On separation axioms in ( L,M )-fuzzy convex structures

Abstract

Different from the separation axioms in the framework of (L, M)-fuzzy convex spaces defined by Liang et al.(2019). In this paper, we give some new investigations on separation axioms in (L, M)-fuzzy convex structures by L-fuzzy hull operators and r-L-fuzzy biconvex. We introduce the concepts of r-LFS_i spaces where i = {0, 1, 2, 3, 4}, and obtain various properties. In particular, we discuss the invariance of these separation properties under subspace and product.

1 Introduction and preliminaries

Separation of sets constitutes one of the fundamental facets of abstract convexity theory in [27, 32] which plays an important role in various branches of mathematics where abstract convexity theory has been applied to many different mathematical research fields, such as topological spaces, lattices, metric spaces and graphs (see, for example, [10 , 33]). In particular, convexity appears naturally in topology and has many topological properties, such as product spaces, convex variables and separation (see, for example, [1–4 , 31]).

Zadeh [36] introduced the notion of a fuzzy subset, which it have been applied to various branches of mathematics. For a generalization of a convex structure, Rosa in 1994 introduced the notion of fuzzy convex structure in [20, 21] which is called I-convex structure. Also, he studied a fuzzy topology together with a fuzzy convexity on the same underlying set X, and introduced fuzzy topology fuzzy convexity spaces and the notion of fuzzy local convexity. Recently, there has been significant research on fuzzy convex structures ([11 , 35]).

Separation axioms constitute one of the facets of the theory of convex structures. Jamison [8] 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. By this motivation, Liang et al. [12] introduced the separation axioms in the framework of (L, M)-fuzzy convex spaces. Sayed et al. [22] defined a new class of L-fuzzy sets called r-L-fuzzy biconvex sets in (L, M)-fuzzy convex structures. The transformation method between L-fuzzy hull operators and (L, M)-fuzzy convex structures were introduced, and a characterization of the product of the L-fuzzy hull operator was obtained. Different from the separation axioms in the framework of (L, M)-fuzzy convex spaces defined by Liang et al. [12], the main contributions of the present paper are to give some new investigations on separation axioms in (L, M)-fuzzy convex structures by L-fuzzy hull operators and r-L-fuzzy biconvex.

Throughout this paper, let X be a non-empty set, both L and M be a completely distributive lattices with order reversing involution ^′ where ⊥_M (⊥ _L) and ⊤_M (⊤ _L) denote the least and the greatest elements in M (L) respectively, and M_{⊥_M} = M - {⊥ _M} (L_{⊥_L} = L - {⊥ _L}) . Recall that an order-reversing involution ^′ on L is a map (-) ′ : L ⟶ L such that for any a, b ∈ L, the following conditions hold: (1) a ≤ b implies b′ ≤ a′ . (2) a^′′ = a. The following properties hold for any subset {b_i : i ∈ I} ∈ L: $(1) (⋁_{i \in I} b_{i})^{'} = ⋀_{i \in I} b_{i}^{'}; (2) (⋀_{i \in I} b_{i})^{'} = ⋁_{i \in I} b_{i}^{'} .$

An L-fuzzy subset of X is a mapping μ : X ⟶ L and the family L^X denoted the set of all fuzzy subsets of a given X [5]. The least and the greatest elements in L^X are denoted by $\underline{0}$ and $\underline{1},$ respectively. For each α ∈ L, let $\underline{α}$ denote the constant L-fuzzy subset of X with the value α . Two L-fuzzy sets are said to be disjoint if their supports are disjoint where support of μ = {x ∈ X : μ (x) >0}. The complementation of a fuzzy subset are defined as μ′ (x) = (μ (x)) ′ for all x ∈ X, (e . g . μ′ (x) =1 - μ (x) inthecaseof L = [0, 1]) . Let $X = \prod_{i \in Γ} X_{i}$ and μ_i ∈ L^{X
_i}, then μ ∈ L^X denote the product of all μ_i ∈ L^{X
_i} are defined as following μ (x) = ∧ _i∈Γμ_i (x_i) for all x ∈ X [28].

Definition 1.1. ([7]) Let ∅ ≠ Y ⊆ X and μ∈ L^X ; the restriction of μ on Y is denoted by μ|Y . An extension of μ ∈ L^Y on X, denoted by μ_X is defined by $\begin{matrix} μ_{X} (x) = {\begin{matrix} μ (x), if x \in Y, \\ ⊥_{L}, if x \in X - Y . \end{matrix} \begin{matrix} \end{matrix} \end{matrix}$

Definition 1.2. ([6, 18]) A fuzzy point x_t for t ∈ L_{⊥_L} is an element of L^X such that $\begin{matrix} x_{t} (y) = {\begin{matrix} t, if y = x, \\ ⊥_{L}, if y \neq x . \end{matrix} \begin{matrix} \end{matrix} \end{matrix}$ The set of all fuzzy points in X is denoted by P_t (X) . A fuzzy point x_t is a fuzzy singleton if t = ⊤ _L and denoted by χ_{x} for all x ∈ X . Two fuzzy points x_t and y_s are distinct if x ≠ y .

Definition 1.3. ([24]) The pair $(X, C)$ is called an (L, M)-fuzzy convex structure ((L, M)-fcs, for short), where $C : L^{X} ⟶ M$ satisfying the following axioms:

(LMC1) $C (\underline{0}) = C (\underline{1}) = ⊤_{M}$ .

(LMC2) If {μ_i : i ∈ Γ} ⊆ L^X is nonempty, then $C (⋀_{i \in Γ} μ_{i}) \geq ⋀_{i \in Γ} C (μ_{i}) .$

(LMC3) If {μ_i : i ∈ Γ} ⊆ L^X is nonempty and totally ordered by inclusion, then $C (⋁_{i \in Γ} μ_{i}) \geq ⋀_{i \in Γ} C (μ_{i}) .$

The mapping $C$ is called an (L, M)-fuzzy convexity on X and $C (μ)$ can be regarded as the degree to which μ is an L-convex fuzzy set.

