M -fuzzifying cotopological spaces and M -fuzzifying convex spaces as M -fuzzifying closure spaces

Abstract

In this paper, the relations between M-fuzzifying cotopologies and M-fuzzifying closure systems, and the relations between M-fuzzifying convex structures and M-fuzzifying closure systems are studied, respectively. A categorical approach is provided to present these relations. It is shown that the category of M-fuzzifying cotopological spaces and the category of M-fuzzifying convex spaces are both bicoreflective subcategories of the category of M-fuzzifying closure spaces.

Keywords

Bicoreflective subcategory

1 Introduction

By a closure system, we usually mean a subset of the power set 2^X which satisfies certain conditions [5]. With the development of fuzzy set theory, closure systems are generalized to different kinds of fuzzy closure systems. Biacino and Gerla [3, 4] defined a kind of fuzzy closure systems by extending 2^X to I^X. Moreover, Kim [16] proved that the lattice of fuzzy closure systems is isomorphic to the lattice of fuzzy closure operators for the respective notions defined in [4]. Later, Bělohlávek [2] outlined a general theory of fuzzy closure operators and fuzzy closure systems (L_K-closure systems), and showed the existence of a one-to-one correspondence between his fuzzy closure operators and fuzzy closure systems.

However, the above-mentioned fuzzy closure systems are crisp families of fuzzy subsets on a universe set X. Following the idea of [11 , 35], Fang [7] proposed the concept of L-fuzzy closure systems. Unluckily, L-fuzzy closure operators in [7] are not equivalent to L-fuzzy closure systems. Later, Luo and Fang [19] introduced the concepts of fuzzifying closure systems and fuzzifying closure operators as a generalization of Birkhoff’s closure operators. Moreover, a one-to-one correspondence between the notions was established. Actually, the relations between closure systems and closure operators are investigated extensively [6 , 40].

In this paper, we will focus on another important aspect of closure systems. As we all know, a closure system is not only a generalization of a cotopology [14] by relaxing the condition that it is closed under finite unions, but also a generalization of a convex structure [36] by relaxing the condition that it is closed under directed unions. In a categorical sense, the category of cotopological spaces and the category of convex spaces are both subcategories of the category of closure spaces. In [21 , 38], convex structures were generalized to the fuzzy case and were widely studied. In the framework of fuzzy topology, categorical relations between different kinds of spatial structures are investigated extensively [12 , 22–24]. Motivated by this, we will provide a categorical approach to investigate the relations between different kinds of spatial structures. In this paper, we will first generalize fuzzifying closure systems in [19] to M-fuzzifying closure systems. Then we attempt to show that M-fuzzifying cotopologies [10, 39] and M-fuzzifying convex structures [33] can be included in the framework of M-fuzzifying closure systems [19] from a categorical aspect.

2 Preliminaries

A complete lattice M is said to be completely distributive if it satisfies the completely distributive law (CD law, in short), i.e., $⋁_{i \in I} ⋀_{j \in J_{i}} a_{i, j} = ⋀_{f \in \prod_{i \in I} J_{i}} ⋁_{i \in I} a_{i, f (i)}$ or $⋀_{i \in I} ⋁_{j \in J_{i}} a_{i, j} = ⋁_{f \in \prod_{i \in I} J_{i}} ⋀_{i \in I} a_{i, f (i)}$ holds for all X_i = {a_i,j ∣ j ∈ J_i} ⊆2^M (i ∈ I).

Throughout this paper, let M denote a completely distributive lattice. The smallest element and the largest element in M are denoted by ⊥ and ⊤, respectively. For a, b ∈ M, we say that a is wedge below b in M, in symbols a ≺ b, if for every subset D ⊆ L, ⋁D ⩾ b implies d ⩾ a for some d ∈ D. A complete lattice M is completely distributive if and only if b = ⋁ {a ∈ M ∣ a ≺ b} for each b ∈ M. For a nonempty set X, 2^X denotes the powerset of X. Let ${A_{j}}_{j \in J} \overset{dir}{\subseteq} 2^{X}$ denote that {A_j} _j∈J is a directed subset of 2^X, which means for each B, C ∈ {A_j} _j∈J, there exists D ∈ {A_j} _j∈J such that B ⊆ D and C ⊆ D. Let ${⋃^{dir}}_{j \in J} A_{j}$ denote the directed union of ${A_{j}}_{j \in J} \overset{dir}{\subseteq} 2^{X}$ .

