Geometric properties of M -fuzzifying convex structures

Abstract

Four properties of M-fuzzifying interval spaces, namely, M-fuzzifying geometric property, M-fuzzifying modular property, M-fuzzifying Pasch property and M-fuzzifying sand-glass property, are introduced and characterized. Those properties together with M-fuzzifying Peano property and M-fuzzifying JHC property are closely related to each other. For example, an M-fuzzifying idempotent Peano-Pasch interval operator possesses M-fuzzifying geometric property and M-fuzzifying sand-glass property. In addition, in an M-fuzzifying modular space, M-fuzzifying Peano property, M-fuzzifying Pasch property, M-fuzzifying geometric property and M-fuzzifying sand-glass property combined with JHC property are equivalent.

1 Introduction

Convexity, being originally inspired by some elementary geometric problems in Euclidean spaces [1], has made great advances in both theory and application [5 , 31]. Nevertheless, the two directions always promote each other. For example, on one hand, several abstract convex structures were studied by Llinares for generalizing some results on the existence of continuous selections and fixed points [13], and geodesic convexity was introduced by Rapcsák for discussing some problems in nonlinear optimizations [22]; on the other hand, some classic results in Euclidean space, obtained by Helly [8], Carathéodory [3] and Radon [21], were later found capturing some combinational features of convexities [12]. Now, convexity has been developed into many mathematical structures such as graphic theories [4], posets [7], metric spaces [22], median algebras [16], lattices [30] and vector spaces [31]. In fact, the development is based on an important fact that all of these mathematical structures share some unified geometrical properties.

Interval operators derived from the above structures not only display the unified geometric properties, but also provide a natural and frequent method of describing or constructing convex structures [2 , 31]. Moreover, interval operators run through the theory of convex structures. For example, interval operators, equipped by one or more special properties of Peano property, Pasch property, sand-glass property and modular property, connect and enhance other properties of convex structures such as ‘arity ≤2’, JHC and separations, invariants and compactness, etc. [31].

Convexity has been extended into fuzzy settings by many means. Rosa introduced the notion of fuzzy convexities [23], which was further extended by Maruyama who introduced the notion of M-convexities [15]. In fact, an M-convexity is actually a crisp family of M-fuzzy sets satisfying certain set of axioms. However, from a totally different point of view, Shi introduced the notion of M-fuzzifying convexities, where each subset of the underling set can be regarded as a convex set to some degree [29]. In addition, an M-fuzzifying closure structure is an M-fuzzifying convex structure iff its closure operator is domain finite [29]. Recently, Pang and Shi introduce the notions of stratified M-convex structures, weakly induced M-convex structures and induced M-convex structures. They further discussed their relations in a category aspect [18]. Xiu defined the notion of M-fuzzifying interval spaces [35]. An M-fuzzifying convex structure is generated by an M-fuzzifying interval space iff it is of arity ≤2. Wu and Bai introduced the notions of M-fuzzifying JHC convexities and M-fuzzifying Peano interval operators. An M-fuzzifying convexity of arity ≤2 is an M-fuzzifying JHC convexity iff its segment operator has M-fuzzifying Peano property [34].

In this paper, we discuss geometric aspects of M-fuzzifying convex structures. Specifically, we introduce M-fuzzifying geometric property and M-fuzzifying modular property of M-fuzzifying interval spaces, and obtain some alternative conditions of the former one. Also, we introduce M-fuzzifying Pasch property and M-fuzzifying sand-galss property of M-fuzzifying interval spaces. We find that (especially, in M-fuzzifying modular spaces) these properties together with M-fuzzifying JHC property and M-fuzzifying Peano property are closely related to each other.

2 Preliminaries

Throughout, X and Y are nonempty sets. The power set of X is denoted by 2^X. The set of all finite subsets of X is denoted by $2_{fin}^{X}$ . Also, we denote $2_{2}^{X} = {{x, y} : x, y \in X}$ . For a subset A of X, the notions 2^A and $2_{fin}^{A}$ are defined analogically.

M is a completely distributive lattice with an inverse inclusion operator ^′. The minimal element and the maximal element in M are denoted by ⊥ and ⊤. An element a ∈ M is called a prime if for all b, c ∈ M, b ∧ c ≤ a implies b ≤ a or c ≤ a. The set of all primes in M ∖ {⊤} is denoted by P (M). An element a ∈ M is called a co-prime if its complement a′ ∈ P (M). The set of all co-primes is denoted by J (M). For each a ∈ M, there exist φ ⊆ P (M) and ψ ∈ J (M) such that a = ⋀ φ = ⋁ ψ [32, 33]. Clearly, for all p, q ∈ M, p ≤ q iff p ≰ r implies q ≰ r for each r ∈ P (M).

A binary relation ≺ on M is defined as for all a, b ∈ M, a ≺ b iff for each φ ⊆ M, b ≤ ⋁ φ always implies the existence of d ∈ φ such that a ≤ d. The mapping β : Mr2^M, defined as β (a) = {b : b ≺ a} for all a ∈ M, satisfies β (⋁ _i∈Ωa_i) = ⋃ _i∈Ωβ (a_i) for all {a_i} _i∈Ω ⊆ M. The opposite relation ≺^op of ≺ is defined as for all a, b ∈ M, a ≺ ^opb iff b′ ≺ a′. The mapping α : Mr2^M, defined as α (a) = {b : a ≺ ^opb} for all a ∈ M, satisfies α (⋀ _i∈Ωa_i) = ⋃ _i∈Ωα (a_i) for all {a_i} _i∈Ω ⊆ M. Clearly, β (⊥) = α (⊤) = ∅ and a = ⋁ β (a) = ⋀ α (a) for all a ∈ M [29, 32]. For convenience, we denote β^* (a) = β (a) ∩ J (M) and α^* (a) = α (a) ∩ P (M) for all a ∈ M.

A mapping U : XrM is called an M-fuzzy set on X. The set of all M-fuzzy sets on X is denoted by M^X. For all x ∈ X and r ∈ M, the M-fuzzy set (or, an M-fuzzy point) x_r ∈ M^X is defined as x_r (y) = r for y = x and x_r (y) =⊥ for all y ≠ x. For U ∈ M^X and r ∈ M, Shi defined the following cut sets [24]:

U_[r] = {x ∈ X : U (x) ≥ r};

U^(r) = {x ∈ X : U (x) ≰ r}.

A function $f_{M}^{\to} : M^{X} \to M^{Y}$ is called an M-fuzzy function, if there exists a function f : XrY such that $f_{M}^{\to} (U) (y) = ⋁ {U (x) : f (x) = y}$ for all U ∈ M^X and y ∈ Y. Its inverse function $f_{M}^{ł} : M^{Y} \to M^{X}$ is defined as $f_{M}^{ł} (V) (x) = V (f (x))$ for all V ∈ M^Y and x ∈ X [25 , 32].

Notions and results of convex structures mentioned in the sequel can be seen in [31]. The following are those of M-fuzzifying convex structures.

Definition 2.1. [29] A mapping 𝒞 : 2^XrM is called an M-fuzzifying convexity and (X, 𝒞) is called an M-fuzzifying convex structure, if 𝒞 satisfies

𝒞 (∅) = 𝒞 (X) = ⊤;

If {U_i} _i∈Ω ⊆ 2^X, then 𝒞 (⋂ _i∈ΩU_i) ≥ ⋀ _i∈Ω𝒞 (U_i);

If {U_k} _i∈Ω ⊆ 2^X is totally ordered byinclusion, then 𝒞 (⋃ _i∈ΩU_i) ≥ ⋀ _i∈Ω𝒞 (U_i).