Definition 1.4. [19] Let f : X ⟶ Y . Then the image f^→ (μ) of μ ∈ L^X and the preimage f^← (ν) of ν ∈ L^Y are defined by: $f^{\to} (μ) (y) = ⋁ {μ (x) : x \in X, f (x) = y}$ and f^← (ν) = ν ∘ f, respectively . It can be verified that the pair (f^→, f^←) is a Galois connection on (L^X, ≤) and (L^Y, ≤).

Definition 1.5. [24] Let $(X, C)$ and $(Y, D)$ be (L, M)-fuzzy convex structures. A function f : X ⟶ Y is called:

(1) An (L, M)-fuzzy convexity preserving function if $C (f^{\leftarrow} (μ)) \geq D (μ)$ for all μ ∈ L^Y .

(2) An (L, M)-fuzzy convex-to-convex function if $D (f^{\to} (μ)) \geq C (μ)$ for all μ ∈ L^X .

Theorem 1.6. ([24]) Let $(X, C)$ be an (L, M)-fuzzy convex structure, ∅ ≠ Y ⊆ X . Then $(Y, C | Y)$ is an (L, M)-fuzzy convex structure on Y where $(C | Y) (μ) = ⋁ {C (ν) : ν \in L^{X}, ν | Y = μ}$ for each μ ∈ L^Y. The pair $(Y, C | Y)$ is called an (L, M)-fuzzy convex substructure of $(X, C) .$

Definition 1.7. ([24]) Let ${(X_{i}, C_{i}) : i \in Γ}$ be a set of (L, M)-fuzzy convex structures. Let X be the product of the sets X_i for i ∈ Γ, and let π_i : X ⟶ X_i the projection for each i ∈ Γ . Define a mapping φ : L^X ⟶ M by $\begin{matrix} φ (μ) = ⋁_{i \in Γ} ⋁_{π_{i}^{\leftarrow} (ν) = μ} C_{i} (ν) for each μ \in L^{X} . \end{matrix}$ Then the product convexity $C$ of X is the one generated by subbase φ . The resulting (L, M)-fuzzy convex structure $(X, C)$ is called the product of ${(X_{i}, C_{i}) : i \in Γ}$ and is denoted by $\prod_{i \in Γ} (X_{i}, C_{i})$ .

Theorem 1.8. ([24]) Let $(X, C)$ be the product of ${(X_{i}, C_{i}) : i \in Γ}$ . Then for all i ∈ Γ, π_i : X ⟶ X_i is an (L, M)-fuzzy convexity preserving function. Moreover, $C$ is the coarsest (L, M)-fuzzy convex structure such that {π_i : i ∈ Γ} are (L, M)-fuzzy convexity preserving functions.

Theorem 1.9. ([22]) Let $(X, C)$ be the product of ${(X_{i}, C_{i}) : i \in Γ}$ . If $π_{i}^{\to} (\prod_{i \in Γ} μ_{i}) = μ_{i}$ for any $μ_{i} \in L^{X_{i}}$ Then for each i ∈ Γ, π_i : X ⟶ X_i is an (L, M)-fuzzy convex-to-convex function.

Throught this paper, we always assume that each projection π_i (i ∈ Γ) is an (L, M)-fuzzy convex-to-convex function. $\forall μ \in L^{X}, \exists μ_{i} \in L^{X_{i}}$ , such that $μ = \prod_{i \in > Γ} μ_{i}$ and $π_{i}^{\to} (μ) = μ_{i}$ for each $i \in Γ$

Definition 1.10. ([22]) Let $(X, C)$ be (L, M)-fuzzy convex structure, r ∈ M_{⊥_M} and μ ∈ L^X . Then μ is called r-L-fuzzy biconvex set if $C (μ) \geq r$ and $C (μ^{'}) \geq r .$

Proposition 1.11. ([22]) Let $(X, C)$ be an (L, M)-fuzzy convex structure, ∅ ≠ Y ⊆ X and μ is an r-L-fuzzy biconvex set in $(X, C) .$ Then μ|Y is an r-L-fuzzy biconvex set in $(Y, C | Y) .$

Theorem 1.12. ([22]) Let $(X, C)$ be an (L, M)-fuzzy convex structure. For each μ ∈ L^X and r ∈ M_{⊥_M} we define a mapping ${CO}_{C} : L^{X} \times M_{⊥_{M}} ⟶ L^{X}$ as follows: $\begin{matrix} {CO}_{C} (μ, r) = ⋀ {ν \in L^{X} : μ \leq ν, C (ν) \geq r} . \end{matrix}$ For μ, ν ∈ L^X and r, s ∈ M_{⊥_M} the operator ${CO}_{C}$ satisfies the following conditions:

(1) ${CO}_{C} (\underline{0}, r) = \underline{0} .$

(2) $μ \leq {CO}_{C} (μ, r) .$

(3) If μ ≤ ν, then ${CO}_{C} (μ, r) \leq {CO}_{C} (ν, r) .$

(4) If r ≤ s, then ${CO}_{C} (μ, r) \leq {CO}_{C} (μ, s) .$

(5) ${CO}_{C} ({CO}_{C} (μ, r), r) = {CO}_{C} (μ, r) .$

(6) For {μ_i : i ∈ Γ} ⊆ L^X is nonempty and totally ordered by inclusion, ${CO}_{C} (⋁_{i \in Γ} μ_{i}, r) = ⋁_{i \in Γ} {CO}_{C} (μ_{i}, r) .$

A mapping ${CO}_{C}$ is called L-fuzzy hull operator.

