Characterizations of M-fuzzifying convex spaces via Galois connections

Abstract

This paper introduces a special Galois connection combined with the wedge-below relation. Furthermore, by using this tool, it is shown that the category of M-fuzzifying betweenness spaces and the category of M-fuzzifying convex spaces are isomorphic and the category of arity-2 M-fuzzifying convex spaces can be embedded in the category of M-fuzzifying interval spaces as a reflective subcategory.

Keywords

Fuzzy convex structure fuzzy betweenness space fuzzy interval space arity-2 fuzzy convexity

1 Introduction

Convex sets widely exist in various research areas of mathematics, such as graphs, metric spaces, matroids, lattices and so on. With the deepening of research, the properties of convex sets are abstracted and an axiomatic convex structure is established. A family of subsets of a set X which is denoted by $C$ is called a convexity [21] provided that $C$ contains the empty set and the universal set, and it is closed under intersections and directed joins (equivalently, totally ordered joins). Then the pair $(X, C)$ is called a convex structure. The relationship between convex structures and matroids is very close. Matroids can be regard as convex structures with exchange law and convex sets are exactly the flat sets in matroids. Polyhedrons are typical convex sets. And betweenness is a relation on X to describe whether a point depends on a polyhedron. Arity-2 convex structures exist widely in various mathematical structures, like pre-ordered sets, Euclidean spaces and so on. Interval spaces abstracted from the intervals play an important role in convexity theory.

With the development of fuzzy mathematics, many structures have been combined with fuzzy set theory, such as fuzzy convergence [6 , 36], fuzzy topology [13 , 38], fuzzy order [22, 23] and so on. Similarly, the combination of convex structures and fuzzy set theory has also been greatly developed. There are generally the following research directions: M-fuzzifying convexities [2 , 33], L-convexities [1 , 34] and (L, M)-fuzzy convexities [8 , 39]. When studying the relationship between convexities and other structures, complement plays an important role. De Morgan algebra can be regarded as a kind of generalization of complement. Under the premise of De Morgan algebra, Shi and Li [17] showed the isomorphism between M-fuzzifying betweenness spaces and M-fuzzifying convex spaces. Xiu and Shi [33] proved that the category of arity-2 M-fuzzifying convex spaces is a reflective subcategory of the category of M-fuzzifying interval spaces. But De Morgan algebra is a very strong condition. In this paper, we will introduce LRG adjunctions combined with the wedge-below relation. By this tool, we use an M-fuzzifying convexity (resp., arity-2 M-fuzzifying convexity) to construct an M-fuzzifying betweenness (resp., M-fuzzifying interval operator). Then we will show the categorical relationship between M-fuzzifying interval spaces and arity-2 M-fuzzifying convex spaces.

This paper is organized as follows. In Section 2, we will recall some necessary concepts of lattices and M-fuzzifying convexities. In Section 3, we will discuss the categorical relationship between M-fuzzifying convex and M-fuzzifying betweenness spaces with a special Galois connection. In Section 4, with this Galois connection, we will consider the categorical relationship between arity-2 M-fuzzifying convex spaces and M-fuzzifying interval spaces.

2 Preliminaries

2.1 Completely distributive lattice

We recall the concepts of completely distributive lattices in this subsection. The pair (L, ≤) is called a lattice [5] provided that for every two elements a, b ∈ L, there exists a supremum a ∨ b and an infimum a ∧ b. A lattice (L, ≤) is complete if for any subset of L the supremum and infimum exist. The largest element and the smallest element in L are denoted by 1 and 0, respectively. For a preordered set P, S ⊆ P is directed provided that for any a, b ∈ S, there exists c ∈ S such that c ≥ a and c ≥ b. A directed set S is always denoted by S^↑.

The binary relation 〈 on L is defined as follows: for a, b ∈ L, a 〈 b if and only if for any D ⊆ L, b ≤ sup D implies that there exists d ∈ D such that a ≤ d [26]. β (b) = {a ∈ L ∣ a 〈 b} is called the greatest minimal family [26] of b. Moreover, the relation 〈^op in L is defined as follows: for a, b ∈ L, b 〈 ^opa if and only if for any D ⊆ L, the relation b ≥ inf D implies that there exists d ∈ D such that a ≥ d. α (b) = {a ∈ L ∣ b 〈 ^opa} is called the greatest maximal family [26] of b. In a completely distributive lattice L, α (b) and β (b) exist, and b = ⋁ β (b) = ⋀ α (b) [26]. If a 〈 b, then there exists c ∈ L such that a 〈 c 〈 b. If a ≤ b 〈 c ≤ d, then a 〈 d. So does 〈^op. Let L₁ and L₂ be two complete lattices. If a pair of mappings f : L₁ → L₂ and g : L₂ → L₁ (f : L₁ ⇄ L₂ : g for short) satisfies f (a) ≥ b if and only if g (b) ≥ a, then we call (f, g) an Galois connection.

Lemma 2.1. [[26]] Let L be a completely distributive lattice and {a_i ∣ i ∈ I} ⊆ L. Then

(1) β (⋁ _i∈Ia_i) = ⋃ _i∈Iβ (a_i), i.e., β is a union-preserving mapping.

(2) α (⋀ _i∈Ia_i) = ⋃ _i∈ωα (a_i), i.e., α is a ∧ -∪ mapping.

Definition 2.2. [16] Let A ∈ L^X, a ∈ L. Define A_[a] = {x ∈ X ∣ A (x) ≥ a} , A^[a] = {x ∈ X ∣ a ∉ α (A (x))} .

