Connectedness degrees in ( L,M )-fuzzy convex structures

Abstract

Connectivity is one of the most essential notions in general topology. Convex structures are topological-like structures. Many properties in topological spaces have been generalized to convex structures, such as separation. However, connectivity has not been studied in convex structures yet. In this paper, firstly, based on the consideration to hull operators, separatedness is defined in classical convex structures, and then we provide the concept of connectivity. Secondly, some equivalent characterizations of connectivity are discussed, and we investigate the related properties of connectivity. In additional, through (L, M)-fuzzy convex hull operators, we propose the separatedness degrees of (L, M)-fuzzy convex structures. Furthermore, the notion of connectedness degrees of (L, M)-fuzzy convex structures is introduced. Finally, many properties of connectivity in general convex structures can be generalized to (L, M)-fuzzy convex structures.

Keywords

Convex structure connectivity separatedness degree connectedness degree

1 Introduction

With the development of fuzzy mathematics, axiomatic convex structures have been endowed with fuzzy set theory. In 1994, Rosa [30] firstly proposed the concept of fuzzy convex structures, and then some scholars began to study fuzzy convex structures. In 2009, Maruyama [22] extended lattice [0, 1] to complete distributive lattices and established L-convex structures. In this framework, Pang provided several features of L-convex structures in [24 –26]. Later, Pang [27] and Jin [10] introduced several subcategories of L-convex spaces and studied the relationship between them. In 2014, Shi and Xiu gave a new method to the fuzzification of convex structures in [34], called M-fuzzifying convex structures. In this case, Shi and Li introduced the restricted M-fuzzifying convex hull operators in [33] and proved that there is a one-to-one correspondence between M-fuzzifying restricted hull operators and M-fuzzifying convex structures. Other works regarding the convexity of M-fuzzifying were presented in [5 , 40]. In 2017, Shi and Xiu [35] proposed a more general method to the fuzzification of convex structures, called (L, M)-fuzzy convex structures, which were the generalization of L-convex structures and M-fuzzifying convex structures. On this basis, many convexities and their induced convex structures were investigated in [17 , 42].

Connectivity is topological invariance and one of the most essential notions in general topology. In 1980, Pu and Liu [28] advanced the first important definition of connectedness of fuzzy topological spaces. Since then, connectedness has been generalized to L-topology in terms of many forms in [1–3 , 45–47]. In fuzzifying topological spaces, Ying [41] introduced a definition of connectivity and Fang [6] proved Fan^′s theorem. In 2009, Shi [32] presented the notions of separatedness degrees and connectedness degrees of L-fuzzy topological spaces. In 2015, J $\ddot{a}$ ger [10] considered connectedness and local connectedness for lattice-valued convergence spaces. In 2021, Fang [8] investigated extensionality and ɛ-connectedness in the category of ⊤-convergence spaces. As topology-like structures, convex structures share many similarities with topological spaces. However, up to now, the combination of connectivity and convex structures has not been studied yet. Naturally, we intend to consider such a question that:

How do we introduce connectivity in convex structures?

Since the convex structures are determined by the hull operators and there are many similarities between the hull operators and the closure operators, we consider using the hull operators to define separatedness. Then, the concept of connectivity is introduced by using separatedness.

This paper is organized as follows. In Section 2, some necessary definitions and results about convex structures and (L, M)-fuzzy convex structures are recalled. In Section 3, the definitions of separatedness and connectedness in convex structures are introduced. Then we provide some equivalent characterizations of connected convex structures and study the related properties of connectivity. In Section 4, we propose the notion of separatedness degrees in (L, M)-fuzzy convex structures by means of (L, M)-fuzzy hull operators. In Section 5, the connectedness degrees are investigated in (L, M)-fuzzy convex structures by separatedness degrees. Many properties of connectivity in general convex structures can be generalized to (L, M)-fuzzy convex structures.

2 Preliminaries

In this section, we refer to [35, 38] for the related notions about convex structures and (L, M)-fuzzy convex structures. Meanwhile, we introduce basic results required for this article.

Throughout this paper, let X be a non-empty set, both L and M be completely distributive lattices with order reversing involution ^′, where ⊥_M (⊥ _L) and⊤_M (⊤ _L) denote the least and the greatest element in M (L), respectively.

We say that a is wedge below b in L, denoted by a ≺ b, if for every subset D ⊆ L, b ≤ ⋁ D implies a ≤ d for some d ∈ D. A complete lattice L is completely distributive if and only if b = ⋁ {a ∈ L : a ≺ b} for each b ∈ L. For any b ∈ L, define β (b) = {a ∈ L : a ≺ b}.

For a nonempty set X, the set of all nonzero coprime elements of L^X is denoted by J (L^X). It is easy to see that J (L^X) is exactly the set of all fuzzy points x_λ (λ ∈ J (L)). The smallest element and the largest element in L^X are denoted by $\underline{⊥}$ and $\underline{⊤}$ , respectively. In the meantime, the smallest element and the largest element in L^{X
_t} are denoted by ⊥_t and ⊤_t, respectively.

Firstly, we recall the definition of convex structures as it was introduced in [38].

Definition 2.1. [38] A subset $C$ of 2^X is called a convexity if it satisfies the following conditions:

(C-1) The empty set ∅ and the universal set X are in $C$ ;

(C-2) $C$ is stable for intersections, that is, if $D \subseteq C$ is non-empty, then $⋂ D$ is in $C$ ;

(C-3) $C$ is stable for nested unions, that is, if $D \subseteq C$ is non-empty and totally ordered by inclusion, then $⋃ D$ is in $C$ .

The pair $(X, C)$ is called a convex structure, the elements in $C$ are called convex sets and their complements are called concave sets. By the axiom (C-1), a subset A of a convex structure X is included in at least one convex set, namely X. In regard to (C-2), A is included in a smallest convex set $co (A) = ⋂ {C ∣ A \subseteq C \in C}$ the (convex) hull of A. A set of type co (F), with F finite, is called a polytope.