Theorem 2.2. [29] The hull operator co_𝒞 : 2^XrM^X (briefly, co) of (X, 𝒞) is defined as: $\begin{matrix} \forall A \in 2^{X}, \forall x \in X, co (A) (x) \\ = ⋀_{x \notin B \supseteq A} (𝒞 (B))^{'} . \end{matrix}$ Then for all A ∈ 2^X and x ∈ X,

co (∅) (x) = ⊥;

co (A) (x) =⊤ whenever x ∈ A;

co (A) (x) = ⋀ _x∉B⊇A ⋁ _y∉Bco (B) (y);

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

Conversely, let co : 2^XrM^X satisfy (MCO1)-(MCO3) and (MFD). Define 𝒞_co : 2^XrM as:

\begin{matrix} \forall A \in 2^{X}, 𝒞_{co} (A) \\ = ⋀_{x \notin A} (co (A) (x))^{'} . \end{matrix}

Then 𝒞_co is an M-fuzzifying convexity with co_{𝒞_co} = co. In addition, if (X, 𝒞) is an M-fuzzifying convex structure, then 𝒞_{co
_𝒞} = 𝒞. The restriction co^seg of co on $2_{2}^{X}$ is called the segment operator of (X, 𝒞). For convenience, co^seg is still denoted by co.

Definition 2.3. [34, 35] An M-fuzzifying convex structure (X, 𝒞) is said to be of arity ≤n, if $\begin{matrix} \forall A \in 2^{X}, 𝒞 (A) \\ = ⋀_{x \notin A} ⋀_{F \in 2_{fin}^{A}, | F | \leq n} (co (F) (x))^{'} . \end{matrix}$

Definition 2.4. [28] An operator $ℋ : 2_{fin}^{X} \to M^{X}$ is called an M-fuzzifying restricted hull operator and (X, ℋ) is called an M-fuzzifying restricted hull space, if for all $F, G \in 2_{fin}^{X}$ and x ∈ X,

ℋ (∅) (x) = ⊥;

ℋ (F) (x) =⊤ whenever x ∈ F;

ℋ (G) (x) ∧ ⋀ _y∈Gℋ (F) (y) ≤ ℋ (F) (x).

Theorem 2.5. [28] If (X, 𝒞) is an M-fuzzifying convex structure, then the restriction co_fin of co on $2_{fin}^{X}$ is an M-fuzzifying restricted hull operator, which is still denoted by co. Conversely, let (X, ℋ) be an M-fuzzifying restricted hull space and define co_ℋ : 2^XrM^X as: $\begin{matrix} \forall A \in 2^{X}, \forall x \in X, {co}_{ℋ} (A) (x) \\ = ⋁_{F \in 2_{fin}^{A}} ℋ (F) (x) . \end{matrix}$ Then co_ℋ is the hull operator of an M-fuzzifying convexity 𝒞_ℋ satisfying (co_ℋ) _fin = ℋ.

Definition 2.6. [34] An M-fuzzifying convex structure (X, 𝒞) is called an M-fuzzifying JHC convex structure (i.e., 𝒞 has M-fuzzifying JHC property), if for all a, z ∈ X and $A \in 2^{X} \underline{{} \emptyset}$ , $co (A) (z) = ⋁_{x \in X} co ({a, x}) (z) \land co (A) (x) .$

Definition 2.7. [34, 35] An operator ℐ : X × XrM^X is called an M-fuzzifying interval operator and (X, ℐ) is called an M-fuzzifying interval space, if for all x, y ∈ X,

ℐ (x, y) (x) = ℐ (x, y) (y) =⊤;

ℐ (x, y) = ℐ (y, x).

Let (X, ℐ) be an M-fuzzifying interval space and r ∈ P (M). Define an operator ℐ^(r) : X × Xr2^X by ℐ^(r) (a, b) = (ℐ (a, b)) ^(r) for all a, b ∈ X. Then (X, ℐ^(r)) is an interval space [34, 35].

Let (X, ℐ_X) and (Y, ℐ_Y) be M-fuzzifying interval spaces. An M-fuzzy function $f_{M}^{\to} : M^{X} \to M^{Y}$ 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 [34, 35].

Theorem 2.8. [34, 35] Let (X, 𝒞) be an M-fuzzifying convex structure. Define ℐ_𝒞 : X × XrM^X as $\forall x, y, z \in X, ℐ_{𝒞} (x, y) (z) = co ({x, y}) (z) .$ Then ℐ_𝒞 is an M-fuzzifying interval operator generated by 𝒞.

Conversely, let (X, ℐ) be an M-fuzzifying interval space and define 𝒞_ℐ : 2^XrM as $\forall A \in 2^{X}, 𝒞_{ℐ} (A) = ⋀_{z \notin A} ⋀_{x, y \in A} (ℐ (x, y) (z))^{'} .$ Then 𝒞_ℐ is an M-fuzzifying convexity generated by ℐ. In addition, $ℐ \leq {co}_{𝒞_{ℐ}}^{seg}$ .

In view of the above theorem, we simply regard the segment operator of an M-fuzzifying convex structure as an M-fuzzifying interval operator.

Theorem 2.9. [34, 35] An M-fuzzifying convexity is generated by an M-fuzzifying interval operator iff it is of arity ≤2.

Definition 2.10. [34] An M-fuzzifying interval space (X, ℐ) is called an M-fuzzifying Peano interval space (i.e., ℐ is an M-fuzzifying Peano interval operator and ℐ has M-fuzzifying Peano property), if for all a, b, c, y, z ∈ X, $\begin{matrix} ℐ (b, c) (y) \land ℐ (a, y) (z) \\ \leq ⋁_{x \in X} ℐ (a, b) (x) \land ℐ (c, x) (z) . \end{matrix}$

Definition 2.11. [34] Let (X, ℐ) be an M-fuzzifying interval space, a, b, c ∈ X and U, V ∈ M^X. Define [a (bc)] _ℐ, (UV) _ℐ, (U/V) _ℐ ∈ M^X respectively as: for all z ∈ X,

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

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

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

In particular, we denote U ∣ _ℐ for (UU) _ℐ.

Theorem 2.12. [34] An M-fuzzifying interval space (X, ℐ) is an M-fuzzifying Peano interval space iff [a (bc)] _ℐ = [(ab) c] _ℐ for all a, b, c ∈ X iff [U (VW)] _ℐ = [(UV) W] _ℐ for all U, V, W ∈ M^X.

Theorem 2.13. [34] An M-fuzzifying JHC convex structure (X, 𝒞) is of arity ≤2. In addition, an M-fuzzifying convex structure of arity ≤2 is an M-fuzzifying JHC convex structure iff its segment operator has M-fuzzifying Peano property.

Theorem 2.14. [34] Let (X, 𝒞) be an M-fuzzifying convex structure and r ∈ P (M). Define co^(r) : 2^Xr2^X as: co^(r) (A) = co (A) ^(r) for all A ∈ 2^X. Then co^(r) is the hull operator of (X, 𝒞_[r′]).

Definition 2.15. Let (V, μ) be a fuzzy vector space over a totally ordered field $𝕂$ [10].

(1) The mapping 𝒞_μ : 2^Vr [0, 1], defined as: 𝒞_μ (A) = ⋁ {a ∈ (0, 1] : A ∈ 𝒞_a} for all A ∈ 2^X, where 𝒞_as2^{μ
_[a]} is the standard convexity on μ_[a] for each a ∈ (0, 1], is an fuzzifying JHC convexity [34].

(2) Let $V_{0} = V \underline{{} 0}$ and μ₀ is the restriction of μ on V₀. The mapping ℒ_{μ
₀} : 2^{V
₀}r [0, 1], defined as: ℒ_{μ
₀} (A) = ⋁ {a ∈ (0, 1] : A ∈ ℒ_a} for all A ∈ 2^X, where ℒ_as2^{μ
_0[a]} is the linear convexity on μ_0[a] for each a ∈ (0, 1], is a fuzzifying JHC convexity [34].

Definition 2.16. [26] $(X, D)$ is called an M-fuzzifying pseudo-metric space, where [0, + ∞) (M) be the set of all M-fuzzy non-negative real numbers and $D : X \times X \to [0, + \infty) (M)$ , if for all x, y, z ∈ X and s, t > 0,

x = y implies $D (x, y) (0 +) = ⊥$ ;

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

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

The operator

{cl}_{D} : 2^{X} \to M^{X}

(X, D)

, defined as:

{cl}_{D} (A) (z) = ⋀_{r > 0} ⋁_{c \in A} (D (z, c) (r))^{'}

for all A ∈ 2^X and z ∈ X, is the M-fuzzifying closure operator of an M-fuzzifying topology [36]. Particularly, if the equality in (MD2) holds, then

(X, D)

is called an M-fuzzifying strong pseudo-metric space.