2.2 M-fuzzifying convexity, M-fuzzifying betweenness and interval operator

Let X be a nonempty set and $2_{fin}^{X}$ be the set {F ⊆ X ∣ |F| isfinite .}, then $F \in 2_{fin}^{X}$ or F ⊆ ^finX indicates that F is finite.

Definition 2.3. [19] A mapping $C : 2^{X} \to M$ is called an M-fuzzifying convexity on X if it satisfies the following conditions:

(MC1) $C (\emptyset) = 1 = C (X)$ ;

(MC2) If {A_i} _i∈I ⊆ 2^X is nonempty, then $C (⋂_{i \in I} A_{i}) \geq ⋀_{i \in I} C (A_{i})$ ;

(MC3) If {A_i} _i∈I ⊆ 2^X is directed by inclusion, then $C (⋃_{i \in I} A_{i}) \geq ⋀_{i \in I} C (A_{i})$ .

If $C$ is an M-fuzzifying convexity on X, then the pair $(X, C)$ is called an M-fuzzifying convex structure or M-fuzzifying convex space (MC for short).

Definition 2.4. [33] A mapping $B : X \times 2^{\underset{fin}{X}} \to M$ is called an M-fuzzifying betweenness on X if it satisfies the following conditions:

(MB1) For each x ∈ X, $B (x, \emptyset) = 0$ ;

(MB2) If $F \in 2_{fin}^{X}$ and x ∈ F, then $B (x, F) = 1$ ;

(MB3) For each $F, G \in 2_{fin}^{X}$ and x ∈ X, $B (x, F) \geq B (x, G) \land [⋀_{u \in G} B (u, F)] .$

If $B$ is an M-fuzzifying betweenness on X, then the pair $(X, B)$ is called an M-fuzzifying betweenness space (MB for short).

Remark 2.5. If $C$ is an M-fuzzifying convexity, then for any a ∈ M, $(X, C_{[a]})$ is a convex space which satisfies (C1), (C2) and (C3) in [21]. Similarly, If $B$ is an M-fuzzifying betweenness, then for any a ∈ M, $(X, B_{[a]})$ is a betweenness space which satisfies (BT1), (BT2) and (BT3) in [21].

Definition 2.6. [33] A mapping $I : X \times X \to M^{X}$ is called an M-fuzzifying interval operator on X if it satisfies the following conditions: for all x, y ∈ X,

(MI1) $I (x, y) (x) = I (x, y) (y) = 1$ ;

(MI2) $I (x, y) = I (y, x)$ .

If $I$ is an M-fuzzifying interval operator on X, then the pair $(X, I)$ is called an M-fuzzifying interval space (MI for short).

In this paper, we need to discuss the Galois connection between two completely distributive lattices and consider the connection between M-fuzzifying betweenness and M-fuzzifying convexity in a broader context. Therefore, if not emphasized, we assume that (P, ≤) and (Q, ≤) are completely distributive lattices, $C$ is a mapping from 2^X to P, $B$ is a mapping from $X \times 2_{fin}^{X}$ to Q and $I$ is a mapping from X × X to Q^X.

3 Relationship between M-fuzzifying betweenness spaces and M-fuzzifying convex spaces via LRG-Galois connections

In this part, in order to discuss the categorical relationship between M-fuzzifying betweenness and M-fuzzifying convex spaces without De Morgan algebra, we propose LRG adjunctions.

Definition 3.1. Assume f : P → Q and g : Q → P are mappings. The pair (f, g) is called an LRG adjunction, if it satisfies the following condition:

(LRG) For any a ∈ P and b ∈ Q, f (a) 〈 ^opb ⇔ g (b) 〈 a .

Example 3.2. Assume P = Q = [0, 1]. (f, g) satisfies (LRG) provided that for any a, b ∈ [0, 1], f (a) < b if and only if g (b) < a (i.e., f (a) ≥ b if and only if g (b) ≥ a). Then (f, g) is a Galois connection.

Proposition 3.3. If a pair of mappings f : P ⇄ Q : g satisfies (LRG). Then f is an antitone mapping and for any A ⊆ P, f (⋁ A) = ⋀ f (A) (f is a ⋁ -⋀ mapping).

Proof. (1) Assume a, b ∈ P with a ≤ b. Take each x ∈ Q such that f (a) 〈 ^opx. By (LRG), we have g (x) 〈 a ≤ b. Then it follows from (LRG) that f (b) 〈 ^opx. This implies f (a) = ⋀ {x ∈ Q ∣ f (a) 〈 ^opx} ≥ ⋀ {x ∈ Q ∣ f (b) 〈 ^opx} = f (b) .

(2) By (1), it follows that $f (⋁_{a \in A} a) \leq ⋀_{a \in A} f (a) .$ In order to show the inverse inequality, take each x ∈ Q such that $f (⋁_{a \in A} a) 〈^{op} x .$ By (LRG), we have $g (x) 〈 ⋁_{a \in A} a .$ Then there exists a₀ ∈ A such that g (x) 〈 a₀, which is equivalent to f (a₀) 〈 ^opx. This imples $⋀_{a \in A} f (a) 〈^{op} x .$ By the arbitrariness of x, we obtain $⋀_{a \in A} f (a) \leq f (⋁_{a \in A} a) .$ This shows f is a ⋁ -⋀ mapping. □

Corollary 3.4. Suppose a pair of mappings f : P ⇄ Q : g satisfies (LRG). Then f (1) =0, f (0) =1.

By (LRG), we can use an M-fuzzifying convexity to construct an M-fuzzifying betweenness.

Theorem 3.5. Suppose f : P ⇄ Q : g are a pair of mappings satisfying (LRG). Let $C : 2^{X} \to P$ be an M-fuzzifying convexity and define $B_{C} : X \times 2^{\underset{fin}{X}} \to Q$ by $B_{C} (x, F) = ⋀_{x \notin B \supseteq^{fin} F} f (C (B)) .$ Then $(X, B_{C})$ is an M-fuzzifying betweenness space.

