Some separation axioms in L -convex spaces

1 Introduction

As we all know, convexity theory plays an important role in solving extremum problems. It has been widely used in many fields of applied mathematics [2]. The study of convexity can be traced back to the mid-18th century or earlier. Initially, the concept of convexity was mainly studied in ℝ n in the pioneering works of Newton, Minkowski and others as described in [2, 8]. In fact, convexities also exist in many different mathematical research areas, such as vector spaces [36], graphs [34], lattices[35, 37], metric spaces [11], median algebras [3] and so on. However, the concept of convex sets have different forms in different mathematical structures. That is to say, in a particular object of study, a convex set has its own characterization. In order to study the generality of convexities in various mathematical structures, three basic axioms of convexities are obtained by means of axiomatic method. This leads to the concept of abstract convexities. More concretely, an abstract convexity on a set X is a collection of subsets of X which contains both X itself and the empty ∅ and which is closed under arbitrary intersections and nested unions [36].

Since Zadeh put forward the concept of fuzzy sets [48], many scholars have applied fuzzy theory to different mathematical structures and obtained many important results, such as fuzzy algebras [16, 49], fuzzy topologies [5, 38], fuzzy orders [47], fuzzy convergence structures [10, 19] and so on. As a new direction of fuzzy theory, the theory of fuzzy convexities have developed rapidly in the past two decades. In 1994, Rosa [27] firstly introduced the concept of fuzzy convex spaces ([0,1]-fuzzy convex spaces). In a [0,1]-fuzzy convex space, the convex sets are fuzzy, but the convexity is crisp. Later, Maruyama [17] generalized the notion of [0,1]-fuzzy convex spaces to L-lattice valued case, where L denotes a completely distributive lattice. Both of fuzzy convex spaces in sense of Rosa and Maruyama are called L-convex spaces nowadays. Recently, L-convex spaces become a popular research direction. Pang and Zhao [25] provided several characterizations of L-convex spaces and discussed some properties of them. Jin and Li [9] investigated the relationships between convex spaces and stratified L-convex spaces from a categorical viewpoint. Specifically, in 2017, Pang and Shi [20] introduced several types L-convex spaces and investigated the categorical relations among these kinds of L-convex spaces. Afterwards, in 2019, Shen and Shi [29] provided some new characterizations of L-convex spaces by using the way-below relation in domain theory. Other studies of L-convex spaces can be found in [7, 50].

As another direction of fuzzy convex spaces, a new approach to fuzzification of convex spaces was provided by Shi and Xiu [32]. Nowadays, this kind of fuzzy convex spaces are called M-fuzzifying convex spaces, where M denotes a completely distributive lattice. In this situation, each subset of X can be regarded as a convex set to some degree. Since the notion of M-fuzzifying convex spaces was put forward, more and more scholars have studied it. Relevant results of M-fuzzifying convex spaces can be seen in [16, 43–46]. Further more, Shi and Xiu [33] proposed the notion of broader convex spaces, which is called (L, M)-fuzzy convex spaces. It contains L-convex spaces and M-fuzzifying convex spaces as special cases. In [13, 42], the authors have given the relevant research contents of (L, M)-fuzzy convex spaces and obtained many good results.

It is generally know that the separation axioms are very important contents in general topology. Many profound results are derived from a separable topological space. As we know, in the framework of L-topological spaces, many experts and scholars have studied the separation axioms in detail and obtained many beautiful results [38]. As a topology-like structure, convexity also have separation axioms associated with it. Similarly, the separations are important part of convex spaces. So far, many scholars have done significant work in this field (see [1, 6]). In the monograph of convex spaces [36], the author gives several definitions of separation axioms in detail and establishes their equivalent characterization. In addition, hereditary properties and productive properties of these separation axioms are also discussed in this monograph. Based on this idea, we consider the properties of separation axioms in L-convex spaces. The purpose of this paper is to discuss some low-level separation axioms in the framework of L-convex spaces.

This paper is organized as follows. In Section 2, we recall some preliminaries on fuzzy sets, lattices and convex spaces. In Section 3, we firstly define S_-1, sub-S₀ and S₀ separation axioms in L-convex spaces, respectively. Then we discuss the relations of these axioms and study their related properties. In Section 4, the definitions of S₁ and S₂ are given. Then the hereditary properties and productive properties of them are discussed. In Section 5, the relations between convex spaces and induced L-convex spaces on some separation axioms are studied.

2 Preliminaries

In this section, we will recall some basic concepts and results on fuzzy sets, lattices and convexities. For undefined notions in this paper, the reader can refer to [36, 48].

Throughout this paper, (L, ∨ , ∧ , ′) is a completely distributive lattice with an order-reversing involution ^′. The smallest element and the largest element in L are denoted by ⊥ and ⊤, respectively. For a, b ∈ L, we say that a is wedge below b in L, in symbols a ≺ b, if for every subset D ⊆ L, b ⩽ ⋁ D implies a ⩽ d for some d ∈ D. The set {a ∈ L ∣ a ≺ b} denoted by β (b) is called the greatest minimal family of b in the sense of [39].

For a non-empty set X, 2 ^X denotes the powerset of X. For any nonempty subset A ∈ 2 ^X , let χ _A denote the characteristic function of A. A family {A _i } _i∈I ⊆ 2 ^X is directed provided for each A₁, A₂ ∈ {A _i } _i∈I there is an element A₃ ∈ {A _i } _i∈I such that A₁ ⊆ A₃ and A₂ ⊆ A₃, in symbols: { A i } i ∈ I ⊆ dir 2 X .

L ^X is the set of all L-fuzzy sets (or L-sets for short) on X. For each a ∈ L, a _ denotes the constant mapping X ⟶ L,  x ↦ a, which is called constant L-set. L ^X is also a completely distributive lattice under the pointwise order. The smallest element and the largest element in L ^X are denoted by ⊥ _ and ⊤ _ , respectively. a ∈ L ∖ {⊥} is said to be a molecule [38] iff a ⩽ b ∨ c implies a ⩽ b or a ⩽ c. An L-fuzzy point, denoted by x _λ  (λ ≠ ⊥), is a special L-set which is defined by ∀ y ∈ L , x λ ( y ) = { λ , y = x ; ⊥ , y ≠ x .