Let X, Y be two nonempty sets and φ : X ⟶ Y be a mapping. Define φ^→ : 2^X ⟶ 2^Y and φ^← : 2^Y ⟶ 2^X as follows: $\begin{matrix} \forall A \in 2^{X}, φ^{\to} (A) & = & {φ (x) ∣ x \in A}; \\ \forall B \in 2^{Y}, φ^{\leftarrow} (B) & = & {x ∣ φ (x) \in B} . \end{matrix}$

Definition 2.1. [19] An M-fuzzifying closure system on X is a mapping $S : 2^{X} ⟶ M$ which satisfies:

$S (\emptyset) = S (X) = ⊤$ ;

$S (⋂_{i \in I} A_{i}) ⩾ ⋀_{i \in I} S (A_{i})$ .

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

A mapping $φ : (X, S_{X}) ⟶ (Y, S_{Y})$ between M-fuzzifying closure spaces is called continuous if it satisfies $\forall B \in 2^{Y}, S_{X} (φ^{\leftarrow} (B)) ⩾ S_{Y} (B) .$

It is easy to check that all M-fuzzifying closure spaces as objects and all continuous mappings as morphisms form a category, denoted by M-FYS.

As we all know, topologies and cotopologies are dual and equivalent. In the situation of M-fuzzifying topology theory, M-fuzzifying topologies are also equivalent to M-fuzzifying cotopologies. In the sequel, we will adopt the concept of M-fuzzifying cotopologies.

Definition 2.2. [10, 39] An M-fuzzifying cotopology on X is a mapping τ : 2^X ⟶ M which satisfies:

τ (∅) = τ (X) = ⊤;

τ (⋂ _i∈IA_i) ⩾ ⋀ _i∈Iτ (A_i);

τ (A ∪ B) ⩾ τ (A) ∧ τ (B).

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

A mapping φ : (X, τ_X) ⟶ (Y, τ_Y) between M-fuzzifying cotopological spaces is called continuous if it satisfies $\forall B \in 2^{Y}, τ_{X} (φ^{\leftarrow} (B)) ⩾ τ_{Y} (B) .$

It is easy to check that all M-fuzzifying cotopological spaces as objects and all continuous mappings as morphisms form a category, denoted by M-FYT.

Definition 2.3. [33] An M-fuzzifying convex structure on X is a mapping $C : 2^{X} ⟶ M$ which satisfies:

$C (\emptyset) = C (X) = ⊤$ ;

$C (⋂_{i \in I} A_{i}) ⩾ ⋀_{i \in I} C (A_{i})$ ;

$C ({⋃^{dir}}_{j \in J} A_{j}) ⩾ ⋀_{j \in J} C (A_{j})$ .

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

A mapping $φ : (X, C_{X}) ⟶ (Y, C_{Y})$ between M-fuzzifying convex spaces is called convexity preserving (CP, in short) if it satisfies $\forall B \in 2^{Y}, C_{X} (φ^{\leftarrow} (B)) ⩾ C_{Y} (B) .$

It is easy to check that all M-fuzzifying convex spaces as objects and all convexity preserving mappings as morphisms form a category, denoted by M-FYC.

Actually, it is easy to see that M-FYT and M-FYC are both full subcategories of M-FYS.

Definition 2.4. [1, 27] A category C is called a topological category over Set with respect to the usual forgetful functor from C to Set if it satisfies (TC1)–(TC3).

(TC1) Existence of final structures: For any set X, any class J, and family ((X_j, ξ_j)) _j∈J of C-object and any family (f_j : X_j ⟶ X) _j∈J of mappings, there exists a unique C-structure ξ on X which is final with respect to the sink (f_j : (X_j, ξ_j) ⟶ X)) _j∈J, this means that for a C-object (Y, η), a mapping g : (X, ξ) ⟶ (Y, η) is a C-morphism iff for all j ∈ J, g ∘ f_j : (X_j, ξ_j) ⟶ (Y, η) is a C-morphism.

