A new characterization of the product of ( L,M )-fuzzy convex structures 1

Abstract

In order to give a characterization of the product of (L, M)-fuzzy convex structures, the notion of convex (L, M)-fuzzy hull operators is presented, it is proved that the category of (L, M)-fuzzy convex structures and the category of convex (L, M)-fuzzy hull operators are isomorphic. In particular, the lattices structure of convex (L, M)-fuzzy hull operators and a new characterization of the product of (L, M)-fuzzy convex structures are given.

Keywords

product

1 Introduction

Convexity theory [18, 20] plays an important role in mathematical research areas. Actually, It exists in many mathematical structures, such as lattices, graphs, and topological spaces, and so on (see, for example, [4 , 19]). With the development of fuzzy mathematics, many mathematics structures have been endowed with fuzzy sets. Convex structures have also been generalized to the fuzzy case. In general, there are three well-established ways to extend convex structures to the fuzzy context, which are L-convex structures (see, for example, [3 , 11–13]), M-fuzzifying convex structures (see, for example, [6 , 24]) and (L, M)-fuzzy convex structures (see, for example, [15 , 29]), respectively. It should be stressed here that the notion of (L, M)-fuzzy convex structures was first introduced by Shi and Xiu in [16], and it was shown that that both L-convex structures and M-fuzzifying convex structures can be regarded as special cases of (L, M)-fuzzy convex structures.

Pang [12] proposed a kind of (L, M)-fuzzy hull operators and established its relationship with (L, M)-fuzzy convex structures. Xiu and Li [23] introduced the notions of (L, M)-fuzzy concave spaces, (L, M)-fuzzy interior spaces, (L, M)-fuzzy interior relations and (L, M)-fuzzy hull relations. They proved that the category of (L, M)-fuzzy concave spaces, the category of (L, M)-fuzzy interior spaces, the category of (L, M)-fuzzy interior spaces and the category of (L, M)-fuzzy hull spaces are isomorphic. They also proved that the category of (L, M)-fuzzy concave spaces and the category of (L, M)-fuzzy convex spaces were isomorphic. Sayed et al. [15] proposed another kind of (L, M)-fuzzy hull operators (which is called Sayed’s (L, M)-fuzzy hull operator), they thought that there is a one-to-one correspondence between Sayed’s (L, M)-fuzzy hull operators and (L, M)-fuzzy convex structures, and ${CO}^{★}$ which constructed by projections and (L, M)-fuzzy convex structures (see Corollary 2.10) can be characterized by Sayed’s (L, M)-fuzzy hull operator induced by the product of (L, M)-fuzzy convex structures, in other words, ${CO}^{★} = {CO}_{\prod_{i \in J} C_{i}}$ . Subsequently, Zhao and Song [27] not only pointed out that the proofs of three results are not true, but also gave correct proofs of these results. Moreover, Zhao and Hu [28] proposed the notions of concave (L, M)-fuzzy interior operators and proved that the category of (L, M)-fuzzy concave structures and the category of concave (L, M)-fuzzy interior operators are isomorphic. Actually, suppose that ${CO}^{★}$ is a Sayed’s (L, M)-fuzzy hull operator on X, ${CO}^{★} = {CO}_{\prod_{i \in J} C_{i}}$ , and $C_{{CO}_{C}} = C$ (here $C$ is an arbitrary (L, M)-fuzzy convex structure), then $\prod_{j \in J} C_{j} = C_{{CO}^{★}},$ i.e., the product of (L, M)-fuzzy convex structures can be characterized by $C_{{CO}^{★}} .$ However, ${CO}^{★}$ need not be a Sayed’s (L, M)-fuzzy hull operator on X, there is no one-to-one correspondence between Sayed’s (L, M)-fuzzy hull operators and (L, M)-fuzzy convex structures, and ${CO}^{★} \neg = {CO}_{\prod_{j \in J} C_{j}}$ (see [28]). So, a natural problem is: how to construct the product of (L, M)-fuzzy convex structures via Sayed’s (L, M)-fuzzy hull operators or ${CO}^{★}$ ? By above motivations, we will further study Sayed’s (L, M)-fuzzy hull operators.

This paper is organized as follow. In Section 2, we will review some of the necessary concepts and give some results on Sayed’s (L, M)-fuzzy hull operators. In Section 3, we will propose the concept of convex (L, M)-fuzzy hull operators, establish its relationship with (L, M)-fuzzy convex structures in a categorical sense, and give a new characterization of the product of (L, M)-fuzzy convex structures.

2 Preliminaries

In this paper, let M be a complete lattice with the smallest element 0_M and the largest element 1_M, respectively, and M_{0_M} = M - {0_M}. An element u in a complete lattice is said to be coprime if u ≤ s ∨ t implies that u ≤ s or u ≤ t. The set of all non-zero coprime elements of M is denoted J (M). For c, d ∈ M, we say that c is wedge below d (denoted by c ≺ d) in M if for all subsets N ⊂ M, d ≤ ⋁ N ⇒ ∃ n ∈ N such that c ≤ n.

A complete lattice M is completely distributive lattice iff b = ⋁ {a ∈ M ∣ a ≺ b} for each b ∈ M. β (b) = {a ∈ M ∣ a ≺ b} is the greatest minimal family of b, β^∗ (b) = β (b) ∩ J (M) is a minimal family of b. α (b) is the greatest maximal family of b. When M is completely distributive lattice, each element b in M has the greatest minimal family. In particular, for any a ∈ J (M), a ∈ β^∗ (b) iff a ≺ b, and b = ⋁ β^∗ (b) for each b ∈ M (see [21]).

In the rest of the paper, L and M always denote completely distributive lattice. We notice that L^X, the set of all L-subsets of X, is also a completely distributive lattice with pointwise order. Its smallest element and the largest element are denoted 0_X and 1_X, respectively. For a directed subset A ⊆ L^X, we use ⋁^dD to denote its supremum. Let X, Y are two nonempty sets and let f : X ⟶ Y be a mapping. Define f^→ : L^X ⟶ L^Y and f^← : L^Y ⟶ L^X as follows: (1) ∀A ∈ L^X, ∀ y ∈ Y, $f_{L}^{\to} (A) (y) = ⋁_{f (x) = y} A (x);$ (2) ∀A ∈ L^Y, ∀ x ∈ X, $f_{L}^{\leftarrow} (A) (x) = A (f (x)) .$ It can be verified that the pair (f^→, f^←) is a Galois connection on (L^X, ≤) and (L^Y, ≤) (see [14, 25]). Some concepts related to category theory can be found in [1].

