The restricted hull operator of M -fuzzifying convex structures 1

Abstract

In this paper, the notion of M-fuzzifying restricted hull operators is introduced and several equivalent characterizations are given. It is shown that there is a one-to-one correspondence between M-fuzzifying restricted hull operators and M-fuzzifying convex structures. As applications, some properties of the cut convex structures of an M-fuzzifying convex structure and of M-fuzzifying convexity preserving functions and of M-fuzzifying convex-to-convex functions are derived. In addition, using M-fuzzifying restricted hull operators, some M-fuzzifying convexities are naturally constructed from M-fuzzy quasi-orders.

Keywords

Convex structure

1 Introduction

Convexity, which is a simple notion inspired by the shape of some figures, such as circles and polyhedrons (convex) in our ordinary 2 or 3-dimensional space, has an old history and numerous applications [2]. Since early decades of last century, the notion of convexity has been extended to more mathematical structures. For example, the well-known Helly’s, Carathéodory’s and Radon’s theorems for convex sets in Euclidean space capture some combinatorial features of convexity and are helpful when handling convex combinations [5, 13]. Order convexities on partially ordered sets and convexities in metric spaces especially in graphs were studied extensively [3 , 24]. The results on convex sets and convex functions play an important role in functional analysis and in optimization theory [1, 17]. From the early fifties of last century on, an axiomatic approach of convexity started to develop. Thisdevelopment was motivated not only by the need for an abstract theory of convexity generalizing the classical theorems in Rⁿ due to Helly, Carathéodory etc., but also by the necessity to unify geometric aspects of all these mathematical structures. The interplay between the axiomatic approach and various structures turned out to be particularly fruitful. A wide variety of general results have been derived under the axiomatic framework (see the monograph of M. van de Vel [25]).

Abstractly, a convexity on a set X is defined to be a family $C$ of subsets of X which contains both X itself and the empty set ∅ and which is closed under arbitrary intersections and nested unions (see Definition 2.2). A convex structure is completely determined by its hull operator, or even by its effect on finite sets (this last because of domain finiteness). Therefore, convexities are often classified by the properties of the hull operator. This is the case, for instance, with matroids, with convex geometries (anti-matroids) and with Join-hull Commutative (JHC) convexities. Further, there is a simple axiomatic characterization of the hull operator on the collection of finite sets [25]. In 1994, Rosa generalized the notion of convex structures to fuzzy setting in [18]. In 2009, Maruyama introduced an L-fuzzy convexity space $(X, C)$ in [14] where $C$ was defined as a crisp family of L-fuzzy subsets of a set X satisfying certain axioms. Also, the classical theorems of Helly, Carathéodory and Radon were extended to the lattice-valued fuzzy case. More recently, a new approach to the fuzzification of convex structures was introduced in [23]. It is called an M-fuzzifying convex structure, in which each subset of X can be regarded as a convex set to some degree. And as expected, M-fuzzifying convex structure can be characterized by its M-fuzzifying hull operator (called closure operator in [23]), or even by its effect on finite sets. In view of the importance of hull operators, in this paper, we study the characterizations of an M-fuzzifying hull operator when it is restricted to the collection of finite sets.

This paper is organized as follows. In Section 2, some basic properties and notions of lattices, convex structures and M-fuzzifying convex structures are provided. In Section 3, M-fuzzifying restricted hull operators are defined and some fundamental properties are discussed. In Section 4, the notion of M-fuzzifying betweennesses is introduced and is used to characterize the M-fuzzifying convexity preserving functions. Finally, conclusions and some perspectives are made.

2 Preliminaries

Throughout this paper, (M, ∨ , ∧ , ′) denotes a completely distributive lattice with an order-reversing involution ^′ . The smallest element and the largest element in M are denoted by ⊥ and ⊤, respectively. The set of non-unit prime elements in M is denoted by P (M) . The binary relation ≺ in M is defined as follows: for a, b ∈ M, a ≺ b if and only if for every subset D ⊆ M, the relation b ≤ sup D always implies the existence of d ∈ D with a ≤ d [8]. {a ∈ M : a ≺ b} is called the greatest minimal family of b in the sense of [27], denoted by β (b) . Moreover, we define a binary relation ≺^op 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 . Then {a ∈ M : a ≺ ^opb} is the greatest maximal family of b in the sense of [27], denoted by α (b) . In a completely distributive lattice M, there exist α (b) and β (b) for each b ∈ M, and b = ⋁ β (b) = ⋀ α (b) (see [26]).

We denote the set of all subsets (resp., all finite subsets) of a set X by 2^X (resp., $2_{fin}^{X}$ ) and denote the interval [0, 1] of real numbers by I .

Theorem 2.1. [26] Let M be a completely distributive lattice and let {a_i : i ∈ Ω} ⊆ M . Then $α (⋀_{i \in Ω} a_{i}) = ⋃_{i \in Ω} α (a_{i}), β (⋁_{i \in Ω} a_{i}) = ⋃_{i \in Ω} β (a_{i}),$ that is, α is a ∧-∪ mapping and β is a union-preserving mapping.

Definition 2.2. [25] A family $C \subseteq 2^{X}$ is called a convexity on a set X if the following conditions hold.

(C1) $\emptyset, X \in C .$

(C2) If ${A_{i} : i \in Ω} \subseteq C$ is non-empty, then $⋂_{i \in Ω} A_{i} \in C .$

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

The pair $(X, C)$ is called a convex structure. The members of $C$ are called convex sets.

Let ${C_{i}}_{i \in Ω}$ be a family of convexities on a set X . Its intersection and join are denoted by $⋂_{i \in Ω} C_{i}$ and $⋁_{i \in Ω} C_{i},$ respectively.

Theorem 2.3. [25] Let $h : 2_{fin}^{X} \to 2^{X}$ be an operator satisfying the following conditions.

(H1) h (∅) = ∅ .

(H2) F ⊆ h (F) for each $F \in 2_{fin}^{X} .$

(H3) For all $G, F \in 2_{fin}^{X},$ if G ⊆ h (F) then h (G) ⊆ h (F) .

Then there is precisely one convexity on X with a hull operator equal to h on $2_{fin}^{X} .$ Conversely, the hull operator of any convexity on X satisfies the conditions (H1) – (H3) .

Remark 2.4. (i) In Theorem 2, the unique convexity $C_{h}$ on X with a hull operator equal to h on $2_{fin}^{X}$ is given by $C_{h} = {C \subseteq X : \forall F \in 2_{fin}^{C}, h (F) \subseteq C} .$ Therefore, once a mapping $h : 2_{fin}^{X} \to 2^{X}$ satisfies the conditions (H1) – (H3) , we have $h = {co}_{C_{h}} |_{2_{fin}^{X}} .$ For this reason, we call such mappings restricted hull operators.

(ii) It is easily seen that $C_{h_{1}} \supseteq C_{h_{2}}$ if and only if h₁ ≤ h₂ (i.e., h₁ (F) ≤ h₂ (F) for all $F \in 2_{fin}^{X}$ ).

Definition 2.5. [25] Betweenness is a binary relation between X and $2_{fin}^{X}$ satisfying the following conditions.

(B1) No point is between the empty set.

(B2) If x ∈ F then x is between F .

(B3) If x is between G and if each element of G is between F, then x is between F .

Definition 2.6. [22] Denote the set of all M-fuzzy sets on X by M^X . Let A ∈ M^X and a ∈ M . Define $\begin{matrix} A_{[a]} & = & {x \in X : A (x) \geq a}, \\ A^{[a]} & = & {x \in X : a \notin α (A (x))}, \\ A_{(a)} & = & {x \in X : a \in β (A (x))}, \\ A^{(a)} & = & {x \in X : A (x) a} . \end{matrix}$

Some properties of these cut sets can be found in [11 , 19–22].

For a ∈ M and D ⊆ X, we define an M-fuzzy set a ∧ D as follows:

$(a \land D) (x) = \{\begin{array}{l} a, & x \in D, \\ ⊥, & x \notin D . \end{array}$

Then for any M-fuzzy set A ∈ M^X, we have [11, 19] $A = ⋁_{a \in M} (a \land A_{[a]}) .$ (1)

Definition 2.7. [23] A mapping $C : 2^{X} \to M$ is called an M-fuzzifying convexity on X if it satisfies the following conditions:

(MYC1) $C (\emptyset) = C (X) = ⊤ .$

(MYC2) If {A_i : i ∈ Ω} ⊆2^X is non-empty, then $C (⋂_{i \in Ω} A_{i}) \geq ⋀_{i \in Ω} C (A_{i}) .$

(MYC3) If {A_i : i ∈ Ω} ⊆2^X is non-empty and totally ordered by inclusion, then $C (⋃_{i \in Ω} A_{i}) \geq ⋀_{i \in Ω} C (A_{i}) .$

If $C$ is an M-fuzzifying convexity on X, then the pair $(X, C)$ is called an M-fuzzifying convex structure.

