Degrees of ( L,M )-fuzzy convexities

Abstract

In this paper, we introduce the degree to which a mapping $C : L^{X} ⟶ M$ is an (L, M)-fuzzy convexity. When $(X, C_{X})$ and $(Y, C_{Y})$ are (L, M)-fuzzy convex spaces to some extent, the degrees to which a mapping f : X ⟶ Y is convexity-preserving or convex-to-convex are defined. Finally, the degree to which a surjective mapping f : X ⟶ Y is a quotient mapping is also defined. We not only give their characterizations but also discuss the relationships among them.

1. Introduction

The convex set, which is an ordinary notion inspired by the shape of some figures, such as circles and polyhedrons in 2 or 3-dimensional Euclidean spaces, has an old history and has been applied to various research areas. In the standard case, a set in an n-dimensional Euclidean space is convex if and only if it contains the whole segments joining any two of its points. By axiomatizing the properties of convex sets in Euclidean spaces, the concept of convexity (or abstract convexity) was introduced in [3 , 30]. Under the axiomatic framework of M. van de Vel, the theory of convexity has been studied widespread from different perspectives, and has been found that it includes many mathematical structures (see the monograph of M. van de Vel [30]). There exist convexities in real vector spaces [28], convexities in lattices [29], convexities in posets [1], convexities in matroids [30], convexities in metric spaces and graphs [27].

With the development of fuzzy mathematics, many mathematical structures have been generalized to the fuzzy case, such as fuzzy topology [2], fuzzy convergence structures [12 , 16] and so on. The notion of convexity has also been combined with fuzzy set theory. In 1994, M.V. Rosa presented the notion of fuzzy convexity in [20]. In 2009, Y. Maruyama generalized it to L-fuzzy setting in [10]. In a Rosa-Maruyama’s fuzzy convexity, convex sets are fuzzy, but the convexity comprising those convex sets is a crisp subset of I^X or L^X. Recently, F.G. Shi and Z.Y. Xiu [25] gave a new approach to fuzzification of convexity in a completely different direction and proposed the concept of M-fuzzifying convexity. Furthermore, F.G. Shi and Z.Y. Xiu introduced the concept of (L, M)-fuzzy convexity in [24]. Now the theory of fuzzy convex structures are generally investigated in the frameworks of L-convex structures [6 , 18], M-fuzzifying convex structures [22 , 32–36] and (L, M)-fuzzy convex structures [8, 24], respectively.

An (L, M)-fuzzy convexity is exactly a mapping from L^X to M satisfying three axioms. Given a mapping $C : L^{X} ⟶ M$ , it is either an (L, M)-fuzzy convexity or not. Then a natural problem is as follows. For any mapping $C : L^{X} ⟶ M$ , is it an (L, M)-fuzzy convexity to some extent? Our aim is to solve this problem.

The main aim of this paper is to define the degree to which a mapping $C : L^{X} ⟶ M$ is an (L, M)-fuzzy convexity and the degree to which an L-subset is an L-convex set with respect to $C$ . Based on these concepts, we will introduce the degrees to which a mappings f : X ⟶ Y is a convexity-preserving, a convex-to-convex or a quotient mapping.

This paper is organized as follows. In Section 2, some necessary concepts of convexity and (L, M)-fuzzy convexity are recalled. In Section 3, the degree to which a mapping $C : L^{X} ⟶ M$ is an (L, M)-fuzzy convexity is introduced. Meanwhile, the degrees to which $C$ is an M-fuzzifying convexity, an (L, M)-fuzzy closure system or an M-fuzzifying closure system are also introduced. Besides, we propose the degree to which an L-subset is an L-convex set with respect to $C$ . In Section 4, when $(X, C_{X})$ and $(Y, C_{Y})$ are (L, M)-fuzzy convex spaces to some extent, the degrees to which a mappings f : X ⟶ Y is convexity-preserving or convex-to-convex are defined. Their characterizations and some properties are given. In Section 5, the degree to which a surjective mapping f : X ⟶ Y is a quotient mapping is also defined. We give not only their characterizations but also discuss the relationships between them.

2. Preliminaries

Throughout this paper, unless otherwise stated, (L, ∨, ∧) denotes a complete lattice and (M, ∨, ∧) denotes a completely distributive lattice. The smallest element and the largest element in M are denoted by ⊥_M and ⊤_M, respectively. X is a non-empty set. We denote the set of all subsets of X by 2^X and denote the set of all L-subsets of X by L^X. L^X is also a complete lattice when it inherits the structure of the lattice L in a natural way, by defining ∧, ∨, ≤ pointwisely. The smallest element and the largest element in L^X are denoted by ⊥_{L
^X} and ⊤_{L
^X}, respectively.

We can define a residual implication in the completely distributive lattice M by $a \to b = ⋁ {c \in M | a \land c \leq b} .$ Also, we denote a ↔ b = (a → b) ∧ (b → a). Some properties of the implication operation are listed in the following lemma.

Lemma 2.1. ([4]) Let (M, ∨, ∧) be a completely distributive lattice and → be the implication operation corresponding to ∧. Then for all a, b, c ∈ M, {a_i} _i∈I, {b_i} _i∈I ⊆ M, the following statements hold.

(1) ⊤_M → a = a.

(2) (a → b) ≥ c ⇔ a ∧ c ≤ b.

(3) a → b = ⊤ _M ⇔ a ≤ b.

(4) a → (⋀ _i∈Ib_i) = ⋀ _i∈I (a → b_i), hence a → b ≤ a → c whenever b ≤ c.

(5) (⋁ _i∈Ia_i) → b = ⋀ _i∈I (a_i → b), hence a → c ≥ b → c whenever a ≤ b.

(6) (a → c) ∧ (c → b) ≤ a → b.

The binary relation≺^op in M is defined as follows: for a, b ∈ M, a ≺ ^opb if and only if for every subset D ⊆ M, the relation inf D ≤b always implies the existence of d ∈ D with d ≤ a. Let α (b) = {a ∈ M | a ≺ ^opb}. We have b = ⋀ α (b) for each b ∈ M and α : M ⟶ 2^M is an ⋀-⋃ and order-reversing mapping (see [21, 31]).

Let f : X ⟶ Y be a mapping. Define f^→ : 2^X ⟶ 2^Y and f^← : 2^Y ⟶ 2^X by f^→ (A) = {f (x) | x ∈ A} for all A ∈ 2^X and f^← (B) = {x ∈ X | f (x) ∈ B} for all B ∈ 2^Y. The forward L-power operator $f_{L}^{\to} : L^{X} ⟶ L^{Y}$ and the backward L-power operator $f_{L}^{\leftarrow} : L^{Y} ⟶ L^{X}$ induced by f are defined by $f_{L}^{\to} (A) (y) = ⋁_{f (x) = y} A (x)$ for all A ∈ L^X, y ∈ Y and $f_{L}^{\leftarrow} (B) = B \circ f$ for all B ∈ L^Y, respectively [19].

Definition 2.2. ([5 , 30]) A subset $C$ of 2^X is called a convexity (or convex structure), if it satisfies the following conditions:

(C1) ∅, $X \in C$ ;

(C2) If ${A_{i} | i \in I} \subseteq C$ is non-empty, then $⋂_{i \in I} A_{i} \in C$ ;

(C3) If ${A_{j} | j \in J} \subseteq C$ is non-empty and totally ordered by inclusion, then $⋃_{j \in J} A_{j} \in C$ .

