Coreflectivities of ( L,M )-fuzzy convex structures and ( L,M )-fuzzy cotopologies in ( L,M )-fuzzy closure systems

Abstract

In this paper, we mainly focus on the relationship between (L, M)-fuzzy closure systems and (L, M)-fuzzy convex structures as well as the relationship between (L, M)-fuzzy closure systems and (L, M)-fuzzy cotopologies. Firstly, we show that there is an adjunction between the category LMFC of (L, M)-fuzzy convex spaces and the category LMFS of (L, M)-fuzzy closure spaces, and there is also an adjunction between the category LMFT of (L, M)-fuzzy cotopological spaces and LMFS. In particular, the categories LMFC and LMFT are both coreflective subcategories of LMFS. Secondly, we prove that there is an adjunction between the category ELFC of extensional L-fuzzy convex spaces and the category LFC of L-fuzzy convex spaces, and there is also an adjunction between the category ELFT of extensional L-fuzzy cotopological spaces and the category LFT of (L, M)-fuzzy cotopological spaces. Specially, ELFC is a coreflective subcategory of LFC and ELFT is a coreflective subcategory of LFT.

1 Introduction

By a closure system, we usually mean a subset of the power set 2^X which satisfies certain conditions [2]. As we all know, a closure system is not only a generalization of a convex structure [34] by relaxing the condition that it is closed under directed unions, but also a generalization of a cotopology (equivalently, topology) by relaxing the condition that it is closed under finite unions. In a categorical sense, the category of convex spaces and the category of cotopological spaces are both full subcategories of the category of closure spaces.

Since Zadeh introduced the concept of fuzzy sets, many mathematical structures have been endowed with fuzzy set theory, such as fuzzy closure systems [1 , 14], fuzzy convex structures [29 , 32], fuzzy cotopological structures [3 , 43] and fuzzy convergence structures [10–12 , 24]. Generally, there are three typical kinds of fuzzy topological structures, which are called L-topology [3], M-fuzzifying topology [43] and (L, M)-fuzzy topology [9, 33]. Adopting the terminology of fuzzy topology, there are also three representative types of fuzzy convex structures, including L-convex structures [7 , 41], M-fuzzifying convex structures [23 , 42] and (L, M)-fuzzy convex structures [13 , 38]. Besides, the concepts of fuzzy closure systems can also be defined in this way. Fang [4] proposed the concept of L-fuzzy closure systems and established its categorical relationship with L-fuzzy closure operators. Later, Luo and Fang [14] introduced the concept of fuzzifying closure systems and showed there was a one-to-one correspondence between fuzzifying closure systems and fuzzifying Birkhoff’s closure operators. More research on the relationship between closure systems and closure operators can be found in [8 , 44].

Category theory is a useful tool in establishing the relationship between different types of mathematical structures. In the fuzzy case, fuzzy convex structures and fuzzy cotopologies can both be considered as special fuzzy closure systems by demanding some fuzzy counterparts of the corresponding required axioms in the classical case. In order to assure the deeper connections between fuzzy closure systems, fuzzy convex structures and fuzzy cotopologies, Xiu and Pang [39] established the categorical relationship between M-fuzzifying closure systems, M-fuzzifying convex structures and M-fuzzifying Alexandrov topologies. Actually, fuzzy mathematical structures in the (L, M)-fuzzy case are more general than those in the M-fuzzifying case. By this motivation, we will consider the relationship between fuzzy closure systems, fuzzy convex structures and fuzzy cotopologies in the (L, M)-fuzzy case.

From a categorical viewpoint, we will establish the relationship between (L, M)-fuzzy closure systems, (L, M)-fuzzy convex structures and (L, M)-fuzzy cotopologies. Using a categorical tool, we will include the categories of (L, M)-fuzzy convex spaces and (L, M)-fuzzy cotopological spaces in the category of (L, M)-fuzzy closure spaces as coreflective subcategories. As we all know, by means of L-equality relations between L-subsets, the concept of extensional L-fuzzy topologies was proposed. Adopting this way, we will propose the concepts of extensional L-fuzzy convex structures and extensional L-fuzzy cotopologies, and then establish their categorical relationship with L-fuzzy convex structures and L-fuzzy cotopologies, respectively.

2 Preliminaries

Let L (resp. M) be a complete lattice with the smallest element ⊥_L (resp. ⊥_M) and the largest element ⊤_L (resp. ⊤_M). For a, b ∈ L, we say that a is way below b in L (in symbols, a ⪡ b) if for all directed subsets D ⊆ L, b ≤ ⋁ D always implies that a ≤ d for some d ∈ D. A complete lattice L is said to be continuous if for all x ∈ L, x = ⋁ ⇓ x, where ⇓x = {y ∈ L ∣ y ⪡ x} ([5]). For a directed subset D ⊆ L, we use ⋁^dirD to denote its supremum. For a, b ∈ M, we say that a is wedge below b in M (in symbols, a ≺ b) if for every subset D ⊆ M, ⋁D ≥ b implies d ≥ a for some d ∈ M. A complete lattice M is completely distributive if and only if b = ⋁ {a ∈ M ∣ a ≺ b} for each b ∈ L.

For a nonempty set X, L^X denotes the set of all L-subsets on X. For each a ∈ L, $\underline{a}$ denotes the constant mapping X ⟶ L, x ⟼ a, which is called a constant L-subset. The operators on L can be translated onto L^X in a pointwise way. The smallest element and the largest element in L^X are denoted by $⊥_{L}^{X}$ and $⊤_{L}^{X}$ , respectively. If L is a continuous lattice, then L^X is also a continuous lattice and the way below relation on L^X is still denoted by “ ⪡ ".

