Separation axioms in ( L,M )-fuzzy convex spaces

Abstract

In this paper, S ₀, S ₁ and S ₂ separation axioms are introduced in (L, M)-fuzzy convex spaces. Each (L, M)-fuzzy convex space can be regarded to be S ₀, S ₁ and S ₂ separated to some degree. Some properties of them are investigated. Moreover, the degrees to which a function is convex preserving, convex-to-convex or isomorphic are defined in (L, M)-fuzzy convex spaces by using implication operation. Their relationships with the degrees of S ₀, S ₁ and S ₂ separation axioms are discussed.

1. Introduction

Abstract convexity theory [27, 29], which is a branch of mathematics, has numerous connections with other areas of mathematics. For different mathematical objects, there are so many collections of sets that can form convex structures, such as convexities in lattices [31], convexities in graphs [26], convexities in real vector spaces [28]. Also, convex structures appeared naturally in topology, especially in the theory of supercompact spaces [30]. As an independent spatial structure, convex structures are investigated from a topological aspect, such as product spaces, convex invariants and so on.

With the development of fuzzy mathematics, many mathematical structures have been generalized to the fuzzy case, such as fuzzy topology [1 , 34] and fuzzy convergence structures [7 , 14]. Motivated by this, convex structures have also been endowed with fuzzy set theory. Weiss [32] considered a convex fuzzy set in a vector space over real or complex number. Maruyama [10] and Rosa [19] independently introduced the concept of fuzzy convex structures, which is called L-convex structures nowadays. Based on L-convex structures, Pang et al. presented several characterizations in a topological approach [11 , 16] and studied the spatial properties a categorical approach [12].

In a completely different direction, Shi and Xiu [23] provided a new approach to fuzzification of convex structures and proposed the notion of M-fuzzifying convex structures. Based on M-fuzzifying convex structures, Shi and Li [25] introduced the notion of (L, M)-fuzzy restricted hull operators and showed that it is equivalent to M-fuzzifying convex structures. Xiu and Pang [33] introduced base axioms and subbase axioms in M-fuzzifying convex spaces. Li and Shi [6] recently generalized the notions of geometric interval spaces, convex geometries, and base-point orders to M-fuzzifying convexity spaces.

In [24], Shi and Xiu proposed the concept of (L, M)-fuzzy convex structures. Among these three kinds of fuzzy convex structures, (L, M)-fuzzy convex structures are the most general one in the sense that it can contain the first two kinds as special cases. Moreover, Li [9] introduced the notion of enriched (L, M)-convex spaces and the category of enriched (L, M)-convex spaces. Up to now, there have not been systematical studies on (L, M)-fuzzy convex structures.

Separation axioms constitute one of the fundamental facets of the theory of convex structures. Jamison [3] introduced the separation axioms and gave a restricted version of the polytope screening characterization in terms of screening with half-spaces. Rosa [20] introduced the separation axioms in L-convex structures. However, separation axioms have not been defined in the setting of (L, M)-fuzzy convex structures. By this motivation, we will aim to introduce the separation axioms in the framework of (L, M)-fuzzy convex spaces.

This paper is organized as follows. In Section 2, we recall some necessary concepts and notations. In Sections 3, 4 and 5, we will present the degree to which an (L, M)-fuzzy convex space is S ₀, S ₁ and S ₂ separated respectively. In Sections 6, we will propose the degrees to which a mapping is convex preserving, convex-to-convex or isomorphic, and discuss their relationships with the degrees of separation axioms in (L, M)-fuzzy convex spaces.

2. Preliminaries

Throughout this paper, both (L, ∧ _L, ∨ _L, ^′L) and (M, ∧ _M, ∨ _M, ^′M) denote completely distributive DeMorgan algebras, i.e., completely distributive lattices with order-reversing involution. For convenience, let (L, ∧, ∨, ′) and (M, ∧, ∨, ′) denote (L, ∧ _L, ∨ _L, ^′L) and (M, ∧ _M, ∨ _M, ^′M), respectively. The smallest element and the largest element in L are denoted by ⊥_L and ⊤_L, respectively. For a, b ∈ L, we say that a is wedge below b in L [18], in symbols a ≺ b, if for every subset D ⊆ L, ⋁D ⩾ b implies d ⩾ a for some d ∈ D. A complete lattice L is completely distributive if and only if b = ⋁ {a ∈ L ∣ a ≺ b} for each b ∈ L. An element a in L is called co-prime if a ⩽ b ∨ c implies a ⩽ b or a ⩽ c. The set of non-zero co-prime elements in L is denoted by J (L) .

For a nonempty set X, L ^X denotes the set of all L-fuzzy subsets on X. L ^X is also a completely distributive De Morgan algebra when it inherits the structure of the lattice L in a natural way, by defining ∨, ∧, ⩽ and ^′ pointwisely. It is easy to see that the set J (L ^X) of non-zero coprimes in L ^X is {x _λ ∣ λ ∈ J (L)}.

Let f : X → Y be a mapping. The forward L-power operator f ^→ : L ^X → L ^Y and the backward L-powerset operator f ^← : L ^Y → L ^X induced by f [21] are defined by f ^→ (A) (y) = ⋁ _f(x)=y A (x) for all A ∈ L ^X and y ∈ Y, and f ^← (B) = B ∘ f for all B ∈ L ^Y, respectively.

In a completely distributive De Morgan algebra L, there exists an implication operation → : L × L → L as the right adjoint for the meet operation ∧ by $a \to b = \lor {c \in L : a \land c \leq b} .$

We list some properties of implication operation.

Lemma 2.1. [2] implication Suppose that (L, ∨, ∧) is a completely distributive lattice and → is the implication operation corresponding to ∧. Then for all a, b, c ∈ L, {a _j} _j∈J, {b _j} _j∈J ⊆L, the following conditions hold:

(a → b) ≥ c ⇔ a ∧ c ≤ b;

a≤ b ⇔ a → b = ⊤;

a → (b → c) = (a ∧ b) → c;

(c → a) ∧ (a → b) ≤ c → b;

c → a ≤ (a → b) → (c → b);

a → ⋀ _j∈J a _j = ⋀ _j∈J (a → a _j), hence a → b ≤ a → c whenever b ≤ c;

⋁_j∈J a _j → b = ⋀ _j∈J (a _j → b), hence a → c ≥ b → c whenever a ≤ b.

Definition 2.2. [29] A subset $C$ of 2^X is called a convex structure if it satisfies the following conditions:

$φ, X \in C$ ;

if ${A_{i} : i \in Ω} \subseteq C$ is nonempty, then $⋂_{i \in Ω} A_{i} \in C$ ;

if ${A_{i} : i \in Ω} \subseteq C$ is nonempty and totally ordered by inclusion, then $⋃_{i \in Ω} A_{i} \in C$ .

