Fuzzy generalized convex spaces

Abstract

On completely distributive lattice, the notion of fuzzy generalized convex space is introduced. It can be characterized by many means including fuzzy generalized hull space, fuzzy generalized restricted hull space, fuzzy generalized convexly enclosed relation space and fuzzy generalized derived hull space.

Keywords

Fuzzy generalized convex space fuzzy generalized hull space fuzzy generalized restricted hull space fuzzy generalized convexly enclosed relation space

1 Introduction

The theory of convex structures is an important mathematical branch that involves many mathematical structures in both theoretic aspects and applied aspects [21]. In this theory, a convex structure is a set-theoretic structure satisfying several axioms that a traditional convex set in Euclidean spaces has.

Convex structure has been extended into fuzzy settings in many ways. Maruyama introduced L-convex structure [6], some of whose properties have been discussed [4 , 28]. Actually, an L-convex structure is a crisp family of L-fuzzy sets satisfying certain set of axioms similar to that an abstract convex structure has. From a totally different point of view, Shi and Xiu introduced M-fuzzifying convex structures [18]. Many subsequent studies have been done [19 , 34]. To generalize L-convex structure and M-fuzzifying convex structure, Shi and Xiu further introduced (L, M)-fuzzy convex structure [20]. Its characterizations and properties have been studied recently [10 , 30].

Completely distributive lattice provides a concise and convenient tool to investigate logic and mathematical structures. Based on completely distributive lattices, Wang introduced the notion of topological molecular lattices and established the theory of topological molecular lattices [23] which is a unified theory of both the theory of topological spaces and the theory of fuzzy topological spaces. Based on this theory, many related theories have been investigated [17 , 35–37]. So, is it suitable to generalized convex structure into completely distributive lattices? Does it can be characterized by related mathematical structures? To solve these problems, we present this paper.

On completely distributive lattice, we introduce fuzzy convex space and obtain some of its characterizations. The arrangement of this paper is as follows. In Section 2, we recall some basic concepts, denotations and results. In Section 3, we introduce fuzzy generalized convex space and characterized it by fuzzy generalized hull space. From Section 4 to Section 6, we respectively introduce fuzzy generalized restricted hull space, fuzzy generalized convexly enclosed relation spaces and fuzzy generalized derived hull spaces. We prove that all of them are categorically isomorphic to fuzzy generalized convex spaces.

2 Preliminaries

In this paper, M is a completely distributive lattice with an inverse involution ^′. Its largest element and smallest elements are respectively denoted by ⊤_M and ⊥_M. L is also a completely distributive lattice whose largest element and smallest element are respectively denoted by ⊥ and ⊤. An element a ∈ L is a molecular, if a ⩽ b ∨ c implies a ⩽ b or a ⩽ c for all b, c ∈ L. The set of all molecular in L ∖ {⊥} is denoted by J (L) [23]. Dually, an element a ∈ L is a prime, if b ∧ c ⩽ a implies b ⩽ a or c ⩽ a for all b, c ∈ L. The set of all primes in L ∖ {⊤} is denoted by P (L). For any φ ⊆ L, we denote ⋁_u∈φu and ⋀_u∈φu by ⋁φ and ⋀φ. Further, we adopt that ⋁∅ = ⊥ and ⋀∅ = ⊤. φ is said to be up-directed, denoted by $φ \overset{dir}{\subseteq} L$ , if for all u, v ∈ φ there is w ∈ φ such that u ∨ v ⩽ w. In this case, we also denote ⋁φ by ⋁^dirφ.

A binary relation ≺ on L is defined by: for all a, b ∈ L, a ≺ b if for all φ ⊆ L, b ⩽ ⋁ φ implies some d ∈ φ such that a ⩽ d. Dually, a binary relation ≺^op on L is defined by: b ≺ ^opa if for all ψ ⊆ L, ⋀ψ ⩽ b implies some d ∈ ψ such that d ⩽ a. Clearly, we have β (⋁ φ) = ⋃ _a∈φβ (a) and α (⋀ φ) = ⋃ _a∈φα (a_i) for all φ ⊆ L, where β (a) = {b : b ≺ a} and α (a) = {b : a ≺ ^opb}. In particular, we have β (⊥) = α (⊤) = ∅. For any a ∈ L, we have a = ⋁ β (a) = ⋁ β^* (a) = ⋀ α (a) = ⋀ α^* (a), where β^* (a) = β (a) ∩ J (L) and α^* (a) = α (a) ∩ P (L) [23].

Let L₁ and L₂ be completely distributive lattices. A mapping F : L₁ → L₂ is called a generalized order homomorphism (GOH), if F (⊥) = ⊥ and both F and F^⊣ are union preserving, where F^⊣ : L₂ → L₁ is defined by F^⊣ (b) = ⋁ {a ∈ L₁ : F (a) ⩽ b} for any b ∈ L₂. It is proved that: (1) F and F^⊣ are order preserving; (2) a ⩽ F^⊣ (F (a)) and F (F^⊣ (b)) ⩽ b for any a ∈ L₁ and any b ∈ L₂; (3) F (a) ⩽ b iff a ⩽ F^⊣ (b) for any a ∈ L₁ and any b ∈ L₂ [23].

X is nonempty, 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 element and smallest element are $\underline{⊤}$ and $\underline{⊥}$ . For A ∈ L^X, we denote β (A) = {x_λ ∈ L^X : λ ∈ β (A (x))}, β^* (A) = β (A) ∩ J (L^X) and $F (A) = {⋁ φ \in L^{X} : φ \in 2_{fin}^{β^{*} (A)}}$ , where J (L^X) = {x_λ ∈ L^X : λ ∈ J (L)} [15, 29].

Definition 2.1. [14] An operator d : 2^X → M^X is called an M-fuzzifying convexly derived operator on X, and the pair (X, d) is called an M-fuzzifying convexly derived space, if

(MCD1) d (∅) (x) = ⊥;

(MCD2) d (A ∖ {x}) (x) = d (A) (x);

(MCD3) d (A) (x) = ⋀ _{x∉B⊇A∖{x}} ⋁ _y∉Bd (B) (y);

(MCD4) $d (⋃_{i \in I}^{dir} A_{i}) = ⋁_{i \in I} d (A_{i})$ .

Definition 2.2. [20] 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) $C (\underline{⊥}) = C (\underline{⊤}) = ⊤_{M}$ ;

(LMC2) $C (⋀ φ) \geq ⋀_{A \in φ} C (A)$ for any φ ⊆ L^X;

(LMC3) $C (⋁^{dir} φ) \geq ⋀_{A \in φ} C (A)$ for any $φ \overset{dir}{\subseteq} L^{X}$ .

Definition 2.3. [29] An operator co : L^X → M^{J(L^X)} is called an (L, M)-fuzzy hull operator and the pair (X, co) is called an (L, M)- fuzzy hull space, if it satisfies

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

(LMCO2) co (A) (x_λ) = ⊤ _M for any x_λ ⩽ A;

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

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

Definition 2.4. [29] An operator $H : F (L^{X}) \to M^{J (L^{X})}$ is called an (L, M)-fuzzy restricted hull operator and (X, ℋ) is called an (L, M)-fuzzy restricted hull space, if for all x_λ ∈ J (L^X) and $F, G \in F (L^{X})$ ,

(LMRH1) $H (\underline{⊥}) (x_{λ}) = ⊥_{M}$ ;

(LMRH2) ℋ (F) (x_λ) = ⊤ _M whenever x_λ ⩽ F;

(LMRH3) ℋ (G) (x_λ) ∧ ⋀ _{y_μ≺G} (x_λ);

(LMRH4) $H (F) (x_{λ}) = ⋀_{μ ≺ λ} ⋁_{G \in F (F)} H (G) (x_{μ})$ .

If L = {⊥ , ⊤} then an (L, M)-fuzzy convex structure (resp. (L, M)-fuzzy hull operator, (L, M)-fuzzy restricted hull operator) reduces to an M-fuzzifying convex structure (resp. M-fuzzifying hull operator, or M-fuzzifying restricted hull operator) [18, 19]. Denotations not mentioned here can be seen in [1 , 16].

3 Fuzzy generalized convex spaces and fuzzy generalized hull spaces

In this section, we introduce fuzzy generalized convex spaces and fuzzy generalized hull spaces. We discuss the one-to-one correspondence between them.

