Separation axioms in fuzzy convex spaces

Abstract

In fuzzy set theory, fuzzy convex structures are important mathematical structures. In this paper, we focus on separation axioms in fuzzy convex spaces. Concretely, we introduce S₀, S₁ and S₂ separation axioms in fuzzy convex spaces and establish their relationships. Furthermore, we investigate their hereditary and productive properties.

Keywords

Fuzzy convex structures separation axioms

1 Introduction

Convexity, which is a branch of geometry, analysis and linear algebra, plays an important role in mathematics. Convexity exists in numerous mathematical environment such as vector spaces, lattices, algebras, graphs and topological spaces. In 1993, M. van de Vel collected the theory of convexity systematically in his famous book [21].

With the development of fuzzy set theory, many mathematical structures have been generalized to fuzzy case, such as fuzzy topologies [1, 4] and fuzzy convergence structures [8, 9]. Following this trend, convex structures have also been endowed with fuzzy set theory. Up to now, there have been three typical kinds of fuzzy convex structures, including L-convex structures [7, 16], M-fuzzifying convex structures [19] and (L, M)-fuzzy convex structures [20]. Many researchers studied fuzzy convex structures from different aspects, such as fuzzy hull operators [10, 13], fuzzy interval operators [14 , 34], categorical properties [12 , 30], convergence structures [29], degree presentations [6 , 31] and geometric properties [26, 27].

Separation is an essential property in the theory of convex spaces. Ellis [2] gave a common generalization of Kakutani Separation Property in real vector spaces and of the Stone Theorem in distributive lattices. Jamison [3] gave a restricted version of the Polytope Characterization in terms of screening with half-spaces to characterize separation properties of general convex spaces. In the fuzzy case, Liang and Li [5] first studied the separation property of M-fuzzifying convex spaces. In a degree approach, they equipped each M-fuzzifying convex space with some degree to be S₀, S₁ and S₂ separated. By this motivation, we will consider separation axioms in fuzzy convex spaces, which is exactly [0,1]-convex spaces. Concretely, we will propose S₀, S₁ and S₂ separation axioms in fuzzy convex spaces. Furthermore, we will study their mutual relationships as well as their hereditary and productive properties.

2 Preliminaries

Throughout this paper, let X be a nonempty set and 2^X be the powerset of X. Each mapping A : X ⟶ [0, 1] is called a fuzzy set of X, the complement of A, denoted by A′, is defined by A′ (x) =1 - A (x) for each x ∈ X. The collection of all fuzzy sets of X is denoted by $F (X)$ . The set of all fuzzy points x_λ(i.e., a fuzzy set $A \in F (X)$ such that A (x) = λ ≠ 0 and A (y) =0 for y ≠ x) is denoted by $pt (F (X))$ . For each a ∈ [0, 1], $\underline{a}$ denotes the constant mapping X ⟶ [0, 1] , x ⟼ a, which is called a constant fuzzy subset. $F (X)$ is also a complete lattice by defining ⩽ on $F (X)$ pointwisely. For each U ⊆ X, its characteristic function $χ_{U} \in F (X)$ is defined as follows: $χ_{U} (x) = {\begin{matrix} 1, & x \in U, \\ 0, & x \notin U . \end{matrix}$ For S ⊆ [0, 1], write ⋁S for the least upper bound of S and ⋀S for the greatest lower bound of S. We say {A_j} _j∈J is a directed subset of $F (X)$ , if for each A_{j
₁}, A_{j
₂} ∈ {A_j} _j∈J, there exists A_{j
₃} ∈ {A_j} _j∈J such that A_{j
₁}, A_{j
₂} ⩽ A_{j
₃}. Also, we adopt the convention that ⋁ ∅ =0 and ⋀ ∅ =1.

Given a mapping f : X ⟶ Y, define $f^{\to} : F (X) ⟶ F (Y)$ and $f^{\leftarrow} : F (Y) ⟶ F (X)$ by f^→ (A) (y) = ⋁ _f(x)=yA (x) for $A \in F (X)$ and y ∈ Y, and f^← (B) = B ∘ f for $B \in F (Y)$ .

Definition 2.1. ([21]) A subset $E$ of 2^X is called a convex structure on X if it satisfies the following conditions:

$\emptyset, X \in E$ ;

if ${U_{i} : i \in I} \subseteq E$ is nonempty, then $⋂_{i \in I} U_{i} \in E$ ;

if ${U_{j} : j \in J} \subseteq E$ is nonempty and totally ordered by inclusion, then $⋃_{j \in J} U_{j} \in E$ .

For a convex structure

E

on X, the pair

(X, E)

is called a convex space and each

U \in E

is called a convex set.

Definition 2.2. ([21]) Let $(X, E)$ be a convex space. A subset H of X is called a biconvex set (half-space, hemispace) provided H is convex and H′ is convex.

Definition 2.3. ([21]) Let $(X, E)$ be a convex space.

$(X, E)$ is said to be S₁ separated provided that all singletons in X are convex.

$(X, E)$ is said to be S₂ separated provided that for all x, y ∈ X with x ≠ y, there is a biconvex set H of X with x ∈ H and y ∉ H.

Definition 2.4. ([7, 16]) A subset $C$ of $F (X)$ is called a fuzzy convex structure on X if it satisfies the following conditions:

$χ_{\emptyset}, χ_{X} \in C$ ;

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

if ${A_{j} ∣ j \in J} \subseteq C$ is directed, then $⋁_{j \in J} A_{j} \in C$ .

For a fuzzy convex structure

C

on X, the pair

(X, C)

is called a fuzzy convex space and each

A \in C

is called a fuzzy convex set.

