( L,M )- fuzzy (restricted) convex derived hull spaces

Abstract

In this paper, the notion of (L, M)- fuzzy convex derived hull spaces is introduced. It is proved that the category of (L, M)- fuzzy convex derived hull spaces is isomorphic to the category of (L, M)- fuzzy convex spaces and the category of (L, M)- fuzzy convex enclosed relation spaces. Based on this, the notion of (L, M)- fuzzy restricted convex derived hull spaces is introduced. It is further proved that the category of (L, M)- fuzzy restricted convex derived hull spaces is isomorphic to the category of (L, M)- fuzzy restricted convex hull spaces.

1 Introduction

An abstract convex structure on a nonempty set is a set-theoretic structure which contains the empty set and the largest set and is closed under arbitrary intersections and nested unions. Its theory involves many mathematical structures [21].

Convex structure has been extended into fuzzy settings. Maruyama introduced L-convex structure [6]. Its characterizations have been discussed [4 , 10]. Latter, Shi and Xiu introduced M-fuzzifying convex structures [17]. Many subsequent studies have been done [18 , 30]. Shi and Xiu further introduced (L, M)- fuzzy convex structure which unifies both of L-convex structure and M-fuzzifying convex structure [19]. Some characterizations and related theories have been discussed [9 , 31].

Derived operator is an important tool in describing mathematical structures. Cantor introduced derived operator which is used to characterize topology [1]. Shi introduced fuzzy derived operator which is used to characterize fuzzy topology [14]. Xin introduced M-fuzzifying matroid derived operator and M-fuzzifying matroid [27]. Shen introduced M-fuzzifying convex derived operator by which they characterize M-fuzzifying convex structure [13].

So, is it possible to apply derived operators to (L, M)- fuzzy convex hull operators or to (L, M)- fuzzy restricted hull operators? If possible, can they be used to characterize (L, M)- fuzzy convex spaces or (L, M)- fuzzy convex hull spaces or (L, M)- fuzzy restricted convex hull spaces?∥The aim of this paper is to solve the above problems. The arrangement of this paper is as follows. In Section 2, we recall some concepts and results related to (L, M)- fuzzy convex spaces. In Section 3, we define (L, M)- fuzzy convex derived hull space by which we characterize both (L, M)- fuzzy convex spaces and (L, M)- fuzzy convex enclosed relation spaces. In Section 4, we define (L, M)- fuzzy restricted convex derived hull spaces by which we characterize of (L, M)- fuzzy restricted convex hull spaces.

2 Preliminaries

In this paper, both L and M are completely distributive lattices and M has an inverse involution^′. The smallest (resp. greatest) element of L or M are respectively denoted by ⊥ (resp. ⊤). An element a ∈ M is a prime, if b ∧ c ≤ a implies b ≤ a or c ≤ a for all b, c ∈ M. The set of all primes in M ∖ {⊤} is denoted by P (M). The set of all co-primes in M ∖ {⊥} is J (M) = {a ∈ M : a′ ∈ P (M)}. For any φ ⊆ M, ⋁_a∈φa and ⋀_a∈φa are simply denoted by ⋁φ and ⋀φ respectively. In particular, we adapt the convention that ⋁∅ = ⊥ and ⋀∅ = ⊤ [26].

A binary relation ≺ on M is defined by: for all a, b ∈ M, a ≺ b if for all φ ⊆ M, b ≤ ⋁ φ implies some d ∈ φ such that a ≤ d. Clearly, β (⋁ _i∈Ia_i) = ⋃ _i∈Iβ (a_i) for all {a_i} _i∈I ⊆ M, where β (a) = {b : b ≺ a} for all a ∈ M. The opposite relation ≺^op of ≺ is defined by: b ≺ ^opa if a′ ≺ b′. Clearly, α (⋀ _i∈Ia_i) = ⋃ _i∈Iα (a_i) for all {a_i} _i∈I ⊆ M, where α (a) = {b : a ≺ ^opb} for all a ∈ M [16]. Also, a = ⋁ β (a) = ⋁ β^* (a) = ⋀ α (a) = ⋀ α^* (a) for all a ∈ M, where β^* (a) = β (a) ∩ J (M) and α^* (a) = α (a) ∩ P (M) [15].

X, Y are nonempty sets. 2^X is the power set of X and $2_{fin}^{X}$ is the set of all finite subsets of X. L^X is the set of all L-fuzzy sets on X, whose greatest (resp. smallest) element is $\underline{⊤}$ (resp. $\underline{⊥}$ ). A subset φ ⊆ L^X is said to be up-directed, denoted by $φ \overset{dir}{\subseteq} L^{X}$ , if for all A_i, A_j ∈ φ there is A_k ∈ φ such that A_i ∨ A_j ≤ A_k. In this case, we denote ⋁φ by ⋁^dirφ. For x ∈ X and λ ∈ L, the L-fuzzy set x_λ ∈ L^X is called an L-fuzzy point which is defined by x_λ (x) = λ and x_λ (y) =⊥ for any y ∈ X ∖ {x}. For any A ∈ L^X, we denote β (A) = {x_λ ∈ L^X : λ ∈ β (A (x))}, β^* (A) = β (A) ∩ J (L^X), where J (L^X) = {x_λ : λ ∈ J (L)} [15]. Also, for any A ∈ L^X and any a ∈ L, we denote A_[a] = {x ∈ X : A (x) ≥ a} and A^[a] = {x ∈ X : a ∉ α (A (x))} [15]. Finally, for any A ∈ L^X, let $F (A) = {F \in L^{X} : \exists φ \in 2_{fin}^{β^{*} (A)}, F = ⋁ φ}$ [24]. In particular, $F (\underline{⊤})$ is written by $F (L^{X})$ . Its properties can be seen in [24].

Definition 2.1 ([19]). A mapping $C : L^{X} \to M$ is called an (L, M)- fuzzy convex structure and the pair $(X, C)$ is called an (L, M)- fuzzy convex space, if

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

(LMC2) $C (⋀_{i \in I} A_{i}) \geq ⋀_{i \in I} C (A_{i})$ for {A_i} _i∈I ⊆ L^X;

(LMC3) $C (⋁_{i \in I}^{dir} A_{i}) \geq ⋀_{i \in I} C (A_{i})$ for ${A_{i}}_{i \in I} \overset{dir}{\subseteq} L^{X}$ .

Let $(X, C_{X})$ and $(Y, C_{Y})$ be (L, M)- fuzzy convex spaces. A mapping f : X → Y is called an (L, M)- fuzzy convexity preserving mapping, if $C_{Y} (B) \leq C_{X} (f_{L}^{ł} (B))$ for any B ∈ L^Y [19]. The category of (L, M)- fuzzy convex spaces and (L, M)- fuzzy convexity preserving mappings is denoted by LMCS [19].

Definition 2.2 ([16, 24]). An operator co : L^X → M^{J(L^X)} is called an (L, M)- fuzzy convex hull operator and the pair (X, co) is called an (L, M)- fuzzy convex hull space, if

(LMCO1) $co (\underline{⊥}) (x_{λ}) = ⊥$ ;

(LMCO2) co (A) (x_λ) =⊤ whenever x_λ ≤ A;

(LMCO3) co (A) (x_λ) = ⋀ _{x_λ≰B≥A} ⋁ _{y_μ≰B}co (B) (y_μ);

(LMCO4) $co (A) (x_{λ}) = ⋀_{μ ≺ λ} ⋁_{F \in F (A)} co (F) (x_{μ})$ .

Let (X, co_X) and (Y, co_Y) be (L, M)- fuzzy convex hull spaces. A mapping f : X → Y is called an (L, M)- fuzzy convex hull preserving mapping, if $\begin{matrix} {co}_{X} (A) (x_{λ}) \leq {co}_{Y} (f_{L}^{\to} (A)) (f_{L}^{\to} (x_{λ})) \end{matrix}$ for A ∈ L^X and x_λ ∈ J (L^X). The category of (L, M)- fuzzy convex hull spaces and (L, M)- fuzzy convex hull preserving mappings is denoted by LMCHS [24].

Theorem 2.3 ([16, 24]). (1) Let $(X, C)$ be an (L, M)- fuzzy convex space. Define an operator ${co}_{C} : L^{X} \to M^{J (L^{X})}$ by ${co}_{C} (A) (x_{λ}) = ⋀_{x_{λ} ≰ B \geq A} [C (B)]^{'}$ for A ∈ L^X and x_λ ∈ J (L^X). Then ${co}_{C}$ is an (L, M)- fuzzy convex hull operator satisfying (LMCO0), where

(LMCO0) ${co}_{C} (A) (x_{λ}) = ⋀_{μ ≺ λ} {co}_{C} (A) (x_{μ})$ .

(2) Let (X, co) be an (L, M)- fuzzy convex hull space. Define a mapping $C_{co} : L^{X} \to M$ by $\forall A \in L^{X}, C_{co} (A) = ⋀_{x_{λ} ≰ A} [co (A) (x_{λ})]^{'} .$ Then $(X, C_{co})$ is an (L, M)- fuzzy convex space.

(3) LMCS is isomorphic to LMCHS.

