Fuzzy counterparts of partial hull operators in the theory of L -convex systems

Abstract

In this paper, we provide some new characterizations of L-convex systems. For this purpose, we first introduce the concept of partial hull operators and establish the categorical relationship between partial hull operators and convex systems. Then we abstract the relationship between a subset and its partially convex hull in convex system to a binary relation, called enclosed relation. Moreover, we prove that the enclosed relations are equivalent to convex systems. Subsequently, we generalize the concept of partial hull operators and enclosed relations to the fuzzy case, which will be called L-partial hull operators and L-enclosed relations respectively. Finally we explore the categorical isomorphisms between them.

1 Introduction

Convexity plays a vital role in many mathematical structures, such as vector spaces, posets, lattices, metric spaces, graphs and median algebras. Generalizing the classical theorems in $ℝ^{n}$ , Helly, Caratheodory et al., an axiomatic convexity (also called a convex structure) is abstracted. A convex structure [21] on a set X is defined to be a subset $E$ of 2^X which contains both the empty set ∅ and X itself and which is closed under arbitrary intersections and directed unions. As a topological-like structure, some topological-like properties, including convex hull operators, subspaces, product spaces and separation axioms in the framework of convex structures are investigated. Further, a convexity and a topology (or uniform structure) on the same set had been comprehensively studied. Meanwhile, the notion of convex systems is proposed. The difference between convex systems and convex structures is simply that the universal set X need not be convex in convex systems. Then, the induced convexity of convex system is defined. However, there has been little follow-up research on convex systems. Therefore, in view of the limited study on convex systems, it motivates us to make a deep research to convex systems considering the methodology in convex structures.

With the development of fuzzy set theory, the notion of convex structures has been extended to the fuzzy case. Up to now, there have been three typical kinds of fuzzy convex structures, including L-convex structures [5,15, 5,15], M-fuzzifying convex structures [19] and (L, M)-fuzzy convex structures [4, 20]. Many researchers characterized fuzzy convex structures by different tools, such as fuzzy (restricted) convex hull operators [12 , 26], fuzzy (fuzzifying) interval operators [9 , 31], categorical properties [8 , 32], convergence properties [6, 28], bases and subbases [7 , 29]. More recently, fuzzy convex structures are being investigated from more aspects [2 , 33]. In particular, Shen and Shi [16] proposed the concepts of L-convex systems, which are generalizations of L-convex structures. In the framework of L-convex systems, they gave the definition of Scott-hull spaces, and also constructed its categorical isomorphisms to L-convex systems.

This paper focus on L-convex systems. The difference between L-convex systems and L-convex structures is simply that the largest element $\underline{⊤}$ need not be convex in L-convex systems. It causes that some L-subsets donot have an L-convex hull. Meanwhile, it leads to the definition of the admissible L-subsets (the sets having an L-convex hull), which is the domain of the L-convex hull operators in L-convex systems. The aforementioned research show that L-convex hull operators play a vital role in L-convex structures, that is, L-convex hull operators and L-convex structures are categorical isomorphism. In view of the importance of L-restricted hull operators (L-convex hull operators effect on finite L-subsets) in L-convex structures, we will consider the partially L-convex hull operators (see[16]) effect on L-finite subsets in L-convex system, which will be called L-partial hull operators.

The contributions of this paper are as follows. First, we present the concept of partial hull operators, that is, the partially convex hull operators effect on the finite sets in the framework of convex systems. We also prove that there is a one-to-one correspondence between partial hull operators and convex systems. Second, we focus on the relationship between the subsets of convex systems and their partially convex hull. And the relationship can be abstracted to a binary relation on the admissible sets of convex systems and the powsets of the universe, called enclosed relation. In addition, we show that partially convex hull are equivalent to enclosed relations. Third, in order to give the definition of L-partial hull operators in L-convex systems, it is indispensable to excavate the conditions that the domain of L-partial hull operators must be satisfied. In particular, it is not only embodies the degree approach of fuzzy mathematics, but also embodies the characteristics of fuzzy mathematics.

The paper is organized as follows. In Section 2, we recall some necessary concepts and results. In Sections 3, the concept of partial hull operators is defined and the categorical isomorphism to convex systems is presented. Then, the notion of enclosed relations is introduced and is used to characterize the convex systems. In Section 4, the concept of partial hull operators and enclosed relations are generalized to the fuzzy case, respectively. Then we explore the relations between them. We conclude the paper in Section 5.

2 Preliminaries

Let L be a complete lattice. The largest element and the smallest element in L are denoted by ⊤ and ⊥, respectively. A nonempty subset D ⊆ L is called directed (in symbols $D \overset{dir}{\subseteq} L$ ) if for each a, b ∈ D, there exists c ∈ D such that a, b ⩽ c. In particular, we use the notation x = ⋁ ^↑D to express that the set D is directed and x is its least upper bound. For x, y ∈ L, x is way below y (in symbols x ⪡ y) if for any $D \overset{dir}{\subseteq} L$ such that ⋁^↑D exists, y ⩽ ⋁ ^↑D always implies the existence of some d ∈ D with x ⩽ d. A complete lattice L is called continuous if it satisfies the axiom of approximation: (∀ x ∈ L) x = ⋁ ^↑ ⇓ x, where ⇓x = {u ∈ L ∣ u ⪡ x} (See [3]).

Throughout this article, L is always assumed to be a continuous lattice.

For a nonempty set X, we write 2^X and 2^(X) for the powerset of X and for the collection of all finite subsets of X, respectively. Each mapping A : X ⟶ L is called an L-subset on X, and the collection of all L-subsets is denoted by L^X. The largest element and the smallest element in L^X are denoted by $\underline{⊤}$ and $\underline{⊥}$ , respectively. We call an L-subset A on X finite if its support set {x ∈ X ∣ A (x) ≠ ⊥} is finite. Let L^(X) denote the collection of all finite L-subsets on X.

Given a mapping f : X ⟶ Y, define $f_{L}^{\to} : L^{X} ⟶ L^{Y}$ and $f_{L}^{\leftarrow} : L^{Y} ⟶ L^{X}$ by $f_{L}^{\to} (A) (y) = ⋁_{f (x) = y} A (x)$ for A ∈ L^X and y ∈ Y, and $f_{L}^{\leftarrow} (B) = B \circ f$ for B ∈ L^Y, respectively.

Proposition 2.1. ([17]) (1) L^X is also a continuous lattice by defining ⩽ on L^X in a pointwise way.

(2) For any A, B ∈ L^X, A ⪡ B if and only if A is finite and A (x) ⪡ B (x) for all x ∈ X.

Next, we recall briefly some basic definitions and results on L-convex spaces and L-convex systems.