The pair $(X, C)$ is called a convex space (or convexity space [30]) and the members of $C$ are called convex sets. A mapping $f : (X, C_{X}) ⟶ (Y, C_{Y})$ is called convexity-preserving if $f^{\leftarrow} (B) \in C_{X}$ for each $B \in C_{Y}$ . A mapping $f : (X, C_{X}) ⟶ (Y, C_{Y})$ is called convex-to-convex if $f^{\to} (A) \in C_{Y}$ for each $A \in C_{X}$ .

In 1994, M.V. Rosa introduced the notion of fuzzy convexity in [20]. In 2009, Y. Maruyama generalized it to L-fuzzy setting in [10]. In 2014, F.G. Shi and Z.Y. Xiu gave the definition of M-fuzzifying convexity in [25]. Recently, F.G. Shi and Z.Y. Xiu [24] further extended it to an M-fuzzy subset of L^X (called an (L, M)-fuzzy convexity) as follows.

Definition 2.3. ([24]) A mapping $C : L^{X} ⟶ M$ is called an (L, M)-fuzzy convexity (or (L, M)-fuzzy convex structure) if it satisfies the following conditions:

(LMC1) $C (⊥_{L^{X}}) = C (⊤_{L^{X}}) = ⊤_{M}$ ;

(LMC2) If {A_i ∣ i ∈ I} ⊆ L^X is non-empty, then $C (⋀_{i \in I} A_{i}) \geq ⋀_{i \in I} C (A_{i})$ ;

(LMC3) If {A_j ∣ j ∈ J} ⊆ L^X is non-empty and totally ordered, then $C (⋁_{j \in J} A_{j}) \geq ⋀_{j \in J} C (A_{j})$ .

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

Denote {0, 1} =2 and [0, 1] = I. Then an (L, 2)-fuzzy convexity is an L-fuzzy convexity in [10], an (I, 2)-fuzzy convexity is a fuzzy convexity in [20], an (2, M)-fuzzy convexity is an M-fuzzifying convexity in [22, 25], an (2, 2)-fuzzy convexity is a convexity in [5 , 30].

Remark 2.4. Denote a directed family {A_j} _j∈J ⊆ L^X by ${A_{j}}_{j \in J}^{dir} \subseteq L^{X}$ . We say {A_j} _j∈J is directed, 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}. Besides, Shi and Xiu show that (LMC3) can be replaced by the following condition in [24]:

(LMC3*) If ${A_{j}}_{j \in J}^{dir} \subseteq L^{X}$ is non-empty, then $C (⋁_{j \in J}^{dir} A_{j}) \geq ⋀_{j \in J} C (A_{j})$ .

Definition 2.5. ([24]) Let $(X, C_{X})$ and $(Y, C_{Y})$ be two (L, M)-fuzzy convex spaces. A mapping f : X ⟶ Y is called a convexity-preserving mapping if $C_{Y} (B) \leq C_{X} (f_{L}^{\leftarrow} (B))$ for all B ∈ L^Y. A mapping f : X ⟶ Y is called a convex-to-convex mapping if $C_{X} (A) \leq C_{Y} (f_{L}^{\to} (A))$ for all A ∈ L^X.

We denote $C_{[a]} = {A \in L^{X} | C (A) \geq a}$ and $C^{[a]} = {A \in L^{X} | a \notin α (C (A))}$ . It is easy to show that $C$ is an (L, M)-fuzzy convexity on X if and only if ∀a ∈ M, $C_{[a]}$ is an L-convexity on X. Excepting this, we have the following theorem.

Theorem 2.6. ([24]) Let $C : L^{X} ⟶ M$ be a mapping. Then the following statements are equivalent.

(1) $C$ is an (L, M)-fuzzy convexity on X.

(2) ∀a ∈ M ∖ {⊥ _M}, $C_{[a]}$ is an L-convexity on X.

(3) ∀a ∈ α (⊥ _M), $C^{[a]}$ is an L-convexity on X.

Finally, we recall the definitions of (L, M)-fuzzy closure system and M-fuzzifying closure system.

Definition 2.7. ([23]) A mapping $C : L^{X} ⟶ M$ is called an (L, M)-fuzzy closure system provided that it satisfies the following conditions:

(LMCS) ∀ {A_i ∣ i ∈ I} ⊆ L^X, $C (⋀_{i \in I} A_{i}) \geq ⋀_{i \in I} C (A_{i})$ .

The pair $(X, C)$ is called an (L, M)-fuzzy closure system space.

3. (L, M)-fuzzy convexity degrees

Now we introduce the notions of the degrees to which a mapping $C : L^{X} ⟶ M$ is an (L, M)-fuzzy convexity and the degrees to which an a mapping $C : L^{X} ⟶ M$ is an (L, M)-fuzzy closure system.

Definition 3.1. D(C) Let $C : L^{X} ⟶ M$ be a mapping. Then $D (C)$ defined by

$\begin{matrix} D (C) = C (⊥_{L^{X}}) \land C (⊤_{L^{X}}) \\ \land ⋀_{{A_{i}}_{i \in I} \subseteq L^{X}} (⋀_{i \in I} C (A_{i}) \to C (⋀_{i \in I} A_{i})) \\ \land ⋀_{{A_{j}}_{j \in J}^{dir} \subseteq L^{X}} (⋀_{j \in J} C (A_{j}) \to C (⋁_{j \in J}^{dir} A_{j})) \end{matrix}$ is called the degree to which $C$ is an (L, M)-fuzzy convexity (or the (L, M)-fuzzy convexity degree of $C$ ).

Remark 3.2. If $D (C) = ⊤_{M}$ , then $C (⊥_{L^{X}}) = C (⊤_{L^{X}}) = ⊤_{M}$ ; ∀ {A_i} _i∈I ⊆ L^X, $⋀_{i \in I} C (A_{i}) \leq C (⋀_{i \in I} A_{i})$ and $\forall {A_{j}}_{j \in J}^{dir} \subseteq L^{X}$ , $⋀_{j \in J} C (A_{j}) \leq C (⋁_{j \in J}^{dir} A_{j})$ It is exactly the definition of (L, M)-fuzzy convexity. Moreover, $C$ is an (L, M)-fuzzy convexity if and only if $D (C) = ⊤_{M}$ . However, there exist examples to show $D (C) \neq ⊤_{M}$ .

Example 3.3. Let M = [0, 1] and $C : L^{X} ⟶ M$ be a mapping which satisfies $C (A) = 0.5$ for all A ∈ L^X. By Definition 3.1, we have $D (C) = 0.5$ . This shows $D (C) \neq ⊤_{M}$ .

Definition 3.4. D(Q) Let $C : L^{X} ⟶ M$ be a mapping. Then $D_{s} (C)$ defined by $D_{s} (C) = \underset{{A_{i}}_{i \in I} \subseteq L^{X}}{⋀} (\underset{i \in I}{⋀} C (A_{i}) \to C (\underset{i \in I}{⋀} A_{i}))$ is called the degree to which $C$ is an (L, M)-fuzzy closure system (or the (L, M)-fuzzy closure system degree of $C$ ).

The following theorem is obvious.

Theorem 3.5. Let $C : L^{X} ⟶ M$ be a mapping. Then $D (C) \leq D_{s} (C)$ .

Remark 3.6. If L = 2 in Definition 3.1 and Definition 3.4, then $D (C)$ (resp., $D_{s} (C)$ ) is the M-fuzzifying convexity degree of $C$ (resp., the M-fuzzifying closure system degree of $C$ ).

From properties of the implication operation in Lemma 2.1, the following lemma is straightforward.