Proof. It suffices to show $B_{C}$ satisfies (MB1)–(MB3).

(MB1) For each x ∈ X, $B_{C} (x, \emptyset) \leq f (C (\emptyset)) = 0$ ;

(MB2) For each x ∈ F, $B_{C} (x, F) = ⋀ \emptyset = 1$ ;

(MB3) By the definition of greatest maximal families, we need only show that for each a ∈ Q, $\begin{matrix} a \in α (B_{C} (x, F)) \\ \Rightarrow & a \in α (B_{C} (x, G) \land ⋀_{u \in G} B_{C} (u, F)) . \end{matrix}$ Suppose a ∈ Q. $\begin{matrix} a \in α (B_{C} (x, F)) \\ \Leftrightarrow & a \in α (f (⋁_{x \notin B \supseteq^{fin} F} C (B))) \\ \Leftrightarrow & f (⋁_{x \notin B \supseteq^{fin} F} C (B)) 〈^{op} a . \end{matrix}$ It follows from (LRG) that $g (a) 〈 ⋁_{x \notin B \supseteq^{fin} F} C (B) . (*)$ (*) holds if and only if there exists B₀ with x ∉ B₀ ⊇ ^finF such that $g (a) 〈 C (B_{0})$ which implies that there exists t ∈ P such that $g (a) 〈 t 〈 C (B_{0})$ . Since $t 〈 C (B_{0})$ implies $t \leq C (B_{0})$ , $B_{0} \in C_{[t]}$ . Hence x ∉ co_t (F) where co_t is the closure operator of $C_{[t]}$ . By properties of closure operator of convexities [21], x ∉ co_t (F) implies that x ∉ co_t (G) or G ⊈ co_t (F), in which G ⊆ ^finX. If x ∉ co_t (G), then there exists a subset A with x ∉ A ⊇ ^finG such that $A \in C_{[t]}$ (i.e., $t \leq C (A)$ ). If G ⊈ co_t (F), then there exists u ∈ G with u ∉ C_u ⊇ ^finF such that $C_{u} \in C_{[t]}$ (i.e., $t \leq C (C_{u})$ ). Furthermore, by g (a) 〈 t, we have $\begin{matrix} ⋁_{x \notin A \supseteq G} C (A) \lor ⋁_{u \in G} ⋁_{u \notin C_{u} \supseteq^{fin} F} C (C_{u}) \\ \Leftrightarrow & f (⋁_{x \notin A \supseteq G} C (A) \lor ⋁_{u \in G} ⋁_{u \notin C_{u} \supseteq^{fin} F} C (C_{u})) 〈^{op} a \end{matrix}$ $\begin{matrix} \Leftrightarrow & a \in α (f (⋁_{x \notin A \supseteq G} C (A) \lor ⋁_{u \in G} ⋁_{u \notin C_{u} \supseteq^{fin} F} C (C_{u}))) \\ \Leftrightarrow & a \in α (B_{C} (x, G) \land ⋀_{u \in G} B_{C} (u, F)) . \end{matrix}$ It follows from $α (B_{C} (x, F)) \subseteq α (B_{C} (x, G) \land ⋀_{u \in G} B_{C} (u, F))$ that $B_{C} (x, F) \geq B_{C} (x, G) \land ⋀_{u \in G} B_{C} (u, F) .$ □

The next theorem shows how to construct an M-fuzzifying convexity by using an M-fuzzifying betweenness.

Theorem 3.6. Let g : Q → P be a mapping with g (0) =1 and $(X, B)$ be an M-fuzzifying betweenness space. Define $C : 2^{X} \to P$ by $C_{B} (A) = ⋀_{F \subseteq^{fin} A x \notin A} g (B (x, F)),$ then $(X, C_{B})$ is an M-fuzzifying convex space.

Proof. It suffices to show $C_{B}$ satisfies (MC1)–(MC3).

(MC1) $C_{B} (X) = ⋀ \emptyset = 1$ . $C_{B} (\emptyset) = ⋀_{x \in X} g (B (x, \emptyset)) = 1$ .

(MC2) We only need show for each a ∈ P and each family of subsets {A_i} _i∈L ∈ 2^X, $a \in α (C_{B} (⋂_{i \in I} A_{i})) \Rightarrow a \in α (⋀_{i \in I} C_{B} (A_{i})) .$ Suppose a ∈ P. $C_{B} (⋂_{i \in I} A_{i}) 〈^{op} a \Leftrightarrow ⋀_{x \notin (⋂_{i \in I} A_{i}) \supseteq^{fin} F} g (B (x, F)) 〈^{op} a,$ which is equivalent to that there exist x₀ and F₀ with x₀ ∉ (⋂ _i∈IA_i) ⊇ ^finF₀ such that $g (B (x, F)) 〈^{op} a$ . This implies that there exist i₀ ∈ I with x₀ ∉ A_{i
₀} ⊇ ^finF₀ such that $g (B (x, F)) 〈^{op} a$ , which means $⋀_{i \in I} ⋀_{x \notin A_{i} \supseteq^{fin} F} g (B (x, F)) 〈^{op} a .$ Hence as we desired, $⋀_{i \in I} C_{B} (A_{i}) 〈^{op} a .$

(MC3) The proof of (MC3) is similar to (MC2). Just need note that F is a finite set, thus for each directed family of subsets $A = {A_{i}}_{i \in I}^{↑}$ with $F \subseteq ⋃^{↑} A$ , there exists an i₁ ∈ I such that F ⊆ A_{i
₁}. And if $x \notin ⋃ A$ , then x ∉ A_i for any i ∈ I. □

Before continuing to the next part, we will define the LRG-Galois connection.