Definition 2.1. ([2]) A mappig $C : L^{X} ⟶ M$ is called an (L, M)-fuzzy closure system on X if it satisfies:

(LMC1) $C (0_{X}) = C (1_{X}) = 1_{M}$ .

(LMC2) If {B_i : i ∈ J} ⊆ L^X is nonempty, then $C (⋀_{i \in J} B_{i}) \geq ⋀_{i \in J} C (B_{i}) .$

Definition 2.2. ([9, 16]) A closure system $C$ is called (L, M)-fuzzy convex structure, if one of the following conditions hold (the second then following as a consequence):

(LMC3) If {B_i : i ∈ J} ⊆ L^X is totally ordered, then $C (⋁_{i \in J} B_{i}) \geq ⋀_{i \in J} C (B_{i}) .$

(LMC3) ^∗ If {B_i : i ∈ J} ⊆ L^X is directed, then $C (⋁_{i \in J} B_{i}) \geq ⋀_{i \in J} C (B_{i}) .$

If $C$ is an (L, M)-fuzzy convex structure on X, then the pair $(X, C)$ is called an (L, M)-fuzzy convex space. Let $(X, C_{X})$ and $(Y, C_{Y})$ be (L, M)-fuzzy convex spaces and let g : X ⟶ Y be a mapping. We say g is (L, M)-convexity-preserving if $C_{Y} (B) \leq C_{X} (g^{\leftarrow} (B))$ for all B ∈ L^Y ((L, M)-CP, for short). It is easy to check that all (L, M)-fuzzy convex spaces as objects and all (L, M)-CP mappings as morphisms form a category, which will be denoted by (L, M)-FC.

Theorem 2.3. ([10, 16]) Let $(\prod_{j \in J} X_{j}, \prod_{j \in J} C_{j})$ be the product of ${(X_{j}, C_{j}) : j \in J}$ . Then $\prod_{j \in J} C_{j}$ is the coarsest (L, M)-fuzzy convex structure on X such that all projection mappings are (L, M)-CP functions.

Theorem 2.4. ([10]) Let ${(X_{j}, C_{j}) : j \in J}$ be a set of (L, M)-fuzzy convex spaces, $X = \prod_{j \in J} X_{j},$ and p_j : X ⟶ X_j the projection for each j ∈ J . Then, for each A ∈ L^X, $\begin{matrix} (\prod_{j \in J} C_{j}) (A) \\ = ⋁_{⋁_{k \in K}^{d} A_{k} = A} ⋀_{k \in K} ⋁_{⋀_{i \in I_{k}} A_{ki} = A_{k}} ⋀_{i \in I_{k}} ⋁_{j \in J} ⋁_{p_{j}^{\leftarrow} (B^{j}) = A_{ki}} C_{j} (B^{j}) . \end{matrix}$

Definition 2.5. A mapping $CO : L^{X} \times M_{0_{M}} ⟶ L^{X}$ is called an (L, M)-fuzzy weak hull operator if it satisfies the following conditions: for any A, B ∈ L^X and r, s ∈ M_{0_M},

(CO1) $CO (0_{X}, r) = 0_{X} .$

(CO2) $B \leq CO (B, r) .$

(CO3) If r ≤ s, then $CO (B, r) \leq CO (B, s) .$

(CO4) If A ≤ B, then $CO (A, r) \leq CO (B, r) .$

(CO5) $CO (CO (B, r), r) = CO (B, r) .$

If $CO$ is an (L, M)-fuzzy weak hull operator on X, then the pair $(X, CO)$ is called an (L, M)-fuzzy weak hull space. Let $(X, {CO}_{X})$ and $(Y, {CO}_{Y})$ be two (L, M)-fuzzy weak hull spaces, then a function f : X ⟶ Y is called an (L, M)-fuzzy weak hull preserving function ((L, M)-WHP, for short) if ${CO}_{Y} (f^{\to} (A), r) \geq f^{\to} ({CO}_{X} (A, r))$ for all A ∈ L^X and r ∈ M_{0_M}.

Definition 2.6. ([15]) If $CO$ is a weak (L, M)-fuzzy hull operator, and satisfies the following conditions:

(CO6) if {B_i : i ∈ J} ⊆ L^X is nonempty and totally ordered by inclusion, then $CO (⋁_{i \in J} B_{i}, r) = ⋁_{i \in J} CO (B_{i}, r) (\forall r \in M_{0_{M}}) .$ Then $CO$ is called a Sayed’s (L, M)-fuzzy hull operator. If $CO$ is a Sayed’s (L, M)-fuzzy hull operator on X, then the pair $(X, CO)$ is called a Sayed’s (L, M)-fuzzy hull space ([15]). Let $(X, {CO}_{X})$ and $(Y, {CO}_{Y})$ be two Sayed’s (L, M)-fuzzy hull spaces, then a function g : X ⟶ Y is called an (L, M)-fuzzy hull preserving function ((L, M)-HP, for short) if $g^{\to} ({CO}_{X} (B, r)) \leq {CO}_{Y} (g^{\to} (B), r)$ for all B ∈ L^X and r ∈ M_{0_M}. The category of all Sayed’s (L, M)-fuzzy hull spaces as objects and their hull preserving functions as morphisms is denoted by (L, M)-FH.

Theorem 2.7. ([15]) (1) Let $(X, C)$ be an (L, M)-fuzzy convex space. Define a mapping ${CO}_{C} : L^{X} \times M_{0_{M}} ⟶ L^{X}$ as follows: ∀A ∈ L^X and ∀s ∈ M_{0_M},

$\begin{matrix} {CO}_{C} (A, s) = ⋀ {B \in L^{X} : A \leq B, C (B) \geq s} . \end{matrix}$ Then ${CO}_{C}$ is a Sayed’s (L, M)-fuzzy hull operator, and therefore an (L, M)-fuzzy weak hull operator.