Theorem 2.8. [23] Let $C : 2^{X} \to M$ be a mapping. Then the following statements are equivalent.

$(X, C)$ is an M-fuzzifying convex structure.

For each $a \in α (⊥), (X, C^{[a]})$ is a convex structure.

For each $a \in M \ {⊥}, (X, C_{[a]})$ is a convexstructure.

Theorem 2.9. [23] Let $(X, C)$ be an M-fuzzifying convex structure. Let ${co}_{C} : 2^{X} \to M^{X}$ be the mapping defined, for all A ⊆ X and x ∈ X, by ${co}_{C} (A) (x) = ⋀_{x \notin B \supseteq A} C (B)^{'} .$ (2)Then, for A ⊆ X, x ∈ X, ${co}_{C}$ has the following properties:

(MCO1) ${co}_{C} (\emptyset) (x) = ⊥ .$

(MCO2) If x ∈ A, then ${co}_{C} (A) (x) = ⊤ .$

(MCO3) If x ∉ A, then

${co}_{C} (A) (x) = ⋀_{x \notin B \supseteq A} [⋁_{y \notin B} {co}_{C} (B) (y)] .$

(MDF) ${co}_{C} (A) (x) = ⋁ {{co}_{C} (F) (x) : F \in 2_{fin}^{A}} .$

Conversely, let co : 2^X → M^X be a mapping satisfying (MCO1) – (MCO3) and (MDF) . Define the mapping $C_{co} : 2^{X} \to M$ by $C_{co} (A) = ⋀_{x \notin A} [co (A) (x)]^{'} .$ Then $(X, C_{co})$ is an M-fuzzifying convex structure. And as expected, ${co}_{C_{co}} = co .$

Remark 2.10. The operator ${co}_{C}$ defined by (2) in Theorem 2.9 is called the M-fuzzifying hull operator for the M-fuzzifying convex structure $(X, C) .$ Based on this theorem, a mapping from 2^X to M^X will be called an M-fuzzifying hull operator if it satisfies the conditions (MCO1) – (MCO3) and (MDF) .

Theorem 2.11. [23] Let $(X, C)$ be an M-fuzzifying convex structure. Then $C_{{co}_{C}} = C .$

Remark 2.12. By Theorems 2.9 and 2.11, we obtain that for any M-fuzzifying convex structure $(X, C)$ and A ⊆ X, $C (A) = ⋀_{F \in 2_{fin}^{A}, x \notin A} [{co}_{C} (F) (x)]^{'} .$ (3)

Definition 2.13. [23] Let $(X, C)$ and $(Y, D)$ be M-fuzzifying convex structures. A function f : X → Y is said to be:

An M-fuzzifying convexity preserving function provided $C (f^{- 1} (B)) \geq D (B)$ for all B∈ 2^Y ;

An M-fuzzifying convex-to-convex function provided $C (A) \leq D (f (A))$ for all A∈ 2^X ; and

An M-fuzzifying isomorphism if f is a bijection, an M-fuzzifying convexity preserving function and an M-fuzzifying convex-to-convex function.

Definition 2.14. [28] Let X be a non-empty set. A mapping e : X × X → M is called an M-fuzzy quasi-order on X if it satisfies the following conditions:

(E1) ∀x∈ X, e (x, x) = ⊤ ;

(E2) ∀x, y, z ∈ X, e (x, y) ∧ e (y, z) ≤ e (x, z) . If e satisfies the additional condition

(E3) ∀x, y ∈ X, e (x, y) = e (y, x) = ⊤ ⇒ x = y,

then it is called an M-fuzzy partial order on X .

The pair (X, e) is called an M-fuzzy partially ordered set provided e is an M-fuzzy partial order on X .

Definition 2.15. [6] A function $d : X \times X \to ℝ$ is called an ultrametric (or non-Archimedean metric) on X, if for all x, y, z ∈ X, there holds:

(UD1) d (x, y) =0 if and only if x = y ;

(UD2) d (x, y) = d (y, x) ;

(UD3) d (x, z) ≤ max {d (x, y) , d (y, z)} .

An ultrametric space (X, d) is a set X equipped with an ultrametric d .

3 M-fuzzifying restricted hull operators

First, we generalize the notion of restricted hull operators to M-fuzzy setting.

Definition 3.1. A mapping $h : 2_{fin}^{X} \to M^{X}$ is called an M-fuzzifying restricted hull operator if it satisfies the following conditions.

(MH1) For all x ∈ X, h (∅) (x) = ⊥ .

(MH2) If $F \in 2_{fin}^{X}, x \in F,$ then h (F) (x) = ⊤ .

(MH3) For all $G, F \in 2_{fin}^{X}$ and x ∈ X, $h (G) (x) ⋀ [⋀_{y \in G} h (F) (y)] \leq h (F) (x) .$

It is easy to show that when M = {⊥ , ⊤} the conditions (MH1) – (MH3) are equivalent to the conditions (H1) – (H3) . Now we give characterizations of an M-fuzzifying restricted hull operator.

Theorem 3.2. The following statements are equivalent for a mapping $h : 2_{fin}^{X} \to M^{X} .$

h is an M-fuzzifying restricted hull operator.

For each $a \in M \ {⊥}, h_{[a]} : 2_{fin}^{X} \to 2^{X}$ is arestricted hull operator, where h_[a] (F) = (h (F)) _[a] for all $F \in 2_{fin}^{X} .$

For each $a \in α (⊥), h^{[a]} : 2_{fin}^{X} \to 2^{X}$ is arestricted hull operator, where h^[a] (F) = (h (F)) ^[a] for all $F \in 2_{fin}^{X} .$

For each $a \in P (M), h^{(a)} : 2_{fin}^{X} \to 2^{X}$ is a res-tricted hull operator, where h^(a) (F) = (h (F)) ^(a) for all $F \in 2_{fin}^{X} .$

If β (a ∧ b) = β (a) ∩ β (b) for any a, b ∈ M, then we can add the following equivalent statement.

For each $a \in β (⊤), h_{(a)} : 2_{fin}^{X} \to 2^{X}$ is arestricted hull operator, where h_(a) (F) = (h (F)) _(a) for all $F \in 2_{fin}^{X} .$

Proof. (i) ⇒ (ii). By (MH1) and (MH2) , it is easy to show that for all a∈ M \ {⊥} , h_[a] (∅) = ∅ and h_[a] (F) ⊇ F . Now let $G, F \in 2_{fin}^{X}$ and G ⊆ h_[a] (F) . Then ⋀_y∈Gh (F) (y) ≥ a . If x ∈ h_[a] (G) , then by (MH3) , h (F) (x) ≥ h (G) (x) ⋀ [⋀ _y∈Gh (F) (y) ] ≥ a, which implies x ∈ h_[a] (F) . Thus h_[a] (G) ⊆ h_[a] (F) , and so h_[a] satisfies (H3) . Therefore, h_[a] is a restricted hull operator.

(ii) ⇒(i). If h (∅) (x) = a ≠ ⊥ for some xinX, then x∈ (h (∅)) _[a] = h_[a] (∅) = ∅ ; a contradiction. To show (MH2) , let x ∈ F, but h (F) (x) = b ≠ ⊤ . Then there exists a ∈ M \ {⊥} such that anleqb . Thus x ∉ h_[a] (F) , contradicting F ⊆ h_[a] (F) . To show (MH3) , let $G, F \in 2_{fin}^{X}, x \in X$ and a ≤ h (G) (x) ⋀ [⋀ _y∈Gh (F) (y)] . Then a ≤ h (F) (y) for all y ∈ G, which means G ⊆ h_[a] (F) . Since h_[a] is a restricted hull operator, it follows that h_[a] (G) ⊆ h_[a] (F) . Note that a ≤ h (G) (x) , that is, x ∈ h_[a] (G) . Thus x ∈ h_[a] (F) , i.e., a ≤ h (F) (x) . Since a is arbitrary, we conclude that h satisfies (MH3) .