(TC2) Fibre-smallness: For any set X, the C-fibre of X, i.e., the class of all C-structures on X, which we denote by C(X), is a set.

(TC3) Terminal separator property: For any set X with cardinality at most one, there exists exactly one C-object with underlying set X (i.e. there exists exactly one C-structure on X).

For more notions related to category theory we refer to [1] and [27].

3 M-FYT as M-FYS

An M-fuzzifying closure system is just an M-fuzzifying cotopology by relaxing (MFYT3). In this sense, M-fuzzifying closure systems are generalizations of M-fuzzifying cotopologies. In this section, we will investigate the relations between M-fuzzifying closure systems and M-fuzzifying cotopologies from a categorical aspect.

Lemma 3.1.Let $(X, S)$ be an M-fuzzifying closure space and define $S^{★} : 2^{X} ⟶ M$ as follows: $\forall A \in 2^{X}, S^{★} (A) = ⋁_{\cup_{j \in F} A_{j} = A} \land_{j \in F} S (A_{j}),$ where F is finite. Then $S^{★}$ satisfies (MFYT3). Moreover, $S^{★} (A) ⩾ S (A) (\forall A \in 2^{X})$ .

Proof. Take any A, B ∈ 2^X. Then $\begin{matrix} S^{★} (A) \land S^{★} (B) \\ = ⋁_{\cup_{i \in F_{1}} A_{i} = A} \land_{i \in F_{1}} S (A_{i}) \land ⋁_{\cup_{j \in F_{2}} B_{j} = B} \land_{j \in F_{2}} S (B_{j}) \\ = ⋁_{\cup_{i \in F_{1}} A_{i} = A} ⋁_{\cup_{j \in F_{2}} B_{j} = B} \land_{i \in F_{1}} \land_{j \in F_{2}} (S (A_{i}) \land S (B_{j})) \\ ⩽ ⋁_{(\cup_{i \in F_{1}} A_{i}) \cup (\cup_{j \in F_{2}} B_{j}) = A \cup B} \land_{i \in F_{1}} \land_{j \in F_{2}} (S (A_{i}) \land S (B_{j})) \\ ⩽ ⋁_{\cup_{k \in F} C_{k} = A \cup B} \land_{k \in F} S (C_{k}) \\ = S^{★} (A \cup B) . \end{matrix}$

Moreover, $S^{★} (A) ⩾ S (A) (\forall A \in 2^{X})$ holds obviously. □

Based on Lemma 3.1, we can construct anM-fuzzifying cotopology from an M-fuzzifying closure system.

Theorem 3.2. Let $(X, S)$ be an M-fuzzifying closure space and define $\bar{S} : 2^{X} ⟶ M$ as follows: $\forall A \in 2^{X}, \bar{S} (A) = ⋁_{⋂_{j \in J} A_{j} = A} ⋀_{j \in J} S^{★} (A_{j}) .$

Then $\bar{S}$ is an M-fuzzifying cotopology on X. Moreover, $\bar{S} (A) ⩾ S (A) (\forall A \in 2^{X})$ .

Proof. By the definition of $\bar{S}$ , it follows that $\bar{S} (A) ⩾ S^{★} (A) ⩾ S (A)$ . Then it suffices to verify that $\bar{S}$ satisfies (MFYT1)–(MFYT3). In fact,

(MFYT1) Since $\bar{S} (A) ⩾ S (A)$ , by (MFYS1), we have $\bar{S} (\emptyset) = \bar{S} (X) = ⊤$ .