Definition 3.7. Suppose a pair of mappings f : P ⇄ Q : g forms a Galois connection. We call (f, g) an LRG-Galois connection if it satisfies (LRG).

Remark 3.8. (1) If a pair of mappings f : P ⇄ Q : g forms a Galois connection, then fg ≥ id_Q and gf ≥ id_P.

(2) In the rest of this paper, we add several conditions that g (1) =0, g (0) =1 and g is a ⋁ -⋀ mapping to LRG-Galois connection.

Example 3.9. (1) Assume P = Q = [0, 1]. Let f (x) =1 - x² and $g (x) = \sqrt{1 - x}$ for each x ∈ [0, 1]. Then (f, g) is an LRG-Galois connection.

(2) Assume (P, ≤) and (Q, ≤) are complete join semi-lattice. P^op (resp., Q^op) is the dual of P (resp., Q). Then the pair (op, op) is an LRG-Galois connection.

Theorem 3.10. Suppose a pair of mappings f : P ⇄ Q : g forms an LRG-Galois connection and $(X, B)$ is an M-fuzzifying betweenness space. Then for any $(x, F) \in X \times 2_{fin}^{X}$ , ${B_{C}}_{B} (x, F) \geq B (x, F)$ .

Proof. For each x ∈ X and each F ⊆ ^finX, we have $\begin{matrix} {B_{C}}_{B} (x, F) \\ = & ⋀_{x \notin A \supseteq^{fin} F} f (C_{B} (A)) \\ = & ⋀_{x \notin A \supseteq^{fin} F} f (⋀_{y \notin A \supseteq^{fin} G} g (B (y, G))) . \end{matrix}$ Suppose $a \in α (B_{C_{B}} (x, F))$ . Then it follows that there exists A₀ with x ∉ A₀ ⊇ ^finF such that $f (⋀_{y \notin A_{0} \supseteq^{fin} G} g (B (y, G))) 〈^{op} a .$ By (LRG), $g (a) 〈 ⋀_{y \notin A_{0} \supseteq^{fin} G} g (B (y, G),$ which implies that for all G and y with y ∉ A₀ ⊇ ^finG, we have $g (a) 〈 g (B (y, G))$ , which is equivalent to that $fg (B (y, G)) 〈^{op} a$ . Since (f, g) is an Galois connection, $B (y, G) \leq fg (B (y, G)) 〈^{op} a$ . Hence for all G and y with y ∉ A₀ ⊇ ^finG, we have $B (y, G) 〈^{op} a$ . Let G = F and y = x, then $B (x, F) 〈^{op} a$ , i.e., $a \in α (B (x, F))$ . It follows from $α ({B_{C}}_{B} (x, F)) \subseteq α (B (x, F))$ that ${B_{C}}_{B} (x, F) \geq B (x, F)$ . □

In Proposition 3.10, if f ∘ g = id, then we can obtain a stronger result as follow.

Theorem 3.11. Let a pair of mappings f : P ⇄ Q : g with f ∘ g = id_Q form an LRG-Galois connection. Assume $(X, B)$ is an M-fuzzifying betweenness space, then for each $(x, F) \in X \times 2_{fin}^{X}$ , ${B_{C}}_{B} (x, F) = B (x, F)$ .

Proof. For each $(x, F) \in X \times 2_{fin}^{X}$ , we only need to show that the left-hand side is smaller than the right-hand side. $\begin{matrix} {B_{C}}_{B} (x, F) \\ = & ⋀_{x \notin A \supseteq^{fin} F} f (⋀_{y \notin A \supseteq^{fin} G} g (B (y, G))) \\ = & ⋀_{x \notin A \supseteq^{fin} F} f \circ g (⋁_{y \notin A \supseteq^{fin} G} B (y, G)) \\ = & ⋀_{x \notin A \supseteq^{fin} F} (⋁_{y \notin A \supseteq^{fin} G} B (y, G)) . \end{matrix}$ Suppose $a 〈 ⋀_{x \notin A \supseteq^{fin} F} ⋁_{y \notin A \supseteq^{fin} G} B (y, G) .$ This is equivalent to that for each A with x ∉ A ⊇ ^finF, there exists (y, G) with y ∉ A ⊇ ^finG such that $a 〈 B (y, G)$ . Suppose that $a ≰ B (x, F)$ . Let $Y_{aF} = {v \in X ∣ a \leq B (v, F)}$ . Then x ∉ Y_aF, otherwise $a \leq B (x, F)$ and this is a contradiction. Since for each u ∈ F, $a \leq 1 = B (u, F)$ . Hence F ⊆ Y_aF. Furthermore, there exists (y, G) with y ∉ Y_aF ⊇ ^finG such that $a 〈 B (y, G)$ . By (BT3), (BT2) and the definition of Y_aF, $\begin{matrix} B (y, F) \\ \geq & B (y, Y_{aF}) \land [⋀_{w \in Y_{aF}} B (w, F)] \\ \geq & B (y, G) \land [⋀_{s \in G} B (s, Y_{aF})] \land [⋀_{w \in Y_{aF}} B (w, F)] \\ \geq & a \land 1 \land a . \end{matrix}$ It follows that y ∈ Y_aF, which is a contradiction. Hence the assumption that $a ≰ B (x, F)$ fails to hold. Thus we have proved that $a 〈 ⋀_{x \notin A \supseteq^{fin} F} ⋁_{y \notin A \supseteq^{fin} G} B (y, G)$ implies that $a \leq B (x, F)$ . By the arbitrariness of a, we obtain $⋀_{x \notin A \supseteq^{fin} F} ⋁_{y \notin A \supseteq^{fin} G} B (y, G) \leq B (x, F),$ as desired. □

Similarly, we can obtain following results.

