On M -fuzzifying JHC convex structures and M -fuzzifying Peano interval spaces

Abstract

In this paper, relations between hull operators which correspond to convex structures formed by cut sets of a given M-fuzzifying convex structure and cut sets of the hull operator of that M-fuzzifying convex structure are discussed. Then concepts of M-fuzzifying Join-Hull Commutativity property and M-fuzzifying Peano property are introduced and characterized, respectively. Also, it is proved that an M-fuzzifying convex structure which has M-fuzzifying JHC property is of arity ≤ 2 and that the segment operator of an M-fuzzifying convex structure of arity ≤2 has M-fuzzifying JHC property iff it has M-fuzzifying Peano property.

1 Introduction

As Berger described in [1], convexity is a very old topic originally inspired by some elementary geometric problems such as the shapes of circles and polytopes in 2 and 3-dimensional Euclidean spaces. Later, this notion has mainly been extended and abstracted along two directions. One is motivated by concrete problems such as the existence of continuous selections and fixed points, or optimization problems [2,8,11,14,16 , 2,8,11,14,16]; the other is based on an axiomatic point of view, where abstract convexity was introduced and its theory was studied [3 , 26].

Convexity has been extended into fuzzy settings since the end of the last century. Rosa introduced the notion of fuzzy convex structures [17], which was further generalized into M-fuzzy settings by Maruyama [12]. Díaz and her colleagues recently introduced notions of H-convex structures and intuitionistic convex structures [5]. They defined convex structures in a common way: each definition is actually a family of non-fuzzy or fuzzy sets satisfying certain set of axioms. However, Shi and Xiu introduced the notion of M-fuzzifying convex structures in a totally different way which was used to define the fuzzification of topologies [30]. Thus, in their definition, each subset can be regarded as a convex set to some degree. They found that an M-fuzzifying closure operator is the hull operator of an M-fuzzifying convex structure iff it is domain finite [21]. Later, Xiu defined the notion of M-fuzzifying interval spaces [29]. He showed that an M-fuzzifying convex structure is generated by an M-fuzzifying interval operator iff it is ofarity ≤2.

It is known that convex structures of arity ≤2 is of vital importance in theory of convex structures. In fact, almost all examples of convex structures investigated by van de Vel in [26] are of this type. Vel showed that a convex structure is of arity ≤2 iff it is generated by an interval operator. In addition, he proved that a Join-Hull Commutativity convex structure is of arity ≤2 and that a convex structure of arity ≤2 has JHC property iff it has Peano property [26].

In this paper, based on some results in [21], we prove that an M-fuzzifying convex structure is of arity ≤n iff convexities of its cut sets are of arity ≤n. Then we define and characterize the notion of M-fuzzifying JHC convex structures. We show that M-fuzzifying JHC convex structures are of arity ≤2 and thatM-fuzzifying JHC convex structures are preserved by M-fuzzifying CP and CC surjective functions. Also, we introduce and characterize M-fuzzifying Peano property in M-fuzzifying interval spaces. We proved that M-fuzzifying Peano property is preserved byM-fuzzifying II surjective functions and that the segment operator of an M-fuzzifying convex structure which is of arity ≤2 has M-fuzzifying Peano property iff it has M-fuzzifying JHC property.

2 Preliminaries

Throughout this paper, X and Y are nonempty sets. The power set of A is denoted by 2^X. The set of all finite subsets of X is denoted by $2_{fin}^{X}$ . The power set of a subset A ∈ 2^X is denoted by 2^A and the set of all finite subsets of A is denoted by $2_{fin}^{A}$ . M is a completely distributive lattice with an inverse inclusion operator ^′. The minimal element and the maximal element in M are denoted respectively by ⊥ and ⊤ [23,27, 23,27]. For φ ⊆ M, ⋁_a∈φa and ⋀_a∈φa are denoted by ⋁φ and ⋀φ, respectively.

An element a ∈ M is called a prime element if for all b, c ∈ M, b ∧ c ≤ a implies b ≤ a or c ≤ a. The set of all prime elements in M ∖ {⊤} is denoted by P (M). An element a ∈ M is called a co-prime element if its complement a′ is a prime element. The set of all co-prime elements in M ∖ {⊥} is denoted by J (M). For any a ∈ M, there exist φ ⊆ P (M) and ψ ∈ J (M) such that a = ⋀ φ = ⋁ ψ [27,28, 27,28].

A binary relation ≺ on M is defined by: for all a, b ∈ M, a ≺ b iff for each φ ⊆ M, the relation b ≤ ⋁ φ always implies the existence of d ∈ φ such that a ≤ d. A mapping β : M → 2^M, defined by: β (a) = {b : b ≺ a} for every a ∈ M, is called the minimal mapping of M. It is a ⋁-⋃ mapping. The opposite relation ≺^op of ≺ is defined by: for all a, b ∈ M, a ≺ ^opb if and only if b′ ≺ a′. A mapping α : M → 2^M, defined by: α (a) = {b : a ≺ ^opb} for every a ∈ M, is called the maximal mapping of M. It is a ⋀-⋃ mapping [19,21,27 , 19,21,27]. For a ∈ M, β^* (a) = β (a) ∩ J (M) and α^* (a) = α (a) ∩ P (M). Clearly, β (⊥) = α (⊤) = ∅ and a = ⋁ β (a) = ⋀ α (a) for each a ∈ M [19].

A mapping U : X → M is called an M-fuzzy set on X. The set of all M-fuzzy sets on X is denoted by M^X. For x ∈ X and r ∈ M, the M-fuzzy set (an M-fuzzy point [28]) x_r ∈ M^X is defined by: $\forall y \in X, x_{r} (y) = {\begin{matrix} r, & y = x, \\ ⊥, & y \neq x . \end{matrix}$

Let U ∈ M^X and r ∈ M. Four types of cut sets of U ∈ M^X are defined as followings [18].

U_[r] = {x ∈ X : U (x) ≥ r}.

U^[r] = {x ∈ X : r ∉ α (U (x))}.

U_(r) = {x ∈ X : r ∈ β (U (x))}.

U^(r) = {x ∈ X : U (x) ≰ r}.

For A ∈ 2^X and r ∈ M, χ_A, r ∨ χ_A ∈ M^X is defined by: for every x ∈ X,

χ_{A} (x) = {\begin{matrix} ⊤, & x \in A, \\ ⊥, & x \notin A, \end{matrix} r \lor χ_{A} (x) = {\begin{matrix} ⊤, & x \in A, \\ r, & x \notin A . \end{matrix}

Definition 2.1. [26] A subset $C$ of 2^X is called a convexity on X if it satisfies the following conditions.

$\emptyset, X \in C$ .

If ${U_{i}}_{i \in Ω} \subseteq C$ is nonempty, then $⋂_{i \in Ω} U_{i} \in C$ .

If ${U_{i}}_{i \in Ω} \subseteq C$ is nonempty and totally ordered by inclusion, then $⋃_{i \in Ω} U_{i} \in C$ .

If $C$ is a convexity on X, then the pair $(X, C)$ is called a convex structure. Elements of $C$ are called convex sets.