Definition 2.5. ([7, 16]) Let $(X, C)$ be a fuzzy convex space. For each $A \in F (X)$ , define $co (A) = ⋀ {B \in C ∣ A ⩽ B} .$ That is, co (A) is the least element of $C$ that contains A, called the fuzzy convex hull of A.

The following definitions and propositions are proposed for a more general lattice background. Here we only use the special case [0, 1] as the table of truth-values.

Definition 2.6. ([20]) Let $(X, C)$ be a fuzzy convex space and ∅ ≠ Y ⊆ X. Then $C |_{Y} = {C |_{Y} ∣ C \in C}$ is a fuzzy convex structure on Y. We call $(Y, C |_{Y})$ a subspace of $(X, C)$ .

Definition 2.7. ([12]) A fuzzy convex structure $C$ on X is called stratified if it satisfies:

∀a ∈ [0, 1], $\underline{a} \in C$ .

For a stratified fuzzy convex structure on X, the pair

(X, C)

is called a stratified fuzzy convex space.

Definition 2.8. ([12]) Let $(X, C_{X})$ and $(Y, C_{Y})$ be fuzzy convex spaces. A mapping f: X ⟶ Y is called

fuzzy convexity preserving (fuzzy CP for short) if $f^{\leftarrow} (B) \in C_{X}$ for each $B \in C_{Y}$ ;

fuzzy convex-to-convex (fuzzy CC for short) if $f^{\to} (A) \in C_{Y}$ for each $A \in C_{X}$ ;

isomorphic if f is a fuzzy CP and fuzzy CC bijection.

Definition 2.9. ([15]) Let ${(X_{i}, C_{i})}_{i \in I}$ be a family of fuzzy convex spaces, $X = \prod_{i \in I} X_{i}$ , and let {p_i : X ⟶ X_i} _i∈I be the family of projection mappings. The fuzzy product convex structure $C$ on X is the one generated by the subbase ${p_{i}^{\leftarrow} (C_{i}) | C_{i} \in C, i \in I} .$ The resulting fuzzy convex space $(X, C)$ is called the product of the spaces ${(X_{i}, C_{i})}_{i \in I}$ , and is denoted by $\prod_{i \in I} (X_{i}, C_{i})$ .

Proposition 2.10. ([15]) Let ${(X_{i}, C_{i})}_{i \in I}$ be a family of stratified fuzzy convex spaces, $X = \prod_{i \in I} X_{i}$ , and let {p_i : X ⟶ X_i} _i∈I be the family of projection mappings. Then $p_{i} : (X, \prod_{i \in I} C_{i}) ⟶ (X_{i}, C_{i})$ is a fuzzy CC and fuzzy CP mapping for each i ∈ I.

3 S₀ separation axiom in fuzzy convex spaces

In this section, we will introduce S₀ separation axiom in fuzzy convex spaces and investigate its hereditary and product property.

Definition 3.1. A fuzzy convex space $(X, C)$ is S₀ separated if it satisfies

for any $x_{λ}, y_{μ} \in pt (F (X))$ with x_λ ≠ y_μ, there exists $A \in C$ such that λ > A (x) and μ ⩽ A (y), or there exists $B \in C$ such that λ ⩽ B (x) and μ > B (y).

In this sense, we call

(X, C)

an S₀ fuzzy convex space.

Remark 3.2. If [0, 1] is replaced by {0, 1}, then S₀ can be reduced to the classical S₀ separation axiom in classical convex spaces. That is,

for any x, y ∈ X with x ≠ y, there exists $U \in E$ such that x ∉ U and y ∈ U or there exists $V \in E$ such that y ∉ V and x ∈ V.

Hence, S₀ fuzzy convex spaces can be considered as the generalizations of S₀ convex spaces.

Given a classical convex space $(X, E)$ , we can induce a fuzzy convex space $(X, C^{E})$ in the following way.

Proposition 3.3. ([28]) Let $(X, E)$ be a classical convex space and define $C^{E} \subseteq F (X)$ as follows: $C^{E} = {χ_{U} ∣ U \in E}$ . Then $(X, C^{E})$ is a fuzzy convex space.

Next we have the following theorem.

Theorem 3.4. Let $(X, E)$ be a classical convex space. If $(X, C^{E})$ is S₀ separated, then so is $(X, E)$ .

Proof. Take any x, y ∈ X with x ≠ y. Then it follows that for each λ ∈ (0, 1], we have $x_{λ}, y_{λ} \in pt (F (X))$ and x_λ ≠ y_λ. Since $(X, C^{E})$ is S₀ separated, there exists $χ_{U} \in C^{E}$ such that λ > χ_U (x) and λ ⩽ χ_U (y), or there exists $χ_{V} \in C^{E}$ such that λ ⩽ χ_V (x) and λ > χ_V (y). This implies that there exists $U \in E$ such that x ∉ U and y ∈ U, or there exists $V \in E$ such that y ∉ V and x ∈ V. Thus, $(X, C)$ is S₀ separated.□

By Theorem 3.4, if $(X, E)$ is not S₀ separated, then $(X, C^{E})$ is not S₀ separated. It means that there really exist some concrete fuzzy convex spaces which are not S₀ separated. Then, we give some examples.

Example 3.5. Let X = {x, y}. We define $C_{1}$ as follows: $A \in C_{1} \Leftrightarrow A (x) = A (y) .$ Then $(X, C_{1})$ is not an S₀ fuzzy convex space.

Proof. It is obvious that $C_{1}$ is a fuzzy convex structure on X. Take $x_{0.5}, y_{0.5} \in pt (F (X))$ satisfying x_0.5 ≠ y_0.5. If there exists $A \in C_{1}$ , such that 0.5 > A (x) and 0.5 ⩽ A (y), then it is a contradiction to A (x) = A (y). If there exists $B \in C_{1}$ , such that 0.5 ⩽ B (x) and 0.5 > B (y), then it is a contradiction to B (x) = B (y). Hence $(X, C_{1})$ is not an S₀ fuzzy convex space.□

Example 3.6. Let X = {x, y}. We define $C_{2}$ as follows: $C_{2} = {B \in F (X) ∣ B (x) ⩾ B (y)} .$ Then $(X, C_{2})$ is an S₀ fuzzy convex space.