Definition 2.2. ([5,15, 5,15]) A subset $E$ of L^X is called an L-convex structure on X if it satisfies the following conditions:

$\underline{⊥}$ , $\underline{⊤} \in E$ ;

if ${C_{i} ∣ i \in I} \subseteq E$ is nonempty, then $⋀_{i \in I} C_{i} \in E$ ;

if ${C_{j} ∣ j \in J} \overset{dir}{\subseteq} E$ , then $⋁_{j \in J}^{↑} C_{j} \in E$ .

For an L-fuzzy convex structure

E

on X, the pair

(X, E)

is called an L-convex space and each

A \in E

is called an L-convex set.

Definition 2.3. ([5,15, 5,15]) Let $(X, E)$ be an L-convex space. For each A ∈ L^X, define $co (A) = ⋀ {B \in E ∣ A ⩽ B} .$ That is, co (A) is the least element of $E$ that contains A, called the L-convex hull of A.

Definition 2.4. ([16]) A subset $C$ of L^X is called a partial L-convexity on X if it satisfies the following conditions:

$\underline{⊥} \in C$ ;

if ${C_{i} ∣ i \in I} \subseteq C$ is nonempty, then $⋀_{i \in I} C_{i} \in C$ ;

if ${C_{j} ∣ j \in J} \overset{dir}{\subseteq} C$ , then $⋁_{j \in J}^{↑} C_{j} \in C$ .

For a partial L-convexity on X, we call the pair

(X, C)

an L-convex system, and each

A \in C

a partially L-convex set.

From the definition of L-convex systems, the differences with L-convex structures is simply that the greatest element $\underline{⊤}$ need not to be L-convex. This has a rather fundamental consequence: not every L-subset has an L-convex hull, that is, $co (A) = ⋀ {B \in C ∣ A ⩽ B}$ may not exists for an L-convex system $(X, C)$ . This leads to the next definition.

Definition 2.5. ([16]) Let $(X, C)$ be an L-convex system. An L-subset A ∈ L^X is called admissible, provided it is included in some partially L-convex sets, and the collection of all admissible L-subsets is denoted by $Ad (X, C)$ , that is, $Ad (X, C) = {A \in L^{X} ∣ \exists B \in C such that A ⩽ B} .$

Definition 2.6. ([16]) Let $(X, C)$ be an L-convex system. For each $A \in Ad (X, C)$ , we define $co (A) = {B \in C ∣ A ⩽ B},$ that is, co (A) is the least element of $C$ that contains A, called the L-convex hull of A. The operator co is called the partially L-convex hull operator on $(X, C)$ .

Proposition 2.7. ([16]) Let $(X, C)$ be an L-convex system and co be the partially L-convex hull operator on $(X, C)$ . Then for any $A, B \in C$ and each directed family ${A_{i} ∣ i \in I} \subseteq Ad (X, C)$ , the operator co satisfies the following properties:

$co (\underline{⊥}) = \underline{⊥}$ ;

A ⩽ co (A);

co (co (A)) = co (A);

$co (⋁_{i \in I}^{↑} A_{i}) = ⋁_{i \in I}^{↑} co (A_{i})$ .

Remark 2.8. ([5, 15]) Let co be a mapping on X satisfying (PCH1)-(PCH3), then (PCH4) is equivalent to the following condition:

$(PCH 4)^{*} co (A) = ⋁_{F ⪡ A}^{↑} co (F) .$

Next two definitions propose the notions of homomorphisms between L-convex systems.

Definition 2.9. ([16]) A mapping $f : (X, C_{X}) ⟶ (Y, C_{Y})$ between two L-convex systems is called partial L-convexity preserving (PCP, for short) if it satisfies the following conditions:

for each $A \in Ad (X, C_{X})$ , $f_{L}^{\to} (A) \in Ad (Y, C_{Y})$ ;

if $B \in C_{Y}$ , then for each $A \in Ad (X, C_{X})$ such that $A ⩽ f_{L}^{\leftarrow} (B)$ , we have ${co}_{X} (A) ⩽ f_{L}^{\leftarrow} (B)$ .

Definition 2.10. ([16]) A mapping $f : (X, C_{X}) ⟶ (Y, C_{Y})$ between two L-convex systems is called strongly partial L-convexity preserving (SPCP, for short) if it satisfies the following conditions:

for each $A \in Ad (X, C_{X})$ , $f_{L}^{\to} (A) \in Ad (Y, C_{Y})$ ;

^∗for each $B \in C_{Y}$ , $f_{L}^{\leftarrow} (B) \in C_{X}$ .

The category whose objects are L-convex systems and whose morphisms are PCP mappings will be denoted by L-PCS. The category whose objects are L-convex systems and whose morphisms are SPCP mappings will be denoted by L- SPCS.

For notions on category theory, we refer to [1, 14]

In order to characterize the L-convex systems, the properties of admissible sets are axiomatized, which will be called scott closed sets.

Definition 2.11. A subset $B$ of L^X is called scott closed on L^X, if it satisfies the following conditions:

$B$ is non-empty;

$B$ is a lower set, that is, if A ⩽ B and $B \in B$ , then $A \in B$ ;

if ${D_{i} ∣ i \in I} \overset{dir}{\subseteq} B$ , then $⋁_{i \in I}^{↑} D_{i} \in B$ .

3 Partial hull operators and enclosed relations in convex systems

3.1 Partial hull operators

In this section, we will introduce the concept of partial hull operators, that is, the partially convex hull operators effect on finite sets, and discuss its relationship with convex systems. Moreover, we are about to introduce the notion of enclosed relations which can characterize the convex systems.

Definition 3.1. Let ∅ ∈ A ⊆ 2^(X). A mapping $h : A ⟶ 2^{X}$ is called a partial hull operator on X if it satisfies:

$h (\emptyset) = \emptyset$ ;

for any F ∈ A, $F \subseteq h (F)$ ;

if F ∈ A and $G \subseteq h (F)$ is finite, then G ∈ A and $h (G) \subseteq h (F)$ .

For a partial hull operator on X, the triple

(X, A, h)

is called a partial hull space.

Proposition 3.2. If $(X, A, h)$ is a partial hull space, then A is a lower set, that is, if F ∈ A and G ⊆ F is finite, then G ∈ A.

Proof. It follows immediately from (PH2) and (PH3).

□ Definition 3.3. A mapping $f : (X, A, h_{X}) ⟶ (Y, B, h_{Y})$ between two partial hull spaces is called partial hull preserving (PHP for short) if it satisfies:

for any A ∈ A, f^→ (A) ∈ B;

for any A ∈ A, $f^{\to} (h_{X} (A)) \subseteq h_{Y} (f^{\to} (A))$ .