Let X, Y be two nonempty sets and let φ : X ⟶ Y be a mapping. Define $φ_{L}^{\to} : L^{X} ⟶ L^{Y}$ and $φ_{L}^{\leftarrow} : L^{Y} ⟶ L^{X}$ as follows: $\forall A \in L^{X}, \forall y \in Y, φ_{L}^{\to} (A) (y) = ⋁_{φ (x) = y} A (x);$ $\forall B \in L^{Y}, \forall x \in X, φ_{L}^{\leftarrow} (B) (x) = B (φ (x)) .$

Lemma 2.1. ([5]) Let L be a continuous lattice and let {a_jk ∣ j ∈ J, k ∈ K (j)} be a nonempty family of elements in L such that {a_jk ∣ k ∈ K (j)} is directed for all j ∈ J. Then the following identity holds: $(DD) ⋀_{j \in J} ⋁_{k \in K (j)}^{dir} a_{jk} = ⋁_{h \in N}^{dir} ⋀_{j \in J} a_{jh (j)},$ where N is the set of all choice functions h : J ⟶ ⋃ _j∈JK (j) with h (j) ∈ K (j) for all j ∈ J.

For a completely distributive lattice L, we can then define a residual implication by $a \to b = ⋁ {λ \in L ∣ a \land λ \leq b} .$ Then we define $a \leftrightarrow b = (a \to b) \land (b \to a) .$ Further, for A, B ∈ L^X, the L-equality relation $E_{L} : L^{X} \times L^{X} ⟶ L$ between L-subsets can be defined by means of “↔" as follows: $\forall A, B \in L^{X}, E_{L} (A, B) = ⋀_{x \in X} (A (x) \leftrightarrow B (x)) .$

We will often use, without explicit proving, the following properties of the L-equality relations.

Lemma 2.2. For each A, B, C, D ∈ L^X, {A_i} _i∈I ⊆ L^X, {B_i} _i∈I ⊆ L^X, the following statements hold.

$E_{L} (A, B) = ⊤ \Leftrightarrow A = B$ .

$E_{L} (A, B) \land E_{L} (C, D) \leq E_{L} (A \lor C, B \lor D)$ .

$⋀_{i \in I} E_{L} (A_{i}, B_{i}) \leq E_{L} (⋀_{i \in I} A_{i}, ⋀_{i \in I} B_{i})$ .

Throughout this paper, L and M denote complete lattices, unless otherwise stated.

Definition 2.3. ([4, 30]). An (L, M)-fuzzy closure system on X is a mapping $S : L^{X} ⟶ M$ which satisfies: (LMS1) $S (⊥_{L}^{X}) = S (⊤_{L}^{X}) = ⊤_{M}$ ; (LMS2) $S (⋀_{i \in I} A_{i}) \geq ⋀_{i \in I} S (A_{i})$ .

For an (L, M)-fuzzy closure system $S$ on X, the pair $(X, S)$ is called an (L, M)-fuzzy closure space.

A mapping $φ : (X, S_{X}) ⟶ (Y, S_{Y})$ between (L, M)-fuzzy closure spaces is called continuous if it satisfies $\forall B \in L^{Y}, S_{X} (φ_{L}^{\leftarrow} (B)) \geq S_{Y} (B) .$

It is easy to check that all (L, M)-fuzzy closure spaces as objects and all continuous mappings as morphisms form a category, denoted by LMFS.

By demanding some more axioms on (L, M)-fuzzy closure systems, another two well-known concepts can be obtained.

Definition 2.4. ([32]). An (L, M)-fuzzy convex structure on X is a mapping $C : L^{X} ⟶ M$ which satisfies: (LMC1) $C (⊥_{L}^{X}) = C (⊤_{L}^{X}) = ⊤_{M}$ ; (LMC2) $C (⋀_{i \in I} A_{i}) \geq ⋀_{i \in I} C (A_{i})$ ; (LMC3) $C (⋁_{j \in J}^{dir} A_{j}) \geq ⋀_{j \in J} C (A_{j})$ .

For an (L, M)-fuzzy convex structure $C$ on X, the pair $(X, C)$ is called an (L, M)-fuzzy convex space.

A mapping $φ : (X, C_{X}) ⟶ (Y, C_{Y})$ between (L, M)-fuzzy convex spaces is called convexity preserving (CP, in short) if it satisfies $\forall B \in L^{Y}, C_{X} (φ_{L}^{\leftarrow} (B)) \geq C_{Y} (B) .$

It is easy to check that all (L, M)-fuzzy convex spaces as objects and all CP mappings as morphisms form a category, denoted by LMFC.

Definition 2.5. ([9, 33]). An (L, M)-fuzzy cotopology on X is a mapping τ : L^X ⟶ M which satisfies: (LMT1) $τ (⊥_{L}^{X}) = τ (⊤_{L}^{X}) = ⊤_{M}$ ; (LMT2) τ (⋀ _i∈IA_i) ≥ ⋀ _i∈Iτ (A_i); (LMT3) τ (A ∨ B) ≥ τ (A) ∧ τ (B).

For an (L, M)-fuzzy cotopology τ on X, the pair (X, τ) is called an (L, M)-fuzzy cotopological space.

A mapping φ : (X, τ_X) ⟶ (Y, τ_Y) between (L, M)-fuzzy cotopological spaces is called continuous if it satisfies $\forall B \in L^{Y}, τ_{X} (φ_{L}^{\leftarrow} (B)) \geq τ_{Y} (B) .$

It is easy to check that all (L, M)-fuzzy cotopological spaces as objects and all continuous mappings as morphisms form a category, denoted by LMFT.

Actually, it is easy to see that LMFC and LMFT are both full subcategories of LMFS.

The subjects of a category A is denoted by ∣A∣. For other notions related to category theory, we refer to [28].

3 Main results

3.1 Coreflectivity of LMFC in LMFS

From the definitions of (L, M)-fuzzy closure systems and (L, M)-fuzzy convex structures, we observe that an (L, M)-fuzzy closure system is an (L, M)-fuzzy convex structure which relaxes the axiom (LMC3). That is to say, (L, M)-fuzzy closure systems are generalizations of (L, M)-fuzzy convex structures. In this subsection, we will further investigate their categorical relationship by requiring L being continuous lattice and M being a completely distributive lattice.

Theorem 3.1.1. Let $(X, S)$ be an (L, M)-fuzzy closure space and define $S^{*} : L^{X} ⟶ M$ as follows: $\forall A \in L^{X}, S^{*} (A) = ⋁_{⋁_{k \in K}^{dir} A_{k} = A} ⋀_{k \in K} S (A_{k}) .$ Then $S^{*}$ is an (L, M)-fuzzy convex structure on X. Moreover, $S^{*} (A) \geq S (A) (\forall A \in L^{X})$ .