Definition 2.2. [26] The hull operator $h_{C} : 2^{X} \to 2^{X}$ (briefly, h) of a convex structure $(X, C)$ is defined by: $\forall A \in 2^{X}, h (A) = ⋂ {B : A \subseteq B \in C} .$ The restriction of h on {{x, y} : x, y ∈ X} is called the segment operator of $C$ and is still denoted by h.

Definition 2.3. [26] A convexity $C$ on X is said to be of arity ≤n, where $n \in ℕ$ and $ℕ$ is the set of all positive natural numbers, if $C = {A \in 2^{X} : \forall F \subseteq A, | F | \leq n, h (F) \subseteq A},$ where |F| is the cardinality of F. In particular, if n = 2, then we say $C$ is of arity ≤2.

Theroem 2.4. [26] Let $(X, C)$ be a convex structure. Then the restriction $h_{fin} : 2_{fin}^{X} \to 2^{X}$ of the hull operator h satisfies the following conditions.

h_fin (∅) = ∅.

For each F ∈ 2^X, F ⊆ h_fin (F).

For all $F, G \in 2_{fin}^{X}$ with G ⊆ h_fin (F), h_fin (G) ⊆ h_fin (F).

Conversely, if an operator $g : 2_{fin}^{X} \to 2^{X}$ satisfies (RH1)-(RH3), then there exists an unique convex structure $(X, C)$ with $(h_{C})_{fin} = g$ . For convenience, we also write h instead of h_fin.

Definition 2.5. [26] A convexity $C$ on X is called a JHC convexity (alternatively, $C$ and its hull operator have JHC property), if its hull operator h satisfies $\forall C \in C ∖ {\emptyset}, \forall a \in X, h ({a} \cup C) = ⋃_{x \in C} h ({a, x}) .$

A convex structure $(X, C)$ has JHC property iff $\forall A \in 2^{X} ∖ {\emptyset}, \forall a \in X, h ({a} \cup A) = ⋃_{x \in h (A)} h ({a, x}) .$

It is clear that the standard convexity and the linear convexity on V are JHC convexities [26].

Definition 2.6. [26] An operator $I : X \times X \to 2^{X}$ is called an interval operator on X if it satisfies the following conditions.

$a, b \in I (a, b)$ .

$I (a, b) = I (b, a)$ .

If $I$ is an interval operator on X, then the pair $(X, I)$ is called an interval space. A ∈ 2^X is called an interval convex set if $I (a, b) \subseteq A$ for all a, b ∈ A. The set $C_{I}$ of interval convex sets is a convexity generated by $I$ .

Let $(X, C)$ be a convex structure. Define an operator $I_{C} : X \times X \to 2^{X}$ by:

$\forall x, y \in X, I_{C} (x, y) = h ({x, y}) .$ Then $I_{C}$ is an interval operator which is called the interval operator generated by the segment operator h. In view of this result, we regard the segment operator of a convex structure as an interval operator [26].

Definition 2.7. [26] An interval space $(X, I)$ is called a Peano interval space, if for all a, b, c, x, y ∈ X, $x \in I (b, c)$ and $y \in I (a, x)$ imply the existence of $z \in I (a, b)$ such that $y \in I (c, z)$ . If $(X, I)$ is a Peano interval space, then we say $I$ has Peano property.

Let $(X, I)$ be an interval space and A, B ∈ M^X. $(AB)_{I} = ⋃ {I (a, b) : a \in A, b \in B}$ is called the join of A and B [26]. Let V be a vector space on a totally ordered field $𝕂$ . Define an operator $I : V \times V \to 2^{V}$ by: $\begin{matrix} \forall a, b \in V, I (a, b) \\ = {ta + (1 - t) b : t \in 𝕂, 0 \leq t \leq 1} . \end{matrix}$ Then $(V, I)$ is a Peano interval space [26].

Definition 2.8. [21] A mapping 𝒞 : 2^X → M is called an M-fuzzifying convexity if it satisfies the following conditions.

𝒞 (∅) = 𝒞 (X) = ⊤.

𝒞 (⋂ _i∈ΩU_i) ≥ ⋀ _i∈Ω 𝒞 (U_i) for all non-empty set {U_i} _i∈Ω ⊆ 2^X.

𝒞 (⋃ _i∈ΩU_i) ≥ ⋀ _i∈Ω 𝒞 (U_i) for all non-empty and totally ordered set {U_k} _i∈Ω ⊆ 2^X.

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

Lemma 2.9. [21] If (X, 𝒞) is an M-fuzzifying convex structure, then for all r ∈ M ∖ {⊥} and s ∈ α (⊥), $𝒞_{[r]} = {A \in 2^{X} : 𝒞 (A) \geq r}$ and $𝒞^{[s]} = {A \in 2^{X} : s \notin α (𝒞 (A))}$ are convexities on X.

Theorem 2.10. [21] Let (X, 𝒞) be an M-fuzzifying convex structure. The hull operator co_𝒞 : 2^X → M^X (briefly, co) of 𝒞 is defined by: $\forall A \in 2^{X}, \forall x \in X, co (A) (x) = ⋀_{x \notin B \supseteq A} (𝒞 (B))^{'} .$ Then co satisfies the following conditions.

co (∅) (x) = ⊥ for every x ∈ X.

If A ∈ 2^X and x ∈ A, then co (A) (x) =⊤.

co (A) (x) = ⋀ _x∉B⊇A ⋁ _y∉Bco (B) (y) for all A ∈ 2^X and x ∈ X.

$co (A) (x) = ⋁_{F \in 2_{fin}^{A}} co (F) (x)$ for all A ∈ 2^X and x ∈ X.

Conversely, let co : 2^X → M^X be an operator satisfying (MCO1)-(MCO3) and (MFD). Define a mapping 𝒞_co : 2^X → M by: $\forall A \in 2^{X}, 𝒞_{c o} (A) = \underset{x \notin A}{\land} {(c o (A) (x))}^{'} .$ Then 𝒞_co is an M-fuzzifying convexity with co as its hull operator. That is, co_{𝒞_co} = co.

Remark 2.11.

If (X, 𝒞) is an M-fuzzifying convex structure, then the restriction of its hull operator on {{x, y} : x, y ∈ X} is called the segment operator of 𝒞 and is still denoted by co.

A mapping 𝒞 : 2^X → M satisfying (MC1) and (MC2) is called an M-fuzzifying closure structure. An operator co : 2^X → M^X satisfying (MCO2) and (MCO3) is called an M-fuzzifying closure operator [20,21, 20,21].

If 𝒞 : 2^X → M is an M-fuzzifying closure structure satisfying 𝒞 (A ∪ B) ≥ 𝒞 (A) ∧ 𝒞 (B) for all A, B ∈ 2^X, then a mapping 𝒯 : 2^X → M, defined by: 𝒯 (A) = 𝒞 (A′) for every A ∈ 2^X, is an M-fuzzifying topology [20,30, 20,30].

Let (X, ℬ) be an M-fuzzifying closure structure. Define a mapping 𝒞_ℬ : 2^X → M by: $\forall A \in 2^{X}, 𝒞_{ℬ} (A) = \underset{⋃_{i \in Ω}^{d i r} B_{i} = A}{\lor} \underset{i \in Ω}{\land} ℬ (B_{i}),$ where {B_i} _i∈Ω ⊆ 2^X is an up-directed family and $⋃_{i \in Ω}^{dir} B_{i}$ is its union. Then 𝒞_ℬ is anM-fuzzifying convexity generated by ℬ [21].