(2) If $CO$ is a Sayed’s (L, M)-fuzzy hull operator, then a mapping $C_{CO} : L^{X} ⟶ M$ defined by $\begin{matrix} C_{CO} (A) = ⋁ {r \in M_{0_{M}} : A = CO (A, r)} \end{matrix}$ is an (L, M)-fuzzy convex structure on X.

Proposition 2.8. ([15],[28]) Let $(X, C)$ and $(Y, D)$ be (L, M)-fuzzy convex spaces. Then $f : (X, C) ⟶ (Y, D)$ is an (L, M)-CP if and only if $f : (X, {CO}_{C}) ⟶ (Y, {CO}_{D})$ is an (L, M)-HP.

Theorem 2.9. ([28]) Let X be any set, ${(X_{j}, {CO}_{j})}_{j \in J}$ be a family of (L, M)-fuzzy weak hull spaces, and f_j : X ⟶ X_j be a function for each j ∈ J. Define ${CO}^{*} : L^{X} \times M_{0_{M}} ⟶ L^{X}$ as follows: for each r ∈ M_{0_M} and A ∈ L^X,

${CO}^{*} (A, r) = ⋀_{j \in J} f_{j}^{\leftarrow} ({CO}_{j} (f_{j}^{\to} (A), r)) .$ Then,

(i) ${CO}^{*}$ is the finest (L, M)-fuzzy weak hull operator on X such that $f_{j} : (X, {CO}^{*}) ⟶ (X_{j}, {CO}_{j})$ is an (L, M)-WHP for each j ∈ J.

(ii) Let $(Y, {CO}_{Y})$ be an (L, M)-fuzzy weak hull space, then $g : (Y, {CO}_{Y}) ⟶ (X, {CO}^{*})$ is an (L, M)-WHP iff $f_{j} \circ g : (Y, {CO}_{Y}) ⟶ (X_{j}, {CO}_{j})$ is an (L, M)-WHP for each j ∈ J .

Corollary 2.10. ([28]) Let $X = \prod_{j \in J} X_{j}$ , p_j : X ⟶ X_j be the j-th projection and $(\prod_{j \in J} X_{j}, \prod_{j \in J} C_{j})$ be the product of ${(X_{j}, C_{j})}_{j \in J} .$ Define ${CO}^{★} : L^{X} \times M_{0_{M}} ⟶ L^{X}$ as follows: ∀A ∈ L^X and ∀r ∈ M_{0_M},

$\begin{matrix} {CO}^{★} (A, r) = ⋀_{j \in J} p_{j}^{\leftarrow} ({CO}_{C_{j}} (p_{j}^{\to} (A), r)) . \end{matrix}$ Then ${CO}^{★}$ is an (L, M)-fuzzy weak hull operator on X, and ${CO}_{\prod_{j \in J} C_{j}} \leq {CO}^{★}$ .

Remark 2.11. Let ${CO}^{★}$ is defined as Corollary 2.10.

(1) If J is a finite index set, then ${CO}^{★}$ is a Sayed’s (L, M)-fuzzy hull operator on X.

(2) If J is an arbitrary index set, then ${CO}^{★}$ need not be a Sayed’s (L, M)-fuzzy hull operator on X. And, there is no ${CO}_{\prod_{i \in J} C_{i}} = {CO}^{★}$ in general.

In order to explain Remark 2.11(1), let’s look at the case where J = {1, 2}, we need to verify the following condition if {A_i : i ∈ I} ⊆ L^X is nonempty and totally ordered by inclusion, then ${CO}^{★} (⋁_{i \in I} A_{i}, r) = ⋁_{i \in I} {CO}^{★} (A_{i}, r) .$ Notice that ${CO}^{★}$ is an (L, M)-fuzzy weak hull operator on X. So, for each i ∈ I, ${CO}^{★} (⋁_{i \in I} A_{i}, r) \geq {CO}^{★} (A_{i}, r) .$ Hence, ${CO}^{★} (⋁_{i \in I} A_{i}, r) \geq ⋁_{i \in I} {CO}^{★} (A_{i}, r) .$

For the inverse inequality, we easily obtain that {p_j (A_i) : i ∈ I} ⊆ L^X is nonempty and totally ordered by inclusion for each j = 1, 2. By Corollary 2.10, we have $\begin{matrix} {CO}^{★} & (⋁_{i \in I} A_{i}, r) \\ = & p_{1}^{\leftarrow} ({CO}_{C_{1}} (p_{1}^{\to} (⋁_{i \in I} A_{i}), r)) \\ ⋀ p_{2}^{\leftarrow} ({CO}_{C_{2}} (p_{2}^{\to} (⋁_{i \in I} A_{i}), r)) \\ = & ⋁_{i \in I} [p_{1}^{\leftarrow} ({CO}_{C_{1}} (p_{1}^{\to} (A_{i}), r))] \\ ⋀ ⋁_{i \in I} [p_{2}^{\leftarrow} ({CO}_{C_{2}} (p_{2}^{\to} (A_{i}), r))] . \end{matrix}$ Let b ∈ M and $b ≺ {CO}^{★} (⋁_{i \in I} A_{i}, r)$ , there esists i₀ ∈ I such that $b \leq p_{1}^{\leftarrow} ({CO}_{C_{1}} (p_{1}^{\to} (A_{i_{0}}), r))$ and there esists j₀ ∈ I such that $b \leq p_{2}^{\leftarrow} ({CO}_{C_{2}} (p_{2}^{\to} (A_{j_{0}}), r))$ . Since {A_i : i ∈ I} ⊆ L^X is nonempty and totally ordered by inclusion, it follows that A_{i
₀} ≤ A_{j
₀} or A_{j
₀} ≤ A_{i
₀}. Without loss of generality, assume that A_{i
₀} ≤ A_{j
₀}. Then, $\begin{matrix} b & \leq & p_{1}^{\leftarrow} ({CO}_{C_{1}} (p_{1}^{\to} (A_{i_{0}}), r)) ⋀ p_{2}^{\leftarrow} ({CO}_{C_{2}} (p_{2}^{\to} (A_{j_{0}}), r)) \\ \leq & p_{1}^{\leftarrow} ({CO}_{C_{1}} (p_{1}^{\to} (A_{j_{0}}), r)) ⋀ p_{2}^{\leftarrow} ({CO}_{C_{2}} (p_{2}^{\to} (A_{j_{0}}), r)) \\ \leq & ⋁_{i \in I} ⋀_{j \in {1, 2}} p_{j}^{\leftarrow} ({CO}_{C_{j}} (p_{j}^{\to} (A_{i}), r)) \\ = & ⋁_{i \in I} {CO}^{★} (A_{i}, r) . \end{matrix}$ Hence, ${CO}^{★} (⋁_{i \in I} A_{i}, r) \leq ⋁_{i \in I} {CO}^{★} (A_{i}, r) .$ It follows that ${CO}^{★}$ is a Sayed’s (L, M)-fuzzy hull operator on X.