Proof. It suffices to show that $S^{*}$ satisfies (LMC1)–(LMC3). Indeed,

(LMC1) Since $S$ satisfies (LMS1), it follows immediately that $S^{*} (⊥_{L}^{X}) = S^{*} (⊤_{L}^{X}) = ⊤_{M}$ .

(LMC2) Take each {A_i ∣ i ∈ I} ⊆ L^X. Then we prove $S^{*} (⋀_{i \in I} A_{i}) \geq ⋀_{i \in I} S^{*} (A_{i})$ , i.e.,

$\begin{matrix} ⋁_{⋁_{k \in K}^{dir} B_{k} = ⋀_{i \in I} A_{i}} ⋀_{k \in K} S (B_{k}) \\ \geq & ⋀_{i \in I} ⋁_{⋁_{j \in J_{i}}^{dir} G_{i, j} = A_{i}} ⋀_{j \in J_{i}} S (G_{i, j}) . \end{matrix}$

Choose each a ∈ M such that $a ≺ ⋀_{i \in I} ⋁_{⋁_{j \in J_{i}}^{dir} G_{i, j} = A_{i}} ⋀_{j \in J_{i}} S (G_{i, j}) .$ Then for each i ∈ I, there exists a directed set {G_i,j ∣ j ∈ J_i} ⊆ L^X such that $⋁_{j \in J_{i}}^{dir} G_{i, j} = A_{i}$ and for each j ∈ J_i, $S (G_{i, j}) \geq a$ . By (DD), we have $⋀_{i \in I} A_{i} = ⋀_{i \in I} ⋁_{j \in J_{i}}^{dir} G_{i, j} = ⋁_{f \in \prod_{i \in I} J_{i}}^{dir} ⋀_{i \in I} G_{i, f (i)}$ and $S (G_{i, f (i)}) \geq a$ . This implies that $\begin{matrix} ⋁_{⋁_{k \in K}^{dir} B_{k} = ⋀_{i \in I} A_{i}} ⋀_{k \in K} S (B_{k}) \\ \geq & ⋀_{f \in \prod_{i \in I} J_{i}} S (⋀_{i \in I} G_{i, f (i)}) \\ \geq & ⋀_{f \in \prod_{i \in I} J_{i}} ⋀_{i \in I} S (G_{i, f (i)}) \\ \geq & a . \end{matrix}$ By the arbitrariness of a, we obtain $S^{*} (⋀_{i \in I} A_{i}) \geq ⋀_{i \in I} S^{*} (A_{i})$ , as desired.

(LMC3) Take each ${A_{j} ∣ j \in J} \overset{dir}{\subseteq} L^{X}$ . Then we prove that $S^{*} (⋁_{j \in J}^{dir} A_{j}) \geq ⋀_{j \in J} S^{*} (A_{j})$ . That is to say, $⋁_{⋁_{k \in K}^{dir} B_{k} = ⋁_{j \in J}^{dir} A_{j}} ⋀_{k \in K} S (B_{k}) \geq ⋀_{j \in J} ⋁_{⋁_{i \in I_{j}}^{dir} G_{j, i} = A_{j}} ⋀_{i \in I_{j}} S (G_{j, i}) .$

Choose each a ∈ M such that $a ≺ ⋀_{j \in J} ⋁_{⋁_{i \in I_{j}}^{dir} G_{j, i} = A_{j}} ⋀_{i \in I_{j}} S (G_{j, i}) .$ Then for each j ∈ J, there exists a directed set {G_j,i ∣ i ∈ I_j} such that $⋁_{i \in I_{j}}^{dir} G_{j, i} = A_{j}$ and for each i ∈ I_j, $S (G_{j, i}) \geq a$ . Put $C = ⋁_{j \in J}^{dir} A_{j}$ . Then $C = ⋁_{j \in J}^{dir} ⋁_{i \in I_{j}}^{dir} G_{j, i}$ . Now define a mapping σ : ⇓C ⟶ L^X as follows: $\forall F \in ⇓ C, σ (F) = ⋀ {G_{j, i} ∣ F \leq G_{j, i}} .$ Then σ is well defined since $F ⪡ ⋁_{j \in J}^{dir} ⋁_{i \in I_{j}}^{dir} G_{j, i}$ implies F ⪡ G_j,i for some j ∈ J and i ∈ I_j. Further, it is easy to check that $C = ⋁_{F \in ⇓ C} F = ⋁_{F \in ⇓ C} σ (F) .$ Since ⇓C is directed and σ is order-preserving, we know {σ (F) ∣ F ∈ ⇓ C} is directed. This means $C = ⋁_{F \in ⇓ C}^{dir} σ (F) .$ By (LMS2), it follows immediately that $S (σ (F)) = S (⋀_{F \leq G_{j, i}} G_{j, i}) \geq ⋀_{F \leq G_{j, i}} S (G_{j, i}) \geq a .$ This implies $⋁_{⋁_{k \in K}^{dir} B_{k} = ⋁_{j \in J}^{dir} A_{j}} ⋀_{k \in K} S (B_{k}) \geq ⋀_{F \in ⇓ C} S (σ (F)) \geq a .$ By the arbitrariness of a, we obtain $⋁_{⋁_{k \in K}^{dir} B_{k} = ⋁_{j \in J}^{dir} A_{j}} ⋀_{k \in K} S (B_{k}) \geq ⋀_{j \in J} ⋁_{⋁_{i \in I_{j}}^{dir} G_{j, i} = A_{j}} ⋀_{i \in I_{j}} S (G_{j, i}),$ as desired. □

Theorem 3.1.2. If $φ : (X, S_{X}) ⟶ (Y, S_{Y})$ is a continuous mapping between (L, M)-fuzzy closure spaces, then $φ : (X, S_{X}^{*}) ⟶ (Y, S_{Y}^{*})$ is a CP mapping between (L, M)-fuzzy convex spaces.