Definition 2.17. [37] A mapping e : X × XrM is called an M-fuzzy partial order and (X, e) is called an M-fuzzy partial order space, if for all x, y, z ∈ X,

e (x, x) =⊤ for all x ∈ X;

e (x, z) ∧ e (z, y) ≤ e (x, y);

e (x, y)∧ e (y, x) = ⊥ whenever x ≠ y.

3 M-fuzzifying geometric interval spaces

Definition 3.1. An M-fuzzifying interval space (X, ℐ) is called

An M-fuzzifying geometric interval space (i.e., ℐ is an M-fuzzifying geometric interval operator and ℐ has M-fuzzifying geometric property), if for all a, b, c, d ∈ X,

ℐ (a, a) = a_⊤;

ℐ (a, b) (c) ∧ ℐ (a, c) (d) ≤ ℐ (a, b) (d);

ℐ (a, b) (c) ∧ ℐ (a, b) (d) ∧ ℐ (a, d) (c) ≤ ℐ (c, b) (d).

An M-fuzzifying modular space, if ℐ is an M-fuzzifying geometric operator and ⋁_x∈Xℳ (a, b, c) (x) =⊤ for all a, b, c ∈ X, where the operator ℳ : X³rM^X, defined as: ℳ (a, b, c) = ℐ (a, b) ∧ ℐ (b, c) ∧ ℐ (c, a) for all a, b, c ∈ X, is called the M-fuzzifying multimedian operator.

An M-fuzzifying interval space is called an M-fuzzifying idempotent interval space, if (MGI1) holds.

Example 3.2. Let $(X, D)$ be an M-fuzzifying pseudo-metric space satisfying $D (c, a) (0 +) = ⊤$ for all c ≠ a. Define an operator $ℐ_{D} : X \times X \to M^{X}$ as: $\begin{matrix} \forall a, b, c \in X, ℐ_{D} (a, b) (c) \\ = {cl}_{D} ({a, b}) (c) . \end{matrix}$ Then $ℐ_{D}$ is an M-fuzzifying geometric interval operator. Moreover, if $(X, D)$ is strong, then $(X, ℐ_{D})$ is an M-fuzzifying modular space.

Example 3.3. Let (V, μ) be a fuzzy vector space over a totally ordered field $𝕂$ . Then $(V, {co}_{𝒞_{μ}}^{seg})$ and $(V, {co}_{ℒ_{μ_{0}}}^{seg})$ are M-fuzzifying modular operators.

Example 3.4. Let (X, e) be an M-fuzzy partial order space and ℐ_e : X × XrM^X be an operator defined as: for all a, b, x ∈ X, $ℐ_{e} (a, b) (x) = [e (a, x) \lor e (b, x)] \land [e (x, a) \lor e (x, b)] .$ Then (X, ℐ_e) is M-fuzzifying geometric.

Theorem 3.5. An M-fuzzifying idempotent interval space (X, ℐ) is an M-fuzzifying geometric interval space iff it satisfies (MGI2)’ and (MGI3)’ below.

ℐ (a, b) (c) ∧ ℐ (b, c) (d) ≤ ℐ (a, b) (d).

ℐ (a, b) (c) ∧ ℐ (b, c) (d) ≤ ℐ (a, d) (c).

Proof. Necessity. Let a, b, c, d ∈ X. (MGI2)’. Exchange a, b in (MGI2). By (MI2), we have $\begin{matrix} ℐ (a, b) (c) \land ℐ (b, c) (d) & = & ℐ (b, a) (c) \land ℐ (b, c) (d) \\ \leq & ℐ (b, a) (d) = ℐ (a, b) (d) . \end{matrix}$

(MGI3)’. Exchange a, b and c, d in (MGI3). By (MGI3) and (MGI2)’, we have $\begin{matrix} ℐ (a, b) (c) \land ℐ (b, c) (d) \\ = ℐ (a, b) (d) \land ℐ (a, b) (c) \land ℐ (b, c) (d) \\ \leq ℐ (a, d) (c) . \end{matrix}$

Sufficiency. We can easily obtain (MGI2) by exchanging a and b in (MGI2)’.

(MGI3). Exchange a, b and c, d in (MGI3)’. We have $\begin{matrix} ℐ (a, b) (c) \land ℐ (a, b) (d) \land ℐ (a, d) (c) \\ \leq ℐ (a, b) (d) \land ℐ (a, d) (c) \leq ℐ (b, c) (d) . \end{matrix}$

Theorem 3.6. An M-fuzzifying idempotent interval space (X, ℐ) is an M-fuzzifying geometric interval space iff for all a, b, c, d ∈ X,

ℐ (a, b) (c) ∧ ℐ (b, c) (d) ≤ ℐ (a, b) (d) ∧ ℐ (a, d) (c).

Proof. It follows directly from Theorem 3.5.

Theorem 3.7. An M-fuzzifying interval space (X, ℐ) is an M-fuzzifying geometric interval space iff it satisfies (MGI2)’, (MGI3)’ and (MGI1)’ below.

ℐ (a, b) (c) ∧ ℐ (b, c) (d) ∧ ℐ (a, c) (d) ≤ c_⊤ (d) for all a, b, c, d ∈ X.

Proof. Let a, b, c, d ∈ X. by (MGI3)’, we have $\begin{matrix} [ℐ (a, b) (c) \land ℐ (b, c) (d)] \land ℐ (a, c) (d) \\ \leq ℐ (c, a) (d) \land ℐ (a, d) (c) \\ \leq ℐ (c, c) (d) = c_{⊤} (d) . \end{matrix}$ Thus (MGI1)’ holds. Conversely, if a, c ∈ X, then ℐ (a, a) (c) = ℐ (a, a) (c) ∧ ℐ (a, c) (a) ≤ c_⊤ (a), which shows ℐ (a, a) ≤ a_⊤. Hence ℐ (a, a) = a_⊤.

Lemma 3.8. Let (X, ℐ) be an M-fuzzifying geometric interval space and r ∈ P (M). Then (X, ℐ^(r)) is a geometric interval space with ℳ^(r) as its multimedian operator, where ℳ^(r) : X × X × Xr2^X is defined as: ℳ^(r) (a, b, c) = ℐ^(r) (a, b) ∩ ℐ^(r) (b, c) ∩ ℐ^(r) (a, c) for all a, b, c ∈ X. In addition, we have c ∈ ℐ^(r) (a, b) iff ℳ^(r) (a, b, c) = {c}.

Theorem 3.9. Let (X, ℐ_i) (i = 1, 2) be M-fuzzifying modular spaces with ℳ_i as their M-fuzzifying multimedian operators. If for all a, b, c ∈ X, $\begin{matrix} ⋁_{z \in X} ℳ_{1} (a, b, c) (z) \land ℳ_{2} (a, b, c) (z) & = & ⊤, \end{matrix}$ then ℐ₁ = ℐ₂ and ℳ₁ = ℳ₂.

Corollary 3.10. Let (X, ℐ₁) be an M-fuzzifying modular space and ℐ₂ be an M-fuzzifying geometric interval operator on X with ℐ₂ ≥ ℐ₁. Then ℐ₁ = ℐ₂.

Theorem 3.11. Let (X, ℐ_X) and (Y, ℐ_Y) be M-fuzzifying interval spaces and $f_{M}^{\to} : M^{X} \to M^{Y}$ be an M-fuzzy II-surjective function. If (X, ℐ_X) is an M-fuzzifying geometric (resp. M-fuzzifying modular) space, then so is (Y, ℐ_Y).