Theorem 3.12. Suppose a pair of mappings f : P ⇄ Q : g forms an LRG-Galois connection and $(X, C)$ is an M-fuzzifying convex space. Then for any A ∈ 2^X, ${C_{B}}_{C} (A) \geq C (A)$ .

Proof. Given an A ∈ 2^X, we have ${C_{B}}_{C} (A) = ⋀_{x \notin A \supseteq^{fin} F} g (⋀_{x \notin B \supseteq^{fin} F} f (C (B))) .$ For any (x, F) with x ∉ A ⊇ ^finF, $⋀_{x \notin B \supseteq^{fin} F} f (C) (B) \leq f (C (A)) .$ Since g is antitone, $g (⋀_{x \notin B \supseteq F} f (C (B))) \geq gf (C (A)) \geq C (A) .$ By the arbitrariness of x and F, ${C_{B}}_{C} (A) \geq C (A)$ .□

Corollary 3.3. Suppose an LRG-Galois connection (f, g) satisfies that g ∘ f = id and $(X, C)$ is an M-fuzzifying convex space. Then ${C_{B}}_{C} = C$ .

Proof. For each $(x, F) \in X \times 2_{fin}^{X}$ , let B_xF = {B ⊆ X ∣ x ∉ B ⊇ ^finF}. By g ∘ f = id and the completely distributive law, given an A ⊆ X, we have $\begin{matrix} {C_{B}}_{C} (A) \\ = & ⋀_{x \notin A \supseteq^{fin} F} g \circ f (⋁_{x \notin B \supseteq^{fin} F} C (B)) \\ = & ⋀_{x \notin A \supseteq^{fin} F} ⋁_{x \notin B \supseteq^{fin} F} C (B) \\ = & ⋁_{p \in \prod_{x \notin A \supseteq F} B_{xF}} (⋀_{x \notin A \supseteq^{fin} F} C (p (x, F))) \\ \leq & ⋁_{p \in \prod_{x \notin A \supseteq F} B_{xF}} (C (⋂_{x \notin A \supseteq^{fin} F} p (x, F))) \end{matrix}$ $\begin{matrix} = & ⋁_{p \in \prod_{x \notin A \supseteq F} B_{xF}} (C (F)) \leq C (A) . \end{matrix}$ Hence ${C_{B}}_{C} \leq C$ . The inverse inequality holds for Theorem 3.12. Thus the conclusion is proved. □

Theorem 3.14. Let a pair of mappings f : P ⇄ Q : g form an LRG-Galois connection. Then

(1) If $(X, C_{1})$ and $(X, C_{2})$ are M-fuzzifying convex spaces with $C_{1} \geq C_{2}$ , then $B_{C_{1}} \leq B_{C_{2}}$ .

(2) If $(X, B_{1})$ and $(X, B_{2})$ are M-fuzzifying betweenness spaces with $B_{1} \geq B_{2}$ , then $C_{B_{1}} \leq C_{B_{2}}$ .

This is because f, g are antitone. On the basis of the above discussion, we will further study the categorical relationship between M-fuzzifying convex spaces and M-fuzzifying betweenness spaces. Definition 3.15 can be found in some researches of fuzzy convex structure such as [17].

Definition 3.15. ([17]). (1) Let $(X, C_{X})$ and $(Y, C_{Y})$ be M-fuzzifying convex spaces. If a mapping h : X → Y satisfies the following condition: for each A ∈ 2^X, $C_{X} (h^{- 1} (A)) \geq C_{Y} (A)$ , then h is called an M-fuzzifying convexity-preserving mapping.

(2) Let $(X, B_{X})$ and $(Y, B_{Y})$ be M-fuzzifying betweenness spaces. If a mapping h : X → Y satisfies the following condition: for each $(x, F) \in X \times 2_{fin}^{X}$ , $B_{X} (x, F) \leq B_{Y} (h (x), h (F))$ . then h is called an M-fuzzifying betweenness-preserving mapping.

The category of M-fuzzifying convex spaces and M-fuzzifying convexity-preserving mappings is denoted by MC. The category of M-fuzzifying betweenness spaces and M-fuzzifying betweenness-preserving mappings is denoted by MB.

Theorem 3.16. There is a pair of functors K: MC⇄MB:H, in which $K (C) = B_{C}$ and $H (B) = C_{B}$ . And there exists a natural isomorphism between the hom-sets ${hom}_{MC} (K (C), B)$ and ${hom}_{MB} (C, H (B))$ .

Proof. (1) Firstly, take any MC-morphism $h : (X, C_{X}) \to (Y, C_{Y})$ . Since f is antitone and h (x) ∉ B ⊇ h (F) if and only if x ∉ h^-1 (B) ⊇ F, $\begin{matrix} {B_{X}}_{C_{X}} (x, F) \\ = & ⋀_{x \notin S \supseteq F} f (C_{X} (S)) \\ \leq & ⋀_{x \notin h^{- 1} (B) \supseteq F} f (C_{X} (h^{- 1} (B))) \\ = & ⋀_{h (x) \notin B \supseteq h (F)} f (C_{X} (h^{- 1} (B))) \\ \leq & ⋀_{h (x) \notin B \supseteq h (F)} f (C_{Y} (B)) \\ = & {B_{Y}}_{C_{Y}} (h (x), h (F)) . \end{matrix}$ Hence $K (h) = h : (X, {B_{X}}_{C_{X}}) \to (Y, {B_{Y}}_{C_{Y}})$ is an MC-morphism.