(i) ⇒(iii). By (MH1) , h (∅) (x) = ⊥ for all x ∈ X . Thus, if a ∈ α (⊥) , then h^[a] (∅) = {x : a ∉ α (h (∅) (x))} = ∅ . Thus h^[a] satisfies (H1) . Noting that α (⊤) = ∅ , it follows from (MH2) that h^[a] (F) ⊇ F . We now prove that h^[a] satisfies (H3) . Let $G, F \in 2_{fin}^{X}$ and G ⊆ h^[a] (F) . Then for all y ∈ G, a ∉ α (h (F) (y)) , and thus a ∉ ⋃ _y∈Gα (h (F) (y)) . Thus, if x ∈ h^[a] (G) , i.e., a ∉ α (h (G) (x)) , then

$\begin{matrix} a & \notin & α (h (G) (x)) ⋃ [⋃_{y \in G} α (h (F) (y))] \\ = & α (h (G) (x) ⋀ [⋀_{y \in G} h (F) (y)]) . \end{matrix}$ By (MH3) , a ∉ α (h (F) (x)) , i.e., x ∈ h^[a] (F) . Hence h^[a] (G) ⊆ h^[a] (F) , and so h^[a] satisfies (H3) .

(iii)⇒(i). If h (∅) (x) = b ≠ ⊥ for some x ∈ X, then there exists a ∈ α (⊥) \ α (b) . Thus x∈ h^[a] (∅) = ∅ ; a contradiction. Thus (MH1) holds. If ⋀_x∈Fh (F) (x) = b ≠ ⊤ , then ∅ ≠ α (b) ⊆ α (⊥) . Let a ∈ α (b) = ⋃ _y∈Fα (h (F) (y)) . Then a ∈ α (h (F) (y)) for some y ∈ F, and hence y ∈ F \ h^[a] (F) , which contradicts the fact that F ⊆ h^[a] (F) . Thus (MH2) holds. Note that a ≤ b if and only if α (b) ⊆ α (a) . To prove (MH3) , it is enough to show that α (h (F) (x)) ⊆ α (h (G) (x) ⋀ [⋀ _y∈Gh (F) (y)]) . Let a ∈ α (h (F) (x)) . Then x ∉ h^[a] (F) . Since h^[a] is a restricted hulloperator, either G ⊈ h^[a] (F) or x∉ h^[a] (G) ; equiva-lently, either a ∈ α (h (F) (y)) for some y ∈ G ora ∈ α (h (G) (x)) . Therefore, a ∈ α (h (G) (x)) ⋃ [⋃ _y∈Gα (h (F) (y))] = α (h (G) (x) ⋀ [⋀ _y∈Gh (F) (y)]) as desired.

(i)⇒(iv). Let a ∈ P (M) . We show that h^(a) is a restricted hull operator. (H1) and (H2) are easy to prove. To show that h^(a) satisfies (H3) , let $G, F \in 2_{fin}^{X}$ and G ⊆ h^(a) (F) . Then h (F) (y) nleqa for all y ∈ G . Now let x ∈ h^(a) (G) (i.e., h (G) (x) nleqa). Since a ∈ P (M) , we have h (G) (x) ⋀ [⋀ _y∈Gh (F) (y) ] nleqa . It follows by (MH3) that h (F) (x) nleqa . Thus h^(a) (G) ⊆ h^(a) (F) , and hence (H3) holds.

(iv)⇒(i). If h (∅) (x) = b ≠ ⊥ = ⋀ P (M) for some x ∈ X, then there exists a ∈ P (M) with bnleqa . Thus x∈ h^(a) (∅) = ∅ ; a contradiction. Thus (MH1) holds. Since h^(a) is a restricted hull operator for every a ∈ P (M) , it follows that F ⊆ h^(a) (F) for all $F \in 2_{fin}^{X}$ and a ∈ P (M) . Therefore, for every a ∈ P (M) , if x ∈ F, then x ∈ h^(a) (F) , i.e., h (F) (x) nleqa . Thus h (F) (x) = ⋀ {a ∈ P (M) : h (F) (x) ≤ a} = ⋀ ∅ = ⊤ , that is, (MH2) holds. To prove that h satisfies (MH3) , we assume the contrary letting G and F be finite subsets of X such that h (G) (x) ⋀ [⋀ _y∈Gh (F) (y)] nleqh (F) (x) for some x ∈ X . Thus there is an a ∈ P (M) such that for all y ∈ G, h (F) (y) nleqa and h (G) (x) nleqa, but h (F) (x) ≤ a . It follows that G ⊆ h^(a) (F) and x ∈ h^(a) (G) but x ∉ h^(a) (F) , which contradicts the fact that h^(a) is a restricted hull operator.

(i)⇒(v). Let a ∈ β (⊤) , we need show that h_(a) is a restricted hull operator. By (MH1) , h (∅) (x) = ⊥ for all x ∈ X . Thus β (h (∅) (x)) = β (⊥) = ∅ . This shows that a ∉ β (h (∅) (x)) (i.e., x ∉ h_(a) (∅)) for all x ∈ X . Hence h_(a) (∅) = ∅ . By (MH2) , h (F) (x) =⊤ for all x ∈ F . Thus a ∈ β (⊤) = β (h (F) (x)) (i.e., x ∈ h_(a) (F)) for all x ∈ F, that is, F ⊆ h_(a) (F) . To show that h_(a) satisfies (H3) , let $G, F \in 2_{fin}^{X}$ and G ⊆ h_(a) (F) . Then a ∈ β (h (F) (y)) for all y ∈ G . Now let x ∈ h_(a) (G) , i.e., a ∈ β (h (G) (x)) . It follows by (MH3) that

$\begin{matrix} a & \in & β (h (G) (x)) ⋂ [⋂_{y \in G} β (h (F) (y))] \\ = & β (h (G) (x) ⋀ [⋀_{y \in G} h (F) (y)]) \\ \subseteq & β (h (F) (x)) . \end{matrix}$ Thus h_(a) (G) ⊆ h_(a) (F) , and so h_(a) satisfies (H3) .

(v)⇒(i). The proof is routine and is omitted. □

Theorem 3.3. Let $(X, C)$ be an M-fuzzifying convex structure and ${co}_{C}$ be its M-fuzzifying hull operator. Then $h_{C} = {co}_{C} |_{2_{fin}^{X}}$ is an M-fuzzifying restricted hull operator.

Proof. By Theorem 2.9, $h_{C}$ satisfies (MH1) and (MH2) . Let $G, F \in 2_{fin}^{X}$ and x ∈ X . It is clear that $h_{C} (G) (x) ⋀ [⋀_{y \in G} h_{C} (F) (y)] \leq h_{C} (F) (x)$ holds for x ∈ G ∪ F . We may now assume that x ∉ G ∪ F . By (MCO3) of Theorem 2.9, $h_{C} (F) (x) = {co}_{C} (F) (x) = ⋀_{x \notin A \supseteq F} [⋁_{z \notin A} {co}_{C} (A) (z)] .$ Observe that, for any A ⊇ F with x ∉ A, if G ⊈ A, then

$\begin{matrix} ⋁_{z \notin A} {co}_{C} (A) (z) & \geq & ⋁_{z \in G \ A} {co}_{C} (A) (z) \\ \geq & ⋁_{z \in G \ A} {co}_{C} (F) (z) (By (MDF)) \\ \geq & ⋀_{y \in G} h_{C} (F) (y) . \end{matrix}$ Otherwise, G ⊆ A, and so $⋁_{z \notin A} {co}_{C} (A) (z) \geq {co}_{C} (A) (x) \geq {co}_{C} (G) (x) = h_{C} (G) (x) .$ Hence by the arbitrariness of A (with x ∉ A ⊇ F), (MH3) holds. □

In what follows, for a given M-fuzzifying convex structure $(X, C),$ we always denote the M-fuzzifying restricted hull operator ${co}_{C} |_{2_{fin}^{X}}$ by $h_{C} .$ The operator $h_{C}$ is called the M-fuzzifying restricted hull operator of $(X, C) .$

Theorem 3.4. Let $h : 2_{fin}^{X} \to M^{X}$ be an M-fuzzifying restricted hull operator. Then there is precisely one M-fuzzifying convexity on X with an M-fuzzifying hull operator equal to h on $2_{fin}^{X} .$

Proof. Let $C_{h}$ be the mapping $C_{h} : 2^{X} \to M$ defined, for all A ⊆ X, by $C_{h} (A) = ⋀_{F \in 2_{fin}^{A}, x \notin A} [h (F) (x)]^{'} .$ (4)

We first show that $C_{h}$ is an M-fuzzifying convexity on X .

(MYC1) Since {x : x ∉ X} = ∅ , we conclude by (4), that $C_{h} (X) = ⊤ .$ By (MH1) ,

C_{h} (\emptyset) = ⋀_{F \in 2_{fin}^{\emptyset}, x \notin \emptyset} [h (F) (x)]^{'} = ⋀_{x \in X} [h (\emptyset) (x)]^{'} = ⊤ .