Definition 3.1. A mapping $C : L \to M$ is called a fuzzy generalized convex structure on L and the pair $(L, C)$ is called a fuzzy generalized convex space if

(FGC1) $C (⊤) = C (⊥) = ⊤_{M}$ ;

(FGC2) $C (⋀ φ) \geq ⋀_{u \in φ} C (u)$ for any φ ⊆ L;

(FGC3) $C (⋁^{dir} φ) \geq ⋀_{u \in φ} C (u)$ for any $φ \overset{dir}{\subseteq} L$ .

Let $(L_{1}, C_{1})$ and $(L_{2}, C_{2})$ be fuzzy generalized convex spaces. A GOH F : L₁ → L₂ is called a fuzzy generalized convexity preserving mapping with respect to $C_{1}$ and $C_{2}$ if $C_{2} (u) ⩽ C_{1} (F^{⊣} (u))$ for any u ∈ L.

The category of fuzzy generalized convex spaces and fuzzy generalized convexity preserving mappings is denoted by FGCS.

Remark 3.2. (1) If $(X, C)$ is an M-fuzzifying (resp. (L, M)-fuzzy) convex space, then $(2^{X}, C)$ (resp. $(L^{X}, C)$ ) is a fuzzy generalized convex space [18, 20], Xiu2017.

(2) If M = {⊥ , ⊤} then a fuzzy generalized convex space $(L, C)$ reduces to a generalized convex space. That is, $C \subseteq L$ is a convex structure on L satisfying

(GC1) $⊥, ⊤ \in C$ ;

(GC2) $⋀ φ \in C$ for any $φ \subseteq C$ ;

(GC3) $⋁^{dir} φ \in C$ for any $φ \overset{dir}{\subseteq} C$ .

Clearly, for an convex space or an L-convex spaces $(X, C)$ , the pair $(2^{X}, C)$ or $(L^{X}, C)$ is a generalized convex space [6, 21].

(4) For any mapping $C : L \to M$ , it is easy to check that the following conditions are equivalent:

(i) $(L, C)$ is a fuzzy generalized convex space;

(ii) $(L, C_{[a]})$ is a generalized convex space for any a ∈ M, where $C_{[a]} = {u \in L : C (u) \geq a}$ ;

(iii) $(L, C^{[a]})$ is a generalized convex space for any a ∈ α (⊥), where $C^{[a]} = {u \in L : a \notin α (C (u))}$ .

To introduce fuzzy generalized hull operators on L, we denote $\begin{matrix} F (u) = {w \in L : \exists φ \subseteq 2_{fin}^{β^{*} (u)}, w = ⋁ φ} \end{matrix}$ for any u ∈ L. In particular, $F (⊤)$ is simply denoted by $F (L)$ . We have the following simple properties.

Proposition 3.3. Let u, v ∈ L and $φ \overset{dir}{\subseteq} L$ . We have

(1) $F (⊥) = {⊥}$ ;

(2) u ⩽ v implies $F (u) \subseteq F (v)$ ;

(3) $β^{*} (u) \subseteq F (u) \overset{dir}{\subseteq} L$ and $⋁^{dir} F (u) = u$ ;

(4) $F (⋁^{dir} φ) = ⋃_{u \in φ}^{dir} F (u)$ .

Definition 3.4. An operator co : L → M^J(L) is called a fuzzy generalized hull operator on L and the pair (L, co) is called a fuzzy generalized hull space if

(FGCO1) co (⊥) (x) = ⊥ _M;

(FGCO2) x ⩽ u implies co (u) (x) = ⊤ _M;

(FGCO3) co (u) (x) = ⋀ _{x ≰ v ≥ u} ⋁ _{y ≰ v}co (v) (y);

(FGCO4) $co (u) (x) = ⋀_{y ≺ x} ⋁_{v \in F (u)} co (v) (y)$ .

Let (L₁, co₁) and (L₂, co₂) be fuzzy generalized hull spaces. A GOH F : L₁ → L₂ is called a fuzzy generalized hull preserving mapping with respect to co₁ and co₂ if co₁ (u) (x) ⩽ co₂ (F (u)) (F (x)) for any u ∈ L₁ and any x ∈ J (L₁).

The category of fuzzy generalized hull spaces and fuzzy generalized hull preserving mappings is denoted by FGHS.

Remark 3.5. (1) If (X, co) is an M-fuzzifying (resp. (L, M)-fuzzy) hull space, then (2^X, co) (resp. (L^X, co)) is a fuzzy generalized hull space [18, 29].

(2) If M = {⊥ , ⊤} then a fuzzy generalized hull space (L, co) reduces to a generalized hull space. That is, co : L → L is a hull operator on L satisfying

(GCO1) co (⊥) = ⊥;

(GCO2) u ⩽ co (u) for any u ∈ L;

(GCO3) co (co (u)) = co (u);

(GCO4) $co (u) = ⋁_{v \in F (u)} co (v)$ .

Clearly, if co : 2^X → 2^X is a hull operator on 2^X then (2^X, co) is a generalized hull space [21]; if co : L^X → L^X is an L-hull operator on L^X then (L^X, co) is a generalized hull space [29].

Theorem 3.6. Let $(L, C)$ be a fuzzy generalized convex space. Define an operator ${co}_{C} : L \to M^{J (L)}$ by $\begin{matrix} {co}_{C} (u) (x) = ⋀_{x ≰ v \geq u} [C (v)]^{'} \end{matrix}$ for u ∈ L and x ∈ J (L). Then $(L, {co}_{C})$ is a fuzzy generalized hull space.

Proof. (FGCO1). It is clear.

(FGCO2). If x ⩽ u then ${co}_{C} (u) (x) = ⋀ \emptyset = ⊤_{M}$ .

(FGCO3). To prove the desired result, we prove that $\begin{matrix} C (u) = ⋀_{x ≰ u} [{co}_{C} (u) (x)]^{'} = ⋀_{x ≰ u} ⋁_{x ≰ v \geq u} C (v) \end{matrix}$ for any u ∈ L.

Actually, $C (u) ⩽ ⋀_{x ≰ u} ⋁_{x ≰ v \geq u} C (v)$ is clear. Conversely, we have

$\begin{matrix} ⋀_{x ≰ u} ⋁_{x ≰ v \geq u} C (v) & = & ⋁_{π \in Π_{x ≰ u} B_{x}} ⋀_{x ≰ u} C (π (x)) \\ ⩽ & ⋁_{π \in Π_{x ≰ u} B_{x}} C (⋀_{x ≰ u} π (x)) = C (u), \end{matrix}$

where ℬ_x = {v ∈ L : x ≰ v ≥ u}. Thus $C (u) = ⋀_{x ≰ u} ⋁_{x ≰ v \geq u} C (v)$ . Hence (FGCO3) holds directly. To prove (FGCO4), we firstly prove (FGCO0).

(FGCO0) ${co}_{C} (u) (x) = ⋀_{y ≺ x} {co}_{C} (u) (y)$ .

Actually, ${co}_{C} (u) (x) ⩽ ⋀_{y ≺ x} {co}_{C} (u) (y)$ is clear. Suppose that $⋀_{y ≺ x} {co}_{C} (u) (y) ≰ {co}_{C} (u) (x)$ . Then there is $a ≺ ⋀_{y ≺ x} {co}_{C} (u) (y)$ such that $a ≰ {co}_{C} (u) (x)$ . Thus there is x ≰ v ≥ u such that $a ≰ [C (v)]^{'}$ . By x ≰ v, there is y ≺ x such that y ≰ v ≥ u. Hence $a ≰ [C (v)]^{'} \geq {co}_{C} (u) (y)$ which is a contradiction. So $⋀_{y ≺ x} {co}_{C} (u) (y) ⩽ {co}_{C} (u) (x)$ . Therefore (FGCO0) holds for ${co}_{C}$ .