Proof. It is obvious that $C_{2}$ is a fuzzy convex structure on X. Hence it suffices to verify that it satisfies S₀ separation axiom. Take any $x_{λ}, y_{μ} \in pt (F (X))$ with x_λ ≠ y_μ. We divide into two cases. Case(1) $x_{λ}, x_{μ} \in pt (F (X))$ with λ ≠ μ. If λ > μ, then there exist λ₁, μ₁ such that λ > λ₁ ⩾ μ₁ > μ. Take $A \in C_{2}$ satisfying A (x) = λ₁, A (x) = μ₁. Hence λ > A (x) and A (x) ⩾ μ. If μ > λ, then there exist λ₂, μ₂ such that μ > λ₂ ⩾ μ₂ > λ. Take $B \in C_{2}$ satisfying B (x) = μ₂, B (x) = λ₂. Hence μ > B (x) and B (x) ⩾ λ. Case(2) x ≠ y. The proof is similar to Case(1). This proves that for any $x_{λ}, y_{μ} \in pt (F (X))$ with x_λ ≠ y_μ, there exists $A \in C_{2}$ such that λ > A (x) and μ ⩽ A (y) or there exists $B \in C_{2}$ such that λ ⩽ B (x) and μ > B (y). Hence $(X, C_{2})$ is an S₀ fuzzy convex space.□

Next let us investigate the hereditary and productive property of S₀ separation axiom.

Theorem 3.7. Let $(X, C)$ be a fuzzy convex space and $(Y, C |_{Y})$ be a subspace of $(X, C)$ . If $(X, C)$ is S₀ separated, then so is $(Y, C |_{Y})$ .

Proof. Take any $x_{λ}, y_{μ} \in pt (F (Y))$ with x_λ ≠ y_μ. Define $x_{λ}^{*}, y_{μ}^{*} : X ⟶ [0, 1]$ as follows: for any z ∈ X, $x_{λ}^{*} (z) = {\begin{matrix} λ, & z = x, \\ 0, & z \neq x . \end{matrix} y_{μ}^{*} (z) = {\begin{matrix} μ, & z = y, \\ 0, & z \neq y . \end{matrix}$ It follows that $x_{λ}^{*}, y_{μ}^{*} \in pt (F (X))$ . Since $(X, C)$ is S₀ separated, there exists $A \in C$ such that λ > A (x^*) and μ ⩽ A (y^*), or there exists $B \in C$ such that λ ⩽ B (x^*) and μ > B (y^*). Let C = A_|Y and D = B_|Y. This implies that there exists $C \in C |_{Y}$ such that λ > C (x) and μ ⩽ C (y), or there exists $D \in C |_{Y}$ such that λ ⩽ D (x) and μ > D (y). Thus $(Y, C |_{Y})$ is S₀ separated.□

In [15], Pang and Xiu proposed the concept of product space of L-convex spaces, where L is a continuous lattice. In order to show the productive property of separation axioms when L = [0, 1], we need the following definitions and theorem.

Definition 3.8. ([15]) Let $(X, C)$ be a fuzzy convex space and $B \subseteq C$ . Then $B$ is called a base of $(X, C)$ provided that for each $C \in C$ , there is a directed family $B_{C} \subseteq B$ such that $C = ⋁ B_{C}$ .

Definition 3.9. ([15]) Let $(X, C)$ be a fuzzy convex space and $U \subseteq C$ . Then $U$ is called a subbase of $(X, C)$ provided that $B_{U}$ is a base of $(X, C)$ , where $B_{U} = {⋀_{i \in I} A_{i} ∣ {A_{i}}_{i \in I} \subseteq U and I \neq \emptyset}$ .

Definition 3.10. ([15]) Let $(X, C)$ be the product space of ${(X_{i}, C_{i})}_{i \in I}$ . Then $B = {⋀_{k \in K} p_{k}^{\leftarrow} (A_{k}) ∣ \forall k \in K, A_{k} \in C_{k}}$ is a base of $C$ . For convenience, we give the following notation with respect to the product space.

Let {X_i} _i∈I be a family of non-empty sets, I≠ ∅, $X = \prod_{i \in I} X_{i}$ , x = {x_i} _i∈I be a fixed point in X, s ∈ I. Denote $\tilde{X_{s}} = {y \in X ∣ y_{s} \in X_{s}, \forall i \in I, if i \neq s, then y_{i} = x_{i}} .$

Definition 3.11. Let $(X, C)$ be the product space of ${(X_{i}, C_{i})}_{i \in I}$ . Then the subspace $(\tilde{X_{s}}, C |_{\tilde{X_{s}}})$ of $(X, C)$ is called a fuzzy plane that is parallel to the factor space $(X_{s}, C_{s})$ through the point x.

Lemma 3.12. Let $(X, C)$ be the product space of ${(X_{i}, C_{i})}_{i \in I}$ , {p_i ∣ X ⟶ X_i} _i∈I be the family of projection mappings, x = {x_i} _i∈I be a fixed point in X. If ( $X_{s}, C_{s})$ is stratified for some s ∈ S, then $p_{s} |_{\tilde{X_{s}}} : (\tilde{X_{s}}, C |_{\tilde{X_{s}}}) ⟶ (X_{s}, C_{s})$ is a fuzzy CC mapping.