Definition 2.4 ([24]). An operator $H : F (L^{X}) \to M^{J (L^{X})}$ is called an (L, M)- fuzzy restricted convex hull operator and the pair $(X, H)$ is called an (L, M)- fuzzy restricted convex hull space, if for all x_λ ∈ J (L^X) and $F, G \in F (L^{X})$ ,

(LMCRH1) $H (\underline{⊥}) (x_{λ}) = ⊥$ ;

(LMCRH2) $H (F) (x_{λ}) = ⊤$ whenever x_λ ≤ F;

(LMCRH3) $H (G) (x_{λ}) \land ⋀_{y_{μ} ≺ G} (x_{λ})$ ;

(LMCRH4) $H (F) (x_{λ}) = ⋀_{μ ≺ λ} ⋁_{G \in F (F)} H (G) (x_{μ})$ .

By (LMCRH4), it is easy to check that $H$ satisfies (LMCRH0), where∥(LMCRH0) $H (F) (x_{λ}) = ⋀_{μ ≺ λ} H (F) (x_{μ})$ .

Let $(X, H_{X})$ and $(Y, H_{Y})$ be (L, M)- fuzzy restricted convex hull spaces. A mapping f : X → Y is called an (L, M)- fuzzy restricted convex hull preserving mapping, if for all $F \in F (L^{X})$ and x_λ ∈ J (L^X), $\begin{matrix} H_{X} (F) (x_{λ}) \leq H_{Y} (f_{L}^{\to} (F)) (f_{L}^{\to} (x_{λ})) . \end{matrix}$ The category of (L, M)- fuzzy restricted convex hull spaces and (L, M)- fuzzy restricted convex hull preserving mappings is denoted by LMRCHS [24].

Theorem 2.5 ([24]) (1) Let $(X, C)$ be an (L, M)- fuzzy convex space. Define an operator $H_{C} : F (L^{X}) \to M^{J (L^{X})}$ by $\begin{matrix} \forall F \in F (L^{X}), H_{C} (F) = co (F) . \end{matrix}$ Then $(X, H_{C})$ is an (L, M)- fuzzy restricted convex hull space.

(2) Let $(X, H)$ be an (L, M)- fuzzy restricted convex hull space. Define a mapping $C_{H} : L^{X} \to M$ by $\forall A \in L^{X}, C_{H} (A) = ⋀_{x_{λ} ≰ A} ⋀_{F \in F (A)} [H (F) (x_{λ})]^{'} .$ Then $(X, C_{H})$ is an (L, M)- fuzzy convex space.

(3) LMRCHS is isomorphic to LMCS.

Definition 2.6 ([31]). A binary relation $R : L^{X} \times L^{X} \to M$ is called an (L, M)- fuzzy convex enclosed relation and the pair $(X, R)$ is called an (L, M)- fuzzy convex enclosed relation space, if

(LMCER1) $R (\underline{⊥}, \underline{⊥}) = ⊤$ ;

(LMCER2) $R (A, B) > ⊥$ implies A ≤ B;

(LMCER3) $R (A, ⋀_{i \in I} B_{i}) = ⋀_{i \in I} R (A, B_{i})$ ;

(LMCER4) $R (A, B) \leq ⋁_{C \in L^{X}} R (A, C) \land R (C, B)$ ;

(LMCER5) $R (⋁_{i \in I}^{dir} A_{i}, B) = ⋀_{i \in I} R (A_{i}, B)$ .

Let $(X, R_{X})$ and $(Y, R_{Y})$ be (L, M)- fuzzy convex enclosed relation spaces. A mapping f : X → Y is called an (L, M)- fuzzy convex enclosed relation preserving mapping if $\begin{matrix} R_{Y} (f^{\to} (A), B) \leq R_{X} (A, f_{L}^{ł} (B)) \end{matrix}$ for all A ∈ L^X and B ∈ L^Y. The category of (L, M)- fuzzy convex enclosed relation spaces and (L, M)- fuzzy convex enclosed relation preserving mappings is denoted by LMCERS.

Theorem 2.7 ([31]). (1) Let $(X, R)$ be an (L, M)- fuzzy convex enclosed relation space. Define an operator ${co}_{R} : L^{X} \to M^{J (L^{X})}$ by $\begin{matrix} \forall A \in L^{X}, \forall x_{λ} \in J (L^{X}), {co}_{R} (A) =_{x_{λ} ≰ B} [R (A, B)]^{'} . \end{matrix}$ Then ${co}_{R}$ is an (L, M)- fuzzy convex hull operator.

(2) Let (X, co) be an (L, M)- fuzzy convex hull space. Define a relation $R_{co} : L^{X} \times L^{X} \to M$ by $\begin{matrix} \forall A, B \in L^{X}, R_{co} (A, B) = ⋀_{x_{λ} ≰ B} [co (A) (x_{λ})]^{'} . \end{matrix}$ Then $R_{co}$ is an (L, M)- fuzzy convex enclosed relation.

(3) LMCERS is isomorphic to LMCHS.

3 (L, M)- Fuzzy convex derived hull spaces

In this section, we introduce (L, M)- fuzzy convex derived hull operator and discuss its relations with (L, M)- fuzzy convex spaces and (L, M)- fuzzy convex enclosed relation spaces.

For all A ∈ L^X and x_λ ∈ J (L^X), we denote $\begin{matrix} A_{x_{λ}}^{-} = ⋁ {y_{μ} \in β^{*} (A) : x_{λ} ≰ y_{μ}} . \end{matrix}$

Proposition 3.1. For all A, B ∈ L^X, {A_i} _i∈I ⊆ L^X and x_λ ∈ J (L^X), we have

(1) x_λ ≰ A implies $A_{x_{λ}}^{-} = A$ ;

(2) A ≤ B implies $A_{x_{λ}}^{-} \leq B_{x_{λ}}^{-}$ ;

(3) $(⋁_{i \in I} A_{i})_{x_{λ}}^{-} = ⋁_{i \in I} (A_{i})_{x_{λ}}^{-}$ ;

(4) $(A_{x_{λ}}^{-})_{x_{λ}}^{-} = A_{x_{λ}}^{-}$ ;

(5) $F (A_{x_{λ}}^{-}) \subseteq {F_{x_{λ}}^{-} : F \in F (A)}$ ;

(6) $A_{x_{λ}}^{-} = ⋁_{F \in F (A)} F_{x_{λ}}^{-}$ .

Proof. (1), (2) and (3) are clear.

(4). Clearly, we have $(A_{x_{λ}}^{-})_{x_{λ}}^{-} \leq A_{x_{λ}}^{-}$ . Conversely, for any $z_{ν} ≺ A_{x_{λ}}^{-}$ , there is z_μ ∈ β^* (A) such that x_λ ≰ z_μ and z_ν ≺ z_μ. Thus $z_{ν} ≺ z_{μ} \leq A_{x_{λ}}^{-}$ and x_λ ≰ z_ν. Hence $z_{ν} \leq (A_{x_{λ}}^{-})_{x_{λ}}^{-}$ . Therefore $A_{x_{λ}}^{-} \leq (A_{x_{λ}}^{-})_{x_{λ}}^{-}$ .

(5). Let $G \in F (A_{x_{λ}}^{-})$ . Then there is ${z_{λ_{i}}^{i}}_{i = 1}^{n} \subseteq 2_{fin}^{β^{*} (A_{x_{λ}}^{-})}$ such that $G = ⋁_{i = 1}^{n} z_{λ_{i}}^{i}$ . For any 1 ≤ i ≤ n, there is $y_{μ_{i}}^{i} \in β^{*} (A)$ such that $z_{λ_{i}}^{i} ≺ y_{μ_{i}}^{i}$ and $x_{λ} ≰ y_{μ_{i}}^{i}$ . This implies that $G \in F (A)$ and x_λ ≰ G. Thus $G = G_{x_{λ}}^{-} \in {F_{x_{λ}}^{-} : F \in F (A)}$ . Hence (6) holds.

(6). Let $F \in F (A)$ . For any y_μ ∈ β^* (F) satisfying x_λ ≰ y_μ, we have y_μ ∈ β^* (A). Thus $y_{μ} \leq A_{x_{λ}}^{-}$ which implies that $F_{x_{λ}}^{-} \leq A_{x_{λ}}^{-}$ . Hence $\begin{matrix} A_{x_{λ}}^{-} = ⋁_{G \in F (A_{x_{λ}}^{-})} G \leq ⋁_{F \in F (A)} F_{x_{λ}}^{-} \leq A_{x_{λ}}^{-} . \end{matrix}$ Therefore (6) holds.□

Definition 3.2. An operator $D : L^{X} \to M^{J (L^{X})}$ is called an (L, M)- fuzzy convex derived hull operator and the pair $(X, D)$ is called an (L, M)- fuzzy convex derived hull space, if for all x_λ ∈ J (L^X) and A ∈ L^X,

(LMCDH1) $D (\underline{⊥}) (x_{λ}) = ⊥$ ;

(LMCDH2) $D (A_{x_{λ}}^{-}) (x_{λ}) = D (A) (x_{λ})$ ;

(LMCDH3) $D (A) (x_{λ}) = ⋀_{x_{λ} ≰ B \geq A_{x_{λ}}^{-}} ⋁_{y_{μ} ≰ B} D (B) (y_{μ})$ ;

(LMCDH4) $D (A) (x_{λ}) = ⋀_{μ ≺ λ} ⋁_{F \in F (A)} D (F) (x_{μ})$ .