To prove (FGCO4), let $a ≺ {co}_{C} (u) (x)$ . By (FGCO0), we have $\begin{matrix} {co}_{C} (u) (x) \geq ⋀_{y ≺ x} ⋁_{w \in F (u)} {co}_{C} (w) (y) . \end{matrix}$ Conversely, we have

$\begin{matrix} a ≺ {co}_{C} (u) (x) \\ Ł & \forall x ≰ v \geq u, v \notin C^{[a^{'}]} \\ \Rightarrow & x ⩽ {co}_{C^{[a^{'}]}} (u) = ⋁_{w \in F (u)} {co}_{C^{[a^{'}]}} (w) \\ Ł & \forall y ≺ x, \exists w \in F (u), s . t . \forall y ≰ v \geq w, v \notin C^{[a^{'}]} \\ Ł & \forall y ≺ x, \exists w \in F (u), s . t . {co}_{C} (w) (y) \geq a \\ \Rightarrow & ⋀_{y ≺ x} ⋁_{w \in F (u)} co (w) (y) \geq a . \end{matrix}$

Hence ${co}_{C} (u) (x) ⩽ ⋀_{y ≺ x} ⋁_{w \in F (u)} {co}_{C} (w) (y)$ .□

Theorem 3.7. If $F : (L_{1}, C_{1}) \to (L_{2}, C_{2})$ is a fuzzy generalized convexity preserving mapping, then $F : (L_{1}, {co}_{C_{1}}) \to (L_{2}, {co}_{C_{2}})$ is a fuzzy generalized hull preserving mapping.

Proof. Let u ∈ L₁ and x ∈ J (L₁). We have $\begin{matrix} {co}_{C_{2}} (F (u)) (F (x)) & \geq & ⋀_{x ≰ F^{⊣} (v) \geq u} [C_{1} (F^{⊣} (v))]^{'} \\ \geq & {co}_{C_{1}} (u) (x) . \end{matrix}$ Therefore F is a fuzzy generalized hull preserving mapping.□

Theorem 3.8. Let (L, co) be a fuzzy generalized hull space. Define a mapping $C_{co} : L \to M$ by $\begin{matrix} C_{co} (u) = ⋀_{x ≰ u} [co (u) (x)]^{'} \end{matrix}$ for any u ∈ L. Then $(L, C_{co})$ is a fuzzy generalized convex space.

Proof. (FGC1). It is direct.

(FGC2). Let φ ⊆ L. We have $\begin{matrix} ⋀_{u \in φ} C_{co} (u) & = & ⋀_{u \in φ} ⋀_{x ≰ u} [co (u) (x)]^{'} \\ ⩽ & ⋀_{x ≰ ⋀ φ} [co (⋀ φ) (x)]^{'} = C_{co} (⋀ φ) . \end{matrix}$

(FGC3). Let $φ \overset{dir}{\subseteq} L$ and u = ⋁ ^dirφ. Let a ∈ M with $a ≰ C_{co} (u)$ . Thus $b ≰ C_{co} (u)$ for some b ≺ a. So $\begin{matrix} C_{co} (u) & = & ⋀_{x ≰ u} [co (u) (x)]^{'} ≱ b \\ \Rightarrow & \exists x ≰ u, s . t . ⋀_{y ≺ x} ⋁_{u \in φ} co (u) (y) ≰ b^{'} \\ \Rightarrow & \exists x ≰ u, s . t . \forall y ≺ x, \exists u_{y} \in φ, \\ [co (u_{y}) (y)]^{'} ≱ b . \end{matrix}$ So a ≰ ⋁ _y≺x ⋀ _u∈φ [co (u) (y)] ′ for some x ≰ u. Thus $\begin{matrix} a & ≰ & ⋀_{x ≰ u} ⋁_{y ≺ x} ⋀_{u \in φ} [co (u) (y)]^{'} \\ \geq & ⋀_{u \in φ} ⋀_{x ≰ u} [co (u) (x)]^{'} = ⋀_{u \in φ} C_{co} (u) . \end{matrix}$ Therefore $⋀_{u \in φ} C_{co} (u) ⩽ C_{co} (u)$ .□

Theorem 3.9. If F : (L₁, co₁) → (L₂, co₂) is a fuzzy generalized hull preserving mapping then $F : (L_{1}, C_{{co}_{1}}) \to (L_{2}, C_{{co}_{2}})$ is a fuzzy generalized convexity preserving mapping.

Proof. Let u ∈ L₂. We have

$\begin{matrix} C_{{co}_{2}} (u) & = & ⋀_{y ≰ u} [{co}_{2} (u) (y)]^{'} ⩽ ⋀_{F (x) ≰ u} [{co}_{2} (u) (F (x))]^{'} \\ ⩽ & ⋀_{x ≰ F^{⊣} (u)} [{co}_{1} (F^{⊣} (u)) (x)]^{'} = C_{{co}_{1}} (F^{⊣} (u)) . \end{matrix}$

Therefore F is a fuzzy generalized convexity preserving mapping.□

Theorem 3.10. We have $C_{{co}_{C}} = C$ for any fuzzy generalized convex space $(L, C)$ . Also, we have $co = {co}_{C_{co}}$ for any fuzzy generalized hull space (L, co).

Proof. Let $(L, C)$ be a fuzzy generalized convex space. For any u ∈ L, it is clear that $C (u) ⩽ C_{{co}_{C}} (u)$ . Conversely, we have $\begin{matrix} C_{{co}_{C}} (u) & = & ⋀_{x ≰ A} ({co}_{C} (u) (x))^{'} = ⋀_{x ≰ u} ⋁_{x ≰ v \geq u} C (v) \\ = & ⋁_{π \in Π_{x ≰ u} B_{x}} ⋀_{x ≰ u} C (π (x)) \\ ⩽ & ⋁_{π \in Π_{x ≰ u} B_{x}} C (⋀_{x ≰ u} π (x)) = C (u), \end{matrix}$ where ℬ_x = {v ∈ L : x ≰ v ≥ u}. Hence $C_{{co}_{C}} = C$ .

Let (L, co) be a fuzzy generalized hull space. For any u ∈ L and x ∈ J (L), we have $\begin{matrix} {co}_{C_{co}} (u) (x) = ⋀_{x ≰ v \geq u} ⋁_{y ≰ v} co (v) (y) = co (u) (x) . \end{matrix}$ Therefore ${co}_{C_{co}} = co$ .□

Based on Theorems 3.6 and 3.7, we define a factor $F$ : FGCS → FGHS by $\begin{matrix} F ((L, C)) = (L, {co}_{C}), F (F) = F . \end{matrix}$ Based on Theorems 3.6–3.10, we find that $F$ is an isomorphic factor. We have the following conclusion.

Theorem 3.11. FGCS is isomorphic to FGHS.

4 Fuzzy generalized restricted hull spaces

In this section, we introduce fuzzy generalized restricted hull space and discuss its relations with fuzzy generalized convex spaces.

Definition 4.1. An operator $h : F (L) \to M^{J (L)}$ is called a fuzzy generalized restricted hull operator on L and the pair (L, h) is called a fuzzy generalized restricted hull space if it satisfies

(FGRH1) h (⊥) (x) = ⊥ _M;

(FGRH2) h (w) (x) = ⊤ _M provided that x ⩽ w;

(FGRH3) h (v) (x) ∧ ⋀ _y≺vh (w) (y) ⩽ h (w) (x);

(FGRH4) $h (w) (x) = ⋀_{y ≺ x} ⋁_{v \in F (w)} h (v) (y)$ .

Remark 4.2. (1) If (X, h) is an M-fuzzifying hull space then (2^X, h) is a fuzzy generalized restricted hull space [5]; if (X, ℋ) is an (L, M)-fuzzy restricted hull space then (L^X, ℋ) is a fuzzy generalized restricted hull space [30].

(2) If M = {⊥ , ⊤} then a fuzzy generalized restricted hull space (L, h) reduces to a generalized restricted hull space. That is, $h : F (L) \to L$ is a generalized restricted hull operator on L satisfying

(RH1) h (⊥) = ⊥;

(RH2) u ⩽ h (u) for any u ∈ L;

(RH3) v ⩽ h (u) implies h (v) ⩽ h (u);

(RH4) $h (u) = ⋁_{v \in F (u)} h (v)$ .

Clearly, if $h : 2_{fin}^{X} \to 2^{X}$ is a restricted hull operator on 2^X then (2^X, h) is a generalized restricted hull space [21]; if $h : F (X) \to L^{X}$ is an L-restricted hull operator on L^X then (L^X, h) is a generalized restricted hull space, where $F (X) = {F \in L^{X} : \exists φ \in 2_{fin}^{β^{*} (\underline{⊤})}, F = ⋁ φ}$ [29].