Proof. Since $φ : (X, S_{X}) ⟶ (Y, S_{Y})$ is continuous, we have $\forall A \in L^{Y}, S_{X} (φ_{L}^{\leftarrow} (A)) \geq S_{Y} (A) .$ Further, by the definition of $S^{*}$ , we obtain $\begin{matrix} S_{Y}^{*} (A) & = & ⋁_{⋁_{k \in K}^{dir} A_{k} = A} ⋀_{k \in K} S_{Y} (A_{k}) \\ \leq & ⋁_{⋁_{k \in K}^{dir} A_{k} = A} ⋀_{k \in K} S_{X} (φ_{L}^{\leftarrow} (A_{k})) \\ \leq & ⋁_{⋁_{k \in K}^{dir} φ_{L}^{\leftarrow} (A_{k}) = φ_{L}^{\leftarrow} (A)} ⋀_{k \in K} S_{X} (φ_{L}^{\leftarrow} (A_{k})) \\ \leq & ⋁_{⋁_{k \in K}^{dir} B_{k} = φ_{L}^{\leftarrow} (A)} ⋀_{k \in K} S_{X} (B_{k}) \\ = & S_{X}^{*} (φ_{L}^{\leftarrow} (A)) . \end{matrix}$ This proves that $φ : (X, S_{X}^{*}) ⟶ (Y, S_{Y}^{*})$ is a CP mapping.

□

By Theorems 3.1.1 and 3.1.2, we obtain a functor $𝔽$ as follows: $\begin{matrix} 𝔽 : {\begin{matrix} LMFS & ⟶ & LMFC \\ (X, S) & ⟼ & (X, S^{*}) \\ φ & ⟼ & φ . \end{matrix} \end{matrix}$

Since LMFC is a full subcategory of LMFS, we denote the inclusion functor from LMFC to LMFS by $𝕀_{1}$ . Then we have the the following result:

Theorem 3.1.3. $(𝕀_{1}, 𝔽)$ is an adjunction between LMFC and LMFS. Moreover, $𝔽$ is a left inverse functor of $𝕀_{1}$ .

Proof. It suffices to verify that

$𝕀_{1} \circ 𝔽 ((X, S)) \leq (X, S)$ for each (L, M)-fuzzy closure space $(X, S)$ , i.e., $S^{*} (A) \geq S (A)$ for each A ∈ L^X.

$𝔽 \circ 𝕀_{1} ((X, C)) = (X, S)$ for each (L, M)-fuzzy convex space $(X, C)$ , i.e., $C^{*} (A) = C (A)$ for each A ∈ L^X.

For (1), it follows from Theorem 3.1.1.

For (2), take each A ∈ L^X. Then $C^{*} (A) = ⋁_{⋁_{k \in K}^{dir} A_{k} = A} ⋀_{k \in K} C (A_{k}) .$ On one hand, let {A_k ∣ k ∈ K} = {A}. Then $⋁_{⋁_{k \in K}^{dir} A_{k} = A} ⋀_{k \in K} C (A_{k}) \geq C (A) .$ On the other hand, $⋁_{⋁_{k \in K}^{dir} A_{k} = A} ⋀_{k \in K} C (A_{k}) \leq ⋁_{⋁_{k \in K}^{dir} A_{k} = A} C (⋁_{k \in K}^{dir} A_{k}) = C (A),$ as desired. □

Corollary 3.1.4. LMFC is a coreflective subcategory of LMFS.

3.2 Coreflectivity of ELFC in LFC

In this subsection, we assume L = M and L is a completely distributive lattice. In this case, (L, M)-fuzzy convex structures will be called L-fuzzy convex structures and the category LLFC be will denoted by LFC. By means of L-equality relations between L-subsets, we first introduced extensional L-fuzzy convex structures by equipping an extensional condition on L-fuzzy convex structures. Then we will establish the relationship between L-fuzzy convex structures and extensional L-fuzzy convex structures from a categorical aspect.

Definition 3.2.1. An L-fuzzy convex structure $C$ on X is called extensional if it satisfies: (ELFC) $C (A) \land E_{L} (A, B) \leq C (B)$ . For an extensional L-fuzzy convex structure $C$ on X, the pair $(X, C)$ is called an extensional L-fuzzy convex space.

The full subcategory of LFC, consisting of extensional L-fuzzy convex spaces, is denoted by ELFC.

For an L-fuzzy convex structure $C$ on X, let $\begin{matrix} F_{C}^{X} = {\bar{C} ∣ (X, \bar{C}) \in ∣ ELFC ∣ and \bar{C} (A) \geq C (A) \\ for each A \in L^{X}} . \end{matrix}$

Then we present the following theorem.

Theorem 3.2.2. Let $(X, C)$ be an L-fuzzy convex structure and define $C^{*} : L^{X} ⟶ L$ by $\forall A \in L^{X}, C^{*} (A) = ⋀_{\bar{C} \in F_{C}^{X}} \bar{C} (A) .$ Then $C^{*}$ is an extensional L-fuzzy convex structure on X.

Proof. It suffices to show that C^* satisfies (LMC1)–(LMC3) and (ELFC). Indeed,

(LMC1) Straightforward.

(LMC2) Take {A_i} ⊆ L^X. Then

$\underset{i \in I}{\land} C * (A_{i}) = \underset{i \in I}{\land} \underset{\bar{C} \in F_{C}^{X}}{\land} \bar{C} (A_{i}) = \underset{\bar{C} \in F_{C}^{X}}{\land} \underset{i \in I}{\land} \bar{C} (A_{i}) \leq \underset{\bar{C} \in F_{C}^{X}}{\land} \bar{C} (\underset{i \in I}{\land} A_{i}) = C * (\underset{i \in I}{\land} A_{i}) .$

(LMC3) Similar to (LMC2).

(ELFC) Take each A ∈ L^X. Then

$\begin{array}{l} C * (A) \land ε_{L} (A,) & = (\underset{\bar{C} \in F_{C}^{X}}{\land} \bar{C} (A)) \land ε_{L} (A, B) \\ = \underset{\bar{C} \in F_{C}^{X}}{\land} (\bar{C} (A) \land ε_{L} (A, B)) \\ \leq \underset{\bar{C} \in F_{C}^{X}}{\land} \bar{C} (B) \\ = C * (B) . \end{array}$

This shows C^* is an extensional L-fuzzy convex structure on X. □

Theorem 3.2.3. If $φ : (X, C_{X}) ⟶ (Y, C_{Y})$ is a CP mapping between L-fuzzy convex spaces, then $φ : (X, C_{X}^{*}) ⟶ (Y, C_{Y}^{*})$ is a CP mapping between extensional L-fuzzy convex spaces.