Let $(X, D_{X})$ and $(Y, D_{Y})$ be (L, M)- fuzzy convex derived hull spaces. A mapping f : X → Y is called an (L, M)- fuzzy convex derived hull preserving mapping, if for all A ∈ L^X and x_λ ∈ J (L^X), $\begin{matrix} D_{X} (A) (x_{λ}) \leq \bar{D_{Y}} (f_{L}^{\to} (A)) (f_{L}^{\to} (x_{λ})), \end{matrix}$ where for any B ∈ L^Y, $\bar{D_{Y}} : L^{Y} \to M^{J (L^{Y})}$ is defined by $\bar{D_{Y}} (B) = D_{Y} (B) \lor Ξ_{B}$ and Ξ_B ∈ M^{J(L^X)} is the M-fuzzy characterization of B defined by

$\begin{matrix} \forall x_{λ} \in J (L^{X}), Ξ_{B} (x_{λ}) = {\begin{matrix} ⊤, & x_{λ} \leq B, \\ ⊥, & x_{λ} ≰ B . \end{matrix} \end{matrix}$

The category of (L, M)- fuzzy convex derived hull spaces and (L, M)- fuzzy convex derived hull preserving mappings is denoted by LMCDHS.

Remark 3.3. (1) If $x_{λ} \leq A_{x_{λ}}^{-}$ , then (LMCDH3) implies that $D (A) (x_{λ}) = ⊤$ .

(2) If L = 2 then it is clear that $A_{x_{λ}}^{-} = A ∖ {x}$ for any A ∈ L^X and x_λ ∈ J (L^X). Thus it is easy to check that an (2, M)-fuzzy convex derived hull space is an M-fuzzifying convex derived hull space [13].

(3) An (L, 2)-fuzzy convex derived hull space $(X, D)$ is an L-convex derived hull space. That is, $D : L^{X} \to L^{X}$ satisfies for A ∈ L^X and x_λ ∈ J (L^X),

(LCDH1) $D (\underline{⊥}) = \underline{⊥}$ ;

(LCDH2) $x_{λ} \leq D (A)$ implies $x_{λ} \leq D (A_{x_{λ}}^{-})$ ;

(LCDH3) $D (A \lor D (A)) \leq A \lor D (A)$ ;

(LCDH4) $D (A) = ⋁_{F \in F (A)} D (F)$ .

Lemma 3.4. Let $D : L^{X} \to M^{J (L^{X})}$ be an operator satisfying (LMCDH4). We have

(1) A ≤ B implies $D (A) \leq D (B)$ ;

(2) $D$ satisfies (LMCDH0), where

(LMCDH0) $D (A) (x_{λ}) = ⋀_{μ ≺ λ} D (A) (x_{μ})$ .

Proof. (1). It directly follows from (LMCDH4).

(2). (LMCDH0) directly follows from (LMCDH4) and the fact that β^* (λ) = ⋃ _{μ∈β^*(λ)}β^* (μ). □

We present some brief characterizations of (LMCDH3) and (LMCDH4) as follows.

Proposition 3.5. An operator $D : L^{X} \to M^{J (L^{X})}$ satisfies (LMCDH4) iff for any ${A_{i}}_{i \in I} \overset{dir}{\subseteq} L^{X}$ , $\begin{matrix} D (⋁_{i \in I}^{dir} A_{i}) (x_{λ}) = ⋀_{μ ≺ λ} ⋁_{i \in I} D (A_{i}) (x_{μ}) . \end{matrix}$

Proof. The sufficiency is clear since $A = ⋁_{F \in F (A)}^{dir} F$ .

Necessity. We have $F (⋁_{i \in I}^{dir} A_{i}) = ⋃_{i \in I} F (A_{i})$ . By (LMCDH4) and Lemma 3.4, we have $\begin{matrix} D (⋁_{i \in I}^{dir} A_{i}) (x_{λ}) & = & ⋀_{μ ≺ λ} ⋁_{i \in I} ⋁_{F \in F (A_{i})} D (F) (x_{μ}) \\ \leq & ⋀_{μ ≺ λ} ⋁_{i \in I} D (A_{i}) (x_{μ}) . \end{matrix}$ Conversely, by (LMCDH0) and (1) of Lemma 3.4, it is clear that $D (⋁_{i \in I}^{dir} A_{i}) (x_{λ}) \geq ⋀_{μ ≺ λ} ⋁_{i \in I} D (A_{i}) (x_{μ})$ . Thus the desired equality holds. □

Proposition 3.6. Let $D : L^{X} \to M^{J (L^{X})}$ be an operator satisfying (LMCDH0), (LMCDH2) and $D (A) (x_{λ}) = ⊤$ whenever $x_{λ} \leq A_{x_{λ}}^{-}$ . Then the following results are equivalent:

(1) $D$ satisfies (LMCDH3);

(2) $⋁ D (A \lor ⋁ D (A)_{[a]})_{[a]} \leq A \lor ⋁ D (A)_{[a]}$ for all A ∈ L^X and a ∈ M;

(3) $⋁ D (A \lor ⋁ D (A)^{[a]})^{[a]} \leq A \lor ⋁ D (A)^{[a]}$ for all A ∈ L^X and a ∈ α (⊥).

Proof. (1) ⇒ (2). For any x_λ ∈ J (L^X) satisfying $x_{λ} ≰ A \lor ⋁ D (A)_{[a]}$ , we have x_λ ≰ A and $a ≰ D (A) (x_{λ})$ . Thus $A_{x_{λ}}^{-} = A$ . Further, by (LMCDH3), $\begin{matrix} [D (A) (x_{λ})]^{'} = ⋁_{x_{λ} ≰ B \geq A} ⋀_{y_{μ} ≰ B} [D (B) (y_{μ})]^{'} ≰ a^{'} . \end{matrix}$ Then there is B ∈ L^X such that x_λ ≰ B ≥ A and $⋀_{y_{μ} ≰ B} [D (B) (y_{μ})]^{'} ≰ a^{'}$ . Thus $y_{μ} \notin D (B)_{[a]}$ for any y_μ ≰ B. Hence $B \geq ⋁ D (B)_{[a]} \geq ⋁ D (A)_{[a]}$ . So $x_{λ} ≰ B \geq A \lor ⋁ D (A)_{[a]}$ . By (LMCDH3) again, $\begin{matrix} [D (A \lor ⋁ D (A)_{[a]}) (x_{λ})]^{'} \\ = & ⋁_{x_{λ} ≰ D \geq A \lor ⋁ D (A)_{[a]}} ⋀_{y_{μ} ≰ D} [D (D) (y_{μ})]^{'} \\ \geq & ⋀_{y_{μ} ≰ B} [D (B) (y_{μ})]^{'} ≰ a^{'} . \end{matrix}$ This implies that $a ≰ D (A \lor ⋁ D (A)_{[a]}) (x_{λ})$ , that is, $x_{λ} \notin D (A \lor ⋁ D (A)_{[a]})_{[a]}$ . By (LMCDH0) again, we have $x_{λ} ≰ ⋁ D (A \lor ⋁ D (A)_{[a]})_{[a]}$ . Therefore $\begin{matrix} ⋁ D (A \lor ⋁ D (A)_{[a]})_{[a]} \leq A \lor ⋁ D (A)_{[a]} . \end{matrix}$

(2) ⇒ (1). Let A ∈ L^X and x_λ ∈ J (L^X). If $x_{λ} \leq A_{x_{λ}}^{-}$ , then (LMCDH3) is trivial. Assume that $x_{λ} ≰ A_{x_{λ}}^{-}$ . By Lemma 3.4 and (LMCDH2), it is clear that $\begin{matrix} D (A) (x_{λ}) \leq ⋀_{x_{λ} ≰ B \geq A_{x_{λ}}^{-}} ⋁_{y_{μ} ≰ B} D (B) (y_{μ}) . \end{matrix}$

To prove the inverse inequality, we prove that $\begin{matrix} [D (A) (x_{λ})]^{'} \leq ⋁_{x_{λ} ≰ B \geq A_{x_{λ}}^{-}} ⋀_{y_{μ} ≰ B} [D (B) (y_{μ})]^{'} . \end{matrix}$

For any b ∈ M satisfying $[D (A) (x_{λ})]^{'} ≰ b$ , there is a ∈ α (b) such that $[D (A) (x_{λ})]^{'} ≰ a$ . Thus $[D (A_{x_{λ}}^{-}) (x_{λ})]^{'} ≰ a$ . By (LMCDH0), we have $x_{λ} ≰ ⋁ D (A_{x_{λ}}^{-})_{[a^{'}]}$ . Let $D = A_{x_{λ}}^{-} \lor ⋁ D (A_{x_{λ}}^{-})_{[a^{'}]}$ . We have $x_{λ} ≰ D \geq A_{x_{λ}}^{-}$ . Also, by (2), we have $⋁ D (D)_{[a]} \leq D$ . So $D (D) (y_{μ}) ≰ a$ for any y_μ ≰ D. Hence $\begin{matrix} ⋁_{x_{λ} ≰ B \geq A_{x_{λ}}^{-}} ⋀_{y_{μ} ≰ B} [D (B) (y_{μ})]^{'} \geq ⋀_{y_{μ} ≰ D} [D (D) (y_{μ})]^{'} ≰ b . \end{matrix}$ By arbitrariness of b ∈ M, we conclude that $\begin{matrix} [D (A) (x_{λ})]^{'} \leq ⋁_{x_{λ} ≰ B \geq A_{x_{λ}}^{-}} ⋀_{y_{μ} ≰ B} [D (B) (y_{μ})]^{'} . \end{matrix}$ Therefore (LMCDH3) holds.