Let (L₁, h₁) and (L₂, h₂) be fuzzy generalized restricted hull spaces. A GOH F : L₁ → L₂ with respect to h₁ and h₂ is called a fuzzy generalized restricted hull preserving mapping if $\begin{matrix} h_{1} (w) (x) ⩽ h_{2} (F (w)) (F (x)) \end{matrix}$ for any $w \in F (L_{1})$ and any x ∈ J (L₁).

The category of fuzzy generalized restricted hull spaces and fuzzy generalized restricted hull preserving mappings is denoted by FGRHS. Next, we discuss the relations between FGRHS and FGCS.

Theorem 4.3. Let $(L, C)$ be a fuzzy generalized convex space. Define an operator $h_{C} : F (L) \to M^{J (L)}$ by $\begin{matrix} h_{C} (w) (x) = {co}_{C} (w) (x), \end{matrix}$ for all $w \in F (L)$ and x ∈ J (L). Then $(L, h_{C})$ is a fuzzy generalized restricted hull space.

Proof. We only check that (FGRH3) holds for $h_{C}$ .

Let $w, v \in F (L)$ and x ∈ J (L). We will prove that $\begin{matrix} h_{C} (v) (x) \land ⋀_{y ≺ v} h_{C} (w) (y) ⩽ h_{C} (w) (x) . \end{matrix}$

Actually, if x ⩽ w, then $h_{C} (w) (x) = ⊤_{M}$ by (LCO2). Thus the desired result holds. If x ≤ v, then $⋀_{y ≺ v} h_{C} (w) (y) ⩽ h_{C} (w) (x)$ . Thus the desired result holds. Assume that x ≰ w ∨ v. For any x ≰ z ≥ w, we have either v ≰ z or v ⩽ z. In the former case, the set φ = {y ≺ v : y ≰ z} is not empty. Thus $\begin{matrix} ⋁_{y ≰ z} {co}_{C} (z) (y) \geq ⋁_{y \in φ} h_{C} (w) (y) \geq ⋀_{y ≺ v} h_{C} (w) (y) . \end{matrix}$ In the latter case, we have $\begin{matrix} ⋁_{y ≰ z} {co}_{C} (z) (y) \geq {co}_{C} (z) (x) \geq {co}_{C} (v) (x) = h_{C} (v) (x) . \end{matrix}$ Combining the these cases into (FGCO3), we find that (FGRH3) holds for $h_{C}$ .□

Theorem 4.4. Let $(L_{1}, C_{1})$ and $(L_{2}, C_{2})$ be fuzzy generalized convex spaces. If F : L₁ → L₂ is a fuzzy generalized convexity preserving mapping then $F : (L_{1}, h_{C_{1}}) \to (L_{2}, h_{C_{2}})$ is a fuzzy generalized restricted hull preserving mapping.

Theorem 4.5. Let (L, h) be a fuzzy generalized restricted hull space. Define $C_{h} : L \to M$ by $\begin{matrix} C_{h} (u) = ⋀_{x ≰ u} ⋀_{z \in F (u)} [h (z) (x)]^{'} \end{matrix}$ for any u ∈ L. Then $(L, C_{h})$ is a fuzzy generalized convex space.

Proof. (FGC1). It is clear.

(FGC2). Let φ ⊆ L and u = ⋀ φ. Then $F (u) \subseteq ⋂_{v \in φ} F (v)$ . Let x ≰ u. Then there is v ∈ φ such that x ≰ v. For any $w \in F (u)$ , we have $w \in F (v)$ and $[h (w) (x)]^{'} \geq C_{h} (v) \geq ⋀_{v \in φ} C_{h} (v)$ . Hence $\begin{matrix} C_{h} (u) = ⋀_{x ≰ u} ⋀_{z \in F (u)} [h (z) (x)]^{'} \geq ⋀_{u \in φ} C_{h} (u) . \end{matrix}$

(FGC3). Let $φ \overset{dir}{\subseteq} L$ and u = ⋁ ^dirφ. If x ≰ u, then x ≰ v for any v ∈ φ. For any $w \in F (u) = ⋃_{v \in φ}^{dir} F (v)$ , there is v ∈ φ such that $w \in F (v)$ . Thus $\begin{matrix} [h (w) (x)]^{'} \geq ⋀_{z \in F (v)} [h (z) (x)]^{'} \geq C_{h} (v) \geq ⋀_{u \in φ} C_{h} (u) . \end{matrix}$ Hence $\begin{matrix} C_{h} (u) = ⋀_{x ≰ u} ⋀_{z \in F (u)} (h (z) (x))^{'} \geq ⋀_{v \in φ} C_{h} (v) . \end{matrix}$ So $(L, C_{h})$ is a fuzzy generalized convex space.□

Theorem 4.6. Let (L₁, h₁) and (L₂, h₂) be fuzzy generalized restricted hull spaces. If F : L₁ → L₂ is a fuzzy generalized restricted hull preserving mapping then $F : (L_{1}, C_{h_{2}}) \to (L_{2}, C_{h_{2}})$ is a fuzzy generalized convexity preesrving mapping.

Proof. Let w ∈ L₂. We have $\begin{matrix} C_{h_{1}} (F^{⊣} (w)) & = & ⋀_{x ≰ F^{⊣} (w)} ⋀_{v \in F (F^{⊣} (w))} [h_{1} (v) (x)]^{'} \\ = & ⋀_{F (x) ≰ w} ⋀_{v \in F (F^{⊣} (w))} [h_{1} (v) (x)]^{'} \\ \geq & ⋀_{F (x) ≰ w} ⋀_{F (v) \in F (w)} [h_{2} (F (v)) (F (x))]^{'} \\ \geq & C_{h_{2}} (w) . \end{matrix}$ Therefore F is a fuzzy generalized convexity preserving mapping.□

Theorem 4.7. We have $C_{h_{C}} = C$ for any fuzzy generalized convex space $(L, C)$ . Also, we have $h_{C_{h}} = h$ for any fuzzy generalized restricted hull space (L, h).

Proof. Let $(L, C)$ be a fuzzy generalized convex space. We have $C ⩽ C_{h_{C}}$ . To prove that $C_{h_{C}} (u) ⩽ C (u)$ , it is sufficient to prove that ${co}_{C} (u) (x) ⩽ {co}_{C_{h_{C}}} (u) (x)$ for all u ∈ L and x ∈ J (L).

Actually, for any x ≰ w ≥ u, there is y_x ≺ x such that y_x ≰ w. By (FGCO4), we have $\begin{matrix} {co}_{C_{h_{C}}} (u) (x) & = & ⋀_{x ≰ w \geq u} ⋁_{y ≰ w} ⋁_{v \in F (w)} h_{C} (v) (y) \\ \geq & ⋀_{x ≰ w \geq u} ⋁_{v \in F (u)} {co}_{C} (v) (y_{x}) \\ \geq & ⋀_{y ≺ x} ⋁_{v \in F (u)} co (v) (y) = {co}_{C} (u) (x) . \end{matrix}$ Hence $C_{h_{C}} (u) ⩽ C (u)$ which shows that $C_{h_{C}} = C$ .

Let (L, h) be a fuzzy generalized restricted hull space. To prove that $h_{C_{h}} = h$ , let $w \in F (L)$ and x ∈ J (L) with x ≰ w. We need to prove that $h_{C_{h}} (w) (x) = h (w) (x)$ .

By (FGRH4) and x = ⋁ _y≺xy, we have h (w) (x) = ⋀ _y≺xh (w) (y). For any v ∈ L with x ≰ v ≥ w, there is y_v ≺ x such that y_v ≰ v. Thus $\begin{matrix} h_{C_{h}} (w) (x) & = & ⋀_{x ≰ v \geq w} ⋁_{y ≰ v} ⋁_{z \in F (v)} h (z) (y) \\ \geq & ⋀_{x ≰ v \geq w} ⋁_{z \in F (v)} h (z) (y_{v}) \\ \geq & ⋀_{y ≺ x} ⋁_{z \in F (w)} h (z) (y) = h (w) (x) . \end{matrix}$ Thus $h_{C_{h}} (w) (x) \geq h (w) (x)$ .