(2) According to the above proof, if J is any index set, we can not obtain the following inequality. $\begin{matrix} ⋀_{j \in J} ⋁_{i \in I} p_{j}^{\leftarrow} ({CO}_{C_{j}} (p_{j}^{\to} (A_{i}), r)) \\ \leq ⋁_{i \in I} ⋀_{j \in J} p_{j}^{\leftarrow} ({CO}_{C_{j}} (p_{j}^{\to} (A_{i}), r)) . \end{matrix}$ So, in this case, ${CO}^{★}$ need not be a Sayed’s (L, M)-fuzzy hull operator on X. In the general framework, an index set J in Theorem 3.4 of [28] should be corrected for a finite index set. A counterexample that ${CO}_{\prod_{i \in J} C_{i}} = {CO}^{★}$ is generally not true can be found in Example 3.6 of [28].

3 Convex (L, M)-fuzzy hull operators

In this section, we will propose the concept of convex (L, M)-fuzzy hull operators, and we will study the relationship between the category of (L, M)-fuzzy convex spaces and that of convex (L, M)-fuzzy hull spaces. In particular, we will give a new characterization of the product of (L, M)-fuzzy convex structures. Meanwhile, we will also give the lattices structure of convex (L, M)-fuzzy hull operators.

Definition 3.1. If $CO : L^{X} \times M_{0_{M}} ⟶ L^{X}$ is a Sayed’s (L, M)-fuzzy hull operator on X, and satisfies the following conditions: for any A ∈ L^X and r, s ∈ M_{0_M},

(CO7) if $r = ⋁ {s \in M_{0_{M}} : A = CO (A, s)}$ , then $CO (A, r) = A .$

Then $CO$ is called a convex (L, M)-fuzzy hull operator on X. If $CO$ is a convex (L, M)-fuzzy hull operator on X, then the pair $(X, CO)$ is called convex (L, M)-fuzzy hull space. The set of all convex (L, M)-fuzzy hull operators on X is denoted by FH(X, L, M). Define a relation ≤ on FH(X, L, M) as follows: ${CO}_{1} \leq {CO}_{2} \Leftrightarrow {CO}_{1} (A, r) \geq {CO}_{2} (A, r),$ for any A ∈ L^X and r ∈ M_{0_M}, then we easily verify that FH(X, L, M) is a poset. Let $(X, {CO}_{X})$ and $(Y, {CO}_{Y})$ be two convex (L, M)-fuzzy hull space and let g : X ⟶ Y be a mapping.We say g is convex (L, M)-fuzzy hull preserving function ((L, M)-CHP, for short) if $g^{\to} ({CO}_{X} (A, a)) \leq {CO}_{Y} (g^{\to} (A), a)$ for all A ∈ L^X and a ∈ M_{0_M}. It is easy to check that all convex (L, M)-fuzzy hull spaces as objects and all corresponding (L, M)-CHP mappings as morphisms form a category, denoted by (L, M)-FCH.

Theorem 3.2. Let $(X, C)$ be a convex (L, M)-fuzzy convex space, ∀A ∈ L^X and a ∈ M_{0_M}, define a mapping ${CO}_{C} : L^{X} \times M_{0_{M}} ⟶ L^{X}$ as follows:

$\begin{matrix} {CO}_{C} (A, a) = ⋀ {B \in L^{X} : A \leq B, C (B) \geq a} . \end{matrix}$ Then ${CO}_{C}$ is a convex (L, M)-fuzzy hull operator, which is called the convex (L, M)-fuzzy hull operator induced by $C$ .

Proof. Notice that ${CO}_{C}$ is a Sayed’s (L, M)-fuzzy hull operator. So, we need to verify the condition (CO7). Indeed, let $r = ⋁ {b \in M_{0_{M}} : B = {CO}_{C} (B, b)},$ then r ∈ M_{0_M}. For each $b \in {b \in M_{0_{M}} : B = {CO}_{C} (B, b)},$ we have $B = {CO}_{C} (B, b)$ , So, $\begin{matrix} C (B) & = C ({CO}_{C} (B, b)) \\ = C (⋀ {D \in L^{X} : B \leq D, C (D) \geq b}) \\ \geq ⋀_{D} C (D) \geq b . \end{matrix}$ Thus, $C (B) \geq ⋁ {b \in M_{0_{M}} : B = {CO}_{C} (B, b)} = r .$ This implies that $\begin{matrix} B & = ⋁_{b} {CO}_{C} (B, b) \leq {CO}_{C} (B, r) \\ = ⋀ {D \in L^{X} : B \leq D, C (D) \geq r} \leq B . \end{matrix}$ Hence, ${CO}_{C} (B, r) = B .$ □

Theorem 3.3. Let $CO$ be a convex (L, M)-fuzzy hull operator, ∀A ∈ L^X and a ∈ M_{0_M}, define a mapping $C_{CO} : L^{X} ⟶ M$ as follows:

$\begin{matrix} C_{CO} (A) = ⋁ {a \in M_{0_{M}} : A = CO (A, a)} . \end{matrix}$ Then $C_{CO}$ is an (L, M)-fuzzy convex structure on X .

Proof. The proof of the theorem is similar to the case of Sayed’s (L, M)-fuzzy hull operator. So, we omit it.□

Proposition 3.4. If $g : (X, {CO}_{X}) ⟶ (Y, {CO}_{Y})$ is an (L, M)-CHP (resp.,(L, M)-HP), then

$g : (X, C_{{CO}_{X}}) ⟶ (Y, C_{{CO}_{Y}})$ is an (L, M)-CP.