2 r-LFS₀ space and r-LFS₁ space

Definition 2.1. Let x_t, y_s ∈ P_t (X) such that x ≠ y and r ∈ M_{⊥_M} . Then an (L, M)-fuzzy convex structure $(X, C)$ is said to be:

(1) r-LFS₀ space if ${CO}_{C} (x_{t}, r) \neq {CO}_{C} (y_{s}, r) .$

(2) r-LFS₁ space if ${CO}_{C} (x_{t}, r) \neq {CO}_{C} (y_{s}, r)$ such that $x_{t} \notin {CO}_{C} (y_{s}, r)$ and $y_{s} \notin {CO}_{C} (x_{t}, r) .$

Proposition 2.2. If $(X, C)$ is an r-LFS₀ space then for distinct fuzzy points x_t and y_s, there exists μ, ν ∈ L^X such that $C (μ) \geq r, C (ν) \geq r$ and μ ≠ ν with x_t ∈ μ and y_s ∈ ν .

Proof. Clear by Definition 2.1.□

The next example shows that the converse of Proposition 2.2 is not true.

Example 2.3. Let L = M = [0, 1] and X = {a, b}. Let μ_i be a fuzzy subsets of X where i = {1, 2} defined as follows: $\begin{matrix} μ_{1} (a) = 0.3, μ_{1} (b) = 0.0, \\ μ_{2} (a) = 0.7, μ_{2} (b) = 1.0 . \end{matrix}$ Define an (L, M)-fuzzy convexity $C : [0, 1]^{X} ⟶ [0, 1]$ on X as follows: $\begin{matrix} C (λ) = {\begin{matrix} 1, if λ \in {\underline{0}, \underline{1}}, \\ \frac{1}{6}, if λ = μ_{1}, \\ \frac{1}{5}, if λ = μ_{2}, \\ 0, otherwise . \end{matrix} \end{matrix}$ For t ∈ [0.7, 1] and s ∈ (0, 1] we obtain only two fuzzy sets which are $\underline{1}$ and μ₂ such that $a_{t}, b_{s} \in \underline{1}$ and a_t, b_s ∈ μ₂ with $C (\underline{1}) \geq \frac{1}{5}$ and $C (μ_{2}) \geq \frac{1}{5}$ but $(X, C)$ is not r-LFS₀ space because ${CO}_{C} (a_{t}, \frac{1}{5}) = {CO}_{C} (b_{s}, \frac{1}{5}) = μ_{2} .$

Proposition 2.4. Let $(X, C)$ be an r-LFS₀ space and ∅ ≠ Y ⊆ X . Then $(Y, C | Y)$ is r-LFS₀ space.

Proof. Let $(X, C)$ be an r-LFS₀ space, x_t, y_s ∈ P_t (Y) such that x ≠ y and $μ = {CO}_{C} (x_{t}, r), ν = {CO}_{C} (y_{s}, r)$ such that μ ≠ ν .

First, we will prove that μ|Y ≠ ν|Y . So, suppose μ|Y = ν|Y . Then, t ≤ (μ|Y) (x) = (ν|Y) (x)ands ≤ (μ|Y) (y) = (ν|Y) (y) forall x, y ∈ Y . It implies that $\begin{matrix} t \leq μ (x), t \leq ν (x), s \leq μ (y) and s \leq ν (y) . \end{matrix}$ Therefore, $t \leq (μ \land ν) (x) \leq μ (x) = {CO}_{C} (x_{t}, r) (x) .$ From Definition 1.3 (2) and Theorem 1.12, we obtain (μ ∧ ν) (x) = μ (x) . Similarly (μ ∧ ν) (y) = ν (y) . So, μ = ν and it is a contradiction for the assumption that μ ≠ ν . Hence, μ|Y ≠ ν|Y .

Second, we will prove that ${CO}_{C | Y} (x_{t}, r) = μ | Y$ and ${CO}_{C | Y} (y_{s}, r) = ν | Y .$ So, suppose that there exist μ₁, ν₁ ∈ L^Y such that $(C | Y) (μ_{1}) \geq r$ and $(C | Y) (ν_{1}) \geq r,$ with t ≤ μ₁ (x) ≤ (μ|Y) (x) ands ≤ ν₁ (y) ≤ (ν|Y) (y) forall x, y ∈ Y . Since, $(C | Y) (μ_{1}) \geq r {and (C | Y) (ν_{1}) \geq r,$ we have μ₁ = λ|Y and ν₁ = ρ|Y where $λ, ρ \in L^{X}, C (λ) \geq r$ and $C (ρ) \geq r .$ It implies that $t \leq (λ | Y) (x) \leq (μ | Y) (x)$ and $s \leq (ρ | Y) (y) \leq (ν | Y) (y) .$ (2.1) Since, $x_{t} \in μ = {CO}_{C} (x_{t}, r)$ and x_t ∈ λ we have μ ≤ λ, Similarly, ν ≤ ρ . Therefore, $t \leq (μ | Y) (x) \leq (λ | Y) (x),$ and $s \leq (ν | Y) (y) \leq (ρ | Y) (y) .$ (2.2) From, Equations (2.1) and (2.2) we obtain $t \leq (μ | Y) (x) = (λ | Y) (x),$ and $s \leq (ν | Y) (y) = (ρ | Y) (y) .$ Which implies that $\begin{matrix} t \leq (μ | Y) (x) = μ_{1} (x), and s \leq (ν | Y) (y) = ν_{1} (y) . \end{matrix}$ Put $μ_{1} = {CO}_{C | Y} (x_{t}, r)$ and $ν_{1} = {CO}_{C | Y} (y_{s}, r)$ . Then, ${CO}_{C | Y} (x_{t}, r) = μ | Y$ and ${CO}_{C | Y} (y_{s}, r) = ν | Y .$ Hence, $(Y, C | Y)$ is an r-LFS₀ space. □

Theorem 2.5. Let $(X, C)$ be the product of ${(X_{i}, C_{i}) : i \in Γ}$ . Then, $(X, C)$ is an r-LFS₀ space if $(X_{i}, C_{i})$ is an r-LFS₀ space for each i ∈ Γ .

Proof. Let $(X_{i}, C_{i})$ is an r-LFS₀ space for each i ∈ Γ and x_t, y_s ∈ P_t (X) such that x ≠ y with $X = \prod_{i \in Γ} X_{i}$ and π_i : X ⟶ X_i be the projection map for each i ∈ Γ . Then for some i ∈ Γ, (x_i) _t and (y_i) _s are distinct fuzzy points in X_i and ${CO}_{C_{i}} ((x_{i})_{t}, r) \neq {CO}_{C_{i}} ((y_{i})_{s}, r)$ (2.3) each i ∈ Γ.

