On the sum of L -convex spaces

Abstract

Considering L be a completely distributive lattice, the notion of the sum of L-convex spaces is introduced and its elementary properties is studied. Firstly, the connections between the sum of L-convex spaces and its factor spaces are established. Secondly, the additivity of separability (S_-1, sub-S₀, S₀, S₁, S₂, S₃ and S₄) are investigated. Finally, the additivity of five types special L-convex spaces are examined.

Keywords

separability additivity

1 Introduction

As we all know, fuzzy set theory has been widely used in many mathematical structures since it was proposed in 1965 [38]. In the past fifty-five years, many fuzzy structures have been studied, such as fuzzy topological structures [27], fuzzy convergence structures [5], fuzzy neighborhood structures [37] and so on. As another important kind of fuzzy structures, the notion of fuzzy convex structures (or, fuzzy convexities) was first presented by Rosa [17] as a natural generalization of the concept of abstract convex structures [26]. In Rosa’s paper, the lattice-valued environment is a real unit interval [0,1] and the object of study is a crisp structure constituted of fuzzy convex sets. Afterwards, Maruyama [10] introduced another more generalized fuzzy convex structures based on a completely distributive lattice, which is a generalization of Rosa’s notion. Nowadays, the definitions of fuzzy convex structures defined by Rosa and Maruyama are collectively called L-convex structures (where is L a complete lattice).

In recent years, the theory of L-convex spaces has gradually become a hot research content, more and more scholars have developed and enriched it. For instance, Jin and Li [4] discussed the relationships between classical convex structures and stratified L-convex structures from the perspective of category. In the setting of a complete Heyting algebra, Zhong and Shi [39] gave the formulations of hull operators in L-ordered convex spaces, L-semilattice convex spaces, L-lattice convex spaces and L-standard convex spaces, respectively. Chen et.al. [1] extended the notions of arity, CUP, JHC and weakly JHC of crisp convex spaces to the L-fuzzy case. Pang and Shi [13] presented axiomatic L-hull operators and L-interval operators to characterize L-convex spaces. Pang and Xiu [15] established the axiom of bases and subbases and gave their applications. Shen and Shi [20] established some novel characterizations of L-convex structures based on way-below relations in a continuous lattice. Zhou and Shi [40] introduced some separability of L-convex spaces and researched their properties. In particular, Pang and Shi [12] provided four kinds of L-convex spaces and discussed the categorical relations among them. There are also some more extensive research on L-convex spaces [9 , 39].

In the setting of a completely distributive lattice M, Shi and Xiu [23] proposed a novel method to the fuzzification of convex structures based on the ideal of multi-valued logic and introduced the concepts of M-fuzzifying convex structures and M-fuzzifying hull operators, and proved that there is a one-to-one correspondence between them. In this situation, a nonempty set equipped with an M-fuzzifying convex structure on it is called an M-fuzzifying convex space. The latest research progress of M-fuzzifying convex spaces can be found in [7 , 33–36]. It is worth noting that a more generalized convex spaces was proposed in [24], which is called (L, M)-fuzzy convex spaces (where L and M are complete lattices). In the framework of (L, M)-fuzzy convex spaces, L-convex spaces and M-fuzzifying convex spaces can be regarded as its special cases. Recently, some scholars have studied it and got many beautiful results [6 , 31].

It is know that there is a common method to study the properties of new spaces by using the properties of initial spaces. Concretely, we usually discuss the properties of subspaces, product spaces and quotient spaces of initial convex spaces. For example, the author systematically discussed the hereditary properties and productive properties of separability in classical convex spaces [26], and the related results are extended to the L-fuzzy case [40]. In the monograph of convex structures [26], many properties of quotient spaces are also studied in detail. However, as an important method of constructing new convex spaces, there are few researches on the sum of convex spaces. Therefore, it is necessary to discuss the sum of L-convex spaces, which can further improve the theory of L-convex spaces. By this motivation, we will aim to introduce the notion of the sum of a family of L-convex spaces and will investigate the relationships between the sum space and its factor spaces.

The structure of the paper is mainly expanded from the following parts. In Section 2, we recall some necessary concepts and notations that are required in the subsequence sections. In Section 3, we will introduce the concept of the sum of a family of L-convex spaces and will study its basic properties. In Section 4, we will consider the additivity of separability in the L-fuzzy case. Especially, we will discuss the connections among S₂, S₃ and S₄ separability with the help of some examples. In Section 5, we will study the additivity of five types special L-convex spaces defined by L-hull operators.

2 Preliminaries

In this section, we recollect some elementary concepts and properties on lattices, L-fuzzy sets and L-convex structures. For concepts not defined in this article, the reader can refer to [2 , 27].

Assume L is a complete lattice. The two elements ⊥_L and ⊤_L are smallest and largest elements of L, respectively. For K ⊆ L, we use ⋁K and ⋀K to denote the supremum and infimum of K, respectively. We say that K is directed subset of L if it is nonempty and each finite subset of K has an upper bound in K. In order to facilitate the writing, we usually use k = ⋁ ^↑K to express that the supremum of directed set K is k.

Definition 2.1 ([2]). Let L be a partial ordered set. For any a, b ∈ L, we say that a is way below b in L (in symbols, a ⪡ b), if for each directed subset K ⊆ L such that ⋁^↑K exists, b ⩽ ⋁ ^↑K always implies the existence of some k ∈ K with a ⩽ k.

Throughout this paper, L denotes a completely distributive lattice with an order-reversing involution ^′, unless otherwise stated. For a nonempty set X, L^X denotes the family of all L-sets on X. Obviously, L^X is also a completely distributive lattice with an order-reversing involution ^′ under the pointwise order. The two L-sets ⊥_{L
^X} and ⊤_{L
^X} are smallest and largest elements of L^X, respectively. We say that an L-set A is finite if its support set SuppA = {x ∈ X : A (x) ≠ ⊥ _L} is finite. We use $L_{fin}^{X}$ denote the family of all finite L-sets on X. The way below relation on L^X is also denoted by “⪡”. Obviously, for any A, B ∈ L^X, A ⪡ B iff $A \in L_{fin}^{X}$ and A (x) ⪡ B (x) for all x ∈ SuppA.

An element a ∈ L is called a co-prime element in L provided that a ⩽ b ∨ c implies a ⩽ b or a ⩽ c for all b, c ∈ L. We denote the family of all non-zero co-prime elements of L (resp., L^X) by J (L) (resp., by J (L^X)). Clearly, x_λ ∈ J (L^X) iff x ∈ X and λ ∈ J (L). For any a ∈ L, there exists the greatest minimal family of a, denoted by β (a). Moreover, we obtain that β^* (a) = β (a) ∩ J (L) is still a minimal family of a [27]. In particular, for any b ∈ J (L), b ∈ β^* (a) iff b ⪡ a.

For each x ∈ X and λ ∈ L, the L-set x_λ, defined by $\forall y \in X, x_{λ} (y) = {\begin{matrix} λ, & y = x; \\ ⊥_{L}, & y \neq x . \end{matrix}$ is called an L-fuzzy point of X.

For a ∈ L and A ∈ L^X, we use the notation A_[a] = {x ∈ X ∣ A (x) ⩾ a}.

Let X and Y be two nonempty sets. For a mapping f : X ⟶ Y, we define $f_{L}^{\to} : L^{X} ⟶ L^{Y}$ and $f_{L}^{\leftarrow} : L^{Y} ⟶ L^{X}$ as follows: $\forall A \in L^{X}, y \in Y, f_{L}^{\to} (A) (y) = ⋁_{f (x) = y} A (x)$ ;

and $\forall B \in L^{Y}, x \in X, f_{L}^{\leftarrow} (B) (x) = B (f (x))$ .

In [3, 21], the notion of L-fuzzy natural numbers was introduced. It is an extension of classical natural number.

Definition 2.2 ([3, 21]). Let $ℕ$ be the set of all natural numbers. An antitone mapping $n : ℕ ⟶ L$ is called an L-fuzzy natural number provided that n (0) = ⊤ _L and $⋀_{n \in ℕ} n (n) = ⊥_{L}$ .

We use $ℕ (L)$ to represent the set of all L-fuzzy natural numbers.

Proposition 2.3 ([3]). For each A ∈ L^X, define a mapping $| A | : ℕ ⟶ L$ as follows: $| A | (n) = ⋁_{| A_{[λ]} | ⩾ n} λ, {where n \in ℕ, λ \in J (L)$ .

Obviously, $| A | \in ℕ (L)$ . We call it the cardinality of A.

For each L-fuzzy natural number n, we write F ⪡ _nA for F ⪡ A, |F| ⩽ n. That is, F ⪡ _nA in L^X iff |F| ⩽ n and F (x) ⪡ A (x) in L for all x ∈ X.

