Degrees of special mappings in the theory of L -convex spaces

Abstract

By using the residual implication on a frame L, a degree approach to special mappings in L-convex spaces and L-interval spaces is introduced. In the framework of L-convex spaces, degrees of L-CP mappings and L-CC mappings between L-convex spaces are defined. Also, in the situation of L-interval spaces, degrees of L-IP mappings and L-AIP mappings are proposed. Moreover, many conclusions with respect to theses mappings are discussed in a degree sense.

1 Introduction

Since Zadeh introduced the concept of fuzzy sets, many mathematical structures have been endowed with fuzzy set theory, such as fuzzy topology [1 , 41] and fuzzy convergence structures [2 , 24]. As a topology-like structure, abstract convex structures (also called convex structures in [34]) have also been generalized to the fuzzy case. Up to now, there are three types of fuzzy convex structures including L-convex structures [11, 27], M-fuzzifying convex structures [31] and (L, M)-fuzzy convex structures [32], and they are discussed from different aspects [4 , 40].

The degree approach that equips each mathematical structure with some degree description is an essential character of fuzzy set theory. In (L, M)-fuzzy topological spaces, Yue and Fang [42] introduced a degree approach to T_i (i = 0, 1, 2) separation property. Shi [28, 29] redefined the degrees of separation axioms which are compatible with (L, M)-fuzzy metric spaces. Afterwards, Li and Shi [9] introduced the degree of compactness in L-fuzzy topological spaces. Inspired by the degree approach, Pang introduced the concept of (L, M)-fuzzy convergence structures, which assigns an (L, M)-fuzzy filter to some degree of converging to a fuzzy point [12] and defined the compact degree of (L, M)-fuzzy convergence spaces [18]. Recently, Pang [15] defined degrees of T_i (i = 0, 1, 2) separation property as well as regular property of stratified L-generalized convergence spaces. Considering the fuzzy inclusion degrees of L-subsets, Fang [2] applied the fuzzy inclusion degrees between stratified L-filters to define stratified L-ordered convergence structures. Pang [14] applied the fuzzy inclusion degrees between stratified L-filters to propose stratified L-ordered filter spaces. In fuzzy rough set theory, Pang et al. [17] provided the concept of L-fuzzifying rough approximate operators to describe the approximating degree of a crisp point in a classical subset of the universe.

All the above-mentioned research mainly equipped some spatial properties with the degree description. Actually, special mappings between structured-spaces can also be endowed with some degrees. In [13], Pang defined degrees of continuous mappings and open mappings between L-fuzzifying topological spaces to describe how a mapping between L-fuzzifying topological spaces becomes a continuous mapping or an open mapping in a degree sense. Liang and Shi [10] further defined the degrees of continuous mappings and open mappings between L-fuzzy topological spaces and investigated their relationship. Compared with continuous mappings and open mappings in topological spaces, CP mappings and CC mappings in convex spaces are defined in a similar way. Motivated by this, Xiu and Pang [38] developed the degree approach to M-fuzzifying convex spaces to define the degrees of M-CP mappings and M-CC mappings. Following this direction, we will focus on the case of L-convex spaces. In this paper, we will consider degrees of L-CP mappings and L-CC mappings in L-convex spaces as well as L-IP mappings and L-AIP mappings in L-interval spaces [20] and will investigate their properties systematically.

2 Preliminaries

In this paper, we consider a complete lattice L where finite meets distribute over arbitrary joins, i.e., a ∧ ⋁ _i∈Ib_i = ⋁ _i∈I (a ∧ b_i) holds for all a, b_i ∈ L (i ∈ I) . These lattices are called frames. The bottom (resp. top) element of L is denoted by ⊥ (resp. ⊤). We can then define a residual implication by $a \to b = ⋁ {λ \in L ∣ a \land λ ⩽ b} .$

Obviously, a completely distributive lattice is a frame. The residual implication in a completely distributive lattice can be defined in the same way. An element a in L is called coprime if a ⩽ b ∨ c implies a ⩽ b or a ⩽ c. The set of nonzero coprime elements in L is denoted by J (L).

For a nonempty set X, let L^X denote the set of all L-subsets on X. L^X is also a complete lattice when it inherits the structure of the lattice L in a natural way, by defining ∨, ∧ and ⩽ pointwisely. The smallest element and the largest element in L^X are denoted by $\underline{⊥}$ and $\underline{⊤}$ , respectively. The set of nonzero coprime elements in L^X is denoted by J (L^X). It is easy to see that J (L^X) is exactly the set of all fuzzy points x_λ (λ ∈ J (L)). For a ∈ L, let $\underline{a} : X ⟶ L$ , x ⟼ a denote the constant L-subset with lattice value a.

Let f : X ⟶ Y be a mapping. Define $f_{L}^{\to} : L^{X} ⟶ L^{Y}$ and $f_{L}^{\leftarrow} : L^{Y} ⟶ L^{X}$ by $f_{L}^{\to} (A) (y) = ⋁_{f (x) = y} A (x)$ for A ∈ L^X and y ∈ Y, and $f_{L}^{\leftarrow} (B) (x) = B (f (x))$ for B ∈ L^Y and x ∈ X, respectively.

Using the residual implication, the concept of fuzzy inclusion order of L-subsets is introduced.

Definition 2.1. ([2]). The mapping $S (-, -) : L^{X} \times L^{X} ⟶ L$ defined by $\forall A, B \in L^{X}, S (A, B) = ⋀_{x \in X} (A (x) \to B (x))$ is called the fuzzy inclusion order of L-subsets.

We will often use, without explicit mentioning, the following properties of fuzzy inclusion orders of L-subsets.