Theorem 2.12. [21] An closure structure 𝒞 : 2^X → M is an M-fuzzifying convexity iff it is domain finite (i.e., its the closure operator (MDF)) iff it is stable for up-directed union.

Definition 2.13. [21] Let (X, 𝒞) and (Y, 𝒟) beM-fuzzifying convex structures. A function f : X → Y is called an M-fuzzifying convexity preserving (briefly, M-fuzzifying CP) function if 𝒞 (f^-1 (B)) ≥ 𝒟 (B) for every B ∈ 2^Y. f is called an M-fuzzifying convex-to-convex (briefly, M-fuzzifying CC) function if 𝒟 (f (A)) ≥ 𝒞 (A) for every A ∈ 2^X.

Definition 2.14. [29] An M-fuzzifying convexity 𝒞 on X is said to be of arity ≤n, if $\forall A \in 2^{X}, {𝒞 (A) = ⋀_{x \notin A} ⋀_{F \in 2_{fin}^{A}, | F | \leq n} (co (F) (x))^{'} .$ In particular, if n = 2, we say 𝒞 is of arity ≤2.

Definition 2.15. [24] An operator $ℋ : 2_{fin}^{X} \to M^{X}$ is called an M-fuzzifying restricted hull operator if it satisfies the following conditions.

ℋ (∅) (x) = ⊥ for every x ∈ X.

ℋ (F) (x) =⊤ for all $F \in 2_{fin}^{X}$ and x ∈ F.

ℋ (G) (x) ∧ ⋀ _y∈G ℋ (F) (y) ≤ ℋ (F) (x) for all $F, G \in 2_{fin}^{X}$ and x ∈ X.

Theorem 2.16. [24] Let (X, 𝒞) be an M-fuzzifying convex structure. Then the restriction ${co}_{fin} : 2_{fin}^{X} \to M^{X}$ of co is an M-fuzzifying restricted hull operator.

Conversely, let $ℋ : 2_{fin}^{X} \to M^{X}$ be an M-fuzzifying restricted hull operator. Define an operator co_ℋ : 2^X → M^X by: $\forall A \in 2^{X}, \forall x \in X, {co}_{ℋ} (A) (x) = ⋁_{F \in 2_{fin}^{A}} ℋ (F) (x) .$ Then co_ℋ is the M-fuzzifying hull operator of an M-fuzzifying convexity with (co_ℋ) _fin = ℋ. For convenience, we still write co instead of co_fin.

Definition 2.17. [29] An operator ℐ : X × X → M^X is called an M-fuzzifying interval operator, if for all x, y ∈ X,

ℐ (x, y) (x) = ℐ (x, y) (y) = ⊤.

ℐ (x, y) = ℐ (y, x).

If ℐ is an M-fuzzifying interval operator on X, then the pair (X, ℐ) is called an M-fuzzifying interval space.

Theorem 2.18. [29] Let (X, 𝒞) be an M-fuzzifying convex structure and co be its segment operator. Then the operator ℐ_𝒞 : X × X → M^X, defined by: $\forall x, y, z \in X, ℐ_{𝒞} (x, y) (z) = c o ({x, y}) (z)$ is an M-fuzzifying interval operator generated by 𝒞.

Conversely, let (X, ℐ) be an M-fuzzifying interval space and let 𝒞_ℐ : 2^X → M be defined by: $\forall A \in 2^{X}, 𝒞_{ℐ} (A) = \underset{z \notin A}{\land} \underset{x, y \in A}{\land} {(ℐ (x, y) (z))}^{'} .$ Then 𝒞_ℐ is an M-fuzzifying convexity generated by ℐ. Moreover, we have ℐ ≤ co, where co is the segment operator of 𝒞_ℐ. That is, ℐ (x, y) ≤ co ({x, y}) for all x, y ∈ X. In view of the is, we simply regard the segment operator of an M-fuzzifying convex structure as an M-fuzzifying interval operator.

Theorem 2.19. [29] An M-fuzzifying convexity is induced by an M-fuzzifying interval operator if and only if it is of arity ≤2.

Definition 2.20. [29] Let (X, ℐ_X) and (Y, ℐ_Y) be M-fuzzifying interval spaces. A function f : X → Y is called an M-fuzzifying II-function if $f_{M}^{\to} (ℐ_{X} (a, b)) = ℐ_{Y} (f (a), f (b))$ for all a, b ∈ X, where $f_{M}^{\to} : M^{X} \to M^{Y}$ is the M-fuzzy (or, M-fuzzy Zadeh’s) function generated by f [20,28, 20,28].

Definition 2.21. [22] Let [0, + ∞) (M) be the set of all non-negative M-fuzzy real numbers. A map $D : X \times X \to [0, + \infty) (M)$ is called an M-fuzzifying pseudo-metric on X if for all x, y, z ∈ X and all s, t > 0,

$x = y \Rightarrow D (x, y) (0 +) = ⊥$ ;

$D (x, z) (r + s) \leq D (x, y) (r) \lor D (y, z) (s)$ ;

$D (x, y) = D (y, x)$ .

If $D$ is an M-fuzzifying pseudo-metric on X, then $(X, D)$ is called an M-fuzzifying pseudo-metric space. If the equality in (MD2) holds, then the pair $(X, D)$ is called an M-fuzzifying strong pseudo-metric space.

Lemma 2.22. [22] Let $(X, D)$ be an M-fuzzifying pseudo-metric space. Define ${cl}_{D} : 2^{X} \to M^{X}$ by: $\forall A \in 2^{X}, \forall z \in X, {cl}_{D} (A) (z) = ⋀_{r > 0} ⋁_{c \in A} (D (z, c) (r))^{'} .$ Then ${cl}_{D}$ is the M-fuzzifying closure operator of an M-fuzzifying topology $𝒯_{D}$ .

Definition 2.23. [5] An H-convex structure is a map H : X × X × [0, 1] → X satisfying the following conditions.

H (x, y, t) = H (y, x, 1 - t) for all x, y ∈ X and t ∈ [0, 1].

H (x, x, t) = x for all x ∈ X and t ∈ [0, 1].

H (x, y, 1) = x for every x ∈ X.

A subset A of X is called an H-convex set if H (x, y, t) ∈ A for all x, y ∈ A and t ∈ [0, 1]. Clearly, the set of all H-convex sets is a convexity on X.

Lemma 2.24. [21] For all p, q ∈ M, the following conditions are equivalent.

p ≤ q.

For each r ∈ M, a ≤ p implies a ≤ q.

For each r ∈ J (M), a ≤ p implies a ≤ q.

For each a ∈ β (⊤), a ≤ p implies a ≤ q.

For each a ∈ α (⊥), a ∉ α (p) implies a ∉ α (q).

3 Relations of cut sets of M-fuzzifying convex structures and cut sets of their hull operators

Lemma 3.1. Let (X, 𝒞) be an M-fuzzifying convex structure and r ∈ P (M). Define an operator co^(r) : 2^X → 2^X by co^(r) (A) = co (A) ^(r) for every A ∈ 2^X. Then its restriction ${co}^{(r)} : 2_{fin}^{X} \to 2^{X}$ is a restricted hull operator.

Proof. (H1) and (H2) are clear.