We denote the set of all molecules of L (resp., L ^X ) by J (L) (resp., by J (L ^X )). Obviously, x _λ  ∈ J (L ^X ) if and only if x ∈ X and λ ∈ J (L).

Given a mapping f : X ⟶ Y, define f L → : L X ⟶ L Y and f L ← : L Y ⟶ L X by f L → ( U ) ( y ) = ⋁ f ( x ) = y U ( x ) , f L ← ( V ) ( x ) = V ( f ( x ) ) for all U ∈ L ^X , y ∈ Y and V ∈ L ^Y , x ∈ X. It is easy to verify that U ⩽ f L ← ∘ f L → ( U ) and f L → ∘ f L ← ( V ) ⩽ V .

For a ∈ L and U ∈ L ^X , we use the following notations [30]

U_[a] = {x ∈ X ∣ U (x) ⩾ a};

It is easy to prove that for any a ∈ L and any U ∈ L ^X , U_(a) ⊆ U_[a]. In particular, when L = [0, 1] , U_(a) = U^(a).

Lemma 2.1. ([30]). Let U ∈ L ^X and a ∈ L, then U^(a) = ⋃ _b≰aU_[b].

Definition 2.2. ([38]). Let U ∈ L ^X and ∅ ≠ Y ⊆ X. The L-set U| _Y  ∈ L ^Y is defined as follows: for any y ∈ Y, (U| _Y ) (y) = U (y). U| _Y is called the restriction of U to Y.

Proposition 2.3. ([38]). For {U _t  ∣ t ∈ T} ⊆ L ^X , U ∈ L ^X , ∅ ≠ Y ⊆ X. We have:

(1) (⋁ _t∈TU _t ) | _Y  = ⋁ _t∈T (U _t | _Y );

(2) (⋀ _t∈TU _t ) | _Y  = ⋀ _t∈T (U _t | _Y );

(3) U′| _Y  = (U| _Y ) ′.

Definition 2.4. ([38]) Let ∅ ≠ Y ⊆ X and U ∈ L ^Y  . The L-set U^* ∈ L ^X is defined as follows: ∀ x ∈ X , U * ( x ) = { U ( x ) , x ∈ Y ; ⊥ , x ∉ Y . U^* is called the extension of U in X.

Especially, if x _λ  ∈ J (L ^Y ), then x λ * ∈ J ( L X ) . Clearly, U^*| _Y  = U.

Definition 2.5. ([36]). A subset C of 2 ^X is called a convexity on X if it satisfies the following conditions:

∅ , X ∈ C ;

The pair ( X , C ) is called a convex space. The members of C are called convex sets and their complements are called concave sets.

Proposition 2.6. ([36]). Let C be a closure system on X, that is, C ⊆ 2 X satisfies (C1) and (C2). Then the following statements are equivalent.

if { A i } i ∈ I ⊆ C is nonempty and totally ordered by inclusion, then ⋃ i ∈ I A i ∈ C ;

Definition 2.7. ([36]) Let ( X , C ) be a convex space. A subset D of X is called a biconvex set provided D is both convex set and concave set.

Definition 2.8. ([14, 36]) A convex space ( X , C ) is said to be

(1) S₀ if for any x, y ∈ X, there exists A ∈ C such that x ∈ A, y ∉ A or B ∈ C such that y ∈ B, x ∉ B;

(2) S₁ if all singletons in X are convex;

(3) S₂ if for any x, y ∈ X with x ≠ y, there exists a biconvex set D of X with x ∈ D, y ∉ D.

In [17], the researcher generalized convex spaces to the concept of L-convex spaces as follows.

Definition 2.9. ([17]). A subset C of L ^X is called an L-convexity on X if it satisfies:

⊥ _ , ⊤ _ ∈ C ;

The pair (X, C) is called an L-convex space. The members of C are called L-convex sets.

Definition 2.10. ([20]). An L-convexity C on X is called stratified if it satisfies: ∀ a ∈ L , a _ ∈ C . For a stratified L-convexity on X, the pair (X, C) is called a stratified L-convex space.

Definition 2.11. ([28]). Let (X, C) be an L-convex space. For each U ∈ L ^X , define co L ( U ) = ⋀ { V ∈ C ∣ U ⩽ V } , that is, co _L is the least element of C that contains U, called the L-convex hull of U.

Definition 2.12. ([20]). A mapping f : (X, C) ⟶ (Y, D) between two L-convex spaces is called L-convexity preserving (L-CP, for short) provided that V ∈ D implies f L ← ( V ) ∈ C .

Definition 2.13. ([33]). Let (X, C) be an L-convex space and ∅ ≠ Y ⊆ X. Then C| _Y  = {U| _Y  ∣ U ∈ C} is an L-convexity on Y. We call (Y, C| _Y ) an L-convex subspace of (X, C).

Definition 2.14. ([33]) Let {(X _t , C _t )} _t∈T be a family of L-convex spaces. Let X be the product of the sets of X _t for t ∈ T, and let P _t  : X ⟶ X _t denote the projection for each t ∈ T. X can be equipped with the L-convexity C generated by the family { ( P t ) L ← ( V ) : V ∈ C t , t ∈ T } as a subbase. Then C is called the product L-convexity for X and (X, C) is called the product L-convex space.

Remark 2.15. From Definition 2 and Definition 2, we know that the mapping P _t  : (X, C) ⟶ (X _t , C _t ) is L-CP for each t ∈ T.

3 S_-1, sub-S₀ and S₀ separation axioms

In this section, we introduce S_-1, sub-S₀ and S₀ separation axioms in L-convex spaces. Some equivalent characterizations of them are given and some related properties are studied. In particular, we discuss the relationships between these separation axioms by some examples.