Definition 2.2. [38] A mapping co : 2^X → 2^X is called a hull operator on X if it satisfies:

(CO-1) co ∅ = ∅ ;

(CO-2) A⊆ co (A) ;

(CO-3) A ⊆ B, then co (A) ⊆ co (B);

(CO-4) co (co (A)) = co (A);

(CO-5) $co (A) = ⋃_{B \in 2_{fin}^{X}} co (B)$ .

Definition 2.3. [38] Let $(X, C)$ be a convex structure. A subset H of X is called a half-space (hemispace, biconvex set) provided H is both convex and concave.

Definition 2.4. [38] Let $(X, C)$ be a convex structure.

$(X, C)$ is said to be S₁ separated if all singletons in X are convex.

$(X, C)$ is said to be S₂ separated if x₁ ≠ x₂ ∈ X then there is a half-space H ⊆ X with x₁ ∈ H ; x₂ ∉ H .

$(X, C)$ is said to be S₃ separated if C ⊆ X is convex and if x ∈ X \ C then there is a half-space H of X with C ⊆ H ; x ∉ H.

$(X, C)$ is said to be S₃ separated if C, D ⊆ X are disjoint convex sets then there is a half-space H of X with C ⊆ H, D ⊆ X \ H.

Clearly, S₂ implies S₁, and under assumption of S₁,

S_{4} \Rightarrow S_{3} \Rightarrow S_{2} .

Definition 2.5. [38] Let $(X, C)$ be a convex structure and let Y be a subset of X. The family of sets $C | Y = {C \cap Y | C \in C}$ is easily seen to be a convexity on Y; it is called the relative convexity of Y. The resulting convex structure $(Y, C | Y)$ is a subspace (substructure) of $(X, C)$ . As a straightforward consequence of the definition, we claim that the hull operator co_Y of a subspace Y of X satisfies $\forall A \subseteq Y : {co}_{Y} (A) = co (A) \cap X .$

Definition 2.6. [38] Let $(X_{i}, C_{i})$ for i ∈ I be a family of convex structures, let X be the product of the sets X_i for i ∈ I, and let $π_{i} : X \to X_{i}$ denote the i^th projection. The product convexity $C$ of X is the one generated by the subbase ${π_{i}^{- 1} (C_{i}) ∣ C_{i} \in C_{i}; i \in I} .$ The resulting convex structure $(X, C)$ is called the product of the spaces $(X_{i}, C_{i})$ for i ∈ I, and is denoted by $\prod_{i \in I} (X_{i}, C_{i})$ , or if $I = 1, 2 \dots n, (X_{1}, C_{1}) \times (X_{2}, C_{2}) \times \dots (X_{n}, C_{n}) .$

Definition 2.7. [38] Let f : X₁ → X₂ be a function between convex structures X₁ and X₂. Then f is said to be a convexity-preserving function (a CP function) provided for each convex set C in X₂, the set f^-1 (C) is convex in X₁.

Definition 2.8. [38] The function f is an isomorphism (explicitly, a CP isomorphism) if it is a bijection and if both f and f^-1 are CP.

Next, we recall the definition of (L, M)-fuzzy convex structures as it was introduced in [35].

Definition 2.9. [35] A mapping $C : L^{X} \to M$ is called an (L, M)-fuzzy convex structure on X if it satisfies:

(LMC1) $C (\underline{⊥}) = C (\underline{⊤}) = ⊤_{M}$ ;

(LMC2) If {A_i ∣ i ∈ I} ⊆ L^X is nonempty, then $⋀_{i \in I} C (A_{i}) \leq C (⋀_{i \in I} A_{i})$ ;

(LMC3) If {A_i ∣ i ∈ I} ⊆ L^X is nonempty and totally ordered by inclusion, then $⋀_{i \in I} C (A_{i}) \leq C (⋁_{i \in I} A_{i})$ .

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

Definition 2.10. [35] Let $(X, C_{X})$ and $(Y, C_{Y})$ be (L, M)-fuzzy convex structures and let f : X → Y be a mapping. We say f is (L, M)-convexity-preserving ((L, M) - CP in short) if $C_{Y} (B) \leq C_{X} (f^{\leftarrow} (B))$ for all B ∈ L^Y.

Definition 2.11. [35] A mapping $H : L^{X} \to M^{J (L^{X})}$ is called an (L, M)-fuzzy hull operator on X if it satisfies:

(LMH1) $H (\underline{⊥}) (x_{λ}) = ⊥_{M}$ ;

(LMH2) $H (A) (x_{λ}) = ⊤_{M}$ for each x_λ ≤ A;

(LMH3) $H (A) (x_{λ}) = ⋀_{x_{λ} ≰ B \geq A} ⋁_{y_{μ} ≰ B} H (B) (y_{μ});$

(LMH4) $H (⋁_{k \in K} A_{K}) (x_{λ}) = ⋀_{μ ⪡ λ} ⋁_{k \in K} H (A_{k}) (x_{μ})$ for each nonempty chain {A_k} _k∈K ⊆ L^X.

For an (L, M)-fuzzy hull operator $H$ on X, the pair $(X, H)$ is called an (L, M)-fuzzy hull space.

Definition 2.12. [23] Let $H : L^{X} \to M^{J (L^{X})}$ be a mapping. Then (LMH3) implies (LMH5) and (LMH6):

(LMH5) $H (A) (x_{λ}) = ⋀_{μ ⪡ λ} H (A) (x_{μ});$

(LMH6) If A ≤ B, then $H (A) \leq H (B) .$

Proposition 2.13. [23] Let $(X, C)$ be an (L, M)-fuzzy convex space and ∀A ∈ L^X, ∀ x_λ ∈ J (L^X). We define $H^{C} : L^{X} \to M^{J (L^{X})}$ as follows: $H^{C} (A) (x_{λ}) = ⋀_{x_{λ} ≰ B \geq A} C (B)^{'} .$ Then $H^{C}$ is an (L, M)-fuzzy hull operator on X.

Proposition 2.14. [23] Let $(X, H)$ be an (L, M)-fuzzy hull space and define $C^{H} : L^{X} \to M$ as follows: $\forall A \in L^{X}, C^{H} (A) = ⋀_{x_{λ} ≰ A} H (A) (x_{λ})^{'} .$ Then $C^{H}$ is an (L, M)-fuzzy convex structure on X.