(H3). Let $F, G \in 2_{fin}^{X}$ with G ⊆ co^(r) (F). Then co (F) (y) ≰ r for each y ∈ G. Thus co (G) (z) ∧ ⋀ _y∈Gco (F) (y) ≰ r for each z ∈ co^(r) (G). By (MRH3) of co, we have co (F) (z) ≰ r. Hence z ∈ co^(r) (F). Therefore co^(r) (G) ⊆ co^(r) (F).

Lemma 3.2. Let (X, 𝒞) be an M-fuzzifying convex structure and r ∈ P (M). Let co^(r) be the operator defined in Lemma 3.1. Then co^(r) (co^(r) (A)) = co^(r) (A) for every A ∈ 2^X.

Proof. Clearly, co^(r) (A) ⊆ co^(r) (co^(r) (A)). Conversely, for each z ∈ X, (MDF) implies that $co (A) (z) = ⋁_{B \in 2_{fin}^{A}} co (B) (z)$ and $co ({co}^{(r)} (A)) (z) = ⋁_{G \in 2_{fin}^{{co}^{(r)} (A)}} co (G) (z)$ Thus ${co}^{(r)} (A) = ⋃_{B \in 2_{fin}^{A}} {co}^{(r)} (B)$ and ${co}^{(r)} ({co}^{(r)} (A)) = ⋃_{G \in 2_{fin}^{{co}^{(r)} (A)}} {co}^{(r)} (G) .$ For each $G \in 2_{fin}^{{co}^{(r)} (A)}$ , there exists ${B_{i}^{G}}_{i = 1}^{n} \subseteq 2_{fin}^{A}$ such that $G \subseteq ⋃_{i = 1}^{n} {co}^{(r)} (B_{i}^{G}) \subseteq {co}^{(r)} (⋃_{i = 1}^{n} B_{i}^{G}) .$ Notice that $⋃_{i = 1}^{n} B_{i}^{G} \in 2_{fin}^{A}$ . By (H3), we have ${co}^{(r)} (G) \subseteq {co}^{(r)} (⋃_{i = 1}^{n} B_{i}^{G}) \subseteq {co}^{(r)} (A) .$ Therefore co^(r) (co^(r) (A)) ⊆ co^(r) (A).

Lemma 3.3. Let (X, 𝒞) be an M-fuzzifying convex structure and r ∈ P (M). Then for every A ∈ 2^X, A ∈ 𝒞_[r′] if and only if co^(r) (A) = A.

Proof. Let A ∈ 2^X. Then we have $\begin{matrix} A \in 𝒞_{[r^{'}]} & \Leftrightarrow & ⋀_{x \notin A} (co (A) (x))^{'} = 𝒞 (A) \geq r^{'} \\ \Leftrightarrow & \forall x \notin A, co (A) (x) \leq r \\ \Leftrightarrow & \forall x \notin A, x \notin {co}^{(r)} (A) \\ \Leftrightarrow & {co}^{(r)} (A) \subseteq A \Leftrightarrow {co}^{(r)} (A) = A . \end{matrix}$ Thus the result holds.

Lemma 3.4. Let (X, 𝒞) be an M-fuzzifying convex structure and r ∈ P (M). Then co^(r) (A) ⊆ h_{𝒞_[r′]} (A) for every A ∈ 2^X.

Proof. Let A ⊆ X and x ∈ X. We have $\begin{matrix} x \in {co}^{(r)} (A) & \Rightarrow & co (A) (x) ≰ r \Rightarrow ⋁_{x \notin B \supseteq A} 𝒞 (B) ≱ r^{'} \\ \Rightarrow & \forall x \notin B \supseteq A, 𝒞 (B) ≱ r^{'} \\ \Rightarrow & \forall x \notin B \supseteq A, B \notin 𝒞_{[r^{'}]} \\ \Rightarrow & x \in h_{𝒞_{[r^{'}]}} (A) . \end{matrix}$ Therefore co^(r′) (A) ⊆ h_{𝒞_[r′]} (A).

Theorem 3.5. Let (X, 𝒞) be an M-fuzzifying convex structure. Then co^(r) is the hull operator of the convex structure (X, 𝒞_[r′]) for every r ∈ P (M).

Proof. By Lemma 3.2, 3.3 and 3.4, the proof is clear.

Theorem 3.6. An M-fuzzifying convex structure (X, 𝒞) is of arity ≤n iff (X, 𝒞_[r′]) is of arity ≤n for each r ∈ P (M).

Proof. Necessity. Let A ∈ 2^X and r ∈ P (M). Then $\begin{matrix} A \in 𝒞_{[r^{'}]} \\ \Leftrightarrow & ⋀_{x \notin A} ⋀_{F \in 2_{fin}^{A}, | F | \leq n} (co (F) (x))^{'} = 𝒞 (A) \geq r^{'} \\ \Leftrightarrow & \forall x \notin A, \forall F \in 2_{fin}^{A}, | F | \leq n, co (F) (x) \leq r \\ \Leftrightarrow & \forall x \notin A, \forall F \in 2_{fin}^{A}, | F | \leq n, x \notin {co}^{(r)} (F) \\ \Leftrightarrow & \forall x \notin A, \forall F \in 2_{fin}^{A}, | F | \leq n, {co}^{(r)} (F) \subseteq A . \end{matrix}$ This shows 𝒞_[r′] is of arity ≤n.

Sufficiency. Let A ∈ 2^X. By Theorem 3.5, $\begin{matrix} 𝒞 (A) \geq r \Leftrightarrow A \in 𝒞_{[r^{'}]} \\ \Leftrightarrow & \forall F \in 2_{fin}^{A}, | F | \leq n, {co}^{(r)} (F) \subseteq A \\ \Leftrightarrow & \forall x \notin A, \forall F \in 2_{fin}^{A}, | F | \leq n, x \notin {co}^{(r)} (F) \\ \Leftrightarrow & \forall x \notin A, \forall F \in 2_{fin}^{A}, | F | \leq n, (co (F) (x))^{'} \geq r^{'} \\ \Leftrightarrow & ⋀_{x \notin A} ⋀_{F \in 2_{fin}^{A}, | F | \leq n} (co (F) (x))^{'} \geq r^{'} . \end{matrix}$ By (3) of Lemma 2.24, $𝒞 (A) = ⋀_{x \notin A} ⋀_{F \in 2_{fin}^{A}, | F | \leq n} (co (F) (x))^{'}$ , which shows that 𝒞 is of arity ≤n.

Corollary 3.7. An M-fuzzifying convex structure (X, 𝒞) is of arity ≤2 iff (X, 𝒞_[r′]) is of arity ≤2 for each r ∈ P (M).

Theorem 3.8. Let β (r ∧ s) = β (r) ∩ β (s) for all r, s ∈ M and (X, 𝒞) be an M-fuzzifying convex structure. Let r ∈ β (⊤). Define an operator co_(r) : 2^X → M^X by co_(r) (A) = co (A) _(r) for every A ∈ 2^X. Then co_(r) is the hull operator of the convex structure (X, 𝒞^[r′]). Moreover, an M-fuzzifying convexity 𝒞 is of arity ≤n iff 𝒞^[r′] is of arity ≤n for each r ∈ β (⊤).