4 M-fuzzifying Pasch interval spaces and M-fuzzifying sand-glass interval spaces

Definition 4.1. An M-fuzzifying interval space (X, ℐ) is called

an M-fuzzifying Pasch interval space, if for all p, a, b, $\bar{a}$ , $\bar{b}$ ∈ X, $\begin{matrix} ℐ (p, a) (\bar{a}) \land ℐ (p, b) (\bar{b}) \\ \leq ⋁_{z \in X} ℐ (a, \bar{b}) (z) \land ℐ (\bar{a}, b) (z); \end{matrix}$

an M-fuzzifying sand-glass interval space, if for all a, b, c, d, p, v ∈ X, $\begin{matrix} ℐ (a, d) (p) \land ℐ (b, c) (p) \land ℐ (a, b) (v) \\ \leq ⋁_{w \in X} ℐ (c, d) (w) \land ℐ (w, v) (p) . \end{matrix}$

If (X, ℐ) is an M-fuzzifying Pasch (resp. M-fuzzifying sand-glass) interval space, then we say ℐ has M-fuzzifying Pasch (resp. M-fuzzifying sand-glass) property.

Example 4.2. Let (V, μ) be a fuzzy vector space over a totally ordered field $𝕂$ . Then $(V, {co}_{𝒞_{μ}}^{seg})$ and $(V, {co}_{ℒ_{μ_{0}}}^{seg})$ are M-fuzzifying Peano-Pasch interval spaces.

Example 4.3. Let (X, e) be an M-fuzzy partial order space. Then (X, ℐ_e) is an M-fuzzifying Peano interval space and an M-fuzzifying sand-glass interval space. In addition, it is follows from Theorem 4.14 that if (X, ℐ_e) is an M-fuzzifying modular interval space, then ℐ_e has M-fuzzifying Pasch property.

Theorem 4.4. Let (X, ℐ) be an M-fuzzifying idempotent interval space.

If ℐ has M-fuzzifying Peano property, then ℐ satisfies (MGI2).

If ℐ has M-fuzzifying Pasch property, then ℐ satisfies (MGI3).

If ℐ has M-fuzzifying Peano property, then ℐ (a, b) ∣ _ℐ ≤ ℐ (a, b) for all a, b ∈ X.

Proof. Let a, b, c, d, w ∈ X.

(1) By (MGI1) and M-fuzzifying Peano property, $\begin{matrix} ℐ (a, b) (c) \land ℐ (a, c) (d) \\ \leq ⋁_{x \in X} ℐ (a, a) (x) \land ℐ (b, x) (d) = ℐ (a, b) (d) . \end{matrix}$

(2) By (MGI1) and M-fuzzifying Pasch property, $\begin{matrix} ℐ (b, c) (d) & = & ⋁_{x \in X} ℐ (b, c) (x) \land ℐ (d, d) (x) \\ \geq & ℐ (a, b) (d) \land ℐ (a, d) (c) \\ \geq & ℐ (a, b) (c) \land ℐ (a, b) (d) \land ℐ (a, d) (c) . \end{matrix}$

(3) By M-fuzzifying Peano property and (MGI2), $\begin{matrix} ℐ (a, b) ∣_{ℐ} (w) \\ = ⋁_{u, v \in X} ℐ (a, b) (u) \land [ℐ (a, b) (v) \land ℐ (u, v) (w)] \\ \leq ⋁_{u, x \in X} ℐ (a, x) (w) \land ℐ (u, b) (x) \land ℐ (a, b) (u) \\ \leq ⋁_{x \in X} ℐ (a, x) (w) \land ℐ (a, b) (x) \leq ℐ (a, b) (w) . \end{matrix}$ Therefore ℐ (a, b) ∣ _ℐ ≤ ℐ (a, b).

Corollary 4.5. An M-fuzzifying idempotent Peano-Pasch interval space is an M-fuzzifying geometric interval space.

Theorem 4.6. If (X, ℐ) is an M-fuzzifying idempotent Peano interval space, then co = ℐ, where co is the segment operator of 𝒞_ℐ.

Proof. By Theorem 2.8, ℐ ≤ co. Conversely, let a, b, z ∈ X. If z ∈ {a, b}, then co ({a, b}) (z) = ⊤ = ℐ (a, b) (z). If z ∉ {a, b}, then by (MGI1) and (3) of Theorem 4.4, we have $\begin{matrix} co ({a, b}) (z) \\ \leq (𝒞_{ℐ} ({a, b}))^{'} = ⋁_{y \notin {a, b}} ⋁_{c, d \in {a, b}} ℐ (c, d) (y) \\ = ⋁_{y \notin {a, b}} ℐ (a, a) (y) \lor ℐ (b, b) (y) \lor ℐ (a, b) (y) \\ = ⋁_{y \notin {a, b}} ℐ (a, b) (y) \leq ℐ (a, b) ∣_{ℐ} (z) \\ \leq ℐ (a, b) (z) . \end{matrix}$ Therefore co = ℐ.

In [31] (Section 4, Chapter 1), Vel concluded that a Peano-Pasch interval operator equals to the segment operator of its generated convexity. In fact, if the interval operator is not idempotent, this conclusion fails. we have the following example.

Example 4.7. Let X = {a, b, c}, M = {⊥ , ⊤} and ℐ : X × XrM^X be defined as: $\begin{matrix} ℐ (a, a) & = & ℐ (c, c) = a_{⊤} \lor c_{⊤}; ℐ (b, b) = b_{⊤}; \\ ℐ (a, c) & = & ℐ (c, a) = a_{⊤} \lor c_{⊤}; \\ ℐ (a, b) & = & ℐ (b, a) = a_{⊤} \lor b_{⊤}; \\ ℐ (b, c) & = & ℐ (c, b) = b_{⊤} \lor c_{⊤} . \end{matrix}$ Then ℐ is an M-fuzzifying Peano-Pasch operator, whereas it not satisfies (MGI1). In addition, co ({a, b}) (c) = ⊤ ≠ ⊥ = ℐ (a, b) (c). Thus co ≠ ℐ.

Theorem 4.8. The segment operator co of an M-fuzzifying convex structure (X, 𝒞) has M-fuzzifying Pasch property iff $\begin{matrix} (co ({a, b}) / p_{⊤})_{co} ∣_{co} \leq (co ({a, b}) / p_{⊤})_{co} \end{matrix}$ for all a, b, p ∈ X.

Proof. Necessity. Let a, b, p, z ∈ X. We have $\begin{matrix} (co ({a, b}) / p_{⊤})_{co} ∣_{co} (z) \\ = ⋁_{c, d \in X} [⋁_{\bar{c} \in X} co ({a, b}) (\bar{c}) \land co ({p, c}) (\bar{c})] \land \\ [⋁_{\bar{d} \in X} co ({a, b}) (\bar{d}) \land co ({p, d}) (\bar{d})] \land co ({c, d}) (z) \\ = ⋁_{c, d, \bar{c}, \bar{d} \in X} [co ({p, c}) (\bar{c}) \land co ({c, d}) (z)] \land \\ co ({p, d}) (\bar{d}) \land co ({a . b}) (\bar{c}) \land co ({a, b}) (\bar{d}) \\ \leq ⋁_{d, \bar{c}, \bar{d}, g \in X} co ({p, z}) (g) \land [co ({\bar{c}, d}) (g) \land \\ co ({p, d}) (\bar{d})] \land co ({a . b}) (\bar{c}) \land co ({a, b}) (\bar{d}) \\ \leq ⋁_{\bar{c}, \bar{d}, g, h \in X} co ({p, z}) (g) \land co ({p, g}) (h) \land \\ co ({\bar{c}, \bar{d}}) (h) \land co ({a . b}) (\bar{c}) \land co ({a, b}) (\bar{d}) \\ = ⋁_{\bar{c}, \bar{d}, g, h \in X} [co ({p, z}) (g) \land co ({p, g}) (h)] \land \\ [co ({\bar{c}, \bar{d}}) (h) \land co ({a . b}) (\bar{c}) \land co ({a, b}) (\bar{d})] \\ \leq ⋁_{g, h \in X} co ({p, z}) (h) \land co ({a, b}) (h) \\ = (co ({a, b}) / p_{⊤})_{co} (z) . \end{matrix}$

Therefore (co ({a, b})/p_⊤) _co ∣ _co ≤ (co ({a, b})/p_⊤) _co.