Since π_i is the projection map, $C_{i} ({CO}_{C_{i}} ((x_{i})_{t}, r)) \geq r$ and $C_{i} ({CO}_{C_{i}} ((y_{i})_{s}, r)) \geq r,$ then by Theorem 1.8, we have $C (π_{i}^{\leftarrow} ({CO}_{C_{i}} ((x_{i})_{t}, r))) \geq r$ $and C (π_{i}^{\leftarrow} ({CO}_{C_{i}} ((y_{i})_{s}, r))) \geq r .$ Moreover, $\begin{matrix} π_{i}^{\leftarrow} ({CO}_{C_{i}} ((x_{i})_{t}, r)) (x) & = {CO}_{C_{i}} ((x_{i})_{t}, r) (π_{i}^{\to} (x)) \\ = {CO}_{C_{i}} ((x_{i})_{t}, r) (x_{i}) \geq t . \end{matrix}$ Therefore, $x_{t} \in π_{i}^{\leftarrow} ({CO}_{C_{i}} ((x_{i})_{t}, r)) .$ Similarly, $y_{s} \in π_{i}^{\leftarrow} ({CO}_{C_{i}} ((y_{i})_{s}, r)) .$

Now we will prove that $π_{i}^{\leftarrow} ({CO}_{C_{i}} ((x_{i})_{t}, r)) \neq π_{i}^{\leftarrow} ({CO}_{C_{i}} ((y_{i})_{s}, r)) .$ So, if possible assume that $\begin{matrix} π_{i}^{\leftarrow} ({CO}_{C_{i}} ((x_{i})_{t}, r)) = π_{i}^{\leftarrow} ({CO}_{C_{i}} ((y_{i})_{s}, r)) . \end{matrix}$ Then, $π_{i}^{\leftarrow} ({CO}_{C_{i}} ((x_{i})_{t}, r)) (x) = π_{i}^{\leftarrow} ({CO}_{C_{i}} ((y_{i})_{s}, r)) (x)$ for all x ∈ X .

Implies that, $\begin{matrix} {CO}_{C_{i}} ((x_{i})_{t}, r) (π_{i}^{\to} (x)) = {CO}_{C_{i}} ((y_{i})_{s}, r) (π_{i}^{\to} (x)) . \end{matrix}$ Therefore, ${CO}_{C_{i}} ((x_{i})_{t}, r) (x_{i}) = {CO}_{C_{i}} ((y_{i})_{s}, r) (x_{i})$ for allx_i ∈ X_i .

So, ${CO}_{C_{i}} ((x_{i})_{t}, r) = {CO}_{C_{i}} ((y_{i})_{s}, r) .$ It is a contradiction for Equation (2.3). Hence, $π_{i}^{\leftarrow} ({CO}_{C_{i}} ((x_{i})_{t}, r)) \neq π_{i}^{\leftarrow} ({CO}_{C_{i}} ((y_{i})_{s}, r)) .$ Now, to prove that $π_{i}^{\leftarrow} ({CO}_{C_{i}} ((x_{i})_{t}, r)) = {CO}_{C} (x_{t}, r)$ and $π_{i}^{\leftarrow} ({CO}_{C_{i}} ((y_{i})_{s}, r)) = {CO}_{C} (y_{s}, r) .$ If possible assume that there exist λ ∈ L^X such that $x_{t} \in λ \leq π_{i}^{\leftarrow} ({CO}_{C_{i}} ((x_{i})_{t}, r))$ with $C (λ) \geq r .$ Then, $π_{i}^{\to} (x_{t}) \in π_{i}^{\to} (λ) \leq {CO}_{C_{i}} ((x_{i})_{t}, r)$ i.e., $(x_{i})_{t} \in π_{i}^{\to} (λ) \leq {CO}_{C_{i}} ((x_{i})_{t}, r) .$ Since, π_i is (L, M)-fuzzy convex-to-convex function, then $C_{i} (π_{i}^{\to} (λ)) \geq r .$ It is a contradiction to assumption that ${CO}_{C_{i}}$ is L-fuzzy hull operator in X_i. Hence, $π_{i}^{\leftarrow} ({CO}_{C_{i}} ((x_{i})_{t}, r)) = {CO}_{C} (x_{t}, r)$ . Similarly, $π_{i}^{\leftarrow} ({CO}_{C_{i}} ((y_{i})_{s}, r)) = {CO}_{C} (y_{s}, r) .$ So, we obtain $π_{i}^{\leftarrow} ({CO}_{C_{i}} ((x_{i})_{t}, r)) \neq π_{i}^{\leftarrow} ({CO}_{C_{i}} ((y_{i})_{s}, r))$ for x_t, y_s ∈ P_t (X) . Hence, $(X, C)$ is an r-LFS₀ space. □

Proposition 2.6. $(X, C)$ is an r-LFS₁ if and only if $C (χ_{{x}}) \geq r$ for all x ∈ X .

Proof. (⇒) Let $(X, C)$ be an r-LFS₁ and assume that there is a x ∈ X such that $C (χ_{{x}}) ≱ r .$ Then, there are y_s ∈ P_t (X) and s ∈ L_{⊥_L} such that $y_{s} \in {CO}_{C} (χ_{{x}}, r) .$ Therefore, the two fuzzy points x_{⊤_L} and y_s, s ∈ L_{⊥_L} cannot be separated by distinct L-fuzzy hull operator which is a contradiction to the assumption that $(X, C)$ is an r-LFS₁. Hence, $C (χ_{{x}}) \geq r .$

(⟸) Clear by Definition. □

Proposition 2.7. An r-LFS₁ space is always r-LFS₀ space.

Proof. Trivial. □

The next example shows that the converse of Proposition 2.7 is not true.