Definition 3.1. An L-convex space (X, C) is said to be

(1) S_-1 if for any x _λ , x _μ  ∈ J (L ^X ) with μ≰λ, there exists U ∈ C such that x _μ ≰U, x _λ  ⩽ U;

(2) sub-S₀ if for any x, y ∈ X with x ≠ y, there exists λ ∈ J (L), U ∈ C such that x _λ ≰U, y _λ  ⩽ U or V ∈ C such that y _λ ≰V, x _λ  ⩽ V;

(3) S₀ if for any x _λ , y _μ  ∈ J (L ^X ) with x _λ  ≠ y _μ , there exists U ∈ C such that x _λ ≰U, y _μ  ⩽ U or V ∈ C such that y _μ ≰V, x _λ  ⩽ V.

Example 3.2. (1) It is easy to check that a stratified L-convex space is S_-1. Let R be the set of all real numbers and n be a natural number. An L-set U on R ⁿ is called convex if and only if U (rx + (1 - r) y) ⩾ U (x) ∧ U (y) for all x, y ∈ R ⁿ , r ∈ [0, 1] . Let C _{R ⁿ} denote the family of all L-convex sets of R ⁿ . Then C _{R ⁿ} is a stratified L-convexity on R ⁿ and (R ⁿ , C _{R ⁿ} ) is a stratified L-convex space. Hence (R ⁿ , C _{R ⁿ} ) is S_-1. (2) Let X = {x, y} and L = [0, 1]. Take C = { 0 _ , 1 _ , U 1 , U 2 , U 1 ∨ U 2 } is defined as follows: U 1 ( x ) = 1 2 , U 1 ( y ) = 0 ; U 2 ( x ) = 0 , U 2 ( y ) = 1 2 . Then (X, C) is an L-convex space. Obviously, (X, C) is sub-S₀. (3) Let X = {x, y, z} and L = [0, 1]. Take C = { 0 _ , 1 _ , U 1 , U 2 , U 3 } is defined as follows: U 1 ( x ) = 1 , U 1 ( y ) = U 1 ( z ) = 0 ; U 2 ( x ) = U 2 ( y ) = 1 , U 2 ( z ) = 0 ; U 3 ( x ) = U 3 ( y ) = 1 , U 3 ( z ) = 1 2 . Then (X, C) is S₀.

Next, we give the equivalent characterizations of sub-S₀ and S₀ respectively, which will be helpful in discussing their related properties later.

Theorem 3.3. An L-convex space (X, C) is sub-S₀ if and only if for any x, y ∈ X with x ≠ y, there exists C ∈ C such that C (x) ≠ C (y).

Proof. Necessity. Let x, y ∈ X with x ≠ y. Since (X, C) is sub-S₀, there exists λ ∈ J (L), U ∈ C such that x _λ ≰U, y _λ  ⩽ U or V ∈ C such that y _λ ≰V, x _λ  ⩽ V. This means λ≰U (x) , λ ⩽ U (y) or λ≰V (y) , λ ⩽ V (x). It implies that U (y) ≰U (x) or V (x) ≰V (y). If U (y) ≰U (x), let C = U, then C (y) ≰C (x). If V (x) ≰V (y), let C = V, then C (x) ≰C (y). Therefore, there exists C ∈ C such that C (x) ≠ C (y).

Sufficiency. Take x, y ∈ X with x ≠ y. Then there exists C ∈ C such that C (x) ≠ C (y), implying that C (x) ≰C (y) or C (y) ≰C (x). If C (x) ≰C (y), then C ( x ) = ⋁ { μ ∈ J ( L ) | μ ⩽ C ( x ) } ≰ C ( y ) .

This means there exists λ ∈ J (L) such that λ ⩽ C (x) and λ≰C (y). It implies that x _λ  ⩽ C and y _λ ≰C. Let U = C, then x _λ  ⩽ U, y _λ ≰U. If C (y) ≰C (x), in similar way, we can prove that there exists V ∈ C such that y _λ  ⩽ V and x _λ ≰V. Hence (X, C) is sub-S₀. □

Theorem 3.4. An L-convex space (X, C) is S₀ if and only if for any x _λ , y _μ  ∈ J (L ^X ) with x _λ  ≠ y _μ , co _L  (x _λ ) ≠ co _L  (y _μ ).

Proof. Necessity. Let x _λ , y _μ  ∈ J (L ^X ) with x _λ  ≠ y _μ . Since (X, C) is S₀, there exists U ∈ C such that x _λ ≰U, y _μ  ⩽ U or V ∈ C such that y _μ ≰V, x _λ  ⩽ V. Without loss of generality, we suppose the former case occurred. Since U ∈ C and x _λ ≰U, y _μ  ⩽ U, we have co _L  (y _μ ) ⩽ co _L  (U) = U. It follows that x _λ ≰co _L  (y _μ ). Thus co _L  (x _λ ) ≰co _L  (y _μ ). This means co _L  (x _λ ) ≠ co _L  (y _μ ).

Sufficiency. Take x _λ , y _μ  ∈ J (L ^X ) with x _λ  ≠ y _μ , then co _L  (x _λ ) ≠ co _L  (y _μ ). This means co _L  (x _λ ) ≰co _L  (y _μ ) or co _L  (y _μ ) ≰co _L  (x _λ ). We suppose that co _L  (x _λ ) ≰co _L  (y _μ ). Then x _λ ≰co _L  (y _μ ) and y _μ  ⩽ co _L  (y _μ ). It implies that (X, C) is S₀. □

The following proposition is obvious.

Proposition 3.5. An S₀ convex space is S_-1 and sub-S₀.

Remark 3.6. (1) In general, S_-1 and sub-S₀ need not be S₀.

(2) There is no necessary connection between S_-1 and sub-S₀. That is to say, S_-1 need not be sub-S₀ and sub-S₀ need not be S_-1.

(3) In a special L-convex space, sub-S₀ and S₀ are equivalent. This fact will be discussed in Section 5.

We can illustrate the problems in Remark 3.6 (1) and (2) with the following examples.