Sufficiency. Let a, b, p, $\bar{a}$ , $\bar{b}$ ∈ X, and let U = (co ({ $\bar{a}$ , b})/a_⊤) _co. Then U ∣ _co ≤ U. Thus $\begin{matrix} U (b) & = & (co ({\bar{a}, b}) / a_{⊤})_{co} (b) \\ = & ⋁_{x \in X} co ({\bar{a}, b}) (x) \land co ({b, a}) (x) \\ \geq & co ({\bar{a}, b}) (b) \land co ({b, a}) (b) = ⊤; \end{matrix}$ $\begin{matrix} U (p) & = & (co ({\bar{a}, b}) / a_{⊤})_{co} (p) \\ = & ⋁_{y \in X} co ({\bar{a}, b}) (y) \land co ({p, a}) (y) \\ \geq & co ({p, a}) (\bar{a}) . \end{matrix}$ Hence $\begin{matrix} co ({p, a}) (\bar{a}) \land co ({p, b}) (\bar{b}) \\ \leq U (p) \land U (b) \land co ({b, p}) (\bar{b}) \\ \leq U ∣_{co} (\bar{b}) \leq U (\bar{b}) = (co ({\bar{a}, b}) / a_{⊤})_{co} (\bar{b}) \\ = ⋁_{z \in X} co ({\bar{a}, b}) (z) \land co ({a, \bar{b}}) (z) . \end{matrix}$ Therefore co has M-fuzzifying Pasch property.

Theorem 4.9. The segment operator co of an M-fuzzifying JHC convex structure (X, 𝒞) has M-fuzzifying sand-glass property iff $\begin{matrix} (p_{⊤} / co (F))_{co} ∣_{co} \leq (p_{⊤} / co (F))_{co} \end{matrix}$ for all p ∈ X and $F \in 2_{fin}^{X} \underline{{} \emptyset}$ .

Proof. Necessity. Let $F \in 2_{fin}^{X} \underline{{} \emptyset}$ and p, v ∈ X and U = (p_⊤/co (F)) _co. Then, by (MRH3) and M-fuzzifying sand-glass property, we have $\begin{matrix} U ∣_{co} (v) \\ = ⋁_{a, b \in X} [⋁_{d \in X} co ({a, d}) (p) \land co (F) (d)] \land \\ [⋁_{c \in X} co ({b, c}) (p) \land co (F) (c)] \land co ({a, b}) (v) \\ = ⋁_{a, b \in X} [⋁_{c, d \in X} co ({a, d}) (p) \land co ({b, c}) (p) \land \\ co ({a, b}) (v)] \land co (F) (c) \land co (F) (d) \\ \leq ⋁_{c, d \in X} [⋁_{w \in X} co ({c, d}) (w) \land co ({v, w}) (p)] \land \\ co (F) (c) \land co (F) (d) \\ = ⋁_{w \in X} [⋁_{c, d \in X} co ({c, d}) (w) \land co (F) (c) \land co (F) (d)] \\ \land co ({v, w}) (p) \\ \leq ⋁_{w \in X} co (F) (w) \land co ({v, w}) (p) = U (v) . \end{matrix}$ Therefore U ∣ _co ≤ U_co.

Sufficiency. Let a, b, c, d, p, v ∈ X. Since co has M-fuzzifying JHC property, $\begin{matrix} co ({a, d}) (p) & \leq & co ({a, c, d}) (p) \\ \leq & ⋁_{x \in X} co ({a, x}) (p) \land co ({c, d}) (x) \\ = & (p_{⊤} / co ({c, d}))_{co} (a) . \end{matrix}$ Similarly, co ({b, c}) (p) ≤ (p_⊤/co ({c, d})) _co (b). Thus $\begin{matrix} co ({a, d}) (p) \land co ({b, c}) (p) \land co ({a, b}) (v) \\ = (p_{⊤} / co ({c, d}))_{co} (a) \land (p_{⊤} / co ({c, d}))_{co} (b) \\ \land co ({a, b}) (v) \\ \leq (p_{⊤} / co ({c, d}))_{co} ∣_{co} (v) \leq (p_{⊤} / co ({c, d}))_{co} (v) \\ = ⋁_{x \in X} co ({v, x}) (p) \land co ({c, d}) (x) . \end{matrix}$ Therefore co has M-fuzzifying sand-glass property.

Theorem 4.10. An M-fuzzifying idempotent Peano-Pasch interval space (X, ℐ) is an M-fuzzifying sand-glass interval space.

Proof. By Corollary 4.5, ℐ is an M-fuzzifying geometric interval operator. Let a, b, c, d, p, v ∈ X. By M-fuzzifying Peano-Pasch property and (MGI2&3), $\begin{matrix} ℐ (a, d) (p) \land ℐ (b, c) (p) \land ℐ (a, b) (v) \\ = ℐ (b, c) (p) \land [ℐ (a, d) (p) \land ℐ (a, b) (v)] \\ \leq ℐ (b, c) (p) \land [⋁_{g \in X} ℐ (b, p) (g) \land ℐ (d, v) (g)] \\ = ⋁_{g \in X} [ℐ (b, c) (p) \land ℐ (b, p) (g)] \land ℐ (d, v) (g) \\ \leq ⋁_{g \in X} [ℐ (b, c) (g) \land ℐ (c, g) (p)] \land ℐ (d, v) (g) \\ \leq ⋁_{g \in X} ℐ (c, g) (p) \land ℐ (d, v) (g) \\ \leq ⋁_{w \in X} ℐ (c, d) (w) \land ℐ (v, w) (p) . \end{matrix}$

Theorem 4.11. If the segment operator of an M-fuzzifying JHC convex structure satisfies (MGI1) and has M-fuzzifying sand-glass property, then it is an M-fuzzifying geometric interval operator.

Proof. Let (X, 𝒞) be an M-fuzzifying JHC convex structure with co as its segment operator.

(GMI2)’. Let a, b, c, d ∈ X. By (MRH3) of co, $\begin{matrix} co ({a, b}) (c) \land co ({b, c}) (d) \\ = co ({b, c}) (d) \land ⋀_{x \in {b, c}} co ({a, b}) (x) \\ \leq co ({a, b}) (d) . \end{matrix}$

(MGI3)’. Let a, b, c, d ∈ X and (c_⊤/co ({a, d})) _co = U. By Theorem 4.9, we have U ∣ _co ≤ U. Thus $\begin{matrix} U (b) & = & ⋁_{p, q \in X} c_{⊤} (p) \land co ({a, d}) (q) \land co ({b, q}) (p) \\ = & ⋁_{q \in X} co ({a, d}) (q) \land co ({b, q}) (c) \\ \geq & co ({a, b}) (c), \end{matrix}$ $\begin{matrix} U (c) & = & ⋁_{p, q \in X} c_{⊤} (p) \land co ({a, d}) (q) \land co ({c, q}) (p) \\ = & ⋁_{q \in X} co ({a, d}) (q) \land co ({c, q}) (c) \\ = & ⋁_{q \in X} co ({a, d}) (q) = ⊤, \end{matrix}$ and by (MRH3) of co, $\begin{matrix} U (d) & = & ⋁_{p, q \in X} c_{⊤} (p) \land co ({a, d}) (q) \land co ({d, q}) (p) \\ = & ⋁_{q \in X} co ({a, d}) (q) \land co ({d, q}) (c) \\ = & ⋁_{q \in X} [co ({d, q}) (c) \land ⋀_{x \in {d, q}} co ({a, d}) (x)] \\ \leq & co ({a, d}) (c) . \end{matrix}$ Hence we have $\begin{matrix} co ({a, b}) (c) \land co ({b, c}) (d) \\ \leq U (b) \land U (c) \land co ({b, c}) (d) \\ \leq ⋁_{p, q \in X} U (p) \land U (q) \land co ({p, q}) (d) \\ = U ∣_{co} (d) \leq U (d) \leq co ({a, d}) (c) . \end{matrix}$ This shows (MGI3)’ holds.

Lemma 4.12. Let (X, ℐ) be an M-fuzzifying interval space and r ∈ P (M). Then (𝒞_ℐ) _[r′] is generated by ℐ^(r). That is, ${(𝒞_{ℐ})}_{[r']} = 𝒞_{ℐ^{(r)}}$ .