Proof. By Definition 3.10, ${⋀_{k \in K} p_{k}^{\leftarrow} (A_{k}) |_{\tilde{X_{s}}}, \forall k \in K, A_{k} \in C_{k}}$ is a base of $(\tilde{X_{s}}, C |_{\tilde{X_{s}}})$ . Then it suffices to verify that $(p_{s} |_{\tilde{X_{s}}})^{\to} (⋀_{k \in K} p_{k}^{\leftarrow} (A_{k}) |_{\tilde{X_{s}}})) \in C_{s}$ for ${A_{k}} \in C_{k}$ and k ∈ K ⊆ I.

Take each y_s ∈ X_s, we have $\begin{matrix} (p_{s} |_{\tilde{X_{s}}})^{\to} (⋀_{k \in K} p_{k}^{\leftarrow} (A_{k}) |_{\tilde{X_{s}}})) (y_{s}) \\ = & ⋁_{p_{s} |_{\tilde{X_{s}}} (z) = y_{s}} ⋀_{k \in K} p_{k}^{\leftarrow} (A_{k}) |_{\tilde{X_{s}}} (z) \\ = & ⋁_{p_{s} (z) = y_{s}, z \in \tilde{X_{s}}} ⋀_{k \in K} A_{k} (p_{k} (z)) . \end{matrix}$

Next we divide into two cases.

Case(1) s ∉ K. Then it follows that $\begin{matrix} ⋁_{p_{s} (z) = y_{s}, z \in \tilde{X_{s}}} ⋀_{k \in K} A_{k} (p_{k} (z)) \\ = & ⋁_{z_{s} = y_{s}, z \in \tilde{X_{s}}} ⋀_{k \in K} A_{k} (z_{k}) \\ \overset{Δ}{=} & a . \end{matrix}$

Case(2) s ∈ K. Then it follows that $\begin{matrix} ⋁_{p_{s} (z) = y_{s}, z \in \tilde{X_{s}}} ⋀_{k \in K} A_{k} (p_{k} (z)) \\ = & ⋁_{z_{s} = y_{s}, z \in \tilde{X_{s}}} (A_{s} (z_{s}) \land ⋁_{k \in K - {s}} A_{k} (z_{k})) \\ = & A_{s} (y_{s}) \land ⋁_{z_{s} = y_{s}, z \in \tilde{X_{s}}} ⋀_{k \in K - {s}} A_{k} (z_{k}) \\ \overset{Δ}{=} & A_{s} (y_{s}) \land b . \end{matrix}$ This implies $\begin{matrix} (p_{s} |_{\tilde{X_{s}}})^{\to} (⋀_{k \in K} p_{k}^{\leftarrow} (A_{k}) |_{\tilde{X_{s}}}) \\ = & {\begin{matrix} \underline{a}, & (s \notin K), \\ A_{s} \land \underline{b}, & (s \in K) . \end{matrix} \end{matrix}$ Since $(X_{s}, C_{s})$ is stratified, it follows that $\underline{a} \in C_{s}$ and $A_{s} \land \underline{b} \in C_{s}$ . So we always have $(p_{s} |_{\tilde{X_{s}}})^{\to} (⋀_{k \in K} p_{k}^{\leftarrow} (A_{k}) |_{\tilde{X_{s}}})) \in C_{s},$ as desired.□

Theorem 3.13. Let $(X, C)$ be the product space of ${(X_{i}, C_{i})}_{i \in I}$ , x = {x_i} _i∈I be a fixed point in X. If $(X_{s}, C_{s})$ is stratified for some s ∈ S, then the fuzzy plane $(\tilde{X_{s}}, C |_{\tilde{X_{s}}})$ is isomorphic to $(X_{s}, C_{s})$ . That is, $(X_{s}, C_{s}) ≅ (\tilde{X_{s}}, C |_{\tilde{X_{s}}}) .$ Proof. Since $(\tilde{X_{s}}, C |_{\tilde{X_{s}}})$ is the fuzzy plane that is parallel to the factor space $(X_{s}, C_{s})$ through the point x, we define a mapping $f : X_{s} ⟶ \tilde{X_{s}}$ by $\forall y_{s} \in X_{s}, f (y_{s}) = z,$ where z_i = x_i for each i ≠ s and z_s = y_s .

It is easy to see that all the coordinates are fixed except for the s^th coordinate, thus f is a bijection. Next it suffices to show that f is fuzzy CP and fuzzy CC.

(1) Take each ${\tilde{C}}_{s} \in (\tilde{X_{s}}, C |_{\tilde{X_{s}}})$ . Then there exists $C \in C$ such that ${\tilde{C}}_{s} = C ∣_{\tilde{X_{s}}}$ . For each y_s ∈ X_s, it follows that $f^{\leftarrow} ({\tilde{C}}_{s}) (y_{s}) = f^{\leftarrow} (C |_{\tilde{X_{s}}}) (y_{s}) = C |_{\tilde{X_{s}}} (f (y_{s})) = C (f (y_{s})) .$ And $\begin{matrix} (p_{s} |_{\tilde{X_{s}}})^{\to} ({\tilde{C}}_{s}) (y_{s}) & = & ⋁_{p_{s} |_{\tilde{X_{s}}} (z) = y_{s}} {\tilde{C}}_{s} (z) \\ = & ⋁_{p_{s} (z) = y_{s}, z \in \tilde{X_{s}}} C |_{\tilde{X_{s}}} (z) \\ = & ⋁_{p_{s} (z) = y_{s}, z \in \tilde{X_{s}}} C (z) \\ = & ⋁_{z_{s} = y_{s}, z \in \tilde{X_{s}}} C (z) . \end{matrix}$ By the definition of f, we have z = f (y_s) for each $z \in \tilde{X_{s}}$ such that z_s = y_s. This implies that $(p_{s} |_{\tilde{X_{s}}})^{\to} ({\tilde{C}}_{s}) (y_{s}) = ⋁_{z_{s} = y_{s}} C (z) = C (f (y_{s})),$ which means $f^{\leftarrow} ({\tilde{C}}_{s}) = (p_{s} |_{\tilde{X_{s}}})^{\to} ({\tilde{C}}_{s}) .$ By Lemma 3.12, we have $f^{\leftarrow} ({\tilde{C}}_{s}) \in C_{s},$ which means f is fuzzy CP.