Definition 2.4 ([27]). Let A ∈ L^X and ∅ ≠ Y ∈ 2^X. The L-set A|_Y ∈ L^Y defined by (A|_Y) (y) = A (y) for all y ∈ Y, is called the restriction of A to Y.

Proposition 2.5 ([27]). For each {A_t} _t∈T ⊆ L^X, A ∈ L^X, ∅ ≠ Y ∈ 2^X. We have the following results:

(⋁ _t∈TA_t) |_Y = ⋁ _t∈T (A_t|_Y).

(⋀ _t∈TA_t) |_Y = ⋀ _t∈T (A_t|_Y).

A′|_Y = (A|_Y) ′.

Definition 2.6 ([27]). Let ∅ ≠ Y ∈ 2^X and A ∈ L^Y . Define two L-sets A^*, A^★ ∈ L^X as follows: $\forall x \in X, A^{*} (x) = {\begin{matrix} A (x), & x \in Y; \\ ⊥_{L}, & x \notin Y, \end{matrix}$ $\forall x \in X, A^{★} (x) = {\begin{matrix} A (x), & x \in Y; \\ ⊤_{L}, & x \notin Y . \end{matrix}$

Clearly, x_λ ∈ J (L^Y) implies $x_{λ}^{*} \in J (L^{X})$ . In particular, A^*|_Y = A and A^★|_Y = A.

In [20], the authors extended the notion of classical convex structures to the notion of L-convex structures as follows.

Definition 2.7 ([20]). A subset $C$ of L^X is called an L-convex structure (or, L-convexity) on X if it fulfills the following assertions:

$⊥_{L^{X}}, ⊤_{L^{X}} \in C$ .

If ${B_{i}}_{i \in I} \subseteq C$ is nonempty, then $⋀_{i \in I} B_{i} \in C$ .

If ${B_{j}}_{j \in J} \subseteq C$ is nonempty and totally ordered, then $⋁_{j \in J} B_{j} \in C$ .

For an L-convex structure on X, we say that the pair $(X, C)$ is an L-convex space. In this case, the elements in $C$ are called L-convex sets of X.

Definition 2.8 ([40]). Let $(X, C)$ be an L-convex space. An L-set H of X is called an L-biconvex set provided that H and H′ are L-convex sets of X.

Proposition 2.9 ([40]). Let $(X, C)$ be an L-convex space and ∅ ≠ Y ∈ 2^X. Then $C |_{Y} = {B |_{Y} : B \in C}$ is an L-convex structure on Y. In this case, we say that $(Y, C |_{Y})$ is a subspace of $(X, C)$ .

Definition 2.10. The pair $(Y, C |_{Y})$ is called an L-convex subspace of the L-convex space $(X, C)$ provided that $χ_{Y} \in C$ .

Definition 2.11. ([19]). Let $(X, C)$ be an L-convex space. For each A ∈ L^X, define ${co}_{X} (A) = ⋀_{A ⩽ B \in C} B .$ We called co_X (A) is the L-convex hull (or, L-hull) of A.

Definition 2.12 ([15 , 40]) Let $f : (X, C) ⟶ (Y, D)$ be a mapping between L-convex spaces. Then

f is called L-convex preserving (or, L-CP) if $D \in D$ implies $f_{L}^{\leftarrow} (D) \in C$ .

f is called L-convex-to-convex (or, L-CC) if $C \in C$ implies $f_{L}^{\to} (C) \in D$ .

f is called L-isomorphic (or, an L-isomorphism) if f is bijective, L-CP and L-CC.

f is called L-embedding if $f |^{f^{\to} (X)} : (X, C) ⟶ (f^{\to} (X), D |_{f^{\to} (X)})$ is an L-isomorphism.

3 The sum of L-convex spaces

In this section, we will establish the relationships between the sum of a family of L-convex spaces and its factor spaces. These results will be useful in the following sections. First of all, we give a lemma to show that a family of L-convex structures can induce a new L-convex structure with respect to a mapping family. Lemma 3.1. Let ${(X_{t}, C_{t})}_{t \in T}$ be a family of L-convex spaces and X be a nonempty set. If f_t : X_t ⟶ X is a mapping for all t ∈ T, then $C \subseteq L^{X}$ defined by $C = {A \in L^{X} : \forall t \in T, (f_{t})_{L}^{\leftarrow} (A) \in C_{t}}$

is an L-convex structure on X.

Proof. It suffices to verify that $C$ satisfies (LC1)-(LC3). In fact,

(LC1) For each t ∈ T and x ∈ X_t, we have $(f_{t})_{L}^{\leftarrow} (⊥_{L^{X}}) (x) = ⊥_{L^{X}} (f_{t} (x)) = ⊥_{L^{X_{t}}} (x)$ and $(f_{t})_{L}^{\leftarrow} (⊤_{L^{X}}) (x) = ⊤_{L^{X}} (f_{t} (x)) = ⊤_{L^{X_{t}}} (x) .$

It implies that $(f_{t})_{L}^{\leftarrow} (⊥_{L^{X}}) = ⊥_{L^{X_{t}}} \in C_{t}$ and $(f_{t})_{L}^{\leftarrow} (⊤_{L^{X}}) = ⊤_{L^{X_{t}}} \in C_{t}$ . This shows that $⊥_{L^{X}}, ⊤_{L^{X}} \in C$ .

(LC2) Let ${A_{j}}_{j \in J} \subseteq C$ . Then $A_{j} \in C$ for all j ∈ J. It implies that $(f_{t})_{L}^{\leftarrow} (A_{j}) \in C_{t}$ for all t ∈ T. It follows that $\forall t \in T, (f_{t})_{L}^{\leftarrow} (⋀_{j \in J} A_{j}) = ⋀_{j \in J} (f_{t})_{L}^{\leftarrow} (A_{j}) \in C_{t},$

and thus $⋀_{j \in J} A_{j} \in C$ .

(LC3) Let {A_k} _k∈K be a totally ordered subset of $C$ . Then $A_{k} \in C$ for all k ∈ K. It implies that $(f_{t})_{L}^{\leftarrow} (A_{k}) \in C_{k}$ for all t ∈ T. Obviously, ${(f_{t})_{L}^{\leftarrow} (A_{k})}_{k \in K}$ is totally ordered. It follows that $\forall t \in T, (f_{t})_{L}^{\leftarrow} (⋁_{k \in K} A_{k}) = ⋁_{k \in K} (f_{t})_{L}^{\leftarrow} (A_{k}) \in C_{t} .$ Hence $⋁_{k \in K} A_{k} \in C$ . This shows that $C$ is an L-convex structure on X. □

Based on the above lemma, we introduce the concept of the sum of a family of L-convex spaces.

Definition 3.2. Let ${(X_{t}, C_{t})}_{t \in T}$ be a family of pairwise disjoint L-convex spaces, i.e., X_{t
₁}∩ X_{t
₂} = ∅ for t₁ ≠ t₂. Consider the set X = ⋃ _t∈TX_t and ∀t ∈ T, j_t : X_t ⟶ X is the usual inclusion mapping (i.e., ∀x ∈ X_t, j_t (x) = x). A subset $C$ of L^X defined by $C = {A \in L^{X} : \forall t \in T, (j_{t})_{L}^{\leftarrow} (A) \in C_{t}} .$

Then $C$ is called the L-sum convex structure of ${C_{t}}_{t \in T}$ and denoted by $\sum_{t \in T} C_{t}$ , briefly $\sum C_{t}$ . The L-convex space $(X, \sum C_{t})$ is called the L-sum convex space of ${(X_{t}, C_{t})}_{t \in T}$ , written as $\sum_{t \in T} (X_{t}, C_{t})$ and briefly $\sum (X_{t}, C_{t})$ .

Example 3.3. Let K be the set of all positive integers. Suppose that X_k = {k} (k ∈ K), L = [0, 1] and $C_{k} = {k_{λ} : λ \in [0, 1]}$ (where k_λ is an L-fuzzy point of X_k). Obviously, X_{k
₁}∩ X_{k
₂} = ∅ for k₁ ≠ k₂, and $(X_{k}, C_{k})$ is an L-convex space. Define $C = {A \in L^{X} : \forall k \in K, (j_{k})_{L}^{\leftarrow} (A) \in C_{k}}$ , where X = ⋃ _k∈KX_k and j_k : X_k ⟶ X is the usual inclusion mapping for all k ∈ K. By Lemma 3.1, we know that $C$ is L-convex structure on X and $C = \sum C_{k}$ .