4 M-fuzzifying Join-Hull Commutativity

Definition 4.1. An M-fuzzifying convexity 𝒞 on X is called an M-fuzzifying Join-Hull Commutativity (briefly, M-fuzzifying JHC) convexity if $co ({a} \cup A) (z) = ⋁_{x \in X} co ({a, x}) (z) \land co (A) (x)$ for all a ∈ X and A ∈ 2^X ∖ {∅}.

If 𝒞 is an M-fuzzifying JHC convexity on X, then the pair (X, 𝒞) is an M-fuzzifying JHC convex structure. In this case, we also say 𝒞 and its hull operator have M-fuzzifying JHC property.

Theorem 4.2.The following conditions are equivalent.

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

(X, 𝒞_[r′]) is a JHC convex structure for each r ∈ P (M).

If β (r ∧ s) = β (r) ∩ β (s) for all r, s ∈ M, then the following equivalence can be added.

(X, 𝒞^[r′]) is a JHC convex structure for each r ∈ β (⊤).

Proof. The proof is direct and omitted.

Example 4.4. Let (V, μ) be a fuzzy vector space and (V, 𝒞_μ) be the fuzzifying convexity generated by μ [21]. Let $C_{a}$ be the standard convexity on μ_[a] for each a ∈ (0, 1] = J ([0, 1]). It is easy to check that $C_{a} = ⋂_{0 < b < a} C_{b}$ and $(𝒞_{μ})_{[a]} = C_{a}$ . Thus $(𝒞_{μ})_{[r^{'}]} = C_{r^{'}}$ is a JHC convexity for each r ∈ P (M). By Theorem 4.2, (V, 𝒞_μ) is an M-fuzzifying JHC convex structure.

Example 4.5. Let (V, μ) be a fuzzy vector space. Define a mapping ⇔1_μ : 2^X → [0, 1] by: $\forall A \in 2^{X}, \Leftrightarrow 1_{μ} (A) = ⋁ {a \in (0, 1] : A \in L_{a} \subseteq μ_{[a]}},$ where $L_{a}$ is the linear convexity on μ_[a]. Similar to Example 18 in [21], we see that (V, ⇔ 1_μ) is an M-fuzzifying convex structure. Similar to Example 4.4, we can prove that (V, ⇔ 1_μ) is an M-fuzzifying JHC convex structure.

Theorem 4.6. Let H : X × X × [0, 1] → X be anH-convex structure and define 𝒞_H : 2^X → [0, 1] by: $\forall A \in 2^{X}, 𝒞_{H} (A) = \underset{t \in (0, 1]}{\land} \underset{x, y \in A}{\land} (t \lor χ_{A}) (H (x, y, t)) .$ Then 𝒞_H is an [0, 1]-fuzzifying convexity. Moreover, a subset A ∈ 2^X is H-convex iff 𝒞_H (A) =1.

Proof. The proof is direct and omitted.

Remark 4.7. By the previous theorem, the set of all H-convex sets is (𝒞_H) _[1]. Thus, if 𝒞_H is an [0, 1]-fuzzifying JHC convexity, then (𝒞_H) _[1] is a JHC convexity. However, even if (𝒞_H) _[1] is a JHC convexity, 𝒞_H fails to be an [0, 1]-fuzzifying JHC convexity.

Example 4.8. Let X = {a, b, d, c, e, f}. Define a map H : X × X × [0, 1] → X by: (1), (2) and (3) below.

For all x, y ∈ X and $t \in [0, 1] ∖ {\frac{1}{4}, \frac{3}{4}}$ , $H (x, y, t) = {\begin{matrix} x, & t \in (\frac{1}{2}, \frac{3}{4}) \cup (\frac{3}{4}, 1], \\ y, & t \in [0, \frac{1}{4}) \cup (\frac{1}{4}, \frac{1}{2}), \\ x, & x = y, t \in {\frac{1}{2}, \frac{1}{4}, \frac{3}{4}}, \\ d, & x \neq y, t = \frac{1}{2} . \end{matrix}$

For all x, y ∈ X,

H (x, y, \frac{1}{4})

is given by the following table.

x∖y	a	b	c	d	e	f
a	a	a	c	a	c	b
b	a	b	d	b	b	b
c	c	d	c	c	c	c
d	a	b	c	d	d	d
e	c	b	c	d	e	e
f	b	b	c	d	e	f

$H (x, y, \frac{3}{4}) = H (y, x, \frac{1}{4})$ for all x, y ∈ X.

Then H is an H-convex structure on X and the set of all H-convex sets is a JHC convexity on X. However,

(𝒞_{H})_{[\frac{1}{2}]}

is not a JHC convexity, which shows that 𝒞_H is not an M-fuzzifying JHC convexity.

Theorem 4.9.Let (X, 𝒞) be an M-fuzzifying convex structure. Then 𝒞 is M-fuzzifying JHC iff for all a, z ∈ X and $F \in 2_{fin}^{X} ∖ {\emptyset}$ , $co ({a} \cup F) (z) = ⋁_{x \in X} co ({a, x}) (z) \land co (F) (x) .$ Proof. Necessity is clear. Sufficiency. Let a, z ∈ X and A ∈ 2^X. By (MDF) and (MRH3) of co, $\begin{matrix} ⋁_{x \in X} co ({a, x}) (z) \land co (A) (x) \\ = & ⋁_{x \in X} (co ({a, x}) (z) \land ⋁_{F \in 2_{fin}^{A}} co (F) (x)) \\ = & ⋁_{F \in 2_{fin}^{A}} ⋁_{x \in X} co ({a, x}) (z) \land co (F) (x) \\ \leq & ⋁_{F \in 2_{fin}^{A}} ⋁_{x \in X} co ({a, x}) (z) \land co ({a} \cup F) (x) \\ \leq & ⋁_{F \in 2_{fin}^{A}} co ({a} \cup F) (z) \leq co ({a} \cup A) (z) . \end{matrix}$

Conversely, by (MDF) of co, we have $\begin{matrix} co ({a} \cup A) (z) \\ = & ⋁_{F \in 2_{fin}^{{a} \cup A}} co (F) (z) = ⋁_{G ∖ {a} \in 2_{fin}^{A ∖ {a}}} co (G) (z) \\ \leq & ⋁_{G ∖ {a} \in 2_{fin}^{A ∖ {a}}} co ({a} \cup (G ∖ {a})) (z) \\ = & ⋁_{G ∖ {a} \in 2_{fin}^{A ∖ {a}}} ⋁_{x \in X} co ({a, x}) (z) \land co (G ∖ {a}) (x) \\ \leq & ⋁_{x \in X} co ({a, x}) (z) \land co (A ∖ {a}) (x) \\ = & ⋁_{x \in X} co ({a, x}) (z) \land co (A) (x) . \end{matrix}$

Theorem 4.10. An M-fuzzifying convex structure (X, 𝒞) is an M-fuzzifying JHC convex structure iff for all sets $F, G \in 2_{fin}^{X} ∖ {\emptyset}$ and each z ∈ X, $\begin{matrix} co (F \cup G) (z) \\ = & ⋁_{a, b \in X} co ({a, b}) (z) \land co (F) (a) \land co (G) (b) . \end{matrix}$

Proof. Let z ∈ X and r = ⋁ _a,b∈Xco ({a, b}) (z) ∧ co (F) (a) ∧ co (G) (b).