Definition 2.15. [35] Let ${(X_{i}, C_{i}) : i \in Γ}$ be a set of (L, M)-fuzzy convex structures. Let X be the product of the sets X_i for i ∈ Γ, and let π_i : X → X_i the projection for each i ∈ Γ. Define a mapping φ : L^X → M by $φ (μ) = ⋁_{i \in Γ} ⋁_{π_{i}^{\leftarrow} (v) = μ} C_{i} (v) for each μ \in L^{X} .$ Then the product convexity $C$ of X is the one generated by subbase φ. The resulting (L, M)-fuzzy convex structure $(X, C)$ is called the product of ${(X_{i}, C_{i}) : i \in Γ}$ and is denoted by $\prod_{i \in Γ} (X_{i}, C_{i})$ .

3 Connectivity in convex structures

In this section, the definitions of separatedness and connectedness in convex structures are introduced. Then we provide some equivalent characterizations of connected convex structures and study the related properties of connectivity.

First, we provide the definitions of separatedness and connectedness in convex structures.

Definition 3.1. Let $(X, C)$ be a convex structure and A, B ⊆ X. Then A and B are called separated if (co (A)∩ B) ∪ (A ∩ co (B)) = ∅.

Definition 3.2. Let $(X, C)$ be a convex structure. Then X is said to be connected if it can not be represented as a union of two separated non-empty sets. Next, we give some equivalent conditions of connectivity.

Theorem 3.3. Let $(X, C)$ be a convex structure. Then the following conditions are equivalent:

X is connected;

X is not the union of two disjoint non-empty convex sets;

X is not the union of two disjoint non-empty concave sets;

The empty set and the whole space are the only half-spaces of the space X;

Every convexity-preserving function f : X → D of the space X to the two-point discrete space D = {0, 1} is constant.

Proof. (1) ⇒ (2): Suppose that X is the union of two disjoint non-empty convex sets, i.e., there exist $A, B \neq \emptyset, A, B \in C, A \cap B = \emptyset, X = A \cup B$ , A = co (A) and B = co (B). Then we have A∩ co (B) = A ∩ B = ∅ and B∩ co (A) = B ∩ A = ∅. Hence, X can be represented as a union of two separated non-empty sets, which leads to a contradiction.

(2) ⇒ (3): Suppose that X is the union of two disjoint non-empty concave sets, i.e., there exist concave sets A, B which are satisfied that A, B≠ ∅, A∩ B = ∅ and X = A ∪ B. Then A = X \ B and B = X \ A are convex sets. So we have (X\ B) ∩ (X \ A) = ∅ and X = (X \ A) ∪ (X \ B). Hence, X is the union of two disjoint non-empty convex sets, which leads to a contradiction.

(3) ⇒ (4): Suppose A is a half-space of the space X. Then A and X \ A are two disjoint non-empty concave sets and are satisfied A ∪ X \ A = X, which leads to a contradiction.

(4) ⇒ (5): Suppose f : X → D is a convexity-preserving function. If f isn’t constant, then there exist x₁, x₂ ∈ X, 0 = f (x₁) ≠ f (x₂) =1. For A = f^-1 (0), A is a nontrivial half-spaces of X, which leads to a contradiction.

(5) ⇒ (1): If X isn’t connected, there exist two separated non-empty sets A and B, such that X = A ∪ B, A∩ co (B) = ∅ and B∩ co (A) = ∅. Then co (A) ⊂ X \ B, co (B) ⊂ X \ A, i.e., co (A) = A, co (B) = B. Hence, we claim that f (x) =0 for x ∈ A and f (x) =1 for x ∈ B, which leads to a contradiction. □ In the following, we provide some examples of connected convex structures.

Example 3.4. Suppose X = {1, 2, 3}, $O = {\emptyset, {1}, {3}, {1, 2, 3}}$ and $P = {\emptyset, {1}, {2}, {2, 3}, {1, 2, 3}}$ . Then

$(X, O)$ is a convex structure, and X is connected.

$(X, P)$ is a convex structure, and X isn’t connected.

Proposition 3.5. Let G be a group, $C$ represents a family of sets consisting of all subgroups on G. Then $C$ is a convex structure on G and G is connected.

Proof. It is obvious that $\emptyset, G \in C$ and the intersections of any subgroups of a group G is still a subgroup of G. Meanwhile, we know the directed unions of subgroups is also a subgroup. Thus, we claim that $C$ is a convex structure. Since the identity element of a subgroup is the identity element of the group G, G is not the union of two disjoint non-empty subgroups, i.e., G is not the union of two disjoint non-empty convex sets. Therefore, G is connected. □

Example 3.6. Let Z_n be the residue class additive group of module n. Then Z₈ is connected.

Proof. Since Z₈ has four subgroups, i.e., ${\bar{0}}, {\bar{0}, \bar{4}}$ , ${\bar{0}, \bar{2}, \bar{4}, \bar{6}}$ and Z₈. Then Z₈ is not the union of two disjoint non-empty subgroups. Thus, Z₈ is connected. □

Proposition 3.7. Let R be a ring, $C$ represents a family of sets consisting of all subrings on R. Then $C$ is a convex structure on R and R is connected.

Proof. It is obvious that $\emptyset, R \in C$ and the intersections of any subrings of a ring R is still a subring of R. Meanwhile, we know the directed unions of subrings is also a subring. Thus, we claim that $C$ is a convex structure. Since the addition of R is made into an additive group and the zero element of the subgroups is the zero element of the additive group, R is not the union of two disjoint non-empty subrings, i.e., R is not the union of two disjoint non-empty convex sets. Therefore, R is connected. □

Corollary 3.8. Let F be a field, $C$ represents a family of sets consisting of all subfields on F. Then $C$ is a convex structure on F and F is connected.

Example 3.9. Let Z_n be the residue class ring of module n. Then Z₆ is connected.

Proof. Since Z₆ has four subrings, i.e., ${\bar{0}}, {\bar{0}, \bar{3}}$ , ${\bar{0}, \bar{2}, \bar{4}}$ and Z₆. Then Z₆ is not the union of two disjoint non-empty subrings. Thus, Z₆ is connected.

□ Next, we discuss the relationships between S₁, ⋯ , S₄ separation axioms and connectedness defined in this paper.