Remark 3.4. The preceding definition requires that X_t (t ∈ T) must be disjoint. In fact, this requirement will not limit us seriously. Let ${(X_{t}, C_{t})}_{t \in T}$ be a family of L-convex spaces. Suppose Y_t = X_t × {t} for all t ∈ T. Then Y_t∩ Y_s = ∅ for t ≠ s. For each t ∈ T, obviously, the usual mapping f_t : Y_t ⟶ X_t, (x, t) ↦ x is bijective. Define $D_{t} \subseteq L^{Y_{t}}$ with $D_{t} = {B \times {t} : B \in C_{t}}$ where (B × {t}) (x, t) = B (x) for all x ∈ X_t. One can readily verify that $D_{t}$ is an L-convex structure on Y_t and $f_{t} : (Y_{t}, D_{t}) ⟶ (X_{t}, C_{t})$ is an L-isomorphism. Therefore, there is no difference between $(Y_{t}, D_{t})$ and $(X_{t}, C_{t})$ from the point of view of isomorphism, and we can define the sum of any family of L-convex spaces (up to an L-isomorphism). For convenience, we still use Definition 3.2 to study the related problems in later discussions.

The following theorem shows the close relationships between the sum of a family of L-convex spaces and its factor spaces.

Theorem 3.5. Let $(X, C) = \sum (X_{t}, C_{t})$ . Then we have the following assertions:

$C = {A \in L^{X} : \forall t \in T, A |_{X_{t}} \in C_{t}}$ .

$C = {A \in L^{X} : \forall t \in T, \exists A_{t} \in C_{t}, A = ⋁_{t \in T} A_{t}^{*}}$ .

For each t ∈ T, if $A_{t} \in C_{t}$ , then $A_{t}^{*} \in C$ .

$C$ is the unique L-convex structure on X with the following properties:

(i) For each t ∈ T, $(X_{t}, C_{t})$ is a subspace of $(X, C)$ , i.e., $C |_{X_{t}} = C_{t}$ .

(ii) For each t ∈ T, if $A_{t} \in C_{t}$ , then $A_{t}^{★} \in C$ .

Proof. (1) For each A ∈ L^X, since $\forall t \in T, (j_{t})_{L}^{\leftarrow} (A) (x) = A (j_{t} (x)) = (A |_{X_{t}}) (x)$

for all x ∈ X_t, we obtain $(j_{t})_{L}^{\leftarrow} (A) = A |_{X_{t}}$ . From Definition 3.2, we know that the equality holds.

(2) Suppose $A \in C$ . Then by (1), it follows that $A |_{X_{t}} \in C_{t}$ for all t ∈ T. For convenience, we denote $D = {A \in L^{X} : \forall t \in T, \exists A_{t} \in C_{t}, A = ⋁_{t \in T} A_{t}^{*}}$ . For each x ∈ X = ⋃ _t∈TX_t, there exists s ∈ T such that x ∈ X_s and x ∉ X_t (t ≠ s). Let A_t = A|_{X
_t} for all t ∈ T. Then $\begin{matrix} (⋁_{t \in T} A_{t}^{*}) (x) & = & (⋁_{t \in T, t \neq s} A_{t}^{*} (x)) \lor A_{s}^{*} (x) \\ = & A_{s} (x) \\ = & (A |_{X_{s}}) (x) \\ = & A (x) \end{matrix}$

Conversely, if $A \in D$ , then there exists $A_{t} \in C_{t}$ such that $A = ⋁_{t \in T} A_{t}^{*}$ for all t ∈ T. For each x ∈ X_t, we have x ∉ X_s (s ≠ t). Since $\begin{matrix} (A |_{X_{t}}) (x) & = & (⋁_{t \in T} A_{t}^{*} |_{X_{t}}) (x) \\ = & ((⋁_{t \in T} A_{t}^{*}) |_{X_{t}}) (x) \\ = & (⋁_{t \in T} A_{t}^{*}) (x) \\ = & A_{t} (x) \end{matrix}$ (3) Suppose $A_{t} \in C_{t}$ for all t ∈ T. Then $A_{t}^{*} |_{X_{r}} = {\begin{matrix} A_{r} \in C_{r}, & r = t; \\ ⊥_{L^{X_{r}}}, & r \neq t . \end{matrix}$

for all r ∈ T. By (1), we obtain $A_{t}^{*} \in C$ .

(4) We first prove that $C$ satisfies properties (i) and (ii). In fact,

(i) For each s ∈ T, we need to show that $C |_{X_{s}} = C_{s}$ . Let $A_{s} \in C |_{X_{s}}$ . Then there exists $A \in C$ such that A_s = A|_{X
_s}. By (1), we can obtain $A_{s} \in C_{s}$ . It implies $C |_{X_{s}} \subseteq C_{s}$ . Conversely, suppose $A_{s} \in C_{s}$ . For each t ≠ s, take $A_{t} = ⊥_{L^{X_{t}}} \in C_{t}$ and $A = ⋁_{t \in T} A_{t}^{*}$ , we have $A \in C$ . It follows that $A_{s} = A |_{X_{s}} \in C |_{X_{s}}$ , i.e., $C_{s} \subseteq C |_{X_{s}}$ . Hence $C |_{X_{t}} = C_{t}$ for all t ∈ T.

(ii) For each s ∈ T, we have $A_{t}^{★} |_{X_{s}} = {\begin{matrix} A_{s} \in C_{s}, & s = t; \\ ⊤_{L^{X_{s}}} \in C_{s}, & s \neq t . \end{matrix}$

On account of (1), one can easily to see that $A_{t}^{★} \in C$ for all t ∈ T.

Next, we show that $C$ is the only L-convex structure on X satisfying conditions (i) and (ii). Suppose that $D$ is an arbitrary L-convex structure on X with properties (i) and (ii). For each $A \in D$ , note that $D |_{X_{t}} = C_{t}$ for all t ∈ T, we obtain that $A |_{X_{t}} \in C_{t}$ , thus $A \in C$ , and hence $D \subseteq C$ . Conversely, suppose $A \in C$ . Then $A_{t} = A |_{X_{t}} \in C_{t}$ for all t ∈ T. On account of (ii), we obtain $A_{t}^{★} \in D$ . For each x ∈ X, there exists s ∈ T such that x ∈ X_s and x ∉ X_t (t ≠ s). It follows that $A (x) = (A |_{X_{s}}) (x) = A_{s}^{★} (x) = (⋀_{t \in T} A_{t}^{★}) (x)$ ,

and thus $A = ⋀_{t \in T} A_{t}^{★} \in D$ . This shows that $C \subseteq D$ , as desired.□

Proposition 3.6. Let $(X, C) = \sum (X_{t}, C_{t})$ . Then we have the following assertions:

$C$ is the finest L-convex structure on X such that j_t is L-CP for all t ∈ T.

For each L-convex space $(Y, D)$ , the mapping $g : (X, C) ⟶ (Y, D)$ is L-CP if and only if $g \circ j_{t} : (X_{t}, C_{t}) ⟶ (Y, D)$ is L-CP for all t ∈ T.

The mapping $j_{t} : (X_{t}, C_{t}) ⟶ (X, C)$ is L-embedding for all t ∈ T.

Let $(Y, D)$ be an L-convex space and ${g_{t} : (X_{t}, C_{t}) ⟶ (Y, D)}$ be a family of L-CP mappings. Then there exists an L-CP mapping $h : (X, C) ⟶ (Y, D)$ such that h ∘ j_t = g_t for all t ∈ T.

Proof. The verifications of (1) and (2) are straightforward.

(3) We need to show that $j_{t} |^{j_{t}^{\to} (X_{t})} : (X_{t}, C_{t}) ⟶ (j_{t}^{\to} (X_{t}), C |_{j_{t}^{\to} (X_{t})})$

is an L-isomorphism. Note that $j_{t}^{\to} (X_{t}) = X_{t}$ , $C |_{X_{t}} = C_{t}$ and ${id}_{X_{t}} : (X_{t}, C_{t}) ⟶ (X_{t}, C_{t})$ is an L-isomorphism, so we obtain that $j_{t} |^{j_{t}^{\to} (X_{t})} = {id}_{X_{t}}$ is an L-isomorphism. Hence $j_{t} : (X_{t}, C_{t}) ⟶ (X, C)$ is L-embedding for all t ∈ T.

(4) For each x ∈ X = ⋃ _t∈TX_t, there exists t ∈ T such that x ∈ X_t. We define h (x) = g_t (x). Obviously, h ∘ j_t = g_t for all t ∈ T. Since $g_{t} : (X_{t}, C_{t}) ⟶ (Y, D)$ is L-CP for all t ∈ T, it follows that $(g_{t})_{L}^{\leftarrow} (D) \in C_{t}$ for all $D \in D$ . Thus, we have $(h \circ j_{t})_{L}^{\leftarrow} (D) = (j_{t})_{L}^{\leftarrow} (h_{L}^{\leftarrow} (D)) = (g_{t})_{L}^{\leftarrow} (D) \in C_{t} .$

By Definition 3.2, we obtain $h_{L}^{\leftarrow} (D) \in C$ . Hence $h : (X, C) ⟶ (Y, D)$ is an L-CP mapping. □

Proposition 3.7. Let $(X, C) = \sum (X_{t}, C_{t})$ . For each t ∈ T, if Y_t ⊆ X_t and Y = ⋃ _t∈TY_t, then $C |_{Y} = \sum_{t \in T} (C_{t} |_{Y_{t}})$ .