(1) ⇔ (3). Similar to (1) ⇔ (2). □

Next, we discuss relations between LMCDHS and LMCS.

Theorem 3.7. Let $(L^{X}, C)$ be an (L, M)- fuzzy convex space. Define an operator $D_{C} : L^{X} \to M^{J (L^{X})}$ by $\begin{matrix} D_{C} (A) (x_{λ}) = {co}_{C} (A_{x_{λ}}^{-}) (x_{λ}) . \end{matrix}$ for any A ∈ L^X and any x_λ ∈ J (L^X). Then $(X, D_{C})$ is an (L, M)- fuzzy convex derived hull space.

Proof. (LMCDH1) and (LMCDH2) can be directly obtain from (1) and (4) of Proposition 3.1.

(LMCDH3). Let A ∈ L^X and x_λ ∈ J (L^X). If $x_{λ} \leq A_{x_{λ}}^{-}$ , then $D_{C} (A) (x_{λ}) = {co}_{C} (A_{x_{λ}}^{-}) (x_{λ}) = ⊤$ . Assume that $x_{λ} ≰ A_{x_{λ}}^{-}$ . By (LMCO3), we have $\begin{matrix} D_{C} (A) (x_{λ}) & = & ⋀_{x_{λ} ≰ B \geq A_{x_{λ}}^{-}} ⋁_{y_{μ} ≰ B} {co}_{C} (B) (y_{μ}) \\ \geq & ⋀_{x_{λ} ≰ B \geq A_{x_{λ}}^{-}} ⋁_{y_{μ} ≰ B} D_{C} (B) (y_{μ}) . \end{matrix}$ In addition, the inverse inequality is clear. Therefore (LMCDH3) holds for $D_{C}$ .

(LMCDH4). By (5) of Proposition 3.1, (1) of Lemma 3.4, (LMCO4) and (LMCO0), we have $\begin{matrix} D_{C} (A) (x_{λ}) & = & ⋀_{μ ≺ λ} ⋁_{F \in F (A_{x_{λ}}^{-})} {co}_{C} (F) (x_{μ}) \\ \leq & ⋀_{μ ≺ λ} ⋁_{F \in F (A)} {co}_{C} (F_{x_{λ}}^{-}) (x_{μ}) \\ = & ⋀_{μ ≺ λ} ⋁_{F \in F (A)} D_{C} (F) (x_{μ}) \\ \leq & ⋀_{μ ≺ λ} D_{C} (A) (x_{μ}) \\ \leq & D_{C} (A) (x_{λ}) . \end{matrix}$ Thus $D_{C} (A) (x_{λ}) = ⋀_{μ ≺ λ} ⋁_{F \in F (A)} D_{C} (F) (x_{μ})$ .

Therefore $(X, D_{C})$ is an (L, M)- fuzzy convex derived hull space. □

Theorem 3.8. Let $(X, C_{X})$ and $(Y, C_{Y})$ be (L, M)- fuzzy convex spaces. If f : X → Y is an (L, M)- fuzzy convexity preserving mapping, then $f : (X, D_{C_{X}}) \to (Y, D_{C_{Y}})$ is an (L, M)- fuzzy convex derived hull preserving mapping.

Proof. If $f_{L}^{\to} (x_{λ}) \leq f_{L}^{\to} (A)$ then $\begin{matrix} D_{C_{X}} (A) (x_{λ}) \leq ⊤ = \bar{D_{C_{Y}}} (f_{L}^{\to} (A)) (f_{L}^{\to} (x_{λ})) . \end{matrix}$ If $f_{L}^{\to} (x_{λ}) ≰ f_{L}^{\to} (A)$ then x_λ ≰ A. Thus $A_{x_{λ}}^{-} = A$ and $f_{L}^{\to} (A)_{f_{L}^{\to} (x_{λ})}^{-} = f_{L}^{\to} (A)$ . Hence $\begin{matrix} D_{C_{X}} (A) (x_{λ}) & \leq & ⋀_{f_{L}^{\to} (x_{λ}) ≰ B \geq f_{L}^{\to} (A)} [C_{Y} (B)]^{'} \\ = & \bar{D_{C_{Y}}} (f_{L}^{\to} (A)) (f_{L}^{\to} (x_{λ})) . \end{matrix}$ Therefore f is an (L, M)- fuzzy convex derived hull preserving mapping. □

Theorem 3.9. Let $(X, D)$ be an (L, M)- fuzzy convex derived hull space. Define a mapping $C_{D} : L^{X} \to M$ by $\begin{matrix} \forall A \in L^{X}, C_{D} (A) = ⋀_{x_{λ} ≰ A} [D (A) (x_{λ})]^{'} . \end{matrix}$ Then $(X, C_{D})$ is an (L, M)- fuzzy convex space.

Proof. (LMC1) is clear.

(LMC2). Let {A_i} _i∈I ⊆ L^X and A = ⋀ _i∈IA_i. For any x_λ ∈ J (L^X) with x_λ ≰ A, there is i₀ ∈ I such that x_λ ≰ A_{i
₀}. Thus $[D (A) (x_{λ})]^{'} \geq [D (A_{i_{0}}) (x_{λ})]^{'}$ . Hence $[D (A) (x_{λ})]^{'} \geq C_{D} (A_{i_{0}}) \geq ⋀_{i \in I} C_{D} (A_{i})$ . Therefore $\begin{matrix} ⋀_{i \in I} C_{D} (A_{i}) \leq ⋀_{x_{λ} ≰ A} [D (A) (x_{λ})]^{'} = C_{D} (A) . \end{matrix}$

(LMC3). Let ${A_{i}}_{i \in I} \overset{dir}{\subseteq} L^{X}$ and $A = ⋁_{i \in I}^{dir} A_{i}$ . For any a ∈ M with $C_{D} (A) ≱ a$ , there is b ≺ a such that $C_{D} (A) ≱ b$ . Thus $\begin{matrix} C_{D} (A) ≱ b & \Rightarrow & \exists x_{λ} ≰ A, ⋀_{μ ≺ λ} ⋁_{F \in F (A)} D (F) (x_{μ}) ≰ b^{'} \\ \Rightarrow & \exists x_{λ} ≰ A s . t . \forall μ ≺ λ, \exists F_{μ} \in F (A), \\ D (F_{μ}) (x_{μ}) ≰ b^{'} \\ \Rightarrow & \exists x_{λ} ≰ A s . t . \forall μ ≺ λ, \exists F_{μ} \in F (A_{i_{μ}}), \\ [D (F_{μ}) (x_{μ})]^{'} ≱ b . \end{matrix}$ Hence $b ≰ ⋁_{μ ≺ λ} ⋀_{F \in F (A_{i_{μ}})} [D (F) (x_{μ})]^{'}$ for some x_λ ≰ A. This implies that $\begin{matrix} ⋀_{i \in I} C_{D} (A_{i}) & = & ⋀_{i \in I} ⋀_{x ≰ A_{i}} [D (A_{i}) (x_{λ})]^{'} \\ \leq & ⋀_{x_{λ} ≰ A} ⋁_{μ ≺ λ} ⋀_{F \in F (A_{i_{μ}})} [D (F) (x_{μ})]^{'} ≱ a . \end{matrix}$

Therefore $⋀_{i \in I} C_{D} (A_{i}) \leq C_{D} (A)$ .

So $(X, C_{D})$ is an (L, M)- fuzzy convex space. □

Corollary 3.10. Let $(X, D)$ be an (L, M)- fuzzy convex derived hull space. We have ${co}_{C_{D}} (A) (x_{λ}) = \bar{D} (A) (x_{λ})$ for any A ∈ L^X and any x_λ ∈ J (L^X).

Proof. If x_λ ≤ A then ${co}_{C_{D}} (A) (x_{λ}) = \bar{D} (A) (x_{λ}) = ⊤$ . If x_λ ≰ A, then $A_{x_{λ}}^{-} = A$ . By (LMCDH3),

$\begin{matrix} {co}_{C_{D}} (A) (x_{λ}) =_{x_{λ} ≰ B \geq A} ⋁_{y_{μ} ≰ B} \bar{D} (A) (x_{λ}) . \end{matrix}$ This proved the result. □

Theorem 3.11. Let $(X, D_{X})$ and $(Y, D_{Y})$ be (L, M)- fuzzy convex derived hull spaces. If f : X → Y is an (L, M)- fuzzy convex derived hull preserving mapping, then $f : (X, C_{D_{X}}) \to (Y, C_{D_{Y}})$ is an (L, M)- fuzzy convexity preserving mapping.

Proof. Let B ∈ L^X. We have $\begin{matrix} C_{D_{Y}} (B) & \leq & ⋀_{f_{L}^{\to} (x_{λ}) ≰ B} [D_{Y} (B) (f_{L}^{\to} (x_{λ}))]^{'} \\ \leq & ⋀_{x_{λ} ≰ f_{L}^{ł} (B)} [D_{X} (f_{L}^{ł} (B)) (x_{λ})]^{'} \\ = & C_{D_{X}} (f_{L}^{ł} (B)) . \end{matrix}$ So f is an (L, M)- fuzzy convexity preserving mapping. □

Theorem 3.12. We have $D_{C_{D}} = D$ for any (L, M)- fuzzy convex derived hull space $(X, D)$ .