(2) Take each $C_{s} \in C_{s}$ . Then for each $y \in {\tilde{C}}_{s}$ , it follows from the definition of f that $\begin{matrix} f^{\to} (C_{s}) (y) & = & ⋁_{f (y_{s}) = y} C_{s} (y_{s}) \\ = & C_{s} (p_{s} (y)) \\ = & p_{s}^{\leftarrow} (C_{s}) (y) \\ = & p_{s}^{\leftarrow} (C_{s}) |_{\tilde{X_{s}}} (y), \end{matrix}$ which means $f^{\to} (C_{s}) = p_{s}^{\leftarrow} (C_{s}) |_{\tilde{X_{s}}} .$ Since $p_{s} : (X, C) ⟶ (X_{s}, C_{s})$ is fuzzy CP, it follows that $p_{s}^{\leftarrow} (C_{s}) \in C .$ This implies that $f^{\to} (C_{s}) = p_{s}^{\leftarrow} (C_{s}) |_{\tilde{X_{s}}} \in C |_{\tilde{X_{s}}} .$ Hence, f is fuzzy CC.□

Theorem 3.14. Let $f : (X, C_{X}) ⟶ (Y, C_{Y})$ be an isomorphic mapping. If $(X, C_{X})$ is S₀ separated, then so is $(Y, C_{Y})$ . Proof. For any $x_{λ}, y_{μ} \in pt (F (Y))$ with x_λ ≠ y_μ, it is easy to check that $f^{\leftarrow} (x_{λ}), f^{\leftarrow} (y_{μ}) \in pt (F (X))$ and f^← (x_λ) ≠ f^← (y_μ). Since $(X, C_{X})$ is S₀ separated, there exists $A \in C_{X}$ such that λ > A (f^-1 (x)) and μ ⩽ A (f^-1 (y)), or there exists $B \in C_{X}$ such that λ ⩽ B (f^-1 (x)) and μ > B (f^-1 (y)). Furthermore, f is isomorphic, then it follows that there exists $f^{\to} (A) \in C_{Y}$ such that λ > f^→ (A) (x) and μ ⩽ f^→ (A) (y), or there exists $f^{\to} (B) \in C_{Y}$ such that λ ⩽ f^→ (B) (x) and μ > f^→ (B) (y). Thus, $(Y, C_{Y})$ is an S₀ fuzzy convex space.□

Theorem 3.15. Let ${(X_{i}, C_{i})}_{i \in I}$ be a family of fuzzy convex spaces and $(X, C)$ be the product space. If $(X_{i}, C_{i})$ is S₀ separated for each i ∈ I, then so is $(X, C)$ . Conversely, if $(X, C)$ is S₀ separated, and $(X_{i}, C_{i})$ is stratified for some i ∈ I, then $(X_{i}, C_{i})$ is S₀separated.

Proof. Take any $x_{λ}, y_{μ} \in pt (F (X))$ such that x_λ ≠ y_λ. Then there exists i₀ ∈ I such that p_{i
₀} (x) _λ ≠ p_{i
₀} (y) _μ. Since $(X_{i_{0}}, C_{i_{0}})$ is S₀ separated, there exists $A_{i_{0}} \in C_{i_{0}}$ such that λ > A_{i
₀} (p_{i
₀} (x)) and μ ⩽ A_{i
₀} (p_{i
₀} (y)), or there exists $B_{i_{0}} \in C_{i_{0}}$ such that λ ⩽ B_{i
₀} (p_{i
₀} (x)) and μ > B_{i
₀} (p_{i
₀} (y)). By the definition of product of fuzzy convex spaces, we can obtain that the projection mapping $p_{i} : (X, C) ⟶ (X_{i}, C_{i})$ is a fuzzy CP mapping. Then $A_{i_{0}} \in C_{i_{0}}$ implies $p_{i_{0}}^{\leftarrow} (A_{i_{0}}) \in C$ . This means that there exists $p_{i_{0}}^{\leftarrow} (A_{i_{0}}) \in C$ such that $λ > p_{i_{0}}^{\leftarrow} (A_{i_{0}}) (x)$ and $μ ⩽ p_{i_{0}}^{\leftarrow} (A_{i_{0}}) (y)$ or there exists $p_{i_{0}}^{\leftarrow} (B_{i_{0}}) \in C$ such that $λ ⩽ p_{i_{0}}^{\leftarrow} (B_{i_{0}}) (x)$ and $μ > p_{i_{0}}^{\leftarrow} (B_{i_{0}}) (y)$ . Thus we obtain that $(X, C)$ is an S₀ fuzzy convex space. Conversely, assume that $(X_{i}, C_{i})$ is stratified for some i ∈ I, it follows from Theorem 3.13 that $(X_{i}, C_{i})$ is isomorphic to $(\tilde{X_{i}}, C |_{\tilde{X_{i}}})$ , which is a subspace of $(X, C)$ . From Theorems 3.7 and 3.14, $(X_{i}, C_{i})$ is S₀ separated.□

4 S₁ separation axiom in fuzzy convex spaces

In this section, we will present the definition of S₁ separation axiom in fuzzy convex spaces and investigate its properties.

Definition 4.1. A fuzzy convex space $(X, C)$ is S₁ separated if it satisfies

for any $x_{λ}, y_{μ} \in pt (F (X))$ with x_λ≰y_μ, there exists $A \in C$ such that λ > A (x) and μ ⩽ A (y).