Example 2.8. Let L = M = [0, 1] and X = {a, b, c}. Let μ_i be fuzzy subsets of X where i = {1, 2, 3} defined as follows: $\begin{matrix} μ_{1} (a) = 0.4, μ_{1} (b) = 0.0, μ_{1} (c) = 0.0, \\ μ_{2} (a) = 0.4, μ_{2} (b) = 1.0, μ_{2} (c) = 0.0, \\ μ_{3} (a) = 0.4, μ_{3} (b) = 1.0, μ_{3} (c) = 1.0, \end{matrix}$ Define an (L, M)-fuzzy convexity $C : [0, 1]^{X} ⟶ [0, 1]$ on X as follows: $\begin{matrix} C (λ) = {\begin{matrix} 1, if λ \in {\underline{0}, \underline{1}}, \\ \frac{1}{4}, if λ = μ_{1}, \\ \frac{1}{3}, if λ = μ_{2}, \\ \frac{1}{2}, if λ = μ_{3}, \\ 0, otherwise . \end{matrix} \end{matrix}$ Then $(X, C)$ is r-LFS₀ space but it is not r-LFS₁ space because $μ_{1} = {CO}_{C} (a_{0.4}, \frac{1}{4})$ and $a_{0.4} \in μ_{2} = {CO}_{C} (b_{1.0}, \frac{1}{4})$ .

Theorem 2.9. Let $(X, C^{1})$ be an r-LFS₁ space and $C^{2}$ be an (L, M)-fuzzy convexity on X such that $C^{1}$ is coarser than $C^{2} .$ Then $(X, C^{2})$ is also r-LFS₁ space.

Proof. By Proposition 2.6, it can be easily proved. □

Proposition 2.10. Let $(X, C)$ be an r-LFS₁ space and ∅ ≠ Y ⊆ X . Then $(Y, C | Y)$ is an r-LFS₁ space.

Proof. Let $(X, C)$ be an r-LFS₁ space, ∅ ≠ Y ⊆ X and x_t, y_s ∈ P_t (Y) . Then ${CO}_{C} (x_{t}, r) \neq {CO}_{C} (y_{s}, r)$ such that $x_{t} \notin {CO}_{C} (y_{s}, r)$ and $y_{s} \notin {CO}_{C} (x_{t}, r) .$ Then we can prove as in the proof of Proposition 2.4 that ${CO}_{C} (x_{t}, r) | Y \neq {CO}_{C} (y_{s}, r) | Y$ and $x_{t} \notin {CO}_{C} (y_{s}, r) | Y, y_{s} \notin {CO}_{C} (x_{t}, r) | Y .$ Hence $(Y, C | Y)$ is an r-LFS₁ space. □

Theorem 2.11. Let $(X, C)$ be the product of ${(X_{i}, C_{i}) : i \in Γ}$ . Then, $(X, C)$ is an r-LFS₁ space if $(X_{i}, C_{i})$ is an r-LFS₁ space for each i ∈ Γ .

Proof. Let $(X_{i}, C_{i})$ is an r-LFS₁ space for each i ∈ Γ and x_t, y_s ∈ P_t (X) such that x ≠ y with $X = \prod_{i \in Γ} X_{i}$ and π_i : X ⟶ X_i be the projection map for all i ∈ Γ . Then for some i ∈ Γ, (x_i) _t and (y_i) _s are distinct fuzzy points in X_i and ${CO}_{C_{i}} ((x_{i})_{t}, r) \neq {CO}_{C_{i}} ((y_{i})_{s}, r)$ for each i ∈ Γ such that $(y_{i})_{s} \notin {CO}_{C_{i}} ((x_{i})_{t}, r) and (x_{i})_{t} \notin {CO}_{C_{i}} ((y_{i})_{s}, r)$ for each i ∈ Γ . Then we can prove as in the proof of Theorem 2.5 that

$\begin{matrix} \begin{matrix} {CO}_{C} (x_{t}, r) = π_{i}^{\leftarrow} ({CO}_{C_{i}} ((x_{i})_{t}, r)) \\ π_{i}^{\leftarrow} ({CO}_{C_{i}} ((y_{i})_{s}, r)) = {CO}_{C} (y_{s}, r), \end{matrix} \end{matrix}$ such that $y_{s} \notin π_{i}^{\leftarrow} ({CO}_{C_{i}} ((x_{i})_{t}, r))$ and $x_{t} \notin π_{i}^{\leftarrow} ({CO}_{C_{i}} ((y_{i})_{s}, r))$ Hence, $(X, C)$ is an r-LFS₁ space. □

3 r-LFS₂ space, r-LFS₃ space and r-LFS₄ space

Definition 3.1. Let $(X, C)$ be an (L, M)-fuzzy convex space and r ∈ M_{⊥_M} . Then, $(X, C)$ is said to be an r-LFS₂ space if for distinct an L-fuzzy points x_t, y_s ∈ P_t (X) , there exists r-L-fuzzy biconvex set μ such that x_t ∈ μ and y_s ∈ μ′ .

Theorem 3.2. Let $(X, C^{1})$ be an r-LFS₂ space and $C^{2}$ be an (L, M)-fuzzy convexity on X such that $C^{1}$ is coarser than $C^{2} .$ Then $(X, C^{2})$ is also an r-LFS₂ space.

Proof. Let $(X, C^{1})$ be an r-LFS₂ space, x_t, y_s ∈ P_t (X) such that x ≠ y, and $C^{2}$ be an (L, M)-fuzzy convexity on X . Then, there exists an r-L-fuzzy biconvex set μ in $(X, C^{1})$ such that x_t ∈ μ and y_s ∈ μ′ . Therefore, $C^{1} (μ) \geq r$ and $C^{1} (μ^{'}) \geq r .$ By the assumption $C^{1}$ is coarser than $C^{2}$ we obtain $C^{2} (μ) \geq r$ and $C^{2} (μ^{'}) \geq r .$ So, μ is an r-L-fuzzy biconvex set in $(X, C^{2}) .$ Hence, $(X, C^{2})$ is an r-LFS₂ space. □

Proposition 3.3. Let $(X, C)$ be an r-LFS₂ space and ∅ ≠ Y ⊆ X . Then $(Y, C | Y)$ is an r-LFS₂ space.