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

Definition 2.3. [29] Let $(X, C)$ be a convex space. A subset H of X is called a biconvex set (half-space) provided H is both convex set and convcave set.

Definition 2.4. [29] Let $(X, C)$ be a convex space and A ∈ 2^X. The (convex) hull of A is defined by $co (A) = \cap {C : A \subseteq C \in C} .$ A set of type co (F), with F finite, is called a polytope.

Definition 2.5. [4, 29] Let $(X, C)$ be a convex space.

$(X, C)$ is said to be S ₀ separated if for all x, y ∈ X with x ¬ = y, then co ({x}) ¬ = co ({y});

$(X, C)$ is said to be S ₁ separated if all singletons in X are convex;

$(X, C)$ is said to be S ₂ separated if for all x, y ∈ X with x ¬ = y, then there is a biconvex set H of X with x ∈ H, y ∉ H .

In [24], Shi and Xiu generalized the concept of convex structures to (L, M)-fuzzy convex structures as follows.

Definition 2.6. [24] A mapping $C : L^{X} \to M$ is called an (L, M)-fuzzy convex structure on X if it satisfies the following conditions:

$C (⊥_{L^{Xitsc}}) = C (⊤_{L^{Xitsc}}) = ⊤_{Mitsc}$ ;

if {A _i : i ∈ Ω} ⊆ L ^X is nonempty, then $C (⋀_{i \in Ω} A_{i}) \geq ⋀_{i \in Ω} C (A_{i})$ ;

if {A _i : i ∈ Ω} ⊆ L ^X is nonempty and totally ordered, then $C (⋁_{i \in Ω} A_{i}) \geq ⋀_{i \in Ω} C (A_{i})$ .

If $C$ is an (L, M)-fuzzy convex structure on X, then the pair $(X, C)$ is called an (L, M)-fuzzy convex space. For A ∈ L ^X, $C (A)$ can be regarded as the degree to which A is a L-convex set.

Definition 2.7. [24] Let $(X, C)$ and $(Y, D)$ be (L, M)-fuzzy convex spaces. A function f : X → Y is called an (L, M)-fuzzy convexity preserving function if $C (f^{\leftarrow} (B)) \geq D (B)$ for all B ∈ L ^Y.

Definition 2.8. [24] Let $(X, C)$ and $(Y, D)$ be (L, M)-fuzzy convex spaces. A function f : X → Y is called an (L, M)-fuzzy convex-to-convex function if $D (f^{\to} (A)) \geq C (A)$ for all A ∈ L ^X.

Definition 2.9. [24] Let $(X, C)$ be an (L, M)-fuzzy convex space and φ ≠ Y ⊆ X. $(Y, C |_{Y})$ is called the subspace of $(X, C)$ , where for each A ∈ L ^Y, $C |_{Yitsc} (A) = ⋁ {C (B) : B \in L^{X}, B |_{Yitsc} = A}$ .

Definition 2.10. [24] Let φ : L ^X → M be a mapping. The (L, M)-fuzzy convex space $(X, C)$ generated by φ is given by $\forall A \in L^{X}, C (A) = \land {D (A) : φ \leq D \in H},$ where $H$ denotes the family of all (L, M)-fuzzy convex structures on X. Then φ is called a subbase of the (L, M)-fuzzy convex structure $C$ . Alternatively, we say that φ generates the convex structure $C$ .

Definition 2.11. [24] Let ${(X_{t}, C_{t})}_{t \in T}$ be a family of (L, M)-fuzzy convex spaces, X be the product of {X _t} _t∈T, and P _t : X → X _t denote the projection for each t ∈ T. Define a mapping φ : L ^X → M by $φ (A) = ⋁_{t \in T} ⋁_{{P_{t}}^{\leftarrow} (B) = A} C_{t} (B)$ for each A ∈ L ^X. Then the product (L, M)-fuzzy convex structure $C$ on X is the one generated by the subbase φ. The resulting (L, M)-fuzzy convex space $(X, C)$ is called the product of (L, M)-fuzzy convex spaces ${(X_{t}, C_{t})}_{t \in T}$ and is denoted by $\prod_{t \in T} (X_{t}, C_{t})$ .

Theorem 2.12. Let ${(X_{t}, C_{t})}_{t \in T}$ be a family of (L, M)-fuzzy convex spaces, $(X, C)$ be the product of ${(X_{t}, C_{t})}_{t \in T}$ . Then for all t ∈ T, P _t : X → X _t is an (L, M)-fuzzy convexity preserving function. Moreover, $C$ is the coarsest (L, M)-fuzzy convex structure such that {P _t : t ∈ T} are (L, M)-fuzzy convexity preserving functions.

Theorem 2.13. Let $(X, C)$ be an (L, M)-fuzzy convex space, ${co}^{C}$ be the (L, M)-fuzzy hull operater induced by $C$ . Then for all a ∈ J (L ^X), A ∈ L ^X, ${co}^{C} (A) (a) = ⋀_{a B \geq A} {C (B)}^{'} .$

Theorem 2.14. Let co : L ^X → M ^{J(L
^X)} be an (L, M)-fuzzy hull operater on X. Define $C^{co} : L^{Xitsc} \to M$ by $C^{co} (A) = ⋀_{a A} co (A) (a)^{'} .$ Then $C^{co}$ is an (L, M)-fuzzy convex structure.

Theorem 2.15. For an (L, M)-fuzzy convex space $(X, C), {C^{co}}^{C} = C .$

3. S ₀ separation axiom

In this section, we introduce the degree to which an (L, M)-fuzzy convex space is S ₀ separated.

Lemma 3.1. A convex space $(X, C)$ is S ₀ separated if and only if for any x, y ∈ X with x ≠ y, there exists $A \in C$ such that x ∈ A, y ∉ A, or $B \in C$ such that y ∈ B, x ∉ B.

Proof. For any x, y ∈ X with x ≠ y, we have $\begin{matrix} co ({x}) \neg = co ({y}) \\ \Leftrightarrow & y \notin co ({x}) or x \notin co ({y}) \\ \Leftrightarrow & there exists A \in C such that x \in A, y \notin A, \\ or B \in C such that y \in B, x \notin B . \end{matrix}$

Thus we complete the proof.□ Next, we generalize the degree to which an (L, M)-fuzzy convex space is S ₀ separated by using the above lemma.

Definition 3.2. For an (L, M)-fuzzy convex space $(X, C)$ , we define the degree to which $(X, C)$ is S ₀ separated as follows: $\begin{matrix} S_{0} (X, C) & = & ⋀ {⋁_{a A \geq b} C (A) \lor ⋁_{a \leq B b} C (B) : \\ a, b \in J (L^{X}), a b} . \end{matrix}$

Remark 3.3. If L and M are replaced by {0, 1}, then Definition 3.2 is equivalent to Definition 2.5 (1). So we can see that Definition 3.2 is reasonable generalization of S ₀ separation axiom.