Proof. Since $φ : (X, C_{X}) ⟶ (Y, C_{Y})$ is CP, it follows that $\forall B \in L^{Y}, C_{X} (φ_{L}^{\leftarrow} (B)) \geq C_{Y} (B) .$ Take each $\bar{C} \in F_{C_{X}}^{X}$ and define ${\bar{C}}_{Y} : L^{Y} ⟶ L$ by ${\bar{C}}_{Y} (B) = \bar{C} (φ_{L}^{\leftarrow} (B))$ for each B ∈ L^Y. Then it is easy to check that ${\bar{C}}_{Y}$ is an extensional L-fuzzy convex structure on Y and ${\bar{C}}_{Y} (B) = \bar{C} (φ_{L}^{\leftarrow} (B)) \geq C_{X} (φ_{L}^{\leftarrow} (B)) \geq C_{Y} (B) .$ This means ${\bar{C}}_{Y} \in F_{C_{Y}}^{Y}$ . Thus, it follows that $C_{Y}^{*} (B) = ⋀_{C \in F_{C_{Y}}^{Y}} C (B) \leq {\bar{C}}_{Y} (B) = \bar{C} (φ_{L}^{\leftarrow} (B)) .$ By the arbitrariness of $\bar{C}$ , we obtain $C_{Y}^{*} (B) \leq ⋀_{C \in F_{C_{X}}^{X}} \bar{C} (φ_{L}^{\leftarrow} (B)) = C_{X}^{*} (φ_{L}^{\leftarrow} (B)),$ as desired. □

By Theorems 3.2.2 and 3.2.3, we obtain a functor $𝔽^{*}$ as follows:

$𝔽*: {\begin{matrix} LFC \to ELFC \\ (X,C) \mapsto (X, C *) \\ ϕ \mapsto ϕ . \end{matrix}$

Since ELFC is a full subcategory of LFC, we denote the inclusion functor from ELFC to LFC by $𝕀_{1}^{*}$ . Then we have the following result:

Theorem 3.2.4. $(𝕀_{1}^{*}, 𝔽^{*})$ is an adjunction between ELFC and LFC. Moreover, $𝔽^{*}$ is a left inverse functor of $𝕀_{1}^{*}$ .

Proof. It suffices to verify that

$𝕀_{1}^{*} \circ 𝔽^{*} ((X, C)) \leq (X, C)$ for each L-fuzzy convex space $(X, C)$ , i.e., $C^{*} (A) \geq C (A)$ for each A ∈ L^X.

$𝔽^{*} \circ 𝕀_{1}^{*} ((X, C_{e})) = (X, C_{e})$ for each extensional L-fuzzy convex space $(X, C_{e})$ , i.e., $C_{e}^{*} (A) = C_{e} (A)$ for each A ∈ L^X.

For (1), take each A ∈ L^X. Then

C^{*} (A) = ⋀_{\bar{C} \in F_{C}^{X}} \bar{C} (A) \geq C (A) .

For (2), take each A ∈ L^X. Since $C_{e} \in F_{C_{e}}^{X}$ , it follows that $C_{e}^{*} (A) = ⋀_{\bar{C} \in F_{C_{e}}^{X}} \bar{C} (A) = C_{e} (A),$ as desired. □

Corollary 3.2.5. ELFC is a coreflective subcategory of LFC.

3.3 Coreflectivity of LMFT in LMFS

It is observed that each (L, M)-fuzzy cotopology is an (L, M)-fuzzy closure system by demanding the axiom (LMT3). In this subsection, we will focus on the categorical relationship between (L, M)-fuzzy closure systems and (L, M)-fuzzy cotopologies in the assumption that L and M are completely distributive lattices.

Lemma 3.3.1. Let $(X, S)$ be an (L, M)-fuzzy closure space and define $S^{★} : L^{X} ⟶ M$ as follows: $\forall A \in L^{X}, S^{★} (A) = ⋁_{\lor_{j \in F} A_{j} = A} \land_{j \in F} S (A_{j}),$ where F is finite. Then $S^{★}$ satisfies (LMT3). Moreover, $S^{★} (A) \geq S (A) (\forall A \in L^{X})$ .

Proof. Take each A, B ∈ L^X. Then $\begin{matrix} S^{★} (A) \land S^{★} (B) \\ = & ⋁_{\lor_{i \in F_{1}} A_{i} = A} \land_{i \in F_{1}} S (A_{i}) \land ⋁_{\lor_{j \in F_{2}} B_{j} = B} \land_{j \in F_{2}} S (B_{j}) \\ = & ⋁_{\lor_{i \in F_{1}} A_{i} = A} ⋁_{\lor_{j \in F_{2}} B_{j} = B} \land_{i \in F_{1}} \land_{j \in F_{2}} (S (A_{i}) \land S (B_{j})) \\ \leq & ⋁_{(\lor_{i \in F_{1}} A_{i}) \lor (\lor_{j \in F_{2}} B_{j}) = A \lor B} \land_{i \in F_{1}} \land_{j \in F_{2}} (S (A_{i}) \land S (B_{j})) \\ \leq & ⋁_{\lor_{k \in F} C_{k} = A \lor B} \land_{k \in F} S (C_{k}) \\ = & S^{★} (A \lor B) . \end{matrix}$ Moreover, $S^{★} (A) \geq S (A) (\forall A \in L^{X})$ holds obviously. □

By Lemma 3.3.1, we can construct an (L, M)-fuzzy cotopology from an (L, M)-fuzzy closure system.

Theorem 3.3.2. Let $(X, S)$ be an (L, M)-fuzzy closure space and define $\bar{S} : L^{X} ⟶ M$ as follows: $\forall A \in L^{X}, \bar{S} (A) = ⋁_{⋀_{j \in J} A_{j} = A} ⋀_{j \in J} S^{★} (A_{j}) .$ Then $\bar{S}$ is an (L, M)-fuzzy cotopology on X. Moreover, $\bar{S} (A) \geq S^{★} (A) (\forall A \in L^{X})$ .