Proof. Let j_t : X_t ⟶ X = ⋃ _t∈TX_t be the usual inclusion mapping and j_t|_{Y
_t} : Y_t ⟶ Y = ⋃ _t∈TY_t be the restriction of j_t to Y_t for all t ∈ T. Then from Definition 3.2, we obtain $\sum_{t \in T} (C_{t} |_{Y_{t}}) = {B \in L^{Y} : \forall t \in T, (j_{t} |_{Y_{t}})_{L}^{\leftarrow} (B) \in C_{t} |_{Y_{t}}}$ .

We shall first prove $C |_{Y} \subseteq \sum_{t \in T} (C_{t} |_{Y_{t}})$ . Suppose $A \in C |_{Y}$ . Then there exists $C \in C$ such that A = C|_Y. Since $\begin{matrix} (j_{t} |_{Y_{t}})_{L}^{\leftarrow} (C |_{Y}) (y) & = & (C |_{Y}) ((j_{t} |_{Y_{t}}) (y)) \\ = & (C |_{Y}) (y) \\ = & C (j_{t} (y)) \\ = & (j_{t})_{L}^{\leftarrow} (C) (y) \\ = & ((j_{t})_{L}^{\leftarrow} (C) |_{Y_{t}}) (y) \end{matrix}$ for all y ∈ Y_t, it follows that $(j_{t} |_{Y_{t}})_{L}^{\leftarrow} (C |_{Y}) = (j_{t})_{L}^{\leftarrow} (C) |_{Y_{t}}$ . By Definition 3.2, we obtain that $(j_{t})_{L}^{\leftarrow} (C) \in C_{t}$ for all t ∈ T, thus $(j_{t})_{L}^{\leftarrow} (C) |_{Y_{t}} \in C_{t} |_{Y_{t}}$ , and hence $A = C |_{Y} \in \sum_{t \in T} (C_{t} |_{Y_{t}})$ . It implies that $C |_{Y} \subseteq \sum_{t \in T} (C_{t} |_{Y_{t}})$ .

Next, we need to show that $\sum_{t \in T} (C_{t} |_{Y_{t}}) \subseteq C |_{Y}$ . Suppose $B \in \sum_{t \in T} (C_{t} |_{Y_{t}})$ . Then $(j_{t} |_{Y_{t}})_{L}^{\leftarrow} (B) \in C_{t} |_{Y_{t}}$ for all t ∈ T. This implies that there exists $A_{t} \in C_{t}$ such that $(j_{t} |_{Y_{t}})_{L}^{\leftarrow} (B) = A_{t} |_{Y_{t}}$ . Let $A = ⋁_{t \in T} A_{t}^{*}$ . By Theorem 3.5(2), we obtain $A \in C$ . For each y ∈ Y, there exists s ∈ T such that y ∈ Y_s and y ∉ Y_r (r ≠ s). It follows that $(A |_{Y}) (y) = ((⋁_{t \in T} A_{t}^{*}) |_{Y}) (y) = A_{s} (y) .$ Note that $(j_{t} |_{Y_{t}})_{L}^{\leftarrow} (B) = A_{t} |_{Y_{t}}$ , so we obtain $\begin{matrix} A_{s} (y) & = & (A_{s} |_{Y_{s}}) (y) \\ = & (j_{s} |_{Y_{s}})_{L}^{\leftarrow} (B) (y) \\ = & B ((j_{s} |_{Y_{s}}) (y)) \\ = & B (j_{s} (y)) \\ = & B (y) \end{matrix}$

This shows B = A|_Y. Therefore, $B \in C |_{Y}$ , and thus $\sum_{t \in T} (C_{t} |_{Y_{t}}) \subseteq C |_{Y}$ . This completes the proof.□

4 The additivity of separability

In the classical convex space theory, separability is a basic and important property. Many profound results are obtained from convex spaces with some separability [26]. In this section, we first study the additivity of some low-level separability (S_-1, sub-S₀, S₀, S₁ and S₂) of L-convex spaces mentioned in [40], and then discuss the heredity and additivity of high-level separability (S₃ and S₄). Further, we give some examples to illustrate the differences and relationships among these separability.

Definition 4.1. ([40]) An L-convex space $(X, C)$ is called

S_-1, if for each x_λ, x_μ ∈ J (L^X) with μ ≰ λ, there exists $A \in C$ such that x_μ ≰ A, x_λ ⩽ A.

sub-S₀, if for each x, y ∈ X with x ≠ y, there exists λ ∈ J (L), $A \in C$ such that x_λ ≰ A, y_λ ⩽ A or $B \in C$ such that y_λ ≰ B, x_λ ⩽ B.

S₀, if for each x_λ, y_μ ∈ J (L^X) with x_λ ≠ y_μ, there exists $A \in C$ such that x_λ ≰ A, y_μ ⩽ A or $B \in C$ such that y_μ ≰ B, x_λ ⩽ B.

S₁, if for each $x_{λ} \in J (L^{X}), x_{λ} \in C$ .

S₂, if for each x_λ, y_μ ∈ J (L^X) with x_λ ⩽ (y_μ) ′, there exists an L-biconvex set H such that x_λ ⩽ H ⩽ (y_μ) ′.

Remark 4.2. In [40], the authors discussed the above-mentioned separability of L-convex spaces in detail. From the paper, we know that S_-1, sub-S₀, S₀, S₁ and S₂ separability are hereditary.

Theorem 4.3. Let $(X, C) = \sum (X_{t}, C_{t})$ . Then we have the following assertions:

$(X, C)$ is S_-1 iff $(X_{t}, C_{t})$ is S_-1 for all t ∈ T.

$(X, C)$ is sub-S₀ iff $(X_{t}, C_{t})$ is sub-S₀ for all t ∈ T.

$(X, C)$ is S₀ iff $(X_{t}, C_{t})$ is S₀ for all t ∈ T.

$(X, C)$ is S₁ iff $(X_{t}, C_{t})$ is S₁ for all t ∈ T.

$(X, C)$ is S₂ iff $(X_{t}, C_{t})$ is S₂ for all t ∈ T.

Proof. The necessity of (1)–(5) are immediate consequences of Theorem 3.5(4) and Remark 4.2. We only prove sufficiency.

(1) Suppose $(X_{t}, C_{t})$ is S_-1 for all t ∈ T. For each x_λ, x_μ ∈ J (L^X) with μ ≰ λ, we have x ∈ X and λ, μ ∈ J (L). Then there exists r ∈ T such that x ∈ X_r and x ∉ X_s (s ≠ t). Since $(X_{r}, C_{r})$ is S_-1, there exists $A_{r} \in C_{r}$ such that x_μ ≰ A_r and x_λ ⩽ A_r. It implies that there exists $B \in C$ such that A_r = B|_{X
_r}. Note that x ∈ X_r, so we obtain that x_μ ≰ B and x_λ ⩽ B. This shows that $(X, C)$ is S_-1.

(2) Let x, y ∈ X with x ≠ y. Then there exists r, s ∈ T such that x ∈ X_r and y ∈ X_s. We consider the following two cases:

Case 1: r = s, x, y ∈ X_r. Since $(X_{r}, C_{r})$ is sub-S₀, there exists λ ∈ J (L), $A_{r} \in C_{r}$ such that x_λ ≰ A_r, y_λ ⩽ A_r or $B_{r} \in C_{r}$ such that y_λ ≰ B_r, x_λ ⩽ B_r. Let $A = A_{r}^{*}$ and $B = B_{r}^{*}$ . By Theorem 3.5(3), we obtain $A, B \in C$ and $x_{λ} = x_{λ}^{*} ≰ A_{r}^{*} = A$ , $y_{λ} = y_{λ}^{*} ⩽ A_{r}^{*} = A$ or $y_{λ} = y_{λ}^{*} ≰ B_{r}^{*} = B$ , $x_{λ} = x_{λ}^{*} ⩽ B_{r}^{*} = B$ .

Thus, x_λ ≰ A, y_λ ⩽ A or y_λ ≰ B, x_λ ⩽ B. It implies that $(X, C)$ is sub-S₀.