Lemma 2.2. Let $S (-, -) : L^{X} \times L^{X} ⟶ L$ be the fuzzy inclusion order of L-subsets and let f : X ⟶ Y be a mapping. Then for each A, B, C ∈ L^X, the following statements hold:

$S (A, B) = ⊤$ iff A ⩽ B.

A ⩽ B implies $S (A, C) ⩾ S (B, C)$ .

B ⩽ C implies $S (A, B) ⩽ S (A, C)$ .

$S (A, B) \land S (B, C) ⩽ S (A, C)$ .

$S (A, B) ⩽ S (f_{L}^{\to} (A), f_{L}^{\to} (B))$ .

Definition 2.3. ([11, 27]). A subset $C$ of L^X is called an L-convex structure on X if it satisfies:

(LC1) $\underline{⊥}, \underline{⊤} \in C$ ;

(LC2) ${A_{k}}_{k \in K} \subseteq C$ implies $⋀_{k \in K} A_{k} \in C$ ;

(LC3) If ${A_{j}}_{j \in J} \subseteq C$ is totally ordered, then $⋁_{j \in J} A_{j} \in C$ . For an L-convex structure $C$ on X, the pair $(X, C)$ is called an L-convex space.

A mapping $f : (X, C_{X}) ⟶ (Y, C_{Y})$ is called L-convexity-preserving (L-CP, in short) provided that $B \in C_{Y}$ implies $f_{L}^{\leftarrow} (B) \in C_{X}$ .

A mapping $f : (X, C_{X}) ⟶ (Y, C_{Y})$ is called L-convex-to-convex (L-CC, in short) provided that $A \in C_{X}$ implies $f_{L}^{\to} (A) \in C_{Y}$ .

Definition 2.4. ([19]). An L-convex structure $C$ on X is called stratified if it satisfies:

(SLC) ∀a ∈ L, $\underline{a} \in C$ .

For a stratified L-convex structure $C$ on X, the pair $(X, C)$ is called a stratified L-convex space.

Definition 2.5. An L-hull operator on X is a mapping h : L^X ⟶ L^X which satisfies:

(LH1) $h (\underline{⊥}) = \underline{⊥}$ ;

(LH2) A ⩽ h (A);

(LH3) h (h (A)) = h (A);

(LH4) $h (⋁_{j \in J}^{↑} A_{j}) = ⋁_{j \in J} h (A_{j})$ , where $⋁_{j \in J}^{↑} A_{j}$ means {A_j ∣ j ∈ J} is a directed subfamily of L^X and $⋁_{j \in J}^{↑} A_{j} = ⋁_{j \in J} A_{j}$ . For an L-hull operator h on X, the pair (X, h) is called an L-hull space.

A mapping f : (X, h_X) ⟶ (Y, h_Y) between L-hull spaces is called L-hull-preserving (L-HP, in short) provided that $\forall A \in L^{X}, f_{L}^{\to} (h_{X} (A)) ⩽ h_{Y} (f_{L}^{\to} (A)) .$

It was proved in [21] that L-convex structures and L-hull operators are conceptually equivalent with transferring process $h^{C} (A) = ⋀ {B \in L^{X} ∣ A ⩽ B \in C}$ for each A ∈ L^X and $C^{h} = {A \in L^{X} ∣ h (A) = A}$ . Correspondingly, L-CP mappings between L-convex spaces and L-HP mappings between L-hull spaces are compatible. In the sequel, we treat L-convex spaces with their L-CP mappings and L-hull operators with their L-HP mappings equivalently. We will use h to represent $h^{C}$ tacitly.

3 Degrees of L-CP mappings and L-CC mappings in L-convex spaces

In this section, we mainly define degrees of L-CP mappings and L-CC mappings to equip each mapping between L-convex spaces with some degree to be an L-CP mapping and an L-CC mapping, respectively. Then we will study their connections in a degree sense.

Definition 3.1. Let $(X, C_{X})$ and $(Y, C_{Y})$ be L-convex spaces, and let f : X ⟶ Y be a mapping. Then

(1) D_cp (f) defined by $D_{cp} (f) = ⋀_{A \in L^{X}} S (f_{L}^{\to} (h_{X} (A)), h_{Y} (f_{L}^{\to} (A)))$ is called the degree to which f is L-CP.

(2) D_cc (f) defined by $D_{cc} (f) = ⋀_{A \in L^{X}} S (h_{Y} (f_{L}^{\to} (A)), f_{L}^{\to} (h_{X} (A)))$ is called the degree to which f is L-CC.

Remark 3.2. (1) If D_cp (f) =⊤, then $f_{L}^{\to} (h_{X} (A)) ⩽ h_{Y} (f_{L}^{\to} (A))$ for all A ∈ L^X, which is exactly the definition of L-HP mappings between L-hull spaces. As we claimed that L-CP mappings and L-HP mappings are compatible, we don’t distinguish them. Therefore, we defined the degree of L-CP mappings between L-convex spaces by using L-HP mappings between their induced L-hull spaces.

(2) If D_cc (f) =⊤, then $h_{Y} (f_{L}^{\to} (A)) ⩽ f_{L}^{\to} (h_{X} (A))$ for all A ∈ L^X. This is exactly the equivalent form of L-CC mappings between L-convex spaces by means of the corresponding L-hull operators.