Proof. By the definition of $\bar{S}$ , it follows that $\bar{S} (A) \geq S^{★} (A) \geq S (A)$ . Then it suffices to verify that $\bar{S}$ satisfies (LMT1)–(LMT3). Indeed,

(LMT1) Since $\bar{S} (A) \geq S (A)$ , by (LMS1), we have $\bar{S} (⊥_{L}^{X}) = \bar{S} (⊤_{L}^{X}) = ⊤_{M}$ .

(LMT2) For {A_j ∣ j ∈ J} ⊆ L^X, we have $\begin{matrix} ⋀_{j \in J} \bar{S} (A_{j}) & = & ⋀_{j \in J} ⋁_{⋀_{k \in K_{j}} A_{k, j} = A_{j}} ⋀_{k \in K_{j}} S^{★} (A_{k, j}) \\ = & ⋁_{f \in \prod_{j \in J} A_{j}} ⋀_{j \in J} ⋀_{B \in f (j)} S^{★} (B) \\ (by CD law, where \\ A_{j} = {{A_{k, j}}_{k \in K_{j}} ∣ ⋀_{k \in K_{j}} A_{k, j} = A_{j}}) \\ \leq & ⋁_{⋀_{m \in M} B_{m} = ⋀_{j \in J} A_{j}} ⋀_{m \in M} S^{★} (B_{m}) \\ = & \bar{S} (⋀_{j \in J} A_{j}) . \end{matrix}$

(LMT3) For each A, B ∈ L^X, it follows Lemma 3.3.1 that $\begin{matrix} \bar{S} (A) \land \bar{S} (B) \\ = & ⋁_{⋀_{j \in J} A_{j} = A} ⋀_{j \in J} S^{★} (A_{j}) \land ⋁_{⋀_{k \in K} B_{k} = B} ⋀_{k \in K} S^{★} (B_{k}) \\ = & ⋁_{⋀_{j \in J} A_{j} = A} ⋁_{⋀_{k \in K} B_{k} = B} ⋀_{j \in J} ⋀_{k \in K} S^{★} (A_{j}) \land S^{★} (B_{k}) \\ \leq & ⋁_{⋀_{j \in J} A_{j} = A} ⋁_{⋀_{k \in K} B_{k} = B} ⋀_{j \in J} ⋀_{k \in K} S^{★} (A_{j} \lor B_{k}) \\ \leq & ⋁_{⋀_{j \in J} ⋀_{k \in K} (A_{j} \lor B_{k}) = A \lor B} ⋀_{j \in J} ⋀_{k \in K} S^{★} (A_{j} \lor B_{k}) \end{matrix}$

$\begin{matrix} \leq & ⋁_{⋀_{m \in M} C_{m} = A \lor B} ⋀_{m \in M} S^{★} (C_{m}) \\ = & \bar{S} (A \lor B) . \end{matrix}$ This shows that $\bar{S}$ is an (L, M)-fuzzy cotopology on X. □

Theorem 3.3.3. If $φ : (X, S_{X}) ⟶ (Y, S_{Y})$ is a continuous mapping between (L, M)-fuzzy closure spaces, then $φ : (X, {\bar{S}}_{X}) ⟶ (Y, {\bar{S}}_{Y})$ is a continuous mapping between (L, M)-fuzzy cotopological spaces.

Proof. Since $φ : (X, S_{X}) ⟶ (Y, S_{Y})$ is continuous, we have $\forall A \in L^{Y}, S_{X} (φ_{L}^{\leftarrow} (A)) \geq S_{Y} (A) .$ Further, by the definition of $\bar{S}$ , we obtain $\begin{matrix} {\bar{S}}_{Y} (A) \\ = & ⋁_{⋀_{j \in J} A_{j} = A} ⋀_{j \in J} S_{Y}^{★} (A_{j}) \\ = & ⋁_{⋀_{j \in J} A_{j} = A} ⋀_{j \in J} ⋁_{\lor_{k \in F} B_{j, k} = A_{j}} \land_{k \in F} S_{Y} (B_{j, k}) \\ \leq & ⋁_{⋀_{j \in J} A_{j} = A} ⋀_{j \in J} ⋁_{\lor_{k \in F} B_{j, k} = A_{j}} \land_{k \in F} S_{X} (φ_{L}^{\leftarrow} (B_{j, k})) \\ \leq & ⋁_{⋀_{j \in J} A_{j} = A} ⋀_{j \in J} ⋁_{\lor_{k \in F} B_{j, k} = A_{j}} \land_{k \in F} S_{X}^{★} (φ_{L}^{\leftarrow} (B_{j, k})) \\ \leq & ⋁_{⋀_{j \in J} A_{j} = A} ⋀_{j \in J} ⋁_{\lor_{k \in F} B_{j, k} = A_{j}} S_{X}^{★} (\lor_{k \in F} φ_{L}^{\leftarrow} (B_{j, k})) \end{matrix}$ $\begin{matrix} = & ⋁_{⋀_{j \in J} A_{j} = A} ⋀_{j \in J} S_{X}^{★} (φ_{L}^{\leftarrow} (A_{j})) \\ \leq & ⋁_{⋀_{j \in J} φ_{L}^{\leftarrow} (A_{j}) = φ_{L}^{\leftarrow} (A)} ⋀_{j \in J} S_{X}^{★} (φ_{L}^{\leftarrow} (A_{j})) \\ \leq & ⋁_{⋀_{j \in J} B_{j} = φ_{L}^{\leftarrow} (A)} ⋀_{j \in J} S_{X}^{★} (B_{j}) \\ = & {\bar{S}}_{X} (φ_{L}^{\leftarrow} (A)) . \end{matrix}$ This proves that $φ : (X, {\bar{S}}_{X}) ⟶ (Y, {\bar{S}}_{Y})$ is continuous. □

By Theorems 3.3.2 and 3.3.3, we obtain a functor $𝔾$ as follows: $𝔾: {\begin{matrix} LMFC \to LMFT \\ (X, S) \mapsto (X, \bar{S}) \\ ϕ \mapsto ϕ . \end{matrix}$

Let us denote the inclusion functor from LMFT to LMFS by $𝕀_{2}$ . Then we have the following result:

Theorem 3.3.4. $(𝕀_{2}, 𝔾)$ is an adjunction between LMFT and LMFS. Moreover, $𝔾$ is a left inverse functor of $𝕀_{2}$ .