To prove $h_{C_{h}} (w) (x) ⩽ h (w) (x)$ , let a ∈ β (⊤) with $a ≺ h_{C_{h}} (w) (x)$ . Suppose that a ≰ h (w) (x) and let v = ⋁ h (w) _[a]. Then x ∉ h (w) _[a]. We say x ≰ v. Otherwise, h (w) (x) ≥ ⋀ _z≺vh (w) (z) ≥ a which is a contradiction. Thus x ≰ v ≥ w. Hence there is y ≺ x such that y ≰ v.

By (FGRH4), $a ≺ h_{C_{h}} (w) (x)$ implies that $a ≺ ⋁_{z \in F (v)} h (z) (y)$ . Thus there is $z \in F (v)$ such that a ⩽ h (z) (y). Further, for any r ≺ z, we have r ≺ v. Hence h (w) (r) ≥ a. By (FGRH3), we have $\begin{matrix} a ⩽ h (z) (y) \land ⋀_{r ≺ z} h (w) (r) ⩽ h (w) (y) . \end{matrix}$ It is a contradiction. Thus $h_{C_{h}} (w) (x) ⩽ h (w) (x)$ .□

Based on Theorems 4.3 and 4.4, we define a factor $G$ : FGCS → FGRHS by $\begin{matrix} G ((L, C)) = (L, h_{C}), G (F) = F . \end{matrix}$ Based on Theorems 4.3–4.7, we find that $G$ is an isomorphic factor. So we have the following conclusion.

Corollary 4.8. FGRHS is isomorphic to FGCS.

5 Fuzzy generalized convexly enclosed relation spaces

In this section, we introduce fuzzy generalized convexly enclosed relation space and discuss its relations with fuzzy generalized convex spaces.

Definition 5.1. A binary relation ℰ : L × L → M is called a fuzzy generalized convex enclosed relation on L and the pair (L, ℰ) is called a fuzzy generalized convex enclosed relation space if

(FGCER1) ℰ (⊥ , ⊥) = ⊤ _M;

(FGCER2) ℰ (u, v) ≠ ⊥ _M implies u ⩽ v;

(FGCER3) ℰ (u, ⋀ φ) = ⋀ _v∈φ ℰ (u, v);

(FGCER4) ℰ (u, v) ⩽ ⋁ _w∈L ℰ (u, w) ∧ ℰ (w, v);

(FGCER5) $E (u, v) = ⋀_{w \in F (u)} E (u, v)$ .

Remark 5.2. (1) For any fuzzy generalized convexly enclosed relation space (L, ℰ), it directly follows from (FGCER3) and (FGCER5) that ℰ (u, v) ⩽ ℰ (u₁, v₁) for all u₁ ⩽ u and v ⩽ v₁.

(2) If M = {⊥ , ⊤} then a fuzzy generalized convexly enclosed relation space (L, ⪕) is called a generalized convexly enclosed space. That is, ⪕ is a convexly enclosed relation on L satisfying

(GCER1) ⊥⪕ ⊥;

(GCER2) u ⪕ v implies u ⩽ v;

(GCER3) u ⪕ ⋀ φ iff u ⪕ v for any v ∈ φ;

(GCER4) u ⪕ v implies a w ∈ L with u ⪕ w ⪕ v;

(GCER5) ⋁^dirφ ⪕ v iff u ⪕ v for any v ∈ φ.

(3) If (X, ⪕) is an L-convexly enclosed relation space then (L^X, ⪕) is a generalized convexly enclosed relation space [5]; if (X, ℰ) is an (L, M)-fuzzy convexly enclosed relation space, then $(L^{X}, C)$ is a generalized convexly enclosed relation space [30].

Let $(L_{1}, C_{1})$ and $(L_{2}, C_{2})$ be fuzzy generalized convexly enclosed relation spaces. A GOH F : L₁ → L₂ is called a fuzzy generalized convex enclosed relation preserving mapping if ℰ₂ (F (u) , v) ⩽ ℰ₁ (u, F^⊣ (v)) for any u ∈ L₁ and any v ∈ L₂.

The category of fuzzy generalized convexly enclosed relation spaces and fuzzy generalized convexly enclosed relation preserving mappings is denoted by FGCERS.

Next, we discuss the relations between FGCERS and FGCS. For this, we present the following lemma.

Lemma 5.3. An operator c : L → M^J(L) satisfies (FGCO3) iff it satisfies (FGCO3*), that is, $\begin{matrix} c (u) (x) = ⋀_{x ≰ v \geq u} [c (v) (x) \lor ⋁_{y ≰ v} c (u) (y)] \end{matrix}$ for any u ∈ L and any x ∈ J (L).

Proof. Either (FGCO3) or (FGCO3*) implies that c is order preserving. Actually, the necessity is easy. For the sufficiency, assume that (FGCO3*) holds for c. Clearly, $\begin{matrix} c (u) (x) ⩽ ⋀_{x ≰ v \geq u} ⋁_{y ≰ v} c (v) (y) . \end{matrix}$ To prove the inverse result, it is sufficient to prove that $\begin{matrix} [c (u) (x)]^{'} ⩽ ⋁_{x ≰ v \geq u} ⋀_{y ≰ v} [c (v) (y)]^{'} . \end{matrix}$

To prove it, let a ∈ β^* (⊤) with a ≺ [c (u) (x)] ′. By (FGCO3*), there is v ∈ L with x ≰ v ≥ u such that a ≺ [c (v) (x)] ′ ∧ ⋀ _{y ≰ v} [c (u) (y)] ′. Let $\begin{matrix} φ_{a} (x, u) & = & {v \in L : x ≰ v \geq u, \\ a ≺ [c (v) (x)]^{'} \land ⋀_{y ≰ v} [c (u) (y)]^{'}} . \end{matrix}$ and let w = ⋀ φ_a (x, u). We have w ∈ φ.

For any y ≰ w, it follows from a ≺ [c (u) (y)] ′ and (FGCO3*) that there is z_y ∈ L such that z_y ∈ φ_a (y, u). In addition, it is easy to check that w ∧ z_y ∈ φ_a (x, u). Hence w ⩽ w ∧ z_y which implies that w ⩽ z_y. This further implies that a ≺ [c (z_y) (y)] ′ ⩽ [c (w) (y)] ′ for any y ≰ w. Therefore $\begin{matrix} a ⩽ ⋀_{y ≰ w} [c (w) (y)]^{'} ⩽ ⋁_{x ≰ v \geq u} ⋀_{y ≰ v} [c (v) (y)]^{'} . \end{matrix}$ By arbitrariness of a, we conclude that $\begin{matrix} [c (u) (x)]^{'} ⩽ ⋁_{x ≰ v \geq u} ⋀_{y ≰ v} [c (v) (y)]^{'} . \end{matrix}$ Therefore (FGCO3) holds for c.□

Theorem 5.4. Let (L, co) be a fuzzy generalized hull space. Define a mapping ℰ_co : L × L → M by $\begin{matrix} E_{co} (u, v) = ⋀_{x ≰ v} [co (u) (x)]^{'} \end{matrix}$ for all u, v ∈ L. Then (L, ℰ_co) is a fuzzy generalized convexly enclosed relation space.

Proof. (FGCER1) and (FGCER2) directly follow from (FGCO1) and (FGCO2). In addition, (FGCER3) is direct.

(FGCER4). We need to prove that $\begin{matrix} E_{co} (u, v) ⩽ ⋁_{w \in L} [E_{co} (u, w) \land E_{co} (w, v)] . \end{matrix}$

Let a ∈ β (⊤) with a ≺ ℰ_co (u, v). Then a ≺ ℰ_co (u, v) = ⋀ _{x ≰ v} [co (u) (x)] ′. Thus a ≺ [co (u) (x)] ′ for any x ≰ v. Hence by (FGCO3*), we have $\begin{matrix} a & ≺ & [co (u) (x)]^{'} \\ = & ⋁_{x ≰ w \geq u} ([co (w) (x)]^{'} \land ⋀_{y ≰ w} [co (u) (y)]^{'}) . \end{matrix}$ So there is w_x ∈ L such that x ≰ w_x ≥ u and $\begin{matrix} a ⩽ [co (w_{x}) (x)]^{'} \land ⋀_{y ≰ w_{x}} [co (u) (y)]^{'} . \end{matrix}$

Let w₀ = ⋀ _{x ≰ v}w_x. We have u ⩽ w₀ ⩽ v. Thus $\begin{matrix} ⋀_{z ≰ w_{0}} [co (u) (z)]^{'} \geq ⋀_{x ≰ v} ⋀_{z ≰ w_{x}} [co (u) (z)]^{'} \geq a \end{matrix}$ and $\begin{matrix} ⋀_{x ≰ v} [co (w_{0}) (x)]^{'} \geq ⋀_{x ≰ v} [co (w_{x}) (x)]^{'} \geq a . \end{matrix}$ By arbitrariness of a, we have $\begin{matrix} E_{co} (u, v) & ⩽ & ⋀_{z ≰ w_{0}} [co (u) (z)]^{'} \land ⋀_{x ≰ v} [co (w_{0}) (x)]^{'} \\ = & E_{co} (u, w_{0}) \land E_{co} (w_{0}, v) \\ ⩽ & ⋁_{w \in L} [E_{co} (u, w) \land E_{co} (w, v)] . \end{matrix}$ Therefore (FCER4) holds for ℰ_co.