Example 3.7. Let X = L = [0, 1]. An L-set U is defined as follows: U ( x ) = { 2 x , x ∈ [ 0 , 1 2 ] ; 1 , x ∈ ( 1 2 , 1 ] . Let C = { 0 _ , 1 _ , U } . It is easy to check that (X, C) is an S_-1 L-convex space. But it is not S₀. Actually, for two fuzzy points ( 1 3 ) 1 2 and ( 3 4 ) λ , where λ ∈ (0, 1]. Obviously, ( 1 3 ) 1 2 ≠ ( 3 4 ) λ and co L ( ( 1 3 ) 1 2 ) = co L ( ( 3 4 ) λ ) = U . By Theorem 3, we know that (X, C) is not S₀.

Example 3.8. Let X = {x, y, z} and L = [0, 1]. We define C = { 0 _ , 1 _ , U 1 , U 2 } , where U 1 ( x ) = 1 , U 1 ( y ) = U 1 ( z ) = 0 ; U 2 ( x ) = 1 , U 2 ( y ) = 1 2 , U 2 ( z ) = 1 4 . Then (X, C) is a sub-S₀ L-convex space. But it is not S₀. In fact, for two fuzzy points x _λ and y _μ , where λ ∈ ( 0 , 1 2 ] , μ ∈ ( 0 , 1 4 ] . Obviously, x _λ  ≠ y _μ and co _L  (x _λ ) = co _L  (y _μ ) = U₂. Hence (X, C) is not S₀.

Example 3.9. For a stratified L-convex space (X, C), where C = { a _ ∣ a ∈ L } . Apparently, (X, C) is S_-1. By Theorem 3, we know that (X, C) is not sub-S₀.

Example 3.10. Consider the L-convex space in Example 3 (2). It is easy to verify that (X, C) is sub-S₀. For x 1 5 , x 1 4 ∈ J ( [ 0 , 1 ] X ) , it is impossible to have such an L-set U to make 1 5 ⩽ U ( x ) ⩽ 1 4 valid. Hence (X, C) is not S_-1.

Now, we consider the hereditary property and productive property of S_-1, sub-S₀ and S₀ separation axioms in L-convex spaces.

Theorem 3.11. If (X, C) is an S_-1 L-convex space and (Y, C| _Y ) is its subspace, then (Y, C| _Y ) is S_-1.

Proof. Let (X, C) be S_-1 and let x _λ , x _μ  ∈ J (L ^Y ) with μ≰λ. Then x λ * , x μ * ∈ J ( L X ) . Since (X, C) is S_-1, there exists U ∈ C such that x μ * ≰ U , x λ * ⩽ U . By Definition 2, we know that U| _Y  ∈ C| _Y and x _μ ≰U| _Y , x _λ  ⩽ U| _Y . Hence (Y, C| _Y ) is S_-1. □

Theorem 3.12. Let (X, C) be the product space of {(X _t , C _t )} _t∈T. If for each t ∈ T, (X _t , C _t ) is S_-1, then so is (X, C).

Proof. Suppose that for each t ∈ T, (X _t , C _t ) is S_-1. Let x = {x ^t } _t∈T ∈ X and μ≰λ, where λ, μ ∈ J (L). Then x _λ , x _μ  ∈ J (L ^X ). For r ∈ T, since (X _r , C _r ) is S_-1, there exists U _r  ∈ C _r such that (x ^r )  _μ ≰U _r and (x ^r )  _λ  ⩽ U _r . Then we have P r → ( x μ ) = P r ( x ) μ = ( x r ) μ ≰ U r and P r → ( x λ ) = P r ( x ) λ = ( x r ) λ ⩽ U r . Hence x μ ≰ P r ← ( U r ) and x λ ⩽ P r ← ( U r ) . Since P _r  : (X, C) → (X _r , C _r ) is an L-CP mapping, we have P r ← ( U r ) ∈ C . Therefore, (X, C) is S_-1. □

The proof of the following theorem is obvious.

Theorem 3.13. If (X, C) is a sub-S₀ L-convex space and (Y, C| _Y ) is its subspace, then (Y, C| _Y ) is sub-S₀.

Theorem 3.14. Let (X, C) be the product space of {(X _t , C _t )} _t∈T. If for each t ∈ T, (X _t , C _t ) is sub-S₀, then so is (X, C).

Proof. Suppose that for each t ∈ T, (X _t , C _t ) is sub-S₀ and x, y ∈ X with x ≠ y, where x = {x ^t } _t∈T, y = {y ^t } _t∈T. Then there exists r ∈ T such that x ^r  ≠ y ^r . Hence there exists an L-convex set U _r of X _r such that U _r  (x ^r ) ≠ U _r  (y ^r ). Since P _r  : (X, C) → (X _r , C _r ) is an L-CP mapping, we know that P r ← ( U r ) is an L-convex set of X. By P r ← ( U r ) ( x ) = U r ( P r ( x ) ) = U r ( x r ) and P r ← ( U r ) ( y ) = U r ( P r ( y ) ) = U r ( y r ) , we have P r ← ( U r ) ( x ) ≠ P r ← ( U r ) ( y ) . Therefore, (X, C) is sub-S₀. □

The proof of the following theorem is obvious.

Theorem 3.15. If (X, C) is an S₀ L-convex space and (Y, C| _Y ) is its subspace, then (Y, C| _Y ) is S₀.

Theorem 3.16. Let (X, C) be the product space of {(X _t , C _t )} _t∈T. If for each t ∈ T, (X _t , C _t ) is S₀, then so is (X, C).

Proof. Suppose that for each t ∈ T, (X _t , C _t ) is S₀ and x _λ , y _μ  ∈ J (L ^X ) with x _λ  ≠ y _μ , where x = {x ^t } _t∈T, y = {y ^t } _t∈T. Then we have x ≠ y or λ ≠ μ. Case 1: If x ≠ y, then there exists r ∈ T such that x ^r  ≠ y ^r . It implies (x ^r )  _λ  ≠ (y ^r )  _μ . Case 2: If λ ≠ μ, then (x ^r )  _λ  ≠ (y ^r )  _μ for all r ∈ T. Hence there exists r ∈ T such that (x ^r )  _λ  ≠ (y ^r )  _μ . Since (X _r , C _r ) is S₀, there exists U _r  ∈ C _r such that (x ^r )  _λ ≰U _r , (y ^r )  _μ  ⩽ U _r or V _r  ∈ C _r such that (y ^r )  _μ ≰V _r , (x ^r )  _λ  ⩽ V _r . Without loss of generality, we suppose the former case occurred.