Proof. Let $(X, C)$ be an r-LFS₂ space, x_t, y_s ∈ P_t (Y) such that x ≠ y. Then, there exists an r-L-fuzzy biconvex set μ ∈ L^X such that x_t ∈ μ and y_s ∈ μ′ . By Proposition 1.11, we have μ|Y is an r-L-fuzzy biconvex set in L^Y such that x_t ∈ μ and y_s ∈ μ′ . Hence, $(Y, C | Y)$ is an r-LFS₂ space. □

Theorem 3.4. Let $(X, C)$ be the product of ${(X_{i}, C_{i}) : i \in Γ}$ . Then, $(X, C)$ is an r-LFS₂ space if $(X_{i}, C_{i})$ is an r-LFS₂ space for each i ∈ Γ .

Proof. Let $(X_{i}, C_{i})$ is an r-LFS₂ space for each i ∈ Γ and x_t, y_s ∈ P_t (X) such that x ≠ y with $X = \prod_{i \in Γ} X_{i}$ and π_i : X ⟶ X_i be the projection map for all i ∈ Γ . Then, for some i ∈ Γ, (x_i) _t, (y_i) _s ∈ P_t (X_i) such that x_i ≠ y_i . Therefore, there exists an r-L-fuzzy biconvex set μ in $(X_{i}, C_{i})$ such that (x_i) _t ∈ μ and (y_i) _s ∈ μ′ . Then, $π_{i}^{\leftarrow} (μ)$ is r-L-fuzzy biconvex set in $(X, C)$ such that $x_{t} \in π_{i}^{\leftarrow} (μ)$ and $y_{s} \in π_{i}^{\leftarrow} (μ^{'}) .$ Hence, $(X, C)$ is an r-LFS₂ space. □

Proposition 3.5. An r-LFS₂ space is always an r-LFS₁ space.

Proof. Clear by Definition. □

The next example shows that the converse of Proposition 3.5 is not true.

Example 3.6. Let L = M = [0, 1] and X = {a, b, c}. Let μ_i be fuzzy subsets of X where i = {1, 2, 3, 4, 5} defined as follows: $\begin{matrix} μ_{1} (a) = 1.0, μ_{1} (b) = 0.0, μ_{1} (c) = 0.0, \\ μ_{2} (a) = 0.0, μ_{2} (b) = 1.0, μ_{2} (c) = 0.0, \\ μ_{3} (a) = 0.0, μ_{3} (b) = 0.0, μ_{3} (c) = 1.0, \\ μ_{4} (a) = 0.5, μ_{4} (b) = 0.0, μ_{4} (c) = 0.0, \\ μ_{5} (a) = 0.5, μ_{5} (b) = 1.0, μ_{5} (c) = 1.0 . \end{matrix}$ Define an (L, M)-fuzzy convexity $C : [0, 1]^{X} ⟶ [0, 1]$ on X as follows: $\begin{matrix} C (λ) = {\begin{matrix} 1, if λ \in {\underline{0}, \underline{1}}, \\ \frac{1}{4}, if λ \in {μ_{1}, μ_{2}, μ_{3}}, \\ \frac{1}{3}, if λ = μ_{4}, \\ \frac{1}{2}, if λ = μ_{5}, \\ 0, otherwise . \end{matrix} \end{matrix}$ Then $(X, C)$ is r-LFS₁ space but it is not r-LFS₂ space because the only $\frac{1}{3}$ -L-fuzzy biconvex set is μ₅ and a_0.5, c_1.0 ∈ μ₅.

Definition 3.7. Let $(X, C)$ be an (L, M)-fuzzy convex space and r ∈ M_{⊥_M} . Then, $(X, C)$ is said to be an r-LFS₃ space if for an L-fuzzy point x_t ∈ P_t (X) and μ ∈ L^X such that $C (μ) \geq r$ with the supports of x_t and μ are disjoint, there exists an r-L-fuzzy biconvex set λ such that μ ≤ λ and x_t ∈ λ′ .

Remark 3.8. If $(X, C^{1})$ is an r-LFS₃ space and $C^{2}$ be an (L, M)-fuzzy convexity on X such that $C^{1}$ is coarser than $C^{2},$ then $(X, C^{2})$ need not be an r-LFS₃ space.

Example 3.9. Let L = M = [0, 1] and X = {a, b, c}. Let μ_i be fuzzy subsets of X where i = {1, 2, 3, 4, 5} defined as follows: $\begin{matrix} μ_{1} (a) = 1.0, μ_{1} (b) = 0.0, μ_{1} (c) = 0.0, \\ μ_{2} (a) = 0.0, μ_{2} (b) = 1.0, μ_{2} (c) = 1.0, \\ μ_{3} (a) = 0.0, μ_{3} (b) = 0.3, μ_{3} (c) = 0.3, \\ μ_{4} (a) = 1.0, μ_{4} (b) = 0.7, μ_{4} (c) = 0.7, \\ μ_{5} (a) = 0.0, μ_{5} (b) = 0.0, μ_{5} (c) = 0.8 . \end{matrix}$ Define two mapings $C^{1}, C^{2} : [0, 1]^{X} ⟶ [0, 1]$ on X as follows: $\begin{matrix} C^{1} (λ) = {\begin{matrix} 1, if λ \in {\underline{0}, \underline{1}}, \\ \frac{1}{5}, if λ \in {μ_{1}, μ_{2}}, \\ \frac{1}{4}, if λ = μ_{3}, \\ \frac{1}{3}, if λ = μ_{4}, \\ 0, otherwise, \end{matrix} \end{matrix}$ $\begin{matrix} C^{2} (λ) = {\begin{matrix} 1, if λ \in {\underline{0}, \underline{1}}, \\ \frac{1}{4}, if λ \in {μ_{1}, μ_{2},}, \\ \frac{1}{3}, if λ = μ_{3}, \\ \frac{1}{2}, if λ = μ_{4}, \\ \frac{1}{2}, if λ = μ_{5}, \\ 0, otherwise . \end{matrix} \end{matrix}$ Then both $C^{1}$ and $C^{2}$ are (L, M)-fuzzy convexities, and $C^{1}$ is coarser than $C^{2}, (X, C^{1})$ is an r-LFS₃ space and $(X, C^{2})$ is not r-LFS₃ space because μ₃ and its complement are $\frac{1}{4}$ -L-fuzzy biconvex sets and $μ_{1} \leq \underline{1} - μ_{3}$ and c_0.8 ∉ μ₃.