(FGCER5). Clearly, $E_{co} (u, v) ⩽ ⋀_{w \in F (u)} E_{co} (w, v)$ . Conversely, let a ∈ M with a ≰ ℰ_co (u, v). Then there is x ≰ v such that a ≰ [co (u) (x)] ′. By x ≰ v, there is y_v ≺ x such that y_v ≰ v. Also, by a ≰ [co (u) (x)] ′, $\begin{matrix} a ≰ [co (u) (x)]^{'} = ⋁_{y ≺ x} ⋀_{w \in F (u)} [co (w) (y)]^{'} . \end{matrix}$ Thus $a ≰ ⋀_{w \in F (u)} [co (w) (y_{v})]^{'}$ . That is, there is $w_{v} \in F (u)$ such that a ≰ [co (w_v) (y_v)] ′. Hence $\begin{matrix} a ≰ [co (w_{v}) (y_{v})]^{'} \geq E_{co} (w_{v}, v) \geq ⋀_{w \in F (u)} E_{co} (w, v) \end{matrix}$ By arbitrariness of a, we have $⋀_{w \in F (u)} E_{co} (w, v) ⩽ E_{co} (u, v)$ . Therefore (FGCER5) holds for ℰ_co.□

Corollary 5.5. Let $(L, C)$ be a fuzzy generalized convex space. Define a mapping $E_{C} : L \times L \to M$ by $\begin{matrix} E_{C} (u, v) = ⋀_{x ≰ v} ⋁_{x ≰ w \geq u} C (w) \end{matrix}$ for all u, v ∈ L. Then $(L, E_{C})$ is a fuzzy generalized convexly enclosed relation space.

Theorem 5.6. Let $(L_{1}, C_{1})$ and $(L_{2}, C_{2})$ be fuzzy generalized convex spaces. If F : L₁ → L₂ is a fuzzy generalized convexity preserving mapping then $F : (L_{1}, E_{C_{1}}) \to (L_{2}, E_{C_{2}})$ is a fuzzy generalized convexly enclosed relation preserving mapping.

Proof. Let u ∈ L₁ and v ∈ L₂. We have $\begin{matrix} E_{C_{2}} (F (u), v) & ⩽ & ⋀_{F (x) ≰ v} ⋁_{F (x) ≰ w \geq F (u)} C_{2} (w) \\ ⩽ & ⋀_{x ≰ F^{⊣} (v)} ⋁_{x ≰ F^{⊣} (w) \geq u} C_{1} (F^{⊣} (w)) \\ ⩽ & E_{1} (u, F^{⊣} (v)) . \end{matrix}$ Thus F is a fuzzy generalized convexly enclosed relation preserving mapping.□

Theorem 5.7. Let (L, ℰ) be a fuzzy generalized convexly enclosed relation space. Define an operator co_ℰ : L → M^J(L) by $\begin{matrix} {co}_{E} (u) (x) = ⋀_{x ≰ v} [E (u, v)]^{'} \end{matrix}$ for u ∈ L and x ∈ J (L). Then (L, co_ℰ) is a fuzzy generalized hull space.

Proof. (FGCO1) and (FGCO2) can be directly obtained from (FGCER1) and (FGCER2).

(FGCO3). By Lemma 5, it is sufficient to prove that (FGCO3*) holds for co_ℰ. That is, we need to prove that $\begin{matrix} {co}_{E} (u) (x) = ⋀_{x ≰ v \geq u} [{co}_{E} (v) (x) \lor ⋁_{y ≰ v} {co}_{E} (u) (y)] . \end{matrix}$ for all u ∈ L and x ∈ J (L). Actually, it is clear that $\begin{matrix} {co}_{E} (u) (x) ⩽ ⋀_{x ≰ v \geq u} [{co}_{E} (v) (x) \lor ⋁_{y ≰ v} {co}_{E} (u) (y)] . \end{matrix}$ Conversely, let a ∈ β^* (⊤) with a ≺ [co_ℰ (u) (x)] ′. Then there is a z ∈ L such that x ≰ z and a ≺ ℰ (u, z). By (FGCER4), there is w ∈ L such that a ≺ ℰ (u, w) ∧ ℰ (w, z). Thus u ⩽ w ⩽ z by (FGCER2). Hence $\begin{matrix} [{co}_{E} (w) (x)]^{'} = ⋁_{x ≰ s} E (w, s) \geq E (w, z) \geq a \end{matrix}$ and ⋀_{y ≰ w} [co_ℰ (u) (y)] ′ ≥ ⋀ _{y ≰ w} ℰ (u, w) ≥ a. By the arbitrariness of a, we conclude that $\begin{matrix} {co}_{E} (u) (x) \geq ⋀_{x ≰ v \geq u} [{co}_{E} (v) (x) \lor ⋁_{y ≰ v} {co}_{E} (u) (y)] . \end{matrix}$ Therefore (FGCO3*) holds.

(FGCO4). By (FGCER2) and (FGCER5), we have $\begin{matrix} ⋀_{y ≺ x} ⋁_{w \in F (u)} {co}_{E} (w) (y) \\ = & ⋀_{y ≺ x} [⋀_{w \in F (u)} ⋁_{y ≰ v \geq w} E (w, v)]^{'} \\ = & ⋀_{y ≺ x} [⋁_{π \in Π_{w \in F (u)} B_{w}} ⋀_{w \in F (u)} E (w, π (w))]^{'} \\ = & ⋀_{y ≺ x} [⋁_{π \in Π_{w \in F (u)} B_{w}} E (u, π (w))]^{'} \geq {co}_{E} (u) (x), \end{matrix}$ where ℬ_w = {v ∈ L : y ≰ v ≥ w}. Conversely, by (FGCER5), we have $\begin{matrix} {co}_{E} (u) (x) & = & ⋀_{y ≺ x} ⋀_{y ≰ v} ⋁_{w \in F (u)} [E (w, v)]^{'} \\ \geq & ⋀_{y ≺ x} ⋁_{w \in F (u)} ⋀_{y ≰ v} [E (w, v)]^{'} \\ = & ⋀_{y ≺ x} ⋁_{w \in F (u)} {co}_{E} (w) (y) . \end{matrix}$ Hence (L, co_ℰ) is a fuzzy generalized hull space.□

Corollary 5.8. Let (L, ℰ) be a fuzzy generalized convexly enclosed relation space. Define a mapping $C_{E} : L \to M$ by $\begin{matrix} C_{E} (u) = ⋀_{x ≰ u} ⋁_{x ≰ v} E (u, v) \end{matrix}$ for any u ∈ L. Then $(L, C_{E})$ is a fuzzy generalized convex space.

Theorem 5.9. Let (L₁, ℰ₁) and (L₂, ℰ₂) be fuzzy generalized convexly enclosed relation space. If F : L₁ → L₂ is a fuzzy generalized convexly enclosed relation preserving mapping then $F : (L_{1}, C_{E_{1}}) \to (L_{2}, C_{E_{2}})$ is a fuzzy generalized convexity preserving mapping.