In this sense, we call

(X, C)

an S₁ fuzzy convex space.

In the classical case, S₁ convex space $(X, E)$ is described as follows: $(X, E)$ is S₁ iff ${x} \in E$ for each x ∈ X.

In the case of fuzzy convex spaces, S₁ separation axiom can also be characterized in this way.

Proposition 4.2. A fuzzy convex space $(X, C)$ is S₁ separated iff it satisfies

$\forall x_{λ} \in pt (F (X))$ , $x_{λ} \in C$ .

Proof. (S₁) ^∗ ⇒ (S₁): Obviously. (S₁) ⇒ (S₁) ^∗: Take each $y_{μ} \in pt (F (X))$ such that y_μ≰x_λ. Then it follows from (S₁) that there exists $A \in C$ such that μ > A (y) and λ ⩽ A (x). Then we have y_μ≰A and x_λ ⩽ A. This implies $y_{μ} ≰ ⋀ {A \in C ∣ x_{λ} ⩽ A} .$ By the arbitrariness of y_μ, we get $⋀ {A \in C ∣ x_{λ} ⩽ A} ⩽ x_{λ} .$ Then we have $x_{λ} ⩽ ⋀ {A \in C ∣ x_{λ} ⩽ A} ⩽ x_{λ} .$ This means $x_{λ} = ⋀ {A \in C ∣ x_{λ} ⩽ A} \in C,$ as desired.□

The next result will show the relationship between S₀ and S₁ separation axioms.

Theorem 4.3. Let $(X, C)$ be a fuzzy convex space. If $(X, C)$ is S₁ separated, then $(X, C)$ is S₀ separated.

Proof. It follows immediately from the definitions of S₀ and S₁ separation axioms.□

In Example (3.6), take x_0.5≰y_0.5, then there is no $A \in C_{2}$ satisfying 0.5 > A (x) and 0.5 ⩽ A (y), but there exists $B \in C_{2}$ such that 0.5 ⩽ B (x) and 0.5 > B (y). Hence we know that $(X, C_{2})$ is an S₀ fuzzy convex space but it is not an S₁ fuzzy convex space.

Next, we will give an example of S₁ fuzzy convex space.

Example 4.4. Let X = {x}. Take $C_{3} = {\underline{a} ∣ a \in [0, 1]}$ . Then $(X, C_{3})$ is a stratified fuzzy convex space. For each $x_{λ} \in pt (F (X))$ , we have $x_{λ} = x_{1} \lor \underline{a} \in C_{3}$ . Hence it is an S₁ fuzzy convex space from Proposition 4.2. In the following, we will show the hereditary and productive property of S₁ separation axiom.

Theorem 4.5. Let $(X, C)$ be a fuzzy convex space and $(Y, C |_{Y})$ be a subspace of $(X, C)$ . If $(X, C)$ is S₁ separated, then so is $(Y, C |_{Y})$ .

Proof. Take each $x_{λ} \in pt (F (Y))$ . Define $x_{λ}^{*} : X ⟶ [0, 1]$ as follows: for any z ∈ X, $x_{λ}^{*} (z) = {\begin{matrix} λ, & z = x, \\ 0, & z \neq x . \end{matrix}$ Then it follows that $x_{λ}^{*} \in pt (F (X))$ . Since $(X, C)$ is S₁ separated, we have $x_{λ}^{*} \in C$ from Proposition 4.2. It is easy to see that $x_{λ} = x_{λ}^{*} |_{Y}$ , which means $x_{λ} \in C |_{Y}$ . Thus we obtain $x_{λ} \in C |_{Y}$ for each $x_{λ} \in pt (F (Y))$ . By Proposition 4.2, we know that $(Y, C |_{Y})$ is S₁ separated.

□

Theorem 4.6. Let $f : (X, C_{X}) ⟶ (Y, C_{Y})$ be an isomorphic mapping. If $(X, C_{X})$ is S₁ separated, then so is $(Y, C_{Y})$ .

Proof. Take each $x_{λ} \in pt (F (Y))$ . For f is a bijection, we have $f^{\leftarrow} (x_{λ}) \in pt (F (X))$ . Since $(X, C_{X})$ is S₁ separated, it follows that $f^{\leftarrow} (x_{λ}) \in C_{X}$ by Proposition 4.2. Furthermore, f is fuzzy CP, hence $x_{λ} = f^{\to} (f^{\leftarrow} (x_{λ})) \in C_{Y}$ , as desired.□

Theorem 4.7. Let ${(X_{i}, C_{i})}_{i \in I}$ be a family of fuzzy convex spaces and $(X, C)$ be the product space. If $(X_{i}, C_{i})$ is S₁ separated for each i ∈ I, then so is $(X, C)$ . Conversely, if $(X, C)$ is S₁ separated, and $(X_{i}, C_{i})$ is stratified for some i ∈ I, then $(X_{i}, C_{i})$ is S₁ separated.

Proof. Take each $x_{λ} \in pt (F (X))$ . Then $p_{i} (x)_{λ} \in pt (F (X_{i}))$ for each i ∈ I. Since $(X_{i}, C_{i})$ is S₁ separated, it follows from Proposition 4.2 that $p_{i} (x)_{λ} \in C_{i}$ for each i ∈ I. By the definition of product of fuzzy convex spaces, we can obtain that the projection mapping $p_{i} : (X, C) ⟶ (X_{i}, C_{i})$ is a fuzzy CP mapping. This means $x_{λ} = ⋀_{i \in I} p_{i}^{\leftarrow} (p_{i} (x))_{λ} \in C$ . Thus $x_{λ} \in C$ for each $x_{λ} \in pt (F (X))$ . By Proposition 4.2, we know that $(X, C)$ is S₁ separated. Conversely, assume that $(X_{i}, C_{i})$ is stratified for some i ∈ I, it follows from Theorem 3.13 that $(X_{i}, C_{i})$ is isomorphic to $(\tilde{X_{i}}, C |_{\tilde{X_{i}}})$ , which is a subspace of $(X, C)$ . From Theorems 4.5 and 4.6, $(X_{i}, C_{i})$ is S₁ separated.□

5 S₂ separation axiom in fuzzy convex spaces

In this section, we will propose S₂ separation axiom in fuzzy convex spaces and investigate its relationships with S₀ and S₁ separation axioms as well as its hereditary and productive properties.