Since (x ^r )  _λ ≰U _r , (y ^r )  _μ  ⩽ U _r and P _r  : (X, C) → (X _r , C _r ) is an L-CP mapping, we have x λ ≰ P r ← ( U r ) , y μ ⩽ P r ← ( U r ) and P r ← ( U r ) ∈ C . Therefore, (X, C) is S₀.

4 S₁ and S₂ separation axioms

In this section, we first introduce the relevant contents of S₁ separation axiom in L-convex spaces. Then, we give the definition of L-biconvex sets. Based on this, the concept of S₂ separation axiom is investigated and some properties of S₂ are studied. Moreover, the relations between S₁ and S₂ are discussed.

Definition 4.1. An L-convex space (X, C) is said to be S₁, if for any x _λ , y _μ  ∈ J (L ^X ) with x _λ ≰y _μ , there exists U ∈ C such that x _λ ≰U, y _μ  ⩽ U.

Example 4.2. Let (X, ⩽) be a poset. An L-set U on X is called convex if and only if x ⩽ z ⩽ y implies U (z) ⩾ U (x) ∧ U (y) for all x, y, z ∈ X. Let C_⩽ denote the family of all L-convex sets of X, then (X, C_⩽) is an L-convex space. For any x _λ  ∈ J (L ^X ), by Theorem 3.3 in [50], we have co _L  (x _λ ) = x _λ . Hence (X, C_⩽) is S₁.

Theorem 4.3. Let (X, C) be an L-convex space, then (X, C) is S₁ if and only if for any y _μ  ∈ J (L ^X ) , co _L  (y _μ ) = y _μ .

Proof. Necessity. y _μ  ⩽ co _L  (y _μ ) is obvious. We need to show co _L  (y _μ ) ⩽ y _μ . Since (X, C) is S₁, for any x _λ , y _μ  ∈ J (L ^X ) with x _λ ≰y _μ , we know that there exists U ∈ C such that x _λ ≰U and y _μ  ⩽ U. Hence co _L  (y _μ ) ⩽ U and x _λ ≰co _L  (y _μ ). This means co _L  (y _μ ) ⩽ y _μ .

Sufficiency. Suppose x _λ , y _μ  ∈ J (L ^X ) with x _λ ≰y _μ . Let U = y _μ . Then by assumption, U is an L-convex set such that x _λ ≰U and y _μ  ⩽ U. Hence (X, C) is S₁. □

The following proposition is obvious.

Proposition 4.4. An S₁ L-convex space is S₀.

The following example shows that the converse of Proposition 2o is not true.

Example 4.5. Consider the L-convex space (X, C) in Example 3 (3). It is S₀ but not S₁. Let x ∈ X and let λ , μ ∈ ( 1 2 , 2 3 ) , μ < λ . Then we can not find U to satisfy μ ⩽ U (x) < λ. Hence (X, C) is not S₁.

Theorem 4.6. If (X, C) is an S₁ L-convex space and (Y, C| _Y ) is its subspace, then (Y, C| _Y ) is S₁.

Proof. Let (X, C) be an S₁ L-convex space and y _μ  ∈ J (L ^Y ). Then y μ * ∈ J ( L X ) . Since (X, C) is S₁, by Theorem 4, we have co L ( y μ * ) = y μ * . Hence y μ = y μ * | Y = co L ( y μ * ) | Y ∈ C | Y . This means y _μ is an L-convex set of Y. Therefore, (Y, C| _Y ) is S₁. □

Theorem 4.7. Let (X, C) be the product space of {(X _t , C _t )} _t∈T. If for each t ∈ T, (X _t , C _t ) is S₁, then so is (X, C).

Proof. Suppose that for each t ∈ T, (X _t , C _t ) is S₁ and x, y ∈ X, λ, μ ∈ J (L) with x _λ ≰y _μ , where x = {x ^t } _t∈T, y = {y ^t } _t∈T. Then we have x ≠ y or λ≰μ. Case 1: If x ≠ y, then there exists r ∈ T such that x ^r  ≠ y ^r . It implies (x ^r )  _λ ≰ (y ^r )  _μ . Case 2: If λ≰μ, then (x ^r )  _λ ≰ (y ^r )  _μ for all r ∈ T. Hence there exists r ∈ T such that (x ^r )  _λ ≰ (y ^r )  _μ . Since (X _r , C _r ) is S₁, there exists U _r  ∈ C _r such that (x ^r )  _λ ≰U _r and (y ^r )  _μ  ⩽ U _r . By P _r  : (X, C) → (X _r , C _r ) is an L-CP mapping, we know that P r ← ( U r ) is an L-convex set of X. By P r → ( x λ ) = P r → ( x ) λ = ( x r ) λ and P r → ( y μ ) = P r → ( y ) μ = ( y r ) μ , we have x λ ≰ P r ← ( U r ) and y μ ⩽ P r ← ( U r ) . Therefore, (X, C) is S₁. □

Definition 4.8. Let (X, C) be an L-convex space. An L-set H of X is called an L-biconvex set if H and H′ are L-convex sets.

Example 4.9. Let X = L = [0, 1]. Take C = { 0 _ , 1 _ , U 1 , U 2 } is defined as follows: U 1 ( x ) = { 1 2 , x ∈ [ 0 , 1 2 ] ; 0 , x ∈ ( 1 2 , 1 ] . and U 2 ( x ) = { 1 2 , x ∈ [ 0 , 1 2 ] ; 1 , x ∈ ( 1 2 , 1 ] .

Obviously, (X, C) is an L-convex space. For any U ∈ C, then U′ ∈ C. This means that all elements in C are L-biconvex sets.

Proposition 4.10. If f : (X, C) ⟶ (Y, D) is an L-CP mapping between L-convex spaces and H is an L-biconvex set of Y, then f^← (H) is an L-biconvex set of X.