Proposition 3.10. Let $(X, C)$ be an r-LFS₃ space and ∅ ≠ Y ⊆ X . Then $(Y, C | Y)$ is an r-LFS₃ space.

Proof. Let $(Y, C | Y)$ be an (L, M)-fuzzy convex subspace of an r-LFS₃ space $(X, C)$ , x_t ∈ P_t (Y) and μ ∈ L^Y such that $(C | Y) (μ) \geq r$ with the supports of x_t and μ are disjoint. Then μ = ν|Y where ν ∈ L^X such that $C (ν) \geq r .$ Since the supports of x_t and μ are disjoint, we have the supports of x_t and ν are disjoint. Since $(X, C)$ is an r-LFS₃ space, there exists an r-L-fuzzy biconvex set λ ∈ L^X such that ν ≤ λ and x_t ∈ λ′ . By Proposition 1.11, we have λ|Y is an r-L-fuzzy biconvex set in L^Y such that ν ≤ λ|Y and x_t ∈ λ′|Y . Hence, $(Y, C | Y)$ is an r-LFS₃ space. □

Theorem 3.11. Let $(X, C)$ be the product of ${(X_{i}, C_{i}) : i \in Γ}$ . Then, $(X, C)$ is an r-LFS₃ space if $(X_{i}, C_{i})$ is an r-LFS₃ space for each i ∈ Γ .

Proof. Let $X = \prod_{i \in Γ} X_{i}, π_{i} : X ⟶ X_{i}$ be the projection map for all i ∈ Γ and $(X_{i}, C_{i})$ is an r-LFS₃ space for each i ∈ Γ . Let x_t ∈ P_t (X) and μ ∈ L^X such that $C (μ) \geq r$ with the supports of x_t and μ are disjoint. Since $C (μ) \geq r$ and π_i is the projection map, we can take μ as $μ = ⋀_{i \in Γ} π_{i}^{\leftarrow} (ν_{i})$ such that $C_{i} (ν_{i}) \geq r .$ For some i, (x_i) _t ∈ P_t (X_i) and the supports of (x_i) _t and ν_i are disjoint. Since $(X_{i}, C_{i})$ is an r-LFS₃ space, there exists an r-L-fuzzy biconvex set λ_i ∈ L^{X
_i} such that ν_i ≤ λ_i and $(x_{i})_{t} \in λ_{i}^{'} .$ Then $λ = π_{i}^{\leftarrow} (λ_{i})$ is an r-L-fuzzy biconvex set in X such that $μ = π_{i}^{\leftarrow} (ν_{i}) \leq π_{i}^{\leftarrow} (λ_{i}) = λ$ and $x_{t} \in π_{i}^{\leftarrow} (λ_{i}^{'}) = λ^{'} .0$ Hence, $(X, C)$ is an r-LFS₃ space. □

Example 3.12. Let L, M, X and μ_i be given as Example 3.9. Define an (L, M)-fuzzy convexity $C = C^{1} : [0, 1]^{X} ⟶ [0, 1]$ on X as Example 3.9. Then,

(1) $(X, C)$ is an r-LFS₃ space but it is not r-LFS₂ space because μ₂ and its complement are $\frac{1}{5}$ -L-fuzzy biconvex sets where b_0.3 ∈ μ₂ and $c_{0.3} \notin \underline{1} - μ_{2} = μ_{1}$ .

(2) $(X, C)$ is an r-LFS₃ space but it is not r-LFS₁ space because $\begin{matrix} μ_{3} = CO (b_{0.3}, \frac{1}{5}) \neq CO (c_{1.0}, \frac{1}{5}) = μ_{2} \end{matrix}$ and $b_{0.3} \in CO (c_{1.0}, \frac{1}{5})$ and $c_{1.0} \notin CO (b_{0.3}, \frac{1}{5}) .$

(3) $(X, C)$ is an r-LFS₃ space but it is not r-LFS₀ space because $\begin{matrix} CO (b_{0.3}, \frac{1}{5}) = CO (c_{0.3}, \frac{1}{5}) = μ_{3} . \end{matrix}$

Definition 3.13. An (L, M)-fuzzy convex structure $(X, C)$ is said to be an r-LFS₄ space if two disjoint L-fuzzy sets μ, ν ∈ L^X such that $C (μ) \geq r$ and $C (ν) \geq r$ there exist an r-L-fuzzy biconvex set λ such that μ ≤ λ and ν ≤ λ′ .

Proposition 3.14. Let $(X, C)$ be an r-LFS₄ space and ∅ ≠ Y ⊆ X . Then $(Y, C | Y)$ is an r-LFS₄ space.

Proof. Let $(X, C)$ be an r-LFS₄ space, $(Y, C | Y)$ be an (L, M)-fuzzy convex subspace of $(X, C)$ and μ, ν ∈ L^Y are disjoint L-fuzzy sets such that $(C | Y) (μ) \geq r$ and $(C | Y) (ν) \geq r .$ Then μ, ν are disjoint L-fuzzy sets in X and there exists an r-L-fuzzy biconvex set λ ∈ L^X such that μ ≤ λ and ν ≤ λ′ . By Proposition 1.11, we have λ|Y is an r-L-fuzzy biconvex set in Y such that μ ≤ λ|Y and ν ≤ (λ|Y) ′ . Hence, $(Y, C | Y)$ is an r-LFS₄ space. □

Theorem 3.15. Let $(X, C)$ be the product of ${(X_{i}, C_{i}) : i \in Γ}$ . Then, $(X, C)$ is an r-LFS₄ space if $(X_{i}, C_{i})$ is an r-LFS₄ space for each i ∈ Γ .