In the classical case, S ₀ separation axiom has the heritability and the productive property. Next we investigate these properties in (L, M)-fuzzy convex spaces.

Theorem 3.4. Let $(X, C)$ be an (L, M)-fuzzy convex space, and $(Y, C |_{Y})$ be the subspace of $(X, C)$ . Then $S_{0} (X, C) \leq S_{0} (Y, C |_{Y}) .$

Proof. Since $C |_{Yitsc} (A) = ⋁ {C (B) : B \in L^{X}, B |_{Yitsc} = A}$ , we know that $C (A) \leq C |_{Yitsc} (A |_{Yitsc}), \forall A \in L^{Xitsc} .$

Then $\begin{matrix} S_{0} (X, C) & = & ⋀ {⋁_{a A \geq b} C (A) \lor ⋁_{a \leq B b} C (B) : \\ a, b \in J (L^{X}), a b} \\ ⩽ & ⋀ {⋁_{a A \geq b} C |_{Y} (A |_{Y}) \lor ⋁_{a \leq B b} C |_{Y} (B |_{Y}) : \\ a, b \in J (L^{X}), a b} \\ ⩽ & ⋀ {⋁_{c A |_{Y} \geq d} C |_{Y} (A |_{Y}) \lor \\ ⋁_{c \leq B |_{Y} d} C |_{Y} (B |_{Y}) : c, d \in J (L^{Y}), c d} \\ = & ⋀ {⋁_{c U \geq d} C |_{Y} (U) \lor ⋁_{c \leq V d} C |_{Y} (V) : \\ c, d \in J (L^{Y}), c d} \\ = & S_{0} (Y, C |_{Y}) . \end{matrix}$

Thus we complete the proof.□

The above theorem can be thought of the many valued interpretation of the classical resualt “a subspace of an S ₀ convex space is S ₀”. As we all know, the product of a family of S ₀ convex spaces is S ₀. Next we give the lattice-valued characterizations of this conclusion.

Theorem 3.5. Let $(X, C)$ be the product of (L, M)-fuzzy convex spaces ${(X_{t}, C_{t})}_{t \in T}$ . Then

⋀_{t \in T} S_{0} (X_{t}, C_{t}) \leq S_{0} (X, C) .

Proof. Take any α ∈ M with $α ≺ ⋀_{t \in T} S_{0} (X_{t}, C_{t})$ , then for any t ∈ T, it holds that $α ≺ S_{0} (X_{t}, C_{t})$ . Take any a, b ∈ J (L ^X) with a b, then there exists a t ∈ T, such that P _t ^→ (a), P _t ^→ (b) ∈ J (L ^{X
_t}) and P _t ^→ (a) nleqP _t ^→ (b).

Then $\begin{matrix} α & ≺ & S_{0} (X_{t}, C_{t}) \\ = & ⋀_{a_{t} b_{t}} (⋁_{a_{t} U_{t} \geq b_{t}} C_{t} (U_{t}) \lor ⋁_{b_{t} V_{t} \geq a_{t}} C_{t} (V_{t})) \\ \leq & ⋁_{{P_{t}}^{\to} (a) U_{t} \geq {P_{t}}^{\to} (b)} C_{t} (U_{t}) \lor \\ ⋁_{{P_{t}}^{\to} (b) V_{t} \geq {P_{t}}^{\to} (a)} C_{t} (V_{t}) \\ \leq & ⋁_{{P_{t}}^{\to} (a) U_{t} \geq {P_{t}}^{\to} (b)} C ({P_{t}}^{\leftarrow} (U_{t})) \lor \\ ⋁_{{P_{t}}^{\to} (b) V_{t} \geq {P_{t}}^{\to} (a)} C ({P_{t}}^{\leftarrow} (V_{t})) \\ \leq & ⋁_{a {P_{t}}^{\leftarrow} (U_{t}) \geq b} C ({P_{t}}^{\leftarrow} (U_{t})) \lor \\ ⋁_{b {P_{t}}^{\leftarrow} (V_{t}) \geq a} C ({P_{t}}^{\leftarrow} (V_{t})) \\ \leq & ⋁_{a A \geq b} C (A) \lor ⋁_{b B \geq a} C (B) . \end{matrix}$

This implies that $α \leq ⋀_{a b} (⋁_{a A \geq b} C (A) \lor ⋁_{b B \geq a} C (B)) = S_{0} (X, C) .$

Since α is arbitrary, it follows that $⋀_{t \in T} S_{0} (X_{t}, C_{t}) \leq S_{0} (X, C)$ .□

4. S ₁ separation axiom

In this section, we introduce the degree to which an (L, M)-fuzzy convex space is S ₁ separated, and discuss its relation with S ₀ in (L, M)-fuzzy convex space.

Definition 4.1. For an (L, M)-fuzzy convex space $(X, C)$ , we define the degree to which $(X, C)$ is S ₁ separated as follows: $S_{1} (X, C) = ⋀_{a \in J (L^{X})} C (a) .$

Remark 4.2. If L and M are replaced by {0, 1}, then Definition 4.1 reduce to Definition 2.5 (2). So we can see that Definition 4.1 is reasonable generalization of S ₁ separation axiom.

Theorem 4.3. Let $(X, C)$ be an (L, M)-fuzzy convex space. Then

S_{1} (X, C) = ⋀_{a b} ⋁_{a F \geq b} C (F) .

Proof. Take any b ∈ J (L ^X), by Theorem 2.13, 2.14 and 2.15, we can obtain that $\begin{matrix} C (b) & = & ⋀_{a b} (co (b) (a))^{'} \\ = & ⋀_{a b} ⋁_{a F \geq b} C (F) . \end{matrix}$

Then $S_{1} (X, C) = ⋀_{b \in J (L^{X})} C (b) = ⋀_{a b} ⋁_{a F \geq b} C (F) .$

Thus we complete the proof.□ By Theorem 4.3, we can obtain the following theorem.

Theorem 4.4. Let $(X, C)$ be an (L, M)-fuzzy convex space. Then $S_{1} (X, C) \leq S_{0} (X, C) .$

Proof. It is straightforward and omitted.□

Example 4.5. Let X = {x, y} and $C : I^{X} \to I$ be defined as follows: $C (A) = {\begin{matrix} 1 / 4, & A (x) > A (y); \\ 1 / 5, & A (x) < A (y); \\ 1, & A (x) = A (y) . \end{matrix}$

Then $(X, C)$ is an (I, I)-fuzzy convex space. From Definition 3.2 and 4.1, we have $\begin{matrix} S_{0} (X, C) & = & ⋀_{a b} (⋁_{a A \geq b} C (A) \lor ⋁_{a \leq B b} C (B)) \\ = & 1 / 4, \end{matrix}$ and $S_{1} (X, C) = ⋀_{a \in J (L^{X})} C (a) = 1 / 5 .$

Hence $S_{1} (X, C) < S_{0} (X, C) .$

In the classical case, S ₁ separation axiom has the heritability and the productive property. Next we investigate these properties in (L, M)-fuzzy convex spaces.