Proposition 3.3. Let $(X, C_{X})$ and $(Y, C_{Y})$ be L-convex spaces, and let f : X ⟶ Y be a mapping. Then $D_{cp} (f) = ⋀_{A \in L^{X}} S (h_{X} (A), f_{L}^{\leftarrow} (h_{Y} (f_{L}^{\to} (A)))) .$

Proof. By the definition of $S (-, -)$ , it follows that $\begin{matrix} S (f_{L}^{\to} (h_{X} (A)), h_{Y} (f_{L}^{\to} (A))) \\ = & ⋀_{y \in Y} (f_{L}^{\to} (h_{X} (A)) (y) \to h_{Y} (f_{L}^{\to} (A)) (y)) \\ = & ⋀_{y \in Y} (⋁_{f (x) = y} h_{X} (A) (x) \to h_{Y} (f_{L}^{\to} (A)) (y)) \\ = & ⋀_{y \in Y} ⋀_{f (x) = y} (h_{X} (A) (x) \to h_{Y} (f_{L}^{\to} (A)) (y)) \\ = & ⋀_{y \in Y} ⋀_{f (x) = y} (h_{X} (A) (x) \to h_{Y} (f_{L}^{\to} (A)) (f (x))) \\ = & ⋀_{y \in Y} ⋀_{f (x) = y} (h_{X} (A) (x) \to f_{L}^{\leftarrow} (h_{Y} (f_{L}^{\to} (A))) (x)) \\ = & ⋀_{x \in X} (h_{X} (A) (x) \to f_{L}^{\leftarrow} (h_{Y} (f_{L}^{\to} (A))) (x)) \\ = & S (h_{X} (A), f_{L}^{\leftarrow} (h_{Y} (f_{L}^{\to} (A)))) . \end{matrix}$

This means $\begin{matrix} D_{cp} (f) & = & ⋀_{A \in L^{X}} S (f_{L}^{\to} (h_{X} (A)), h_{Y} (f_{L}^{\to} (A))) \\ = & ⋀_{A \in L^{X}} S (h_{X} (A), f_{L}^{\leftarrow} (h_{Y} (f_{L}^{\to} (A)))), \end{matrix}$ as desired. □

Proposition 3.4. (1) If $id : (X, C_{X}) ⟶ (X, C_{X})$ is the identity mapping, then D_cp (id) =⊤ and D_cc (id) =⊤.

(2) If $y_{0} : (X, C_{X}) ⟶ (Y, C_{Y})$ is a constant mapping between stratified L-convex spaces with the constant y₀ ∈ Y, then D_cp (y₀) =⊤.

Proof. (1) Straightforward.

(2) It follows immediately from the definition of D_cp (y₀) that $\begin{matrix} D_{cp} (y_{0}) & = & ⋀_{A \in L^{X}} S (h_{X} (A), (y_{0})_{L}^{\leftarrow} (h_{Y} ((y_{0})_{L}^{\to} (A)))) \\ = & ⋀_{A \in L^{X}} ⋀_{x \in X} (h_{X} (A) (x) \to h_{Y} ((y_{0})_{L}^{\to} (A)) (y_{0})) . \end{matrix}$

Since $(X, C_{X})$ is stratified, we know $\underline{⋁_{z \in X} A (z)} \in C_{X}$ . Further, we have $A ⩽ \underline{⋁_{z \in X} A (z)}$ . Then for each A ∈ L^X and x ∈ X, it follows that $h_{X} (A) (x) = ⋀ {B (x) ∣ A ⩽ B \in C_{X}} ⩽ ⋁_{z \in X} A (z) .$

Then we have $\begin{matrix} h_{Y} ((y_{0})_{L}^{\to} (A)) (y_{0}) & ⩾ & (y_{0})_{L}^{\to} (A) (y_{0}) \\ = & ⋁_{y_{0} (z) = y_{0}} A (z) \\ = & ⋁_{z \in X} A (z) ⩾ h_{X} (A) (x) . \end{matrix}$

This implies that $h_{X} (A) (x) ⩽ h_{Y} ((y_{0})_{L}^{\to} (A)) (y_{0})$ for each A ∈ L^X and x ∈ X. Therefore, we have D_cp (y₀) =⊤. □

Next we give another characterization of degrees of L-CP mappings.

Proposition 3.5. Let $(X, C_{X})$ and $(Y, C_{Y})$ be L-convex spaces, and let f : X ⟶ Y be a mapping. Then $D_{cp} (f) = ⋀_{B \in L^{Y}} S (f_{L}^{\to} (h_{X} (f^{\leftarrow} (B))), h_{Y} (B)) .$

Proof. It follows from the definition of D_cp (f) that $\begin{matrix} D_{cp} (f) \\ = & ⋀_{A \in L^{X}} S (f_{L}^{\to} (h_{X} (A)), h_{Y} (f_{L}^{\to} (A))) \\ ⩽ & ⋀_{B \in L^{Y}} S (f_{L}^{\to} (h_{X} (f_{L}^{\leftarrow} (B))), h_{Y} (f_{L}^{\to} (f_{L}^{\leftarrow} (B)))) \\ ⩽ & ⋀_{B \in L^{Y}} S (f_{L}^{\to} (h_{X} (f_{L}^{\leftarrow} (B))), h_{Y} (B)) \end{matrix}$ $\begin{matrix} ⩽ & ⋀_{A \in L^{X}} S (f_{L}^{\to} (h_{X} (f_{L}^{\leftarrow} (f_{L}^{\to} (A)))), h_{Y} (f_{L}^{\to} (A))) \\ ⩽ & ⋀_{A \in L^{X}} S (f_{L}^{\to} (h_{X} (A)), h_{Y} (f_{L}^{\to} (A))) \\ = & D_{cp} (f) . \end{matrix}$

This implies

$D_{cp} (f) = ⋀_{B \in L^{Y}} S (f_{L}^{\to} (h_{X} (f^{\leftarrow} (B))), h_{Y} (B)),$ as desired. □

In L-convex spaces, compositions of L-CP mappings (resp. L-CC mappings) are still L-CP mappings (resp. L-CC mappings). Now let us give a degree representation of this result.