It is easy to check that all partial hull spaces as objects and all corresponding PHP mappings as morphisms form a category, denoted by PHS.

Next, we will establish the relationship between PHS and PCS.

Proposition 3.4. Let $(X, E)$ be a convex system. Define $A^{E}$ and $h^{E}$ as follows: $A^{E} = Ad (X, E) ⋂ 2^{(X)} and$ $\forall F \in A^{E}, h^{E} (F) = ⋂ {A \in 2^{X} ∣ F \subseteq A \in E} .$ Then $(X, A^{E}, h^{E})$ is a partial hull space.

Proof. It is straightforward that $\emptyset \in A^{E} \subseteq 2^{(X)}$ . So it suffices to verify that $h^{E}$ satisfies (PH1)-(PH3).

(PH1) and (PH2) are straightforward.

(PH3) Suppose that $F \in A^{E}$ and $G \subseteq h^{E} (F)$ is finite. By (PC2), $h^{E} (F) = ⋂ {C \in E ∣ F \subseteq C} \in E$ . It follows that $h^{E} (G) = ⋂ {A \in 2^{X} ∣ G \subseteq A \in E} \subseteq h^{E} (F)$ , as desired. □

Proposition 3.5. If $f : (X, E_{X}) ⟶ (Y, E_{Y})$ is PCP, then $f : (X, A^{E_{X}}, h^{E_{X}}) ⟶ (Y, B^{E_{Y}}, h^{E_{Y}})$ is PHP.

Proof. It is straightforward and is omitted. □ By Propositions 3.4 and 3.5, we obtain a functor $ℍ_{ℂ} : PCS ⟶ PHS$ by

$ℍ_{ℂ} : (X, E) \mapsto (X, A^{E}, h^{E})$ and f ↦ f.

Conversely, we can construct a partial convexity from a partial hull operator.

Proposition 3.6. Let $(X, A, h)$ be a partial hull space and define $E^{(A, h)}$ as follows:

$\begin{matrix} E^{(A, h)} = {A \in 2^{X} ∣ \forall F \in 2^{(A)}, F \in A \\ and h (F) \subseteq A} . \end{matrix}$

Then $E^{(A, h)}$ is a partial convexity on X.

Proof. It suffices to show that $E^{(A, h)}$ satisfies (PC1)-(PC3).

(PC1) It is straightforward from (PH1).

(PC2) For any ${A_{i}}_{i \in I} \subseteq E^{(A, h)}$ is nonempty and F ∈ 2^{(⋂_i∈IA_i)}, it follows that F ∈ 2^(A_i) for all i ∈ I. Since $A_{i} \in E^{(A, h)}$ for all i ∈ I, we have F ∈ A and $h (F) \subseteq A_{i}$ for all i ∈ I. Thus $h (F) \in ⋂_{i \in I} A_{i}$ . Hence $⋂_{i \in I} A_{i} \in E^{(A, h)}$ .

(PC3) For any ${D_{j}}_{j \in J} \subseteq^{dir} E^{(A, h)}$ and $F \in 2^{(⋃_{j \in J}^{↑} D_{j})}$ , it follows that F ∈ 2^{(D_{j
₀})} for some j₀ ∈ J. Since $D_{j_{0}} \subseteq E^{(A, h)}$ , we have F ∈ A and $h (F) \subseteq D_{j_{0}} \subseteq ⋃_{j \in J}^{↑} D_{j}$ . Thus $⋃_{j \in J}^{↑} D_{j} \in E^{(A, h)}$ . □

Proposition 3.7. If $f : (X, A, h_{X}) ⟶ (Y, B, h_{Y})$ is PHP, then $f : (X, E^{(A, h_{X})}) ⟶ (Y, E^{(B, h_{Y})})$ is PCP.

Proof. It is straightforward and is omitted. □ By Propositions 3.6 and 3.7, we obtain a functor $ℂ_{ℍ} : PHS ⟶ PCS$ by

$ℂ_{ℍ} : (X, A, h) \mapsto (X, E^{(A, h)})$ and f ↦ f.

In order to establish the categorical relationship between partial hull operators and convex systems, we first give the following two lemmas.

Lemma 3.8. Let $(X, A, h)$ be a partial hull space and $(X, E^{(A, h)})$ be its induced convex system. Then $h (F) \subseteq E^{(A, h)}$ for all F ∈ A.

Proof. For any F ∈ A and $G \subseteq h (F)$ is finite, we have G ∈ A and $h (G) \subseteq h (F)$ by (PH3). Thus $h (F) \in E^{(A, h)}$ for all F ∈ A. □

Lemma 3.9. Let $(X, E^{(A, h)})$ be a convex system induced by a partial hull space $(X, A, h)$ . Then $Ad (X, E^{(A, h)}) = {A \in 2^{X} ∣ \forall F \in 2^{(A)}, F \in A} .$

Proof.

$(♯) : = {A \in 2^{X} ∣ \exists C \in E^{(A, h)} such that A \subseteq C} .$

(♮) :={A ∈ 2^X ∣ ∀ F ∈ 2^(A), F ∈ A}.

(1) ♯⊆ ♮: Let A∈ ♯. For each F ∈ 2^(A), since A∈ ♯, there exists $C \in E^{(A, h)}$ such that A ⊆ C. Thus F ∈ A. Hence A∈ ♮.

(2) ♮⊆ ♯: Let A∈ ♮. Then ∀F ∈ 2^(A), F ∈ A, we have $h (F) \in E^{(A, h)}$ by Lemma 3.8. It follows that $A \subseteq ⋃_{F \in 2^{(X)}} h (F) \in E^{(A, h)}$ . Hence A∈ ♯.

Thus $Ad (X, E^{(A, h)}) = {A \in 2^{X} ∣ \exists C \in E^{(A, h)}, A \in C} = {A \in 2^{X} ∣ \forall F \in 2^{(A)}, F \in A} .$

□ Theorem 3.10. PHS and PCS are isomorphic.

Proof. It suffices to verify that (1) $E^{(A^{E}, h^{E})} = E$ , (2) $A^{E^{(A, h)}} = A$ and $h^{E^{h}} = h$ .