(MFYT2) For{A_j ∣ j ∈ J} ⊆2^X, we have $\begin{matrix} ⋀_{j \in J} \bar{S} (A_{j}) & = & ⋀_{j \in J} ⋁_{⋂_{k \in K_{j}} A_{k, j} = A_{j}} ⋀_{k \in K_{j}} S^{★} (A_{k, j}) \\ = & ⋁_{F \in \prod_{j \in J} A_{j}} ⋀_{j \in J} ⋀_{B \in F (j)} S^{★} (B) \\ (by CD law, where \\ A_{j} = {{A_{k, j}}_{k \in K_{j}} ∣ ⋂_{k \in K_{j}} A_{k, j} = A_{j}}) \\ ⩽ & ⋁_{⋂_{m \in M} B_{m} = ⋂_{j \in J} A_{j}} ⋀_{m \in M} S^{★} (B_{m}) \\ = & \bar{S} (⋂_{j \in J} A_{j}) . \end{matrix}$

(MFYT3) For each A, B ∈ 2^X, by Lemma 3.1, it follows that $\begin{matrix} \bar{S} (A) \land \bar{S} (B) \\ = ⋁_{⋂_{j \in J} A_{j} = A} ⋀_{j \in J} S^{★} (A_{j}) \land ⋁_{⋂_{k \in K} B_{k} = B} ⋀_{k \in K} S^{★} (B_{k}) \\ = ⋁_{⋂_{j \in J} A_{j} = A} ⋁_{⋂_{k \in K} B_{k} = B} ⋀_{j \in J} ⋀_{k \in K} S^{★} (A_{j}) \land S^{★} (B_{k}) \\ ⩽ ⋁_{⋂_{j \in J} A_{j} = A} ⋁_{⋂_{k \in K} B_{k} = B} ⋀_{j \in J} ⋀_{k \in K} S^{★} (A_{j} \cup B_{k}) \\ ⩽ ⋁_{⋃_{j \in J} ⋃_{k \in K} A_{j} \cup B_{k} = A \cup B} ⋀_{j \in J} ⋀_{k \in K} S^{★} (A_{j} \cup B_{k}) \\ ⩽ ⋁_{⋂_{m \in M} C_{m} = A \cup B} ⋀_{m \in M} S^{★} (C_{m}) \\ = \bar{S} (A \cup B) . \end{matrix}$

This shows that $\bar{S}$ is an M-fuzzifying cotopology on X. □

Lemma 3.3. Let (Y, τ_Y) be an M-fuzzifying cotopological space and φ : Y ⟶ X be a mapping. Define τ_X : 2^X ⟶ M as follows: $\forall A \in 2^{X}, τ_{X} (A) = τ_{Y} (φ^{\leftarrow} (A)) .$

Then τ_X is an M-fuzzifying cotopology on X.

Proof. It suffices to show that τ_X satisfies (MFYT1)–(MFYT3).

(MFYT1) τ_X (∅) = τ_Y (φ^← (∅)) = τ_Y (∅) = ⊤ and τ_X (X) = τ_Y (φ^← (X)) = τ_Y (Y) =⊤.

(MFYT2) For {A_j ∣ j ∈ J} ⊆2^X, it follows that $\begin{matrix} τ_{X} (⋂_{j \in J} A_{j}) & = & τ_{Y} (φ^{\leftarrow} (⋂_{j \in J} A_{j})) = τ_{Y} (⋂_{j \in J} φ^{\leftarrow} (A_{j})) \\ ⩾ & ⋀_{j \in J} τ_{Y} (φ^{\leftarrow} (A_{j})) \\ = & ⋀_{j \in J} τ_{X} (A_{j}) . \end{matrix}$

(MFYT3) Take any A, B ∈ 2^X. Then $\begin{matrix} τ_{X} (A \cup B) & = & τ_{Y} (φ^{\leftarrow} (A \cup B)) \\ = & τ_{Y} (φ^{\leftarrow} (A) \cup φ^{\leftarrow} (B)) \\ ⩾ & τ_{Y} (φ^{\leftarrow} (A)) \land τ_{Y} (φ^{\leftarrow} (B)) \\ = & τ_{X} (A) \land τ_{X} (B) . □ \end{matrix}$

Theorem 3.4. M-MFYT is a bicoreflective subcategory of M-MFYS.

Proof. Let $(X, S)$ be an M-fuzzifying closure space. By Theorem 3.2, we obtain $\bar{S}$ is an M-fuzzifying cotopology on X. Next we claim that ${id}_{X} : (X, \bar{S}) ⟶ (X, S)$ is the M-MFYT-bicoreflector.

For this it suffices to show:

${id}_{X} : (X, \bar{S}) ⟶ (X, S)$ is continuous.