We note that S₁ separation axiom is independent of connectivity. Here is an example to illustrateit.

Example 3.10. Suppose X = {1, 2, 3}, $F = {\emptyset, {1}, {2}, {3}, {1, 2, 3}}$ and $Q = {\emptyset, {1}, {3}, {1, 3}, {1, 2, 3}}$ . Then

$(X, O)$ is a convex structure, and X is an S₁ convex structure.

$(X, P)$ is a convex structure, and X isn’t an S₁ convex structure.

Theorem 3.11. Let $(X, C)$ be a convex structure. If X is connected, then X isn’t an S₂ convex structure.

Proof. Suppose that X is an S₂ convex structure, i.e., if x₁ ≠ x₂ ∈ X, then there is a half-space H ⊆ X with x₁ ∈ H ; x₂ ∉ H . Then X = H ∪ X \ H, i.e., X is not the union of two disjoint non-empty convex sets, which leads to a contradiction. □

Corollary 3.12. Let $(X, C)$ be a convex structure. If X is connected, then X isn’t S₃ (S₄) convex structure.

Proof. Since S₄ ⇒ S₃ ⇒ S₂, then X isn’t an S₂ convex structure ⇒ X isn’t an S₃ convex structure ⇒ X isn’t an S₄ convex structure. Therefore, the conclusion is valid. □

Example 3.13. Suppose X = {1, 2, 3}, $F = {\emptyset, {1}, {2}, {3}, {1, 2, 3}}$ and $R = {\emptyset, {1}, {2}, {3}, {1, 3}, {1, 2, 3}}$ . Then

$(X, F)$ is a convex structure. X isn’t an S₂, S₃ and S₄ convex structure, and X is connected.

$(X, R)$ is a convex structure. X isn’t an S₂, S₃ and S₄ convex structure, and X isn’t connected.

Now, we can apply the relevant examples from reference [38] to this paper.

Example 3.14.

A symmetric H-convexity admits a subbase of half-spaces and hence is S₃, so it isn’t connected.

A semilattice has the separation property S₄, so it isn’t connected.

A distributive lattice has the separation property S₄, so it isn’t connected.

Below, we give the concept of connected subsets and discuss their related properties.

Definition 3.15. Let Y be a subset of convex structure X. If Y is a connected space as a subspace of X, then Y is a connected subset of X.

Theorem 3.16. Let Y be a subset of convex structure X, A, B ⊆ Y. Then A and B are separated subsets of the subspace Y if and only if they are separated subsets of the convex structure X. Therefore, Y is not a connected subset of X if and only if there are two non-empty separated sets A and B in X such that A ∪ B = Y .

Proof. Necessity: It is known that A and B are separated subsets of Y. Then we have A∩ co_Y (B) = ∅ and B∩ co_Y (A) = ∅. Since co_Y (B) = co (B) ∩ X, then A∩ co (B) ∩ X = ∅ and B∩ co (A) ∩ X = ∅, i.e., A∩ co (B) = ∅ and B∩ co (A) = ∅ are holding. Therefore, A and B are separated subsets in the convex structure X.

Sufficiency: Since A and B are separated subsets in X, such thatA∩ co (B) = ∅ and B∩ co (A) = ∅, i.e., A∩ co (B) ∩ X = ∅ and B∩ co (A) ∩ X = ∅ are holding. So, we have A∩ co_Y (B) = ∅ and B∩ co_Y (A) = ∅. Therefore, A and B are separated subsets in the subspace Y. □

Theorem 3.17. A subspace Y of a convex structure X is connected if and only if for every pair A, B of separated subsets of X such that Y = A ∪ B we have either A =∅ or B =∅.

Proof. Necessity: Suppose Y is connected and Y = A ∪ B, where A ∩ co (B) = ∅ = B ∩ co (A). Clearly, the sets A and B are separated in the space Y, so that by Theorem 3.16 one of them is empty.

Sufficiency: If Y is not connected, there exist A and B of non-empty disjoint convex subsets of Y such that Y = A ∪ B. Clearly, the sets A and B are separated in the space X, which leads to a contradiction. □

Corollary 3.18. Let Y be a connected subset of convex structure X, and if there are separated sets A and B in X such that Y ⊆ A ∪ B. Then either Y ⊆ A or Y ⊆ B.

Proof. The sets Y ∩ A and Y ∩ B are separated in X and their union is equal to Y. By Theorem 3.17 one of them must be empty, so that Y is contained in the other. □

Theorem 3.19. If a subspace Y of X is connected, then every subspace Z of X satisfying Y ⊆ Z ⊆ co (Y) is also connected.

Proof. Suppose that Z is not a connected subset of X , according to Theorem 3.16, there are non-empty separated sets A and B in X, such that Z = A ∪ B. So Y ⊆ A ∪ B. Since Y is connected, according to Corollary 3.18, either Y ⊆ A or Y ⊆ B. If Y ⊆ A, then we have Z ⊆ co (Y) ⊆ co (A) and Z∩ B ⊆ co (A) ∩ B = ∅. Thus B = Z∩ B = ∅. Similarly, if Y ⊆ B, then A =∅. □

Theorem 3.20. Let {A_i} _i∈I be a family of connected subsets of a convex structure X and ⋂_i∈IA_i≠ ∅. Then the union ⋃_i∈IA_i is connected.

Proof. We prove that if $f : ⋃_{i \in I} A_{i} \to {0, 1}$ is a convexity-preserving mapping, then f is a constant mapping. Since A_i is connected for any i ∈ I, then f limit to A_i is a constant mapping. By ⋂_i∈IA_i≠ ∅, we claim that f is a constant mapping.

□

Theorem 3.21. Let $(X, C)$ be a connected convex structure and f be a convexity- preserving function of X into a convex structure Y. Then f (X) is connected.

Proof. Assuming that f (X) is an unconnected subset of Y, then there exists A ⊆ f (X) that A is a nontrivial half-space. Therefore, f^-1 (A) is a nontrivial half-space of X, which leads to a contradiction. □

Theorem 3.22. Every product of connected convex structures is connected.