Proof. Let A ∈ L^Y, assume that $b \in β (C_{{CO}_{Y}} (A))$ , then $b ≺ C_{{CO}_{Y}} (A) = ⋁ {a \in M_{0_{M}} : A = {CO}_{Y} (A, a)},$ there exists a₀ ∈ M_{0_M} such that $A = {CO}_{Y} (A, a_{0}),$ and b ≤ a₀. If $g : (X, {CO}_{X}) ⟶ (Y, {CO}_{Y})$ is an (L, M)-CHP, then $\begin{matrix} g^{\leftarrow} (A) & = g^{\leftarrow} ({CO}_{Y} (A, a_{0})) \\ \geq g^{\leftarrow} ({CO}_{Y} (g^{\to} (g^{\leftarrow} (A)), a_{0})) \\ \geq g^{\leftarrow} g^{\to} ({CO}_{X} (g^{\leftarrow} (A), a_{0})) \\ \geq {CO}_{X} (g^{\leftarrow} (A), a_{0}) \geq g^{\leftarrow} (A) . \end{matrix}$ Thus, ${CO}_{X} (g^{\leftarrow} (A), a_{0}) = g^{\leftarrow} (A)$ . So, $\begin{matrix} C_{{CO}_{X}} & (g^{\leftarrow} (A)) \\ = ⋁ {s \in M_{0_{M}} : g^{\leftarrow} (A) = {CO}_{X} (g^{\leftarrow} (A), s)} \\ \geq a_{0} \geq b . \end{matrix}$ Hence, $C_{{CO}_{X}} (g^{\leftarrow} (A)) \geq C_{{CO}_{Y}} (A) (\forall A \in L^{Y}) .$ Therefore $g : (X, C_{{CO}_{X}}) ⟶ (Y, C_{{CO}_{Y}})$ is an (L, M)-CP.□

Proposition 3.5. If $g : (X, C_{X}) ⟶ (Y, C_{Y})$ is an (L, M)-CP, then $g : (X, {CO}_{C_{X}}) ⟶ (Y, {CO}_{C_{Y}})$ is an (L, M)-CHP (resp.,(L, M)-HP).

Proof. Since $g : (X, C_{X}) ⟶ (Y, C_{Y})$ is (L, M)-CP, we have $C_{X} (g^{\leftarrow} (B)) \geq C_{Y} (B)$ for each B ∈ L^Y . So, for each A ∈ L^X and a ∈ M_{0_M}, $\begin{matrix} g^{\leftarrow} ({CO}_{C_{Y}} (g^{\to} (A), a)) \\ = g^{\leftarrow} (⋀ {B \in L^{Y} : g^{\to} (A) \leq B, C_{Y} (B) \geq a}) \\ = ⋀ {g^{\leftarrow} (B) \in L^{X} : g^{\to} (A) \leq B, C_{Y} (B) \geq a} \\ \geq ⋀ {g^{\leftarrow} (B) \in L^{X} : A \leq g^{\leftarrow} (B), C_{X} (g^{\leftarrow} (B)) \geq a} \\ \geq ⋀ {C \in L^{X} : A \leq C, C_{X} (C) \geq a} \\ = {CO}_{C_{X}} (A, a) . \end{matrix}$ Therefore, $\begin{matrix} {CO}_{C_{Y}} (g^{\to} (A), a) & \geq g^{\to} (g^{\leftarrow} ({CO}_{C_{Y}} (g^{\to} (A), a))) \\ \geq g^{\to} ({CO}_{C_{X}} (A, a)) . \end{matrix}$ It follows that $g : (X, {CO}_{C_{X}}) ⟶ (Y, {CO}_{C_{Y}})$ is an (L, M)-CHP.□

By Theorems 3.2 and 3.3, Propositions 3.4 and 3.5, we can obtion two concrete functors as follows: $\begin{matrix} {CO}_{ℂ} : {\begin{matrix} (L, M) - FC ⟶ (L, M) - FCH, \\ (X, C) ⟼ (X, {CO}_{C}), \\ g ⟼ g, \end{matrix} \end{matrix}$

$\begin{matrix} ℂ_{CO} : {\begin{matrix} (L, M) - FCH ⟶ (L, M) - FC, \\ (X, CO) ⟼ (X, C_{CO}), \\ g ⟼ g . \end{matrix} \end{matrix}$

Now, we will show that ${CO}_{ℂ}$ and $ℂ_{CO}$ are isomorphic functors.

Theorem 3.6. (L, M)-FCH is isomorphic to (L, M)-FC.

Proof. It suffices to show that $ℂ_{CO} \circ {CO}_{ℂ} = I_{(L, M) - FC}$ and ${CO}_{ℂ} \circ ℂ_{CO} = {CO}_{(L, M) - FCH}$ . That is, for each (L, M)-fuzzy convex space $(X, C)$ and for each convex (L, M)-fuzzy hull space $(X, CO)$ , it follows that

(1) $C_{{CO}_{C}} = C .$

(2) ${CO}_{C_{CO}} = CO$ .

For (1), if $C (A) = 0_{M}$ , we easily obtain $C_{{CO}_{C}} (A) \geq 0_{M} = C (A) .$ If $C (A) \in M_{0_{M}}$ , then $\begin{matrix} A & \leq {CO}_{C} (A, C (A)) \\ = ⋀ {B \in L^{X} : A \leq B, C (B) \geq C (A)} \leq A . \end{matrix}$ So, we have $A = {CO}_{C} (A, C (A)) .$ By the definition of $C_{{CO}_{C}},$ we obtain $C_{{CO}_{C}} (A) \geq C (A) .$

Conversely, suppose that $\begin{matrix} b & \in β (C_{{CO}_{C}} (A)) \\ = β (⋁ {a \in M_{0_{M}} : A = {CO}_{C} (A, a)}) . \end{matrix}$ Then $b ≺ ⋁ {a \in M_{0_{M}} : A = {CO}_{C} (A, a)},$ there exists a₀ ∈ M_{0_M} such that $A = {CO}_{C} (A, a_{0}),$ and b ≤ a₀. So, $C (A) = C ({CO}_{C} (A, a_{0})) \geq a_{0} \geq b .$ Hence, $C_{{CO}_{C}} (A) \leq C (A) .$ Therefore, $C_{{CO}_{C}} (A) = C (A) .$