Proof. Let A ∈ L^X and x_λ ∈ J (L^X). If $x_{λ} \leq A_{x_{λ}}^{-}$ then $D_{C_{D}} (A) (x_{λ}) = D (A) (x_{λ}) = ⊤$ . Assume that $x_{λ} ≰ A_{x_{λ}}^{-}$ . By (LMCDH3) and (LMCDH2), we have $\begin{matrix} D_{C_{D}} (A) (x_{λ}) & = & {co}_{C_{D}} (A_{x_{λ}}^{-}) (x_{λ}) \\ = & ⋀_{x_{λ} ≰ B \geq A_{x_{λ}}^{-}} [C_{D} (B)]^{'} \\ = & ⋀_{x_{λ} ≰ B \geq A_{x_{λ}}^{-}} ⋁_{y_{μ} ≰ B} D (B) (y_{μ}) \\ = & D (A_{x_{λ}}^{-}) (x_{λ}) = D (A) (x_{λ}) . \end{matrix}$ Therefore $D_{C_{D}} = D$ . □

Theorem 3.13. We have $C_{D_{C}} = C$ for any (L, M)- fuzzy convex space $(X, C)$ .

Proof. Since $A_{x_{λ}}^{-} = A$ for any x_λ ≰ A, we have $\begin{matrix} C_{D_{C}} (A) = ⋀_{x_{λ} ≰ A} [{co}_{C} (A) (x_{λ})]^{'} = C_{{co}_{C}} (A) = C (A) . \end{matrix}$ Thus $C_{D_{C}} = C$ . □

Based on Theorems 3.7 and 3.8, we present the functor $𝔽 :$ LMCS → LMCDS by $\begin{matrix} 𝔽 ((X, C)) = (X, D_{C}), 𝔽 (f) = f . \end{matrix}$ Based on Theorems 3.8–3.13, $𝔽$ is an isomorphism. Thus we have the following conclusion.

Theorem 3.14. LMCDHS is isomorphic to LMCS.

Further, we discuss relations between LMCDHS and LMCERS.

Theorem 3.15. Let $(X, R)$ be an (L, M)- fuzzy convex enclosed relation space. Define an operator $D_{R} : L^{X} \to M^{J (L^{X})}$ by $\begin{matrix} D_{R} (A) (x_{λ}) = ⋀_{x_{λ} ≰ B} [R (A_{x_{λ}}^{-}, B)]^{'} \end{matrix}$ for all A ∈ L^X and x_λ ∈ J (L^X). Then $(X, D_{R})$ is an (L, M)- fuzzy convex derived hull space.

Proof. Since $D_{R} (A) (x_{λ}) = {co}_{R} (A_{x_{λ}}^{-}) (x_{λ})$ , it follows from Theorems 2.7 and 3.7 that $(X, D_{R})$ is an (L, M)- fuzzy convex derived hull space. □

Theorem 3.16. Let $(X, R_{X})$ and $(X, R_{Y})$ be an (L, M)- fuzzy convex enclosed relation spaces. If f : X → Y is an (L, M)- fuzzy convex enclosed relation preserving mapping, then $f : (X, D_{R_{X}}) \to (Y, D_{R_{Y}})$ is an (L, M)- fuzzy convex derived hull preserving mapping.

Proof. Let A ∈ L^X and x_λ ∈ J (L^X). If $f_{L}^{\to} (x_{λ}) \leq f_{L}^{\to} (A)$ then $\begin{matrix} D_{R_{X}} (A) (x_{λ}) \leq ⊤ = D_{R_{Y}} (f_{L}^{\to} (A)) (f_{L}^{\to} (x_{λ})) . \end{matrix}$ If $f_{L}^{\to} (x_{λ}) ≰ f_{L}^{\to} (A)$ then x_λ ≰ A. Thus $A_{x_{λ}}^{-} = A$ and $f_{L}^{\to} (A)_{f_{L}^{\to} (x_{λ})}^{-} = f_{L}^{\to} (A)$ . Hence $\begin{matrix} D_{R_{X}} (A) (x_{λ}) & \leq & ⋀_{x_{λ} ≰ f_{L}^{ł} (B)} [R_{X} (A, f_{L}^{ł} (B))]^{'} \\ \leq & ⋀_{f_{L}^{\to} (x_{λ}) ≰ B} [R_{Y} (f_{L}^{\to} (A), B)]^{'} \\ = & ⋀_{f_{L}^{\to} (x_{λ}) ≰ B} [R_{Y} (f_{L}^{\to} (A)_{f_{L}^{\to} (x_{λ})}^{-}, B)]^{'} \\ = & D_{R_{Y}} (f_{L}^{\to} (A)) (f_{L}^{\to} (x_{λ})) . \end{matrix}$ Therefore f is an (L, M)- fuzzy convex derived hull preserving mapping. □

Theorem 3.17 Let $(X, D)$ be an (L, M)- fuzzy convex derived hull space. Define a relation $R_{D} : L^{X} \times L^{X} \to M$ by $\begin{matrix} \forall A, B \in L^{X}, R_{D} (A, B) = ⋀_{x_{λ} ≰ B} [\bar{D} (A) (x_{λ})]^{'} . \end{matrix}$ Then $(X, R_{D})$ is an (L, M)- fuzzy convex enclosed relation space.

Proof. Notice that $R_{D} (A, B) = R_{{co}_{D}} (A, B)$ for all A, B ∈ L^X. By Theorem 2.7 and Corollary 3.10, $R_{D}$ is an (L, M)- fuzzy convex enclosed relation. □

Theorem 3.18. Let $(X, D_{X})$ and $(X, D_{Y})$ be an (L, M)- fuzzy convex derived hull spaces. If f : X → Y is an (L, M)- fuzzy convex derived hull preserving mapping, then $f : (X, R_{D_{X}}) \to (Y, R_{D_{Y}})$ is an (L, M)- fuzzy convex enclosed relation preserving mapping.

Proof. Let A ∈ L^X and B ∈ L^Y. We have $\begin{matrix} R_{D_{Y}} (f_{L}^{\to} (A), B) \\ \leq & ⋀_{f_{L}^{\to} (x_{λ}) ≰ B} [\bar{D_{Y}} (f_{L}^{\to} (A)) (f_{L}^{\to} (x_{λ}))]^{'} \\ \leq & ⋀_{x_{λ} ≰ f_{L}^{ł} (B)} [D_{X} (A) (x_{λ})]^{'} \\ = & R_{D_{X}} (A, f_{L}^{ł} (B)) . \end{matrix}$ Thus f is an (L, M)- fuzzy convex enclosed relation preserving mapping. □

Theorem 3.19. We have $D_{R_{D}} = D$ for any (L, M)- fuzzy convex derived hull space $(X, D)$ .

Proof. Let A ∈ L^X and x_λ ∈ J (L^X). By (1) of Lemma 3.4 and (LMCDH3), we have $\begin{matrix} D_{R_{D}} (A) (x_{λ}) & = & ⋀_{x_{λ} ≰ B \geq A_{x_{λ}}^{-}} ⋁_{y_{μ} ≰ B} D (A_{x_{λ}}^{-}) (y_{μ}) \\ \leq & ⋀_{x_{λ} ≰ B \geq A_{x_{λ}}^{-}} ⋁_{y_{μ} ≰ B} D (B) (y_{μ}) \\ = & D (A) (x_{λ}) . \end{matrix}$ Conversely, by (LMCER2) of $R_{D}$ and (LMCDH2) of $D$ , we have $\begin{matrix} D_{R_{D}} (A) (x_{λ}) & = & ⋀_{x_{λ} ≰ B \geq A_{x_{λ}}^{-}} [R_{D} (A_{x_{λ}}^{-}, B)]^{'} \\ = & ⋀_{x_{λ} ≰ B \geq A_{x_{λ}}^{-}} ⋁_{y_{μ} ≰ B} D (A_{x_{λ}}^{-}) (y_{μ}) \\ \geq & ⋀_{x_{λ} ≰ B \geq A_{x_{λ}}^{-}} D (A_{x_{λ}}^{-}) (x_{λ}) \\ = & D (A) (x_{λ}) . \end{matrix}$ Thus $D_{R_{D}} = D$ . □

Theorem 3.20. We have $R_{D_{R}} = R$ for any (L, M)- fuzzy convex enclosed relation space $(X, R)$ .

Proof. Let A, B ∈ L^X. If A ≰ B then $R_{D_{R}} (A, B) = R (A, B) = ⊥$ . Assume that A ≤ B. We have $\begin{matrix} R_{D_{R}} (A, B) & = & ⋀_{x_{λ} ≰ B} [\bar{D_{R}} (A) (x_{λ})]^{'} \\ = & ⋀_{x_{λ} ≰ B} [D_{R} (A) (x_{λ})]^{'} \\ = & ⋀_{x_{λ} ≰ B} ⋁_{x_{λ} ≰ D} R (A_{x_{λ}}^{-}, D) \\ = & ⋀_{x_{λ} ≰ B} ⋁_{x_{λ} ≰ D} R (A, D) \\ = & ⋀_{x_{λ} ≰ B} [{co}_{R} (A) (x_{λ})]^{'} \\ = & R (A, B) . \end{matrix}$ Therefore $R_{D_{R}} = R$ . □

Based on Theorems 3.16 and 3.17, we define a functor $𝔾 :$ LMCDHS → LMCERS by $\begin{matrix} 𝔾 ((X, D)) = (X, R_{D}), 𝔾 (f) = (f) . \end{matrix}$ Based on Theorems 3.14–3.20, we find that $𝔾$ is an isomorphism. Thus we have the following conclusion.