Secondly, take any MB-morphism $h : (X, B_{X}) \to (Y, B_{Y})$ . Since g is antitone and h (x) ∉ B ⊇ h (F) if and only if x ∉ h^-1 (B) ⊇ F, $\begin{matrix} {C_{Y}}_{B_{Y}} (B) \\ = & ⋀_{y \notin B \supseteq G} g (B_{Y} (y, G)) \\ \leq & ⋀_{h (x) \notin B \supseteq h (F)} g (B_{Y} (h (x), h (F))) \\ \leq & ⋀_{h (x) \notin B \supseteq h (F)} g (B_{X} (x, F)) \\ = & ⋀_{x \notin h^{- 1} (B) \supseteq F} g (B_{X} (x, F)) \\ = & {C_{X}}_{B_{X}} (h^{- 1} (B)) . \end{matrix}$ Hence $H (h) = h : (X, {C_{X}}_{B_{X}}) \to (Y, {C_{Y}}_{B_{Y}})$ is an MC-morphism.

Lastly, it is not difficult to check that H and K preserve the unit morphism and composition.

(2) The naturality of the transformation between ${hom}_{MC} (K (C), B)$ and ${hom}_{MB} (C, H (B))$ is established by compositions of mappings. Then need only show that for any M-fuzzifying convex space $(X, C_{X})$ , any M-fuzzifying betweenness space $(Y, B_{Y})$ , and any mapping h : X → Y, ${C_{X}}_{B_{X}} (h^{- 1} (B)) \geq C_{Y} (B)$ if and only if $B_{X} (x, F) \leq {B_{Y}}_{C_{Y}} (h (x), h (F))$ .

On one hand, as h (x) ∉ B ⊇ h (F) ⇔ x ∉ h^-1 (B) ⊇ F, we have $\begin{matrix} {C_{X}}_{B_{X}} (h^{- 1} (B)) \geq C_{Y} (B) \\ \Rightarrow & f ({C_{X}}_{B_{X}} (h^{- 1} (B))) \leq f (C_{Y} (B)) \\ \Rightarrow & ⋁_{x \notin h^{- 1} (B) \supseteq F} B_{X} (x, F) \\ \leq ⋁_{x \notin h^{- 1} (B) \supseteq F} f (g (B_{X} (x, F))) \\ = f (⋀_{x \notin h^{- 1} (B) \supseteq F} g (B_{X} (x, F))) \\ \leq f (C_{Y} (B)) \\ \Rightarrow & B_{X} (x, F) \leq ⋀_{h (x) \notin B \supseteq h (F)} f (C_{Y} (B)) \\ = {B_{Y}}_{C_{Y}} (h (x), h (F)) . \end{matrix}$

On the other hand, as h (x) ∉ B ⊇ h (F) ⇔ x ∉ h^-1 (B) ⊇ F, we have:

$\begin{matrix} B_{X} (x, F) \leq {B_{Y}}_{C_{Y}} (h (x), h (F)) \\ = ⋀_{h (x) \notin B \supseteq h (F)} f (C_{Y} (B)) \\ \Rightarrow & g (B (x, F)) \\ \geq g (⋀_{h (x) \notin B \supseteq h (F)} f (C_{Y} (B))) \\ = gf (⋁_{h (x) \notin B \supseteq h (F)} (C_{Y} (B))) \end{matrix}$ $\begin{matrix} \geq ⋁_{h (x) \notin B \supseteq h (F)} C_{Y} (B) \\ \Rightarrow & {C_{X}}_{B_{X}} (h^{- 1} (B)) \\ = ⋀_{x \notin h^{- 1} (B) \supseteq F} g (B_{X} (x, F)) \geq C_{Y} (B) . \end{matrix}$ □

Corollary 3.17. (1) If there exists an LRG-Galois connection (f, g) between P and Q, then there exists a adjoint pair between MC and MB;

(2) Suppose (f, g) is an LRG-Galois connection between P and Q. If fg = id and gf = id, then MC and MB are isomorphic.

4 Relationship between M-fuzzifying interval spaces and arity-2 M-fuzzifying convex spaces via LRG-Galois connections

A convexity is of arity-n provided that it is precisely a collection of such sets C which satisfies the property that if |F| ≤ n, then F ⊆ C implies co (F) ⊆ C. Arity-2 convexities exist widely in various mathematical structures (like convexity in Rⁿ, poset, convex graph and so on). In this part, we will discuss the relationship between M-fuzzifying arity-2 convexities and M-fuzzifying interval operators.

Example 4.1. For the number axis $ℝ$ , let M = {0, 1}, $I (x, y) = [x, y]$ , Then M-fuzzifying interval space $(ℝ, I)$ is all the close intervals in the general sense.

Theorem 4.2. Suppose $(X, I)$ is an M-fuzzifying interval space and g : Q → P is a mapping. Define $C : 2^{X} \to P$ by $C_{I} (A) = ⋀_{z \notin A \supseteq {x, y}} g (I (x, y) (z)) .$ Then $(X, C_{I} (A))$ is an M-fuzzifying convex space.

Proof. It suffices to show $C_{I}$ satisfies (MC1)–(MC3).

(MC1) $C_{B} (\emptyset) = C_{B} (X) = ⋀ \emptyset = 1$ .