Proposition 3.6. Let $f : (X, C_{X}) ⟶ (Y, C_{Y})$ and $g : (Y, C_{Y}) ⟶ (Z, C_{Z})$ be mappings between L-convex spaces. Then

(1) D_cp (f) ∧ D_cp (g) ⩽ D_cp (g ∘ f) .

(2) D_cc (f) ∧ D_cc (g) ⩽ D_cc (g ∘ f) .

Proof. (1) By Proposition 3.3, we have $\begin{matrix} D_{cp} (f) \land D_{cp} (g) \\ = & ⋀_{A \in L^{X}} S (f_{L}^{\to} (h_{X} (A)), h_{Y} (f_{L}^{\to} (A))) \\ \land ⋀_{B \in L^{Y}} S (g_{L}^{\to} (h_{Y} (B)), h_{Z} (g_{L}^{\to} (B))) \\ = & ⋀_{A \in L^{X}} S (f_{L}^{\to} (h_{X} (A)), h_{Y} (f_{L}^{\to} (A))) \\ \land ⋀_{B \in L^{Y}} S (h_{Y} (B), g_{L}^{\leftarrow} (h_{Z} (g_{L}^{\to} (B)))) \\ ⩽ & ⋀_{A \in L^{X}} (S (f_{L}^{\to} (h_{X} (A)), h_{Y} (f_{L}^{\to} (A))) \\ \land S (h_{Y} (f_{L}^{\to} (A)), g_{L}^{\leftarrow} (h_{Z} (g_{L}^{\to} (f_{L}^{\to} (A)))))) \\ ⩽ & ⋀_{A \in L^{X}} S (f_{L}^{\to} (h_{X} (A)), g_{L}^{\leftarrow} (h_{Z} ((g \circ f)_{L}^{\to} (A)))) \\ = & ⋀_{A \in L^{X}} S ((g \circ f)_{L}^{\to} (h_{X} (A)), h_{Z} ((g \circ f)_{L}^{\to} (A))) \\ = & D_{cp} (g \circ f) . \end{matrix}$ (2) Adopting the proof of (1), it can be verified directly. □

Next we investigate the connections between degrees of L-CP mappings and that of L-CC mappings.

Proposition 3.7. Let $f : (X, C_{X}) ⟶ (Y, C_{Y})$ and $g : (Y, C_{Y}) ⟶ (Z, C_{Z})$ be mappings between L-convex spaces. If f is surjective, then $D_{cc} (g \circ f) \land D_{cp} (f) ⩽ D_{cc} (g) .$

Proof. Since f is surjective, we have $f_{L}^{\to} (f_{L}^{\leftarrow} (B)) = B$ for all B ∈ L^Y. Then it follows that $\begin{matrix} ⋀_{B \in L^{Y}} S (h_{Z} (g_{L}^{\to} (B)), g_{L}^{\to} (h_{Y} (B))) \\ = & ⋀_{B \in L^{Y}} S (h_{Z} (g_{L}^{\to} (f_{L}^{\to} (f_{L}^{\leftarrow} (B)))), g_{L}^{\to} (h_{Y} (f_{L}^{\to} (f_{L}^{\leftarrow} (B))))) \\ ⩾ & ⋀_{A \in L^{X}} S (h_{Z} ((g \circ f)_{L}^{\to} (A)), g_{L}^{\to} (h_{Y} (f_{L}^{\to} (A)))) \\ ⩾ & ⋀_{B \in L^{Y}} S (h_{Z} (g_{L}^{\to} (B)), g_{L}^{\to} (h_{Y} (B))) . \end{matrix}$

This shows $\begin{matrix} ⋀_{B \in L^{Y}} S (h_{Z} (g_{L}^{\to} (B)), g_{L}^{\to} (h_{Y} (B))) \\ = & ⋀_{A \in L^{X}} S (h_{Z} ((g \circ f)_{L}^{\to} (A)), g_{L}^{\to} (h_{Y} (f_{L}^{\to} (A)))) . \end{matrix}$

Then we have

$\begin{matrix} D_{cc} (g \circ f) \land D_{cp} (f) \\ = & ⋀_{A \in L^{X}} S (h_{Z} ((g \circ f)_{L}^{\to} (A)), (g \circ f)_{L}^{\to} (h_{X} (A))) \\ \land ⋀_{A \in L^{X}} S (f_{L}^{\to} (h_{X} (A)), h_{Y} (f_{L}^{\to} (A))) \\ ⩽ & ⋀_{A \in L^{X}} S (h_{Z} ((g \circ f)_{L}^{\to} (A)), (g \circ f)_{L}^{\to} (h_{X} (A))) \\ \land ⋀_{A \in L^{X}} S ((g \circ f)_{L}^{\to} (h_{X} (A)), g_{L}^{\to} (h_{Y} (f_{L}^{\to} (A)))) \\ ⩽ & ⋀_{A \in L^{X}} S (h_{Z} ((g \circ f)_{L}^{\to} (A)), g_{L}^{\to} (h_{Y} (f_{L}^{\to} (A)))) \end{matrix}$ $\begin{matrix} = & ⋀_{B \in L^{Y}} S (h_{Z} (g_{L}^{\to} (B)), g_{L}^{\to} (h_{Y} (B))) \\ = & D_{cc} (g), \end{matrix}$ as desired. □

Proposition 3.8. Let $f : (X, C_{X}) ⟶ (Y, C_{Y})$ and $g : (Y, C_{Y}) ⟶ (Z, C_{Z})$ be mappings between L-convex spaces. If g is injective, then $D_{cc} (g \circ f) \land D_{cp} (g) ⩽ D_{cc} (f) .$