Theorem 3.21. LMCDHS is isomorphic to LMCERS.

4 (L, M)- Fuzzy restricted convex derived hull spaces

In this section, we introduce (L, M)- fuzzy restricted convex derived hull spaces by which we characterize (L, M)- fuzzy restricted convex hull spaces. Before this, we firstly introduce the following denotations.

For any $F \in F (L^{X})$ , we denote $\begin{matrix} \nabla_{F} = {φ \subseteq 2_{fin}^{β^{*} (\underline{⊤})} : F = ⋁ φ} . \end{matrix}$ Clearly, ∇_F is nonempty by the definition of $F \in F (L^{X})$ . Take any φ ∈ ∇ _F. If there are x_λ, y_μ ∈ φ such that x_λ ≤ y_μ, then $φ \underline{{} x_{λ}} \in \nabla_{F}$ . Notice that φ is finite. After some finite process as above, we obtain a subset ψ_φ ⊆ φ such that ψ_φ ∈ ∇ _F and elements of ψ_φ are incomparable with each other.

Proposition 4.1. We have ψ_φ = ⋂ _{φ∈∇_F}φ for any $F \in F (L^{X})$ and any φ ∈ ∇ _F.

Proof. Let $ψ_{φ} = {x_{λ_{i}}^{i}}_{i = 1}^{n}$ . Take any $φ = {y_{μ_{j}}^{j}}_{j = 1}^{m} \in \nabla_{F}$ . For each 1 ≤ i ≤ n, by $x_{λ_{i}}^{i} \leq F = ⋁_{j = 1}^{m} y_{μ_{j}}^{j}$ , there is 1 ≤ j ≤ m such that $x_{λ_{i}}^{i} \leq y_{μ_{j}}^{j}$ . Conversely, by $y_{μ_{j}}^{j} \leq F = ⋁_{t = 1}^{n} x_{λ_{k}}^{k}$ , there is 1 ≤ k ≤ n such that $y_{μ_{j}}^{j} \leq x_{λ_{k}}^{k}$ . Thus $x_{λ_{i}}^{i} \leq y_{μ_{j}}^{j} \leq x_{λ_{k}}^{k}$ . Since elements in ψ_φ are incomparable, we have $x_{λ_{i}}^{i} = x_{λ_{k}}^{k}$ . Thus $x_{λ_{i}}^{i} = y_{μ_{j}}^{j} \in φ$ . This implies that ψ_φ ⊆ φ. By arbitrariness of φ ∈ ∇ _F, we have ψ_φ = ⋂ _{φ∈∇_F}φ. □

For any $F \in F (L^{X})$ and all φ, φ ∈ ∇ _F, we have ψ_φ = ψ_φ by Proposition 4. Thus we denote this unique set conveniently by [F]. Clearly, we have [⊥] = {∅} and [y_μ] = {y_μ} for any y_μ ∈ β^* (L^X).

For all $F \in F (L^{X})$ and x_λ ∈ J (L^X), we denote $\begin{matrix} F_{x_{λ}}^{\land} = ⋁ {y_{μ} \in [F] : x_{λ} ≰ y_{μ}} . \end{matrix}$

Proposition 4.2. Let A ∈ L^X, $F \in F (L^{X})$ and x_λ ∈ J (L^X). We have

(1) x_λ ≰ F implies $F_{x_{λ}}^{\land} = F$ ;

(2) $(F_{x_{λ}}^{\land})_{x_{λ}}^{\land} = F_{x_{λ}}^{\land} \in F (L^{X})$ ;

(3) $A_{x_{λ}}^{-} = ⋁_{F \in F (A)} F_{x_{λ}}^{\land}$ ;

(4) $G_{x_{λ}}^{\land} = G$ for any $G \in F (F_{x_{λ}}^{\land})$ .

Proof.

(1) is clear.

(2). Clearly, ${y_{μ} \in [F] : x_{λ} ≰ y_{μ}} \in \nabla_{F_{x_{λ}}^{\land}}$ whose elements are incomparable. Thus $[F_{x_{λ}}^{\land}] = {y_{μ} \in [F] : x_{λ} ≰ y_{μ}}$ . Hence $(F_{x_{λ}}^{\land})_{x_{λ}}^{\land} = F_{x_{λ}}^{\land} \in F (L^{X})$ .

(3). If $y_{η} ≺ A_{x_{λ}}^{-}$ , then there is y_μ ∈ β^* (A) such that y_η ≺ y_μnotgeqx_λ. Thus $y_{μ} \in F (A)$ and so $y_{η} \leq y_{μ} = (y_{μ})_{x_{λ}}^{\land} \leq ⋁_{F \in F (A)} F_{x_{λ}}^{\land}$ . Hence $A_{x_{λ}}^{-} \leq ⋁_{F \in F (A)} F_{x_{λ}}^{\land}$ .

Conversely, let $F \in F (A)$ . For any y_μ ∈ [F] with x_λ ≰ y_μ, we have y_μ ∈ β^* (A). Thus $y_{μ} \leq A_{x_{λ}}^{-}$ . Hence $F_{x_{λ}}^{\land} \leq A_{x_{λ}}^{-}$ which shows that $⋁_{F \in F (A)} F_{x_{λ}}^{\land} \leq A_{x_{λ}}^{-}$ . Therefore $A_{x_{λ}}^{-} = ⋁_{F \in F (A)} F_{x_{λ}}^{\land}$ .

(4). By $G \in F (F_{x_{λ}}^{\land})$ , there is ${y_{μ_{i}}^{i}}_{i = 1}^{n} \subseteq 2_{fin}^{β^{*} (F_{x_{λ}}^{\land})}$ such that $G = ⋁_{i = 1}^{n} y_{μ_{i}}^{i}$ . Thus, for any 1 ≤ i ≤ n, there is $y_{η_{i}}^{i} \in [F]$ such that $y_{μ_{i}}^{i} ≺ y_{η_{i}}^{i}$ and $x_{λ} ≰ y_{η_{i}}^{i}$ . This implies that $x_{λ} ≰ y_{μ_{i}}^{i}$ for any 1 ≤ i ≤ n. Hence x_λ ≰ G. Therefore $G = G_{x_{λ}}^{\land}$ . □

Proposition 4.3. For any operator $H : F (L^{X}) \to M^{J (L^{X})}$ satisfying (LMRCH0), $H$ satisfies (LMRCH3) iff it satisfies (LMRCH3*), where

(LMRCH3*) $\forall F \in F (L^{X})$ , ∀x_λ ∈ J (L^X), $\begin{matrix} H (G) (x_{λ}) \land ⋀_{y_{μ} \in [G]} H (F) (y_{μ}) \leq H (F) (x_{λ}) . \end{matrix}$

Proof. We have $\begin{matrix} ⋀_{y_{μ} \in [G]} H (F) (y_{μ}) & = & ⋀_{y_{μ} \in [G]} ⋀_{η ≺ μ} H (F) (y_{η}) \\ = & ⋀_{y_{η} ≺ G} H (F) (y_{η}) . \end{matrix}$ So (LMRCH3) and (LMRCH3*) are equivalent. □

Definition 4.4. An operator $E : F (L^{X}) \to M^{J (L^{X})}$ is called an (L, M)- fuzzy restricted convex derived hull operator and the pair $(X, E)$ is an (L, M)- fuzzy restricted convex derived hull space, if

(LMRCDH1) $E (\underline{⊥}) (x_{λ}) = ⊥$ ;

(LMRCDH2) $E (F) (x_{λ}) = E (F_{x_{λ}}^{\land}) (x_{λ})$ ;

(LMRCDH3) x_λ ≰ F implies $\begin{matrix} E (G) (x_{λ}) \land ⋀_{y_{μ} \in [G], y_{μ} ≰ F} E (F) (y_{μ}) \leq E (F) (x_{λ}); \end{matrix}$

(LMRCDH4) x_λ ≰ F implies $\begin{matrix} E (F) (x_{λ}) = ⋀_{μ ≺ λ, x_{μ} ≰ F} ⋁_{G \in F (F)} E (G) (x_{μ}) . \end{matrix}$

Let $(X, E_{X})$ and $(Y, E_{Y})$ be (L, M)- fuzzy restricted convex derived hull spaces. A mapping f : X → Y is an (L, M)- fuzzy restricted convex derived hull preserving mapping if $\begin{matrix} E_{X} (F) (x_{λ}) \leq \bar{E_{Y}} (f_{L}^{\to} (F)) (f_{L}^{\to} (x_{λ})) \end{matrix}$ for any $F \in F (L^{X})$ and any x_λ ∈ J (L^X), where $\bar{E_{Y}} (f_{L}^{\to} (F)) (f_{L}^{\to} (x_{λ})) = E_{Y} (f_{L}^{\to} (F)) \lor Ξ_{f_{L}^{\to} (F)}$ .