Lemma 3.7. D(C) Let $C : L^{X} ⟶ M$ be a mapping. For any a ∈ M, $a \leq D (C)$ if and only if $a \leq C (⊥_{L^{X}})$ , $a \leq C (⊤_{L^{X}})$ , $⋀_{i \in I} C (A_{i}) \land a \leq C (⋀_{i \in I} A_{i})$ for all {A_i} _i∈I ⊆ L^X and $⋀_{j \in J} C (A_{j}) \land a \leq C (⋁_{j \in J}^{dir} A_{j})$ for all ${A_{j}}_{j \in J}^{dir} \subseteq L^{X}$ .

Theorem 3.8. Let $C : L^{X} ⟶ M$ be a mapping. Then

$\begin{matrix} D (C) = ⋁ {a \in M ∣ \begin{matrix} a \leq C (⊥_{L^{X}}), a \leq C (⊤_{L^{X}}), \\ ⋀_{i \in I} C (A_{i}) \land a \leq C (⋀_{i \in I} A_{i}), \\ \forall {A_{i}}_{i \in I} \subseteq L^{X}, \\ ⋀_{j \in J} C (A_{j}) \land a \leq C (⋁_{j \in J}^{dir} A_{j}), \\ \forall {A_{j}}_{j \in J}^{dir} \subseteq L^{X} \end{matrix}} . \end{matrix}$

Proof. By Lemma 3.7, it is easy to be proved.□

The next two theorems give other characterizations about (L, M)-fuzzy convexity degrees by means of its two kinds of cut sets.

Theorem 3.9. D(C) Let $C : L^{X} ⟶ M$ be a mapping. Then

$\begin{matrix} D (C) = ⋁ {a \in M | \forall b \leq a, C_{[b]} is anL - convexity} . \end{matrix}$

Proof. Let RHS denote the right hand side of the above equality.

Suppose that $a \leq C (⊥_{L^{X}}), a \leq C (⊤_{L^{X}})$ , ⋀_i∈I $C (A_{i}) \land a \leq C (⋀_{i \in I} A_{i})$ , ∀ {A_i} _i∈I ⊆ L^X, ⋀_j∈J $C (A_{j}) \land a \leq C (⋁_{j \in J}^{dir} A_{j})$ , $\forall {A_{j}}_{j \in J}^{dir} \subseteq L^{X}$ . For any b ≤ a, ${A_{i}}_{i \in I} \subseteq C_{[b]}$ , ${A_{j}}_{j \in J}^{dir} \subseteq C_{[b]}$ , we have $C (⊥_{L^{X}}) \geq b$ , $C (⊤_{L^{X}}) \geq b$ and $C (⋀_{i \in I} A_{i}) \geq (⋀_{i \in I} C (A_{i})) \land b = b$ , $C (⋁_{j \in J}^{dir} A_{j})$ $\geq (⋀_{j \in J} C (A_{j})) \land b = b .$ This shows $⊥_{L^{X}}, ⊤_{L^{X}} \in C_{[b]}$ , $⋀_{i \in I} A_{i} \in C_{[b]}$ and $⋁_{j \in J}^{dir} A_{j} \in C_{[b]}$ . By Theorem 3.8, $D (C) \leq RHS$ .

Conversely, assume that $\forall b \leq a, C_{[b]}$ is an L-convexity. For any {A_i} _i∈I ⊆ L^X and ${A_{j}}_{j \in J}^{dir} \subseteq L^{X}$ . Let b = a. Then ⊥_{L
^X}, $⊤_{L^{X}} \in C_{[a]}$ , which means $C (⊥_{L^{X}}) \geq a$ , $C (⊤_{L^{X}}) \geq a$ . Let $b = ⋀_{i \in I} C (A_{i}) \land a$ . Then b ≤ a and ${A_{i}}_{i \in I} \subseteq C_{[b]}$ . Thus $⋀_{i \in I} A_{i} \in C_{[b]}$ , i.e., $C (⋀_{i \in I} A_{i}) \geq b = (⋀_{i \in I} C (A_{i})) \land a$ . Let $b = ⋀_{j \in J} C (A_{j}) \land a$ . Then b ≤ a and ${A_{j}}_{j \in J}^{dir} \subseteq C_{[b]}$ . Thus $⋁_{j \in J}^{dir} A_{j} \in C_{[b]}$ , i.e., $C (⋁_{j \in J}^{dir} A_{j}) \geq b = (⋀_{j \in J} C (A_{j})) \land a$ . Combining this with Theorem 3.8, $D (C) \geq RHS$ .□

Theorem 3.10. D(C) Let $C : L^{X} ⟶ M$ be a mapping. Then $\begin{matrix} D (C) \\ = & ⋁ {a \in M ∣ \forall b \notin α (a), C^{[b]} is an L - convexity} . \end{matrix}$ Let RHS denote the right hand side of the above equality.

Suppose that $a \leq C (⊥_{L^{X}}), a \leq C (⊤_{L^{X}})$ , $⋀_{i \in I} C (A_{i}) \land a \leq C (⋀_{i \in I} A_{i})$ , ∀ {A_i} _i∈I ⊆ L^X, $⋀_{j \in J} C (A_{j}) \land a \leq C (⋁_{j \in J}^{dir} A_{j})$ , $\forall {A_{j}}_{j \in J}^{dir} \subseteq L^{X}$ . For any b ∉ α (a), ${A_{i}}_{i \in I} \subseteq C^{[b]}$ , ${A_{j}}_{j \in J}^{dir} \subseteq C^{[b]}$ , we have $b \notin α (a) \cup ⋃_{i \in i} α (C (A_{i}))$ and $b \notin α (a) \cup ⋃_{j \in J}^{dir} α (C (A_{j}))$ . Since α is an ⋀-⋃ map and α is order-reversing, we have $α (a) \cup ⋃_{i \in I} α (C (A_{i})) = α (a \land ⋀_{i \in I} C (A_{i})) \supseteq α (C (⋀_{i \in I} A_{i}))$ , and α (a)∪ $⋃_{j \in J}^{dir} α (C (A_{j})) \supseteq α (C (⋁_{j \in J}^{dir} A_{j}))$ . This implies $b \notin α (C (⋀_{i \in I} A_{i}))$ and $b \notin α (C (⋁_{j \in J}^{dir} A_{j}))$ , which means $⋀_{i \in I} A_{i} \in C^{[b]}$ and $⋁_{j \in J}^{dir} A_{j} \in C^{[b]}$ . Since $α (a) \supseteq α (C (⊥_{L^{X}}))$ and $α (a) \supseteq α (C (⊤_{L^{X}}))$ , we know $b \notin α (C (⊥_{L^{X}}))$ and $b \notin α (C (⊤_{L^{X}}))$ , which shows $⊥_{L^{X}}, ⊤_{L^{X}} \in C^{[b]}$ . By Theorem 3.8, $D (C) \leq RHS$ .