For (1), $E^{(A^{E}, h^{E})} = {A \in 2^{X} ∣ \forall F \in 2^{(A)} \Rightarrow F \in A^{E} and h^{E} (F) \subseteq A} .$

For any C ∈ 2^X, we have

$C \in E^{(A^{E}, h^{E})}$ $\begin{matrix} \Leftrightarrow & \forall F \in 2^{(C)} \Rightarrow F \in A^{E} and h^{E} (F) \subseteq C \\ \Leftrightarrow & \forall F \in 2^{(C)} \Rightarrow F \in Ad (X, E) and co (F) \subseteq C \\ \Leftrightarrow & C \in E . \end{matrix}$

For (2), $A^{E^{(A, h)}} = Ad (X, E^{(A, h)}) ⋂ 2^{(X)} = A .$ (By Lemma 3.9)

Take any F ∈ A. Then

$\begin{matrix} h^{E^{h}} (F) & = & ⋂ {A \in 2^{X} ∣ F \in 2^{(A)}, A \in E^{(A, h)}} \\ = & ⋂ {A \in 2^{X} ∣ F \in 2^{(A)} \in A, h (F) \subseteq A} \\ \subseteq & h (F) (By Lemma 3.8) . \end{matrix}$

And $h (F) \subseteq h^{E^{h}} (F)$ is trivial.

Thus $h^{E^{h}} = h$ . □

3.2 Enclosed systems

The relationship between the subsets of convex system $(X, E)$ and their partially convex hull operator co can be abstracted by a binary relation on $Ad (X, E) \times 2^{X}$ as follows: for any $A \in Ad (X, C)$ and B ∈ 2^X $co (A) \subseteq B \Leftrightarrow A ⋞^{co} B .$

Then ^⋞co satisfies the following properties.

Proposition 3.11. Let $(X, E)$ be a convex system and co be the partially convex hull operator on it and define ^⋞co as follows: for any $A \in Ad (X, C)$ and B ∈ 2^X $A ⋞^{co} B \Leftrightarrow co (A) \subseteq B .$ Then ^⋞co satisfies the following conditions:

∅^⋞co∅;

A^⋞coB implies A ⊆ B;

A^⋞co ⋂ {B_i ∈ 2^X ∣ i ∈ I} ⇔ ∀ i ∈ I, A^⋞coB_i;

A^⋞coB ⇔ ∀ F ∈ 2^(A), F^⋞coB;

if A^⋞coB, then there exists $C \in Ad (X, E)$ such that A^⋞coC^⋞coB.

Proof.

(1) and (2) are straightforward.

(3) For any $A \in Ad (X, C)$ and {B_i} _i∈I ⊆ 2^X, it follows that $\begin{matrix} A ⋞^{co} ⋂ {B_{i} \in 2^{X} ∣ i \in I} \\ \Leftrightarrow & co (A) \subseteq ⋂_{i \in I} B_{i} \\ \Leftrightarrow & \forall i \in I, co (A) \subseteq B_{i} \\ \Leftrightarrow & \forall i \in I, A ⋞^{co} B . \end{matrix}$

(4) For any $A \in Ad (X, C)$ and B ∈ 2^X, $\begin{matrix} A ⋞^{co} B & \Leftrightarrow & co (A) \subseteq B \\ \Leftrightarrow & ⋃_{F \in 2^{(A)}} co (F) \subseteq B \\ \Leftrightarrow & \forall F \in 2^{(A)}, co (F) \subseteq B \\ \Leftrightarrow & \forall F \in 2^{(A)}, F ⋞^{co} B . ∥ \end{matrix}$ (5) ∀A, B ∈ 2^X, it follows that A^⋞coB ⇔ co (A) ⊆ B.

Let $C = co (A) \in Ad (X, E)$ . We have A^⋞coC. Further, co (C) = co (co (A)) = co (A) ⊆ B, it follows that C^⋞coB. Hence, A^⋞coC^⋞coB. □ By means of (1)-(5), we will introduce the concept of enclosed relation system. Now we give the axiomatic definition.

Definition 3.12. Let $B$ be a scott closed set on X. An enclosed relation on X is a subset $⋞ \subseteq B \times 2^{X}$ which satisfies (ER1)–(ER5).

∅⋞∅;

A⋞B implies A ⊆ B;

A⋞ ⋂ {B_i ∈ 2^X ∣ i ∈ I} ⇔ ∀ i ∈ I, A⋞B_i;

A⋞B ⇔ ∀ F ∈ 2^(A), F⋞B;

if A⋞B, then $\exists C \in B$ such that A⋞C⋞B.

For an enclosed relation on X, the triple

(X, B, ⋞)

is called an enclosed system.

Lemma 3.13. Let $(X, B, ⋞)$ be an enclosed system. Then

∅⋞A for all A ∈ 2^X;

if A ⊆ B⋞C ⊆ D, then A⋞D;

if A⋞B, then there exists $C \in B$ such that A ⊆ C⋞C ⊆ B.

Proof. (1) It is straightforward.

(2) Suppose A ⊆ B⋞C ⊆ D. By (ER3), we have B⋞D. For any F ∈ 2^(A), it follows that F ∈ 2^(B). By (ER4), we have F⋞D. Hence A⋞D.

(3) Suppose A⋞B. Let $D = {D \in 2^{X} ∣ A ⋞ D}$ and let $C = ⋂ {D \in 2^{X} ∣ D \in D}$ . It follows $C \subseteq B \in D$ . By (ER3), A⋞C. By (ER5), there exists $E \in B$ such that A⋞E⋞C, which implie A ⊆ E ⊆ C. By the definition of C, we have C ⊆ E. Thus E = C. Hence A ⊆ C⋞C ⊆ B.

□ Proposition 3.14. A binary relation on X satisfies (ER4) if and only if it satisfies the following condition:

$(ER 4)^{*} if {B_{i}}_{i \in I} \overset{dir}{\subseteq} B$ , then $⋃_{i \in I}^{↑} B_{i} ⋞ A$ iff B_i⋞A for all i ∈ I.

Proof. It suffices to show (ER4) ⇒ (ER4) ^∗.

(⇒) Suppose ${B_{i}}_{i \in I} \overset{dir}{\subseteq} B, ⋃_{i \in I}^{↑} B_{i} ⋞ A .$ For each i ∈ I, $F \in 2^{(B_{i})} \subseteq ⋃_{i \in I}^{↑} B_{i} ⋞ A$ , then by (ER4) we have B_i⋞A for all i ∈ I.

(⇐) Conversely, suppose B_i⋞A for all i ∈ I. Then for each F ∈ 2^{(⋃^↑B_i)}, there exists some i₀ ∈ I, F ∈ 2^{(F_{i
₀})}. Since B_{i
₀}⋞A, we have F⋞A. Hence $⋃_{i \in I}^{↑} B_{i} ⋞ A$ by (ER4). □ Now we consider the mappings between enclosed systems.

Definition 3.15. Let $(X, B, ⋞_{X})$ and $(Y, F, ⋞_{Y})$ be two enclosed systems. A mapping f : X ⟶ Y is called enclosed dual-preserving (EDP, in short) provided that:

for any $A \in B$ , $f^{\to} (A) \in F$

for any $U \in F, V \in 2^{Y}$ , U_⋞YV implies f^← (U) _⋞Xf^← (V).