Necessity. For a, b ∈ X, by (MRH3) of co, $\begin{matrix} co ({a, b}) (z) \land co (F) (a) \land co (G) (b) \\ \leq & co ({a, b}) (z) \land ⋀_{w \in {a, b}} co (F \cup G) (w) \\ \leq & co (F \cup G) (z) . \end{matrix}$ Thus r ≤ co (F ∪ G) (z).

Conversely, let n = |F| and m = |G|. We prove co (F ∪ G) (z) ≤ r by induction on n + m.

(1) If n + m = 2 (i.e., n = m = 1), then it is easy to check that the desired inequality holds.

(2) n + m > 2. Assume that the inequality holds for values that <n + m. We have the following subcases.

(2a) n = 1. Let F = {c}. By M-fuzzifying JHC property and (MRH3) of co, we have $co (F \cup G) (z) = ⋁_{b \in X} co ({c, b}) (z) \land co (G) (b) \leq r .$ Thus the desired inequality holds in this case.

(2b) n > 1. Fix a point q ∈ F. By the induction hypothesis and M-fuzzifying JHC property and (MRH3), $\begin{matrix} co (F \cup G) (z) \\ = co ({q} \cup (G \cup F ∖ {q})) (z) \\ = ⋁_{y \in X} co ({q, y}) (z) \land co (G \cup F ∖ {q}) (y) \\ \leq ⋁_{y \in X} co ({q, y}) (z) \land (⋁_{\bar{a}, b \in X} co ({\bar{a}, b}) (y) \\ \land co (F ∖ {q}) (\bar{a}) \land co (G) (b)) \\ = ⋁_{\bar{a}, b \in X} (⋁_{y \in X} co ({q, y}) (z) \land co ({\bar{a}, b}) (y)) \\ \land co (F ∖ {q}) (\bar{a}) \land co (G) (b) \\ = ⋁_{\bar{a}, b \in X} co ({q, \bar{a}, b}) (z) \land co (F ∖ {q}) (\bar{a}) \land co (G) (b) \\ = ⋁_{\bar{a}, b \in X} ⋁_{a \in X} co ({b, a}) (z) \land co ({q, \bar{a}}) (a) \\ \land co (F ∖ {q}) (\bar{a}) \land co (G) (b) \\ = ⋁_{a, b \in X} (⋁_{\bar{a} \in X} co ({q . \bar{a}}) (a) \land co (F ∖ {q}) (\bar{a})) \\ \land co ({b, a}) (z) \land co (G) (b) \\ = ⋁_{a, b \in X} co ({b, a}) (z) \land co (F) (a) \land co (G) (b) . \end{matrix}$ Therefore the desired inequality holds for n > 1.

Sufficiency. Let a, z ∈ X and $F \in 2_{fin}^{X} ∖ {\emptyset}$ . By (MRH3), $\begin{matrix} co ({a} \cup F) (z) \\ = & ⋁_{x, y \in X} co ({x, y}) (z) \land co ({a}) (x) \land co (F) (y) \\ \leq & ⋁_{x, y \in X} co ({x, y}) (z) \land co ({a, y}) (x) \land co (F) (y) \\ \leq & ⋁_{y \in X} co ({a, y}) (z) \land co (F) (y) . \end{matrix}$ The inverse inclusion directly follows from (MRH3). Therefore the equality holds. By Theorem 4.9, 𝒞 has M-fuzzifying JHC property.

Theorem 4.11.An M-fuzzifying convex structure (X, 𝒞) is an M-fuzzifying JHC convex structure iff for ${F_{i}}_{i = 1}^{n} \subseteq 2_{fin}^{X} ∖ {\emptyset}$ and z ∈ X, $\begin{matrix} co (\cup_{i = 1}^{n} F_{i}) (z) \\ = & ⋁_{a_{1}, \cdot \cdot \cdot, a_{n} \in X} co ({a_{i}}_{i = 1}^{n}) (z) \land ⋀_{i = 1}^{n} co (F_{i}) (a_{i}) . \end{matrix}$ Proof. By Theorem 4.10, the result can be proved directly by induction on n.

Theorem 4.12. An M-fuzzifying convex structure (X, 𝒞) is an M-fuzzifying JHC convex structure iff for ${A_{i}}_{i = 1}^{n} \subseteq 2^{X} ∖ {\emptyset}$ and z ∈ X, $\begin{matrix} co (\cup_{i = 1}^{n} A_{i}) (z) \\ = & ⋁_{a_{1}, \cdot \cdot \cdot, a_{n} \in X} co ({a_{i}}_{i = 1}^{n}) (z) \land ⋀_{i = 1}^{n} co (A_{i}) (a_{i}) . \end{matrix}$

Proof. By Theorem 4.11 and (MDF) of co, the proof is direct.

Theorem 4.13. Let (X, 𝒞) and (Y, 𝒟) be M-fuzzifying convex structures. Then a function f : X → Y is an M-fuzzifying CP function iff co_𝒞 (f^-1 (B)) (x) ≤ co_𝒟 (B) (f (x)) for B ∈ 2^Y and x ∈ X.

Proof. The proof directly follows from definition.

Theorem 4.14. Let (X, 𝒞) and (Y, 𝒟) be M-fuzzifying convex structures. A surjective function f : X → Y is an M-fuzzifying CC function iff co_𝒞 (A) (x) ≥ co_𝒟 (f (A)) (f (x)) for A ∈ 2^X and x ∈ X.

Proof. The proof directly follows from definition.

Corollary 4.15. Let (X, 𝒞) and (Y, 𝒟) be M-fuzzifying convex structures and f : X → Y be a surjective function. Then the following conditions are equivalent

f is an M-fuzzifying CP and CC function.

co_𝒞 (f^-1 (B)) (x) = co_𝒟 (B) (f (x)) for B ∈ 2^Y and x ∈ X.

co_𝒞 (A) (x) = co_𝒟 (f (A)) (f (x)) for A ∈ 2^X and x ∈ X.

Theorem 4.16. Let (X, 𝒞) and (Y, 𝒟) be M-fuzzifying convex structures and f : X → Y be an M-fuzzifying CP and CC surjective function. If (X, 𝒞) is anM-fuzzifying JHC convex structure, then so is (Y, 𝒟).

Proof. The proof is direct and omitted.

Theorem 4.17. An M-fuzzifying JHC convex structure is of arity ≤2.

Proof. Let (X, 𝒞) be an M-fuzzifying JHC convex structure and A ∈ 2^X. By (MDF), we have $𝒞 (A) = ⋀_{z \notin A} (co (A) (z))^{'} = ⋀_{z \notin A} ⋀_{F \in 2_{fin}^{A}} (co (F) (z))^{'} .$ By Theorem 2.18, ℐ_𝒞 is an M-fuzzifying interval operator. Let 𝒟_co = 𝒞_{ℐ_𝒞}. That is, $\forall A \in 2^{X}, 𝒟_{co} (A) = ⋀_{z \notin A} ⋀_{x, y \in A} (co ({x, y}) (z))^{'} .$ Then 𝒟_co is an M-fuzzifying convexity of arity ≤2. Next, we prove 𝒞 (A) = 𝒟_co (A). Clearly, 𝒞 (A) ≤ 𝒟_co (A). In order to prove that 𝒞 (A) ≥ 𝒟_co (A), we prove that 𝒟_co (A) ≤ (co (F) (z)) ′ for all z ∉ A and $F \in 2_{fin}^{A}$ . We prove this by induction on n = |F|.