For each M-fuzzifying cotopological space (Y, τ_Y) and each mapping φ : Y ⟶ X, the continuity of $φ : (Y, τ_{Y}) ⟶ (X, S)$ implies the continuity of $φ : (Y, τ_{Y}) ⟶ (X, \bar{S})$ .

For (1), by Theorem 3.2, we know for each A ∈ 2^X, it follows that $\bar{S} (A) ⩾ S (A)$ . This implies that ${id}_{X} : (X, \bar{S}) ⟶ (X, S)$ is continuous.

For (2), for each M-fuzzifying cotopological space (Y, τ_Y) and each mapping φ : Y ⟶ X, define τ_X as in Lemma 3.3. Then (X, τ_X) is an M-fuzzifying cotopological space. Since $φ : (Y, τ_{Y}) ⟶ (X, S)$ is continuous, we have $\forall A \in 2^{X}, τ_{X} (A) = τ_{Y} (φ^{\leftarrow} (A)) ⩾ S (A) .$

Further, by the definition of $\bar{S}$ , we obtain $\begin{matrix} \bar{S} (A) & = & ⋁_{⋂_{j \in J} A_{j} = A} ⋀_{j \in J} S^{★} (A_{j}) \\ ⩽ & ⋁_{⋂_{j \in J} A_{j} = A} ⋀_{j \in J} ⋁_{\cup_{k \in F} B_{j, k} = A_{j}} \land_{k \in F} S (B_{j, k}) \\ ⩽ & ⋁_{⋂_{j \in J} A_{j} = A} ⋀_{j \in J} ⋁_{\cup_{k \in F} B_{j, k} = A_{j}} \land_{k \in F} τ_{X} (B_{j, k}) \\ ⩽ & ⋁_{⋂_{j \in J} A_{j} = A} ⋀_{j \in J} ⋁_{\cup_{k \in F} B_{j, k} = A_{j}} τ_{X} (\cup_{k \in F} B_{j, k}) \\ = & ⋁_{⋂_{j \in J} A_{j} = A} ⋀_{j \in J} τ_{X} (A_{j}) \\ ⩽ & ⋁_{⋂_{j \in J} A_{j} = A} τ_{X} (⋂_{j \in J} A_{j}) \\ = & τ_{X} (A) \\ = & τ_{Y} (φ^{\leftarrow} (A)) . \end{matrix}$

This proves that $φ : (Y, τ_{Y}) ⟶ (X, \bar{S})$ is continuous. □

4 M-FYC as M-FYS

In this section, we will investigate the relations between M-fuzzifying closure systems and M-fuzzifying convex structures. By Definitions 2.1 and 2.3, it is easy to see that an M-fuzzifying closure system is exactly an M-fuzzifying convex structure by relaxing (MFYC3). In the sequel, we will investigate their categorical relations.

Theorem 4.1. Let $(X, S)$ be an M-fuzzifying closure space and define $S^{*} : 2^{X} ⟶ M$ as follows: $\forall A \in 2^{X}, S^{*} (A) = ⋁_{{⋃^{dir}}_{k \in K} A_{k} = A} ⋀_{k \in K} S (A_{k}) .$

Then $S^{*}$ is an M-fuzzifying convex structure on X. Moreover, $S^{*} (A) ⩾ S (A) (\forall A \in 2^{X})$ .

Proof. It suffices to show that $S^{*}$ satisfies (MFYC1)–(MFYC3). In fact,

(MFYC1) Since $S$ satisfies (MFYS1), it follows that $S^{*} (\emptyset) = S^{*} (X) = ⊤$ .