Theorem 4.6. Let $(X, C)$ be an (L, M)-fuzzy convex space, and $(Y, C |_{Y})$ be the subspace of $(X, C)$ . Then $S_{1} (X, C) \leq S_{1} (Y, C |_{Y}) .$

Proof. Since $C |_{Yitsc} (A) = ⋁ {C (B) : B \in L^{Xitsc}, B |_{Yitsc} =$ A } , we have $\begin{matrix} S_{1} (X, C) & = & ⋀_{a \in J (L^{X})} C (a) \\ \leq & ⋀_{b \in J (L^{Y})} C (b) \\ \leq & ⋀_{b \in J (L^{Y})} ⋁ {C (B) : B \in L^{X}, B |_{Y} = b} \\ = & ⋀_{b \in J (L^{Y})} C |_{Y} (b) = S_{1} (Y, C |_{Y}) . \end{matrix}$

This implies $S_{1} (X, C) \leq S_{1} (Y, C |_{Y}) .$ □

Theorem 4.7. Let $(X, C)$ be the product of (L, M)-fuzzy convex spaces ${(X_{t}, C_{t})}_{t \in T}$ . Then

⋀_{t \in T} S_{1} (X_{t}, C_{t}) \leq S_{1} (X, C) .

Proof. Take any α ∈ M with $α ≺ ⋀_{t \in T} S_{1} (X_{t}, C_{t})$ , then for any t ∈ T, it holds that $α ≺ S_{1} (X_{t}, C_{t})$ . Let a, b ∈ J (L ^X) with a b. Then there exists a t ∈ T, such that P _t ^→ (a), P _t ^→ (b) ∈ J (L ^{X
_t}) and P _t ^→ (a) nleqP _t ^→ (b). Therefore, $\begin{matrix} α & ≺ & S_{1} (X_{t}, C_{t}) \\ = & ⋀_{a_{t} b_{t}} {⋁_{a_{t} U_{t} \geq b_{t}} C_{titsc} (U_{titsc}) : a_{titsc} b_{titsc}, a_{titsc}, b_{titsc} \in J (L^{X_{titsc}})} \\ \leq & ⋁_{{P_{t}}^{\to} (a) U_{t} \geq {P_{t}}^{\to} (b)} C (U_{t}) \\ \leq & ⋁_{a {P_{t}}^{\leftarrow} (U_{t}) \geq b} C ({P_{t}}^{\leftarrow} (U_{t})) \\ = & ⋁_{a A \geq b} C (A) . \end{matrix}$

This implies that $α \leq ⋀_{a b} ⋁_{a A \geq b} C (A) = S_{1} (X, C) .$

Since α is arbitrary, we have $⋀_{t \in T} S_{1} (X_{t}, C_{t}) \leq S_{1} (X, C)$ .□

5. S ₂ separation axiom

In this section, we introduce the degree to which an (L, M)-fuzzy convex space is S ₂ separated, and discuss its relation with the degree to which an (L, M)-fuzzy convex space is S ₁ separated.

Firstly, we introduce the degree to which a set is biconvex set in (L, M)-fuzzy convex spaces.

Definition 5.1. Let $(X, C)$ be an (L, M)-fuzzy convex space and A ∈ L ^X. The degree $H_{C} (A)$ to which A is biconvex set (half-space, hemispace) is defined by $H_{C} (A) = C (A) \land C (A^{'}) .$

Remark 5.2. If L and M are replaced by {0, 1}, then Definition 5.1 reduce to Definition 2.3. So we can see that Definition 5.1 is reasonable generalization of biconvex set.

Theorem 5.3. Let $(X, C)$ and $(Y, D)$ be (L, M)-fuzzy convex spaces, $f : (X, C) \to (Y, D)$ be (L, M)-fuzzy convex preserving function. Then

H_{D} (A) \leq H_{C} (f^{\leftarrow} (A)) .

Proof. By Definition 1, we have $\begin{matrix} H_{D} (A) & = & D (A) \land D (A^{'}) \\ \leq & C (f^{\leftarrow} (A)) \land C (f^{\leftarrow} (A^{'})) \\ = & C (f^{\leftarrow} (A)) \land C ((f^{\leftarrow} (A))^{'}) \\ = & H_{C} (f^{\leftarrow} (A)) . \end{matrix}$

Thus we complete the proof.□

Theorem 5.4. Let $(X, C)$ be an (L, M)-fuzzy convex space, $(Y, C |_{Y})$ be the subspace of $(X, C)$ , A ∈ L ^X. Then $H_{C} (A) \leq H_{C |_{Yitsc}} (A |_{Yitsc}) .$

Proof. From $\begin{matrix} C |_{Y} (A |_{Y}) & = & ⋁ {C (B) : B \in L^{X}, B |_{Y} = A |_{Y}} \\ ⩾ & C (A), \end{matrix}$ and $\begin{matrix} C |_{Y} ((A |_{Y})^{'}) & = & ⋁ {C (B) : B \in L^{X}, B |_{Y} = (A |_{Y})^{'}} \\ \geq & C (A^{'}), \end{matrix}$ we can obtain $C (A) \land C (A^{'}) \leq C |_{Yitsc} (A |_{Yitsc}) \land C |_{Yitsc} ((A |_{Yitsc})^{'}) .$

Therefore, $H_{C} (A) \leq H_{C |_{Yitsc}} (A |_{Yitsc}) .$ □ Next we introduce the degree to which an (L, M)-fuzzy convex space is S ₂ separated by using biconvex set.

Definition 5.5. For an (L, M)-fuzzy convex space $(X, C)$ , we define the degree to which $(X, C)$ is S ₂ separated as follows: $S_{2} (X, C) = ⋀_{a b} ⋁ {H_{C} (A) : A \in L^{X}, a A, b \leq A} .$

Remark 5.6. If L and M are replaced by {0, 1}, then Definition 5.5 reduce to Definition 2.5 (3). So we can see that Definition 5.5 is reasonable generalization of S ₂ separation axiom.