(1) If n ≤ 2, then the result is trivial.

(2) Let n > 2. Assume that the result holds for all values that <n. Fix a point p ∈ F. Since 𝒞 hasM-fuzzifying JHC property, we have $\begin{matrix} co (F) (z) & = & co ({p} \cup F ∖ {p}) (z) \\ = & ⋁_{a \in X} co ({p, a}) (z) \land co (F ∖ {p}) (a) . \end{matrix}$ Thus $\begin{matrix} (co (F) (z))^{'} \\ = ⋀_{a \in X} [(co ({p, a}) (z))^{'} \lor (co (F ∖ {p}) (a))^{'}] \end{matrix}$ $\begin{matrix} \geq & [⋀_{a \notin A} (co (F ∖ {p}) (a))^{'}] \lor [⋀_{a \in A} (co ({p, a}) (z))^{'}] . \end{matrix}$

If a ∉ A, then 𝒟_co (A) ≤ (co (F ∖ {p}) (a)) ′ by induction hypothesis. If a ∈ A, then p, a ∈ Aand 𝒟_co (A) ≤ (co ({p, a}) (z)) ′. Thus 𝒟_co (A) ≤ (co (F) (z)) ′ for each z ∉ A. Hence 𝒟_co (A) ≤ 𝒞 (A).

5 M-fuzzifying Peano property

Let (X, ℐ) be an M-fuzzifying interval space, U, V ∈ M^X and a, b, c ∈ X. Define three M-fuzzy sets [(ab) c] _ℐ, U ∣ _ℐ, (UV) _ℐ ∈ M^X by: for z ∈ X,

[(ab) c] _ℐ (z) = ⋁ _x∈X ℐ (a, b) (x) ∧ ℐ (c, x) (z);

U ∣ _ℐ (z) = ⋁ _a,b∈XU (a) ∧ U (b) ∧ ℐ (a, b) (z);

(UV) _ℐ (z) = ⋁ _a,b∈XU (a) ∧ V (b) ∧ ℐ (a, b) (z).

We won’t distinguish [(ab) c] _ℐ and [c (ab)] _ℐ. It is easy to check that U ≤ U ∣ _ℐ = (UU) _ℐ.

Definition 5.1. An M-fuzzifying interval space (X, ℐ) is called an M-fuzzifying Peano interval space, if for all a, b, c, y, z ∈ X, $ℐ (b, c) (y) \land ℐ (a, y) (z) \leq ⋁_{x \in X} ℐ (a, b) (x) \land ℐ (c, x) (z) .$

If (X, ℐ) is an M-fuzzifying Peano interval space, then we say ℐ has M-fuzzifying Peano property.

According to Theorem 5.8 below, segment operators of M-fuzzifying convexities defined in Example 4.3, 4.4 and 4.5 have M-fuzzifying Peano property.

Theorem 5.2.Let (X, ℐ) be an M-fuzzifying interval space. Then the following conditions are equivalent.

ℐ has M-fuzzifying Peano property.

(X, ℐ^(r)) is a Peano interval space for each r ∈ P (M), where ℐ^(r) : X × X → 2^X is defined by: ℐ^(r) (x, y) = ℐ (x, y) ^(r) = {z ∈ X : ℐ (x, y) (z) ≰ r} for all x, y ∈ X.

If β (t ∧ s) = β (t) ∩ β (s) for all t, s ∈ M, then the following equivalence can be added.

(X, ℐ_(r)) is a Peano interval space for each r ∈ β (⊤), where ℐ_(r) : X × X → 2^X is defined by: ℐ_(r) (x, y) = ℐ (x, y) _(r) = {z ∈ X : r ∈ β (ℐ (x, y) (z))} for all x, y ∈ X.

Proof. The proof is direct and omitted.

Theorem 5.3. An M-fuzzifying interval space (X, ℐ) is an M-fuzzifying Peano interval space if and only if [(ab) c] _ℐ = [a (bc)] _ℐ for all a, b, c ∈ X.

Proof. Necessity. Let a, b, c, z ∈ X. Since ℐ hasM-fuzzifying Peano property, we have $\begin{matrix} ℐ (b, c) (y) \land ℐ (a, y) (z) \\ \leq & ⋁_{x \in X} ℐ (a, b) (x) \land ℐ (c, x) (z) = [(ab) c]_{ℐ} (z) \end{matrix}$ for every y ∈ X. Thus $\begin{matrix} [a (bc)]_{ℐ} (z) & = & ⋁_{y \in X} ℐ (b, c) (y) \land ℐ (a, y) (z) \\ \leq & [(ab) c]_{ℐ} (z) . \end{matrix}$ Hence [a (bc)] _ℐ ≤ [(ab) c] _ℐ. Similarly, we have[(ab) c] _ℐ ≤ [a (bc)] _ℐ. Therefore [a (bc)] _ℐ = [(ab) c] _ℐ.

Sufficiency. Let a, b, c, y, z ∈ X. Then $\begin{matrix} ℐ (b, c) (y) \land ℐ (a, y) (z) \leq [a (bc)]_{ℐ} (z) \\ = & [(ab) c]_{ℐ} (z) = ⋁_{x \in X} ℐ (a, b) (x) \land ℐ (c, x) (z) . \end{matrix}$ Therefore ℐ has M-fuzzifying Peano property.

Lemma 5.4. Let (X, ℐ) be an M-fuzzifying interval space. Then for all U, V, W ∈ M^X and z ∈ X, $\begin{matrix} [(UV) W]_{ℐ} (z) \\ = & ⋁_{a, b, c \in X} [(ab) c]_{ℐ} (z) \land U (a) \land V (b) \land W (c) . \end{matrix}$

Proof. By definition, we have $\begin{matrix} [(UV) W]_{ℐ} (z) \\ = & ⋁_{x, c \in X} (⋁_{a, b \in X} U (a) \land V (b) \land ℐ (a, b) (x)) \\ \land W (c) \land ℐ (x, c) (z) \\ = & ⋁_{a, b, c \in X} (⋁_{x \in X} ℐ (a, b) (x) \land ℐ (x, c) (z)) \\ \land U (a) \land V (b) \land W (c) \end{matrix}$ $\begin{matrix} = & ⋁_{a, b, c \in X} [(ab) c]_{ℐ} (z) \land U (a) \land V (b) \land W (c) . \end{matrix}$ Therefore the result holds.

Theorem 5.5. An M-fuzzifying interval space (X, ℐ) is an M-fuzzifying Peano interval space if and only if [(UV) W] _ℐ = [U (VW)] _ℐ.

Proof. Necessity. By Theorem 5.3 and Lemma 5.4, $\begin{matrix} [(UV) W]_{ℐ} (z) \\ = & ⋁_{a, b, c \in X} [(ab) c]_{ℐ} (z) \land U (a) \land V (b) \land W (c) \\ = & ⋁_{a, b, c \in X} [a (bc)]_{ℐ} (z) \land U (a) \land V (b) \land W (c) \\ = & [U (VW)]_{ℐ} (z) . \end{matrix}$ Thus [(UV) W] _ℐ = [U (VW)] _ℐ.