Case 2: r ≠ s, x ∈ X_r and y ∈ X_s. Suppose λ ∈ J (L) and $A_{r} \in C_{r}$ with x_λ ≰ A_r. Let $A = A_{r}^{*} \lor ⊤_{L^{X_{s}}}^{*}$ . Then we obtain $A |_{X_{t}} = (A_{r}^{*} \lor ⊤_{L^{X_{s}}}^{*}) |_{X_{t}} = {\begin{matrix} A_{t} \in C_{t}, & t = r \neq s; \\ ⊤_{L^{X_{t}}} \in C_{t}, & r \neq s = t; \\ ⊥_{L^{X_{t}}} \in C_{t}, & r \neq s \neq t . \end{matrix}$ for all t ∈ T. This shows that $A \in C$ . One can readily obtain that x_λ ≰ A and y_λ ⩽ A. If there exists $B_{s} \in C_{s}$ with y_λ ≰ B_s, then $B = B_{s}^{*} \lor ⊤_{L^{X_{r}}}^{*} \in C$ , thus y_λ ≰ B and x_λ ⩽ B. Hence $(X, C)$ is sub-S₀.

(3) Let x_λ, y_μ ∈ J (L^X) with x_λ ≠ y_μ. Then there exists r, s ∈ T such that x ∈ X_r and y ∈ X_s. We consider the following two cases:

Case 1: r = s, x, y ∈ X_r. Since $(X_{r}, C_{r})$ is sub-S₀, there exists $A_{r} \in C_{r}$ such that x_λ|_{X
_r} = x_λ ≰ A_r, y_μ|_{X
_r} = y_μ ⩽ A_r or exists $B_{r} \in C_{r}$ such that y_μ|_{X
_r} = y_μ ≰ B_r, x_λ|_{X
_r} = x_λ ⩽ B_r.

Let $A = A_{r}^{*}, B = B_{r}^{*}$ . Then $A, B \in C$ . Note that $(x_{λ} |_{X_{r}})^{*} = x_{λ}^{*} |_{X_{r}} = x_{λ}, (y_{μ} |_{X_{r}})^{*} = y_{μ}^{*} |_{X_{r}} = y_{μ}$ , so we can obtain x_λ ≰ A, y_μ ⩽ A or y_μ ≰ B, x_λ ⩽ B. It implies that $(X, C)$ is S₀.

Case 2: r ≠ s, x ∈ X_r and y ∈ X_s. Obviously, x_λ = x_λ|_{X
_r} ∈ J (L^{X
_r}) and y_μ = y_μ|_{X
_s} ∈ J (L^{X
_s}). Suppose $A_{r} \in C_{r}$ with x_λ|_{X
_r} ≰ A_r. Let $A = A_{r}^{*} \lor ⊤_{L^{X_{s}}}^{*}$ . Then $A \in C$ . It is easy to check that x_λ ≰ A and y_μ ⩽ A. If there exists $B_{s} \in C_{s}$ with y_μ|_{X
_s} ≰ B_s, then $B = B_{s}^{*} \lor ⊤_{L^{X_{r}}}^{*} \in C$ , thus y_μ ≰ B and x_λ ⩽ B. Hence $(X, C)$ is S₀.

(4) For each x_λ ∈ J (L^X), there exists r ∈ T such that x ∈ X_r and x ∉ X_s (s ≠ r), thus x_λ = x_λ|_{X
_r} ∈ J (L^{X
_r}). Since $(X_{r}, C_{r})$ is S₁, we obtain $x_{λ} |_{X_{r}} \in C_{r}$ . Note that $x_{λ} |_{X_{s}} = ⊥_{L^{X_{s}}} \in C_{s} (s \neq r)$ , so we have $x_{λ} |_{X_{t}} \in C_{t}$ for all t ∈ T. It implies that $x_{λ} \in C$ . Hence $(X, C)$ is S₁.

(5) Let x_λ, y_μ ∈ J (L^X) with x_λ ⩽ (y_μ) ′. Then x ≠ y or x = y, λ ⩽ μ′. Obviously, there exists r, s ∈ T such that x ∈ X_r, y ∈ X_s. We consider the following two cases:

Case 1: x ≠ y. Suppose r = s and x, y ∈ X_r. Since $(X_{r}, C_{r})$ is S₂, there exists an L-biconvex set H_r of X_r such that x_λ = x_λ|_{X
_r} ⩽ H_r ⩽ (y_μ|_{X
_r}) ′ = (y_μ) ′ .

Let $H = H_{r}^{*}$ . Then H is an L-biconvex set of X. In fact, $H_{r}^{*} \in C$ is obvious. Since $(H_{r}^{*})^{'} |_{X_{r}} = (H_{r}^{*} |_{X_{r}})^{'} = H_{r}^{'} \in C_{r}$ , we know that $(H_{r}^{*})^{'} \in C$ from Theorem 3.5(1). It follows that $x_{λ} = x_{λ}^{*} = (x_{λ} |_{X_{r}})^{*} ⩽ H_{r}^{*} ⩽ ((y_{μ} |_{X_{r}})^{'})^{*} = (y_{μ})^{'} .$

This shows that $(X, C)$ is S₂.

Suppose r ≠ s and x ∈ X_r, y ∈ X_s. Let H_r be an L-biconvex set of X_r with x_λ ⩽ H_r. Then x_λ|_{X
_r} ⩽ H_r, and thus $x_{λ} = x_{λ}^{*} = (x_{λ} |_{X_{r}})^{*} ⩽ H_{r}^{*}$ . Note that $H_{r}^{*} ⩽ ⊥_{L^{X_{s}}}^{★} ⩽ (y_{μ})^{'}$ and $H_{r}^{*}$ is an L-biconvex set of X, so we have $x_{λ} ⩽ H_{r}^{*} ⩽ (y_{μ})^{'}$ . Hence $(X, C)$ is S₂.

Case 2: x = y, λ ⩽ μ′. The verification is straightforward. □

Next, we will propose the notions of S₃ and S₄ separability in L-convex spaces and discuss their related properties.

Definition 4.4. An L-convex space $(X, C)$ is called

S₃, if for each $A \in C$ and x_λ ∈ J (L^X) with x_λ ⩽ A′, there exists an L-biconvex set H of X such that A ⩽ H ⩽ (x_λ) ′.

S₄, if for each $A, B \in C$ with A ⩽ B′, there exists an L-biconvex set H of X such that A ⩽ H ⩽ B′.

Example 4.5. (1) Let X be the set of all positive integers and L = [0, 1]. We define $C = {⊥_{L^{X}}, C_{1}, C_{2}, C_{3}, C_{4}, C_{5}, ⊤_{L^{X}}}$ , where $C_{1} (x) = {\begin{matrix} 1, & x = 1; \\ 0, & x ⩾ 2 . \end{matrix}, C_{2} (x) = {\begin{matrix} 0, & x = 1; \\ 1, & x ⩾ 2 . \end{matrix}$

$C_{3} (x) = {\begin{matrix} 0, & x = 1; \\ \frac{1}{2}, & x is even; \\ 1, & x ⩾ 3 and x is odd . \end{matrix}$

$C_{4} (x) = {\begin{matrix} 1, & x = 1; \\ \frac{1}{2}, & x is even; \\ 0, & x ⩾ 3 and x is odd . \end{matrix}$

$C_{5} (x) = {\begin{matrix} \frac{1}{2}, & x is even; \\ 0, & x is odd . \end{matrix}$

By the constructions of $C$ and L, we have J (L) = (0, 1] and the set of all L-biconvex sets H = {⊥ _{L
^X}, C₁, C₂, C₃, C₄, ⊤ _{L
^X}}. It is verify that $(X, C)$ is an S₃ L-convex space.

(2) Let X = {a, b} and L = {⊥ _L, p, q, ⊤ _L} (where p′ = q, q′ = p). The Hasse diagram of L is shown in the following figure:

Suppose $C = {⊥_{L^{X}}, C_{1}, C_{2}, C_{3}, C_{4}, C_{5}, C_{6}, C_{7}, C_{8}, ⊤_{L^{X}}}$ is defined as follows: $C_{1} (x) = {\begin{matrix} p, & x = a; \\ q, & x = b . \end{matrix}, C_{2} (x) = {\begin{matrix} q, & x = a; \\ p, & x = b . \end{matrix}$

$C_{3} (x) = {\begin{matrix} p, & x = a; \\ ⊥_{L}, & x = b . \end{matrix}, C_{4} (x) = {\begin{matrix} q, & x = a; \\ ⊥_{L}, & x = b . \end{matrix}$

$C_{5} (x) = {\begin{matrix} ⊥_{L}, & x = a; \\ p, & x = b . \end{matrix}, C_{6} (x) = {\begin{matrix} ⊥_{L}, & x = a; \\ q, & x = b . \end{matrix}$

$C_{7} (x) = {\begin{matrix} p, & x = a; \\ p, & x = b . \end{matrix}, C_{8} (x) = {\begin{matrix} q, & x = a; \\ q, & x = b . \end{matrix}$

Obviously, the set of all L-biconvex sets H = {⊥ _{L
^X}, C₁, C₂, C₇, C₈, ⊤ _{L
^X}} and J (L) = {p, q}. One can readily verify that $(X, C)$ is an S₄ L-convex space.