Proof. Since g is injective, we have $g_{L}^{\leftarrow} (g_{L}^{\to} (B)) = B$ for all B ∈ L^Y. Then it follows that $\begin{matrix} D_{cc} (g \circ f) \land D_{cp} (g) \\ = & ⋀_{A \in L^{X}} S (h_{Z} ((g \circ f)_{L}^{\to} (A)), (g \circ f)_{L}^{\to} (h_{X} (A))) \\ \land ⋀_{B \in L^{Y}} S (g_{L}^{\to} (h_{Y} (B)), h_{Z} (g_{L}^{\to} (B))) \\ ⩽ & ⋀_{A \in L^{X}} S (h_{Z} ((g \circ f)_{L}^{\to} (A)), (g \circ f)_{L}^{\to} (h_{X} (A))) \\ \land ⋀_{A \in L^{X}} S (g_{L}^{\to} (h_{Y} (f_{L}^{\to} (A))), h_{Z} (g_{L}^{\to} (f_{L}^{\to} (A)))) \\ ⩽ & ⋀_{A \in L^{X}} S (h_{Z} ((g \circ f)_{L}^{\to} (A)), (g \circ f)_{L}^{\to} (h_{X} (A))) \\ \land ⋀_{A \in L^{X}} S (g_{L}^{\to} (h_{Y} (f_{L}^{\to} (A))), h_{Z} ((g \circ f)_{L}^{\to} (A))) \\ ⩽ & ⋀_{A \in L^{X}} S (g_{L}^{\to} (h_{Y} (f_{L}^{\to} (A))), (g \circ f)_{L}^{\to} (h_{X} (A))) \\ = & ⋀_{A \in L^{X}} S (h_{Y} (f_{L}^{\to} (A)), g_{L}^{\leftarrow} ((g \circ f)_{L}^{\to} (h_{X} (A)))) \\ = & ⋀_{A \in L^{X}} S (h_{Y} (f_{L}^{\to} (A)), f_{L}^{\to} (h_{X} (A))) \\ = & D_{cc} (f), \end{matrix}$ as desired. □

Next, we will equip the isomorphic mappings between L-convex spaces with some degrees and present the corresponding conclusion in a degree sense.

Definition 3.9. Let $f : (X, C_{X}) ⟶ (Y, C_{Y})$ be a bijective mapping between L-convex spaces. Then $D_{iso} (f) = D_{cp} (f) \land D_{cp} (f^{- 1})$ is called the degree to which f is an L-isomorphic mapping, where f^-1 denotes the inverse mapping of f.

Definition 3.9 can be thought of as the lattice-valued generalization of the classical definition “a bijective mapping f between L-convex spaces is an isomorphism provided that both f and f^-1 are L-CP mappings”.

Proposition 3.10. Let $f : (X, C_{X}) ⟶ (Y, C_{Y})$ be a bijective mapping between L-convex spaces. Then $D_{cp} (f^{- 1}) = D_{cc} (f) .$

Proof. Since f is bijective, we have $(f^{- 1})_{L}^{\leftarrow} (A) = f_{L}^{\to} (A)$ for all A ∈ L^X. By Proposition 3.3, it follows that $\begin{matrix} D_{cp} (f^{- 1}) \\ = & ⋀_{B \in L^{Y}} S (h_{Y} (B), (f^{- 1})_{L}^{\leftarrow} (h_{X} ((f^{- 1})_{L}^{\to} (B)))) \\ = & ⋀_{B \in L^{Y}} S (h_{Y} (B), f_{L}^{\to} (h_{X} ((f^{- 1})_{L}^{\to} (B)))) \\ = & ⋀_{A \in L^{X}} S (h_{Y} (f_{L}^{\to} (A)), f_{L}^{\to} (h_{X} ((f^{- 1})_{L}^{\to} (f_{L}^{\to} (A))))) \\ = & ⋀_{A \in L^{X}} S (h_{Y} (f_{L}^{\to} (A)), f_{L}^{\to} (h_{X} (A))) \\ = & D_{cc} (f), \end{matrix}$ as desired. □

Corollary 3.11. Let $f : (X, C_{X}) ⟶ (Y, C_{Y})$ be a bijective mapping between L-convex spaces. Then $D_{iso} (f) = D_{cp} (f) \land D_{cc} (f) .$

The above corollary gives an logical interpretation to the result that “a bijective mapping f between convex spaces is an isomorphism provided that f is a CP mapping and a CC mapping in the classical convex spaces”.

4 Degrees of L-IP mappings in L-interval spaces

Considering L being a completely distributive lattice, Pang and Shi [20] introduced the concept of L-interval spaces and established its relationship with L-convex spaces. In this section, we will give each L-IP mapping between L-interval spaces some measures to describe how a mapping between L-interval spaces become an L-IP mapping in some degree. Now let us recall the definitions of L-interval spaces and L-IP mappings.

Definition 4.1. ([20]). An L-interval operator on X is a mapping $I : J (L^{X}) \times J (L^{X}) ⟶ L^{X}$ which satisfies:

(LI1) $x_{λ}, y_{μ} ⩽ I (x_{λ}, y_{μ})$ ;

(LI2) $I (x_{λ}, y_{μ}) = I (y_{μ}, x_{λ})$ ;

(LI3) $I (x_{λ}, y_{μ}) = ⋁_{ν ⊲ λ} ⋁_{γ ⊲ μ} I (x_{ν}, y_{γ})$ , where “⊲” denotes the wedge below relation on L. For an L-interval operator $I$ on X, the pair $(X, I)$ is called an L-interval space.