Proof. Let v ∈ L₂. We have $\begin{matrix} C_{E_{2}} (w) & ⩽ & ⋀_{F (x) ≰ w} ⋁_{F (x) ≰ v} E_{2} (w, v) \\ ⩽ & ⋀_{F (x) ≰ w} ⋁_{F (x) ≰ v} E_{2} (F (F^{⊣} (w)), v) \\ ⩽ & ⋀_{x ≰ F^{⊣} (w)} ⋁_{x ≰ F^{⊣} (v)} E_{1} (F^{⊣} (w), F^{⊣} (v)) \\ ⩽ & ⋀_{x ≰ u} ⋁_{x ≰ z} E_{1} (F^{⊣} (w), z) = C_{E_{1}} (F^{⊣} (w)) . \end{matrix}$ Therefore F is a fuzzy generalized convexity preserving mapping.□

Theorem 5.10. We have $C_{E_{C}} = C$ for any fuzzy generalized convex space $(L, C)$ and $E_{C_{E}} = E$ for any fuzzy generalized convexly enclosed relation space (L, ℰ).

Proof. Let $(L, C)$ be a fuzzy generalized convex space and u ∈ L. Clearly, we have $C_{E_{C}} (u) \geq C (u)$ . Conversely, by (FGCER2) of $E_{C}$ and (FGCO3) of ${co}_{C}$ , $\begin{matrix} C_{E_{C}} (u) & = & ⋀_{x ≰ u} [⋀_{x ≰ v \geq u} ⋁_{y ≰ v} ⋀_{y ≰ w \geq u} [C (w)]^{'}]^{'} \\ = & ⋀_{x ≰ u} [⋀_{x ≰ v \geq u} ⋁_{y ≰ v} {co}_{C} (u) (y)]^{'} \\ ⩽ & ⋀_{x ≰ u} [⋀_{x ≰ v \geq u} ⋁_{y ≰ v} {co}_{C} (v) (y)]^{'} \\ = & ⋀_{x ≰ u} [{co}_{C} (u) (x)]^{'} = C (u) . \end{matrix}$ Thus $C_{E_{C}} (u) = C (u)$ which shows that $C_{E_{C}} = C$ .

Let (L, ℰ) be a fuzzy generalized convexly enclosed relation space, and let u, v ∈ L. It is clear that $E_{C_{E}} (u, v) \geq E (u, v)$ . Conversely, by (FGCO3) of co_ℰ, $\begin{matrix} E_{C_{E}} (u, v) & = & ⋀_{x ≰ v} [⋀_{x ≰ w \geq u} ⋁_{y ≰ w} {co}_{E} (w) (y)]^{'} \\ = & ⋀_{x ≰ v} [{co}_{E} (u) (x)]^{'} = ⋀_{x ≰ v} ⋁_{x ≰ w} E (u, w) \\ = & ⋁_{π \in Π_{x ≰ v} B_{x}} ⋀_{x ≰ v} E (u, π (x)) \\ = & ⋁_{π \in Π_{x ≰ v} B_{x}} E (u, ⋀_{x ≰ v} π (x)) \\ ⩽ & ⋁_{π \in Π_{x ≰ v} B_{x}} E (u, v) = E (u, v) . \end{matrix}$ where ℬ_x = {w ∈ L : x ≰ w}. Thus $E_{C_{E}} = E$ .□

Based on Corollary 5 and Theorem 5, we define a factor $ℍ$ : FGCS → FGCERS by $\begin{matrix} ℍ ((L, C)) = (L, E_{C}), ℍ (F) = F . \end{matrix}$ Based on Corollaries 5.5 and 5.8 and Theorems 5.6, 5.9 and 5.10, we find that $ℍ$ is an isomorphic factor. So we have the following conclusion.

Corollary 5.11. FGCERS is isomorphic to FGCS.

6 Fuzzy generalized derived hull spaces

In this paper, we introduce fuzzy generalized derived hull space and discuss its relations with fuzzy generalized convex space.

For any u ∈ L and any x ∈ J (L), we denote $\begin{matrix} u_{x}^{-} = ⋁ {y \in β (u) : x ≰ y} . \end{matrix}$

Proposition 6.1. Let u, v ∈ L, {u_i} _i∈I ⊆ L and x ∈ J (L). We have

(1) x ≰ u implies $u_{x}^{-} = u$ ;

(2) v ⩽ u implies $v_{x}^{-} ⩽ u_{x}^{-}$ ;

(3) $(⋁_{i \in I} u_{i})_{x}^{-} = ⋁_{i \in I} (u_{i})_{x}^{-}$ ;

(4) $(u_{x}^{-})_{x}^{-} = u_{x}^{-}$ ;

(5) $F (u_{x}^{-}) \subseteq {f_{x}^{-} : f \in F (u)}$ .

Proof. (1)–(3) are easy.

(4). Clearly, $(u_{x}^{-})_{x}^{-} ⩽ u_{x}^{-}$ . Conversely, if $z ≺ u_{x}^{-}$ , there is y ⩽ u such that x ≰ y and z ⩽ y. Thus $z ⩽ u_{x}^{-}$ and x ≰ z. Hence $z ⩽ (u_{x}^{-})_{x}^{-}$ . Therefore $u_{x}^{-} ⩽ (u_{x}^{-})_{x}^{-}$ .

(5). Let $g \in F (u_{x}^{-})$ . That is, there is ${z_{i}}_{i = 1}^{n} \subseteq 2_{fin}^{β^{*} (u_{x}^{-})}$ such that $g = ⋁_{i = 1}^{n} z_{i}$ . For each 1 ⩽ i ⩽ n, there is y_i ∈ β^* (u) such that z_i ⩽ y_i and x ≰ y_i. Let $y = ⋁_{i = 1}^{n} y_{i}$ . Then $y \in F (u)$ and x ≰ y. Thus $F (u_{x}^{-}) \subseteq {f_{x}^{-} : f \in F (u)}$ .□

Definition 6.2. An operator $D$ : L → M^J(L) is called a fuzzy generalized derived hull operator and (L, $D$ ) is called a fuzzy generalized derived hull space if

(FGDH1) $D$ (⊥) (x) = ⊥ _M;

(FGDH2) $D (u_{x}^{-}) (x) = D (u) (x)$ ;

(FGDH3) $D (u) (x) = ⋀_{x ≰ v \geq u_{x}^{-}} ⋁_{y ≰ v} D (v) (y)$ ;

(FGDH4) $D (u) (x) = ⋀_{y ≺ x} ⋁_{f \in F (u)} D (f) (y)$ .

Let (L₁, $D$ ₁) and (L₂, $D$ ₂) be fuzzy generalized derived hull spaces. A GOH F : L₁ → L₂ is called a fuzzy generalized derived hull preserving mapping if $D_{1} (u) (x) ⩽ \bar{D_{2}} (F (u)) (F (x))$ for any u ∈ L₁ and any x ∈ J (L₁), where $\bar{D_{2}} : L \to M$ is defined by $\bar{D_{2}} (v) = D_{2} (v) \lor χ_{v}$ for any v ∈ L₂.

The category of fuzzy generalized derived hull spaces and fuzzy generalized derived hull preserving mappings is denoted by FGDHS.

Remark 6.3. (1) If (X, d) is an M-fuzzifying derived hull space then (2^X, d) is a fuzzy generalized derived hull space [14].

(2) If M = {⊥ , ⊤} then a fuzzy generalized derived hull space (L, $D$ ) reduces to a generalized derived hull space. That is, $D : F (L) \to L$ is a generalized restricted hull operator on L satisfying

(GDH1) $D$ (⊥) = ⊥;

(GDH2) x ≤ $D$ (u) implies $x \leq D (u_{x}^{-})$ ;

(GDH3) $D$ (u ∨ $D$ (u)) ≤ u ∨ $D$ (u);

(GDH4) $D (u) = ⋁_{v \in F (u)} D (v)$ .

Theorem 6.4. Let $(L, C)$ be a fuzzy generalized convex space. Define an operator $D_{C} : L \to M^{J (L)}$ by $\begin{matrix} \forall u \in L, \forall x \in J (L), D_{C} (u) (x) = co (u_{x}^{-}) (x) . \end{matrix}$ Then $(L, D_{C})$ is a fuzzy generalized derived hull space.

Proof. (FGDH1) and (FGDH2) direct follow from (1) and (4) of Proposition 6.

(FGDH3). By (FGCO4), we have $\begin{matrix} D_{C} (u) (x) & = & ⋀_{x ≰ v \geq u_{x}^{-}} ⋁_{y ≰ v} co (v) (y) \\ \geq & ⋀_{x ≰ v \geq u_{x}^{-}} ⋁_{y ≰ v} D_{C} (v) (y) \geq D_{C} (u) (x) . \end{matrix}$ Thus $D_{C} (u) (x) = ⋀_{x ≰ v \geq u_{x}^{-}} ⋁_{y ≰ v} D_{C} (v) (y)$ .