Proof. Let A ∈ 2^X and r ∈ P (M). Thus $\begin{matrix} A \in (𝒞_{ℐ})_{[r^{'}]} & \Leftrightarrow & 𝒞_{ℐ} (A) \geq r^{'} \\ \Leftrightarrow & ⋀_{z \notin A} ⋀_{x, y \in A} (ℐ (x, y) (z))^{'} \geq r^{'} \\ \Leftrightarrow & \forall z \notin A, \forall x, y \in A, z \notin ℐ^{(r)} (x, y) \\ \Leftrightarrow & \forall x, y \in A, ℐ^{(r)} (x, y) \in A \\ \Leftrightarrow & A \in ℐ^{(r)} . \end{matrix}$ Therefore (𝒞_ℐ) _[r′] is generated by ℐ^(r).

Lemma 4.13. Let (X, ℐ) be an M-fuzzifying interval space, co be the segment operator of (X, 𝒞_ℐ) and U ∈ M^X. Then the following areequivalent:

$U^{(r)} \in ℐ^{(r)}$ for each r ∈ P (M);

U ∣ _co ≤ U;

U ∣ _ℐ ≤ U.

Proof. The proof follows directly from Theorem 2.14 and Lemma 4.12.

Theorem 4.14. Let (X, ℐ) be an M-fuzzifying modular space and co be the segment operator of (X, 𝒞_ℐ). Then the following conditions are equivalent:

ℐ has M-fuzzifying Pasch property;

ℐ has M-fuzzifying Peano property;

ℐ (a, b) ∣ _ℐ ≤ ℐ (a, b) for all a, b ∈ X;

ℐ (a, b) ∣ _co ≤ ℐ (a, b) for all a, b ∈ X;

co is M-fuzzifying geometric;

co = ℐ;

$ℐ^{(r)} (a, b) \in ℐ^{(r)}$ for all a, b ∈ X and r ∈ P (M);

ℳ^(r) (a, b, c) is a singleton for each r ∈ P (M);

𝒞_ℐ is an M-fuzzifying JHC convexity and co has M-fuzzifying sand-glass property.

Proof. (1) ⇒ (2). Let a, b, p, x, y ∈ X. In order to show M-fuzzifying Peano property, we have to show ℐ (p, b) (x) ∧ ℐ (a, x) (y) ≤ ⋁ _z∈Xℐ (p, a) (z) ∧ ℐ (b, z) (y). Let r ∈ P (M) with ℐ (p, b) (x) ∧ ℐ (a, x) (y) ≰ r. We have the following two cases.

(i) The case ℐ (b, y) (x) ≰ r. Since (X, ℐ) is M-fuzzifying modular, there exists z₀ ∈ X such that M (a, y, p) (z₀) ≰ r. Then ℐ (p, a) (z₀) ≰ r. Next, we show that ℐ (b, z₀) (y) ≰ r.

By ℐ (p, y) (z₀) ∧ ℐ (p, b) (x) ≰ r and M-fuzzifying Pasch property, we have $\begin{matrix} ℐ (p, y) (z_{0}) \land ℐ (p, b) (x) \\ \leq ⋁_{s \in X} ℐ (b, z_{0}) (s) \land ℐ (x, y) (s) ≰ r . \end{matrix}$ Thus there is s ∈ X such that ℐ (b, z₀) (s) ∧ ℐ (x, y) (s) ≰ r. Hence ℐ (x, y) (s) ≰ r.

Further, ℐ (x, y) (s) ∧ ℐ (b, y) (x) ≰ r and M-fuzzifying Pasch property yield that $\begin{matrix} ℐ (y, x) (s) \land ℐ (y, b) (x) \\ \leq ⋁_{t \in X} ℐ (x, x) (t) \land ℐ (b, s) (t) = ℐ (b, s) (x) ≰ r . \end{matrix}$ Thus, by (MGI2), we have $ℐ (b, z_{0}) (x) \geq ℐ (b, z_{0}) (s) \land ℐ (b, s) (x) ≰ r .$

Similarly, ℐ (a, y) (z₀) ∧ ℐ (a, x) (y) ≰ r. Thus $\begin{matrix} r & ≱ & ℐ (a, y) (z_{0}) \land ℐ (a, x) (y) \\ \leq & ⋁_{w \in X} ℐ (y, y) (w) \land ℐ (x, z_{0}) (w) \\ = & ℐ (x, z_{0}) (y) . \end{matrix}$ Hence, by (MGI2), $ℐ (b, z_{0}) (x) \land ℐ (x, z_{0}) (y) \leq ℐ (b, z_{0}) (y),$ showing that ℐ (b, z₀) (y) ≰ r.

Finally, ℐ (p, a) (z₀) ∧ ℐ (b, z₀) (y) ≰ r which implies ⋁_z∈Xℐ (p, a) (z) ∧ ℐ (b, z) (y) ≰ r. By arbitrariness of r ∈ P (M), ℐ (p, b) (x) ∧ ℐ (a, x) (y) ≤ ⋁ _z∈Xℐ (p, a) (z) ∧ ℐ (b, z) (y). Therefore ℐ has M-fuzzifying Peano property.

(ii) The case ℐ (b, y) (x) ≤ r. Let $\bar{x}$ ∈ X with ℳ (y, b, x) ( $\bar{x}$ ) ≰ r. Then ℐ (b, y) ( $\bar{x}$ ) ≰ r and ℐ (x, y) ( $\bar{x}$ ) ≰ r. Further, ℐ (a, x) (y) ∧ ℐ (x, y) ( $\bar{x}$ ) ≰ r and (MGI2&3) yield that $\begin{matrix} ℐ (a, x) (y) \land ℐ (x, y) (\bar{x}) \\ \leq ℐ (a, x) (\bar{x}) \land ℐ (a, \bar{x}) (y) ≰ r . \end{matrix}$ Thus ℐ (a, $\bar{x}$ ) (y) ≰ r. Replace x with $\bar{x}$ in (i). Thus ℐ has M-fuzzifying Peano property.

(2) ⇒ (3). It follows from (3) of Theorem 4.4.

(3) ⇒ (1). Let p, a, b, $\bar{a}$ , $\bar{b}$ ∈ X.

Let r ∈ P (M) with ℐ (p, a) ( $\bar{a}$ ) ∧ ℐ (p, b) ( $\bar{b}$ ) ≰ r. We have the following two cases.

(i) The case ℐ (a, b) (p) ≰ r. By (MGI3)’, we have ℐ (a, b) (p) ∧ ℐ (b, p) ( $\bar{b}$ ) ≤ ℐ (a, $\bar{b}$ ) (p) and ℐ (a, b) (p) ∧ ℐ (b, p) ( $\bar{a}$ ) ≤ ℐ (a, $\bar{a}$ ) (p).

Thus ℐ (a, $\bar{b}$ ) (p) ≰ r and ℐ (b, $\bar{a}$ ) (p) ≰ r. Hence $\begin{matrix} ℐ (a, \bar{b}) (p) \land ℐ (b, \bar{a}) (p) \\ \leq ⋁_{z \in X} ℐ (a, \bar{b}) (z) \land ℒ (b, \bar{a}) (z) ≰ r . \end{matrix}$ Therefore we obtain the desired inequality.

(ii) The case ℐ (a, b) (p) ≤ r. Assume that $⋁_{z \in X} ℒ (a, \bar{b}) (z) \land ℒ (b, \bar{a}) (z) \leq r .$ (iia) That is, ℐ (a, $\bar{b}$ ) (z) ∧ ℐ (b, $\bar{a}$ ) (z) ≤ r for all z ∈ X. Since (X, ℐ) is an M-fuzzifying modular space, there exist w_a, v_b ∈ X such that ℳ ( $\bar{a}$ , b, p) (w_a) ∧ ℳ ( $\bar{b}$ , a, p) (v_b) ≰ r. Thus $ℒ (p, b) (w_{a}) \land ℒ (p, a) (v_{b}) ≰ r .$ (iib) By (MGI2), ℐ (p, a) ( $\bar{a}$ ) ∧ ℐ (p, $\bar{a}$ ) (w_a) ≤ ℐ (p, a) (w_a) and ℐ (p, b) ( $\bar{b}$ ) ∧ ℐ (p, $\bar{b}$ ) (v_a) ≤ ℐ (p, b) (v_a). Thus ℐ (a, p) (w_a) ≰ r and ℐ (p, b) (v_b) ≰ r.