(MFYC2) Take {A_i ∣ i ∈ I} ⊆2^X. Then we prove $S^{*} (⋂_{i \in I} A_{i}) ⩾ ⋀_{i \in I} S^{*} (A_{i})$ , i.e., $\begin{matrix} ⋁_{{⋃^{dir}}_{k \in K} B_{k} = ⋂_{i \in I} A_{i}} ⋀_{k \in K} S (B_{k}) \\ ⩾ ⋀_{i \in I} ⋁_{{⋃^{dir}}_{j \in J_{i}} G_{i, j} = A_{i}} ⋀_{j \in J_{i}} S (G_{i, j}) . \end{matrix}$

Take any a ∈ M such that $a ≺ ⋀_{i \in I} ⋁_{{⋃^{dir}}_{j \in J_{i}} G_{i, j} = A_{i}} ⋀_{j \in J_{i}} S (G_{i, j}) .$

Then for each i ∈ I, there exists a directed set {G_i,j ∣ j ∈ J_i} ⊆2^X such that ${⋃^{dir}}_{j \in J_{i}} G_{i, j} = A_{i}$ and for each j ∈ J_i, $S (G_{i, j}) ⩾ a$ . Thus, $⋂_{i \in I} A_{i} = ⋂_{i \in I} ⋃_{j \in J_{i}} G_{i, j} = ⋃_{f \in \prod_{i \in I} J_{i}} ⋂_{i \in I} G_{i, f (i)}$ and $S (G_{i, f (i)}) ⩾ a$ . Since {G_i,j ∣ j ∈ J_i} is directed, it is trivial to check that ${⋂_{i \in I} G_{i, f (i)} ∣ f \in \prod_{i \in I} J_{i}}$ is directed. This implies that $\begin{matrix} ⋁_{{⋃^{dir}}_{k \in K} B_{k} = ⋂_{i \in I} A_{i}} ⋀_{k \in K} S (B_{k}) \\ ⩾ ⋀_{f \in \prod_{i \in I} J_{i}} S (⋂_{i \in I} G_{i, f (i)}) \\ ⩾ ⋀_{f \in \prod_{i \in I} J_{i}} ⋀_{i \in I} S (G_{i, f (i)}) \\ ⩾ a . \end{matrix}$

By the arbitrariness of a, we obtain $S^{*} (⋂_{i \in I} A_{i}) ⩾ ⋀_{i \in I} S^{*} (A_{i})$ , as desired.

(MFYC3) Take ${A_{j} ∣ j \in J} \overset{dir}{\subseteq} 2^{X}$ . Then we prove that $S^{*} ({⋃^{dir}}_{j \in J} A_{j}) ⩾ ⋀_{j \in J} S^{*} (A_{j})$ . That is to say, $\begin{matrix} ⋁_{{⋃^{dir}}_{k \in K} B_{k} = {⋃^{dir}}_{j \in J} A_{j}} ⋀_{k \in K} S (B_{k}) \\ ⩾ ⋀_{j \in J} ⋁_{{⋃^{dir}}_{i \in I_{j}} G_{j, i} = A_{j}} ⋀_{i \in I_{j}} S (G_{j, i}) . \end{matrix}$

Take any a ∈ M such that $a ≺ ⋀_{j \in J} ⋁_{{⋃^{dir}}_{i \in I_{j}} G_{j, i} = A_{j}} ⋀_{i \in I_{j}} S (G_{j, i}) .$

Then for each j ∈ J, there exists a directed set {G_j,i ∣ i ∈ I_j} such that ${⋃^{dir}}_{i \in I_{j}} G_{j, i} = A_{j}$ and for each i ∈ I_j, $S (G_{j, i}) ⩾ a$ . Put $C = {⋃^{dir}}_{j \in J} A_{j}$ . Then $C = {⋃^{dir}}_{j \in J} {⋃^{dir}}_{i \in I_{j}} G_{j, i}$ . Now define a mapping $σ : 2_{fin}^{C} ⟶ 2^{X}$ as follows: $\forall F \in 2_{fin}^{C}, σ (F) = ⋂ {G_{j, i} ∣ F \subseteq G_{j, i}},$ where $2_{fin}^{C}$ denotes the set of all finite subsets of C. Then it is easy to check that $C = ⋃_{F \in 2_{fin}^{C}} F = ⋃_{F \in 2_{fin}^{C}} σ (F) = ⋃_{F \in 2_{fin}^{C}} ⋂_{F \subseteq G_{j, i}} G_{j, i} .$