Conversely, assume that $\forall b \notin α (a), C^{[b]}$ is an L-convexity. For any ∀b ∉ α (a), $⊥_{L^{X}}, ⊤_{L^{X}} \in C^{[b]}$ , we have $b \notin α (C (⊥_{L^{X}})), b \notin α (C (⊤_{L^{X}}))$ . This implies $a \leq C (⊥_{L^{X}})$ and $a \leq C (⊤_{L^{X}})$ . For any {A_i} _i∈I ⊆ L^X, suppose that $b \notin α (⋀_{i \in I} C (A_{i}) \land a)$ . Since $α (⋀_{i \in I} C (A_{i}) \land a) = ⋃_{i \in I} α (C (A_{i})) \cup α (a)$ . it follows that b ∉ α (a) and $b \notin α (C (A_{i}))$ , which means $A_{i} \in C^{[b]}$ for any i ∈ I. Note that $C^{[b]}$ is an L-convexity. Then $⋀_{i \in I} A_{i} \in C^{[b]}$ , i.e., $b \notin α (C (⋀_{i \in I} A_{i}))$ . By the arbitrariness of b, we obtain $⋀_{i \in I} C (A_{i}) \land a \leq C (⋀_{i \in I} A_{i})$ . Similarly, we can prove $⋀_{j \in J} C (A_{j}) \land a \leq C (⋁_{j \in J}^{dir} A_{j})$ for all ${A_{j}}_{j \in J}^{dir} \subseteq L^{X}$ Combining this with Theorem 3.8, $D (C) \geq RHS$ . □

Analogously, we can obtain characterizations of (L, M)-fuzzy closure systems in the following theorem.

Theorem 3.11. D(Q) Let $C : L^{X} ⟶ M$ be a mapping. Then

$\begin{matrix} D_{s} (C) \\ = & ⋁ {a \in M ∣ \begin{matrix} ⋀_{i \in I} C (A_{i}) \land a \leq C (⋀_{i \in I} A_{i}), \\ \forall {A_{i}}_{i \in I} \subseteq L^{X} \end{matrix}} . \\ = & ⋁ {a \in M ∣ \forall b \leq a, C_{[b]} is an L - closure system} . \\ = & ⋁ {a \in M ∣ \forall b \notin α (a), C^{[b]} is an L - closure system} . \end{matrix}$

As we know, $D (C)$ can also be treated as a mapping D : M^{L
^X} ⟶ M defined by $C \mapsto D (C)$ . Then we have the following theorem.

Theorem 3.12. D(C)is an (L, M)-fuzzy pretopology Let ${C_{λ} : L^{X} ⟶ M}_{λ \in Λ}$ be a family of mappings. Then $⋀_{λ \in Λ} D (C_{λ}) \leq D (⋀_{λ \in Λ} C_{λ})$ .

Proof. By Lemma 2.1, we have $\begin{array}{l} D (\underset{λ \in Λ}{\land} ℭ_{λ}) = \underset{Λ λ \in}{\land} ℭ_{λ} (⊥_{L^{X}}) \land \underset{λ \in Λ}{\land} ℭ_{λ} (⊤_{L^{X}}) \\ \land \underset{{A_{i}}_{i \in I} \subseteq L^{X}}{\land} (\underset{i \in I}{\land} \underset{λ \in Λ}{\land} ℭ_{λ} (A_{i}) \to \underset{λ \in Λ}{\land} ℭ_{λ} (\underset{i \in I}{\land} A_{i})) \\ \land \underset{{A_{j}}_{j \in J}^{d i r} \subseteq L^{X}}{\land} (\underset{j \in J}{\land} \underset{λ \in Λ}{\land} ℭ_{λ} (A_{j}) \to \underset{λ \in Λ}{\land} ℭ_{λ} (\lor_{j \in J}^{d i r} A_{j})) \\ = \underset{λ \in Λ}{\land} ℭ_{λ} (⊥_{L^{X}}) \land \underset{λ \in Λ}{\land} ℭ_{λ} (⊤_{L^{X}}) \\ \land \underset{{A_{i}}_{i \in I} \subseteq L^{X}}{\land} \underset{λ \in Λ}{\land} (\underset{i \in I}{\land} \underset{λ \in Λ}{\land} ℭ_{λ} (A_{i}) \to ℭ_{λ} (\underset{i \in I}{\land} A_{i})) \\ \land \underset{{A_{j}}_{j \in J}^{d i r} \subseteq L^{X}}{\land} \underset{λ \in Λ}{\land} (\underset{j \in J}{\land} \underset{λ \in Λ}{\land} ℭ_{λ} (A_{j}) \to ℭ_{λ} (\lor_{j \in J}^{d i r} A_{j})) \\ \geq \underset{λ \in Λ}{\land} ℭ_{λ} (⊥_{L^{X}}) \land \underset{λ \in Λ}{\land} ℭ_{λ} (⊤_{L^{X}}) \\ \land \underset{{A_{i}}_{i \in I} \subseteq L^{X}}{\land} \underset{λ \in Λ}{\land} (\underset{i \in I}{\land} ℭ_{λ} (A_{i}) \to ℭ_{λ} (\underset{i \in I}{\land} A_{i})) \\ \land \underset{{A_{j}}_{j \in J}^{d i r} \subseteq L^{X}}{\land} \underset{λ \in Λ}{\land} (\underset{j \in J}{\land} ℭ_{λ} (A_{j}) \to ℭ_{λ} (\lor_{j \in J}^{d i r} A_{j})) \\ = \underset{λ \in Λ}{\land} ℭ_{λ} (⊥_{L^{X}}) \land \underset{λ \in Λ}{\land} ℭ_{λ} (⊤_{L^{X}}) \\ \land \underset{λ \in Λ}{\land} \underset{{A_{i}}_{i \in I} \subseteq L^{X}}{\land} (\underset{i \in I}{\land} ℭ_{λ} (A_{i}) \to ℭ_{λ} (\underset{i \in I}{\lor} A_{i})) \\ \land \underset{λ \in Λ}{\land} \underset{{A_{j}}_{j \in J}^{d i r} \subseteq L^{X}}{\land} (\underset{j \in J}{\land} ℭ_{λ} (A_{j}) \to ℭ_{λ} (\lor_{j \in J}^{d i r} A_{j})) \\ = \underset{λ \in Λ}{\land} D (ℭ_{λ}) . \end{array}$ □

In what follows, we introduce the degree to which an L-subset is an L-convex set with respect to $C$ , which is a generalization of L-convex set degree in [24].

Definition 3.13. Con(A) Given a mapping $C : L^{X} ⟶ M$ and a fuzzy set A ∈ L^X, we define ${Con}_{C} (A)$ by ${Con}_{C} (A) = D (C) \land C (A) .$ Then the value ${Con}_{C} (A)$ is called the degree to which A is an L-convex set with respect to $C$ (or the L-convex set degree of A with respect to $C$ ).

Remark 3.14. If $D (C) = ⊤_{M}$ , which means $C$ is an (L, M)-fuzzy convexity, then ${Con}_{C} (A) = C (A)$ , which can be regarded as a generalization of L-convex set degree in [24].

Theorem 3.15. L-convex set Given a mapping $C : L^{X} ⟶ M$ and a fuzzy set A ∈ L^X. Then

(1) $⋀_{i \in I} {Con}_{C} (A_{i}) \leq {Con}_{C} (⋀_{i \in I} A_{i})$ , ∀ {A_i ∣ i ∈ I} ⊆ L^X.

(2) $⋀_{j \in J} {Con}_{C} (A_{j}) \leq {Con}_{C} (⋁_{j \in J}^{dir} A_{j})$ , ∀ {A_j ∣ j ∈ J} ^dir ⊆ L^X.

If ${Con}_{C} (A)$ is viewed as a mapping ${Con}_{C} : L^{X} ⟶ M$ defined by $A \mapsto {Con}_{C} (A)$ , then ${Con}_{C}$ satisfies the conditions (LMC2) and (LMC3) inDefinition 2.3.