(MC2) Given a family of subsets {A_I} _i∈I, for each a ∈ P, $C (⋂_{i \in I} A_{i}) = ⋀_{z \notin (⋂_{i \in I} A_{i}) \supseteq {x, y}} g (I (x, y) (z)) 〈^{op} a$ holds if and only if there exist z₀, y₀, z₀ with z₀ ∉ (⋂ _i∈IA_i) ⊇ {x₀, y₀} such that $g (I (x_{0}, y_{0}) (z_{0})) 〈^{op} a$ . It follows that there exist A₀ with z₀ ∉ A₀ ⊇ {x₀, y₀} such that $g (I (x_{0}, y_{0}) (z_{0})) 〈^{op} a$ . This is equivalent to that $⋀_{i \in I} C (A_{i}) = ⋀_{i \in I} ⋀_{z \notin A_{i} \supseteq {x, y}} g (I (x, y) (z)) 〈^{op} a .$ Then it follows from $α (C (⋂_{i \in I} A_{i})) \subseteq α (⋀_{i \in I} C (A_{i}))$ that $C (⋂_{i \in I} A_{i}) \geq ⋀_{i \in I} C (A_{i})$ .

(MC3) Given a directed family of subsets ${A_{i}}_{i \in I}^{↑}$ , for each a ∈ P, $C ({⋃_{i \in I}}^{↑} A_{i}) = ⋀_{z \notin (⋃_{i \in I}^{↑} A_{i}) \supseteq {x, y}} g (I (x, y) (z)) 〈^{op} a$ holds if and only if there exist z₀, y₀, z₀ with $z_{0} \notin ({⋃_{i \in I}}^{↑} A_{i}) \supseteq {x_{0}, y_{0}}$ such that $g (I (x_{0}, y_{0}) (z_{0})) 〈^{op} a$ . It follows that there exists A₀ with z ∉ A₀ ⊇ {x₀, y₀} such that $g (I (x_{0}, y_{0}) (z_{0})) 〈^{op} a$ . This is equivalent to that $⋀_{i \in I} C (A_{i}) = ⋀_{i \in I} ⋀_{z \notin A_{i} \supseteq {x, y}} g (I (x, y) (z)) 〈^{op} a$ It follows from $α (C (⋃_{i \in I} A_{i})) \subseteq α (⋀_{i \in I} C (A_{i}))$ that $C (⋃_{i \in I} A_{i}) \geq ⋀_{i \in I} C (A_{i})$ . □

Corollary 4.3. If $(X, I)$ is an M-fuzzifying interval space, then $(X, B_{C_{I}})$ is an M-fuzzifying betweenness space.

By Theorem 4.2, we have a following definition:

Definition 4.4. We call the M-fuzzifying convex space $(X, C_{I})$ an arity-2 M-fuzzifying convex space, if $(X, I)$ is an M-fuzzifying interval space.

The following theorem show the rationality of Definition 4.

Theorem 4.5. Suppose $(X, I)$ is an M-fuzzifying interval space and $(X, C_{I})$ is an arity-2 M-fuzzifying convex space. Then for any a ∈ Q, ${C_{I}}_{[a]}$ is an arity-2 convexity.

Proof. By Remark 2.2, for any a ∈ Q, ${C_{I}}_{[a]}$ is a convexity. We only need to show that for a given subset D, if {s, t} ⊆ D, then $⋂_{A_{i} \in C_{I} {s, t} \subseteq A_{i}} A_{i} \subseteq D \Rightarrow D \in {C_{I}}_{[a]} .$ $D \in {C_{I}}_{[a]}$ is equivalent to that $C_{I} (D) = ⋀_{z \notin D \supseteq {x, y}} g (I (x, y) (x)) \geq a . (* 1)$ And (*1) holds if and only if for any x, y, z ∈ X with z ∉ D ⊇ {x, y}, we have $g (I (x, y) (z)) \geq a$ . Take each s, t, r ∈ X such that r ∉ D ⊇ {s, t}. If for any $A_{i} \in {C_{I}}_{[a]}$ , A_i ⊉ {s, t}, then $⋂_{A_{i} \in C_{I} {s, t} \subseteq A_{i}} A_{i} = ⋂ \emptyset = X \subseteq D .$ Then $C_{I} (D) = 1 \geq a$ . Suppose there exists $A_{i} \in {C_{I}}_{[a]}$ with A_i ⊇ {s, t}. It follows that ${s, t} \subseteq A = ⋂_{A_{i} \in C_{I} {s, t} \subseteq A_{i}} A_{i} .$ By A ⊆ D, we can know r ∉ D implies r ∉ A. By (C2) in Remark 2.2, we can know $A \in {C_{I}}_{[a]}$ . it follows from $g (I (s, t) (r)) \geq C_{I} (A) = ⋀_{z \notin A \supseteq {x, y}} g (I (x, y) (x)) \geq a$ that $g (I (s, t) (r)) \geq a$ . By arbitrariness of r, s, t, we can know (*1) hold.□

Theorem 4.6. If $(X, B)$ is an M-fuzzifying betweenness space, define mapping $I_{B} : X \times X \to Q^{X}$ by $I_{B} (x, y) (z) = B (z, {x, y})$ . Then $(X, I_{B})$ is an M-fuzzifying interval space.

Proof. It is not difficult to know that (MI1) holds for (MB1) and (MI2) holds for (MB2). □

Theorem 4.7. Let a pair of mappings f : P ⇄ Q : g form an LRG-Galois connection and $(X, I)$ be an M-fuzzifying interval space. Then for any s, t ∈ X, $I_{B_{C_{I}}} (s, t) (x) \geq I (s, t) (x)$ .