Theorem 3.16. If $(X, B, ⋞_{X}) ⟶ (Y, F, ⋞_{Y})$ and $(Y, F, ⋞_{Y}) ⟶ (Z, K, ⋞_{Z})$ are two EDP mappings, then the composition mapping g ∘ f is also EDR mapping. It is easy to check that all enclosed spaces as objects and all corresponding EDP mappings as morphisms form a category, denoted by ES.

Proposition 3.17. If $f : (X, C_{X}) ⟶ (Y, C_{Y})$ is a SPCP mapping, then $f : (X, Ad (X, C_{X}), ⋞_{C_{X}}) ⟶ (Y, Ad (Y, C_{Y}), ⋞_{C_{Y}})$ is an EDP mapping.

Proof. (EDPI) is straightforward from (PCP1).

(EDP2) For any $U \in Ad (Y, C_{Y}), V \in 2^{Y}$ with $U ⋞_{C_{Y}} V$ . Then co_Y (U) ⊆ V. It follows that f^← (co_Y (U)) ⊆ f^← (V). Since $f : (X, C_{X}) ⟶ (Y, C_{Y})$ is a PCP mapping, we have co_X (f^← (U)) ⊆ f^← (co_Y (U)) ⊆ f^← (V). Hence, $f^{\leftarrow} (U) ⋞_{C_{X}} f^{\leftarrow} (V)$ . Thus $f : (X, Ad (X, C_{X}) ⋞_{X}) ⟶ (Y, Ad (Y, C_{Y}) ⋞_{Y})$ is an EDP mapping. □

Conversely, we will induce a convex system by means of an enclosed relation.

Proposition 3.18. Let $(X, B, ⋞)$ be an enclosed system and define co^⋞ as follows:

$\forall A \in B, {co}^{⋞} (A) = ⋂ {B \in 2^{X} ∣ A ⋞ B} .$

Then (1) ∀B ∈ 2^X, co^⋞ (A) ⊆ B ⇔ A⋞B.

(2) co^⋞ is a partially convex hull operator. Hence it induces a convex system on X, denoted by $C^{⋞}$ .

Proof. (1)The sufficient is obvious.

Necessity. If co^⋞ (A) ⊆ B, by (ER3), we have A⋞co^⋞ (A) ⊆ B . Thus A⋞B by Lemma 3.13.

(2) (PCH1) and (PCH2) are straightforward from (ER1) and (ER2), so it suffices to show that co^⋞ satisfies (PCH3) and (PCH4) ^∗.

(PCH3) We only need to prove co^⋞ (co^⋞ (A)) ⊆ co^⋞ (A). Let co^⋞ (A) = D. Then A⋞D. By (ER5), there exists $C \in B$ such that A⋞C⋞D. This implies co^⋞ (A) ⊆ C and co^⋞ (C) ⊆ D. Thus we have co^⋞ (co^⋞ (A)) ⊆ co^⋞ (C) ⊆ D. Therefore co^⋞ (co^⋞ (A)) ⊆ co^⋞ (A).

(PCH4) ^∗ The direction ⊇ holds naturally from (ER4), so we just to prove the other direction ⊆. Let M = ⋃ _F∈2^(A)co^⋞ (F). Then co^⋞ (F) ⊆ M for any F ∈ 2^(A). This implies co^⋞ (A) ⊆ M. Hence ${co}^{⋞} (A) \subseteq ⋃_{F \in 2^{(A)}}^{↑} {co}^{⋞} (F) .$ Therefore ${co}^{⋞} (A) = ⋃_{F \in 2^{(A)}}^{↑} {co}^{⋞} (F) .$

□ Proposition 3.19. If $f : (X, B, ⋞_{X}) ⟶ (Y, F, ⋞_{Y})$ is an EDP mapping, then $f : (X, C_{⋞_{X}}) ⟶ (Y, C_{⋞_{Y}})$ is a SPCP mapping.

Proof. Take any $A \in C_{⋞_{Y}}$ . Then A = ⋂ {B ∈ 2^Y ∣ A_⋞YB}. It follows that f^← (A) = ⋂ {f^← (B) ∣ B ∈ 2^Y, A_⋞YB} . Since f is an EDP mapping, we have

$\begin{matrix} f^{\leftarrow} (A) & = & ⋂ {f^{\leftarrow} (B) ∣ B \in 2^{Y}, A ⋞_{Y} B} \\ \supseteq & ⋂ {E \in 2^{X} ∣ f^{\leftarrow} (A) ⋞_{X} E} \\ \supseteq & f^{\leftarrow} (A) \end{matrix}$

Hence f^← (A) = ⋂ {E ∈ 2^X ∣ f^← (A) _⋞XE} = co_{_⋞X} (f^← (A)). □ The following is straightforward by Proposition 3.18

Corollary 3.20. For an enclosed relation ⋞ on X, it follows that $C = {A \in B ∣ A ⋞ A} .$

By Propositions 3.11 and 3.18, the following theorems are straightforward.

Theorem 3.21. For an enclosed space $(X, B, ⋞)$ , $⋞^{C^{⋞}} = ⋞$ .

Theorem 3.22. For a convex system $(X, C)$ , $C^{⋞^{C}} = C .$

By Theorems 3.21, 3.22 and Propositions 3.17, 3.19, we obtain the following theorem.

Theorem 3.23. SPCS is isomorphic to ES.

4 The fuzzy counterpart of partial hull operators and enclosed relations in L-convex systems

4.1 L-partial hull operators

In this section, we will generalize the concept of partial hull operators to L-partial hull operators and show that the resulting category is isomorphic to L-PCS.

In order to introduce the concept of L-partial hull operators, we first give the following notation, which is different from the classical case.

Denote $A \subseteq L^{(X)}$ which satisfies the following conditions:

$\underline{⊥} \in A$ ;

∀F ⪡ A, $F \in A \Rightarrow A \in A$ .

Definition 4.1. A mapping $h : A ⟶ L^{X}$ is called an L-partial hull operator on X if it satisfies the following conditions:

$h (\underline{⊥}) = \underline{⊥}$ ;

for any $F \in A, F ⩽ h (F)$ ;

if $F \in A$ and G ⪡ h (F), then $G \in A$ and h (G) ⩽ h (F);

for any $F \in A$ , $h (F) = ⋁_{G ⪡ F}^{↑} h (G)$ .

For an L-partial hull operator on X, the triple

(X, A, h)

is called an L-partial hull space.

Proposition 4.2. If $(X, A, h)$ is an L-partial hull space, then $A$ is a lower set. That is, if $F \in A$ and G ⩽ F, then $G \in A$ .

Proof.