Proof. It suffices to verify that

$𝕀_{2} \circ 𝔾 ((X, S)) \leq (X, S)$ for each (L, M)-fuzzy closure space $(X, S)$ , i.e., $\bar{S} (A) \geq S (A)$ for each A ∈ L^X.

$𝔾 \circ 𝕀_{2} ((X, τ)) = (X, τ)$ for each (L, M)-fuzzy cotopological space (X, τ), i.e., $\bar{τ} (A) = τ (A)$ for each A ∈ L^X.

For (1), it follows immediately from Lemma 3.3.1 and Theorem 3.3.2.

For (2), take each A ∈ L^X. Then $\bar{τ} (A) = ⋁_{⋀_{j \in J} A_{j} = A} ⋀_{j \in J} ⋁_{\lor_{i \in F_{j}} A_{i, j} = A_{j}} \land_{i \in F_{j}} τ (A_{i, j}) .$ On one hand, let {A_j ∣ j ∈ J} = {A} and {A_i,j ∣ i ∈ F_j} = {A}. Then $\bar{τ} (A) = ⋁_{⋀_{j \in J} A_{j} = A} ⋀_{j \in J} ⋁_{\lor_{i \in F_{j}} A_{i, j} = A_{j}} \land_{i \in F_{j}} τ (A_{i, j}) \geq τ (A) .$ On the other hand, $\begin{matrix} \bar{τ} (A) & = & ⋁_{⋀_{j \in J} A_{j} = A} ⋀_{j \in J} ⋁_{\lor_{i \in F_{j}} A_{i, j} = A_{j}} \land_{i \in F_{j}} τ (A_{i, j}) \\ \leq & ⋁_{⋀_{j \in J} A_{j} = A} ⋀_{j \in J} ⋁_{\lor_{i \in F_{j}} A_{i, j} = A_{j}} τ (\lor_{i \in F_{j}} A_{i, j}) \\ = & ⋁_{⋀_{j \in J} A_{j} = A} ⋀_{j \in J} ⋁_{\lor_{i \in F_{j}} A_{i, j} = A_{j}} τ (A_{j}) \\ = & ⋁_{⋀_{j \in J} A_{j} = A} ⋀_{j \in J} τ (A_{j}) \\ \leq & ⋁_{⋀_{j \in J} A_{j} = A} τ (⋀_{j \in J} A_{j}) = τ (A), \end{matrix}$ as desired. □

Corollary 3.3.5. LMFT is a coreflective subcategory of LMFS.

3.4 Coreflectivity of ELFT in LFT

In [6], the authors introduced the concept of extensional L-fuzzy ((L, L)-fuzzy) topological spaces and showed the resulting category is a coreflective subcategory of the category of L-fuzzy topological spaces. The similar way can be used to deal with L-fuzzy cotopological spaces. In this subsection, we will adopt a new way to show the coreflectivity of extensional L-fuzzy cotopological spaces in L-fuzzy cotopological spaces based on a completely distributive lattice L.

Now let us give the concept of extensional L-fuzzy cotopological spaces, which is equivalent to the notion of extensional L-fuzzy topological spaces in [6] when L is a completely distributive De Morgan algebra.

Definition 3.4.1. An L-fuzzy cotopology τ on X is called extensional if it satisfies (ELFT) $τ (A) \land E_{L} (A, B) \leq τ (B)$ . For an extensional L-fuzzy cotopology τ on X, the pair (X, τ) is called an extensional L-fuzzy cotopological space.

The full subcategory of LFT, consisting of extensional L-fuzzy cotopological spaces, is denoted by ELFT.

Theorem 3.4.2. Let (X, τ) be an L-fuzzy cotopological space and define τ^* : L^X ⟶ L by $\forall A \in L^{X}, τ^{*} (A) = ⋁_{B \in L^{X}} (τ (B) \land E_{L} (B, A)) .$ Then τ^* is an extensional L-fuzzy cotopology on X.

Proof. It suffices to show that τ^* satisfies (LMT1)–(LMT3) and (ELFT). Indeed,

(LMT1) Straightforward.

(LMT2) Take each A, B ∈ L^X. Then $\begin{matrix} τ^{*} (A) \land τ^{*} (B) & = & ⋁_{C \in L^{X}} (τ (C) \land E_{L} (C, A)) \land \\ ⋁_{D \in L^{X}} (τ (D) \land E_{L} (D, B)) \end{matrix}$

$\begin{matrix} = & ⋁_{C \in L^{X}} ⋁_{D \in L^{X}} (τ (C) \land τ (D) \land \\ E_{L} (C, A) \land E_{L} (D, B)) \\ \leq & ⋁_{C \in L^{X}} ⋁_{D \in L^{X}} (τ (C \lor D) \land \\ E_{L} (C \lor D, A \lor B)) (by Lemma 2.2) \\ \leq & ⋁_{E \in L^{X}} (τ (E) \land E_{L} (E, A \lor B)) \\ = & τ^{*} (A \lor B) . \end{matrix}$

(LMT3) Take each {A_i} _i∈I ⊆ L^X. Then $\begin{matrix} ⋀_{i \in I} τ^{*} (A_{i}) & = & ⋀_{i \in I} ⋁_{B_{i} \in L^{X}} (τ (B_{i}) \land E_{L} (B_{i}, A_{i})) \\ = & ⋁_{f \in \prod_{i \in I} L^{X}} ⋀_{i \in I} (τ (f (i)) \land E_{L} (f (i), A_{i})) \\ = & ⋁_{f \in \prod_{i \in I} L^{X}} ((⋀_{i \in I} τ (f (i))) \land \\ (⋀_{i \in I} E_{L} (f (i), A_{i}))) \\ \leq & ⋁_{f \in \prod_{i \in I} L^{X}} (τ (⋀_{i \in I} f (i)) \land \\ E_{L} (⋀_{i \in I} f (i), ⋀_{i \in I} A_{i})) (by Lemma 2.2) \\ \leq & ⋁_{C \in L^{X}} (τ (C) \land E_{L} (C, ⋀_{i \in I} A_{i})) \\ = & τ^{*} (⋀_{i \in I} A_{i}) . \end{matrix}$