Proof. Given s, t ∈ X, since (f, g) is an LRG-Galois connection, we have $\begin{matrix} I_{B_{C_{I}}} (s, t) (x) \\ = & ⋀_{x \notin B \supseteq {s, t}} f (C_{I} (B)) \\ = & ⋀_{x \notin B \supseteq {s, t}} f (⋀_{w \notin B \supseteq {u, v}} g (I (u, v) (w))) \\ \geq & ⋀_{x \notin B \supseteq {s, t}} ⋁_{w \notin B \supseteq {u, v}} fg (I (u, v) (w)) \\ \geq & ⋀_{x \notin B \supseteq {s, t}} ⋁_{w \notin B \supseteq {u, v}} I (u, v) (w) \\ \geq & I (s, t) (x) □ \end{matrix}$

Theorem 4.8. Suppose a pair of mappings f : P ⇄ Q : g forms an LRG-Galois connection and $(X, C)$ is an M-fuzzifying convex space, then for any A ∈ 2^X, $C_{I_{B_{C}}} (A) \geq C (A)$ .

Proof. Given an A ∈ 2^X, since (f, g) is an LRG-Galois connection, we have $\begin{matrix} C_{I_{B_{C}}} (A) & = ⋀_{z \notin A \supseteq {x, y}} g (I_{B_{C}} (x, y) (z)) \\ = ⋀_{z \notin A \supseteq {x, y}} g (B_{C} (z, {x, y})) \\ = ⋀_{z \notin A \supseteq {x, y}} g (⋀_{z \notin B \supseteq {x, y}} f (C (B))) \\ \geq ⋀_{z \notin A \supseteq {x, y}} ⋁_{z \notin B \supseteq {x, y}} gf (C (B))) \\ \geq ⋀_{z \notin A \supseteq {x, y}} ⋁_{z \notin B \supseteq {x, y}} C (B)) \\ \geq C (A) \end{matrix}$ □

Furthermore, an arity-2 M-fuzzifying convex space satisfies the following property.

Theorem 4.9. Suppose a pair of mappings f : P ⇄ Q : g forms an LRG-Galois connection and $(X, B)$ is an M-fuzzifying betweenness space. Assume $(X, C_{I})$ is an arity-2 M-fuzzifying convex space, then $C_{I_{B_{C_{I}}}} (A) = C_{I} (A)$ for any A ∈ 2^X.

Proof. Similarly to Theorem 4.8, by Theorem 3.5, Theorem 4.2 and Theorem 4.6, $C_{I_{B_{C_{I}}}} (A) \geq C_{I} (A)$ . By Theorem 3.14 and Theorem 4.7, $C_{I_{B_{C_{I}}}} (A) \leq C_{I} (A)$ . □

The next definition can be found in some articles of fuzzy convex structures such as [25].

Definition 4.10. ([25]). (1) Let $(X, C_{X})$ and $(Y, C_{Y})$ be arity-2 M-fuzzifying convex spaces. If a mapping h : X → Y satisfies the following condition: for each A ∈ 2^X, $C_{X} (h^{- 1} (A)) \geq C_{Y} (A)$ , then h is called an arity-2 M-fuzzifying convexity-preserving mapping.

(2) Let $(X, I_{X})$ and $(Y, I_{Y})$ be M-fuzzifying interval spaces. If a mapping h : X → Y satisfies the following condition: for each $(x, F) \in X \times 2_{fin}^{X}$ , $I_{X} (x, F) \leq I_{Y} (h (x), h (F))$ . then h is called an M-fuzzifying interval-preserving mapping.

The category of arity-2 M-fuzzifying convex spaces and arity-2 M-fuzzifying convexity-preserving mappings is denoted by MCa2. The category of M-fuzzifying interval spaces and M-fuzzifying interval-preserving mappings is denoted by MI.

Remark 4.11. MCa2 is a full subcategory of MC.

Next, we will discuss the categorical relationship between MCa2 and MI.

Theorem 4.12. There exists a pair of functors I :MCa2⇄MB :R, in which $I (C) = {I_{B}}_{C}$ , $R (I) = C_{I}$ . And the functors satisfies:

(1) R ∘ I = Id;

(2) I is a full functor;

(3) ${hom}_{MC} (R (C), B) ≅ {hom}_{MB} (C, I (B))$ is a natural isomorphism.

Proof. Similarly to Theorem 3.16, I and R are functors. By Theorem 4.9, we can know that on Ob(MCa2), R ∘ I = Id. Similarly to Theorem 3.16, for a mapping h : X → Y, $C_{X} (h^{- 1} (A)) \geq C_{Y} (A)$ if and only if ${I_{X}}_{C_{X}} (x, F) \geq {I_{Y}}_{C_{Y}} (h (x), h (F))$ . Hence (1) and (2) are established. (3) ia a corollary of Theorem 3.16.

Corollary 4.13. If there exists an LRG-Galois connection (f, g) between P and Q, then MCa2 is a reflective subcategory of MI.

By Theorem 4, we have $I_{B_{C_{I}}} \geq I$ , then $C_{I_{B_{C_{I}}}} \leq C_{I}$ . By Theorem 4.9, we have $C_{B_{C_{I}}} \geq C_{I}$ . In these inductions, we need not fg = id or gf = id. Hence fg = id or gf = id is not necessary in Corollary 4.13.

5 Conclusion

In this paper, we propose LRG-Galois connection. The categorical relationship between MC and MB is discussed. And MCa2 is proved to be a reflective subcategory of MI. In fact, we can further consider the role of Galois connections.

(1) A Galois connection is a generalization of De Morgan law. In many situations that complement sets are needed, we can use special Galois connections instead of complements. Similarly, a Galois adjunction is a generalization of the inclusion relation. In many situations that complement sets are not needed, we can use a special Galois adjunctions instead of containments.

(2) In theory of structures on lattices (such as [26]), a generalized order-homomorphisms [26] h and its adjunctions h^⊣ are often used. (h, h^⊣) is a Galois adjunction, so if we regard MCs and MBs as objects, then morphism between an MC and MB could be the left Galois adjunctions.

Footnotes

Acknowledgments

This work is supported by the Postdoctoral Science Foundation of China (No. 2020M670142).

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