(FGDH4). By (5) of Proposition 6, (FGCO4) and (FGCO0), we have $\begin{matrix} D_{C} (u) (x) & = & ⋀_{y ≺ x} ⋁_{f \in F (u_{x}^{-})} co (f) (y) \\ ⩽ & ⋀_{y ≺ x} ⋁_{f \in F (u)} D_{C} (f) (y) \\ ⩽ & ⋀_{y ≺ x} D_{C} (u) (y) = D_{C} (u) (x) . \end{matrix}$ Thus $D_{C} (u) (x) = ⋀_{y ≺ x} ⋁_{f \in F (u)} D_{C} (f) (y)$ .

Therefore $(L, D_{C})$ is a fuzzy generalized derived hull space.□

Theorem 6.5. Let $(L_{1}, C_{1})$ and $(L_{2}, C_{2})$ be fuzzy generalized convex spaces. If F : L₁ → L₂ is a fuzzy generalized convexity preserving mapping then $F : (L_{1}, D_{C_{1}}) \to (L_{2}, D_{C_{2}})$ is a fuzzy generalized derived hull preserving mapping.

Proof. Let u ∈ L₁ and x ∈ J (L₁). It is sufficient to prove that $D_{C_{1}} (u) (x) ⩽ \bar{D_{C_{2}}} (F (u)) (F (x))$ .

We can assume that F (x) ≰ F (u) since the result holds trivially otherwise. Then x ≰ u and $u_{x}^{-} = u$ and $F (u)_{F (x)}^{-} = F (u)$ . Hence $\begin{matrix} D_{C_{1}} (u) (x) \leq F (u) \end{matrix} = \bar{D_{C_{2}}} (F (u)) (F (x)) .$ Therefore F is a fuzzy generalized derived hull preserving mapping.□

Theorem 6.6. Let (L, $D$ ) be a fuzzy generalized derived hull space. Define a mapping $C_{D} : L \to M$ by $\begin{matrix} \forall u \in L, C_{D} (u) = ⋀_{x ≰ u} [D (u) (x)]^{'} . \end{matrix}$ Then $(L, C_{D})$ is a fuzzy generalized convex space.

Proof. (FGC1). It is clear.

(FGC2). Let {u_i} _i∈I ⊆ L and u = ⋀ _i∈Iu_i. Let x ∈ J (L) with x ≰ u. Then there is i₀ ∈ I such that x ≰ u_{i
₀}. Thus [ $D$ (u) (x)] ′ ≥ [ $D$ (u_{i
₀}) (x)] ′ and so $[D (u) (x)]^{'} \geq C_{D} (u_{i_{0}}) \geq ⋀_{i \in I} C_{D} (u_{i})$ . By arbitrariness of x ≰ u, we have $\begin{matrix} ⋀_{i \in I} C_{D} (u_{i}) ⩽ ⋀_{x ≰ u} [D (u) (x)]^{'} = C_{D} (u) . \end{matrix}$

(FGC3). Let {u_i} _i∈I ⊆ L be up-directed and $u = ⋁_{i \in I}^{dir} u_{i}$ . If a ∈ M with $C (u) ≱ a$ , then there exists b ≺ a such that $C (u) ≱ b$ . Thus

$\begin{matrix} C (u) & = & ⋀_{x ≰ u} [D (u) (x)]^{'} ≱ b \\ \Rightarrow & \exists x ≰ u, ⋀_{y ≺ x} ⋁_{f \in F (u)} D (f) (y) ≰ b^{'} \\ \Rightarrow & \exists x ≰ u, s . t . \forall y ≺ x, \exists f_{y} \in F (u), \\ D (f_{y}) (y) ≰ b^{'} \\ \Rightarrow & \exists x ≰ u, s . t . \forall y ≺ x, \exists f_{y} \in F (u_{i_{y}}) (i_{y} \in I), \\ [D (f_{y}) (y)]^{'} ≱ b . \end{matrix}$

So $⋁_{y ≺ x} ⋀_{f \in F (u_{i_{y}})} [D (f) (y)]^{'} ≱ a$ for some x ≰ u. Thus we have $\begin{matrix} ⋀_{i \in I} C (u_{i}) & = & ⋀_{i \in I} ⋀_{x ≰ u_{i}} [D (u_{i}) (x)]^{'} \\ ⩽ & ⋀_{x ≰ u} ⋁_{y ≺ x} ⋀_{f \in F (u_{i_{y}})} [D (f) (y)]^{'} ≱ a . \end{matrix}$ Hence $⋀_{i \in I} C_{D} (u_{i}) ⩽ C_{D} (u)$ .

So $(L, C_{D})$ is a fuzzy generalized convex space.□

Theorem 6.7. Let (L₁, $D$ ₁) and (L₂, $D$ ₂) be fuzzy generalized derived hull spaces. If F : L₁ → L₂ is a fuzzy generalized derived hull preserving mapping, then $F : (L_{1}, C_{D_{1}}) \to (L_{2}, C_{D_{2}})$ is a fuzzy generalized convexity preserving mapping.

Proof. Let v ∈ L₂. We have $\begin{matrix} C_{D_{2}} (v) & = & ⋀_{y ≰ v} [D_{2} (v) (y)]^{'} \\ ⩽ & ⋀_{F (x) ≰ v} [D_{2} (F (F^{⊣} (v))) (F (x))]^{'} \\ ⩽ & ⋀_{x ≰ F^{⊣} (v)} [D_{1} (F^{⊣} (v)) (x)]^{'} = C_{D_{1}} (F^{⊣} (v)) . \end{matrix}$ Thus F is a fuzzy generalized convexity preserving mapping.□

Theorem 6.8. We have $D_{C_{D}} = D$ for any fuzzy generalized derived hull space (L, $D$ ) and $C_{D_{C}} = C$ for any fuzzy generalized convex space $(L, C)$ .

Proof. Let (L, $D$ ) be a fuzzy generalized derived hull space. We have $\begin{matrix} D_{C_{D}} (u) (x) & = & {co}_{C_{D}} (u_{x}^{-}) (x) = ⋀_{x ≰ v \geq u_{x}^{-}} [C_{D} (v)]^{'} \\ = & ⋀_{x ≰ v \geq u_{x}^{-}} ⋁_{y ≰ v} D (v) (y) = D (u) (x) . \end{matrix}$ Thus $D_{C_{D}} = D$ .

Let $(L, C)$ be a fuzzy generalized convex space. Then $\begin{matrix} C_{D_{C}} (u) & = & ⋀_{x ≰ u} [D_{C} (u) (x)]^{'} = ⋀_{x ≰ u} [{co}_{C} (u_{x}^{-}) (x)]^{'} \\ = & ⋀_{x ≰ u} [{co}_{C} (u) (x)]^{'} = C (u) . \end{matrix}$ Therefore $C_{D_{C}} = C$ .□

Based on Theorems 6.6 and 6.7, we define a factor $U$ : FGDHS → FGCS by $\begin{matrix} U ((L, D)) = (L, C_{D}), U (F) = F . \end{matrix}$ Based on Theorems 6.4–6.8, we find that $U$ is an isomorphic factor. We have the following conclusion.

Theorem 6.9. FGDHS is isomorphic to FGCS.

7 Conclusions

Completely distributive lattice is a basic tool to study some mathematical structures such as topology, uniformity, proximity, filter, algebra, convergence etc. In the framework of completely distributive lattice, we introduce fuzzy generalized convex space and characterize it by several approaches. As we can see, fuzzy generalized convex space is a generalization of both M-fuzzifying convex space and (L, M)-fuzzy convex space. Based on such space we may investigate its properties and its relations with fuzzy topological molecular lattice and fuzzy uniform space.

Footnotes

Acknowledgment

The author deeply thanks the editor for handling the paper and the anonymous reviewers for their valuable comments and suggestions. This work is supported by the key project of Hunan Educational Committee (18A474) and the Youth Science Foundation of Hunan Province (2018JJ3192,2019JJ40089) and Doctor Scientific Research Foundation of Anhui Normal University (751966).

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