In order to give the definition of S₂ fuzzy convex spaces, we first give the following concept.

Definition 5.1. Let $(X, C)$ be a fuzzy convex space and $A \in F (X)$ . A is called a fuzzy biconvex subset (fuzzy half-space, fuzzy hemispace) provided that $A \in C$ and $A^{'} \in C$ .

For convenience, the family of all fuzzy biconvex subsets in $(X, C)$ is denoted by $H (X, C)$ .

Remark 5.2. If [0, 1] is replaced by {0, 1}, then A′ reduces to the complement of A. Then fuzzy biconvex subsets in fuzzy convex spaces will reduce to classical biconvex sets in classical convex spaces.

In [35], Yang and Shi proposed L-biconvex sets on some fuzzy algebraic substructures, where L is a complete residuated lattice. Here we list some results when L = [0, 1].

Example 5.3. ([35])

Let $(X, C_{S})$ be a fuzzy meet semilattice convex space. Denote $H_{S} = {C^{↑} \in F (X) ∣ C \in C_{S}} .$ Then H_S is the family of fuzzy biconvex subsets of $(X, C_{S})$ .

Let $(X, C_{S})$ be a fuzzy join semilattice convex space. Denote $H_{S} = {C^{↓} \in F (X) ∣ C \in C_{S}} .$ Then H_S is the family of fuzzy biconvex subsets of $(X, C_{S})$ .

Let X be a lattice. Denote $H_{L} = {H \in F (X) ∣ H isafuzzyprimeideal (filter)} .$ Then H_L is the family of fuzzy biconvex subsets of X.

Now let us present the definition of S₂ fuzzy convex spaces.

Definition 5.4. A fuzzy convex space $(X, C)$ is S₂ separated if it satisfies

for any $x_{λ}, y_{μ} \in pt (F (X))$ with x_λ≰y_μ, there exists $A \in H (X, C)$ such that λ > A (x) and μ ⩽ A (y).

In this case, we call

(X, C)

an S₂ fuzzy convex space.

Remark 5.5. If [0, 1] is replaced by {0, 1}, then S₂ separation axiom in Definition 5.3 will reduce to S₂ separation axiom in classical convex spaces. That is,

for any x, y ∈ X with x ≠ y, there exists a biconvex set A such that x ∉ A and y ∈ A.

This shows that S₂ fuzzy convex spaces are exactly the generalizations of S₂ convex spaces.

The following result shows the relationship between S₁ and S₂ separation axiom.

Proposition 5.6. Let $(X, C)$ be a fuzzy convex space. If $(X, C)$ is S₂ separated, then $(X, C)$ is S₁ separated.

Proof. Since $H (X, C) \subseteq C$ , that is, fuzzy biconvex subsets are fuzzy convex subsets. It follows immediately from the definitions of S₁ and S₂ separation axioms that S₂ fuzzy convex spaces must be S₁ fuzzy convex spaces.□

Example 5.7. It is easy to check that the fuzzy convex space $(X, C_{3})$ in Example (4.4) is an S₂ fuzzy convex space.

Next we show the hereditary of S₂ separation axiom. We first give the following lemma.

Lemma 5.8. Let $(X, C)$ be a fuzzy convex space and let $(Y, C |_{Y})$ be a subspace of $(X, C)$ . If $A \in H (X, C)$ , then $A |_{Y} \in H (Y, C |_{Y})$ .

Proof. Since $A \in C$ and $A^{'} \in C$ , it follows that $A |_{Y} \in C |_{Y}$ and $A^{'} |_{Y} \in C |_{Y}$ . Furthermore, it is easy to check that (A_|Y) ′ = A′_|Y. This implies $(A |_{Y})^{'} \in C |_{Y}$ . Thus we have $A |_{Y} \in C |_{Y}$ and $(A |_{Y})^{'} \in C |_{Y}$ , which means $A |_{Y} \in H (Y, C |_{Y})$ .□

Theorem 5.9. Let $(X, C)$ be a fuzzy convex space and $(Y, C |_{Y})$ be a subspace of $(X, C)$ . If $(X, C)$ is S₂ separated, then so is $(Y, C |_{Y})$ .

Proof. Take any $x_{λ}, y_{μ} \in pt (F (Y))$ with x_λ≰y_μ. Define $x_{λ}^{*}, y_{μ}^{*} : X ⟶ [0, 1]$ as follows: for any z ∈ X, $x_{λ}^{*} (z) = {\begin{matrix} λ, & z = x, \\ 0, & z \neq x . \end{matrix} y_{μ}^{*} (z) = {\begin{matrix} μ, & z = y, \\ 0, & z \neq y . \end{matrix}$ It follows that $x_{λ}^{*}, y_{μ}^{*} \in pt (F (X))$ . By S₂ separation axiom of $(X, C)$ , there exists $A \in H (X, C)$ such that λ > A (x^*) and μ ⩽ A (y^*). This implies λ > A_|Y (x) and μ ⩽ A_|Y (y). By Lemma 5.8, we have $A |_{Y} \in H (Y, C |_{Y})$ . This proves that for any $x_{λ}, y_{μ} \in pt (F (Y))$ with x_λ≰y_μ, there exists $A |_{Y} \in H (Y, C |_{Y})$ such that λ > A_|Y (x) and μ ⩽ A_|Y (y), as desired.□

In order to show the productive property of S₂ separation axiom, the following lemma and theorem are necessary.