Remark 4.6. Based on Definition 4.4, we have the following assertions:

In general, S₃ ≱ S₂ and S₄ ≱ S₃.

If under the assumption of S₁, then S₄ ⇒ S₃ ⇒ S₂

We can illustrate the relationships in Remark 4.6(1) with the following examples.

Example 4.7. Let X = {a, b, c} and L = [0, 1]. We define $C = {⊥_{L^{X}}, C_{1}, C_{2}, C_{3}, C_{4}, ⊤_{L^{X}}}$ , where $C_{1} (x) = {\begin{matrix} 1, & x = a; \\ 0, & x = b; \\ 0, & x = c . \end{matrix}, C_{2} (x) = {\begin{matrix} 0, & x = a; \\ \frac{1}{2}, & x = b; \\ \frac{1}{2}, & x = c . \end{matrix}$

$C_{3} (x) = {\begin{matrix} 1, & x = a; \\ \frac{1}{2}, & x = b; \\ \frac{1}{2}, & x = c . \end{matrix}, C_{4} (x) = {\begin{matrix} 0, & x = a; \\ 1, & x = b; \\ 1, & x = c . \end{matrix}$

Obviously, the set of all L-biconvex sets $H = C$ and $(X, C)$ is an S₃ L-convex space. But it is not S₂. In fact, for $a_{\frac{1}{2}} ⩽ (a_{\frac{1}{3}})^{'}$ , we can not find an L-biconvex set H to satisfy $a_{\frac{1}{2}} ⩽ H ⩽ (a_{\frac{1}{3}})^{'}$ .

Example 4.8. Let X = {a, b, c} and L = [0, 1]. We define $C = {⊥_{L^{X}}, C_{1}, C_{2}, C_{3}, C_{4}, ⊤_{L^{X}}}$ , where $C_{1} (x) = {\begin{matrix} 1, & x = a; \\ 0, & x = b; \\ 0, & x = c . \end{matrix}, C_{2} (x) = {\begin{matrix} \frac{1}{2}, & x = a; \\ 0, & x = b; \\ 0, & x = c . \end{matrix}$

$C_{3} (x) = {\begin{matrix} 0, & x = a; \\ \frac{1}{2}, & x = b; \\ 0, & x = c . \end{matrix}, C_{4} (x) = {\begin{matrix} 0, & x = a; \\ 1, & x = b; \\ 1, & x = c . \end{matrix}$

It is verify that the set of all L-biconvex sets H = {⊥ _{L
^X}, C₁, C₄, ⊤ _{L
^X}} and $(X, C)$ is an S₄ L-convex space. But it is not S₃. In fact, for $a_{\frac{1}{3}} ⩽ C_{2}^{'}$ , we can not find an L-biconvex set H to satisfy $C_{2} ⩽ H ⩽ (a_{\frac{1}{3}})^{'}$ .

The following lemma is trivial and the proof is omitted here.

Lemma 4.9. Let $(X, C)$ be an L-convex space and $(Y, C |_{Y})$ a subspace of $(X, C)$ . If H is an L-biconvex set of X, then H|_Y is an L-biconvex set of Y.

Proposition 4.10. Let $(X, C)$ be an L-convex space and ∅ ≠ Y ∈ 2^X. If $(X, C)$ is S₃, then so is $(Y, C |_{Y})$ .

Proof. Let $A \in C |_{Y}$ and x_λ ∈ J (L^Y) with x_λ ⩽ A′. Then there exists $C \in C$ such that A = C|_Y and x_λ ⩽ (C|_Y) ′ = C′|_Y. Thus, we obtain $x_{λ} = x_{λ}^{*} = (x_{λ} |_{Y})^{*} ⩽ (C^{'} |_{Y})^{*} ⩽ C^{'} .$

Obviously, x_λ ∈ J (L^X). Since $(X, C)$ is S₃, there exists an L-biconvex set H of X such that C ⩽ H ⩽ (x_λ) ′. It implies that C|_Y ⩽ H|_Y ⩽ (x_λ) ′|_Y. Note that H|_Y is an L-biconvex set of Y, so we obtain A ⩽ H|_Y ⩽ (x_λ) ′. Hence $(Y, C |_{Y})$ is S₃. □

Proposition 4.11. Let $(X, C)$ be an L-convex space and $(Y, C |_{Y})$ be an L-convex subspace of $(X, C)$ . If $(X, C)$ is S₄, then so is $(Y, C |_{Y})$ .

Proof. Let $A, B \in C |_{Y}$ with A ⩽ B′. Then there exists $C, D \in C$ such that A = C|_Y and B = D|_Y. Note that C ∧ χ_Y = (C|_Y) ^*, so we obtain C ∧ χ_Y = A^* ⩽ (B′) ^* = (D′|_Y) ^* ⩽ D′ .

Obviously, $C \land χ_{Y} \in C$ . Since $(X, C)$ is S₄, there exists an L-biconvex set H of $(X, C)$ such that C ∧ χ_Y ⩽ H ⩽ D′.

It follows that A = C|_Y = (C ∧ χ_Y) |_Y ⩽ H|_Y ⩽ D′|_Y = B′ .

From Lemma 4.9, we know that H|_Y is an L-biconvex set of Y. Hence $(Y, C |_{Y})$ is S₄. □

Theorem 4.12. Let $(X, C) = \sum (X_{t}, C_{t})$ . Then $(X, C)$ is S₃ iff $(X_{t}, C_{t})$ is S₃ for all t ∈ T.

Proof. It is obvious that the necessity from Theorem 3.5(4) and Proposition 4.10. We need to prove sufficiency. Let $C \in C$ and x_λ ∈ J (L^X) with x_λ ⩽ C′. Then there exists r ∈ T such that x ∈ X_r, and thus x_λ = x_λ|_{X
_r} ⩽ C′|_{X
_r} = (C|_{X
_r}) ′ . By Theorem 3.5(1), we know that C|_{X
_r} is an L-convex set of $(X_{r}, C_{r})$ . Since $(X_{r}, C_{r})$ is S₃, there exists an L-biconvex set H_r of $(X_{r}, C_{r})$ such that C|_{X
_r} ⩽ H_r ⩽ (x_λ) ′(x_λ ∈ J (L^{X
_r})). It follows that $C ⩽ (C |_{X_{r}})^{★} ⩽ H_{r}^{★} ⩽ ((x_{λ})^{'})^{★} = (x_{λ})^{'} .$ Let $H = H_{r}^{★}$ . Then H is an L-biconvex set of $(X, C)$ . In fact, $H_{r}^{★} \in C$ from Theorem 3.5(4). Since $(H_{r}^{★})^{'} |_{X_{r}} = (H_{r}^{★} |_{X_{r}})^{'} = (H_{r})^{'} \in C_{r},$ we obtain that $(H_{r}^{★})^{'} \in C$ from Theorem 3.5(1). Hence $(X, C)$ is S₃. □

Theorem 4.13. Let $(X, C) = \sum (X_{t}, C_{t})$ . If $(X_{t}, C_{t})$ is S₄ for all t ∈ T, then $(X, C)$ is S₄.

Proof. Let $C, D \in C$ with C ⩽ D′. Note that D ∈ L^X, so we can obtain D = ⋁ _{x_λ⩽D}x_λ and x_λ ∈ J (L^X), and thus, D′ = (⋁ _{x_λ⩽D}x_λ) ′ = ⋁ _{x_λ⩽D} (x_λ) ′.

Obviously, there exists r ∈ T such that x ∈ X_r. By Theorem 3.5(4), we obtain that $C |_{X_{r}}, D |_{X_{r}} \in C_{r} and C |_{X_{r}} ⩽ D^{'} |_{X_{r}} = (D |_{X_{r}})^{'} .$ Since $(X_{r}, C_{r})$ is S₄, there exists an L-biconvex set H_r of $(X_{r}, C_{r})$ such that C|_{X
_r} ⩽ H_r ⩽ D′|_{X
_r} = ⋁ _{x_λ⩽D} (x_λ|_{X
_r}) ′ .

Note that x ∈ X_r, so we obtain x_λ|_{X
_r} = x_λ. Thus, C|_{X
_r} ⩽ H_r ⩽ ⋁ _{x_λ⩽D} (x_λ) ′ .

It implies that $C ⩽ (C |_{X_{r}})^{★} ⩽ H_{r}^{★} ⩽ (⋁_{x_{λ} ⩽ D} (x_{λ})^{'})^{★} = D^{'} .$

Similar to the proof in Theorem 4.12, we know that $H_{r}^{★}$ is an L-biconvex set of $(X, C)$ . Hence $(X, C)$ is S₄. □

5 The additivity of five kinds of special L-convex spaces

In this section, we will discuss the additivity of five types special L-convex spaces, which are defined by L-hull operators and called arity ⩽n, dense, CUP, JHC and weakly JHC, respectively. Firstly, we give a very useful lemma that will be used later.