Proof. (1) By Definition 3.13, it suffices to show that $D (C) \land ⋀_{i \in I} C (A_{i}) \leq D (C)$ and $D (C) \land ⋀_{i \in I} C (A_{i}) \leq C (⋀_{i \in I} A_{i})$ for any {A_i ∣ i ∈ I} ⊆ L^X. By Definition 3.1 and Lemma 2.1, we have $D (C) \land ⋀_{i \in I} C (A_{i}) \leq (⋀_{i \in I} C (A_{i}) \to C (⋀_{i \in I} A_{i})) \land ⋀_{i \in I} C (A_{i}) \leq C (⋀_{i \in I} A_{i})$ .

(2) can be proved similarly. □

4. Degrees of convexity-preserving and convex-to-convex mappings

In this section, when $(X, C_{X})$ and $(Y, C_{Y})$ are (L, M)-fuzzy convex spaces to some extent, the degrees to which a mappings f : X ⟶ Y is convexity-preserving or convex-to-convex with respect to $C_{X}$ and $C_{Y}$ are defined. Their characterizations and some properties are given.

Definition 4.1. CP, CC Given two mappings $C_{X} : L^{X} ⟶ M$ and $C_{Y} : L^{Y} ⟶ M$ . Let f : X ⟶ Y be a mapping. Then

a) the convexity-preserving degree of f with respect to $C_{X}$ and $C_{Y}$ is defined by $CP (f) = \underset{B \in L^{Y}}{⋀} ({Con}_{C_{Y}} (B) \to {Con}_{C_{X}} (f_{L}^{\leftarrow} (B))) .$

b) the convex-to-convex degree of f with respect to $C_{X}$ and $C_{Y}$ is defined by $CC (f) = \underset{A \in L^{X}}{⋀} ({Con}_{C_{X}} (A) \to {Con}_{C_{Y}} (f_{L}^{\to} (A))) .$

In the following, characterizations about the convexity-preserving degree, the convex-to-convex degree of f with respect to $C_{X}$ and $C_{Y}$ will be presented.

Theorem 4.2. characterizationsof convexity-preserving degree Given two mappings $C_{X} : L^{X} ⟶ M$ and $C_{Y} : L^{Y} ⟶ M$ . Let $D (C_{X})$ and $D (C_{Y})$ denote the (L, M)-fuzzy convexity degrees of $C_{X}$ and $C_{Y}$ , respectively. Then

$\begin{array}{l} CP(f) \\ = \lor {a \in M \begin{matrix} D (ℭ_{Y}) \land ℭ_{Y} (B) \land a \\ ≰ D (ℭ_{X}) \land ℭ_{X} (f_{L}^{\leftarrow} (B)), \\ \forall B \in L^{Y} \end{matrix}} . \\ = \lor {a \in M ∣ \begin{matrix} \forall b \leq D (ℭ_{Y}) \land a, \forall B \in {(ℭ_{Y})}_{[b]}, \\ b \leq D (ℭ_{X}), f_{L}^{\leftarrow} (B) \in {(ℭ_{X})}_{[b]}, \end{matrix}} . \\ = \lor {a \in M ∣ \begin{matrix} \forall b \notin α (D (ℭ_{Y}) \land a), \forall B \in {(ℭ_{Y})}^{[b]}, \\ b \notin α (D (ℭ_{X})), f_{L}^{\leftarrow} (B) \in {(ℭ_{X})}^{[b]} \end{matrix}} . \end{array}$

Proof. (1) For any a ∈ M, a ≤ CP (f) if and only if $a \land {Con}_{C_{Y}} (B) \leq {Con}_{C_{X}} (f_{L}^{\leftarrow} (B))$ for all B ∈ L^Y, if and only if $D (C_{Y}) \land C_{Y} (B) \land a \leq D (C_{X}) \land C_{X} (f_{L}^{\leftarrow} (B))$ for all B ∈ L^Y.

(2) Let RHS denote the right hand side of equality. Suppose that $D (C_{Y}) \land C_{Y} (B) \land a \leq D (C_{X}) \land C_{X} (f_{L}^{\leftarrow} (B))$ for all B ∈ L^Y. For any $b \leq D (C_{Y}) \land a$ and $B \in (C_{Y})_{[b]}$ , i.e., $b \leq (C_{Y}) (B)$ , we have $b \leq D (C_{X}) \land C_{X} (f_{L}^{\leftarrow} (B))$ . Thus $b \leq D (C_{X})$ and $b \leq C_{X} (f_{L}^{\leftarrow} (B))$ , i.e., $f_{L}^{\leftarrow} (B) \in (C_{X})_{[b]}$ . By the result of (1), CP (f) ≤ RHS.

Conversely, take any $b \leq D (C_{Y}) \land C_{Y} (B) \land a$ . Then $b \leq D (C_{Y}) \land a$ and $b \leq C_{Y} (B)$ , i.e., $B \in (C_{Y})_{[b]}$ . Thus $D (C_{X}) \geq b$ and $f_{L}^{\leftarrow} (B) \in (C_{X})_{[b]}$ , i.e., $C_{X} (f_{L}^{\leftarrow} (B)) \geq b$ . This implies $D (C_{X}) \land C_{X} (f_{L}^{\leftarrow} (B)) \geq b$ . By the arbitrariness of b, we obtain $D (C_{Y}) \land C_{Y} (B) \land a \leq D (C_{X}) \land C_{X} (f_{L}^{\leftarrow} (B))$ . Combining this with (1), CP (f) ≥ RHS.

(3) Let RHS denote the right hand side of equality. Suppose that $D (C_{Y}) \land C_{Y} (B) \land a \leq D (C_{X}) \land C_{X} (f_{L}^{\leftarrow} (B))$ for all B ∈ L^Y. For any $b \notin α (D (C_{Y}) \land a)$ and $B \in (C_{Y})^{[b]}$ , i.e., $b \notin α (C_{Y} (B))$ , we have $b \notin α (C_{Y} (B)) \cup α (D (C_{Y}) \land a)$ . Since α is an ⋀-⋃ map, it follows $\begin{matrix} α (C_{Y} (B)) \cup α (D (C_{Y}) \land a) \\ = & α (D (C_{Y}) \land C_{Y} (B) \land a) \\ \supseteq & α (D (C_{X}) \land C_{X} (f_{L}^{\leftarrow} (B))) \\ = & α (D (C_{X})) \cup α (C_{X} (f_{L}^{\leftarrow} (B))) . \end{matrix}$ This implies $b \notin α (D (C_{X}))$ and $b \notin α (C_{X} (f_{L}^{\leftarrow} (B)))$ , i.e., $f_{L}^{\leftarrow} (B) \in (C_{X})^{[b]}$ . By the result of (1), CP (f) ≤ RHS.

Conversely, take any $b \notin α (D (C_{Y}) \land C_{Y} (B) \land a)$ . Since $α (D (C_{Y}) \land C_{Y} (B) \land a) = α (D (C_{Y}) \land a) \cup α (C_{Y} (B))$ , we have $b \notin α (D (C_{Y}) \land a)$ and $b \notin α (C_{Y} (B))$ , i.e., $B \in (C_{Y})^{[b]}$ . Thus $b \notin α (D (C_{X}))$ and $f_{L}^{\leftarrow} (B) \in (C_{X})^{[b]}$ , i.e., $b \notin α (C_{X} (f_{L}^{\leftarrow} (B))$ . This implies $b \notin α (D (C_{X})) \cup α (C_{X} (f_{L}^{\leftarrow} (B))) = α (D (C_{X}) \land C_{X} (f_{L}^{\leftarrow} (B)))$ . By the arbitrariness of b, we obtain $D (C_{Y}) \land C_{Y} (B) \land a \leq D (C_{X}) \land C_{X} (f_{L}^{\leftarrow} (B))$ . Combining this with (1), CP (f) ≥ RHS. □

Analogous, we can obtain characterizations about the convex-to-convex degree of f with respect to $C_{X}$ and $C_{Y}$ in the followingtheorem.