It is trivial and is omitted. □ Definition 4.3. A mapping $f : (X, A, h_{X}) ⟶ (Y, B, h_{Y})$ is called an L-partial hull preserving (L-PHP) if it satisfies:

for any $A \in A$ , $f_{L}^{\to} (A) \in B$ ;

for any $A \in A$ , $f_{L}^{\to} (h_{X} (A)) ⩽ h_{Y} (f_{L}^{\to} (A))$ .

It is easy to check that all L-partial hull spaces as objects and all corresponding L-PHP mappings as morphisms form a category, denoted by L-PHS.

Proposition 4.4. Let $(X, C)$ be an L-convex system. Define $A^{C}$ and $h^{C}$ as follows: $A^{C} = Ad (X, C) ∣_{L^{(X)}} and$ $\forall F \in A, h^{C} (F) = ⋀ {A \in L^{X} ∣ F ⩽ A \in C} .$ Then $(X, A^{C}, h^{C})$ is an L-partial hull space.

Proof. (LPH1) and (LPH2) are straightforward.

(LPH3) Suppose $F \in A^{C}$ and $G ⪡ h^{C} (F)$ . By (PC2), $h^{C} (F) = ⋀ {A \in C ∣ F ⩽ A} \in C$ . Then $h^{C} (G) = ⋀ {C \in C ∣ G ⩽ C \in C} ⩽ h^{C} (F) .$

(LPH4) Take any $F \in A^{C}$ . Since ⇓F is directed and $h^{C}$ is monotone, ${h^{C} (G) ∣ G ⪡ F} \subseteq C$ is directed. Thus $⋁_{G ⪡ F}^{↑} h^{C} (G) \in C$ . Then it follows that $F = ⋁_{G ⪡ F}^{↑} G ⩽ ⋁_{G ⪡ F}^{↑} h^{C} (G) \in C$ . Therefore $h^{C} (F) ⩽ ⋁_{G ⪡ F}^{↑} h^{C} (G)$ . The inverse inequality is trivial.

Hence $h^{C} (F) = ⋁_{G ⪡ F}^{↑} h^{C} (G) .$ □

Proposition 4.5. If $f : (X, C_{X}) ⟶ (Y, C_{Y})$ is L-PCP, then $f : (X, A^{C_{X}}, h^{C_{X}}) ⟶ (Y, B^{C_{Y}}, h^{C_{Y}})$ is L-PHP.

Proof. It is straightforward. □

By Propositions 4.4 and 4.5, we obtain a functor $ℍ_{ℂ} : L - PCS ⟶ L - PHS$ defined by

$ℍ_{ℂ} : (X, C) \mapsto (X, A^{C}, h^{C})$ and f ↦ f.

Conversely, we can construct an L-convex system from an L-partial hull operator.

Proposition 4.6. Let $(X, A, h)$ be an L-partial hull space, and define $C^{(A, h)}$ as follows:

$C^{(A, h)} = {A \in L^{X} ∣ \forall F ⪡ A, F \in A, h (F) ⩽ A} .$ Then $(X, C^{(A, h)})$ is an L-convex system.

Proof. (PC1) Since $⇓ \underline{⊥} = {\underline{⊥}}$ , $\underline{⊥} \in C$ and $h (\underline{⊥}) = \underline{⊥} .$ That $\underline{⊥} \in C^{(A, h)}$ is trivial.

(PC2) Take each ${A_{i}}_{i \in I} \subseteq C^{(A, h)}$ . If F ⪡ ⋀ _i∈IA_i, it follows that F ⪡ A_i for all i ∈ I. Since ${A_{i}}_{i \in I} \subseteq C^{(A, h)}$ , it follows that $F \in A$ and h (F) ⩽ A_i for all i ∈ I. Thus $F \in A$ and h (F) ⩽ ⋀ _i∈IA_i. Hence $⋀_{i \in I} \in C^{(A, h)}$ .

(PH3) Take each ${A_{j}}_{j \in J} \overset{dir}{\subseteq} C^{(A, h)}$ . If $F ⪡ ⋁_{j \in J}^{↑} A_{j}$ , then F ⪡ A_{j
₀} for some j₀ ∈ J. Since $A_{j_{0}} \in C^{(A, h)}$ , it follows that $F \in A$ and $h (F) ⩽ A_{j_{0}} ⩽ ⋁_{j \in J}^{↑} A_{j}$ . Hence $⋁_{j \in J}^{↑} A_{j} \in C^{(A, h)} .$ □

Proposition 4.7. If $(X, A, h_{X}) ⟶ (Y, B, h_{Y})$ is L-PHP, then $f : (X, C^{(A, h_{X})}) ⟶ (Y, C^{(B, h_{Y})})$ is L-PCP.

Proof.

It is straightforward. □ By Propositions 4.6 and 4.7, we obtain a functor $ℂ_{ℍ} : L - PHS ⟶ L - PCS$ defined by

$ℂ_{ℍ} : (X, A, h) \mapsto (X, C^{(A, h)})$ and f ↦ f.

Proposition 4.8. Let $(X, A, h)$ be an L-partial hull space and let $(X, C^{(A, h)}$ be the L-convex system induced by $(X, A, h)$ . Then

for any $F \in A, h (F) \in C^{(A, h)}$ .

$Ad (X, C^{(A, h)}) = {A \in L^{X} ∣ \forall F ⪡ A, F \in A}$ .

Proof. (1) For any $F \in A$ and G ⪡ h (F), we have $G \in A$ and h (G) ⩽ h (F) by (LPH3). Thus $h (F) \in C^{h}$ .

(2) We need to prove ${A \in L^{X} ∣ \exists C \in C^{(A, h)} such that A ⩽ C} = {A \in L^{X} ∣ \forall F ⪡ A, A \in A} .$

$♯ : = {A \in L^{X} ∣ \exists C \in C^{(A, h)} such that A ⩽ C}$ .

$♮ : = {A \in L^{X} ∣ \forall F ⪡ A, A \in A}$ .

♯ ⊆ ♮ : Take each A∈ ♯, it follows that there exists $C \in C^{(A, h)}$ such that A ⩽ C. For any F ⪡ A, we have F ⪡ C. Since $C \in C^{(A, h)}$ , $F \in A$ . Thus A∈ ♮.

♮⊆ ♯: Take each A∈ ♮. For any F ⪡ A and $F \in A$ , $h (F) \in C^{(A, h)}$ by (1). It follows that $⋁_{F ⪡ A}^{↑} h (F) \in C^{(A, h)}$ . Since $A = ⋁_{F ⪡ A}^{↑} F ⩽ ⋁_{F ⪡ A}^{↑} h (F)$ . Thus A∈ ♯. □

Theorem 4.9. L-PHS and L-PCS are isomorphic.