Proof. Since f is an an L-CP mapping and H is an L-biconvex set of Y, we have f^← (H) and f^← (H′) are L-convex sets of X. For any x ∈ X, we know that f ← ( H ′ ) ( x ) = H ′ ( f ( x ) ) = ( H ( f ( x ) ) ) ′ and ( f ← ( H ) ) ′ ( x ) = ( f ← ( H ) ( x ) ) ′ = ( H ( f ( x ) ) ) ′ . This means (f^← (H)) ′ = f^← (H′). Hence f^← (H) is an L-biconvex set of X.

The following proposition is obvious.

Proposition 4.11. 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.

Definition 4.12. An L-convex space (X, C) is said to be S₂, if for any 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 ) ′.

Example 4.13. Let X = {x, y} and L = {⊥ , a, b, ⊤} where a′ = b, b′ = a. The Hasse diagram of L is shown in the following figure:

Take C = { ⊥ _ , a _ , b _ , U 1 , U 2 , U 3 , U 4 , U 5 , U 6 , ⊤ _ } is defined as follows: U₁ (x) = a, U₁ (y) = b; U₂ (x) = b, U₂ (y) = a; U₃ (x)= a, U₃ (y) =⊥; U₄ (x) = ⊥ , U₄ (y) = a; U₅ (x) = ⊥ , U₅ (y) = b; U₆ (x) = b, U₆ (y) =⊥.

By the constructions of L and C, we have J (L) = {a, b} and the set of all L-biconvex sets H = { ⊥ _ , a _ , b _ , U 1 , U 2 , ⊤ _ } . It is verify that (X, C) is an S₂ L-convex space.

Remark 4.14. In [38], the author discusses in detail the related content of separation axioms (T_-1, T₀, sub-T₀, T₁ and T₂) in L-topological spaces. The separation axioms (S_-1, S₀, sub-S₀, S₁ and S₂) in L-convex spaces defined here are different from that of L-topological spaces. We use an example to illustrate the difference between T₂ and S₂. Similarly, the differences between other separation axioms can be illustrated by other examples.

In Example 4.13, take τ = { ⊥ _ , V 1 , V 2 , ⊤ _ } , where V₁ and V₂ are defined by: V₁ (x) = a, V₁ (y) =⊥; V₂ (x) = a, V₂ (y) = b. Then τ is an L-topology on X. (X, τ) is not T₂. In fact, for x _a , y _a  ∈ J (L ^X ), there are no P and Q satisfying that x ≠ y with x _a ≰P and y _a ≰Q such that P ∨ Q = ⊤ _ . But the L-convex space (X, C) is S₂.

Proposition 4.15. An S₂ L-convex space is S₁.

Proof. Let (X, C) be an S₂ L-convex space and x _λ , y _μ  ∈ J (L ^X ) with x _λ ≰y _μ . Then we have (y _μ ) ′≰ (x _λ ) ′ and (y _μ ) ′ ∈ L ^X . It follows that ( y μ ) ′ = ⋁ { e ∈ J ( L X ) | e ⩽ ( y μ ) ′ } ≰ ( x λ ) ′ . This means there exists z _γ  ∈ J (L ^X ) such that z _γ  ⩽ (y _μ ) ′ and z _γ ≰ (x _λ ) ′. Hence x _λ ≰ (z _γ ) ′. Since (X, C) is S₂, there exists an L-biconvex set H such that z _γ  ⩽ H ⩽ (y _μ ) ′. Hence H′ ⩽ (z _γ ) ′ and y _μ  ⩽ H′. By x _λ ≰ (z _γ ) ′, we have x _λ ≰H′. Therefore, (X, C) is S₁. □

The following example shows that the converse of Proposition 2o4 is not true.

Example 4.16. Let X = [0, 1] and L = [0, 1] ^[0,1]. Define U ∈ L ^X is a convex set if U = ⊥ _ , or U = ⊤ _ , or |supp (U) | < ω and for any x ∈ supp (U), |supp (U (x)) | < ω (considering U (x) ∈ L = [0, 1] ^[0,1] as a [0,1]-fuzzy set on [0,1]). Let C denote all this kind of convex sets, then C is closed to arbitrary meets and totally ordered joins, so C is an L-convexity on X. For any λ ∈ J (L) = J ([0, 1] ^[0,1]), λ is a [0,1]-fuzzy point on [0,1], so every x _λ  ∈ J (L ^X ) is a convex set in (X, C) and (X, C) is S₁. For the L-convex space (X, C), it has only two L-biconvex sets ⊥ _ and ⊤ _ . Therefore, for any x _λ , y _μ  ∈ J (L ^X ) with x _λ  ⩽ (y _μ ) ′, we can not find an L-biconvex set H to satisfy x _λ  ⩽ H ⩽ (y _μ ) ′. Hence (X, C) is not S₂.

Theorem 4.17. If (X, C) is an S₂ L-convex space and (Y, C| _Y ) is its subspace, then (Y, C| _Y ) is S₂.

Proof. Let (X, C) be S₂ and let x _λ , y _μ  ∈ J (L ^Y ) with x _λ  ⩽ (y _μ ) ′. Then x λ * , y μ * ∈ J ( L X ) and x λ * ⩽ ( ( y μ ) ′ ) * ⩽ ( y μ * ) ′ . Since (X, C) is S₂, there exists an L-biconvex set H such that x λ * ⩽ H ⩽ ( y μ * ) ′ . It follows that x λ = x λ * | Y ⩽ H | Y ⩽ ( y μ * ) ′ | Y = ( y μ * | Y ) ′ = ( y μ ) ′ .

Hence x _λ  ⩽ H| _Y  ⩽ (y _μ ) ′. By Proposition 2o3, we know that H| _Y is an L-biconvex set of Y. Therefore, (Y, C| _Y ) is S₂. □

Theorem 4.18. Let (X, C) be the product space of {(X _t , C _t )} _t∈T. If for each t ∈ T, (X _t , C _t ) is S₂, then so is (X, C).