Theorem 4.3. Given two mappings $C_{X} : L^{X} ⟶ M$ and $C_{Y} : L^{Y} ⟶ M$ . Let $D (C_{X})$ and $D (C_{Y})$ denote the (L, M)-fuzzy convexity degrees of $C_{X}$ and $C_{Y}$ , respectively. Then

$\begin{array}{l} C C (f) \\ = \lor {a \in M ∣ \begin{matrix} D (ℭ_{X}) \land ℭ_{X} (A) \land a \\ \leq & D (ℭ_{Y}) \land ℭ_{Y} (f_{L}^{\to} (A)), \\ \forall A \in L^{X} \end{matrix}} . \\ = \lor {a \in M ∣ \begin{matrix} \forall b \leq D (ℭ_{X}) \land a, \forall A \in {(ℭ_{X})}_{[b]}, \\ b \leq D (ℭ_{Y}), f_{L}^{\to} (A) \in {(ℭ_{Y})}_{[b]} \end{matrix}} . \\ = \lor {a \in M ∣ \begin{matrix} \forall b \notin α (D (ℭ_{X}) \land a), A \in {(ℭ_{X})}^{[b]}, \\ b \notin α (D (ℭ_{Y})), f_{L}^{\to} (A) \in {(ℭ_{Y})}^{[b]} \end{matrix}} \end{array}$

Definition 4.4. Given two mappings $C_{X} : L^{X} ⟶ M$ and $C_{Y} : L^{Y} ⟶ M$ . Let f : X ⟶ Y be a bijective mapping. Then the isomorphism degree of f with respect to $C_{X}$ and $C_{Y}$ is defined by $ISO (f) = CP (f) \land CP (f^{- 1}),$ where f^-1 is the inverse mapping of f.

Theorem 4.5. Given two mappings $C_{X} : L^{X} ⟶ M$ and $C_{Y} : L^{Y} ⟶ M$ . Let f : X ⟶ Y be a bijective mapping. Then CP (f^-1) = CC (f) and

$ISO (f) = CP (f) \land CP (f^{- 1}) = CP (f) \land CC (f) .$

Proof. Since f is bijective, we have $(f^{- 1})_{L}^{\leftarrow} (A) = f_{L}^{\to} (A)$ for all A ∈ L^X. Then $\begin{matrix} CP (f^{- 1}) \\ = & \underset{A \in L^{X}}{⋀} (\underset{C_{X}}{Con} (A) \to {Con}_{C_{Y}} ((f^{- 1})_{L}^{\leftarrow} (A))) \\ = & \underset{A \in L^{X}}{⋀} ({Con}_{C_{X}} (A) \to {Con}_{C_{Y}} (f_{L}^{\to} (A))) \\ = & CC (f) . \end{matrix}$ Hence ISO (f) = CP (f)∧ CP (f^-1) = CP (f) ∧ CC (f). □

Proposition 4.6. Given a mapping $C_{X} : L^{X} ⟶ M$ . Let id : X ⟶ X be the identity mapping. Then $CP (id) = CC (id) = ISO (id) = ⊤_{M} .$

Proof. Obviously. □

Proposition 4.7. Given three mappings $C_{X} : L^{X} ⟶ M$ , $C_{Y} : L^{Y} ⟶ M$ and $C_{Z} : L^{Z} ⟶ M$ . Let f : X ⟶ Y and g : Y ⟶ Z be mappings.

(1) If f is surjective, then CP (f) ∧ CC (g ∘ f) ≤ CC (g).

(2) If g is injective, then CP (g) ∧ CC (g ∘ f) ≤ CC (f).

Proof. (1) Since f is surjective, it follows that $f_{L}^{\to} (f_{L}^{\leftarrow} (B)) = B$ for any B ∈ L^Y. Then $(g \circ f)_{L}^{\to} (f_{L}^{\leftarrow} (B)) = g_{L}^{\to} (f_{L}^{\to} (f_{L}^{\leftarrow} (B))) = g_{L}^{\to} (B)$ . By Lemma 2.1 (6), we have $\begin{array}{l} C P (f) \land C C (g ° f) \\ = (\underset{B \in L^{Y}}{\land} (C o n_{ℭ_{Y}} (B) \to C o n_{ℭ_{X}} (f_{L}^{\leftarrow} (B)))) \\ \land (\underset{A \in L^{X}}{\land} (C o n_{ℭ_{X}} (A) \to C o n_{ℭ_{Z}} ({(g ° f)}_{L}^{\to} (A)))) \\ \leq (\underset{B \in L^{Y}}{\land} (C o n_{ℭ_{Y}} (B) \to C o n_{ℭ_{X}} (f_{L}^{\leftarrow} (B)))) \\ \land (\underset{B \in L^{Y}}{\land} (C o n_{ℭ_{X}} (f_{L}^{\leftarrow} (B)) \\ \to C o n_{ℭ_{Z}} ({(g ° f)}_{L}^{\to} (f_{L}^{\leftarrow} (B))))) \\ = \underset{B \in L^{Y}}{\land} ((C o n_{ℭ_{Y}} (B) \to C o n_{ℭ_{X}} (f_{L}^{\leftarrow} (B))) \\ \land (C o n_{ℭ_{X}} (f_{L}^{\leftarrow} (B)) \to C o n_{ℭ_{Z}} (g_{L}^{\to} (B)))) \\ \leq \underset{B \in L^{Y}}{\land} (C o n_{ℭ_{Y}} (B) \to C o n_{ℭ_{Z}} (g_{L}^{\to} (B))) = C C (g) . \end{array}$

(2) Since g is injective, we have $g_{L}^{\leftarrow} (g_{L}^{\to} (B)) = B$ for all B ∈ L^Y. Then $g_{L}^{\leftarrow} ((g \circ f)_{L}^{\to} (A)) = f_{L}^{\to} (A)$ for all A ∈ L^X. By Lemma 2.1 (6), we have

$\begin{array}{l} C P (g) \land C C (g ° f) \\ = (\underset{C \in L^{Z}}{\land} (C o n_{ℭ_{Z}} (C) \to C o n_{ℭ_{Y}} (g_{L}^{\leftarrow} (C)))) \\ \land (\underset{A \in L^{X}}{\land} (C o n_{ℭ_{X}} (A) \to C o n_{ℭ_{Z}} ({(g ° f)}_{L}^{\to} (A)))) \\ \leq (\underset{A \in L^{X}}{\land} (C o n_{ℭ_{Z}} ({(g ° f)}_{L}^{\to} (A)) \\ \to C o n_{ℭ_{Y}} (g_{L}^{\leftarrow} ({(g ° f)}_{L}^{\to} (A))))) \\ \land (\underset{A \in L^{X}}{\land} (C o n_{ℭ_{X}} (A) \to C o n_{ℭ_{Z}} ({(g ° f)}_{L}^{\to} (A)))) \\ = \underset{A \in L^{X}}{\land} ((C o n_{ℭ_{Z}} ((g_{L}^{\to} (f_{L}^{\to} (A)) \to C o n_{ℭ_{Y}} (f_{L}^{\to} (A))) \\ \land (C o n_{ℭ_{X}} (A) \to C o n_{ℭ_{Z}} (g_{L}^{\to} (f_{L}^{\to} (A)))) \\ \leq \underset{A \in L^{X}}{\land} (C o n_{ℭ_{X}} (A) \to C o n_{ℭ_{Y}} (f_{L}^{\to} (A))) = C C (f) . □ \end{array}$

Proposition 4.8. Given three mappings $C_{X} : L^{X} ⟶ M$ , $C_{Y} : L^{Y} ⟶ M$ and $C_{Z} : L^{Z} ⟶ M$ . Let f : X ⟶ Y and g : Y ⟶ Z be mappings. Then

(1) CP (f) ∧ CP (g) ≤ CP (g ∘ f).