For (2), for each A ∈ L^X and a ∈ M_{0_M}, we obtain $\begin{matrix} {CO}_{C_{CO}} (A, a) & = ⋀ {B \in L^{X} : A \leq B, C_{CO} (B) \geq a} \\ \leq ⋀ {B \in L^{X} : A \leq B = CO (B, a)} \\ \leq CO (A, a) . \end{matrix}$

Conversely, for each $B \in {B \in L^{X} : A \leq B, C_{CO} (B) \geq a},$ we have A ≤ B, and $C_{CO} (B) \geq a .$ So, by (CO7), we have $\begin{matrix} B & \leq & CO (B, a) \leq CO (B, C_{CO} (B)) \\ = & CO (B, ⋁ {s \in M_{0_{M}} : B = CO (B, s)}) = B . \end{matrix}$ It implies that $B = CO (B, a) \geq CO (A, a) .$ Hence, $\begin{matrix} {CO}_{C_{CO}} (A, a) & = & ⋀ {B \in L^{X} : A \leq B, C_{CO} (B) \geq a} \\ \geq & CO (A, a) . \end{matrix}$ Therefore, ${CO}_{C_{CO}} = CO .$ □

Theorem 3.7. (FH(X, L, M) , ≤) is a complete lattice.

Proof. We easily obtain ${CO}^{1} : L^{X} \times M_{0_{M}} ⟶ L^{X}$ is the greatest element in (FH(X, L, M) , ≤), where ${CO}^{1} (A, a) = A$ for any A ∈ L^X and a ∈ M_{0_M}. Next, we only need to prove that it’s closed for non-empty intersection operation in (FH(X, L, M) , ≤).

Let ∅≠ {CO_i} _i∈I ⊆FH (X, L, M) , and I is an index set. Define $CO : L^{X} \times M_{0_{M}} ⟶ L^{X}$ as follows: $\begin{matrix} CO & (A, a) \\ = & ⋀ {B \in L^{X} : A \leq B, ⋀_{i \in I} ⋁_{{CO}_{i} (B, r) = B} r \geq a} . \end{matrix}$ Then $CO$ is the greatest lower bound of {CO_i} _i∈I . Indeed, firstly, by the definition of $CO,$ we easily obtain (CO1), (CO2), (CO3) and (CO4).

(CO5) By (CO2), for each B ∈ L^X and a ∈ M_{0_M}, we have $CO (CO (B, a), a) \geq CO (B, a) .$

On the other hand, notice that $\begin{matrix} CO & (B, a) \\ = & ⋀ {C \in L^{X} : B \leq C, ⋀_{i \in I} ⋁_{{CO}_{i} (C, r) = C} r \geq a} . \end{matrix}$ Let B ≤ C, and $⋀_{i \in I} ⋁_{{CO}_{i} (C, r) = C} r \geq a,$ then, for each i ∈ I, $\begin{matrix} CO & i (CO (B, a), a) \\ = & {CO}_{i} (⋀_{B \leq C} {C \in L^{X} : ⋀_{i \in I} ⋁_{{CO}_{i} (C, r) = C} r \geq a}, a) \\ \leq & {CO}_{i} (⋀_{B \leq C} {C \in L^{X} : ⋀_{i \in I} ⋁_{{CO}_{i} (C, r) = C} r \geq a}, \\ ⋀_{i \in I} ⋁_{{CO}_{i} (C, r) = C} r) \\ \leq & {CO}_{i} (C, ⋀_{i \in I} ⋁_{{CO}_{i} (C, r) = C} r) \\ \leq & {CO}_{i} (C, ⋁_{{CO}_{i} (C, r) = C} r) = C . \end{matrix}$ So, for each i ∈ I, $CO (B, a) \leq {CO}_{i} (CO (B, a), a) \leq CO (B, a) .$ It implies that ${CO}_{i} (CO (B, a), a) = CO (B, a) .$ Hence $⋀_{i \in I} ⋁_{{CO}_{i} (CO (B, a), r) = CO (B, a)} r \geq a .$ Therefore, $\begin{matrix} CO & (CO (B, a), a) \\ = & ⋀ {C \in L^{X} : CO (B, a) \leq C, \\ ⋀_{i \in I} ⋁_{{CO}_{i} (C, r) = C} r \geq a} \\ \leq & CO (B, a) . \end{matrix}$

It follows that $CO (CO (B, a), a) = CO (B, a) .$

(CO6) Let {A_j : j ∈ J} ⊆ L^X is nonempty and totally ordered by inclusion. By (CO4), we obtain $CO (⋁_{j \in J} A_{j}, a) \geq ⋁_{j \in J} CO (A_{j}, a) .$ and ${CO (A_{j}, a)}_{j \in J}$ is nonempty and totally ordered by inclusion. So, $⋁_{j \in J} A_{j} \leq ⋁_{j \in J} CO (A_{j}, a),$ and, for each i ∈ I, $\begin{matrix} {CO}_{i} (⋁_{j \in J} CO (A_{j}, a), a) & = & ⋁_{j \in J} {CO}_{i} (CO (A_{j}, a), a) \\ = & ⋁_{j \in J} CO (A_{j}, a) . \end{matrix}$ It implies that $⋀_{i \in I} ⋁_{{CO}_{i} (⋁_{j \in J} CO (A_{j}, a), r) = ⋁_{j \in J} CO (A_{j}, a)} r \geq a .$ Hence $\begin{matrix} CO & (⋁_{j \in J} A_{j}, a) \\ = & ⋀ {B \in L^{X} : ⋁_{j \in J} A_{j} \leq B, \\ ⋀_{i \in I} ⋁_{{CO}_{i} (B, r) = B} r \geq a} \\ \leq & ⋁_{j \in J} CO (A_{j}, a) . \end{matrix}$ So, $CO (⋁_{j \in J} A_{j}, a) \leq ⋁_{j \in J} CO (A_{j}, a) .$ Therefore, $CO (⋁_{j \in J} A_{j}, a) = ⋁_{j \in J} CO (A_{j}, a) .$