Proof. Suppose that for each t ∈ T, (X _t , C _t ) is S₂ and x _λ , y _μ  ∈ J (L ^X ) with x _λ  ⩽ (y _μ ) ′, where x = {x ^t } _t∈T, y = {y ^t } _t∈T. Then we have x ≠ y or x = y, λ ⩽ μ′. Case 1: If x ≠ y, then there exists r ∈ T such that x ^r  ≠ y ^r . It implies (x ^r )  _λ  ⩽ ((y ^r )  _μ ) ′. Case 2: If x = y, λ ⩽ μ′, then (x ^r )  _λ  ⩽ ((y ^r )  _μ ) ′ for all r ∈ T.

Hence there exists r ∈ T such that (x ^r )  _λ  ⩽ ((y ^r )  _μ ) ′. Since (X _r , C _r ) is S₂, there exists an L-biconvex set H _r such that (x ^r )  _λ  ⩽ H _r  ⩽ ((y ^r )  _μ ) ′. It follows that P r → ( x λ ) = P r ( x ) λ = ( x r ) λ ⩽ H r and P r → ( y μ ) = P r ( y ) μ = ( y r ) μ ⩽ ( H r ) ′ .

Hence x λ ⩽ P r ← ( H r ) and y μ ⩽ P r ← ( ( H r ) ′ ) . Since ( P r ← ( H r ) ) ′ = P r ← ( ( H r ) ′ ) , we have P r ← ( H r ) = ( P r ← ( ( H r ) ′ ) ) ′ . Hence x λ ⩽ P r ← ( H r ) ⩽ ( y μ ) ′ . By Proposition 2o2, we know that P r ← ( H r ) is an L-biconvex set of X. Therefore, (X, C) is S₂.□

Next, we consider the problem of the above separation axioms preserved under L-isomorphisms. First, we give the concept of L-isomorphisms.

Definition 4.19. A mapping f : (X, C) ⟶ (Y, D) between two L-convex spaces is called an L-isomorphisms provided that f is an L-CP and L-CC bijection.

The following proposition is obvious.

Proposition 4.20. Let f : (X, C) ⟶ (Y, D) be an L-isomorphism between two L-convex spaces. If (X, C) is S _i (sub-S₀), then (Y, D) is S _i (sub-S₀), where (i = -1, 0, 1, 2).

Remark 4.21. The condition that f is an L-isomorphisms is necessary for maintaining the separation axioms between two L-convex spaces. That is to say, if f does not satisfy any of the conditions of L-CP, L-CC and bijection, then these separation axioms will not be maintained under f. We illustrate S₀ by the following example. Other separation axioms can be similarly illustrated by examples.

Example 4.22. Let L = [0, 1] and X = {a, b, c}. Take C = { 0 _ , U 1 , U 2 . U 3 , 1 _ } , where U₁ (a) =1, U₁ (b) = U₁ (c) =0; U₂ (a) =0, U₂ (b) =1, U₂ (c) =0; U 3 ( a ) = U 3 ( b ) = 1 , U 3 ( c ) = 1 2 . Then (X, C) is S₀.

Consider Y = {m, n} and D = { 0 _ , V , 1 _ } , where V ( m ) = 1 , V ( n ) = 1 2 . Define f : X → Y as f (a) = f (b) = m and f (c) = n. It is easy to verify that f is L-CP and L-CC. But (Y, D) is not S₀. Since co (m _λ ) = co (n _μ ) = V for 0 < λ ≤ 1 and 0 < μ ≤ 1 2 .

5 Relations between convex spaces and induced L-convex spaces on separations

In [20], Pang and Shi proved that a convex space can induce a stratified L-convex space under the condition that β (a ∧ b) = β (a) ∩ β (b) for all a, b ∈ L. In this section, we discuss the relations between S₀, S₁, S₂ separation axioms of a convex space and sub-S₀, S₀, S₁, S₂ separation axioms of the induced L-convex space.

Proposition 5.1. ([20]). Let ( X , C ) be a convex space and define C C as follows: C C = { U ∈ L X ∣ ∀ a ∈ L , U [ a ] ∈ C } . Then C C is a stratified L-convexity on X and ( X , C C ) is a stratified L-convex space.

Theorem 5.2. ( X , C C ) is sub-S₀ if and only if ( X , C ) is S₀.

Proof. Necessity. Let x, y ∈ X with x ≠ y. Since ( X , C C ) is sub-S₀, there exists λ ∈ J ( L ) , U ∈ C C such that x _λ ≰U, y _λ  ⩽ U or V ∈ C C such that y _λ ≰V, x _λ  ⩽ V. This means λ≰U (x) , λ ⩽ U (y) or λ≰V (y) , λ ⩽ V (x). Hence x ∉ U_[λ], y ∈ U_[λ] or y ∉ V_[λ], x ∈ V_[λ]. Obviously, U [ λ ] , V [ λ ] ∈ C . Therefore, ( X , C ) is S₀.

Sufficiency. Let x, y ∈ X with x ≠ y. Since ( X , C ) is S₀, there exists P ∈ C such that x ∈ P, y ∉ P or Q ∈ C such that y ∈ Q, x ∉ Q. Let λ ∈ J (L), then (χ _P ) _[λ] = {z ∈ X ∣ (χ _P ) (z) ⩾ λ} = P. This means χ P ∈ C C . Similarly, χ Q ∈ C C . Hence x _λ ≰χ _Q , y _λ  ⩽ χ _Q or y _λ ≰χ _P , x _λ  ⩽ χ _P . Therefore, ( X , C C ) is sub-S₀. □

Theorem 5.3. ( X , C C ) is S₀ if and only if ( X , C ) is S₀.

Proof. Necessity. Let ( X , C C ) be S₀ and let x, y ∈ X with x ≠ y. For any λ ∈ J (L), then x _λ , y _λ  ∈ J (L ^X ) and x _λ  ≠ y _μ . Since ( X , C C ) is S₀, it follows that there exists U ∈ C C such that x _λ ≰U, y _λ  ⩽ U or V ∈ C C such that y _λ ≰V, x _λ  ⩽ V. Case 1: If there exists U ∈ C C such that x _λ ≰U, y _λ  ⩽ U, then U [ λ ] = { z ∈ X ∣ U ( z ) ⩾ λ } ∈ C and λ≰U (x) , λ ⩽ U (y). This means x ∉ U_[λ], y ∈ U_[λ] . Case 2: If there exists V ∈ C C such that y _λ ≰V, x _λ  ⩽ V, then V [ λ ] = { z ∈ X ∣ V ( z ) ⩾ λ } ∈ C . This means x ∈ V_[λ], y ∉ V_[λ]. Hence ( X , C ) is S₀.