A mapping $f : (X, I_{X}) ⟶ (Y, I_{Y})$ between L-interval spaces is called L-interval-preserving (L-IP, in short) provided that for each x_λ, y_μ ∈ J (L^X), $f_{L}^{\to} (I_{X} (x_{λ}, y_{μ})) ⩽ I_{Y} (f (x)_{λ}, f (y)_{μ}) .$

Definition 4.2. Let $(X, I_{X})$ and $(X, I_{Y})$ be L-interval spaces, and let f : X ⟶ Y be a mapping. Then D_ip (f) defined by

$D_{ip} (f) = ⋀_{x_{λ}, y_{μ} \in J (L^{X})} S (f_{L}^{\to} (I_{X} (x_{λ}, y_{μ})), I_{Y} (f (x)_{λ}, f (y)_{μ}))$ is called the degree to which f is L-IP.

Dually, we can give another definition in the following.

Definition 4.3. Let $(X, I_{X})$ and $(Y, I_{Y})$ be L-interval spaces, and let f : X ⟶ Y be a mapping. Then D_aip (f) defined by $D_{aip} (f) = ⋀_{x_{λ}, y_{μ} \in J (L^{X})} S (I_{Y} (f (x)_{λ}, f (y)_{μ}), f_{L}^{\to} (I_{X} (x_{λ}, y_{μ})))$ is called the degree to which f is an L-anti-interval preserving (L-AIP, in short) mapping.

Proposition 4.4. (1) If $id : (X, I_{X}) ⟶ (X, I_{X})$ is the identity mapping, then D_ip (id) =⊤ and D_aip (id) =⊤.

(2) Let $(X, I_{X})$ and $(Y, I_{Y})$ be L-interval spaces, and let f : X ⟶ Y be a mapping. Then $D_{ip} (f) = ⋀_{x_{λ}, y_{μ} \in J (L^{X})} S (I_{X} (x_{λ}, y_{μ}), f_{L}^{\leftarrow} (I_{Y} (f (x)_{λ}, f (y)_{μ}))) .$

Proof. (1) Straightforward.

(2) Adopting the proof of Proposition 3.3, it is obvious. □

Proposition 4.5. Let $f : (X, I_{X}) ⟶ (Y, I_{Y})$ and $g : (Y, I_{Y}) ⟶ (Z, I_{Z})$ be mappings between L-convex spaces. Then

(1) D_ip (f) ∧ D_ip (g) ⩽ D_ip (g ∘ f) .

(2) D_aip (f) ∧ D_aip (g) ⩽ D_aip (g ∘ f) .

Proof. (1) By Proposition 3.3, we have $\begin{matrix} D_{ip} (f) \land D_{ip} (g) \\ = & ⋀_{x_{λ}^{1}, x_{μ}^{2} \in J (L^{X})} S (f_{L}^{\to} (I_{X} (x_{λ}^{1}, x_{μ}^{2})), I_{Y} (f (x^{1})_{λ}, f (x^{2})_{μ})) \\ \land ⋀_{y_{ν}^{1}, y_{υ}^{2} \in J (L^{Y})} S (I_{Y} (y_{ν}^{1}, y_{υ}^{2}), g_{L}^{\leftarrow} (I_{Z} (g (y^{1})_{ν}, g (y^{2})_{υ}))) \\ ⩽ ⋀_{x_{λ}^{1}, x_{μ}^{2} \in J (L^{X})} S (f_{L}^{\to} (I_{X} (x_{λ}^{1}, x_{μ}^{2})), I_{Y} (f (x^{1})_{λ}, f (x^{2})_{μ})) \\ \land ⋀_{x_{ν}^{3}, x_{υ}^{4} \in J (L^{X})} S (I_{Y} (f (x^{3})_{ν}, f (x^{4})_{υ}), \\ g_{L}^{\leftarrow} (I_{Z} (g (f (x^{3}))_{ν}, g (f (x^{4}))_{υ}))) \\ ⩽ ⋀_{x_{λ}^{1}, x_{μ}^{2} \in J (L^{X})} S (f_{L}^{\to} (I_{X} (x_{λ}^{1}, x_{μ}^{2})), \\ g_{L}^{\leftarrow} (I_{Z} (g (f (x^{1}))_{λ}, g (f (x^{2}))_{μ}))) \\ = ⋀_{x_{λ}^{1}, x_{μ}^{2} \in J (L^{X})} S ((g \circ f)_{L}^{\to} (I_{X} (x_{λ}^{1}, x_{μ}^{2})), I_{Z} ((g \circ f) (x^{1})_{λ}, \\ (g \circ f) (x^{2})_{μ})) \\ = D_{ip} (g \circ f) . \end{matrix}$

(2) Adopting the proof of (1), it can be verified directly. □

Next we investigate the connections between degrees of L-IP mappings and that of L-AIP mappings.

Proposition 4.6. Let $f : (X, I_{X}) ⟶ (Y, I_{Y})$ and $g : (Y, I_{Y}) ⟶ (Z, I_{Z})$ be mappings between L-interval spaces. If f is surjective, then