(CO7) Let s ∈ M_{0_M} and $A = CO (A, s),$ then $A \leq {CO}_{i} (A, s) \leq {CO}_{i} (CO (A, s), s) = CO (A, s) = A$ for each i ∈ I . Denote $a = ⋁ {s \in M_{0_{M}} : A = CO (A, s)},$ $b = ⋁ {s \in M_{0_{M}} : A = {CO}_{i} (A, s)},$ then a ≤ b . Thus, $A \leq {CO}_{i} (A, a) \leq {CO}_{i} (A, b) = A$ for each i ∈ I . So, $A = {CO}_{i} (A, a)$ (∀ i ∈ I) . Further, $⋀_{i \in I} ⋁_{{CO}_{i} (A, r) = A} r \geq a,$ It follows that

$\begin{matrix} A & \leq & CO (A, a) \\ = & ⋀ {B \in L^{X} : A \leq B, ⋀_{i \in I} ⋁_{{CO}_{i} (B, r) = B} r \geq a} \\ \leq & A, \end{matrix}$ i.e., $A = CO (A, a) .$ So, if $a = ⋁ {s \in M_{0_{M}} : A = CO (A, s)},$ then $CO (A, a) = A .$

According to the above proof, we know that the mapping $CO : L^{X} \times M_{0_{M}} ⟶ L^{X}$ is a convex (L, M)-fuzzy hull operator on X .

Secondly, for each A ∈ L^X and a ∈ M_{0_M} . Let A ≤ B, and $⋀_{i \in I} ⋁_{{CO}_{i} (B, r) = B} r \geq a .$ Then, ∀i ∈ I, $\begin{matrix} B & = & {CO}_{i} (B, ⋁_{{CO}_{i} (B, r) = B} r) \\ \geq & {CO}_{i} (B, ⋀_{i \in I} ⋁_{{CO}_{i} (B, r) = B} r) \\ \geq & {CO}_{i} (B, a) \geq {CO}_{i} (A, a) . \end{matrix}$ Hence, $\begin{matrix} CO & (A, a) \\ = & ⋀ {B \in L^{X} : A \leq B, ⋀_{i \in I} ⋁_{{CO}_{i} (B, r) = B} r \geq a} \\ \geq & {CO}_{i} (A, a), \end{matrix}$ i.e., $CO \leq {CO}_{i} .$ It implies that $CO$ is a lower bound of {CO_i} _i∈I .

Finally, suppose that ${CO}^{circledS} : L^{X} \times M_{0_{M}} ⟶ L^{X}$ is another lower bound of {CO_i} _i∈I, then ∀A ∈ L^X and ∀a ∈ M_{0_M}, there must be ${CO}^{circledS} (A, a) \geq {CO}_{i} (A, a) .$ Thus ${CO}^{circledS} (A, a) \geq A,$ and $\begin{matrix} {CO}^{circledS} (A, a) & = & {CO}^{circledS} ({CO}^{circledS} (A, a), a) \\ \geq & {CO}_{i} ({CO}^{circledS} (A, a), a) \\ \geq & {CO}^{circledS} (A, a) . \end{matrix}$ So, ${CO}^{circledS} (A, a) = {CO}_{i} ({CO}^{circledS} (A, a), a),$ it follows that $⋀_{i \in I} C_{{CO}_{i}} ({CO}^{circledS} (A, a)) \geq a .$ Hence, $\begin{matrix} {CO}^{circledS} & (A, a) \\ \geq & ⋀ {B \in L^{X} : A \leq B, ⋀_{i \in I} C_{{CO}_{i}} (B) \geq a} \\ = & ⋀ {B \in L^{X} : A \leq B, ⋀_{i \in I} ⋁_{{CO}_{i} (B, r) = B} r \geq a} \\ = & CO (A, a), \end{matrix}$ i.e., ${CO}^{circledS} \leq CO .$ Hence, $CO$ is the greatest lower bound of {CO_i} _i∈I .□

Finally, we will give a characterization of the product of (L, M)-fuzzy convex structures.

Theorem 3.8. Let J be an arbitrary index set, $X = \prod_{j \in J} X_{j}$ , p_j : X ⟶ X_j be the j-th projection and $(\prod_{j \in J} X_{j}, \prod_{j \in J} C_{j})$ be the product of ${(X_{j}, C_{j})}_{j \in J} .$ Then, for each A ∈ L^X and a ∈ M_{0_M},

$(\prod_{j \in J} C_{j}) (A) = ⋁_{⋁_{i \in I}^{d} A_{i} = A} ⋀_{i \in I} ⋁_{{CO}^{★} (A_{i}, a) = A_{i}} a,$ where ${CO}^{★} (A_{i}, a) = \prod_{j \in J} {CO}_{C_{j}} (p_{j}^{\to} (A_{i}), a)$ .

Proof. Firstly, define a mapping $S : L^{X} ⟶ M$ as follows: for each A ∈ L^X, $S (A) = ⋁_{{CO}^{★} (A, a) = A} a .$ Then $S$ is an (L, M)-fuzzy closure system on X. Indeed,

(LMC1) Since ${CO}^{★}$ is an (L, M)-fuzzy weak hull operator on X, we easily obtain $S (0_{X}) = S (1_{X}) = 1_{M} .$

(LMC2) Let {B_i : i ∈ I} be a nonempty subset of L^X. Notice that $S (⋀_{i \in I} B_{i}) = ⋁_{{CO}^{★} (⋀_{i \in I} B_{i}, r) = ⋀_{i \in I} B_{i}} r,$ and $⋀_{i \in I} S (B_{i}) = ⋀_{i \in I} ⋁_{{CO}^{★} (B_{i}, s) = B_{i}} s .$ Suppose that $b \in β^{*} (⋀_{i \in I} S (B_{i}))$ , then $b ≺ ⋁_{{CO}^{★} (B_{i}, s) = B_{i}} s$ for each i ∈ I, and b ∈ J (M). So, there exists $a_{0}^{i} \in M_{0_{M}}$ such that ${CO}^{★} (B_{i}, a_{0}^{i}) = B_{i}$ and $b ≺ a_{0}^{i}$ . Put $a_{0} = ⋀_{i \in I} a_{0}^{i}$ , then b ≤ a₀ and a₀ ∈ M_{0_M}. Thus, $\begin{matrix} ⋀_{i \in I} B_{i} & \leq & {CO}^{★} (⋀_{i \in I} B_{i}, a_{0}) \\ \leq & ⋀_{i \in I} {CO}^{★} (B_{i}, a_{0}) \\ \leq & ⋀_{i \in I} {CO}^{★} (B_{i}, a_{0}^{i}) = ⋀_{i \in I} B_{i} . \end{matrix}$ It implies that ${CO}^{★} (⋀_{i \in I} B_{i}, a_{0}) = ⋀_{i \in I} B_{i} .$ Therefore, $S (⋀_{i \in I} B_{i}) \geq a_{0} \geq b$ . Hence, $S (⋀_{i \in I} B_{i}) \geq ⋀_{i \in I} S (B_{i}) .$