(MYC2) Let {A_i : i ∈ Ω} ⊆2^X be non-empty. If x ∉ ⋂ _i∈ΩA_i, then there exists i_x ∈ Ω such that x ∉ A_{i
_x} . Thus

$\begin{matrix} ⋀_{F \in 2_{fin}^{⋂_{i \in Ω} A_{i}}} [h (F) (x)]^{'} & \geq & ⋀_{F \in 2_{fin}^{A_{i_{x}}}} [h (F) (x)]^{'} \\ \geq & ⋀_{F \in 2_{fin}^{A_{i_{x}}}, y \notin A_{i_{x}}} [h (F) (y)]^{'} \\ = & C_{h} (A_{i_{x}}) \\ \geq & ⋀_{i \in Ω} C_{h} (A_{i}) . \end{matrix}$ It follows from 4 that $C_{h} (⋂_{i \in Ω} A_{i}) \geq ⋀_{i \in Ω} C_{h} (A_{i}) .$

(MYC3) Let {A_i : i ∈ Ω} ⊆2^X be non-empty and totally ordered by inclusion. Note that for any $F \in 2_{fin}^{⋃_{i \in Ω} A_{i}}$ and x ∉ ⋃ _i∈ΩA_i, there exists i_F ∈ Ω such that $F \in 2_{fin}^{A_{i_{F}}}$ and x ∉ A_{i
_F} . Thus

$\begin{matrix} [h (F) (x)]^{'} & \geq & ⋀_{G \in 2_{fin}^{A_{i_{F}}}} [h (G) (x)]^{'} \\ \geq & ⋀_{G \in 2_{fin}^{A_{i_{F}}}, y \notin A_{i_{F}}} [h (G) (y)]^{'} \\ = & C_{h} (A_{i_{F}}) \\ \geq & ⋀_{i \in Ω} C_{h} (A_{i}) . \end{matrix}$ It follows from 4 that $C_{h} (⋃_{i \in Ω} A_{i}) \geq ⋀_{i \in Ω} C_{h} (A_{i}) .$

We show next that ${co}_{C_{h}} |_{2_{fin}^{X}} = h,$ i.e. $, {co}_{C_{h}} (A) (x) = h (A) (x)$ for all $A \in 2_{fin}^{X}$ and x ∈ X .

In case x ∈ A, the equality holds obviously. Let x ∉ A . Then by Theorem 2.9 and (4), we have

$\begin{matrix} {co}_{C_{h}} (A) (x) & = & ⋀_{x \notin B \supseteq A} [C_{h} (B)]^{'} \\ = & ⋀_{x \notin B \supseteq A} {⋀_{F \in 2_{fin}^{B}, y \notin B} [h (F) (y)]^{'}}^{'} \\ = & ⋀_{x \notin B \supseteq A} [⋁_{F \in 2_{fin}^{B}, y \notin B} h (F) (y)] . \end{matrix}$ By taking y = x, F = A for each B with x ∉ B ⊇ A, we see ${co}_{C_{h}} (A) (x) \geq h (A) (x) .$ On the other hand, let $a ≺ {co}_{C_{h}} (A) (x) .$ We show that a ≤ h (A) (x) . If not, let $\tilde{B} = h (A)_{[a]} .$ Then $x \notin \tilde{B} .$ From the above, we know $a ≺ ⋁_{F \in 2_{fin}^{\tilde{B}}, y \notin \tilde{B}} h (F) (y) .$ Therefore, there exist $\tilde{y} \notin \tilde{B}$ and $\tilde{F} \in 2_{fin}^{\tilde{B}}$ such that $a \leq h (\tilde{F}) (\tilde{y}) .$ It follows from $\tilde{F} \subseteq h (A)_{[a]}$ that $⋀_{z \in \tilde{F}} h (A) (z) \geq a .$ By (MH3) , $h (A) (\tilde{y}) \geq h (\tilde{F}) (\tilde{y}) ⋀ [⋀_{z \in \tilde{F}} h (A) (z)] \geq a,$ contradicting that $\tilde{y} \notin \tilde{B} .$ This contradiction shows that a ≤ h (A) (x) . Since a is arbitrary, we have ${co}_{C_{h}} (A) (x) \leq h (A) (x) .$

Finally, the proof of the uniqueness part follows easily from (MDF) . □

In what follows, for a given M-fuzzifying restricted hull operator h, the unique M-fuzzifying convexity given in Theorem 3.4 (that is, the one defined by 4) is always denoted by $C_{h} .$

Remark 3.5. By Theorems 3.3 and 3.4, there is a one-to-one correspondence between M-fuzzifying restricted hull operators and M-fuzzifying convex structures. More precisely, for any M-fuzzifying convex structure $(X, C)$ there is an M-fuzzifying restricted hull operator $h_{C}$ such that $C = C_{h_{C}} .$ Conversely, for any M-fuzzifying restricted hull operator h, there is an M-fuzzifying convex structure $(X, C_{h})$ such that $h = h_{C_{h}} .$

Corollary 3.6. A mapping $h : 2_{fin}^{X} \to M^{X}$ is an M-fuzzifying restricted hull operator if and only if it satisfies conditions (MH1) , (MH2) and the following condition. (MH3) ′ If $F \in 2_{fin}^{X}$ and x ∈ X \ F, then $h (F) (x) = ⋀_{x \notin A \supseteq F} [⋁_{G \in 2_{fin}^{A}, y \notin A} h (G) (y)] .$

Proof. Necessity. We need only show that (MH3) ′ holds. By Theorem 3.4 we know $h = {co}_{C_{h}} |_{2_{fin}^{X}} .$ Thus by Theorem 2.9,

$\begin{matrix} h (F) (x) & = & {co}_{C_{h}} (F) (x) \\ = & ⋀_{x \notin A \supseteq F} [⋁_{y \notin A} {co}_{C_{h}} (A) (y)] \\ = & ⋀_{x \notin A \supseteq F} [⋁_{G \in 2_{fin}^{A}, y \notin A} {co}_{C_{h}} (G) (y)] \\ = & ⋀_{x \notin A \supseteq F} [⋁_{G \in 2_{fin}^{A}, y \notin A} h (G) (y)] . \end{matrix}$

Sufficiency. The proof of (MH3) is analogous to that in Theorem 3.3. □

Lemma 3.7. A mapping cl : 2^X → M^X is an M-fuzzifying hull operator if and only if it satisfies the conditions (MCO1) , (MCO2) , (MDF) and the following condition.

(MCO3) ′ For all A ∈ 2^X and a ∈ M, $cl (cl (A)_{[a]})_{[a]} = cl (A)_{[a]} .$

Proof. Necessity. We need only prove that (MCO3) ′ holds. It is clear from (MCO2) that cl (cl (A) _[a]) _[a] ⊇ cl (A) _[a] . Conversely, by (MCO3) ,

$\begin{matrix} x \notin cl (A)_{[a]} \\ \Rightarrow & a cl (A) (x) \\ \Rightarrow & [cl (A) (x)]^{'} a^{'} \\ \Rightarrow & ⋁_{x \notin B \supseteq A} {⋀_{y \notin B} [cl (B) (y)]^{'}} a^{'} \\ \Rightarrow & there exsits \tilde{B} such that x \notin \tilde{B} \supseteq A, and \\ ⋀_{y \notin \tilde{B}} [cl (\tilde{B}) (y)]^{'} a^{'} \\ \Rightarrow & \forall y \notin \tilde{B}, [cl (\tilde{B}) (y)]^{'} a^{'} (i . e ., cl (\tilde{B}) (y) a) \\ \Rightarrow & \forall y \notin \tilde{B}, y \notin cl (\tilde{B})_{[a]} \\ \Rightarrow & cl (\tilde{B})_{[a]} \subseteq \tilde{B} (By (MCO 2), cl (\tilde{B})_{[a]} \supseteq \tilde{B}) \\ \Rightarrow & cl (\tilde{B})_{[a]} = \tilde{B} \\ \Rightarrow & cl (A)_{[a]} \subseteq \tilde{B} (By (MDF), cl (A) \leq cl (\tilde{B})) \\ \Rightarrow & cl (cl (A)_{[a]})_{[a]} \subseteq \tilde{B} \\ \Rightarrow & x \notin cl (cl (A)_{[a]})_{[a]} . \end{matrix}$ Therefore, cl (cl (A) _[a]) _[a] ⊆ cl (A) _[a] and hence (MCO3) ′ holds.