Proof. Firstly, let us consider the product X × Y of two connected convex structures, and then the product of any finite connected convex structures has proven. Suppose X and Y are connected, and set(a, b) ∈ X × Y. For any x ∈ X, {x} × Y and X × {b} are connected subsets of X × Y since {x} × Y and X × {b} are respectively isomorphic to Y and X. Simultaneously, since (x, b) ∈ ({x} × Y) ∪ (X × {b}), we have $T_{x} : = (x, b) \in ({x} \times Y) \cup (X \times {b})$ is connected by Theorem 3.20. Therefore, if X × Y = ⋃ _x∈XT_x and (a, b) ∈ ⋂ _x∈XT_x, then X × Y is connected.

Secondly, the product of any family of connected convex structures has proven. Let {X_i} _i∈I is a family of connected convex structures, and X represents the product space $\prod_{i \in I} X_{i}$ . We prove that for any given convexity-preserving functions f : X → {0, 1} and x, y ∈ X, such that f (x) =1, then we have f (y) =1.

Since f (x) =1, i.e., f^-1 (1) is non-empty convex set of X. So there exists a w ∈ f^-1 (1) which satisfies $J : = {i \in I ∣ w_{i} \neq y_{i}}$ is finite. Let $Y = \prod_{j \in J} X_{j}; Z = \prod_{i \notin J} X_{i} .$ Then Y is connected, and X is isomorphic to the product space Y × Z. Now we equate Y × Z to X. Suppose $h : Y \to Y \times Z$ stands for a identity mapping 1_Y : Y → Y, and there exist a constant mapping c : Y → Z of value (w_i) _i∉J and a diagonal mapping 1_Y △ c, then the composite mapping $f \circ h : Y \to Y \times Z \to {0, 1}$ is a convexity-preserving function, in the meantime, which is a constant mapping. Consider two points y₁ = (y_i) _i∈J, y₂ = (w_i) _i∈J of Y. Since h (y₁) = y and h (y₂) = w are holding. Therefore, f (y) = f ∘ h (y₁) = f ∘ h (y₂) = f (w) =1. □

4 Separatedness degrees in (L, M)-fuzzy convex structures

In this section, we propose the notion of separatedness degrees in (L, M)-fuzzy convex structures by means of (L, M)-fuzzy hull operators.

Definition 4.1. Let $(X, C)$ be an (L, M)-fuzzy convex structure and A, B ∈ L^X. Define $\begin{array}{l} S e p (A, B) = (\land {{(ℋ (B) (x_{λ}))}^{'} : x_{λ} \leq A}) \\ \land (\land {{(ℋ (A) (y_{μ}))}^{'} : y_{μ} \leq B}) \\ = (\underset{x_{λ} \leq A}{\land} {(ℋ (B) (x_{λ}))}^{'}) \\ \land (\underset{y_{μ} \leq B}{\land} {(ℋ (A) (y_{μ}))}^{'}) . \end{array}$ Then Sep (A, B) is said to be the separatedness degree of A and B.

We can easily gain the following result.

Proposition 4.2. Let $(X, C) : L^{X} \to {⊥_{M}, ⊤_{M}}$ be an (L, M)-fuzzy convex structure and A, B ∈ L^X. Then Sep (A, B) = ⊤ _M if and only if A and B are separated in $(X, C)$ .

Lemma 4.3. Let $(X, C)$ be an (L, M)-fuzzy convex structures and A, B ∈ L^X. If $A \land B \neq \underline{⊥}$ , then Sep (A, B) = ⊥ _M.

Proof. By $A \land B \neq \underline{⊥}$ , we take z_r ∈ J (L^X) such that z_γ ≤ A ∧ B. Thus we have $\begin{matrix} Sep (A, B) & = (⋀_{x_{λ} \leq A} (H (B) (x_{λ}))^{'}) \\ \land (⋀_{y_{μ} \leq B} (H (A) (y_{μ}))^{'}) \\ \leq {(H (B) (z_{γ}))}^{'} \land {(H (A) (z_{γ}))}^{'} \\ = ⊤_{M}^{'} \land ⊤_{M}^{'} = ⊥_{M} . \end{matrix}$

□

Lemma 4.4. Let $(X, C)$ be an (L, M)-fuzzy convex structure and A, B, C, D ∈ L^X. If C ≤ A, D ≤ B, then Sep (A, B) ≤ Sep (C, D).

Proof. If C ≤ A, D ≤ B, then $H (C) \leq H (A)$ , $H (D) \leq H (B)$ . Hence, we have $\begin{array}{l} S e p (A, B) \\ = (\underset{x_{λ} \leq A}{\land} {(ℋ (B) (x_{λ}))}^{'}) \land (\underset{y_{μ} \leq B}{\land} {(ℋ (A) (y_{μ}))}^{'}) \\ \leq (\underset{x_{λ} \leq A}{\land} {(ℋ (D) (x_{λ}))}^{'}) \land (\underset{y_{μ} \leq B}{\land} {(ℋ (C) (y_{μ}))}^{'}) \\ \leq (\underset{x_{λ} \leq C}{\land} {(ℋ (D) (x_{λ}))}^{'}) \land (\underset{y_{μ} \leq D}{\land} {(ℋ (C) (y_{μ}))}^{'}) \\ = S e p (C, D) . \end{array}$ □

Lemma 4.5. Let $(X, C)$ be an (L, M)-fuzzy convex structure, A, B ∈ L^X and a ∈ J (L). Then Sep (A, B) ≱a if and only if there exist D, E ∈ L^X such that $D \geq A, E \geq B, D \land B = E \land A = \underline{⊥}, C {(D)}^{'} \lor C {(E)}^{'} ≱ a .$