Lemma 5.10. Let f: $(X, C_{X}) ⟶ (Y, C_{Y})$ be a fuzzy CP mapping. Then $A \in H (Y, C_{Y})$ implies $f^{\leftarrow} (A) \in H (X, C_{X}) .$

Proof. Take each $A \in H (Y, C_{Y})$ . Then $A \in C_{Y}$ and $A^{'} \in C_{Y}$ . Since f is a fuzzy CP mapping, it follows that $f^{\leftarrow} (A) \in C_{X}$ and $f^{\leftarrow} (A)^{'} = f^{\leftarrow} (A^{'}) \in C_{X}$ . This implies $f^{\leftarrow} (A) \in H (X, C_{X})$ , as desired.□

Theorem 5.11. Let $f : (X, C_{X}) ⟶ (Y, C_{Y})$ be an isomorphic mapping. If $(X, C_{X})$ is S₂ separated, then so is $(Y, C_{Y})$ .

Proof. Take any $x_{λ}, y_{μ} \in pt (F (Y))$ with x_λ≰y_μ. Then $f^{\leftarrow} (x_{λ}), f^{\leftarrow} (y_{μ}) \in pt (F (X))$ and f^← (x_λ) ≰f^← (y_μ). By S₂ separation axiom of $(X, C_{X})$ , there exists $A \in H (X, C_{X})$ such that λ > A (f^-1 (x)) and μ ⩽ A (f^-1 (y)). Since f is isomorphic, this implies that there exists $f^{\to} (A) \in H (Y, C_{Y})$ such that λ > f^→ (A) (x) and μ ⩽ f^→ (A) (y). Hence, $(Y, C_{Y})$ is an S₂ fuzzy convex space.□

Theorem 5.12. Let ${(X_{i}, C_{i})}_{i \in I}$ be a family of fuzzy convex spaces and $(X, C)$ be the product space. If $(X_{i}, C_{i})$ is S₂ separated for each i ∈ I, then so is $(X, C)$ . Conversely, if $(X, C)$ is S₂ separated, and $(X_{i}, C_{i})$ is stratified for some i ∈ I, then $(X_{i}, C_{i})$ is S₂ separated.

Proof. Take any $x_{λ}, y_{μ} \in pt (F (X))$ with x_λ≰y_μ. Then there exists i₀ ∈ I such that p_{i
₀} (x) _λ≰p_{i
₀} (y) _μ. Since $(X_{i_{0}}, C_{i_{0}})$ is S₂ separated, there exists $A_{i_{0}} \in H (X_{i_{0}}, C_{i_{0}})$ such that λ > A_{i
₀} (p_{i
₀} (x)) and μ ⩽ A_{i
₀} (p_{i
₀} (y)). Then $λ > p_{i_{0}}^{\leftarrow} (A_{i_{0}}) (x)$ and $μ ⩽ p_{i_{0}}^{\leftarrow} (A_{i_{0}}) (y)$ . Since p_{i
₀} is a fuzzy CP mapping, we have $p_{i_{0}}^{\leftarrow} (A_{i_{0}}) \in H (X, C)$ by Lemma 5.10. This shows that for any $x_{λ}, y_{μ} \in pt (F (X))$ with x_λ≰y_μ, there exists $p_{i_{0}}^{\leftarrow} (A_{i_{0}}) \in H (X, C)$ such that $λ > p_{i_{0}}^{\leftarrow} (A_{i_{0}}) (x)$ and $μ ⩽ p_{i_{0}}^{\leftarrow} (A_{i_{0}}) (y)$ . Thus $(X, C)$ is S₂ separated. Conversely, assume that $(X_{i}, C_{i})$ is stratified for some i ∈ I, it follows from Theorem 3.13 that $(X_{i}, C_{i})$ is isomorphic to $(\tilde{X_{i}}, C |_{\tilde{X_{i}}})$ , which is a subspace of $(X, C)$ . From Theorems 5.9 and 5.11, we have that $(X_{i}, C_{i})$ is S₂ separated.□

6 Conclusions

In this paper, we mainly introduced S₀, S₁ and S₂ separation axioms in fuzzy convex spaces and discussed their hereditary and productive properties.

As we all know, in classical convex spaces, S₃ and S₄ separation axioms are also important properties and have close connections with other properties. S₃ separation property can be characterized by Sand-glass Property under Join-hull Commutativity Property. The convexity of a Pasch-Peano space can be characterized by S₄ separation property and Join-hull Commutativity Property. In [27], the Sand-glass Property and Pasch-Peano Property have already been generalized to the fuzzy case, therefore our further exploration is to study the high-level separation axioms in fuzzy convex spaces. That is to say, we will consider S₃ and S₄ separation axioms in fuzzy convex spaces and then investigate the relationships among the multiple properties mentioned above.

Footnotes

Acknowledgments

The author would like to express the sincere thanks to referees for their careful reading and constructive comments. The project is supported by the National Natural Science Foundation of China (12071033, 11871097).

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Separation axioms in fuzzy convex spaces

Abstract

Keywords

1 Introduction

2 Preliminaries

3 S0 separation axiom in fuzzy convex spaces

4 S1 separation axiom in fuzzy convex spaces

5 S2 separation axiom in fuzzy convex spaces

6 Conclusions

Footnotes

Acknowledgments

References

3 S₀ separation axiom in fuzzy convex spaces

4 S₁ separation axiom in fuzzy convex spaces

5 S₂ separation axiom in fuzzy convex spaces