Theorem 5.7. Let $(X, C)$ be an (L, M)-fuzzy convex space. Then $S_{2} (X, C) \leq S_{1} (X, C) .$

Proof. Since $S_{1} (X, C) = ⋀_{a b} ⋁_{a F \geq b} C (F),$ and $\begin{matrix} S_{2} (X, C) \\ = ⋀_{a b} ⋁ {C (A) \land C (A^{'}) : A \in L^{X}, a A, b \leq A}, \end{matrix}$ we can obtain that $S_{2} (X, C) \leq S_{1} (X, C) .$ □

Example 5.8. In Example 1, $S_{1} (X, C) = 1 / 5 .$ From Definition 5.5, we can obtain $S_{2} (X, C) = C (x_{λ}) \land C (y_{μ}) = 1 / 5 .$

Hence, $S_{2} (X, C) = S_{1} (X, C) .$

In the classical case, S ₂ separation axiom has the heritability and the productive property. Next we investigate these properties in (L, M)-fuzzy convex spaces.

Theorem 5.9. Let $(X, C)$ be an (L, M)-fuzzy convex space, and $(Y, C |_{Y})$ be the subspace of $(X, C)$ . Then $S_{2} (X, C) ⩽ S_{2} (Y, C |_{Y}) .$

Proof. By Theorem 1, we can obtain $\begin{matrix} S_{2} (X, C) \\ = & ⋀_{\binom{a b}{a, b \in J (L^{X})}} ⋁ {H_{C} (A) : A \in L^{X}, a \leq A, b A} \\ ⩽ & ⋀_{\binom{a b}{a, b \in J (L^{X})}} ⋁ {H_{C |_{Yitsc}} (A |_{Y}) : \\ A \in L^{X}, a \leq A |_{Y}, b A |_{Y}} \\ ⩽ & ⋀_{\binom{c d}{c, d \in J (L^{Y})}} ⋁ {H_{C |_{Yitsc}} (B) : B \in L^{Y}, c \leq B, d B} \\ = & S_{2} (Y, C |_{Y}) . \end{matrix}$

Thus we complete the proof.□

Theorem 5.10. Let $(X, C)$ be the product of (L, M)-fuzzy convex spaces ${(X_{t}, C_{t})}_{t \in T}$ . Then

⋀_{t \in T} S_{2} (X_{t}, C_{t}) ⩽ S_{2} (X, C) .

Proof. Take any α ∈ M with $α ≺ ⋀_{t \in T} S_{2} (X_{t}, C_{t})$ , then for any t ∈ T, it holds that $α ≺ S_{2} (X_{t}, C_{t})$ . Let a, b ∈ J (L ^X) and a b. Then there exists a t ∈ T, such that P _t ^→ (a), P _t ^→ (b) ∈ J (L ^{X
_t}) and P _t ^→ (a) nleqP _t ^→ (b).

Therefore, $\begin{matrix} α & ≺ & S_{2} (X_{t}, C_{t}) \\ = & ⋀_{\binom{a_{t} b_{t}}{a_{t}, b_{t} \in J (L^{X_{t}})}} ⋁_{a_{t} U_{t} \geq b_{t}} C_{t} (U_{t}) \land C_{t} (U_{t}^{'}) \\ ⩽ & ⋁_{{P_{t}}^{\to} (a) U_{t} \geq {P_{t}}^{\to} (b)} C_{t} (U_{t}) \land C_{t} (U_{t}^{'}) \\ ⩽ & ⋁_{{P_{t}}^{\to} (a) U_{t} \geq {P_{t}}^{\to} (b)} C ({P_{t}}^{\leftarrow} (U_{t})) \land C ({P_{t}}^{\leftarrow} (U_{t}^{'})) \\ ⩽ & ⋁_{a U \geq b} C (U) \land C (U^{'}) \\ = & ⋁_{a U \geq b} H_{C} (U) . \end{matrix}$

This implies that

$\begin{matrix} α & \leq & ⋀_{\binom{a b}{a, b \in J (L^{X})}} ⋁ {H_{C} (A) : A \in L^{X}, a A, b \leq A} \\ = & S_{2} (X, C) . \end{matrix}$

Since α is arbitrary, it follows that $⋀_{t \in T} S_{2} (X_{t}, C_{t}) \leq S_{2} (X, C)$ .□

6. convex-to-convex and convexity preserving function

In this section, we define the degrees to which a function is convex-to-convex or convexity preserving in (L, M)-fuzzy convex spaces, and discuss their relationships with the degrees of S ₀, S ₁ and S ₂ separation axioms in (L, M)-fuzzy convex spaces.

Definition 6.1. Let $f : (X, C) \to (Y, D)$ be a function between two (L, M)-fuzzy convex spaces. Then

the degree CC (f) to which f is convex-to-convex is defined by $CC (f) = ⋀_{A \in L^{X}} (C (A) \to D (f^{\to} (A)));$

the degree CP (f) to which f is convexity preserving is defined by $CP (f) = ⋀_{B \in L^{Y}} (D (B) \to C (f^{\leftarrow} (B))) .$

Remark 6.2. (1) If CC (f) = ⊤ _M, by Lemma 2.1, we know $C (A) \leq D (f^{\to} (A))$ for all A ∈ L ^X . This is just the definition of convex-to-convex function between two (L, M)-fuzzy convex spaces. (2) If CP (f) = ⊤ _M, by Lemma 2.1, we know $D (B) \leq C (f^{\leftarrow} (B))$ for all B ∈ L ^Y . This is just the definition of convexity preserving function between two (L, M)-fuzzy convex spaces.

Definition 6.3. Let $f : (X, C) \to (Y, D)$ be a bijective mapping between two (L, M)-fuzzy convex spaces. Then the isomorphism degree of f is defined by $ISO (f) = CC (f) \land CP (f) .$

Theorem 6.4. Let $(X, C)$ , $(Y, D)$ and $(Z, E)$ be (L, M)-fuzzy convex spaces, f : X → Y and g : Y → Z be two mappings. Then

CP (f)∧ CP (g) ≤ CP (g ∘ f);

CC (f) ∧ CC (g) ≤ CC (g ∘ f) .

Proof. We only prove (1). The proof of (2) is similar. By Lemma 2.1 (4), we have $\begin{matrix} CP (f) \land CP (g) \\ = & ⋀_{B \in L^{Y}} (D (B) \to C (f^{\leftarrow} (B))) \\ \land ⋀_{D \in L^{Z}} (E (D) \to D (g^{\leftarrow} (D))) \\ \leq & ⋀_{D \in L^{Z}} (D (g^{\leftarrow} (D)) \to C (f^{\leftarrow} (g^{\leftarrow} (D)))) \\ \land ⋀_{D \in L^{Z}} (E (D) \to D (g^{\leftarrow} (D))) \\ = & ⋀_{D \in L^{Z}} ((D (g^{\leftarrow} (D)) \to C ((g \circ f)^{\leftarrow} (D))) \\ \land (E (D) \to D (g^{\leftarrow} (D)))) \\ \leq & ⋀_{D \in L^{Z}} (E (D) \to C ((g \circ f)^{\leftarrow} (D))) \\ = & CP (g \circ f) . \end{matrix}$ Thus we complete the proof.□ By Definition 6.3 and Theorem 6.4, we can get the following result.