Proof. It need to verify that (1) $C^{(A^{C}, h^{C})} = C$ and (2) $Ad (X, C^{(A, h)}) ∣_{L^{(X)}} =$ $A^{C^{(A, h)}} = A$ and $h^{C^{(A, h)}} = h$

For (1), take each A ∈ L^X, it follows that

$A \in C^{(A^{C}, h^{C})}$ $\begin{matrix} \Leftrightarrow & \forall F ⪡ A \Rightarrow F \in A^{C} and h^{C} (F) ⩽ A \\ \Leftrightarrow & \forall F ⪡ A, \Rightarrow F \in Ad (X, C) ∣_{L^{(X)}}, h^{C} (F) ⩽ A . \\ \Leftrightarrow & \forall F ⪡ A \Rightarrow F \in Ad (X, C) ∣_{L^{(X)}}, co (F) ⩽ A \\ \Leftrightarrow & A = ⋁_{F ⪡ A}^{↑} F \in Ad (X, C), A = ⋁_{F ⪡ A}^{↑} co (F) \\ \Leftrightarrow & A \in C . \end{matrix}$

For (2), take each A ∈ L^(X) such that F ⪡ A and $F \in A^{C^{(A, h)}}$ , then $⇓ A \subseteq Ad (X, C)$ is directed. It follows that ${co (F) ∣ F ⪡ A} \subseteq C$ is directed. We have $A = ⋁_{F ⪡ A}^{↑} F ⩽ ⋁_{F ⪡ A}^{↑} co (F) \in C$ . Thus $A \in A^{C^{(A, h)}}$ .

Take any $F \in A$ . Then

$h^{C^{(A, h)}} (F)$ $\begin{matrix} = & ⋀ {A \in L^{X} ∣ F ⩽ A \in C^{(A, h)}} \\ = & ⋀ {A \in L^{X} ∣ F ⩽ A, \forall G \\ ⪡ A, G \in A, h (G) ⩽ A} . \end{matrix}$ On one hand, since $h (F) \in C^{(A, h)}$ , it follows that $h^{C^{(A, h)}} (F) ⩽ h (F)$ .

On the other hand, if $A \in C^{(A, h)}$ and F ⩽ A, then for any G ⪡ F, we have G ⩽ A. Hence h (G) ⩽ A. By (LRH4), $h (F) = ⋁_{G ⪡ F}^{↑} h (G) ⩽ A$ . Thus $h (F) ⩽ h^{C^{(A, h)}} (F)$ .

Hence, $h^{C^{(A, h)}} = h$ . □

4.2 L-enclosed systems

In this section, we will generalize the concept of enclosed relations to the fuzzy case. Subsequently, we will discuss its relationship with L-convex systems.

Definition 4.10. We call a subset $⋞ \subseteq B \times L^{X}$ an L-enclosed relation, if it satisfies the following conditions: $\forall A \in B, B \in L^{X}$ ,

$\underline{⊥} ⋞ \underline{⊥}$ ,

A⋞B ⇒ A ⩽ B,

A⋞ ⋀ {B_i ∈ L^X ∣ i ∈ I} ⇔ ∀ i ∈ I, A⋞B_i,

A⋞B ⇔ ∀ F ⪡ A, F⋞B,

if A⋞B, then there exists $C \in B$ such that A⋞C⋞B.

For an L-enclosed relation ⋞ on X, the pair

(X, B, ⋞)

is called an L-enclosed space.

Lemma 4.11. Let $(X, B, ⋞)$ be an L-enclosed space. Then

$\underline{⊥} ⋞ A$ for A ∈ L^X;

if A ⩽ B⋞C ⩽ D, then A⋞D;

if A⋞B, then there exists $C \in B$ such that A ⩽ C⋞C ⩽ B.

Proof. (1) It is straightforward.

(2) By (LER3), we have E⋞D. Take any F ⪡ A. Since A ⩽ B, F ⪡ B. By (LER4), we have F⋞D. Thus A⋞D.

(3) Suppose A⋞B. Let $D = {D \in L^{X} ∣ A ⋞ D}$ and let $C = ⋀ {D \in L^{X} ∣ D \in D}$ . Then $C ⩽ B \in D$ . By (LER3), we have A⋞C. And by (LER5), there exists $E \in B$ such that A⋞E⋞C. It implies A ⩽ E ⩽ C. By the definition of C we have C ⩽ E. Thus E = C. Hence A⋞C⋞C⋞B. □

Proposition 4.12. A binary relation on X satisfies (LER4) if and only if it satisfies the following condition:

(LER4) ^∗ : if ${B_{i} ∣ i \in I} \overset{dir}{\subseteq} B$ , then $⋁_{i \in I}^{↑} B_{i} ⋞ A$ iff B_i⋞A for all i ∈ I .

Proof. It suffices to prove (LER4)⇒(LER4) ^∗ .

(⇒) Suppose $⋁_{i \in I}^{↑} B_{i} ⋞ A$ . By (LER4), it suffices to show F⋞A for all F ⪡ B_j. Since $F ⪡ ⋁_{i \in I}^{↑} B_{i} ⋞ A$ . By (LER4), F⋞A, as desired.

(⇐) Conversely, B_i⋞A for all i ∈ I and for any $F ⪡ ⋁_{i \in I}^{↑} B_{i}$ . Then there exists i₀ ∈ I, such that F ⪡ B_{i
₀}. Since B_{i
₀}⋞A, we have F⋞A. Therefore, $⋁_{i \in I}^{↑} B_{i} ⋞ A$ by (LER4). □ The following proposition shows that every L-convex system can induce an L-enclosed relation. Its proof is straightforward.

Proposition 4.13. Let $(X, C)$ be an L-convex system and co be the partially L-convex hull operator on it. Define a binary relation ^⋞co as follows: $\forall A \in Ad (X, C) and B \in L^{X},$

$A ⋞^{co} B iff co (A) ⩽ B .$ Then ^⋞co is an L-enclosed relation on X.

For an L-enclosed relation, we can also induce an L-convex system.

Proposition 4.14. Let $(X, B, ⋞)$ be an L-enclosed system. Define an operator on X as follows: for all $A \in B$ and B ∈ L^X,

${co}^{⋞} (A) = ⋀ {B \in L^{X} ∣ A ⋞ B} .$

co^⋞ (A) ⩽ B if and only if A⋞B.

The operator co_⋞ is a partially L-convex hull operator. Hence it induces an L-convex system on X, denoted by $C^{⋞}$

Proof. (1) It is straightforward that A⋞B implies co^⋞ (A) ⩽ B. Conversely, suppose co^⋞ (A) ⩽ B. Then by (LER3), we have A⋞co^⋞ (A) ⩽ B. Hence A⋞B by Lemma 4.11.

(2) (PCH1) and (PCH2) are obvious.