The category of (L, M)- fuzzy restricted convex derived hull spaces and (L, M)- fuzzy restricted convex derived hull preserving mappings is denoted by LMRCDHS.

Remark 4.5 (1) If L = 2 then an (L, M)- fuzzy restricted convex derived hull space $(X, E)$ is an M-fuzzifying restricted convex derived hull space [13].

(2) An (L, 2)-fuzzy restricted convex derived hull space $(X, E)$ is called an L-restricted convex derived hull space, that is, $E : F (L^{X}) \to L^{X}$ satisfies

(LRCDH1) $E (\underline{⊥}) = \underline{⊥}$ ;

(LRCDH2) $x_{λ} \leq E (F)$ implies $x_{λ} \leq E (F_{x_{λ}}^{\land})$ ;

(LRCDH3) $G \in F (L^{X})$ and $G \leq F \lor E (F)$ imply $E (G) \leq F \lor E (F)$ ;

(LRCDH4) $F \lor E (F) = F \lor ⋁_{G \in F (F)} E (G)$ .

Next, we discuss the relations between LMRCDHS and LMRCHS.

Theorem 4.6. Let $(X, H)$ be an (L, M)- fuzzy restricted convex hull space. Define an operator $E_{H} : F (L^{X}) \to M^{J (L^{X})}$ by $\begin{matrix} E_{H} (F) (x_{λ}) = H (F_{x_{λ}}^{\land}) (x_{λ}) \end{matrix}$ for all $F \in F (L^{X})$ and x_λ ∈ J (L^X). Then $(X, E_{H})$ is an (L, M)- fuzzy restricted convex derived hull space.

Proof. (LMRCDH1) and (LMRCDH2) are direct.

(LMRCDH3). Let A ∈ L^X and x_λ ∈ J (L^X) with x_λ ≰ F. By (LMRCH2), we have $H (F) (y_{μ}) = ⊤$ for any y_μ ≤ F. Further, since $H$ satisfies (LMCRH0), we obtain from (LMRCH3*) that $\begin{matrix} E_{H} (F) (x_{λ}) & = & H (F) (x_{λ}) \\ \geq & H (G) (x_{λ}) \land ⋀_{y_{μ} \in [G]} H (F) (y_{μ}) \\ \geq & H (G_{x_{λ}}^{\land}) (x_{λ}) \land ⋀_{y_{μ} \in [G], y_{μ} ≰ F} & = & E_{H} (G) (x_{λ}) \land ⋀_{y_{μ} \in [G], y_{μ} ≰ F} E_{H} (F) (y_{μ}) . \end{matrix}$ Therefore (LMRCDH3) holds for $E_{H}$ .

(LMRCDH4). Let A ∈ L^X and x_λ ∈ J (L^X) with x_λ ≰ F. By (LMRCH4), we have $\begin{matrix} E_{H} (F) (x_{λ}) & = & H (F) (x_{λ}) \\ = & ⋀_{μ ≺ λ} ⋁_{G \in F (F)} H (G) (x_{μ}) \\ \leq & ⋀_{μ ≺ λ, x_{μ} ≰ F} ⋁_{G \in F (F)} H (G) (x_{μ}) \\ = & ⋀_{μ ≺ λ, x_{μ} ≰ F} ⋁_{G \in F (F)} E_{H} (G) (x_{μ}) . \end{matrix}$ Conversely, by (LMRCH0) and (LMRCH2), we have $\begin{matrix} E_{H} (F) (x_{λ}) & = & H (F) (x_{λ}) \\ = & ⋀_{μ ≺ λ} H (F) (x_{μ}) \\ = & ⋀_{μ ≺ λ, x_{μ} ≰ F} H (F) (x_{μ}) \\ \geq & ⋀_{μ ≺ λ, x_{μ} ≰ F} ⋁_{G \in F (F)} H (G) (x_{μ}) \\ = & ⋀_{μ ≺ λ, x_{μ} ≰ F} ⋁_{G \in F (F)} E_{H} (G) (x_{μ}) . \end{matrix}$ Therefore (LMRCDH4) holds for $E_{H}$ . □

Theorem 4.7. Let $(X, H_{X})$ and $(Y, H_{Y})$ be (L, M)- fuzzy restricted convex hull spaces. If f : X → Y is an (L, M)- fuzzy restricted convex hull preserving mapping, then $f : (X, E_{H_{X}}) \to (Y, E_{H_{Y}})$ is an (L, M)- fuzzy restricted convex derived hull preserving mapping.

Proof. Let $F \in F (L^{X})$ and x_λ ∈ J (L^X). If $f_{L}^{\to} (x_{λ}) \leq f_{L}^{\to} (F)$ then $\begin{matrix} E_{H_{X}} (F) (x_{λ}) \leq ⊤ = \bar{E_{H_{Y}}} (f_{L}^{\to} (F)) (f_{L}^{\to} (x_{λ})) . \end{matrix}$ If $f_{L}^{\to} (x_{λ}) ≰ f_{L}^{\to} (F)$ then $f_{L}^{\to} (F) = f_{L}^{\to} (F)_{f_{L}^{\to} (x_{λ})}^{\land}$ and $x_{λ} ≰ F = F_{x_{λ}}^{\land}$ . Thus $\begin{matrix} E_{H_{X}} (F) (x_{λ}) & = & H_{X} (F) (x_{λ}) \\ \leq & H_{Y} (f_{L}^{\to} (F)) (f_{L}^{\to} (x_{λ})) \\ = & E_{H_{Y}} (f_{L}^{\to} (F)) (f_{L}^{\to} (x_{λ})) . \end{matrix}$ Therefore f is an (L, M)- fuzzy restricted convex derived hull preserving mapping. □

Lemma 4.8. Let $(X, E)$ be an (L, M)- fuzzy restricted convex derived hull space. For all $F, G \in F (L^{X})$ , x_λ ∈ J (L^X) with x_λ ≰ F ≥ G, we have

(1) $E (G) (x_{λ}) \leq E (F) (x_{λ})$ ;

(2) $E (F) (x_{λ}) = ⋀_{μ ≺ λ, x_{μ} ≰ F} E (F) (x_{μ})$ .

Proof. (1). It directly follows from (LMRCH4) and $F (G) \subseteq F (F)$ .

(2). By (LMRCDH3) and (LMRCDH4), we have $\begin{matrix} E (F) (x_{λ}) & = & ⋀_{μ ≺ λ, x_{μ} ≰ F} ⋁_{G \in F (F)} E (G) (x_{μ}) \\ \leq & ⋀_{μ ≺ λ, x_{μ} ≰ F} E (F) (x_{μ}) \\ = & ⋀_{μ ≺ λ, x_{μ} ≰ F} ⋀_{η ≺ μ, x_{η} ≰ F} ⋁_{G \in F (F)} E (G) (x_{η}) \\ = & ⋀_{η ≺ λ, x_{η} ≰ F} ⋁_{G \in F (F)} E (G) (x_{η}) \\ = & E (F) (x_{λ}) . \end{matrix}$ Therefore $E (F) (x_{λ}) = ⋀_{μ ≺ λ, x_{μ} ≰ F} E (F) (x_{μ})$ . □

Theorem 4.9. Let $(X, E)$ be an (L, M)- fuzzy restricted convex derived hull space. Define an operator $H_{E} : F (L^{X}) \to M^{J (L^{X})}$ by $\begin{matrix} \forall F \in F (L^{X}), H_{E} (F) = \bar{E} (F) = E (F) \lor Ξ_{F} . \end{matrix}$ Then $H_{E}$ is an (L, M)- fuzzy restricted convex hull operator which induces an (L, M)- fuzzy convex structure $C_{E}$ .

Proof. (LMRCH1) and (LMRCH2) are direct. Next, we check that (LMRCH0) holds for $H_{E}$ .

(LMRCH0). Let $F \in F (L^{X})$ and x_λ ∈ J (L^X). If x_λ ≤ F then $H_{E} (F) (x_{λ}) = ⊤$ . Also, we have x_μ ≺ F for any μ ≺ λ. Thus $x_{μ} \in F (F)$ and $\begin{matrix} H_{E} (F) (x_{λ}) = ⊤ = H_{E} (y_{μ}) (y_{μ}) \leq ⋁_{G \in F (F)} \end{matrix}$ So (LMRCH0) holds. Assume that x_λ ≰ F. By (2) of Lemma 4.8, we have $H_{E} (F) (x_{λ}) = E (F) (x_{λ})$ and $\begin{matrix} E (F) (x_{λ}) = ⋀_{μ ≺ λ, x_{μ} ≰ F} E (F) (x_{μ}) = ⋀_{μ ≺ λ} H_{E} (F) (x_{μ}) . \end{matrix}$ Hence (LMRH0) also holds.

(LMRCH3*). Let $F, G \in F (L^{X})$ and x_λ ∈ J (L^X). If x_λ ≤ F then $H_{E} (F) (x_{λ}) = \bar{E} (F) (x_{λ}) = ⊤$ . If x_λ ≤ G then there is x_μ ∈ [G] such that x_λ ≤ x_μ. Thus, by (LMRCH0), we have $\begin{matrix} H_{E} (F) (x_{λ}) & \geq & H_{E} (F) (x_{μ}) \\ \geq & H_{E} (G) (x_{λ}) \land ⋀_{z_{η} \in [G]} H_{E} (F) (z_{η}) . \end{matrix}$ Assume that x_λ ≰ F ∨ G. By (LMRCH3), we have $\begin{matrix} H_{E} (F) (x_{λ}) & = & E (F) (x_{λ}) \\ \geq & E (G) (x_{λ}) \land ⋀_{y_{μ} \in [G], y_{μ} ≰ F} E (F) (y_{μ}) \\ = & H_{E} (G) (x_{λ}) \land ⋀_{y_{μ} \in [G]} H_{E} (F) (y_{μ}) . \end{matrix}$ Hence (LMRCH3*) holds.