Sufficiency. It suffices to show that (MCO3) holds. Let x ∉ A . Then for any B ⊇ A with x ∉ B, by (MDF) , cl (A) (x) ≤ cl (B) (x) ≤ ⋁ _y∉Bcl (B) (y) . Therefore, $cl (A) (x) \leq ⋀_{x \notin B \supseteq A} [⋁_{y \notin B} cl (B) (y)] .$ Conversely, let a ≺ ⋀ _x∉B⊇A [⋁ _y∉Bcl (B) (y)] . If anleqcl (A) (x) , then x ∉ cl (A) _[a] . Let $\tilde{B} = cl (A)_{[a]} .$ By $(MCO 2), \tilde{B} \supseteq A,$ and so $a ≺ ⋁_{y \notin \tilde{B}} cl (\tilde{B}) (y) .$ Hence there exists $y_{0} \notin \tilde{B}$ with $a \leq cl (\tilde{B}) (y_{0}) .$ It follows, by ${(MCO 3)}^{'}, y_{0} \in cl (\tilde{B})_{[a]} = \tilde{B};$ a contradiction. Hence a ≤ cl (A) (x) . By the arbitrariness of a, ⋀ _x∉B⊇A [⋁ _y∉Bcl (B) (y)] ≤ cl (A) (x) . Thus (MCO3) holds. □

Theorem 3.8. A mapping $h : 2_{fin}^{X} \to M^{X}$ is an M-fuzzifying restricted hull operator if and only if it satisfies conditions (MH1) , (MH2) and the following condition.

(MH3) ″ If $F \in 2_{fin}^{X},$ then $h (F) = ⋁_{a \in M, G \in 2_{fin}^{h (F)_{[a]}}} (a \land h (G)_{[a]}) .$

Proof. Necessity. Let h be an M-fuzzifying restricted hull operator. We need only verify that (MH3) ″ holds.

If p ≤ h (F) (x) , then x ∈ h (F) _[p] . Let G = F ∪ {x} . By (MH2) , x ∈ h (G) _[p] and $G \in 2_{fin}^{h (F)_{[p]}} .$ Thus $p = (p \land h (G)_{[p]}) (x) \leq ⋁_{a \in M, G \in 2_{fin}^{h (F)_{[a]}}} (a \land h (G)_{[a]}) (x) .$ Since p, x are all arbitrary, we have $h (F) \leq ⋁_{a \in M, G \in 2_{fin}^{h (F)_{[a]}}} (a \land h (G)_{[a]}) .$

Conversely, by Lemma 3.7 and Theorem 3.4, for all $a \in M, G \in 2_{fin}^{h (F)_{[a]}},$

$\begin{matrix} a \land h (G)_{[a]} & = & a \land {co}_{C_{h}} (G)_{[a]} \\ \leq & a \land {co}_{C_{h}} (h (F)_{[a]})_{[a]} \\ = & a \land {co}_{C_{h}} ({co}_{C_{h}} (F)_{[a]})_{[a]} \\ = & a \land {co}_{C_{h}} (F)_{[a]} . \end{matrix}$ Therefore, by (1),

$\begin{matrix} h (F) & = & {co}_{C_{h}} (F) \\ = & ⋁_{a \in M} (a \land {co}_{C_{h}} (F)_{[a]}) \\ \geq & ⋁_{a \in M, G \in 2_{fin}^{h (F)_{[a]}}} (a \land h (G)_{[a]}) . \end{matrix}$

Sufficiency. To prove that h satisfies (MH3) , let p ≤ h (G) (x) ⋀ [⋀ _y∈Gh (F) (y)] . It follows that $x \in h (G)_{[p]}, G \in 2_{fin}^{h (F)_{[p]}}$ and hence

$\begin{matrix} h (F) (x) & = & ⋁_{a \in M, G \in 2_{fin}^{h (F)_{[a]}}} (a \land h (G)_{[a]}) (x) \\ \geq & (p \land h (G)_{[p]}) (x) \\ = & p . \end{matrix}$ From the arbitrariness of p, we deduce that h satisfies (MH3) . □

The following theorem gives the relations between the cut convex structures of an M-fuzzifying convex structure and the convex structures determined by the cut operators of its M-fuzzifying restricted hull operator. More specifically, we have

Theorem 3.9. Let $(X, C)$ be an M-fuzzifying convex structure and $h_{C} = {co}_{C} |_{2_{fin}^{X}} .$ Then

$\forall a \in α (⊥), C^{[a]} = ⋂_{a \in α (b^{'})} C_{(h_{C})_{[b]}};$

$\forall a \in M \ {⊥}, C_{[a]} = ⋂_{b \in α (a^{'})} C_{(h_{C})^{[b]}};$

$\forall a \in α (⊥), C_{(h_{C})^{[a]}} = ⋁_{a \in α (b^{'})} C_{[b]};$

$\forall a \in M \ {⊥}, C_{(h_{C})_{[a]}} = ⋁_{b \in α (a^{'})} C^{[b]};$

$\forall a \in P (M), C_{(h_{C})^{(a)}} = ⋂_{b \in α (a)} C_{(h_{C})^{[b]}};$ and

If β (a ∧ b) = β (a) ∩ β (b) for any a, b ∈ M, then $\forall a \in β (⊤), C_{(h_{C})_{(a)}} = ⋂_{a \in β (b)} C_{(h_{C})_{[b]}} .$

Proof. (i). By (3) and Theorem 2.1,

$\begin{matrix} A \in C^{[a]} & \Leftrightarrow & a \notin α (C (A)) \\ \Leftrightarrow & a \notin ⋃_{F \in 2_{fin}^{A}, x \notin A} α (h_{C} (F) (x)^{'}) . \end{matrix}$ Hence if $A \in C^{[a]},$ then $a \notin α (h_{C} (F) (x)^{'})$ for all $F \in 2_{fin}^{A}, x \notin A .$ Thus for any b with a ∈ α (b′) , $h_{C} (F) (x)^{'} b^{'},$ i.e. $, x \notin (h_{C})_{[b]} (F) .$ Thus $(h_{C})_{[b]} (F) \subseteq A$ for all $F \in 2_{fin}^{A} .$ Therefore, $A \in C_{(h_{C})_{[b]}}$ for all b with a ∈ α (b′) , and so $C^{[a]} \subseteq ⋂_{a \in α (b^{'})} C_{(h_{C})_{[b]}} .$ Conversely, if $A \in C_{(h_{C})_{[b]}}$ for all b with a ∈ α (b′) , we show that $A \in C^{[a]} .$ If not, then there exist $\tilde{x} \notin A$ and $\tilde{F} \in 2_{fin}^{A}$ such that $a \in α (h_{C} (\tilde{F}) (\tilde{x})^{'}) .$ Let $b = h_{C} (\tilde{F}) (\tilde{x}) .$ Then $\tilde{x} \in (h_{C})_{[b]} (\tilde{F}) \ A,$ contradicting $(h_{C})_{[b]} (\tilde{F}) \subseteq A .$ Therefore, $A \in C^{[a]},$ and so $C^{[a]} \supseteq ⋂_{a \in α (b^{'})} C_{(h_{C})_{[b]}} .$

(ii). The proof is similar to that of (i).

(iii). It is not difficult to show that $A \in C_{[b]} \Leftrightarrow \forall F \in 2_{fin}^{A}, x \notin A, h_{C} (F) (x) \leq b^{'},$ and $A \in C_{(h_{C})^{[a]}} \Leftrightarrow \forall F \in 2_{fin}^{A}, x \notin A, a \in α (h_{C} (F) (x)) .$ Note that $h_{C} (F) (x) \leq b^{'}$ is equivalent to $α (b^{'}) \subseteq α (h_{C} (F) (x)) .$ It follows that $C_{(h_{C})^{[a]}} \supseteq ⋁_{a \in α (b^{'})} C_{[b]} .$

Conversely, let $A \in C_{(h_{C})^{[a]}} (A \neq X) .$ We shall find an up-directed collection in $⋁_{a \in α (b^{'})} C_{[b]}$ and show its union is equal to A . First, let us consider the mapping $η : 2_{fin}^{A} \to 2^{A}$ defined by $η (F) = ⋂_{x \notin A} {B : x \notin B \supseteq F, a \in α (⋁_{y \notin B} {co}_{C} (B) (y))}$ for all $F \in 2_{fin}^{A} .$ By Theorems 2.1 and 2.9 we know $α (h_{C} (F) (x)) = ⋃_{{B : x \notin B \supseteq F}} α (⋁_{y \notin B} {co}_{C} (B) (y)) .$ Hence if $A \in C_{(h_{C})^{[a]}},$ then for any $x \notin A, F \in 2_{fin}^{A},$ there exists B such that x ∉ B ⊇ F and $a \in α (⋁_{y \notin B} {co}_{C} (B) (y)) .$ Thus η (F) ⊆ A, and so it is well defined. It is also clear that F ⊆ η (F) and η (F) ⊆ η (G) whenever F ⊆ G . Note that for any $B \in {B : x \notin B \supseteq F, a \in α (⋁_{y \notin B} {co}_{C} (B) (y))},$ $a \in α (⋁_{y \notin B} {co}_{C} (B) (y)),$ and for all $z \notin B, G \in 2_{fin}^{B}, h_{C} (G) (z) \leq {co}_{C} (B) (z) \leq ⋁_{y \notin B} {co}_{C} (B) (y) .$ Let $b^{'} = ⋁_{y \notin B} {co}_{C} (B) (y) .$ Then $a \in α (b^{'}), B \in C_{[b]} .$ It follows that $η (F) \in ⋁_{a \in α (b^{'})} C_{[b]} .$ Now it is easily seen that $𝒜 = {η (F) : F \in 2_{fin}^{A}}$ is an up-directed collection in $⋁_{a \in α (b^{'})} C_{[b]}$ and A = ⋃𝒜 Thus $A \in ⋁_{a \in α (b^{'})} C_{[b]},$ and so $C_{(h_{C})^{[a]}} \subseteq ⋁_{a \in α (b^{'})} C_{[b]}$ as desired.