Corollary 6.5. Let $(X, C)$ , $(Y, D)$ and $(Z, E)$ be (L, M)-fuzzy convex spaces, f : X → Y and g : Y → Z be bijective mappings. Then ISO (f) ∧ ISO (g) ≤ ISO (g ∘ f) .

Proof. It is straightforward and omitted.□

Theorem 6.6. Let $(X, C)$ , $(Y, D)$ and $(Z, E)$ be (L, M)-fuzzy convex spaces, f : X → Y and g : Y → Z be two mappings.

If f : X → Y is surjective, then CC (g∘ f) ∧ CP (f) ≤ CC (g);

If g : Y → Z is injective, then CC (g ∘ f) ∧ CP (g) ≤ CC (f) .

Proof. We only prove (1). The proof of (2) is similar. Since f is a surjective mapping, we obtain (g ∘ f) ^→ (f ^← (B)) = g ^→ (B), ∀ B ∈ L ^Y . By Lemma 2.1 (4), we have $\begin{matrix} CC (g \circ f) \land CP (f) \\ = & ⋀_{A \in L^{X}} (C (A) \to E ((g \circ f)^{\to} (A))) \\ \land ⋀_{B \in L^{Y}} (D (B) \to C (f^{\leftarrow} (B))) \\ \leq & ⋀_{B \in L^{Y}} (C (f^{\leftarrow} (B)) \to E ((g \circ f)^{\to} (f^{\leftarrow} (B)))) \\ \land ⋀_{B \in L^{Y}} (D (B) \to C (f^{\leftarrow} (B))) \\ = & ⋀_{B \in L^{Y}} ((C (f^{\leftarrow} (B)) \to E (g^{\to} (B))) \\ \land (D (B) \to C (f^{\leftarrow} (B)))) \\ \leq & ⋀_{B \in L^{Y}} (D (B) \to E (g^{\to} (B))) = CC (g) . \end{matrix}$ The proof is completed.□

Next, we discuss relationships among the degrees to which a function is convex-to-convex or convexity preserving, and the degrees of S ₀, S ₁ and S ₂ separation axioms in (L, M)-fuzzy convex spaces.

Theorem 6.7. Let $(X, C), (Y, D)$ be two (L, M)-fuzzy convex spaces, and f : X → Y be a bijective mapping. Then

$S_{0} (X, C) \land CC (f) \leq S_{0} (Y, D);$

$S_{1} (X, C) \land CC (f) \leq S_{1} (Y, D);$

$S_{2} (X, C) \land CC (f) \leq S_{2} (Y, D)$ .

Proof. We only prove (1) and (3). The proof of (2) is similar to (3).

(1) Take any α ∈ M such that $α ≺ S_{0} (X, C) \land CC (f),$ by Definition 3.2 we know $\begin{matrix} α ≺ S_{0} (X, C) \\ = ⋀_{\binom{a b}{a, b \in L^{X}}} (⋁_{a A \geq b} C (A) \lor ⋁_{a \leq B b} C (B)), \end{matrix}$ and

$α ≺ CC (f) = ⋀_{A \in L^{X}} (C (A) \to D (f^{\to} (A))) .$

Then for any a, b ∈ J (L ^X) with a b, there exists A ∈ L ^X with a A ≥ b such that $α \leq C (A),$ or there exists B ∈ L ^X with b B ≥ a such that $α \leq C (B) .$ For any $F \in L^{X}, α \leq C (F) \to D (f^{\to} (F)),$ by Lemma 2.1, we have $α \land C (F) \leq D (f^{\to} (F)) .$

In order to prove

$\begin{matrix} α & \leq & S_{0} (Y, D) \\ = & ⋀_{\binom{c d}{c, d \in J (L^{Y})}} (⋁_{c C \geq d} D (C) \lor ⋁_{c \leq D \geq d} D (D)), \end{matrix}$ let c, d ∈ J (L ^Y) with c d . Since f is bijective mapping, there exist a, b ∈ J (L ^X) with a b, such that c = f ^→ (a) and d = f ^→ (b) . From a b, we know there exists A ∈ L ^X with a A ≥ b such that $α \leq C (A),$ or there exists B ∈ L ^X with b B ≥ a such that $α \leq C (B) .$ This implies, $\begin{matrix} α & \leq & (α \land C (A)) \lor (α \land C (B)) \\ \leq & D (f^{\to} (A)) \lor D (f^{\to} (B)) \\ \leq & ⋁_{a A \geq b} D (f^{\to} (A)) \lor ⋁_{b B \geq a} D (f^{\to} (B)) \\ = & ⋁_{c f^{\to} (A) \geq d} D (f^{\to} (A)) \lor \\ ⋁_{d f^{\to} (B) \geq c} D (f^{\to} (B)) \\ = & ⋁_{c E \geq d} D (E) \lor ⋁_{d F \geq c} D (F) . \end{matrix}$

Therefore, $\begin{matrix} α & \leq & ⋀_{\binom{c d}{c, d \in J (L^{Y})}} (⋁_{c E \geq d} D (E) \lor ⋁_{c \leq F d} D (F)) \\ = & S_{0} (Y, D) . \end{matrix}$

Since α is arbitrary, we know $S_{0} (X, C) \land CC (f) \leq S_{0} (Y, D)$ .

(3) Take any α ∈ M with $α ≺ S_{2} (X, C) \land CC (f),$ by Definition 5.5 we know $α ≺ S_{2} (X, C) = ⋀_{\binom{a b}{a, b \in J (L^{X})}} ⋁_{a F \geq b} (C (F) \land C (F^{'})),$ and $α ≺ CC (f) = ⋀_{A \in L^{X}} (C (A) \to D (f^{\to} (A))) .$

Then for any a, b ∈ J (L ^X) with a b, there exists F ∈ L ^X with a F ≥ b such that $α \leq C (F) \land C (F^{'})$ . For any $A \in L^{X}, α \leq C (A) \to D (f^{\to} (A)),$ by Lemma 2.1, we have $α \land C (A) \leq D (f^{\to} (A)) . (1)$

In order to prove $α \leq S_{2} (Y, D) = ⋀_{\binom{c d}{c, d \in J (L^{Y})}} ⋁_{c E \geq d} (C (E) \land C (E^{'})),$ let c, d ∈ J (L ^Y) with c d . Since f is bijective mapping, there exist a, b ∈ J (L ^X) with a b, such that c = f ^→ (a) and d = f ^→ (b) . From a b, we can obtain there exists F ∈ L ^X such that a F ≥ b and $α \leq C (F) \land C (F^{'}) .$ Then c = f ^→ (a) nleqf ^→ (F) ≥ f ^→ (b) = d . By (1), we know $α = α \land C (F) \leq D (f^{\to} (F)),$ and $α = α \land C (F^{'}) \leq D (f^{\to} (F^{'})) = D ((f^{\to} (F))^{'}) .$