Lemma 5.1. Let $(X, C) = \sum (X_{t}, C_{t})$ . Then

For each $A \in C$ , ∀x_λ ∈ J (L^X), x_λ ∈ β^* (A) iff x_λ ∈ β^* (A|_{X
_t}) for some t ∈ T.

For each A ∈ L^X, co_X (A) = ⋁ _t∈Tco_{X
_t} (A|_{X
_t}).

Proof (1) It is trivial.

(2) For each x ∈ X, there exists s ∈ T such that x ∈ X_s and x ∉ X_t (t ≠ s). Since $\begin{matrix} (j_{s})_{L}^{\leftarrow} (⋁_{t \in T} {co}_{X_{t}} (A |_{X_{t}})) (x) \\ = ⋁_{t \in T} {co}_{X_{t}} (A |_{X_{t}}) (x) \\ = {co}_{X_{s}} (A |_{X_{s}}) (x) \end{matrix}$ and ${co}_{X_{s}} (A |_{X_{s}}) \in C_{s}$ , we obtain $⋁_{t \in T} {co}_{X_{t}} (A |_{X_{t}}) \in C$ from Definition 3.2.

Let C be an L-convex set of $(X, C)$ with A ⩽ C. Then A|_{X
_t} ⩽ C|_{X
_t} for all t ∈ T. By Theorem 3.5(1), we can obtain that C|_{X
_t} is an L-convex set of $(X_{t}, C_{t})$ . It implies that co_{X
_t} (A|_{X
_t}) ⩽ C|_{X
_t}. Note that A = ⋁ _t∈T (A|_{X
_t}), so we obtain A ⩽ ⋁ _t∈Tco_{X
_t} (A|_{X
_t}) ⩽ ⋁ _t∈T (C|_{X
_t}) = C . Hence co_X (A) = ⋁ _t∈Tco_{X
_t} (A|_{X
_t}). □

Definition 5.2. ([1]) An L-convex space $(X, C)$ is called arity ⩽n, if its L-convex sets are precisely the L-subsets A with the property that co_X (F) ⩽ A for all F ⪡ _nA.

From the above definition in [1], we can know that an L-convex space $(X, C)$ is arity ⩽n iff $C = {A \in L^{X} : \forall F ⪡_{n} A {impiles {co}_{X} (F) ⩽ A}$ , where F ⪡ _nA iff |F| ⩽ n and F (x) ⪡ A (x) for all x ∈ X.

Lemma 5.3 ([20]) Let A ∈ L^X. Then the following assertions are equivalent:

F ⪡ _nA.

for each a ∈ L - {⊥ _L}, F_[a] ⪡ _{n
_[a]}A_[a] in 2^X.

Theorem 5.4. Let $(X, C) = \sum (X_{t}, C_{t})$ . If $(X_{t}, C_{t})$ is arity ⩽n_t for all t ∈ T, then $(X, C)$ is arity ⩽ ⋀ _t∈Tn_t.

Proof. Let A ∈ L^X with co_X (F) ⩽ A for all F ⪡ _{⋀_t∈Tn_t}A. We need to verity that A is an L-convex set of $(X, C)$ . By Lemma 5.3, we obtain F_[a] ⪡ _{(⋀_t∈Tn_t)_[a]}A_[a]

for all a ∈ L - {⊥ _L}. Note that (⋀ _t∈Tn_t) _[a] = ⋂ _t∈T (n_t) _[a], so we obtain F_[a] ⪡ _{(n_t)_[a]}A_[a] for all t ∈ T. This shows that F_[a] is finite and F_[a] ⊆ A_[a]. It follows that F_[a] ∩ X_t is finite and F_[a] ∩ X_t ⊆ A_[a] ∩ X_t for all t ∈ T, and thus (F|_{X
_t}) _[a] = F_[a] ∩ X_t ⪡ _{(n_t)_[a]}A_[a] ∩ X_t = (A|_{X
_t}) _[a] .

Hence (F|_{X
_t}) ⪡ _{n
_t} (A|_{X
_t}) for all t ∈ T. By Lemma 5.1(2) and co_X (F) ⩽ A, we have ⋁_t∈Tco_{X
_t} (F|_{X
_t}) ⩽ A . Furthermore, we obtain co_{X
_t} (F|_{X
_t}) = co_{X
_t} (F|_{X
_t}) |_{X
_t} ⩽ A|_{X
_t} .

Since $(X_{t}, C_{t})$ is arity ⩽n_t for all t ∈ T, we obtain that A|_{X
_t} is an L-convex set of $(X_{t}, C_{t})$ , i.e., $A |_{X_{t}} \in C_{t}$ . It implies $A \in C$ from Theorem 3.5(1). Hence $(X, C)$ is arity ⩽ ⋀ _t∈Tn_t. □

Corollary 5.5. Let $(X, C) = \sum (X_{t}, C_{t})$ . If $(X_{t}, C_{t})$ is arity ⩽n for all t ∈ T, then $(X, C)$ is arity ⩽n.

Definition 5.6 ([1]). An L-convex space $(X, C)$ is called dense, if for each A ∈ L^X, ${co}_{X} (A) = ⋁_{F ⪡ {co}_{X} (F) ⩽ A}^{↑} {co}_{X} (F)$

Theorem 5.7. Let $(X, C) = \sum (X_{t}, C_{t})$ . If $(X_{t}, C_{t})$ is dense for all t ∈ T (where T is finite), then $(X, C)$ is dense.

Proof. For each A ∈ L^X, we obtain co_X (A) = ⋁ _t∈Tco_{X
_t} (A|_{X
_t}) from Lemma 5.1(2). Since $(X_{t}, C_{t})$ is dense for all t ∈ T (T is finite), it follows that $\begin{matrix} {co}_{X} (A) \\ = ⋁_{t \in T} {co}_{X_{t}} (A |_{X_{t}}) \\ = ⋁_{t \in T} ⋁_{E_{t} ⪡ {co}_{X_{t}} (E_{t}) ⩽ (A |_{X_{t}})}^{↑} {co}_{X_{t}} (E_{t}) \\ ⩽ ⋁_{t \in T} ⋁_{⋁_{t \in T} E_{t} ⪡ ⋁_{t \in T} {co}_{X_{t}} (E_{t}) ⩽ ⋁_{t \in T} (A |_{X_{t}})}^{↑} {co}_{X_{t}} (E_{t}) \\ = ⋁_{t \in T} ⋁_{⋁_{t \in T} E_{t} ⪡ ⋁_{t \in T} {co}_{X_{t}} ((⋁_{s \in T} E_{s}) |_{X_{t}}) ⩽ A}^{↑} {co}_{X_{t}} ( \\ (⋁_{s \in T} E_{s}) |_{X_{t}}) \\ ⩽ ⋁_{t \in T} ⋁_{F ⪡ ⋁_{t \in T} {co}_{X_{t}} (F |_{X_{t}}) ⩽ A}^{↑} {co}_{X_{t}} (F |_{X_{t}}) \\ = ⋁_{t \in T} ⋁_{F ⪡ {co}_{X} (F) ⩽ A}^{↑} {co}_{X_{t}} (F |_{X_{t}}) \\ ⩽ ⋁_{t \in T} ⋁_{F ⪡ {co}_{X} (F) ⩽ A}^{↑} {co}_{X} (F) \\ = ⋁_{F ⪡ {co}_{X} (F) ⩽ A}^{↑} {co}_{X} (F) \end{matrix}$ The inequality $⋁_{F ⪡ {co}_{X} (F) ⩽ A}^{↑} {co}_{X} (F) ⩽ {co}_{X} (A)$

holds obviously. Hence ${co}_{X} (A) = ⋁_{F ⪡ {co}_{X} (F) ⩽ A}^{↑} {co}_{X} (F)$ . This shows that $(X, C)$ is dense. □

Definition 5.8. ([1]). An L-convex space $(X, C)$ is called

cone-union property (CUP), if for each L-fuzzy point x_λ ∈ L^X and any L-convex sets C, C₁, C₂, ⋯ , C_n with C ⩽ ⋁ _1⩽i⩽nC_i, it satisfies co_X (x_λ ∨ C) ⩽ ⋁ _{i∈{1,⋯,n}}co_X (x_λ ∨ C_i) .

join-hull commutativity (JHC), if for each L-fuzzy point x_λ ∈ L^X and each non-empty L-convex set C, it satisfies co_X (x_λ ∨ C) = ⋁ _{y_μ∈β^*(C)}co_X (x_λ ∨ y_μ) .

weakly join-hull commutativity (weakly JHC), if for each pair of non-empty L-convex sets C, D ∈ L^X with C D′, it satisfies co_X (C ∨ D) = ⋁ _{x_λ∈β^*(C),y_μ∈β^*(D)}co_X (x_λ ∨ y_μ) .