Proof. Necessity: Suppose that (Sep (A, B)) ′≱a. Then (Sep (A, B)) ′≱b for some b ∈ β^* (a). This implies that $⋁_{x_{λ} \leq A} H (B) (x_{λ}) \lor ⋁_{y_{μ} \leq B} H (A) (y_{μ}) ≱ b .$ Furthermore, we have $⋁_{x_{λ} \leq A} ⋀_{x_{λ} ≰ E \geq B} C {(E)}^{'} \lor ⋁_{y_{μ} \leq B} ⋀_{y_{μ} ≰ D \geq A} C {(D)}^{'} ≱ b .$ Hence, for any x_λ ≤ A and y_μ ≤ B, there exist D_{y
_μ}, E_{x
_λ} ∈ L^X such that x_λ≰E_{x
_λ} ≥ B, y_μ≰D_{y
_μ} ≥ A and $C {(D_{y_{μ}})}^{'} \lor C {(E_{x_{λ}})}^{'} ≱ b$ . Let E = ⋀ _{x_λ≤A}E_{x
_λ}, D = ⋀ _{y_μ≤B}D_{y
_μ}. Then obviously, we have that $D \geq A, E \geq B, D \land B = E \land A = \underline{⊥}$ and $\begin{array}{l} C (D)' \lor C (E)' = (C (\underset{y_{μ} \leq B}{\land} D_{y_{μ}}))' \lor (C (\underset{x_{λ} \leq A}{\land} E_{x_{λ}}))' \\ \leq \underset{y_{μ} \leq B}{\lor} C (D_{y_{μ}})' \lor \underset{x_{λ} \leq A}{\lor} C (E_{x_{λ}})' \\ ≱ a . \end{array}$ Sufficiency: If there exist D, E ∈ L^X, such that $D \geq A, E \geq B, D \land B = E \land A = \underline{⊥}, C {(D)}^{'} \lor C {(E)}^{'} ≱ a .$ Then since $\begin{array}{l} (S e p (A, B))' \\ = \underset{x_{λ} \leq A}{\lor} ℋ (B) (x_{λ}) \lor \underset{y_{μ} \leq B}{\lor} ℋ (A) (y_{μ}) \\ = \underset{x_{λ} \leq A}{\lor} \underset{x_{λ} ≰ G \geq B}{\land} C (G)' \lor \underset{y_{μ} \leq B}{\lor} \underset{y_{μ} ≰ H \geq A}{\land} C (H)' \\ \leq C (D)' \lor C (E)', \end{array}$ we can obtain that (Sep (A, B)) ′≱a . □

5 Connectedness degrees in (L, M)-fuzzy convex structures

In this section, the notion of connectedness degrees is investigated in (L, M)-fuzzy convex structures by separatedness degrees. Many properties of connectivity in general convex structures can be generalized to (L, M)-fuzzy convex structures.

Definition 5.1. Let $(X, C)$ be an (L, M)-fuzzy convex structure and G ∈ L^X. Define $\begin{array}{l} C o n (G) \\ = \land {S e p (A, B)^{'} : A, B \in L^{X} \ {\underline{⊥}}, G = A \lor B} \\ = \land {\underset{x_{λ} \leq A}{\lor} ℋ (B) (x_{λ}) \lor \underset{y_{μ} \leq B}{\lor} ℋ (A) (y_{μ}) : A, B \\ \in L^{X} \ {\underline{⊥}}, G = A \lor B} . \end{array}$ Then Con (G) is said to be the connectedness degree of G.

We can easily gain the following result.

Proposition 5.2. Let $C : L^{X} \to {⊥_{M}, ⊤_{M}}$ be an (L, M)-fuzzy convex structure and G ∈ L^X. Then Con (G) = ⊤ _M if and only if G is connected in $(X, C)$ .

Next, we provide some equivalent characterizations of connectedness degrees.

Theorem 5.3. Let $(X, C)$ be an (L, M)-fuzzy convex structure and G ∈ L^X, then $\begin{array}{l} C o n (G) = \land {C {(A)}^{'} \lor C {(B)}^{'} : G \land A \neq \underline{⊥}, G \land B \\ \neq \underline{⊥}, G \land A \land B = \underline{⊥}, G \leq A \lor B} . \end{array}$

Proof. On one hand, we have the following inequality: $\begin{array}{l} C o n (G) = \land {\underset{x_{λ} \leq A}{\lor} ℋ (B) (x_{λ}) \lor \underset{y_{μ} \leq B}{\lor} ℋ (A) (y_{μ}) : A, B \\ \in L^{X} \ {\underline{⊥}}, G = A \lor B} \\ = \land {\underset{x_{λ} \leq A}{\lor} \underset{x_{λ} ≰ D \geq B}{\land} C (D)' \lor \underset{y_{μ} \leq B}{\lor} \underset{y_{μ} ≰ E \geq A}{\land} C (E)' \\ : A, B \in L^{X} \ {\underline{⊥}}, G = A \lor B} \\ = \land {\underset{x_{λ} \leq G \land A}{\lor} \underset{x_{λ} ≰ D \geq G \land B}{\land} C (D)' \lor \underset{y_{μ} \leq G \land B}{\lor} \underset{y_{μ} ≰ E \geq G \land A}{\land} C (E)' : G \land A \neq \underline{⊥}, G \land B \neq \underline{⊥}, G \leq A \lor B} \\ \leq \land {\underset{x_{λ} \leq G \land A}{\lor} C {(B)}^{'} \lor \underset{y_{μ} \leq G \land B}{\lor} C {(A)}^{'} : G \land A \neq \underline{⊥}, G \land B \neq \underline{⊥}, G \land A \land B = \underline{⊥}, G \leq A \lor B} \\ = \land {C {(B)}^{'} \lor C {(A)}^{'} : G \land A \neq \underline{⊥}, G \land B \neq \underline{⊥}, G \land A \land B = \underline{⊥}, G \leq A \lor B} . \end{array}$ On the other hand, we suppose that Con (G) ≱a (a ∈ J (L)), then there exist $A, B \in L^{X} ∖ {\underline{⊥}}$ , such that G = A ∨ B and (Sep (A, B)) ′≱a. By Lemma 4.5, we know that there exist D, E ∈ L^X such that $D \geq A, E \geq B, D \land B = E \land A = \underline{⊥}, C {(D)}^{'} \lor C {(E)}^{'} ≱ a .$ Obviously, $G \land D \neq \underline{⊥}, G \land E \neq \underline{⊥}, G \land D \land E = \underline{⊥}, G \leq D \lor E$ . Hence, we have $\begin{matrix} ⋀ {C {(B)}^{'} \lor C {(A)}^{'} : G \land A \neq \underline{⊥}, G \land B \neq \underline{⊥}, \\ G \land A \land B = \underline{⊥}, G \leq A \lor B} ≱ a . \end{matrix}$ Therefore $\begin{matrix} Con (G) \geq ⋀ {C {(B)}^{'} \lor C {(A)}^{'} : G \land A \neq \underline{⊥}, \\ G \land B \neq \underline{⊥}, G \land A \land B = \underline{⊥}, G \leq A \lor B} . \end{matrix}$ □