(iv). The proof is similar to that of (iii).

(v) and (vi). The proofs are straightforward. □

Theorem 3.10. Let h be an M-fuzzifying restricted hull operator. Then the following statements hold.

For any $a \in P (M), C_{h^{(a)}} = (C_{h})_{[a^{'}]} .$

If β (a ∧ b) = β (a) ∩ β (b) for any a, b ∈ M, then for any $a \in α (⊥), C_{(h_{(a^{'})})} = (C_{h})^{[a]} .$

Proof. Let $C = C_{h}$ in Theorem 3.9. □

In order to discuss the relation between the restricted hull operators of the cut convex structures of an M-fuzzifying convex structure and the cut operators of its M-fuzzifying restricted hull operator, we need the following lemma.

Lemma 3.11. Let {h_i : i ∈ Ω} be a collection of restricted hull operators defined on $2_{fin}^{X} .$ Define ⋀_i∈Ωh_i and ⋁_i∈Ωh_i by (⋀ _i∈Ωh_i) (A) = ⋂ _i∈Ωh_i (A) and $(⋁_{i \in Ω} h_{i}) (A) = ⋂ {B : A \subseteq B = ⋃_{i \in Ω, F \in 2_{fin}^{B}} h_{i} (F)}$ for all $A \in 2_{fin}^{X},$ respectively. Then

both ⋀_i∈Ωh_i and ⋁_i∈Ωh_i are restricted hull operators;

$⋂_{i \in Ω} C_{h_{i}} = C_{(⋁_{i \in Ω} h_{i})};$

$⋁_{i \in Ω} C_{h_{i}} = C_{(⋀_{i \in Ω} h_{i})} .$

Proof. (i). It is routine to check that ⋀_i∈Ωh_i is a restricted hull operator. As for ⋁_i∈Ωh_i, (H1) and (H2) obviously hold. To show (H3) , let finite set S ⊆ (⋁ _i∈Ωh_i) (A) . By the definition of ⋁_i∈Ωh_i, for any $\tilde{B} \in {B : A \subseteq B = ⋃_{i \in Ω, F \in 2_{fin}^{B}} h_{i} (F)},$ we have $S \subseteq \tilde{B}$ and $⋃_{i \in Ω, F \in 2_{fin}^{\tilde{B}}} h_{i} (F) = \tilde{B} .$ It follows that $(⋁_{i \in Ω} h_{i}) (S) \subseteq \tilde{B} .$ By the arbitrariness of $\tilde{B},$ we conclude that (⋁ _i∈Ωh_i) (S) ⊆ (⋁ _i∈Ωh_i) (A) .

(ii). It is not difficult to show that h_i ≤ ⋁ _i∈Ωh_i . Thus $C_{h_{i}} \supseteq C_{(⋁_{i \in Ω} h_{i})},$ and so $⋂_{i \in Ω} C_{h_{i}} \supseteq C_{(⋁_{i \in Ω} h_{i})} .$ Conversely, let $A \in ⋂_{i \in Ω} C_{h_{i}} .$ Then for all $i \in Ω, G \in 2_{fin}^{A}, h_{i} (G) \subseteq A .$ Thus $A = ⋃_{i \in Ω, G \in 2_{fin}^{A}} h_{i} (G)$ and hence for any finite set F ⊆ A, (⋁ _i∈Ωh_i) (F) ⊆ A . Therefore, $A \in C_{(⋁_{i \in Ω} h_{i})},$ and so $⋂_{i \in Ω} C_{h_{i}} \subseteq C_{(⋁_{i \in Ω} h_{i})} .$

(iii). It is easily seen that h_i ≥ ⋀ _i∈Ωh_i . Thus $C_{h_{i}} \subseteq C_{(⋀_{i \in Ω} h_{i})}$ and hence $⋁_{i \in Ω} C_{h_{i}} \subseteq C_{(⋀_{i \in Ω} h_{i})} .$ Conversely, for any k ∈ Ω,

$\begin{matrix} h_{(⋁_{i \in Ω} C_{h_{i}})} (A) & = & ⋂ {B : A \subseteq B, B \in ⋁_{i \in Ω} C_{h_{i}}} \\ \subseteq & ⋂ {B : A \subseteq B, B \in C_{h_{k}}} \\ = & h_{k} (A) . \end{matrix}$ Thus $h_{(⋁_{i \in Ω} C_{h_{i}})} \leq ⋀_{i \in Ω} h_{i},$ and so $⋁_{i \in Ω} C_{h_{i}} \supseteq C_{(⋀_{i \in Ω} h_{i})} .$ □

Remark 3.12. Let {co_i} _i∈Ω be a family of hull operators defined on 2^X . If we define ⋀_i∈Ωco_i by (⋀ _i∈Ωco_i) (A) = ⋂ _i∈Ωco_i (A) for all A ⊆ X . Then $⋁_{i \in Ω} C_{{co}_{i}} = C_{(⋀_{i \in Ω} {co}_{i})}$ holds for finite index set Ω . But, in general, ⋀_i∈Ωco_i need not even be a hull operator. For example, let $X = ℕ, Ω = ℤ_{+}$ and

$c o_{i} (A) = \{\begin{array}{l} A, & if | A | \leq i, \\ X, & otherwise . \end{array}$

Let $B = {n = 2 k + 1 : k \in ℕ} .$ Then (⋀ _i∈Ωco_i) (B) = ⋂ _i∈Ωco_i (B) = X . But $⋃_{F \in 2_{fin}^{B}} (⋀_{i \in Ω} {co}_{i}) (F) = ⋃_{F \in 2_{fin}^{B}} F = B .$ So ⋀_i∈Ωco_i does not satisfy the condition of domain finiteness.

Now by Theorem 3.9 and Lemma 3.11, we have

Theorem 3.13. Let $(X, C)$ be an M-fuzzifying convex structure. Then

$\forall a \in α (⊥), h_{(C^{[a]})} = ⋁_{a \in α (b^{'})} (h_{C})_{[b]};$

$\forall a \in M \ {⊥}, h_{(C_{[a]})} = ⋁_{b \in α (a^{'})} (h_{C})^{[b]};$

$\forall a \in α (⊥), (h_{C})^{[a]} = ⋀_{a \in α (b^{'})} h_{(C_{[b]})};$

$\forall a \in M \ {⊥}, (h_{C})_{[a]} = ⋀_{b \in α (a^{'})} h_{(C^{[b]})};$

$\forall a \in P (M), (h_{C})^{(a)} = ⋁_{b \in α (a)} (h_{C})^{[b]};$ and

If β (a ∧ b) = β (a) ∩ β (b) for any a, b ∈ M, then $\forall a \in β (⊤), (h_{C})_{(a)} = ⋁_{a \in β (b)} (h_{C})_{[b]} .$

Theorem 3.14. Let $(X, C)$ be an M-fuzzifying convex structure. Then the following statements hold.