This implies, $\begin{matrix} α & \leq & D (f^{\to} (F)) \land D ((f^{\to} (F))^{'}) \\ \leq & ⋁_{c E \geq d} (D (E) \land D (E^{'})) . \end{matrix}$

Therefore, $α \leq ⋀_{\binom{c d}{c, d \in J (L^{Y})}} ⋁_{c E \geq d} (D (E) \land D (E^{'})) = S_{2} (Y, D) .$

Since α is arbitrary, we know $S_{2} (X, C) \land CC (f) \leq S_{2} (Y, D)$ .□

Theorem 6.8. Let $(X, C), (Y, D)$ be two (L, M)-fuzzy convex spaces, and f : X → Y be a bijective mapping. Then

$S_{0} (Y, D) \land CP (f) \leq S_{0} (X, C)$ ;

$S_{1} (Y, D) \land CP (f) \leq S_{1} (X, C)$ ;

$S_{2} (Y, D) \land CP (f) \leq S_{2} (X, C)$ .

Proof. We only prove (1) and (3). The proof of (2) is similar to (3).

(1) Take any α ∈ M with $α ≺ S_{0} (Y, D) \land CP (f),$ by Definition 3.2 we know $\begin{matrix} α & ≺ & S_{0} (Y, D) \\ = & ⋀_{\binom{c d}{c, d \in J (L^{Y})}} (⋁_{c C \geq b} D (C) \lor ⋁_{c \leq D d} D (D)), \end{matrix}$ and $α ≺ CP (f) = ⋀_{E \in L^{Y}} (D (E) \to C (f^{\leftarrow} (E))) .$

Then for any c, d ∈ J (L ^Y) with c d, there exists C ∈ L ^Y with c C ≥ d such that $α \leq D (C),$ or there exists D ∈ L ^Y with d D ≥ c such that $α \leq D (D) .$ For any $E \in L^{Y}, α \leq D (E) \to C (f^{\leftarrow} (E)) .$ By Lemma 2.1, we have $α \land D (E) \leq C (f^{\leftarrow} (E)) .$

In order to prove $\begin{matrix} α & \leq & S_{0} (X, C) \\ = & ⋀_{\binom{a b}{a, b \in J (L^{X})}} (⋁_{a A \geq b} C (A) \lor ⋁_{a \leq B \geq b} C (B)), \end{matrix}$ let a, b ∈ J (L ^X) with a b . Since f is bijective mapping, there exist c, d ∈ J (L ^Y) with c d, such that a = f ^← (c) and b = f ^← (d) . From c d, we have there exists C ∈ L ^Y with c C ≥ d such that $α \leq D (C),$ or there exists D ∈ L ^Y with d D ≥ c such that $α \leq D (D) .$ This implies, $\begin{matrix} α & = & (α \land D (C)) \lor (α \land D (D)) \\ \leq & C (f^{\leftarrow} (C)) \lor C (f^{\leftarrow} (D)) \\ \leq & ⋁_{c C \geq d} C (f^{\leftarrow} (C)) \lor ⋁_{d D \geq c} C (f^{\leftarrow} (D)) \\ = & ⋁_{a f^{\leftarrow} (C) \geq b} C (f^{\leftarrow} (C)) \lor \\ ⋁_{b f^{\leftarrow} (D) \geq a} C (f^{\leftarrow} (D)) \\ = & ⋁_{a A \geq b} C (A) \lor ⋁_{b B \geq a} C (B) . \end{matrix}$

Therefore, $\begin{matrix} α & \leq & ⋀_{\binom{a b}{a, b \in J (L^{X})}} (⋁_{a A \geq b} C (A) \lor ⋁_{a \leq B b} C (B)) \\ = & S_{0} (X, C) . \end{matrix}$

Since α is arbitrary, we know $S_{0} (Y, D) \land CP (f) \leq S_{0} (X, C)$ .

(3) Take any α ∈ M such that $α ≺ S_{2} (Y, D) \land CP (f),$ by Definition 5.5 we know $α ≺ S_{2} (Y, D) = ⋀_{\binom{c d}{c, d \in J (L^{Y})}} ⋁_{c E \geq d} (D (E) \land D (E^{'})),$ and $α ≺ CP (f) = ⋀_{C \in L^{Y}} (D (C) \to C (f^{\leftarrow} (C))) .$ Then for any c, d ∈ J (L ^Y) with c d, there exists E ∈ L ^Y with c E ≥ d such that $α \leq D (E) \land D (E^{'}) .$ For any $C \in L^{Y}, α \leq D (C) \to C (f^{\leftarrow} (C)) .$ By Lemma 2.1, we have $α \land D (C) \leq C (f^{\leftarrow} (C)) . (1)$

In order to prove $α \leq S_{2} (X, C) = ⋀_{\binom{a b}{a, b \in J (L^{X})}} ⋁_{a F \geq b} (C (F) \land C (F^{'})),$ let a, b ∈ J (L ^X) with a b . Since f is bijective mapping, there exist c, d ∈ J (L ^Y) with c d such that a = f ^← (c) and b = f ^← (d) . From c d, there exists E ∈ L ^Y with c E ≥ d such that $α \leq D (E) \land D (E^{'}) .$ Then a = f ^← (c) nleqf ^← (E) ≥ f ^← (d) = b . By (1), we know $\begin{matrix} α & \leq & α \land (D (E) \land D (E^{'})) \\ = & (α \land (D (E)) \land (α \land D (E^{'})) \\ \leq & C (f^{\leftarrow} (E)) \land C (f^{\leftarrow} (E^{'})) \\ \leq & ⋁_{a F \geq b} (C (F) \land C (F^{'})) . \end{matrix}$

This implies, $α \leq ⋀_{\binom{a b}{a, b \in J (L^{X})}} ⋁_{a F \geq b} (C (F) \land C (F^{'})) = S_{2} (X, C) .$

Since α is arbitrary, we know $S_{2} (Y, D) \land CP (f) \leq S_{2} (X, C)$ .□

By Theorem 6.7 and 6.8, we can obtain the following theorem.

Theorem 6.9. Let $(X, C), (Y, D)$ be two (L, M)-fuzzy convex spaces, f : X → Y be a bijective mapping. Then

$S_{i} (X, C) \land ISO (f) \leq S_{i} (Y, D), (i = 0, 1, 2);$

$S_{i} (Y, D) \land ISO (f) \leq S_{i} (X, C), (i = 0, 1, 2) .$

Proof. It is straightforward and omitted.□ By the above theorem, we can easily obtain the following corollary.