(PCH3) Obviously, co^⋞ (A) ⩽ co^⋞ (co^⋞ (A)). We need to prove co^⋞ (co^⋞ (A)) ⩽ co^⋞ (A). Let co^⋞ (A) ⩽ B.Then A⋞B. By (LER5), there exists $C \in B$ such that A⋞C⋞B. This implies co^⋞ (C) ⩽ B and co^⋞ (A) ⩽ C. Thus co^⋞ (co^⋞ (A)) ⩽ co^⋞ (C) ⩽ B. Hence, co^⋞ (co^⋞ (A)) ⩽ co^⋞ (A).

(PCH4)^∗ By (LER4), for any F ⪡ A, F⋞B implies A⋞B, then co^⋞ (F) ⩽ B implies co^⋞ (A) ⩽ B. It follows that $⋁_{F ⪡ A}^{↑} {co}^{⋞} (F) ⩽ {co}^{⋞} (A)$ . It remains to show ${co}^{⋞} (A) ⩽ ⋁_{F ⪡ A}^{↑} {co}^{⋞} (F) .$ Let $D = ⋁_{F ⪡ A}^{↑} {co}^{⋞} (F)$ . Then for any F ⪡ A, co^⋞ (F) ⩽ D. Thus A⋞D. Hence ${co}^{⋞} (A) ⩽ D = ⋁_{F ⪡ A}^{↑} {co}^{⋞} (F) .$ □

Corollary 4.15. For an L-enclosed relation on X, it follows that

$C^{⋞} = {A \in B ∣ A ⋞ A} .$

By Propositions 4.13 and 4.14, we have the following theorems.

Theorem 4.16. For an L-enclosed relation on X, $⋞^{C^{⋞}} = ⋞ .$

Theorem 4.17. For an L-convex system $(X, C)$ , $C^{⋞^{C}} = C$ .

Now, we will consider the mappings between two L-enclosed spaces.

Let $(X, B, ⋞_{X})$ and $(Y, G, ⋞_{Y})$ be two L-enclosed systems. A mapping f : X ⟶ Y is called an L-enclosed dual-preserving (LEDP, in short) providing that:

$\forall A \in B, f_{L}^{\to} (A) \in G;$

$\forall U \in G, V \in 2^{Y}$ , U_⋞YV implies $f_{L}^{\leftarrow} (U) ⋞_{X} f_{L}^{\leftarrow} (V)$ .

Proposition 4.18. If $f : (X, B, ⋞_{X}) ⟶ (Y, G, ⋞_{Y})$ is an L-EDP mapping, then $f : (X, C^{(B, ⋞_{X})}) ⟶ (Y, C^{(G, ⋞_{Y})})$ is a SPCP mapping.

Proof. Take each $V \in C^{(G, ⋞_{Y})}$ . Then V = ⋀ {G ∈ L^Y ∣ V_⋞YG}. Since $f : (X, B, ⋞_{X}) ⟶ (Y, G, ⋞_{Y})$ is an L-EDP mapping, it follows that

$\begin{matrix} f_{L}^{\leftarrow} (V) & = & ⋀ {f_{L}^{\leftarrow} (G) ∣ G \in L^{Y}, V ⋞_{Y} G} \\ ⩾ & ⋀ {E \in L^{X} ∣ f_{L}^{\leftarrow} (V) ⋞_{X} E} \\ ⩾ & f_{L}^{\leftarrow} (V) . \end{matrix}$

Thus $f_{L}^{\leftarrow} (V) = ⋀ {E \in L^{X} ∣ f_{L}^{\leftarrow} (V) ⋞_{X} E} = {co}_{⋞_{X}} (f_{L}^{\leftarrow} (V))$ . That is $f_{L}^{\leftarrow} (V) \in C_{⋞_{X}} .$

Hence, $f : (X, C^{(B, ⋞_{X})}) ⟶ (Y, C^{(G, ⋞_{Y})})$ is a SPCP mapping. □

Proposition 4.19. If $f : (X, C_{X}) ⟶ (Y, C_{Y})$ is a SPCP mapping, then $f : (X, Ad (X, C_{X}), ⋞_{X}^{C}) ⟶ (Y, Ad (Y, C_{Y}), ⋞_{Y}^{C})$ is an L-EDP mapping.

Proof. Take each U, V ∈ L^Y with $U ⋞_{C_{Y}} V$ . Then co_Y (U) ⩽ V. It follows that $f_{L}^{\leftarrow} ({co}_{Y} (U)) ⩽ f_{L}^{\leftarrow} (V)$ . Since $f : (X, C_{X}) ⟶ (Y, C_{Y})$ is a PCP mapping, we have ${co}_{X} (f_{L}^{\leftarrow} (U)) ⩽ f_{L}^{\leftarrow} ({co}_{Y} (U)) ⩽ f_{L}^{\leftarrow} (V)$ . Thus $f_{L}^{\leftarrow} (U) ⋞_{C_{X}} f_{L}^{\leftarrow} (V)$ . Hence $f : (X, Ad (X, C_{X}), ⋞_{X}^{C}) ⟶ (Y, Ad (Y, C_{Y}), ⋞_{Y}^{C})$ is an L-EDP mapping. □ By Theorems 4.16, 4.17 and Propositions 4.18, 4.19, we obtain the following theorem.

Theorem 4.20. L-SPCS is isomorphic to L-ES.

5 Conclusions

In this paper, we discuss the relations between (L-)partial hull operators, (L-)enclosed relations and (L-)convex systems, respectively. First, it is proved that the categories of partial hull spaces and convex systems are isomorphic. Second, it is established the categorical isomorphism between enclosed spaces and strong convex systems. And last, the concepts of partial hull operators and enclosed relations are generalized to the fuzzy case, then the categorical isomorphism between them are established. But in this paper, the truth-value table is a continuous lattice, and the research is just under the framework of L-convex systems, whether the results still hols under more generalized case. Meanwhile, in the classical case, the problems remain unsolved whether uniform convex systems can be extended to uniform convex structures. Therefore, as the future work, we will consider the following problems:

•As a generalization of L-convex structures and M-fuzzifying convex structures, the notion of (L, M)-fuzzy convex structures was introduced in [20]. Thus, it will be interesting to generalize the convex system to (L, M)-fuzzy case and study the properties of (L, M)-fuzzy convex systems by using residual implication.

•In the theory of convex structures, the relations of the uniform structures, topological structures and convex systems were discussed. Moreover, Wang and Shi [23] have already give the notion of M-fuzzifying topological convex spaces. This motivates us to consider the compatibility of convex systems with topological structures and uniform structures on the same underlying set.

Footnotes

Acknowledgement

This work is supported by the National Natural Science Foundation of China (11971065).

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