{B \in L^{Y}} (⋀_{f (x)_{λ} ≺ B} ⋀_{F \in F_{L} (X)} ({lim}_{τ_{Y}} (f^{\Rightarrow} (F)) (f (x)_{λ}) \to \\ f^{\Rightarrow} (F) (B)) \to ⋀_{x_{λ} ≺ f^{\leftarrow} (B)} ⋀_{F \in F_{L} (X)} ({lim}_{τ_{X}} (F) (x_{λ}) \\ \to F (f^{\leftarrow} (B))) \\ \geq ⋀_{B \in L^{Y}} ⋀_{f (x)_{λ} ≺ B} ⋀_{F \in F_{L} (X)} (({lim}_{τ_{Y}} (f^{\Rightarrow} (F)) (f (x)_{λ}) \to \\ f^{\Rightarrow} (F) (B)) \to ({lim}_{τ_{X}} (F) (x_{λ}) \to f^{\Rightarrow} (F) (B))) \\ \geq ⋀_{B \in L^{Y}} ⋀_{f (x)_{λ} ≺ B} ⋀_{F \in F_{L} (X)} ({lim}_{τ_{X}} (F) (x_{λ}) \to \\ {lim}_{τ_{Y}} (f^{\Rightarrow} (F)) (f (x)_{λ})) \\ \geq ⋀_{x_{λ} \in J (L^{X})} ⋀_{F \in F_{L} (X)} ({lim}_{τ_{X}} (F) (x_{λ}) \to \\ {lim}_{τ_{Y}} (f^{\Rightarrow} (F)) (f (x)_{λ})) . \end{matrix}$ This completes the proof. □

7 Conclusion

In this paper, we proposed the concepts of L-fuzzy N-convergence spaces, pretopological L-fuzzy N-convergence spaces and topological L-fuzzy N-convergence spaces. Based on L-fuzzy neighborhood spaces [25], we discussed the relations among these new kinds of lattice-valued convergence spaces and L-fuzzy topological spaces. Since L-fuzzy neighborhood spaces is the main tool, we only require that L is a completely distributive lattice. This is different from our previous works in [20 –24], where L is required to be a completely distributive De Morgan algebra. However, this way also leads to some deficiencies which deserve further discussions in our future work.

In classical sense, the category Top of topological spaces can be embedded in the category Conv of generalized convergence spaces as a reflective subcategory. However, in this paper, we can only prove the category L-FTop of L-fuzzy topological spaces can be embedded in the category L-FNC of L-fuzzy N-convergence spaces. Hence, we will consider the reflectivity of L-FTop in L-FNC.

In general topology, the category Conv has a nicer categorical property than Top, that is, Conv is Cartesian closed. Therefore we will also consider the Cartesian-closeness of L-FNC.

Footnotes

Acknowledgments

The authors would like to express their sincerethanks to the anonymous reviewers for their careful reading and constructive comments. This work is supported by National Nature Science Foundation Committee (NSFC) of China (No. 61573119), China Postdoctoral Science Foundation (No. 2015M581434), Fundamental Research Project of Shenzhen (No. JCYJ20120613144110654 and No. JCYJ20140417172417109).

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