(LMRCH4). Let $F \in F (L^{X})$ and x_λ ∈ J (L^X). If x_λ ≤ F then $H_{E} (F) (x_{λ}) = \bar{E} (F) = ⊤$ . Also, we have x_μ ≺ F for any x_μ ≺ x_λ. Thus there is $G_{x_{μ}} \in F (F)$ such that x_μ ≤ G_{x
_μ} for any x_μ ≺ x_λ. Hence $\begin{matrix} ⊤ = H_{E} (G_{y_{μ}}) (x_{μ}) \leq ⋁_{G \in F (F)} H_{E} (G) (x_{μ}) . \end{matrix}$ So (LMRCH4) holds in this case. Assume that x_λ ≰ F. Then $H_{E} (F) (x_{λ}) = E (F) (x_{λ})$ . Further, by (2) of Lemma 4.8, we have $\begin{matrix} E (F) (x_{λ}) & = & ⋀_{μ ≺ λ, x_{μ} ≰ F} ⋁_{G \in F (F)} E (G) (x_{μ}) \\ = & ⋀_{μ ≺ λ, x_{μ} ≰ F} ⋁_{G \in F (F)} H_{E} (G) (x_{μ}) . \end{matrix}$ Thus $\begin{matrix} E (F) (x_{λ}) & \geq & ⋀_{μ ≺ λ} ⋁_{G \in F (F)} H_{E} (G) (x_{μ}) \\ = & ⋀_{μ ≺ λ, x_{μ} ≰ F} ⋁_{G \in F (F)} H_{E} (G) (x_{μ}) \land \\ ⋀_{μ ≺ λ, x_{μ} \leq F} ⋁_{G \in F (F)} H_{E} (G) (x_{μ}) \\ = & E (F) (x_{λ}) \land ⋀_{μ ≺ λ, x_{μ} \leq F} ⋁_{G \in F (F)} H_{E} (G) (x_{μ}) \\ \geq & E (F) (x_{λ}) \land ⋀_{μ ≺ λ, x_{μ} \leq F} χ_{F} (x_{μ}) \\ = & E (F) (x_{λ}) . \end{matrix}$ Hence $\begin{matrix} H_{E} (F) (x_{λ}) = E (F) (x_{λ}) = ⋀_{μ ≺ λ} ⋁_{G \in F (F)} H_{E} (G) (x_{μ}) . \end{matrix}$ Therefore (LMRCH4) holds for $H_{F}$ . □

Theorem 4.10. Let $(X, E_{X})$ and $(Y, E_{Y})$ be (L, M)- fuzzy restricted convex derived hull spaces. If f : X → Y is an (L, M)- fuzzy restricted convex derived hull preserving mapping then $f : (X, H_{E_{X}}) \to (Y, H_{E_{Y}})$ is an (L, M)- fuzzy restricted convex hull preserving mapping.

Proof. Let $F \in F (L^{X})$ and x_λ ∈ J (L^X). If $f_{L}^{\to} (x_{λ}) \leq f_{L}^{\to} (F)$ then $\begin{matrix} H_{E_{X}} (F) (x_{λ}) \leq ⊤ = H_{E_{Y}} (f_{L}^{\to} F) (f_{L}^{\to} (x_{λ})) . \end{matrix}$ Assume that $f_{L}^{\to} (x_{λ}) ≰ f_{L}^{\to} (F)$ . Then x_λ ≰ F, $F_{x_{λ}}^{\land} = F$ and $f_{L}^{\to} (F)_{f_{L}^{\to} (x_{λ})}^{\land} = f_{L}^{\to} (F)$ . Thus $\begin{matrix} H_{E_{X}} (F) (x_{λ}) & = & E_{X} (F) (x_{λ}) \\ \leq & E_{Y} (f_{L}^{\to} (F)) (f_{L}^{\to} (x_{λ})) \\ = & H_{E_{Y}} (f_{L}^{\to} (F)) (f_{L}^{\to} (x_{λ})) . \end{matrix}$ Therefore f is an (L, M)- fuzzy restricted convex hull preserving mapping. □

Theorem 4.11. We have $H_{E_{H}} = H$ for any (L, M)- fuzzy restricted convex hull space $(X, H)$ .

Proof. Let $F \in F (L^{X})$ and x_λ ∈ J (L^X). If x_λ ≤ F then $H_{E_{H}} (F) (x_{λ}) = H (F) (x_{λ}) = ⊤$ . If x_λ ≰ F then $F_{x_{λ}}^{\land} = F$ . Thus $\begin{matrix} H_{E_{H}} (F) (x_{λ}) & = & H_{E_{H}} (F_{x_{λ}}^{\land}) (x_{λ}) = E_{H} (F) (x_{λ}) \\ = & H (F_{x_{λ}}^{\land}) (x_{λ}) = H (F) (x_{λ}) . \end{matrix}$ Hence $H_{E_{H}} = H$ . □

Theorem 4.12. We have $E_{H_{E}} = E$ for any (L, M)- fuzzy restricted convex derived space $(X, E)$ .

Proof. For any $F \in F (L^{X})$ and any x_λ ∈ J (L^X), we have $x_{λ} ≰ F_{x_{λ}}^{\land}$ . Thus $\begin{matrix} E_{H_{E}} (F) (x_{λ}) & = & H_{E} (F_{x_{λ}}^{\land}) (x_{λ}) \\ = & E (F_{x_{λ}}^{\land}) (x_{λ}) = E (F) (x_{λ}) . \end{matrix}$ Therefore $E_{H_{E}} = E$ . □

Based on Theorems 4.6 and 4.7, we present a functor $ℍ :$ LMRCHS → LMRCDHS by $\begin{matrix} ℍ ((X, H)) = (X, E_{H}), ℍ (f) = f . \end{matrix}$ Based on Theorems 4.6, 4.7, 4.9–4.12, we find that $ℍ$ is an isomorphism. So we have the following result.

Theorem 4.13. LMRCDHS is isomorphic to LMRCHS.

Corollary 4.14. LMRCDHS is isomorphic to LMCS and LMCERS.

5 Conclusions

Derived operators exist in many mathematical structures which are useful tools to describe some related mathematical structures. In this paper, we introduce (L, M)- fuzzy convex derived hull space by which we characterize (L, M)- fuzzy convex space and (L, M)- fuzzy convex enclosed relation space. Based on this, we further introduce (L, M)- fuzzy restricted convex derived hull space by which we characterize (L, M)- fuzzy restricted convex hull spaces.

There is a direct one-to-one correspondence between (L, M)- fuzzy convex derived hull operators and (L, M)- fuzzy restricted convex derived hull operators. It can be realized as follows.

For an (L, M)- fuzzy restricted convex derived hull space $(X, E)$ , we obtain an (L, M)- fuzzy convex derived hull operator $D_{E} : L^{X} \to M^{J (L^{X})}$ by defining $\begin{matrix} D_{E} (A) (x_{λ}) = ⋀_{μ ≺ λ} ⋁_{F \in F (A_{x_{λ}}^{-})} \bar{E} (F) (x_{μ}) \end{matrix}$ for any A ∈ L^X and any x_λ ∈ J (L^X). Conversely, for an (L, M)- fuzzy convex derived hull space $(X, D)$ , we obtain an (L, M)- fuzzy restricted convex derived hull operator $E_{D} : F (L^{X}) \to M^{J (L^{X})}$ by defining $\begin{matrix} E_{D} (F) (x_{λ}) = D (F_{x_{λ}}^{\land}) (x_{λ}) \end{matrix}$ for any $F \in F (L^{X})$ and any x_λ ∈ J (L^X). It is routine to prove that (L, M)- fuzzy convex derived operators and (L, M)- fuzzy restricted convex derived operators are one-to-one correspondent.

(L, M)- fuzzy hull operators and (L, M)- fuzzy convex enclosed relations are used to characterize (L, M)- fuzzy topological-convex spaces [5, 25]. Then it may be worth to consider how to characterize (L, M)- fuzzy topological-convex spaces by (L, M)- fuzzy convex derived hull operators accompany with (L, M)- fuzzy topological derived operators. Also, as (L, M)- fuzzy convergence spaces closely related to (L, M)- fuzzy topological spaces and (L, M)- fuzzy convex spaces [2 , 32], it may also be worth to consider possible characterizations of (L, M)- fuzzy convergence spaces by (L, M)- fuzzy convex derived hull spaces.

Footnotes

Acknowledgment

We sincerely thank the editor for handling our manuscript. And we deeply appreciate comments and suggestions from the reviewers.

This work is supported by the University Natural Science Research Project of Anhui Province (KJ2020A0056), Doctoral Scientific Research Foundation of Anhui Normal University (751966), Hunan Educational Committee Projects (18A474, 19C0822) and Natural Science Foundations of Hunan Province (2018JJ3192, 2019JJ40089).

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