$D_{aip} (g \circ f) \land D_{ip} (f) ⩽ D_{aip} (g) .$

Proof. By Definitions 4.2 and 4.3, it follows that $\begin{matrix} D_{aip} (g \circ f) \land D_{ip} (f) \\ = ⋀_{x_{λ}^{1}, x_{μ}^{2} \in J (L^{X})} S (I_{Z} (g (f (x^{1}))_{λ}, g (f (x^{2}))_{μ}), \\ (g \circ f)_{L}^{\to} (I_{X} (x_{λ}^{1}, x_{μ}^{2}))) \\ \land ⋀_{x_{ν}^{3}, x_{υ}^{4} \in J (L^{X})} S (f_{L}^{\to} (I_{X} (x_{ν}^{3}, x_{υ}^{4})), I_{Y} (f (x^{3})_{ν}, f (x^{4})_{υ})) \\ ⩽ ⋀_{x_{λ}^{1}, x_{μ}^{2} \in J (L^{X})} S (I_{Z} (g (f (x^{1}))_{λ}, g (f (x^{2}))_{μ}), \\ (g \circ f)_{L}^{\to} (I_{X} (x_{λ}^{1}, x_{μ}^{2}))) \\ \land ⋀_{x_{ν}^{3}, x_{υ}^{4} \in J (L^{X})} S ((g \circ f)_{L}^{\to} (I_{X} (x_{ν}^{3}, x_{υ}^{4})), \\ g_{L}^{\to} (I_{Y} (f (x^{3})_{ν}, f (x^{4})_{υ}))) \\ ⩽ ⋀_{x_{λ}^{1}, x_{μ}^{2} \in J (L^{X})} S (I_{Z} (g (f (x^{1}))_{λ}, g (f (x^{2}))_{μ}), \\ g_{L}^{\to} (I_{Y} (f (x^{1})_{λ}, f (x^{2})_{μ}))) \\ = ⋀_{y_{λ}^{1}, y_{μ}^{2} \in J (L^{Y})} S (I_{Z} (g (y^{1})_{λ}, g (y^{2})_{μ}), g_{L}^{\to} (I_{Y} (y_{λ}^{1}, y_{μ}^{2}))) \\ (Since f is surjective) \\ = D_{aip} (g) . \end{matrix}$ □

Proposition 4.7. Let $f : (X, I_{X}) ⟶ (Y, I_{Y})$ and $g : (Y, I_{Y}) ⟶ (Z, I_{Z})$ be mappings between L-interval spaces. If g is injective, then

$D_{aip} (g \circ f) \land D_{ip} (g) ⩽ D_{aip} (f) .$

Proof. Since g is injective, we have $g_{L}^{\leftarrow} (g_{L}^{\to} (B)) = B$ for all B ∈ L^Y. Then it follows that $\begin{matrix} D_{aip} (g \circ f) \land D_{ip} (g) \\ = ⋀_{x_{λ}^{1}, x_{μ}^{2} \in J (L^{X})} S (I_{Z} (g (f (x^{1}))_{λ}, g (f (x^{2}))_{μ}), \\ (g \circ f)_{L}^{\to} (I_{X} (x_{λ}^{1}, x_{μ}^{2}))) \\ \land ⋀_{y_{ν}^{1}, y_{υ}^{2} \in J (L^{Y})} S (g_{L}^{\to} (I_{Y} (y_{ν}^{1}, y_{υ}^{2})), I_{Z} (g (y^{1})_{ν}, g (y^{2})_{υ})) \end{matrix}$ $\begin{matrix} ⩽ ⋀_{x_{λ}^{1}, x_{μ}^{2} \in J (L^{X})} S (I_{Z} (g (f (x^{1}))_{λ}, g (f (x^{2}))_{μ}), \\ (g \circ f)_{L}^{\to} (I_{X} (x_{λ}^{1}, x_{μ}^{2}))) \\ \land ⋀_{x_{ν}^{3}, x_{υ}^{4} \in J (L^{X})} S (g_{L}^{\to} (I_{Y} (g (x^{3})_{ν}, g (x^{4})_{υ})), \\ I_{Z} ((g \circ f) (x^{3})_{ν}, (g \circ f) (x^{4})_{υ})) \\ ⩽ ⋀_{x_{λ}^{1}, x_{μ}^{2} \in J (L^{X})} S (g_{L}^{\to} (I_{Y} (g (x^{1})_{λ}, g (x^{2})_{μ})), \\ (g \circ f)_{L}^{\to} (I_{X} (x_{λ}^{1}, x_{μ}^{2}))) \\ = ⋀_{x_{λ}^{1}, x_{μ}^{2} \in J (L^{X})} S (I_{Y} (g (x^{1})_{λ}, g (x^{2})_{μ}), \\ g_{L}^{\leftarrow} ((g \circ f)_{L}^{\to} (I_{X} (x_{λ}^{1}, x_{μ}^{2})))) \\ = ⋀_{x_{λ}^{1}, x_{μ}^{2} \in J (L^{X})} S (I_{Y} (g (x^{1})_{λ}, g (x^{2})_{μ}), \\ g_{L}^{\leftarrow} (g_{L}^{\to} (f_{L}^{\to} (I_{X} (x_{λ}^{1}, x_{μ}^{2}))))) \\ = ⋀_{x_{λ}^{1}, x_{μ}^{2} \in J (L^{X})} S (I_{Y} (g (x^{1})_{λ}, g (x^{2})_{μ}), f_{L}^{\to} (I_{X} (x_{λ}^{1}, x_{μ}^{2}))) \\ = D_{aip} (g) . □ \end{matrix}$

Definition 4.8. Let $(X, I_{X})$ and $(X, I_{Y})$ be L-interval spaces, and let f : X ⟶ Y be a mapping. Then D_ii (f) defined by $D_{ii} (f) = D_{ip} (f) \land D_{aip} (f)$ is called the degree to which f is an L-II mapping.

Remark 4.9. If D_ii (f) =⊤, then $f^{\to} (I_{X} (x_{λ}, y_{μ})) = I_{Y} (f (x)_{λ}, f (y)_{μ})$ for each x_λ, y_μ ∈ J (L^X). This is exactly the definition of L-II mapping in L-interval spaces. Here we equip each mapping between L-interval spaces with some degree to measure how a mapping becomes an L-II mapping.