Theorem 5.4. Let $(X, C)$ be an (L, M)-fuzzy convex structure and G ∈ L^X. Then $\begin{matrix} Con (G) = & ⋀_{A, B \in D} (C {(A)}^{'} \lor C {(B)}^{'}) \\ = & ⋀_{A, B \in D} (C {(A^{'})}^{'} \lor C {(B^{'})}^{'}), \end{matrix}$ where $D = {(A, B) \in L^{X} \times L^{X} : A \neq \underline{⊥}, B \neq \underline{⊥}, A \lor B = G, A \land B = \underline{⊥}}$ .

Remark 5.5. If L = M = {0, 1}, then we can obtain Theorem 3.3, which means that Theorem 5.4 is a generalization of Theorem 3.3.

In what follows, we provide some examples of connectedness degrees in (L, M)-fuzzy convex structures.

Example 5.6. Let X = {x, y} and L = M = [0, 1]. Define B ∈ [0, 1] ^X by B (x) =0.5 and B (y) =0, and define C ∈ [0, 1] ^X by C (x) =0 and C (y) =0.5, respectively. Let $C : [0, 1]^{X} \to [0, 1]$ be defined as follows: $C (A) = {\begin{matrix} 1, & A \in {\underline{⊥}, \underline{⊤}, \underline{0.5}}, \\ 0.5, & A \in {B^{'}, C^{'}}, \\ 0, & others . \end{matrix}$ Then $C$ is an (L, M)-fuzzy convex structure on X. We claim that $Con (\underline{a}) = 1$ for any a ∈ (0, 0.5].

Example 5.7. Let L = M = [0, 1] and μ_i be fuzzy subsets of X = {a, b, c} where i = {1, 2, 3} is defined as follows: $μ_{1} (a) = 1.0, μ_{1} (b) = 1.0, μ_{1} (c) = 0.0,$ $μ_{2} (a) = 0.2, μ_{2} (b) = 0.2, μ_{2} (c) = 1.0,$ $μ_{3} (a) = 0.0, μ_{3} (b) = 0.0, μ_{3} (c) = 1.0 .$ Define $C : [0, 1]^{X} \to [0, 1]$ on X as follows: $C (ν) = {\begin{matrix} 1, & ν \in {\underline{⊥}, \underline{⊤}}, \\ 0.25, & ν = μ_{1}, \\ 0.25, & ν = \underline{⊤} - μ_{2}, \\ 0.33, & ν = μ_{3}, \\ 0, & others . \end{matrix}$ Then $C$ is an (L, M)-fuzzy convex structure on X. We claim that Con (G) =0.75 for any $G = μ_{1}, μ_{3}, \underline{⊤}$ . Con (G) =0.33 for any G = μ₂ and $Con (\underline{a}) = 0.33$ for any a ∈ (0, 1].

Example 5.8. Let X = {x} be a single set and L = {0_L, a, b, 1_L} be the diamond lattice, i.e, a ∨ b = 1_L, a ∧ b = 0_L, a′ = b and b′ = a. Then $L^{X} = {\underline{⊥}, \underline{a}, \underline{b}, \underline{⊤}}$ . Define a mapping $C : L^{X} \to M$ on X as follows: $C (A) = {\begin{matrix} 1_{L}, & A = {\underline{⊥}, \underline{⊤}}, \\ b, & A = \underline{a}, \\ a, & A = \underline{b} . \end{matrix}$ Then $C$ is an (L, M)-fuzzy convex structure on X. We claim that $Con (\underline{⊤}) = 1$ .

Corollary 5.9. Let $(X, C)$ be an (L, M)-fuzzy convex structure. Then $\begin{matrix} Con (\underline{⊤}) & = ⋀ {C {(A)}^{'} \lor C {(B)}^{'} : A \neq \underline{⊥}, B \neq \underline{⊥}, \\ A \land B = \underline{⊥}, \underline{⊤} = A \lor B} \\ = ⋀ {C {(A^{'})}^{'} \lor C {(B^{'})}^{'} : A \neq \underline{⊥}, B \neq \underline{⊥}, \\ A \land B = \underline{⊥}, \underline{⊤} = A \lor B} . \end{matrix}$

Theorem 5.10. For any G, H ∈ L^X, we have $Con (G \lor H) \geq (Sep (G, H))^{'} \land Con (G) \land Con (H) .$

Proof. Let a ∈ J (L) and a ≤ (Sep (G, H)) ′ ∧ Con (G) ∧ Con (H). Now we prove that Con (G ∨ H) ≥ a. Suppose that Con (G ∨ H) ≱a, then by Theorem 5.3 there exist A, B ∈ L^X such that $(G \lor H) \land A \neq \underline{⊥}, (G \lor H) \land B \neq \underline{⊥}, (G \lor H) \land A \land B = \underline{⊥}, G \lor H \leq A \lor B, C {(A)}^{'} \lor C {(B)}^{'} ≱ a .$ By ( $G \lor H) \land A \neq \underline{⊥}$ , we know that one of $G \land A \neq \underline{⊥}$ and $H \land A \neq \underline{⊥}$ must be true.