By the definition of σ, we know σ is order preserving. Since ${F ∣ F \in 2_{fin}^{C}}$ is directed, we know ${σ (F) ∣ F \in 2_{fin}^{C}}$ is directed. Thus, it follows that $S (σ (F)) = S (⋂_{F \subseteq G_{j, i}} G_{j, i}) ⩾ ⋀_{F \subseteq G_{j, i}} S (G_{j, i}) ⩾ a .$

This implies that $⋁_{{⋃^{dir}}_{k \in K} B_{k} = {⋃^{dir}}_{j \in J} A_{j}} ⋀_{k \in K} S (B_{k}) ⩾ ⋀_{F \subseteq G_{j, i}} S (G_{j, i}) ⩾ a .$

By the arbitrariness of a, we obtain $\begin{matrix} ⋁_{{⋃^{dir}}_{k \in K} B_{k} = {⋃^{dir}}_{j \in J} A_{j}} ⋀_{k \in K} S (B_{k}) \\ ⩾ ⋀_{j \in J} ⋁_{{⋃^{dir}}_{i \in I_{j}} G_{j, i} = A_{j}} ⋀_{i \in I_{j}} S (G_{j, i}), \end{matrix}$ as desired. □

Lemma 4.2. Let $(Y, C_{Y})$ be an M-fuzzifying convex space and φ : Y ⟶ X be a mapping. Define $C_{X} : 2^{X} ⟶ M$ as follows: $\forall A \in 2^{X}, C_{X} (A) = C_{Y} (φ^{\leftarrow} (A)) .$

Then $C_{X}$ is an M-fuzzifying convex structure on X.

Proof. It suffices to show that τ_X satisfies (MFYC1)–(MFYC3).

(MFYC1) $C_{X} (\emptyset) = C_{Y} (φ^{\leftarrow} (\emptyset)) = C_{Y} (\emptyset) = ⊤$ and $C_{X} (X) = C_{Y} (φ^{\leftarrow} (X)) = C_{Y} (Y) = ⊤$ .

(MFYC2) For {A_i ∣ i ∈ I} ⊆2^X, it follows that $\begin{matrix} C_{X} (⋂_{i \in I} A_{i}) & = & C_{Y} (φ^{\leftarrow} (⋂_{i \in I} A_{i})) = C_{Y} (⋂_{i \in I} φ^{\leftarrow} (A_{i})) \\ ⩾ & ⋀_{i \in I} C_{Y} (φ^{\leftarrow} (A_{i})) \\ = & ⋀_{i \in I} C_{X} (A_{i}) . \end{matrix}$

(MFYC3) For ${A_{j} ∣ j \in J} \overset{dir}{\subseteq} 2^{X}$ , it follows that $\begin{matrix} C_{X} ({⋃^{dir}}_{j \in J} A_{j}) & = & C_{Y} (φ^{\leftarrow} ({⋃^{dir}}_{j \in J} A_{j})) \\ = & C_{Y} ({⋃^{dir}}_{j \in J} φ^{\leftarrow} (A_{j})) \\ ⩾ & ⋀_{j \in J} C_{Y} (φ^{\leftarrow} (A_{j})) \\ = & ⋀_{j \in J} C_{X} (A_{j}) . \end{matrix}$

This shows that $C_{X}$ is an M-fuzzifying convex structure on X. □

Theorem 4.3. M-MFYC is a bicoreflective subcategory of M-MFYS.

Proof. Let $(X, S)$ be an M-fuzzifying closure space. By Theorem 4.1, we obtain $S^{*}$ is an M-fuzzifying convex structure on X. Next we claim that ${id}_{X} : (X, S^{*}) ⟶ (X, S)$ is the M-MFYC-bicoreflector.

For this it suffices to show:

${id}_{X} : (X, S^{*}) ⟶ (X, S)$ is continuous.

For each M-fuzzifying convex space $(Y, C_{Y})$ and each mapping φ : Y ⟶ X, if $φ : (Y, C_{Y}) ⟶ (X, S)$ is continuous, then $φ : (Y, C_{Y}) ⟶ (X, S^{*})$ is CP.

For (1), by Theorem 4.1, we know for each A ∈ 2^X, it follows that $S^{*} (A) ⩾ S (A)$ . This implies that ${id}_{X} : (X, S^{*}) ⟶ (X, S)$ is continuous.