Proposition 4.10. Let $(X, I_{X})$ and $(Y, I_{Y})$ be L-interval spaces, and let f : X ⟶ Y be a bijective mapping. Then $D_{ip} (f) = D_{aip} (f^{- 1}),$ where f^-1 denotes the inverse of f.

Proof. For each B ∈ L^Y and x ∈ X, it follows that $\begin{matrix} (f^{- 1})_{L}^{\to} (B) (x) \\ = ⋁_{f^{- 1} (y) = x} B (y) = ⋁_{f (x) = y} B (y) = B (f (x)) = f_{L}^{\leftarrow} (B) (x) . \end{matrix}$

This means $(f^{- 1})_{L}^{\to} (B) = f_{L}^{\leftarrow} (B)$ . Then we have $\begin{matrix} D_{aip} (f^{- 1}) \\ = ⋀_{y_{λ}^{1}, y_{μ}^{2} \in J (L^{Y})} S (I_{X} (f^{- 1} (y^{1})_{λ}, f^{- 1} (y^{2})_{μ}), (f^{- 1})_{L}^{\to} (I_{Y} (y_{λ}^{1}, y_{μ}^{2}))) \\ = ⋀_{y_{λ}^{1}, y_{μ}^{2} \in J (L^{Y})} S (I_{X} (f^{- 1} (y^{1})_{λ}, f^{- 1} (y^{2})_{μ}), f_{L}^{\leftarrow} (I_{Y} (y_{λ}^{1}, y_{μ}^{2}))) \\ = ⋀_{x_{λ}^{1}, x_{μ}^{2} \in J (L^{X})} S (I_{X} (x_{λ}^{1}, x_{μ}^{2}), f_{L}^{\leftarrow} (I_{Y} (f (x^{1})_{λ}, f (x^{2})_{μ}))) \\ (Since f is bijective) \\ = ⋀_{x_{λ}^{1}, x_{μ}^{2} \in J (L^{X})} S (f_{L}^{\to} (I_{X} (x_{λ}^{1}, x_{μ}^{2})), I_{Y} (f (x^{1})_{λ}, f (x^{2})_{μ})) \\ = D_{ip} (f), \end{matrix}$ as desired. □

Corollary 4.11. Let $(X, I_{X})$ and $(Y, I_{Y})$ be L-interval spaces, and let f : X ⟶ Y be a bijective mapping. Then $D_{ii} (f) = D_{ip} (f) \land D_{ip} (f^{- 1}) .$

In [20], Pang and Shi showed each L-convex space $(X, C)$ can induce an L-interval space $(X, I^{C})$ in the following way. $\forall x_{λ}, y_{μ} \in J (L^{X}), I^{C} (x_{λ}, y_{μ}) = h^{C} (x_{λ} \lor y_{μ}) .$

Further, each L-CC mapping between L-convex spaces is an L-IP mapping between their induced L-interval spaces. That is,

Proposition 4.12. ([20]). If $f : (X, C_{X}) ⟶ (Y, C_{Y})$ is an L-CP mapping, then $f : (X, I^{C_{X}}) ⟶ (Y, I^{C_{Y}})$ is an L-IP mapping.

Next we give a degree representation of the above-mentioned result.

Proposition 4.13. Let $(X, C_{X})$ and $(Y, C_{Y})$ be L-interval spaces, and let f : X ⟶ Y be a mapping. Then $D_{cp} (f) ⩽ D_{ip} (f) .$

Proof. By the definition of $I^{C}$ , it follows that

$\begin{matrix} D_{cp} (f) \\ = ⋀_{A \in L^{X}} S (f_{L}^{\to} (h^{C_{X}} (A)), h^{C_{Y}} (f_{L}^{\to} (A))) \\ ⩽ ⋀_{x_{λ}, y_{μ} \in J (L^{X})} S (f_{L}^{\to} (h^{C_{X}} (x_{λ} \lor y_{μ})), h^{C_{Y}} (f (x)_{λ} \lor f (y)_{μ})) \\ = ⋀_{x_{λ}, y_{μ} \in J (L^{X})} S (f_{L}^{\to} (I^{C_{X}} (x_{λ}, y_{μ})), I^{C_{Y}} (f (x)_{λ}, f (y)_{μ})) \\ = D_{ip} (f), \end{matrix}$ as desired. □

5 Conclusions

In this paper, we equipped each mapping between L-convex spaces with some degree to be an L-CP mapping and an L-CC mapping, and equipped each mapping between L-interval spaces with some degree to be an L-IP mapping and an L-AIP mapping. From this aspect, we could consider the L-CP degree, the L-CC degree, the L-IP degree and the L-AIP degree of a mapping in the theory of L-convex spaces even if the mapping is not a CP mapping, a CC mapping, an IP mapping or an AIP mapping. By means of these definitions, we proved that the L-CP degree, the L-CC degree, the L-IP degree and the L-AIP degree naturally suggest lattice-valued logical extensions of properties related to CP mappings, CC mappings, IP mappings and AIP mappings in classical convex spaces to L-convex spaces. As a future work, we will consider applying the degree approach to (L, M)-fuzzy convex spaces [32], which is a more general framework of fuzzy convex spaces.

Footnotes

Acknowledgments

The authors would like to express their sincere thanks to the anonymous reviewers and the editor for their careful reading and constructive comments. This work is supported by the Project of Shandong Province Higher Educational Science and Technology Program (NO. J18KA245), the Natural Science Foundation of China (NOs. 11701189, 11871097), Fujian Natural Science Foundation (NO. 2018J01422), the Scientific Research Project of Minnan Normal University (NO. MK201715) and the project funded by China Postdoctoral Science Foundation (NO. 2017M622563).

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