Suppose that $G \land A \neq \underline{⊥}$ (the case of $H \land A \neq \underline{⊥}$ is similar). Then $G \land B = \underline{⊥}$ , otherwise if $G \land B \neq \underline{⊥}$ , then by $G \land A \neq \underline{⊥}, G \land B \neq \underline{⊥}, G \land A \land B = \underline{⊥}, G \leq A \lor B, C {(A)}^{'} \lor C {(B)}^{'} ≱ a,$ we claim that Con (G) ≱a, which leads to a contradiction. In this case by $(G \lor H) \land B \neq \underline{⊥}$ , we know that $H \land B \neq \underline{⊥}$ . Analogously we can prove $H \land A = \underline{⊥}$ . Therefore by G ∨ H ≤ A ∨ B, we can obtain that G ≤ A and H ≤ B. Hence, by $G \leq A, H \leq B, G \land B = H \land A = \underline{⊥}, C {(A)}^{'} \lor C {(B)}^{'} ≱ a$ and Lemma 4.5 we claim that (Sep (G, H)) ′≱a, which leads to a contradiction. This shows that Con (G ∨ H) ≥ a. It is proved that Con (G ∨ H) ≥ (Sep (G, H)) ′ ∧ Con (G) ∧ Con (H). □

Corollary 5.11. Let $(X, C)$ be an (L, M)-fuzzy convex structure, G, H ∈ L^X. If $G \land H \neq \underline{⊥}$ , then Con (G ∨ H) ≥ Con (G) ∧ Con (H).

Theorem 5.12. Let $(X, C)$ be an (L, M)-fuzzy convex structure and G ∈ L^X. Then

$\begin{matrix} Con (G) & = ⋀_{x_{λ}, y_{μ} \leq G} ⋁ {Con (D_{x_{λ} y_{μ}}) : x_{λ}, y_{μ} \\ \leq D_{x_{λ} y_{μ}} \leq G} . \end{matrix}$

Proof. It is obvious that $\begin{matrix} Con (G) & \leq ⋀_{x_{λ}, y_{μ} \leq G} ⋁ {Con (D_{x_{λ} y_{μ}}) : x_{λ}, y_{μ} \\ \leq D_{x_{λ} y_{μ}} \leq G} . \end{matrix}$ Now we prove that $\begin{matrix} Con (G) & \geq ⋀_{x_{λ}, y_{μ} \leq G} ⋁ {Con (D_{x_{λ} y_{μ}}) : x_{λ}, y_{μ} \\ \leq D_{x_{λ} y_{μ}} \leq G} . \end{matrix}$ Suppose that $\begin{matrix} ⋀_{x_{λ}, y_{μ} \leq G} ⋁ {Con (D_{x_{λ} y_{μ}}) : x_{λ}, y_{μ} \leq D_{x_{λ} y_{μ}} \\ \leq G} \geq a . \end{matrix}$ Take a fixed x_λ ≤ G. Then for any y_μ ≤ G, there exists a D_{x
_λ
y
_μ} ∈ L^X, such that x_λ, y_μ ≤ D_{x
_λ
y
_μ} ≤ G and Con (D_{x
_λ
y
_μ}) ≥ a. Let D_{x
_λ} = ⋁ _{y_μ≤G}D_{x
_λ
y
_μ}. Obviously D_{x
_λ} = G and $⋀_{y_{μ} \leq G} D_{x_{λ} y_{μ}} \neq \underline{⊥}$ . By Corollary 5.11 we easily obtain $Con (G) = Con (D_{x_{λ}}) \geq ⋀_{y_{μ} \leq G} Con (D_{x_{λ} y_{μ}}) \geq a .$ This shows that $\begin{matrix} Con (G) & \geq ⋀_{x_{λ}, y_{μ} \leq G} ⋁ {Con (D_{x_{λ} y_{μ}}) : x_{λ}, y_{μ} \\ \leq D_{x_{λ} y_{μ}} \leq G} . \end{matrix}$

□

Theorem 5.13. If ${f_{L}}^{\to} : (X, C_{1}) \to (Y, C_{2})$ is an (L, M)-convexity-preserving function. Then $Con (G) \leq Con ({f_{L}}^{\to} (G)) .$

Proof. It suffices to show $⋁_{(C, D) \in D_{Y}} C_{2} (C) \land C_{2} (D) \leq ⋁_{(A, B) \in D_{X}} C_{1} (A) \land C_{1} (B) .$ Let $(C, D) \in D_{Y}$ and define A^* = f^-1 (C) and B^* = f^-1 (D). Then we have $(A^{*}, B^{*}) \in D_{X}$ . Since ${f_{L}}^{\to} : (X, C_{1}) \to (Y, C_{2})$ is an (L, M)-convexity-preserving function, $C_{2} (C) \leq C_{1} (A^{*})$ and $C_{2} (D) \leq C_{1} (B^{*})$ . Therefore, $\begin{matrix} C_{2} (C) \land C_{2} (D) & \leq C_{1} (A^{*}) \land C_{1} (B^{*}) \\ \leq ⋁_{(A, B) \in D_{X}} C_{1} (A) \land C_{1} (B) . \end{matrix}$ Hence we gain that $⋁_{(C, D) \in D_{Y}} C_{2} (C) \land C_{2} (D) \leq ⋁_{(A, B) \in D_{X}} C_{1} (A) \land C_{1} (B) .$ □ Corollary 5.14. Let ${(X_{t}, C_{t})}_{t \in T}$ be a family of (L, M)-fuzzy convex structures and $(X, C)$ be the product space of ${(X_{t}, C_{t})}_{t \in T}$ , G ∈ L^X. Then $Con (\underline{⊤}) \leq ⋀_{t \in T} (Con (⊤_{t})) .$

6 Conclusion

In this paper, we give the definitions of separatedness and connectivity in classical convex structures, which based on the consideration to hull operators. Meanwhile, we discussed the related properties of connectivity in classical convex structures and provided some examples. Besides, the connectivity of classical convex structures is extended to the (L, M) case. We then propose the separatedness degree of (L, M)-fuzzy convex structures by taking (L, M)-fuzzy convex hull operators. Furthermore, the notion of connectedness degrees of (L, M)- fuzzy convex structures is introduced. Finally, many properties of connectivity in general convex structures can be generalized to (L, M)-fuzzy convex structures. Our research fills the gap in connectivity research in convex results. Meaningfully, our work has important theoretical significance and application value and will be of great influence in the convex structures and subsequent researches.

Footnotes

Acknowledgments

The authors express their sincere thanks to the editors and anonymous reviewers for their most valuable comments and suggestions for improving this article greatly. This work was supported by the Natural Science Foundation of China (No. 11761055), the Natural Science Foundation of Inner Mongolia (No.2023MS01009) and Fundamental Research Funds for the Inner Mongolia Normal University (No. 2023JBQN044).

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