Theorem 5.9. Let $(X, C) = \sum (X_{t}, C_{t})$ . If $(X_{t}, C_{t})$ is CUP for all t ∈ T, then $(X, C)$ is CUP.

Proof. Let x_λ ∈ J (L^X) and $C, C_{1}, C_{2}, \dots, C_{n} \in C$ with C ⩽ ⋁ _1⩽i⩽nC_i. Then there exists s ∈ T such that x ∈ X_s, x ∉ X_t (t ≠ s). Obviously, C_t = C|_{X
_t} ⩽ (⋁ _1⩽i⩽nC_i) |_{X
_t} = ⋁ _1⩽i⩽n (C_i|_{X
_t})

for all t ∈ T. Since $(X_{t}, C_{t})$ is CUP for all t ∈ T, by Lemma 5.1(2), we obtain $\begin{matrix} {co}_{X} (x_{λ} \lor C) \\ = ⋁_{t \in T} {co}_{X_{t}} ((x_{λ} \lor C) |_{X_{t}}) \\ = {co}_{X_{s}} (x_{λ} \lor C_{s}) \lor ⋁_{t \neq s} {co}_{X_{t}} ((x_{λ} \lor C) |_{X_{t}}) \\ ⩽ ⋁_{1 ⩽ i ⩽ n} {co}_{X_{s}} (x_{λ} \lor (C_{i} |_{X_{s}})) \lor ⋁_{t \neq s} {co}_{X_{t}} (C_{t}) \\ ⩽ ⋁_{1 ⩽ i ⩽ n} {co}_{X_{s}} (x_{λ} \lor (C_{i} |_{X_{s}})) \lor ⋁_{t \neq s} C_{t} \\ ⩽ ⋁_{1 ⩽ i ⩽ n} {co}_{X_{s}} (x_{λ} \lor (C_{i} |_{X_{s}})) \lor ⋁_{t \neq s} ⋁_{1 ⩽ i ⩽ n} (C_{i} |_{X_{t}}) \\ ⩽ ⋁_{1 ⩽ i ⩽ n} {co}_{X_{s}} (x_{λ} \lor (C_{i} |_{X_{s}})) \lor \\ ⋁_{t \neq s} ⋁_{1 ⩽ i ⩽ n} {co}_{X_{t}} (x_{λ} \lor (C_{i} |_{X_{t}})) \\ = ⋁_{1 ⩽ i ⩽ n} ⋁_{t \in T} {co}_{X_{t}} (x_{λ} \lor (C_{i} |_{X_{t}})) \\ = ⋁_{1 ⩽ i ⩽ n} ⋁_{t \in T} {co}_{X_{t}} ((x_{λ} \lor C_{i}) |_{X_{t}}) \\ = ⋁_{1 ⩽ i ⩽ n} {co}_{X} (x_{λ} \lor C_{i}) \end{matrix}$

This shows co_X (x_λ ∨ C) ⩽ ⋁ _1⩽i⩽nco_X (x_λ ∨ C_i). Hence $(X, C)$ is CUP. □

Theorem 5.10. Let $(X, C) = \sum (X_{t}, C_{t})$ . If $(X_{t}, C_{t})$ is JHC for all t ∈ T, then $(X, C)$ is JHC.

Proof. Let x_λ ∈ J (L^X), i.e., x ∈ ⋃ _t∈TX_t and λ ∈ J (L), and let C be an L-convex set of $(X, C)$ . We may let x ∈ X_s. Note that (x_λ ∨ C) |_{X
_t} = C_t when t ≠ s (where C_t = C|_{X
_t}). Since $(X_{t}, C_{t})$ is JHC for all t ∈ T, by Lemma 5.1(2), we obtain $\begin{matrix} {co}_{X} (x_{λ} \lor C) \\ = ⋁_{t \in T} {co}_{X_{t}} ((x_{λ} \lor C) |_{X_{t}}) \\ = {co}_{X_{s}} (x_{λ} \lor C |_{X_{s}}) \lor ⋁_{t \neq s} {co}_{X_{t}} (C |_{X_{t}}) \\ = {co}_{X_{s}} (x_{λ} \lor C_{s}) \lor ⋁_{t \neq s} C_{t} \\ = {co}_{X_{s}} (x_{λ} \lor C_{s}) \lor ⋁_{t \neq s} ⋁_{z_{γ} \in β^{*} (C_{t})} z_{γ} \\ ⩽ ⋁_{y_{μ} \in β^{*} (C_{s})} {co}_{X_{s}} (x_{λ} \lor y_{μ}) \lor \\ ⋁_{t \neq s} ⋁_{z_{γ} \in β^{*} (C_{t})} {co}_{X_{t}} (z_{γ}) \\ ⩽ ⋁_{y_{μ} \in β^{*} (C_{s})} {co}_{X_{s}} (x_{λ} \lor y_{μ}) \lor \\ ⋁_{t \neq s} ⋁_{z_{γ} \in β^{*} (C_{t})} {co}_{X_{t}} (x_{λ} \lor z_{γ}) \\ ⩽ ⋁_{y_{μ} \in β^{*} (C)} {co}_{X} (x_{λ} \lor y_{μ}) \\ ⩽ {co}_{X} (x_{λ} \lor C) . \end{matrix}$ This shows co_X (x_λ ∨ C) = ⋁ _{y_μ∈β^*(C)}co_X (x_λ ∨ y_μ). Hence $(X, C)$ is JHC. □

Theorem 5.11. Let $(X, C) = \sum (X_{t}, C_{t})$ . If $(X_{t}, C_{t})$ is weakly JHC for all t ∈ T, then $(X, C)$ is weakly JHC.

Proof. Let C, D be two non-empty L-convex sets of $(X, C)$ with C D′. Then $C |_{X_{t}}, D |_{X_{t}} \in C_{t}$ with C|_{X
_t} (D|_{X
_t}) ′ for all t ∈ T. Since $(X_{t}, C_{t})$ is weakly JHC for all t ∈ T, by Lemma 5.1(2), we obtain $\begin{matrix} {co}_{X} (C \lor D) \\ = ⋁_{t \in T} {co}_{X_{t}} ((C \lor D) |_{X_{t}}) \\ = ⋁_{t \in T} {co}_{X_{t}} (C |_{X_{t}} \lor D |_{X_{t}}) \\ = ⋁_{t \in T} ⋁_{x_{λ} \in β^{*} (C |_{X_{t}}), y_{μ} \in β^{*} (D |_{X_{t}})} {co}_{X_{t}} (x_{λ} \lor y_{μ}) \\ ⩽ ⋁_{t \in T} ⋁_{x_{λ} \in β^{*} (C), y_{μ} \in β^{*} (D)} {co}_{X_{t}} ((x_{λ} \lor y_{μ}) |_{X_{t}}) \\ = ⋁_{x_{λ} \in β^{*} (C), y_{μ} \in β^{*} (D)} ⋁_{t \in T} {co}_{X_{t}} ((x_{λ} \lor y_{μ}) |_{X_{t}}) \\ = ⋁_{x_{λ} \in β^{*} (C), y_{μ} \in β^{*} (D)} {co}_{X} (x_{λ} \lor y_{μ}) . \end{matrix}$ The inequality ⋁_{x_λ∈β^*(C),y_μ∈β^*(D)}co_X (x_λ ∨ y_μ) ⩽ co_X (C ∨ D)

holds obviously. This shows that co_X (C ∨ D) = ⋁ _{x_λ∈β^*(C),y_μ∈β^*(D)}co_X (x_λ ∨ y_μ) .

Hence $(X, C)$ is weakly JHC. □

6 Conclusions

As we all know, the sum of a family of convex spaces is a basic and very useful operation [26]. In this paper, we first generalized the sum of a family of convex spaces to the L-fuzzy case. Then we discussed the relationships between the sum of a family of L-convex spaces and its factor spaces. Secondly, we introduced the notion of S₃, S₄ separability and established their connections with S₂ separability by some examples. We showed that these separability satisfy additivity. Furthermore, we considered the additivity of five types special L-convex spaces based on L-hull operators.

Following this paper, we will consider the following problems in the future.

In the crisp setting, screening is an important concept for describing S₃ and S₄ separability [26]. We will extend it to the L-fuzzy case and characterize S₃ and S₄ separability of L-convex spaces.

The notion of the sum of M-fuzzifying convex spaces has been proposed [23], but there are few researches on it. Therefore, we will study more of its properties.

Convex spaces have been generalized to (L, M)-fuzzy convex spaces [24], which can be seen as a broader form of L-convex spaces. Thus, we will consider the sum of a family of (L, M)-fuzzy convex spaces.

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