Consider the configurations a, b, p, w_a, v_b. For each z ∈ X, by the assumption and (MGI2), we have $\begin{matrix} [ℒ (b, \bar{a}) (w_{a}) \land ℒ (b, w_{a}) (z)] \\ \land [ℒ (a, \bar{b}) (v_{b}) \land ℒ (a, v_{a}) (z)] \\ \leq ℒ (b, \bar{a}) (z) \land ℒ (a, \bar{b}) (z) \leq r . \end{matrix}$ Thus we have ℐ (b, w_a) (z) ∧ ℐ (a, v_a) (z) ≤ r since ℐ (b, $\bar{a}$ ) (w_a) ∧ ℐ (a, $\bar{b}$ ) (v_b) ≰ r. Hence we have ⋁_z∈Xℐ (b, w_a) (z) ∧ ℐ (a, v_a) (z) ≤ r. Thus by (iia), $\begin{matrix} ℒ (p, a) (w_{a}) \land ℒ (p, b) (v_{b}) \\ ≰ ⋁_{z \in X} ℒ (b, w_{a}) (z) \land ℒ (a, v_{a}) (z) . \end{matrix}$ This shows that M-fuzzifying Pasch property fails for the configurations a, b, p, w_a, v_b.

Again, since (X, ℐ) is an M-fuzzifying modular space, there exist x_a, y_b ∈ X such that ℳ ( $\bar{a}$ , a, b) (x_a) ∧ ℳ ( $\bar{b}$ , a, b) (y_b) ≰ r.

Consider the configurations a, b, p, x_a, y_b. By (MGI2), ℐ (a, p) ( $\bar{a}$ ) ∧ ℐ (a, $\bar{a}$ ) (x_a) ≤ ℐ (a, p) (x_a) and ℐ (b, p) ( $\bar{b}$ ) ∧ ℐ (b, $\bar{b}$ ) (y_a) ≤ ℐ (b, p) (y_b). Thus $ℒ (a, p) (x_{a}) \land ℒ (b, p) (y_{b}) ≰ r .$ (iic)

By the assumption and (MGI2), for each w ∈ X, $\begin{matrix} [ℒ (b, \bar{a}) (x_{a}) \land ℒ (b, x_{a}) (w)] \\ \land [ℒ (a, \bar{b}) (y_{b}) \land ℒ (a, y_{b}) (w)] \\ \leq ℒ (b, \bar{a}) (w) \land ℒ (a, \bar{b}) (w) \leq r . \end{matrix}$

Since ℐ (b, $\bar{a}$ ) (x_a) ∧ ℐ (a, $\bar{b}$ ) (y_b) ≰ r, we have ℐ (b, x_a) (w) ∧ ℐ (a, y_b) (w) ≤ r. Hence $⋁_{w \in X} ℒ (b, x_{a}) (w) \land ℒ (a, y_{b}) (w) \leq r .$ (iid)

(iic) and (iid) imply that M-fuzzifying Pasch property fails for the configurations a, b, p, x_a, y_b.

Let q ∈ X with ℳ (x_a, y_b, p) (q) ≰ r and consider the configurations a, b, q, x_a, y_b. Then ℐ (x_a, y_b) (q) ≰ r. In addition, by (MGI3)’, we have $\begin{matrix} ℒ (a, p) (x_{a}) \land ℒ (p, x_{a}) (q) \leq ℒ (a, q) (x_{a}); \\ ℒ (b, p) (y_{b}) \land ℒ (p, y_{b}) (q) \leq ℒ (b, q) (y_{b}) . \end{matrix}$ Thus, by (iic), ℐ (a, q) (x_a) ∧ ℐ (b, q) (y_b) ≰ r.

From (iid), we assert that $ℒ (a, b) (q) \leq r .$ (iie) Otherwise, ℐ (a, b) (q) ≰ r. Repeat the process of (i) with the configuration a, b, q, x_a, y_b. So $\begin{matrix} ℒ (a, q) (x_{a}) \land ℒ (b, q) (y_{b}) \\ \leq ⋁_{w \in X} ℒ (b, x_{a}) (w) \land ℒ (a, y_{b}) (w) ≰ r, \end{matrix}$ which contradicts (iid). Hence (iie) must be true.

By (3), we have $\begin{matrix} ℒ (a, b) (x_{a}) \land ℒ (a, b) (y_{b}) \land ℒ (x_{a}, y_{b}) (q) \\ \leq ℒ (a, b) ∣_{ℒ} (q) \leq ℒ (a, b) (q) . \end{matrix}$ However, by ℐ (a, b) (x_a) ∧ ℐ (a, b) (y_b) ∧ ℐ (x_a, y_b) (q) ≰ r, we obtain that ℐ (a, b) (q) ≰ r, which contradicts to (iie). Consequently, (iia) can not be true. That is, ⋁_z∈Xℐ (a, $\bar{b}$ ) (z) ∧ ℐ (b, $\bar{a}$ ) (z) ≰ r. Hence the desired inequality is obtained. Therefore ℐ has M-fuzzifying Pasch property.

(3) ⇔ (4) ⇔ (7) follows from Lemma 4.13.

(3) ⇒ (5). Let a, b ∈ X. By (3), ℐ (a, b) ∣ _ℐ ≤ ℐ (a, b). By Lemma 4.13, $ℒ (a, b)^{(r)} \in ℒ^{(r)}$ for each r ∈ P (M). Thus ℐ (a, b) ^(r)sco^(r) ({a, b}) ∈ℐ (a, b) ^(r) for each r ∈ P (M). Hence co ({a, b}) = ℐ (a, b). Thus co = ℐ is M-fuzzifying geometric.

(5) ⇒ (3). ℐ ≤ co follows from Theorem 2.8. Thus, by Corollary 3.10, we have ℐ = co. Let a, b ∈ X. By Theorem 2.14 and Lemma 4.12, ${co}^{(r)} ({a, b}) \in ℒ^{(r)}$ for each r ∈ P (M). Hence, by Lemma 4.13, ℐ (a, b) ∣ _ℐ = co ({a, b}) ∣ _co ≤ co ({a, b}) = ℐ (a, b).

(3) ⇔ (6) is implied by the proof of (3) ⇔ (5).

(1) ⇒ (8). Let a, b, c, x, y ∈ X with x, y ∈ ℳ^(r) (a, b, c). Then, by M-fuzzifying Pasch property and (MGI2), we have $\begin{matrix} [(ℒ (a, c) (x) \land ℒ (a, b) (y)) \land ℒ (c, b) (x)] \\ \land ℒ (b, c) (y) \\ \leq [⋁_{w \in X} ℒ (c, y) (w) \land (ℒ (b, x) (w) \land ℒ (b, c) (x))] \\ \land ℒ (b, c) (y) \\ \leq [⋁_{w \in X} ℒ (c, y) (w) \land (⋁_{v \in X} ℒ (x, x) (v) \\ \land ℒ (c, w) (v))] \land ℒ (b, c) (y) \\ = [⋁_{w \in X} ℒ (c, y) (w) \land ℒ (c, w) (x)] \land ℒ (c, b) (y) \\ \leq ℒ (c, y) (x) \land ℒ (c, b) (y) \\ \leq ⋁_{t \in X} ℒ (y, y) (t) \land ℒ (b, x) (t) = ℒ (b, x) (y) . \end{matrix}$ This shows ℐ (b, x) (y) ≰ r. Similarly, we have $\begin{matrix} [(ℒ (a, b) (x) \land ℒ (a, c) (y)) \land ℒ (c, b) (x)] \\ \land ℒ (b, c) (y) \\ \leq [⋁_{w \in X} ℒ (b, y) (w) \land (ℒ (c, x) (w) \land ℒ (c, b) (x))] \\ \land ℒ (b, c) (y) \\ \leq [⋁_{w \in X} ℒ (b, y) (w) \land (⋁_{v \in X} ℒ (x, x) (v) \\ \land ℒ (b, w) (v))] \land ℒ (b, c) (y) \\ = [⋁_{w \in X} ℒ (b, y) (w) \land ℒ (b, w) (x)] \land ℒ (b, c) (y) \\ \leq ℒ (b, y) (x) \land ℒ (b, c) (y) \\ \leq ⋁_{t \in X} ℒ (y, y) (t) \land ℒ (c, x) (t) = ℒ (c, x) (y) . \end{matrix}$ Thus ℐ (c, x) (y) ≰ r. Hence, by (MGI1)’, we have $ℒ (b, c) (x) \land ℒ (b, x) (y) \land ℒ (c, x) (y) = x_{⊤} (y) .$ This shows x_⊤ (y) ≰ r and x = y. Therefore ℳ^(r) (a, b, c) is a singleton.