(2) CC (f) ∧ CC (g) ≤ CC (g ∘ f).

(3) if f and g are bijective, then ISO (f) ∧ ISO (g) ≤ ISO (g ∘ f).

Proof. We only prove (1), (2) and (3) can be proved similarly. By Definition 4.1 and Lemma 2.1 (6), we have $\begin{array}{l} C P (f) \land C P (g) \\ = (\underset{B \in L^{Y}}{\land} (C o n_{ℭ_{Y}} (B) \to C o n_{ℭ_{X}} (f_{L}^{\leftarrow} (B)))) \\ \land (\underset{C \in L^{Z}}{\land} (C o n_{ℭ_{Z}} (C) \to C o n_{ℭ_{Y}} (g_{L}^{\leftarrow} (C)))) \\ \leq (\underset{D \in L^{Z}}{\land} (C o n_{ℭ_{Y}} (g_{L}^{\leftarrow} (D)) \to C o n_{ℭ_{X}} (f_{L}^{\leftarrow} (g_{L}^{\leftarrow} (D))))) \\ \land (\underset{C \in L^{Z}}{\land} (C o n_{ℭ_{Z}} (C) \to C o n_{ℭ_{Y}} (g_{L}^{\leftarrow} (C)))) \\ = (\underset{D \in L^{Z}}{\land} (C o n_{ℭ_{Y}} (g_{L}^{\leftarrow} (D)) \to C o n_{ℭ_{X}} ({(g ° f)}_{L}^{\leftarrow} (D)))) \\ \land (\underset{C \in L^{Z}}{\land} (C o n_{ℭ_{Z}} (C) \to C o n_{ℭ_{Y}} (g_{L}^{\leftarrow} (C)))) \\ = \underset{C \in L^{Z}}{\land} ((C o n_{ℭ_{Y}} (g_{L}^{\leftarrow} (C)) \to C o n_{ℭ_{X}} ({(g ° f)}_{L}^{\leftarrow} (C))) \\ \land (C o n_{ℭ_{Z}} (C) \to C o n_{ℭ_{Y}} (g_{L}^{\leftarrow} (C)))) \\ \leq \underset{C \in L^{Z}}{\land} (C o n_{ℭ_{Z}} (C) \to C o n_{ℭ_{X}} ({(g ° f)}_{L}^{\leftarrow} (C))) \\ C P (g ° f) . □ \end{array}$

Lemma 4.9. Given two mappings $C_{X} : L^{X} ⟶ M$ and $C_{Y} : L^{Y} ⟶ M$ . Let f : X ⟶ Y be a bijective mapping. Then

$CP (f) = ⋀_{A \in L^{X}} ({Con}_{C_{Y}} (f_{L}^{\to} (A)) \to {Con}_{C_{X}} (A))$ .

$CC (f) = ⋀_{B \in L^{Y}} ({Con}_{C_{X}} (f_{L}^{\leftarrow} (B)) \to {Con}_{C_{Y}} (B))$ .

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

Since f is bijective, we have $f_{L}^{\leftarrow} (f_{L}^{\to} (A)) = A$ for all A ∈ L^X and $f_{L}^{\to} (f_{L}^{\leftarrow} (B)) = B$ for all B ∈ L^Y. Then $\begin{array}{l} \underset{A \in L^{X}}{\land} (C o n_{ℭ_{Y}} (f_{L}^{\to} (A)) \to C o n_{ℭ_{X}} (A)) \\ = \underset{A \in L^{X}}{\land} (C o n_{ℭ_{Y}} (f_{L}^{\to} (A)) \to C o n_{ℭ_{X}} (f_{L}^{\leftarrow} (f_{L}^{\to} (A)))) \\ \geq \underset{B \in L^{Y}}{\land} (C o n_{ℭ_{Y}} (B) \to C o n_{ℭ_{X}} (f_{L}^{\leftarrow} (B))) = C P (f) \\ = \underset{B \in L^{Y}}{\land} (C o n_{ℭ_{Y}} (f_{L}^{\to} (f_{L}^{\leftarrow} (B))) \to C o n_{ℭ_{X}} (f_{L}^{\leftarrow} (B))) \\ \geq \underset{A \in L^{X}}{\land} (C o n_{ℭ_{Y}} (f_{L}^{\to} (A)) \to C o n_{ℭ_{X}} (A)) . \end{array}$ Hence $CP (f) = ⋀_{A \in L^{X}} ({Con}_{C_{Y}} (f_{L}^{\to} (A)) \to {Con}_{C_{X}}$ (A)). □

Theorem 4.10. Given two mappings $C_{X} : L^{X} ⟶ M$ and $C_{Y} : L^{Y} ⟶ M$ . Let f : X ⟶ Y be a bijective mapping. Then

$\begin{array}{l} I S O (f) \\ = \underset{A \in L^{X}}{\land} (C o n_{ℭ_{X}} (A) \leftrightarrow C o n_{ℭ_{Y}} (f_{L}^{\to} (A))) \\ = \underset{B \in L^{Y}}{\land} (C o n_{ℭ_{Y}} (B) \leftrightarrow C o n_{ℭ_{X}} (f_{L}^{\leftarrow} (B))) . \end{array}$

5. Degrees of quotient mappings

In this section, when $(X, C_{X})$ and $(Y, C_{Y})$ are (L, M)-fuzzy convex spaces to some extent, the degree to which a surjective mapping f : X ⟶ Y is a quotient mapping is defined and the relationships between the quotient degree, the convexity-preserving degree and the convex-to-convex degree of mappings are discussed.

Definition 5.1. Given two mappings $C_{X} : L^{X} ⟶ M$ and $C_{Y} : L^{Y} ⟶ M$ . Let f : X ⟶ Y be a surjective mapping. Then QU (f) defined by $QU (f) = \underset{B \in L^{Y}}{⋀} ({Con}_{C_{Y}} (B) \leftrightarrow {Con}_{C_{X}} (f_{L}^{\leftarrow} (B)))$ is called the quotient degree of f with respect to $C_{X}$ and $C_{Y}$ .

Theorem 5.2. Given two mappings $C_{X} : L^{X} ⟶ M$ and $C_{Y} : L^{Y} ⟶ M$ . Let f : X ⟶ Y be a surjective mapping. Then QU (f) ≤ CP (f).

Proof. Obviously. □

Finally, we discuss the relationships among QU (f), CP (f) and CC (f).

Theorem 5.3. Given two mappings $C_{X} : L^{X} ⟶ M$ and $C_{Y} : L^{Y} ⟶ M$ . Let f : X ⟶ Y be a surjective mapping. Then CP (f) ∧ CC (f) ≤ QU (f).