Secondly, define a mapping $C : L^{X} ⟶ M$ as follows: for each A ∈ L^X, $C (A) = ⋁_{⋁_{i \in I}^{d} A_{i} = A} ⋀_{i \in I} S (A_{i}) .$ Then $C$ is the coarsest (L, M)-fuzzy convex structure on X such that $p_{j} : (\prod_{j \in J} X_{j}, C) ⟶ (X_{j}, C_{j})$ is an (L, M)-CP for each j ∈ J . Indeed, according to the proof of Theorem 3.12 in [10], we immediately know that $C$ is an (L, M)-fuzzy convex structure on X.

For all B^j ∈ L^{X
_j}, $\begin{matrix} C & (p_{j}^{\leftarrow} (B^{j})) \\ = & ⋁_{⋁_{i \in I}^{d} A_{i} = p_{j}^{\leftarrow} (B^{j})} ⋀_{i \in I} S (A_{i}) \\ \geq & S (p_{j}^{\leftarrow} (B^{j})) \\ = & ⋁_{{CO}^{★} (p_{j}^{\leftarrow} (B^{j}), a) = p_{j}^{\leftarrow} (B^{j})} a \\ = & ⋁ {a \in M_{0_{M}} : ⋀_{k \in J} p_{k}^{\leftarrow} ({CO}_{C_{k}} (p_{k}^{\to} (p_{j}^{\leftarrow} (B^{j})), a)) \\ = p_{j}^{\leftarrow} (B^{j})} \\ \geq & ⋁ {a \in M_{0_{M}} : p_{j}^{\leftarrow} ({CO}_{C_{j}} (p_{j}^{\to} (p_{j}^{\leftarrow} (B^{j})), a)) \\ = p_{j}^{\leftarrow} (B^{j})} \\ = & ⋁ {a \in M_{0_{M}} : p_{j}^{\leftarrow} ({CO}_{C_{j}} (B^{j}, a)) = p_{j}^{\leftarrow} (B^{j})} \\ \geq & ⋁ {a \in M_{0_{M}} : {CO}_{C_{j}} (B^{j}, a) = B^{j}} \\ = & C_{{CO}_{C_{j}}} (B^{j}) = C_{j} (B^{j}) . \end{matrix}$ Hence, $p_{j} : (\prod_{j \in J} X_{j}, C) ⟶ (X_{j}, C_{j})$ is an (L, M)-CP for each j ∈ J.

If $C^{★}$ is another (L, M)-fuzzy convex structure on X such that $p_{j} : (\prod_{j \in J} X_{j}, C^{★}) ⟶ (X_{j}, C_{j})$ is an (L, M)-CP for each j ∈ J, then $p_{j} : (\prod_{j \in J} X_{j}, {CO}_{C}^{★}) ⟶ (X_{j}, {CO}_{C}_{j})$ is an (L, M)-CHP for each j ∈ J . So, $p_{j}^{\to} ({CO}_{C}^{★} (A, a)) \leq {CO}_{C}_{j} (p_{j}^{\to} (A), a)$ for all A ∈ L^X and a ∈ M_{0_M}. Further, $\begin{matrix} {CO}_{C}^{★} (A, a) & \leq & ⋀_{j \in J} p_{j}^{\leftarrow} ({CO}_{C}_{j} (p_{j}^{\to} (A), a)) \\ = & \prod_{j \in J} {CO}_{C_{j}} (p_{j}^{\to} (A), a) \\ = & {CO}^{★} (A, a) . \end{matrix}$ So, $\begin{matrix} C^{★} (A) & = & C_{{CO}_{C^{★}}} (A) \\ = & ⋁ {a \in M_{0_{M}} : A = {CO}_{C^{★}} (A, a)} \\ \geq & ⋁ {a \in M_{0_{M}} : A = {CO}^{★} (A, a)} \\ = & S (A) . \end{matrix}$ Thus, for each A ∈ L^X, if $⋁_{i \in I}^{d} A_{i} = A$ , then $C^{★} (A) = C^{★} (⋁_{i \in I}^{d} A_{i}) \geq ⋀_{i \in I} C^{★} (A_{i}) \geq ⋀_{i \in I} S (A_{i}) .$ It follows that $C^{★} (A) \geq ⋁_{⋁_{i \in I}^{d} A_{i} = A} ⋀_{i \in I} S (A_{i}) = C (A) .$ Hence, $\begin{matrix} (\prod_{j \in J} C_{j}) (A) & = & ⋁_{⋁_{i \in I}^{d} A_{i} = A} ⋀_{i \in I} S (A_{i}) \\ = & ⋁_{⋁_{i \in I}^{d} A_{i} = A} ⋀_{i \in I} ⋁_{{CO}^{★} (A_{i}, a) = A_{i}} a . \end{matrix}$ □

4 Conclusions

As a special (L, M)-fuzzy weak hull operators, convex (L, M)-fuzzy hull operators play an important role in (L, M)-fuzzy convex structures. Actually, the category of (L, M)-fuzzy convex structures is isomorphic to that of convex (L, M)-fuzzy hull operators. Based on this, we not only gave the lattices structure of convex (L, M)-fuzzy hull operators, but also gave a good characterization of the product of (L, M)-fuzzy convex structures. In [29], we gave a reasonable definition with respect to the coproduct of (L, M)-fuzzy convex structures. But in this paper, how to characterize the coproduct of (L, M)-fuzzy convex structures by using convex (L, M)-fuzzy hull operators in our sense is not given, it will remain for us to solve this problem in future.

Acknowledgments

We would like to give heartfelt thanks to the referees for their very careful reading of the paper and for their very valuable comments and suggestions which improved the paper.

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