Sufficiency. Let ( X , C ) be S₀ and let x _λ , y _μ  ∈ J (L ^X ) with x _λ  ≠ y _μ . Then we have x ≠ y or λ ≠ μ. Case 1: If x ≠ y, then there exists P ∈ C such that x ∈ P, y ∉ P or Q ∈ C such that x ∉ Q, y ∈ Q. If the former holds, let U = χ _P . Then U ∈ C C and x _λ  ⩽ U, y _μ ≰U. If the latter holds, let V = χ _Q . Then V ∈ C C and x _λ ≰V, y _μ  ⩽ V. Hence ( X , C C ) is S₀. Case 2: If λ ≠ μ, then μ≰λ or λ≰μ. Since ( X , C C ) is stratified, we have λ _ , μ _ ∈ C C . This means y μ ≰ λ _ , x λ ⩽ λ _ or x λ ≰ μ _ , y μ ⩽ μ . Hence ( X , C C ) is S₀. □

According to Theorem 5 and Theorem 5, we have the following fact.

Remark 5.4. In general L-convex spaces, S₀ is stronger than sub-S₀. But in the special L-convex space of Proposition 2ro, S₀ and sub-S₀ are equivalent.

Theorem 5.5. ( X , C C ) is S₁ if and only if ( X , C ) is S₁.

Proof. Necessity. Let ( X , C C ) be S₁. For any x ∈ X, λ ∈ J (L), then x _λ  ∈ J (L ^X ) and we have co _L  (x _λ ) = x _λ . Hence x λ ∈ C C and ( x λ ) [ λ ] ∈ C . Since ( x λ ) [ λ ] = { y ∈ X | ( x λ ) ( y ) ⩾ λ } = { x } ∈ C , it follows that ( X , C ) is S₁.

Sufficiency. Let ( X , C ) be S₁ and x _λ  ∈ J (L ^X ). For any a ∈ L, since (x_⊤) _[a] = {y ∈ X| (x_⊤) (y) ⩾ a} = {x} or ∅, it follows that ( x ⊤ ) [ a ] ∈ C . By Proposition 2ro, we know that ( X , C C ) is stratified, then λ _ ∈ C C . Hence x λ = x ⊤ ∧ λ _ . This means co _L  (x _λ ) = x _λ . Therefore, ( X , C C ) is S₁ .

Theorem 5.6. ( X , C C ) is S₂ if and only if ( X , C ) is S₂.

Proof. Necessity. Let x, y ∈ X with x ≠ y and λ ∈ J (L). By ⊤ = ⋁ {γ ∣ γ ∈ J (L)} and λ≰⊥, we have λ≰ ⋀ {γ′ ∣ γ ∈ J (L)}. Then there exists μ ∈ J (L) such that λ≰μ′. Obviously, x _λ  ⩽ (y _μ ) ′. Since ( X , C C ) is S₂, there exists an L-biconvex set H such that x _λ  ⩽ H ⩽ (y _μ ) ′. Hence λ ⩽ H (x) and H (y) ⩽ μ′. It follows from λ≰μ′ that λ≰H (y). This means x ∈ H_[λ] and y ∉ H_[λ]. For any a ∈ L, H_[λ] ∈ C and ( H ′ ) [ λ ] ∈ C . By Lemma 2, we have ( H [ λ ] ) ′ = ( H ′ ) ( λ ′ ) = ⋃ a ≰ λ ′ ( H ′ ) [ a ] . It is easy to verify that {(H′) _[a] ∣ a≰λ′, a ∈ L} is directed. Hence ( H [ λ ] ) ′ ∈ C . This means H_[λ] is a biconvex set of X. Therefore, ( X , C ) is S₂

Sufficiency. Let x _λ , y _μ  ∈ J (L ^X ) with x _λ  ⩽ (y _μ ) ′. Then x ≠ y or x = y, λ ⩽ μ′. Case 1: If x ≠ y, then there exists a biconvex set P such that x ∈ P and y ∉ P. For any a ∈ L, (χ _P ) _[a] = {z ∈ X ∣ (χ _P ) (z) ⩾ a} = P or X and (χ_P′) _[a] = {z ∈ X ∣ (χ_P′) (z) ⩾ a} = P′ or X. Hence χ P , χ P ′ ∈ C C . By χ_P′ = (χ _P ) ′, we know that χ _P is an L-biconvex set of X. Hence x _λ  ⩽ χ _P  ⩽ (y _μ ) ′. This means ( X , C C ) is S₂.

Case 2: If x = y, λ ⩽ μ′. Since ( X , C C ) is a stratified L-convex space, it follows that H = λ _ is an L-biconvex set of X. Hence x λ ⩽ λ _ ⩽ ( x μ ) ′ . Therefore, ( X , C C ) is S₂. □

6 Conclusions

It is well know that separation plays an important role in the study of convex spaces. In the paper, we first introduced the definitions of some low-level separation axioms (S_-1, sub-S₀, S₀, S₁ and S₂) of L-convex spaces, which is a generalization of the separability of classical convex spaces. Then we provided some characterizations of them and discussed the relationships among these separation axioms with the help of some examples. In addition, the hereditary properties and productive properties of them are also studied. Finally, with the idea that a convex space can induce a stratified L-convex space as mentioned in [20], we discussed the relationships between convex spaces and induced stratified L-convex spaces on some separation axioms. All the concepts and the relevant relationship in the framework of L-convex spaces are shown to be proper generalizations of those in the classical case. Following this paper, we will consider the following two problems in the future.

(1) In the case of L-convex spaces, we will consider the separation of S₃ and S₄ and study their related properties.

(2) Convex spaces have been generalized to (L, M)-fuzzy convex spaces [33], which can be seen as a broader form of L-convex spaces. Thus, we will consider the separation axioms of (L, M)-fuzzy convex spaces.