Proof. Since f is surjective, we have $f_{L}^{\to} (f_{L}^{\leftarrow} (D)) = D$ for all D ∈ L^Y. Then $\begin{array}{l} C P (f) \land C C (f) \\ = (\underset{B \in L^{Y}}{\land} (C o n_{ℭ_{Y}} (B) \to C o n_{ℭ_{X}} (f_{L}^{\leftarrow} (B)))) \\ \land (\underset{A \in L^{X}}{\land} (C o n_{ℭ_{X}} (A) \to C o n_{ℭ_{Y}} (f_{L}^{\to} (A)))) \\ \leq (\underset{B \in L^{Y}}{\land} (C o n_{ℭ_{Y}} (B) \to C o n_{ℭ_{X}} (f_{L}^{\leftarrow} (B)))) \\ \land (\underset{D \in L^{Y}}{\land} (C o n_{ℭ_{X}} (f_{L}^{\leftarrow} (D)) \to C o n_{ℭ_{Y}} (f_{L}^{\to} (f_{L}^{\leftarrow} (D))))) \\ = (\underset{B \in L^{Y}}{\land} (C o n_{ℭ_{Y}} (B) \to C o n_{ℭ_{X}} (f_{L}^{\leftarrow} (B)))) \\ \land (\underset{D \in L^{Y}}{\land} (C o n_{ℭ_{X}} (f_{L}^{\leftarrow} (D)) \to C o n_{ℭ_{Y}} (D))) \\ = \underset{B \in L^{Y}}{\land} ((C o n_{ℭ_{Y}} (B) \to C o n_{ℭ_{X}} (f_{L}^{\leftarrow} (B))) \\ \land (C o n_{ℭ_{X}} (f_{L}^{\leftarrow} (B)) \to C o n_{ℭ_{Y}} (B))) \\ = \underset{B \in L^{Y}}{\land} (C o n_{ℭ_{Y}} (B) \leftrightarrow C o n_{ℭ_{X}} (f_{L}^{\leftarrow} (B))) \\ = Q U (f) . □ \end{array}$

Theorem 5.4. Given three mappings $C_{X} : L^{X} ⟶ M$ , $C_{Y} : L^{Y} ⟶ M$ and $C_{Z} : L^{Z} ⟶ M$ . Let f : X ⟶ Y and g : Y ⟶ Z be surjective mappings. Then

(1) QU (f) ∧ QU (g) ≤ QU (g ∘ f).

(2) QU (g ∘ f) ∧ CP (f) ∧ CP (g) ≤ QU (g).

Proof. (1) By Lemma 2.1, we have $\begin{array}{l} Q U (f) \land Q U (g) \\ = (\underset{B \in L^{Y}}{\land} (C o n_{ℭ_{Y}} (B) \leftrightarrow C o n_{ℭ_{X}} (f_{L}^{\leftarrow} (B)))) \\ \land (\underset{C \in L^{Z}}{\land} (C o n_{ℭ_{Z}} (C) \leftrightarrow C o n_{ℭ_{Y}} (g_{L}^{\leftarrow} (C)))) \\ \leq (\underset{D \in L^{Z}}{\land} (C o n_{ℭ_{Y}} (g_{L}^{\leftarrow} (D)) \leftrightarrow C o n_{ℭ_{X}} (f_{L}^{\leftarrow} (g_{L}^{\leftarrow} (D))))) \\ \land (\underset{C \in L^{Z}}{\land} (C o n_{ℭ_{Z}} (C) \leftrightarrow C o n_{ℭ_{Y}} (g_{L}^{\leftarrow} (C)))) \\ = (\underset{D \in L^{Z}}{\land} (C o n_{ℭ_{Y}} (g_{L}^{\leftarrow} (D)) \leftrightarrow C o n_{ℭ_{X}} ({(g ° f)}_{L}^{\leftarrow} (D)))) \\ \land (\underset{C \in L^{Z}}{\land} (C o n_{ℭ_{Z}} (C) \leftrightarrow C o n_{ℭ_{Y}} (g_{L}^{\leftarrow} (C)))) \\ = \underset{C \in L^{Z}}{\land} ((C o n_{ℭ_{Y}} (g_{L}^{\leftarrow} (C)) \leftrightarrow C o n_{ℭ_{X}} ({(g ° f)}_{L}^{\leftarrow} (C))) \\ \land (C o n_{ℭ_{Z}} (C) \leftrightarrow C o n_{ℭ_{Y}} (g_{L}^{\leftarrow} (C)))) \\ \leq \underset{C \in L^{Z}}{\land} (C o n_{ℭ_{Z}} (C) \leftrightarrow C o n_{ℭ_{X}} ({(g ° f)}_{L}^{\leftarrow} (C))) \\ = Q U (g ° f) . \end{array}$

(2) Firstly, we know $\begin{matrix} QU (g \circ f) \\ = & \underset{C \in L^{Z}}{⋀} ({Con}_{C_{Z}} (C) \leftrightarrow {Con}_{C_{X}} ((g \circ f)_{L}^{\leftarrow} (C))) \\ \leq & \underset{C \in L^{Z}}{⋀} ({Con}_{C_{X}} ((g \circ f)_{L}^{\leftarrow} (C)) \to {Con}_{C_{Z}} (C)), \end{matrix}$ and

$\begin{array}{l} C P (f) = \underset{B \in L^{Y}}{\land} (C o n_{ℭ_{Y}} (B) \to C o n_{ℭ_{X}} (f_{L}^{\leftarrow} (B))) \\ \leq \underset{D \in L^{Z}}{\land} (C o n_{ℭ_{Y}} (g_{L}^{\leftarrow} (D)) \to C o n_{ℭ_{X}} (f_{L}^{\leftarrow} (g_{L}^{\leftarrow} (D)))) \\ = \underset{D \in L^{Z}}{\land} (C o n_{ℭ_{Y}} (g_{L}^{\leftarrow} (D)) \to C o n_{ℭ_{X}} ({(g ° f)}_{L}^{\leftarrow} (D))) . \end{array}$ Then $\begin{matrix} QU (g \circ f) \land CP (f) \\ \leq & \underset{D \in L^{Z}}{⋀} ({Con}_{C_{Y}} (g_{L}^{\leftarrow} (D)) \to {Con}_{C_{Z}} (D)) . \end{matrix}$ Hence $\begin{matrix} QU (g \circ f) \land CP (f) \land CP (g) \\ \leq & (⋀_{D \in L^{Z}} ({Con}_{C_{Y}} (g_{L}^{\leftarrow} (D)) \to {Con}_{C_{Z}} (D))) \\ \land (⋀_{C \in L^{Z}} ({Con}_{C_{Z}} (C) \to {Con}_{C_{Y}} (g_{L}^{\leftarrow} (C)))) \\ = & ⋀_{C \in L^{Z}} (({Con}_{C_{Y}} (g_{L}^{\leftarrow} (C)) \to {Con}_{C_{Z}} (C)) \\ \land ({Con}_{C_{Z}} (C) \to {Con}_{C_{Y}} (g_{L}^{\leftarrow} (C)))) \\ = & ⋀_{C \in L^{Z}} ({Con}_{C_{Z}} (C) \leftrightarrow {Con}_{C_{Y}} (g_{L}^{\leftarrow} (C))) \\ = & QU (g) . \end{matrix}$

6. Conclusion

In this paper, the degree to which a mapping $C : L^{X} ⟶ M$ is an (L, M)-fuzzy convexity and the degree to which an L-subset is an L-convex set with respect to $C$ were defined. Afterwards, we defined the degrees to which a mapping f : X ⟶ Y is convexity-preserving, convex-to-convex and a quotient mapping with respect to $C_{X}$ and $C_{Y}$ and discussed the relationships between them.

Motivated by these, many properties and results in classical convex theory can be reconsidered under (L, M)-fuzzy convexity degrees, such as convex invariance, topological convex structures etc., which will be the subjects of our future research.

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