(ELFT) Take each A ∈ L^X. Then $\begin{matrix} τ^{*} (A) \land E_{L} (A, B) \\ = & (⋁_{C \in L^{X}} (τ (C) \land E_{L} (C, A))) \land E_{L} (A, B) \\ = & ⋁_{C \in L^{X}} (τ (C) \land E_{L} (C, A) \land E_{L} (A, B)) \\ \leq & ⋁_{C \in L^{X}} (τ (C) \land E_{L} (C, B)) (by Lemma 2.2) \\ = & τ^{*} (B) . \end{matrix}$ This shows τ^* is an extensional L-fuzzy cotopology on X. □

Theorem 3.4.3. If φ : (X, τ_X) ⟶ (Y, τ_Y) is a continuous mapping between L-fuzzy topological spaces, then $φ : (X, τ_{X}^{*}) ⟶ (Y, τ_{Y}^{*})$ is a continuous mapping between extensional L-fuzzy topological spaces.

Proof. Since φ : (X, τ_X) ⟶ (Y, τ_Y) is continuous, it follows that $\forall B \in L^{Y}, τ_{X} (φ_{L}^{\leftarrow} (B)) \geq τ_{Y} (B) .$ Then for A ∈ L^Y, we have $\begin{matrix} τ_{Y}^{*} (A) & = & ⋁_{B \in L^{Y}} (τ_{Y} (B) \land E_{L} (B, A)) \\ \leq & ⋁_{B \in L^{Y}} (τ_{X} (φ_{L}^{\leftarrow} (B)) \land E_{L} (B, A)) \\ \leq & ⋁_{B \in L^{Y}} (τ_{X} (φ_{L}^{\leftarrow} (B)) \land E_{L} (φ_{L}^{\leftarrow} (B), φ_{L}^{\leftarrow} (A))) \\ \leq & ⋁_{C \in L^{X}} (τ_{X} (C) \land E_{L} (C, φ_{L}^{\leftarrow} (A))) \\ = & τ_{X}^{*} (φ_{L}^{\leftarrow} (A)), \end{matrix}$ as desired. □

By Theorems 3.4.2 and 3.4.3, we obtain a functor $𝔾^{*}$ as follows: $𝔾*: {\begin{matrix} LFT \to ELFT \\ (X, τ) \mapsto (X, τ *) \\ ϕ \mapsto ϕ . \end{matrix}$

Since ELFT is a full subcategory of LFT, we denote the inclusion functor from ELFT to LFT by $𝕀_{2}^{*}$ . Then we have the the following result:

Theorem 3.4.4. $(𝕀_{2}^{*}, 𝔾^{*})$ is an adjunction between ELFT and LFT. Moreover, $𝔾^{*}$ is a left inverse functor of $𝕀_{2}^{*}$ .

Proof. It suffices to verify that

$𝕀_{2}^{*} \circ 𝔾^{*} ((X, τ)) \leq (X, τ)$ for each L-fuzzy cotopological space (X, τ), i.e., τ^* (A) ≥ τ (A) for each A ∈ L^X.

$𝔾^{*} \circ 𝕀_{2}^{*} ((X, τ_{e})) = (X, τ_{e})$ for each extensional L-fuzzy cotopological space (X, τ_e), i.e., $τ_{e}^{*} (A) = τ_{e} (A)$ for each A ∈ L^X.

For (1), take each A ∈ L^X. Then

τ^{*} (A) = ⋁_{B \in L^{X}} (τ (B) \land E_{L} (B, A)) \geq τ (A) \land E_{L} (A, A) = τ (A) .

For (2), take each A ∈ L^X. Then $τ_{e}^{*} (A) = ⋁_{B \in L^{X}} (τ_{e} (B) \land E_{L} (B, A)) .$ By (1), we get $τ_{e}^{*} (A) \geq τ_{e} (A)$ . Since τ_e is extensional, we also have $τ_{e}^{*} (A) = ⋁_{B \in L^{X}} (τ_{e} (B) \land E_{L} (B, A)) \leq ⋁_{B \in L^{X}} τ_{e} (A) = τ_{e} (A),$ as desired. □

Corollary 3.4.5. ELFT is a coreflective subcategory of LFT.

4 Conclusions

In this paper, we mainly used a categorical tool to demonstrate the relationship between (L, M)-fuzzy closure systems and (L, M)-fuzzy convex structures, and the relationship between (L, M)-fuzzy closure systems and (L, M)-fuzzy cotopologies. We showed that the categories of (L, M)-fuzzy convex spaces and (L, M)-fuzzy cotopological spaces are both coreflective subcategories of the category of (L, M)-fuzzy closure spaces. Further, by equipping an extensional condition on L-fuzzy convex structures and L-fuzzy cotopologies, we introduced extensional L-fuzzy convex structures and extensional L-fuzzy cotopologies, and established their categorical relationship with L-fuzzy convex structures and L-fuzzy cotopologies, respectively.

Compared with (L, M)-fuzzy convex structures and (L, M)-fuzzy cotopologies, the notion of (L, M)-fuzzy Alexandroff topological structures is a more strong mathematical structure since it is closed under arbitrary meets and arbitrary unions. Following the idea in this paper, we will consider the relationship between (L, M)-fuzzy convex structures, (L, M)-fuzzy cotopological structures and (L, M)-fuzzy Alexandroff topological structures in the future.

Footnotes

Acknowledgments

The authors would like to express their sincere thanks to the anonymous reviewers and the editor for their careful reading and constructive comments. The first author is supported by the Natural Science Foundation of China (No. 11871097) and the funding from Heilongjiang Education Department (Nos.1354ZD009 and 1352MSYYB008). The second author is supported by the Natural Science Foundation of China (No. 11701122), the Natural Science Foundation of Guangdong Province (No. 2017A030310584) and Beijing Institute of Technology Research Fund Program for Young Scholars (2019CX04111).

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