(8) ⇒ (1). Assume that M-fuzzifying Pasch property fails for ℐ. That is, there exist p, a, b, $\bar{a}$ , $\bar{b}$ ∈ X such that ℐ (p, a) ( $\bar{a}$ ) ∧ ℐ (p, b) ( $\bar{b}$ ) ≰ ⋁ _z∈Xℐ (a, $\bar{b}$ ) (z) ∧ ℐ (b, $\bar{a}$ ) (z). Thus there exists r ∈ P (M) such that ℐ (p, a) ( $\bar{a}$ ) ∧ ℐ (p, b) ( $\bar{b}$ ) ≰ r and ⋁_z∈Xℐ (a, $\bar{b}$ ) (z) ∧ ℐ (b, $\bar{a}$ ) (z) ≤ r.

Clearly, (X, ^ℐ (r)) is modular. By $\bar{a}$ ∈ ^ℐ (r) (p, a), $\bar{b}$ ∈ ^ℐ (r) (p, b) and ^ℐ (r) ( $\bar{a}$ , b) ∩ ^ℐ (r) (a, $\bar{b}$ ) = ∅, we know that ^ℐ (r) doesn’t have Pasch property.

Let x_a ∈ ℳ^(r) ( $\bar{a}$ , b, p) and y_b ∈ ℳ^(r) ( $\bar{b}$ , a, p). Then x_a ∈ ^ℐ (r) ( $\bar{a}$ , b) and y_b ∈ ^ℐ (r) (a, $\bar{b}$ ). Since ^ℐ (r) is geometric, it is follows from (GI2) that $\begin{matrix} ℒ^{(r)} (x_{a}, b) \cap ℒ^{(r)} (a, y_{b}) \\ \in ℒ^{(r)} (\bar{a}, b) \cap ℒ^{(r)} (a, \bar{b}) = \emptyset . \end{matrix}$

Since x_a ∈ ^ℐ (r) (p, $\bar{a}$ ) and y_b ∈ ^ℐ (r) (p, $\bar{b}$ ), we see that p, x_a, y_b, x_a, y_b don’t satisfy Pasch property.

Let $\bar{x}$ _a ∈ ℳ^(r) (x_a, a, b) and $\bar{y}$ _b ∈ ℳ^(r) (y_b, a, b). Then $\bar{x}$ _a, $\bar{y}$ _b ∈ ^ℐ (r) (a, b). Since $\bar{x}$ _a ∈ ^ℐ (r) (x_a, b) and $\bar{y}$ _b ∈ ^ℐ (r) (y_b, a), we have ^ℐ (r) ( $\bar{x}$ _a, b) ∩ ^ℐ (r) ( $\bar{y}$ _b, a) ∈^ℐ (r) (x_a, b) ∩ ^ℐ (r) (y_b, a) = ∅. Thus, the configurations p, $\bar{x}$ _a, $\bar{y}$ _b, x_a, y_b don’t satisfy Pasch property.

Let ${\hat{x}}_{a} \in ℒ^{(r)} ({\bar{x}}_{a}, b, p)$ and ${\hat{y}}_{b} \in ℒ^{(r)} ({\bar{y}}_{b}, a, p)$ . Then ${\hat{x}}_{a} \in ℒ^{(r)} ({\bar{x}}_{a}, b) \in ℒ^{(r)} (a, b)$ and ${\hat{y}}_{b} \in ℒ^{(r)} ({\bar{y}}_{b}, a) \in ℒ^{(r)} (a, b)$ . Notice that ${\hat{x}}_{a} \in ℒ^{(r)} ({\bar{x}}_{a}, b)$ and ${\hat{y}}_{b} \in ℒ^{(r)} ({\bar{y}}_{b}, a)$ . Thus $ℒ^{(r)} ({\hat{x}}_{a}, b) \cap ℒ^{(r)} (a, {\hat{y}}_{b}) \in ℒ^{(r)} ({\bar{x}}_{a}, b) \cap ℒ^{(r)} ({\bar{y}}_{b}, a) = \emptyset$ . Hence ${\hat{x}}_{a} \neq {\hat{y}}_{b}$ .

Since x_a ∈ ^ℐ (r) ( $\bar{a}$ , p) ∈^ℐ (r) (a, p), we have $\bar{x}$ _a ∈ ^ℐ (r) (x_a, a) ∈^ℐ (r) (a, p). Thus ${\hat{x}}_{a} \in ℒ^{(r)} ({\bar{x}}_{a}, p) \in ℒ^{(r)} (a, p)$ . Hence ${\hat{x}}_{a} \in ℒ^{(r)} (a, p) \cap ℒ^{(r)} (b, p) \cap ℒ^{(r)} (a, b) = ℳ^{(r)} (p, a, b)$ . Similarly, we have ${\hat{y}}_{b} \in ℒ^{(r)} (a, p) \cap ℒ^{(r)} (b, p) \cap ℒ^{(r)} (a, b) = ℳ^{(r)} (p, a, b)$ .Hence ${\hat{x}}_{a}, {\hat{y}}_{b} \in ℳ^{(r)} (p, a, b)$ which is a contradiction. Therefore ℐ has M-fuzzifying Pasch property.

(2) ⇒ (9). By (2) ⇔ (5), we see that co is idempotent. By Theorem 2.9 and 2.13, 𝒞_ℐ is of arity ≤2 and co has M-fuzzifying JHC property. Thus, by Theorem 4.10, co has M-fuzzifying sand-glass property.

(9) ⇒ (5). Let a, z ∈ X (z ≠ a). By (MGI1) of ℐ, $\begin{matrix} co ({a, a}) (z) & \leq & (𝒞_{ℒ} ({a}))^{'} \\ = & ⋁_{y \notin {a}} ⋁_{x, w \in {a}} ℒ (x, w) (y) = ⊥ . \end{matrix}$ Hence co ({a, a}) = a_⊤. Therefore, by Theorem 4.11, co has M-fuzzifying geometric.

Theorem 4.15. Let (X, _ℐX) and (Y, _ℐY) be M-fuzzifying interval spaces and $f_{M}^{\to} : M^{X} \to M^{Y}$ be an M-fuzzy surjective function. If (X, _ℐX) is an M-fuzzifying Pasch interval space (resp. M-fuzzifying sand-glass interval space), then so is (Y, _ℐY).

5 Conclusions

The aim of this paper is to discuss the geometric aspect of M-fuzzifying convex structures. Specifically, four properties, M-fuzzifying geometric property, M-fuzzifying modular property, M-fuzzifying Pasch property and M-fuzzifying sand-glass property, are introduced and characterized. These properties, together with “arity ≤2”, M-fuzzifying Peano property and M-fuzzifying JHC property, are closely related (see Theorems 2.9, 2.13, 4.10, 4.11 and 4.14 and Corollary 4.5). Those relations reveal the geometric features of M-fuzzifying convex structures. We hope that the results in the paper will be useful for discussing further properties of M-fuzzifying convex structures.

Footnotes

Acknowledgments

The authors sincerely thank Prof. Violeta Fotea for his 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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