Corollary 6.10. Let $(X, C), (Y, D)$ be two (L, M)-fuzzy convex spaces, and f : X → Y be a bijective mapping. Then

\begin{matrix} S_{i} (X, C) \land ISO (f) \\ = S_{i} (Y, D) \land ISO (f), (i = 0, 1, 2) . \end{matrix}

Proof. It is straightforward and omitted.□

7. Conclusions

The theory of fuzzy convex structures is a new branch of fuzzy mathematics. In this paper, we firstly introduced the degrees to which an (L, M)-fuzzy convex space is S ₀, S ₁ and S ₂ separated. We discussed the relations among them, and give a lot of examples to show the relations. Also, we investigated the degrees to which a function is convex-to-convex or convexity preserving in (L, M)-fuzzy convex spaces, and their relationships with the degrees of S ₀, S ₁ and S ₂ separation axioms in (L, M)-fuzzy convex spaces are discussed.

Footnotes

Acknowledgement

The authors would like to express their sincere thanks to the anonymous reviewers for their careful reading and constructive comments. This work is supported by Beijing Natural Science Foundation (1174014), the Outstanding Talents Program of Beijing (2016000020124G017), and the Yuyou Young Talent Program of North China University of Technology.

References

C.L.

Chang , Fuzzy topological spaces, J Math Anal Appl 24 (1968), 182–190.

Höhle and

A.P.

šostak , Axiomatic foudations of fixed-basis fuzzy topology, in:

Höhle and

S.E.

Rodabaugh (Eds.), Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, Handbook Series, vol.3, Kluwer Academic Publishers, Boston, Dordrecht, London, 1999, pp. 123–173.

R.E.

Jamison , A General Theory of Convexity, Dissertation, University of Washington, Seattle, Washington, 1974.

R.E.

Jamison-Waldner , A Perspective on Abstract Convexity: Classifying Alignments by varieties in “Convexity and related Combinatorial Geometry” (

Kay and

Breen Eds), Decker, New York, 1982, pp. 113- 150.

Kubiak , On fuzzy topologies, Ph. D.Thesis, Adam Mickiewicz, Poznan, Poland, 1985.

E.Q.

Li and

F.-G.

Shi , Some properties of M-fuzzifying convexities induced by M-orders, Fuzzy Sets Syst 350 (2018), 41–54.

L.Q.

Li ,

Jin and

Hu , Lattice-valued convergence associated with CNS spaces, Fuzzy Sets Syst (2018). https://doi.org/10.1016/j.fss.2018.05.023

L.Q.

Li ,

Jin ,

Hu and

F.F.

Zhao , The axiomatic characterizations on L-fuzzy covering-based approximation operators, International Journal of General Systems 46(4) (2017), 332–353.

L.Q.

Li , On the category of enriched (L;M)-convex spaces, JIntell Fuzzy Syst 33 (2017), 3209–3216.

10.

Maruyama , Lattice-valued fuzzy convex geometry, RIMS Kokyuroku 164 (2009), 22–37.

11.

Pang and

Zhao , Characterizations of L-convex spaces, Iran J Fuzzy Syst 13 (2016), 51–61.

12.

Pang and

F.-G.

Shi , Subcategories of the category of L-convex spaces, Fuzzy Sets Syst 313 (2017), 61–74.

13.

Pang , Degrees of separation properties in stratified L-generalized convergence spaces using residual implication, Filomat 31(20) (2017), 6293–6305.

14.

Pang , Stratified L-ordered filter spaces, Quaest Math 40(5) (2017), 661–678.

15.

Pang and

Z.-Y.

Xiu , An axiomatic approach to bases and subbases in L-convex spaces and their applications, Fuzzy Sets and Systems (2018). https://doi.org/ 10.1016/j.fss.2018.08.002

16.

Pang and

F.-G.

Shi , Strong inclusion orders between L-subsets and its applications in L-convex spaces, Quaest Math 41(8) (2018), 1021–1043.

17.

Pang , (L, M)-fuzzy convex spaces: (restricted) hull operators, bases and subbases, submitted.

18.

G.N.

Raney , A subdirect-union representation for completely distibutive complete lattices, Proc Amer Math Soc 4 (1953), 518–522.

19.

M.V.

Rosa , On fuzzy topology fuzzy convexity spaces and fuzzy local convexity, Fuzzy Sets Syst 62 (1994), 97–100.

20.

M.V.

Rosa , A study of fuzzy convexity with special reference to separation properties [Ph. D. thesis], Cochin University of Science and Technology, 1994.

21.

S.E.

Rodabaugh , Powerset operator foundations for poslat fuzzy set theories and topologies, in:

Höhle and

S.E.

Rodabaugh (Eds.), Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, Handbook Series, vol.2, Kluwer Academic Publishers, Boston, Dordrecht, London, 1999, pp. 91–116.

22.

A.P.

Sostak , On a fuzzy topological structure, Rend Circ Mat Palermo (Suppl Ser II) 11 (1985), 89–103.

23.

F.-G.

Shi and

Z.-Y.

Xiu , A new approach to the fuzzification of convex structures, J Appl Math 2014 (2014), 1–12.

24.

F.-G.

Shi and

Z.-Y.

Xiu , (L, M)-fuzzy convex structures, J Nolinear Sci Appl 10(7) (2017), 3655–3669.

25.

F.-G.

Shi and

E.Q.

Li , The restricted hull operator of M-fuzzifying convex structures, J Intell Fuzzy Syst 30 (2015), 409–421.

26.

V.P.

Soltan , D-convexity in graphs, Soviet Math Dokl 28 (1983), 419–421.

27.

V.P.

Soltan , Introduction to the axiomatic theory of convexity, (Russian) Shtiinca, Kishinev, 1984.

28.

F.A.

Valentine , Convex sets, McGraw-Hill, 1964.

29.

Van De Vel , Theory of convex structures, North-Holland, Amsterdam, 1993.

30.

Van Mill , Supercompactness and wallman spaces, Math Cent Tracts 85 (1977), Amsterdam.

31.

J.C.

Varlet , Remarks on distributive lattices, Bull Acad Pol Sci 23 (1975), 1143–1147.

32.

M.D.

Weiss , Fixed points, separation and induced topologies for fuzzy sets, J Math Anal Appl 50 (1975), 142–150.

33.

Z.-Y.

Xiu and

Pang , Base axioms and subbase axioms in M-fuzzifying convex spaces, Iran J Fuzzy Syst 15(2) (2018), 75–87.

34.

M.-S.

Ying , A new approach for fuzzy topology (I), Fuzzy Sets Syst 39 (1991), 303–321.

Separation axioms in ( L,M )-fuzzy convex spaces

Abstract

1. Introduction

2. Preliminaries

3. S 0 separation axiom

4. S 1 separation axiom

5. S 2 separation axiom

6. convex-to-convex and convexity preserving function

7. Conclusions

Footnotes

Acknowledgement

References

3. S ₀ separation axiom

4. S ₁ separation axiom

5. S ₂ separation axiom