Proof. Let $X = \prod_{i \in Γ} X_{i}$ and π_i : X ⟶ X_i be the projection map for all $i \in Γ, (X_{i}, C_{i})$ is an r-LFS₄ space for each i ∈ Γ and μ, ν ∈ L^X are disjoint L-fuzzy sets such that $C (μ) \geq r$ and $C (ν) \geq r .$ Then, $\begin{matrix} μ = ⋀_{i \in Γ} π_{i}^{\leftarrow} (λ_{i}) and ν = ⋀_{i \in Γ} π_{i}^{\leftarrow} (ρ_{i}) \end{matrix}$ there exist λ_i, ρ_i ∈ L^{X
_i} are disjoint L-fuzzy sets such that $C_{i} (λ_{i}) \geq r$ and $C_{i} (ρ_{i}) \geq r .$ for some i∈Γ. Since $(X_{i}, C_{i})$ is an r-LFS₄ space for each i ∈ Γ, there exists an r-L-fuzzy biconvex set $A_{i} \in L^{X_{i}}$ such that $λ_{i} \leq A_{i}$ and $ρ_{i} \leq A_{i}^{'} .$ Then $π_{i}^{\leftarrow} (A_{i})$ is an r-L-fuzzy biconvex set in X such that $μ \leq π_{i}^{\leftarrow} (A_{i})$ and $ν \leq π_{i}^{\leftarrow} (A_{i}^{'}) .$ Hence $(X, C)$ is an r-LFS₄ space. □

The next example shows that

(1) An r-LFS₄ space need not be r-LFS₃ space.

(2) If $(X, C^{1})$ is an r-LFS₄ space and $C^{2}$ be an (L, M)-fuzzy convexity on X such that $C^{1}$ is coarser than $C^{2},$ then $(X, C^{2})$ need not be an r-LFS₄ space.

Example 3.16. Let L = M = [0, 1] and X = {a, b, c}. Let μ_i be fuzzy subsets of X where i = {1, 2, 3, 4, 5, 6} defined as follows: $\begin{matrix} μ_{1} (a) = 0.0, μ_{1} (b) = 0.0, μ_{1} (c) = 1.0, \\ μ_{2} (a) = 1.0, μ_{2} (b) = 1.0, μ_{2} (c) = 0.0, \\ μ_{3} (a) = 0.0, μ_{3} (b) = 0.0, μ_{3} (c) = 0.5, \\ μ_{4} (a) = 0.0, μ_{4} (b) = 0.5, μ_{4} (c) = 0.0, \\ μ_{5} (a) = 1.0, μ_{5} (b) = 0.0, μ_{5} (c) = 0.0, \\ μ_{6} (a) = 1.0, μ_{6} (b) = 0.0, μ_{6} (c) = 0.5 . \end{matrix}$ Define two mappings $C^{1}, C^{2} : [0, 1]^{X} ⟶ [0, 1]$ on X as follows: $\begin{matrix} C^{1} (λ) = {\begin{matrix} 1, if λ \in {\underline{0}, \underline{1}}, \\ \frac{1}{3}, if λ \in {μ_{1}, μ_{2}}, \\ \frac{1}{2}, if λ = μ_{3}, \\ \frac{1}{2}, if λ = μ_{4}, \\ 0, otherwise, \end{matrix} \end{matrix}$ $\begin{matrix} C^{2} (λ) = {\begin{matrix} 1, if λ \in {\underline{0}, \underline{1}}, \\ \frac{1}{3}, if λ \in {μ_{1}, μ_{2},}, \\ \frac{1}{2}, if λ = μ_{3}, \\ \frac{1}{2}, if λ = μ_{4}, \\ \frac{1}{2}, if λ = μ_{5}, \\ \frac{1}{2}, if λ = μ_{6}, \\ 0, otherwise . \end{matrix} \end{matrix}$ Then both $C^{1}$ and $C^{2}$ are (L, M)-fuzzy convexities, but the only $\frac{1}{3}$ -L-fuzzy biconvex set is μ₂. So,

(1) $(X, C^{1})$ is an r-LFS₄ space but it is not r-LFS₃ space because $C^{1} (μ_{4}) \geq \frac{1}{2}$ and for t ∈ (0, 1] we obtain μ₄ ≤ μ₂ and $a_{t} \notin μ_{1} = \underline{1} - μ_{2} .$

(2) $C^{1}$ is coarser than $C^{2}, (X, C^{1})$ is an r-LFS₄ space and $(X, C^{2})$ is not r-LFS₄ space because $C^{2} (μ_{5}) \geq \frac{1}{2}$ and $C^{2} (μ_{6}) \geq \frac{1}{2}$ where μ₅ ≤ μ₂ and $μ_{6} ≰ μ_{1} = \underline{1} - μ_{2}$ .

4 Conclusion

Following the notion of r-L-fuzzy biconvex sets and L-fuzzy hull operators in (L, M)-fuzzy convex structures introduced by Sayed et al.(2019), we gave some new investigations on separation axioms in (L, M)-fuzzy convex structures. Specifically, we introduced the concepts of r-LFS_i spaces where i = {0, 1, 2, 3, 4}. We discussed the relations among them, and gave a lot of examples to show the relations. In particular, we discussed the invariance of these separation properties under subspace and product.

Footnotes

Acknowledgments

Authors would like to express their sincere thanks to the referees and the editors for giving valuable comments which helped to improve the presentation of this paper.

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On separation axioms in ( L,M )-fuzzy convex structures

Abstract

1 Introduction and preliminaries

2 r-LFS0 space and r-LFS1 space

4 Conclusion

Footnotes

Acknowledgments

References

2 r-LFS₀ space and r-LFS₁ space