Sufficiency. Let a, b, c ∈ X. Then a_⊤, b_⊤, c_⊤ ∈ M^X are M-fuzzy points. Moreover, we have $[(ab) c]_{ℐ} = [(a_{⊤} b_{⊤}) c_{⊤}]_{ℐ} = [a_{⊤} (b_{⊤} c_{⊤})]_{ℐ} = [a (bc)]_{ℐ} .$ Therefore ℐ has M-fuzzifying Peano property.

Lemma 5.6. Let (X, 𝒞) be an M-fuzzifying convex structure of arity ≤2. Then co (A) ∣ _co ≤ co (A) for every A ∈ 2^X.

Proof. Since 𝒞 is of arity ≤2, then 𝒞 (A) = ⋀ _x∉A ⋀ _a,b∈A (co ({a, b}) (x)) ′ for every A ∈ 2^X. Let z ∈ X. If z ∈ A, then co (A) (z) =⊤ and co (A) ∣ _co (z) ≤ co (A) (z). If z ∉ A, then $\begin{matrix} co (A) ∣_{co} (z) \\ = & ⋁_{a, b \in X} co (A) (a) \land co (A) (b) \land co ({a, b}) (z) \end{matrix}$ and $\begin{matrix} co (A) (z) & = & ⋀_{z \notin B \supseteq A} (𝒞 (B))^{'} \\ = & ⋀_{z \notin B \supseteq A} (⋁_{y \notin B} ⋁_{c, d \in B} co ({a, b}) (y))^{'} . \end{matrix}$ In order to prove co (A) ∣ _co (z) ≤ co (A) (z), we prove $co (A) (a) \land co (A) (b) \land co ({a, b}) (z) \leq (𝒞 (B))^{'}$ for all a, b ∈ X and B ∈ 2^X with z ∉ B ⊇ A.

In fact, if a ∉ B, then $(𝒞 (B))^{'} \geq ⋀_{a \notin D \supseteq A} (𝒞 (D))^{'} = co (A) (a) .$ Similarly, if b ∉ B, then (𝒞 (B)) ′ ≥ ⋀ _b∉D⊇A (𝒞 (D)) ′ = co (A) (b). If a, b ∈ B, then (𝒞 (B)) ′ ≥ co ({a, b}) (z).

Therefore co (A) ∣ _co (z) ≤ co (A) (z).

Lemma 5.7. Let (X, 𝒞) be an M-fuzzifying convex structure of arity ≤2 and U ∈ M^X with U ∣ _co ≤ U. If A ∈ 2^X with U (x) =⊤ for each x ∈ A, then co (A) ≤ U.

Proof. Let r ∈ P (M). By Corollary 3.7, 𝒞_[r′] is of arity ≤2. By Theorem 3.5, co^(r) is the hull operator of 𝒞_[r′]. Next, we prove that U^(r) ∈ 𝒞_[r′].

In fact, if a, b ∈ U^(r) and z ∈ co^(r) ({a, b}), then U ∣ _co (z) = U (a) ∧ U (b) ∧ co ({a, b}) (z) ≰ r. Thus U ∣ _co (z) ≰ r. Hence U (z) ≰ r and z ∈ U^(r). Therefore co^(r) ({a, b}) ⊆ U^(r) and U^(r) ∈ 𝒞_[r′].

Finally, we prove that co (A) ≤ U. Clearly, A ⊆ U^(r). Thus co (A) ^(r) = co^(r) (A) ⊆ co^(r) (U^(r)) = U^(r). By arbitrariness of r ∈ P (M), co (A) ≤ U.

Theorem 5.8. An M-fuzzifying convexity of arity ≤2 is an M-fuzzifying JHC convexity iff its segment operator has M-fuzzifying Peano property.

Proof. Necessity. Let (X, 𝒞) be an M-fuzzifying JHC convex structure and a, b, c, z ∈ X. Then $\begin{matrix} co ({a, b, c}) (z) & = & ⋁_{x \in X} co ({a, x}) (z) \land co ({b, c}) (x) \\ = & [a (bc)]_{co} (z) . \end{matrix}$ Similarly, co ({a, b, c}) (z) = [(ab) c] _co (z). Thus we have [a (bc)] _co = [(ab) c] _co.

Sufficiency. Let U = co ({a}) and V = co (F). For convenience, we write UV instead of (UV) _co. By Theorem 5.5 and Lemma 5.6, we have $\begin{matrix} (UV) (UV) & = & [U (V (UV))] = [U ((VU) V)] \\ = & [U ((UV) V)] = [(U (UV)) V] \\ = & [((UU) V) V] = (UU) (VV) \\ = & (U ∣_{co}) (V ∣_{co}) \leq UV . \end{matrix}$ This shows (UV) ∣ _co = (UV) (UV) ≤ UV. Moreover, it is clear that (UV) (y) =⊤ for each y ∈ {a} ∪ F. Thus, by Lemma 5.7, co ({a} ∪ F) ≤ (UV) _co.

Let z ∈ X. By M-fuzzifying Peano property and (MRH3) of co, we have $\begin{matrix} co ({a} \cup F) (z) \leq (UV)_{co} (z) \\ = & ⋁_{x, y \in X} co ({a}) (x) \land co (F) (y) \land co ({x, y}) (z) \\ \leq & ⋁_{y, g \in X} co ({a, y}) (g) \land co ({a, g}) (z) \land co (F) (y) \\ \leq & ⋁_{y \in X} co ({a, y}) (z) \land co (F) (y) \end{matrix}$ The inverse inclusion directly follows from (MRH3). Therefore 𝒞 has M-fuzzifying JHC property.

Theorem 5.9. Let (X, ℐ_X) and (Y, ℐ_Y) be M-fuzzifying interval spaces and f : X → Y be an M-fuzzifying II surjective function. If ℐ_X has M-fuzzifying Peano property, then so does ℐ_Y.

Proof. The proof is direct and omitted.

6 Conclusion

In this paper, it is showed that hull operators which correspond to convex structures formed by cut sets of a given M-fuzzifying convex structure equal to cut sets of the hull operator of that M-fuzzifying convex structure. With this result, the relations between theories of classic convex structures and their corresponding fuzzy theories can be discussed freely and directly. Two properties, namely, M-fuzzifying JHC property and M-fuzzifying Peano property are introduced. The former is preserved by M-fuzzifying CP and CC surjective functions and the latter by M-fuzzifying II surjective functions. Both properties have many characterizations. The segment operator of an M-fuzzifying convex structure of arity ≤2 has M-fuzzifying JHC property iff it has M-fuzzifying Peano property.

It is known that Pasch property, sand-glass property and modular property of interval spaces are as important as peano property. They are explicit tools to reveal the geometric properties of convex structures and are closely related to JHC property and “arity ≤2” [26]. Thus, methods and results in this paper are useful to help further investigations.

Footnotes

Acknowledgment

The authors sincerely thank Prof. Susana Montes for her tremendous help and the referees for their valuable suggestions. The first author thank Prof. F.G. Shi and his colleagues for their help and encouragement during his visiting study in Beijing Institute of Technology.

This work is supported by the National Natural Science Foundation of China (No. 11471202) and the Educational Commission Foundation of Hunan province (No. 15C0586).

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