For any a ∈ P (M) and $A \subseteq X, {co}_{(C_{[a^{'}]})} (A) = ({co}_{C} (A))^{(a)} .$

If β (a ∧ b) = β (a) ∩ β (b) for any a, b ∈ M, then for any a ∈ α (⊥) and A ⊆ X, ${co}_{(C^{[a]})} (A) = {co}_{C} (A)_{(a^{'})} .$

Proof. (i). Combining the results of Theorem 3.13 (ii) and (v), we get $h_{(C_{[a^{'}]})} = (h_{C})^{(a)}$ for any a ∈ P (M) . By (MDF) , ${co}_{C} (A) = ⋁_{F \in 2_{fin}^{A}} {co}_{C} (F)$ for any A ⊆ X . Thus ${co}_{C} (A)^{(a)} = ⋃_{F \in 2_{fin}^{A}} {co}_{C} (F)^{(a)} = ⋃_{F \in 2_{fin}^{A}} {co}_{(C_{[a^{'}]})} (F) = {co}_{(C_{[a^{'}]})} (A) .$

(ii). The proof is similar to that of (i). □

Example 3.15. Let X = {x, y, z} . Consider the I-fuzzifying convex structure illustrated in Table 1.

It is easy to check that

$𝒞_{[a]} = \{\begin{array}{l} 2^{X}, & a \in (0, 0.3], \\ 2^{X} \ {{x, y}}, & a \in (0.3, 0.4], \\ 2^{X} \ {{y}, {x, y}}, & a \in (0.4, 0.5], \\ {\emptyset, {x}, {z}, {y, z}, X}, & a \in (0.5, 0.61], \\ {\emptyset, {z}, {y, z}, X}, & a \in (0.61, 0.71], \\ {\emptyset, {z}, X}, & a \in (0.71, 0.8], \\ {\emptyset, X}, & a \in (0.8, 1] . \end{array}$

Note that for $a = 0.4, ⋁_{a \in α (b^{'})} C_{[b]} = ⋁_{b > 0.6} C_{[b]} = C_{[a^{'}]};$ for $a = 0.5, ⋁_{b > 0.5} C_{[b]} \neq C_{[0 . 5^{'}]} .$ Thus $C_{(h_{C})^{[0.4]}} = C_{[0.6]}, C_{(h_{C})^{[0.5]}} \neq C_{[0.5]}$ as Theorem 3.9 (iii) stated. Moreover, for any a ∈ (0, 1] , by Theorems 3.10 and 3.13, $C_{(h_{C})^{(a^{'})}} = C_{[a]}$ and $(h_{C})^{[a]} = ⋀_{b > a^{'}} h_{(C_{[b]})}$ as one can check.

Theorem 3.16. Let $(X, C)$ and $(Y, D)$ be M-fuzzifying convex structures. A function f : X → Y is an M-fuzzifying convexity preserving function if and only if $h_{C} (F) (x) \leq h_{D} (f (F)) (f (x))$ for all $F \in 2_{fin}^{X}$ and x ∈ X .

Proof. Necessity. Let f (F) = G, f (x) = y . If y ∈ G, then the conclusion is obvious. Now assume y ∉ G . By (MCO3) of Theorem 2.9,

$\begin{matrix} h_{C} (F) (x) \leq h_{D} (G) (y) \\ \Leftrightarrow & {co}_{C} (F) (x) \leq ⋀_{y \notin B \supseteq G} [⋁_{z \notin B} {co}_{D} (B) (z)] . \end{matrix}$ Since f is an M-fuzzifying convexity preserving function, for any B with y ∉ B ⊇ G, by (3),

$\begin{matrix} C (f^{- 1} (B)) \geq D (B) (with f^{- 1} (B) written as N) \\ \Rightarrow & ⋀_{Q \in 2_{fin}^{N}, p \notin N} [{co}_{C} (Q) (p)]^{'} \geq ⋀_{S \in 2_{fin}^{B}, z \notin B} [{co}_{D} (S) (z)]^{'} \\ \Rightarrow & ⋁_{Q \in 2_{fin}^{N}, p \notin N} {co}_{C} (Q) (p) \leq ⋁_{S \in 2_{fin}^{B}, z \notin B} {co}_{D} (S) (z) \\ \Rightarrow & {co}_{C} (F) (x) \leq ⋁_{z \notin B} {co}_{D} (B) (z) (since F \subseteq N, x \notin N) . \end{matrix}$ By the arbitrariness of B, we get the desired inequality. Sufficiency. By (3), for any B ⊆ Y, $C (f^{- 1} (B)) = ⋀_{F \in 2_{fin}^{f^{- 1} (B)}, x \notin f^{- 1} (B)} [h_{C} (F) (x)]^{'} .$ Note that if $F \in 2_{fin}^{f^{- 1} (B)}, x \notin f^{- 1} (B),$ then f (F) ⊆ B, f (x) ∉ B . It follows from $h_{C} (F) (x) \leq h_{D} (f (F)) (f (x))$ that

$\begin{matrix} D (B) & = & ⋀_{G \in 2_{fin}^{B}, y \notin B} [h_{D} (G) (y)]^{'} \\ \leq & [h_{D} (f (F)) (f (x))]^{'} \\ \leq & [h_{C} (F) (x)]^{'} . \end{matrix}$ Thus $C (f^{- 1} (B)) \geq D (B)$ as required. □

Theorem 3.17. Let $(X, C)$ and $(Y, D)$ be M-fuzzifying convex structures and f : X → Y be a bijection. Then

f is an M-fuzzifying convex-to-convex function if and only if for all $F \in 2_{fin}^{X}$ and x ∈ X, $h_{C} (F) (x) \geq h_{D} (f (F)) (f (x));$

f is an M-fuzzifying isomorphism if and only if for all $F \in 2_{fin}^{X}$ and x ∈ X, $h_{C} (F) (x) = h_{D} (f (F)) (f (x)) .$

Proof. The straightforward proof is omitted. □

Remark 3.18. (i) In Theorem 3.17 (i), the condition ‘ f is a bijection ’ is not superfluous. For instance, let $(X, C)$ as in Example 3.15 and let $(Y, D)$ be the convex structure given by Y = {1, 2} and $D (A) = 1$ for all A ⊆ Y . If we define f : X → Y by f (x) =1, f (y) =2, f (z) =1, then f is clearly an M-fuzzifying convex-to-convex function. But $h_{C} ({x, y}) (z) h_{D} (f ({x, y})) (f (z)) .$ Let $(Y, E)$ be the convex structure given by Y = {1, 2} and $E ({1}) = 0.6, E ({2}) = 0.7, E (Y) = E (\emptyset) = 1 .$ If we define f : Y → X by f (1) = x, f (2) = y, then f satisfies $h_{E} (F) (y) \geq h_{C} (f (F)) (f (y))$ for all $F \in 2_{fin}^{Y}$ and y ∈ Y . But f is not an M-fuzzifying convex-to-convex function since $E (Y) C (f (Y)) .$ (ii) The result (ii) of Theorem 3.17 implies that a bijection f : X → Y is an M-fuzzifying isomorphism if and only if the following diagram is commutative.

4 M-fuzzifying betweenness

In this section, we generalize the notion of betweenness (see Definition 2.5) to M-fuzzy setting and show that it can be interpreted in terms of M-fuzzifying restricted hull operator and vice versa.

Definition 4.1. A mapping $δ : X \times 2_{fin}^{X} \to M$ is called an M-fuzzifying betweenness if it satisfies the following conditions.

(MB1) For all x ∈ X, δ (x, ∅) = ⊥ .

(MB2) If $F \in 2_{fin}^{X}, x \in F,$ then δ (x, F) = ⊤ .

(MB3) For all $G, F \in 2_{fin}^{X}$ and x ∈ X, $δ (x, G) ⋀ [⋀_{y \in G} δ (y, F)] \leq δ (x, F) .$

Definition 4.2. Let $δ : X \times 2_{fin}^{X} \to M$ and $η : Y \times 2_{fin}^{Y} \to M$ be M-fuzzifying betweennesses. A mapping f : X → Y is called an M-fuzzifying betweenness preserving function if δ (x, F) ≤ η (f (x) , f (F)) for all x ∈ X and $F \in 2_{fin}^{X} .$ The following theorem shows that there is a natural one-to-one correspondence between M-fuzzifying betweennesses and M-fuzzifying restricted hulloperators.

Theorem 4.3. If $δ : X \times 2_{fin}^{X} \to M$ is an M-fuzzifying betweenness, then $h_{δ} : 2_{fin}^{X} \to M^{X}$ defined by h_δ (F) (x) = δ (x, F) is an M-fuzzifying restricted hull operator. Conversely, if $h : 2_{fin}^{X} \to M^{X}$ is an M-fuzzifying restricted hull operator, then $δ_{h} : X \times 2_{fin}^{X} \to M$ defined by δ_h (x, F) = h (F) (x) is an M-fuzzifying betweenness.