For (2), for each M-fuzzifying convex space $(Y, C_{Y})$ and each mapping φ : Y ⟶ X, define $C_{X}$ as in Lemma 4.2. Then $(X, C_{X})$ is an M-fuzzifying convex space. Since $φ : (Y, C_{Y}) ⟶ (X, S)$ is continuous, we have $\forall A \in 2^{X}, C_{X} (A) = C_{Y} (φ^{\leftarrow} (A)) ⩾ S (A) .$

Further, by the definition of $S^{*}$ , we obtain $\begin{matrix} S^{*} (A) & = & ⋁_{{⋃^{dir}}_{j \in J} A_{j} = A} ⋀_{j \in J} S (A_{j}) \\ ⩽ & ⋁_{{⋃^{dir}}_{j \in J} A_{j} = A} ⋀_{j \in J} C_{X} (A_{j}) \\ ⩽ & ⋁_{{⋃^{dir}}_{j \in J} A_{j} = A} C_{X} ({⋃^{dir}}_{j \in J} A_{j}) \\ = & C_{X} (A) \\ = & C_{Y} (φ^{\leftarrow} (A)) . \end{matrix}$

This proves that $φ : (Y, C_{Y}) ⟶ (X, S^{*})$ is CP, as desired. □

In a categorical sense, the category M-FYT of M-fuzzifying cotopological spaces is a topological category over Set. In the final, we will show that the M-FYC of M-fuzzifying convex spaces is also a topological category over Set.

Theorem 4.4. The category M-FYC is topological over Set.

Proof. We first prove the existence of final structures. Let $((X_{λ}, C_{λ}))_{λ \in Λ}$ be a family of M-fuzzifying convex spaces and let X be a nonempty set. Let further $((φ_{λ} : (X_{λ}, C_{λ}) ⟶ X))_{λ \in Λ}$ be a sink. Define $C : 2^{X} ⟶ M$ by $C (A) = ⋀_{λ \in Λ} C_{λ} (φ_{λ}^{\leftarrow} (A)) .$

Since $φ_{λ}^{\leftarrow}$ preserves arbitrary meets and directed joins, we can verify that $C$ is an M-fuzzifying convex structure on X.

Let further $(Y, C_{Y})$ be an M-fuzzifying convex space and ψ : X ⟶ Y be a mapping. Assume that ψ ∘ φ_λ is CP for all λ ∈ Λ. We have for all B ∈ 2^Y, $\begin{matrix} C (ψ^{\leftarrow} (B)) & = & ⋀_{λ \in Λ} C_{λ} (φ_{λ}^{- 1} (ψ^{\leftarrow} (B))) \\ = & ⋀_{λ \in Λ} C_{λ} ((ψ \circ φ_{λ})^{\leftarrow} (B))) \\ ⩾ & ⋀_{λ \in Λ} C_{Y} (B) \\ = & C_{Y} (B) . \end{matrix}$

This implies that $ψ : (X, C) ⟶ (Y, C_{Y})$ is CP, as desired.

Secondly, the class of all M-fuzzifying convex structures on a fixed set X is a subset of 2^{(M^{(2^X)})}, which means that the M-FYC fibre of X is aset.

Finally, for a one point set X = {x}, there exists only one M-fuzzifying convex structure $C$ such that $C ({x}) = ⊤$ . Hence, M-FYC satisfies the terminal separator property. Therefore, M-FYC is a topological category in the sense of [27], that is, a well-fibred topological category in the terminology of [1]. □

5 Conclusions

In this paper, we studied the categorical relations between M-fuzzifying closure systems, M-fuzzifying cotopologies and M-fuzzifying convex structures. The category of M-fuzzifying cotopological spaces and the category of M-fuzzifying convex spaces are both bicoreflective subcategories of the category of M-fuzzifying closure spaces. Based on these conclusions, we know both M-fuzzifying cotopologies and M-fuzzifying convex structures can be treated as special cases of M-fuzzifying closure systems in a categorical sense.

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 Scientific Research Foundation for Introduced Talents of CUIT (KYTZ201631), the 2016-2017 Scientific Research Foundation of CUIT (CRF201611) and the National Natural Science Foundation of China (11371002).

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