Proof. The straightforward proof is omitted. □

Remark 4.4. For a given M-fuzzifying betweenness $δ : X \times 2_{fin}^{X} \to M,$ there is an M-fuzzifying convex structure $(X, C_{h_{δ}})$ (or written simply as $(X, C_{δ})$ ). Conversely, for an M-fuzzifying convex structure $(X, C)$ there is an M-fuzzifying betweenness $δ_{h_{C}}$ (or written simply as $δ_{C}$ ). Under these notations, it is easily seen that $δ_{C_{δ}} = δ$ and $C_{δ_{C}} = C .$

Theorem 4.5. Let $(X, C)$ and $(Y, D)$ be M-fuzzifying convex structures. A function f : X → Y is an M-fuzzifying convexity preserving function if and only if f is an M-fuzzifying betweenness preserving function, that is, $δ_{C} (x, F) \leq δ_{D} (f (x), f (F))$ for all x ∈ X and $F \in 2_{fin}^{X} .$

Proof. This follows immediately from Theorem 3.16. □

Theorem 4.6. Let e : X × X → M be an M-fuzzy quasi-order. Define three mappings from $X \times 2_{fin}^{X}$ to M by $δ_{e}^{↓} (x, F) = ⋁_{y \in F} e (x, y), δ_{e}^{↑} (x, F) = ⋁_{y \in F} e (y, x)$ and $δ_{e} (x, F) = δ_{e}^{↓} (x, F) ⋀ δ_{e}^{↑} (x, F)$ for all $(x, F) \in X \times 2_{fin}^{X},$ respectively. Then

$δ_{e}^{↓}, δ_{e}^{↑}, δ_{e}$ are M-fuzzifying betweennesses; and

The M-fuzzifying convexities $C_{δ_{e}^{↓}}$ and $C_{δ_{e}^{↑}}$ satisfy $C_{δ_{e}^{↓}} (⋃_{i \in Ω} A_{i}) \geq ⋀_{i \in Ω} C_{δ_{e}^{↓}} (A_{i})$ and $C_{δ_{e}^{↑}} (⋃_{i \in Ω} A_{i}) \geq ⋀_{i \in Ω} C_{δ_{e}^{↑}} (A_{i}),$ for any {A_i : i ∈ Ω} ⊆2^X, respectively.

Proof. (i). Take $δ_{e}^{↓}$ as an example. (MB1) and (MB2) are easily obtained. To show (MB3) , we may assume that G≠ ∅ as it holds trivially when G = ∅ . Thus

$\begin{matrix} δ_{e}^{↓} (x, G) ⋀ [⋀_{y \in G} δ_{e}^{↓} (y, F)] \\ = & [⋁_{g \in G} e (x, g)] ⋀ {⋀_{y \in G} [⋁_{z \in F} e (y, z)]} \\ = & ⋁_{g \in G} {e (x, g) ⋀ {⋀_{y \in G} [⋁_{z \in F} e (y, z)]}} \\ = & ⋁_{g \in G} {⋀_{y \in G} {e (x, g) ⋀ [⋁_{z \in F} e (y, z)]}} \end{matrix}$

$\begin{matrix} = & ⋁_{g \in G} {⋀_{y \in G} ⋁_{z \in F} [e (x, g) \land e (y, z)]} \\ \leq & ⋁_{g \in G} {⋁_{z \in F} [e (x, g) \land e (g, z)]} \\ \leq & ⋁_{g \in G} [⋁_{z \in F} e (x, z)] \\ = & δ_{e}^{↓} (x, F) . \end{matrix}$ (ii). Let A = ⋃ _i∈ΩA_i . Then

$\begin{matrix} C_{δ_{e}^{↓}} (A) & = & ⋀_{x \notin A, F \in 2_{fin}^{A}} [δ_{e}^{↓} (x, F)]^{'} \\ = & ⋀_{x \notin A, F \in 2_{fin}^{A}} [⋀_{y \in F} e (x, y)^{'}] . \end{matrix}$ Let $a ≺ ⋀_{i \in Ω} C_{δ_{e}^{↓}} (A_{i}) .$ We claim that $a \leq C_{δ_{e}^{↓}} (A) .$ Otherwise, there exist $\tilde{x} \notin A, \tilde{F} \in 2_{fin}^{A}$ and $\tilde{y} \in \tilde{F}$ such that $a [e (\tilde{x}, \tilde{y})]^{'} .$ Let $\tilde{y} \in A_{\tilde{j}}, \tilde{j} \in Ω .$ Note that $a ≺ ⋀_{i \in Ω} C_{δ_{e}^{↓}} (A_{i})$ implies $a ≺ C_{δ_{e}^{↓}} (A_{\tilde{j}}) = ⋀_{x \notin A_{\tilde{j}}, F \in 2_{fin}^{A_{\tilde{j}}}} ⋀_{y \in F} [e (x, y)]^{'} .$ Since $\tilde{x} \notin A_{\tilde{j}}, \tilde{y} \in \tilde{F} \cap A_{\tilde{j}} \in 2_{fin}^{A_{\tilde{j}}},$ we have $a \leq [e (\tilde{x}, \tilde{y})]^{'};$ a contradiction. Thus $a \leq C_{δ_{e}^{↓}} (A)$ as claimed. Since a is arbitrary, we conclude that $C_{δ_{e}^{↓}} (⋃_{i \in Ω} A_{i}) \geq ⋀_{i \in Ω} C_{δ_{e}^{↓}} (A_{i}) .$ □

Remark 4.7. One can view a partially ordered set (X, ≤) as an M-fuzzy partially ordered set (X, e_X) in which M = 2 and e_X (x, y) =⊤ if and only if x ≤ y . Then it is easily seen that the M-fuzzifying convexities $C_{δ_{e_{X}}^{↓}} (C_{δ_{e_{X}}^{↑}})$ and $C_{δ_{e_{X}}}$ are the lower (upper) convexity and the order convexity of X, respectively.

Corollary 4.8. [9] Let R be a fuzzy preorder (i.e., R is a reflexive and transitive fuzzy binary relation on X). Then both $(X, C_{δ_{R}^{↓}})$ and $(X, C_{δ_{R}^{↑}})$ are Alexanderoff fuzzifying topological spaces.

Corollary 4.9. Let (X, d) be an ultrametric space and let $e_{d} (x, y) = \frac{1}{1 + d (x, y)} .$ Then all the M-fuzzifying convexities $C_{δ_{e_{d}}^{↓}}, C_{δ_{e_{d}}^{↑}}$ and $C_{δ_{e_{d}}}$ are equal and $(X, C_{δ_{e_{d}}})$ is an Alexanderoff fuzzifying topological space.

Example 4.10. Note that e : I × I → I defined by e (x, y) = x → y is an I-fuzzy quasi-order. The I-fuzzifying convexity $C_{δ_{e}}$ on I is computed as

$𝒞_{δ_{e}} (A) = \{\begin{array}{l} 1, & if A = \emptyset or I, \\ 1 - b, & if A = [a, b) or (a, b), \\ 1 - b, & if A = [a, b] or (a, b], where b < 1, \\ 1 - a, & if A = [a, 1] or (a, 1] or {a}, \\ 0, & otherwise . \end{array}$

Example 4.11. (i) Let d be the discrete metric on X, that is, for all x, y ∈ X, if x = y, then d (x, y) =0 ; otherwise, d (x, y) =1 . Then

$𝒞_{δ_{e_{d}}} (A) = \{\begin{array}{l} 1, & if A \in {\emptyset, X}, \\ 0.5, & otherwise, \end{array}$

which is the same as Example 20 given in [23].

(ii) Consider the p-adic distance d_p (x, y) = |x - y|_p on the field Q of rational numbers, where | · |_p is the p-adic norm [12]. Since d_p is an ultrametric on Q, it induces an I-fuzzifying convexity $C_{δ_{e_{d_{p}}}}$ on Q .

5 Conclusions and future work

In this paper, the notion of M-fuzzifying restricted hull operator was defined and several equivalent characterizations were given. The notion of M-fuzzifying betweennesses was also proposed. As applications, some properties of the cut convex structures of an M-fuzzifying convex structure and of M-fuzzifying convexity preserving functions and of M-fuzzifying convex-to-convex functions were derived.

As we have seen, an M-fuzzifying convex structure $(X, C)$ is determined by its M-fuzzifying restricted hull operator $h_{C} .$ Therefore, one can use M-fuzzifying restricted hull operators to construct M-fuzzifying convexities, as shown in Example 4.10. From the viewpoint of theory generalization, there are many work to do. For instance, by Remark 4.7, the M-fuzzifying convexity $C_{δ_{e}}$ induced by an M-fuzzy partial order e is a reasonable generalization of the order convexity of a partially ordered set. As the order convexity of a partially ordered set is JHC, S₃ and its segment operator is geometric, we need to consider how to generalize these properties to M-fuzzifying convexity C_{δ
_e} .

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