M -fuzzifying Bryant-Webster spaces and M -fuzzifying join spaces

Abstract

In M-fuzzifying interval spaces, some properties, including M-fuzzifying dense property, M-fuzzifying decomposable property, M-fuzzifying ramification property and M-fuzzifying straight property, are introduced and their relations are studied. Based on these properties, notions of M-fuzzifying Bryant-Webster spaces and M-fuzzifying join spaces are introduced. It is proved that the category of M-fuzzifying Bryant-Webster spaces and M-fuzzifying convexity preserving mappings can be characterized by the category of M-fuzzifying join spaces and M-fuzzifying join preserving mappings.

1 Introduction

The study of convexity, being originally inspired by some elementary geometric problems such as characterizations of circles and polytopes in low dimensional Euclidean spaces [2], has made great advances both in theoretical and practical aspects [3 , 29]. Now, a rather complete theory of convex spaces had been established [29], where many mathematical structures are involved such as graphic theories [6], posets [8], metric spaces [18], median algebras [12], lattices [28] and vector spaces [29], etc. All these involvements are based on a fact that these mathematical structures share some unified geometric properties.

Interval operator is a convenient and common tool to study convexities. In fact, interval operators derived from the above mathematical structures can not only display these unified geometric properties, but also provide a natural and frequent method of describing or constructing convex structures [5, 29]. In addition, interval operators, equipped by some special properties such as decomposable property, straight property, Peano-Pasch property, not only implicate many other properties such as JHC property, ramification property and geometric property, etc., but also enhance separation properties and invariant properties, etc [29].

Interval operators can be constructed from various mathematical structures. Vel listed many interval operators via vector spaces, posets, metric spaces, lattices, semilattices, modular spaces and median algebras, etc [29]. Also, Ryan constructed interval operators by mixture sets, which is closely related to antimatroids [20] and is useful to capture the combination properties of Utility Thoery and Convex Theory [1].

Interval spaces are usually interpreted as (closed) ‘join’ spaces. Nevertheless, join spaces as ‘open’ interval operators are also introduced, where the join of two points is the set of points that strictly between the end points [4, 17]. In addition, there is a one-to-one correspondence between join spaces and Bryant-Webster spaces, where a Bryant-Webster space is a JHC convex space whose segment operator is dense, decomposable and straight [29]. Bryant-Webster spaces are quite useful in studying of convex spaces. For example, relations among convex invariants are greatly enhanced in a Bryant-Webster space, and a complete Bryant-Webster space without boundary can be turned into a topological convex space which is local convex, Hausdorff and closure-interior stable [29].

Convexity has been extended into fuzzy settings in several ways. Fuzzy convexities defined by Rosa [19] was further extended into M-convexities by Maruyama [11]. Later, some characterizations of L-convex spaces have been obtain [14–16 , 36] Actually, a fuzzy convexity or an M-convexity is a crisp family of fuzzy sets or M-fuzzy sets satisfying certain set of axioms that a classic convex space has. However, from a totally different point of view, Shi and Xiu introduced M-fuzzifying convexities where each subset of the underling set can be regarded as a convex set to some degree [26]. In the framework of M-fuzzifying convex spaces or L-convex spaces, many properties had been studied [25 , 39]. Xiu and Shi also defined M-fuzzifying interval spaces [37], based on which, Wu et al defined and studied M-fuzzifying geometric (resp. Peano, Pasch, sand-glass) interval spaces [32 , 35].

This paper is arranged as follows. In Section 2, we recall some basic definitions and results of both convex spaces and M-fuzzifying convex spaces. In Section 3, we introduce some properties of M-fuzzifying interval spaces. In addition, we show that these properties together with M-fuzzifying geometric property and M-fuzzifying convex matroids are closely related (Proposition 3.3 and Theorem 3.4). Based on these properties, we define M-fuzzifying Bryant-Webster spaces. Then we give some examples of M-fuzzifying Bryant-Webster spaces (Example 3.8), and prove that M-fuzzifying Bryant-Webster spaces are M-fuzzifying convex geometries (Theorem 3.5). In Section 4, we introduce M-fuzzifying join spaces. We verify that M-fuzzifying Bryant-Webster spaces and M-fuzzifying join spaces are mutually induced (Theorems 4.7 and 4.8). Based on this, we establish the categorical relations between M-fuzzifying Bryant-Webster spaces and M-fuzzifying join spaces (Corollary 4.11).

2 Preliminaries

Throughout this paper, 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 $2_{fin}^{X}$ . Clearly, $2_{2}^{X} = {{x, y} : x, y \in X} \subseteq 2_{fin}^{X}$ .

(M, ∨, ∧, ′) is a completely distributive lattice with an inverse involution ^′. The least (resp. largest) element in M is denoted by ⊥ (resp. ⊤). An element a ∈ M is called a prime if for all b, c ∈ M, b ∧ c ≤ a implies b ≤ a or c ≤ a. P (M) is the set of all primes in M ∖ {⊤}, and J (M) = {a ∈ M : a′ ∈ P (M)}. For each a ∈ M, there exist φ ⊆ P (M) and ψ ∈ J (M) such that a = ⋀ φ = ⋁ ψ [31, 30]. For p, q ∈ M, p ≤ q iff p ≰ r implies q ≰ r for all r ∈ P (M) [33]. For t ∈ M, t ≰ ^*r implies r ∈ P (M) with t ≰ r.

M^X is the set of all M-fuzzy sets on X. For U ∈ M^X and r ∈ M, U_[r] = {x ∈ X : U (x) ≥ r} and U^(r) = {x ∈ X : U (x) ≰ r} are subsets of X [22]. If A ∈ 2^X and r ∈ M, the M-fuzzy set r ∨ A is defined by: (r∨ A) (y) = ⊤ for all y ∈ A and (r ∨ A) (x) = r for all x ∈ X ∖ A [22]. For a mapping f : X → Y, $f_{M}^{\to} : M^{X} \to M^{Y}$ is defined by: $f_{M}^{\to} (U) (y) = ⋁ {U (x) : f (x) = y}$ for U ∈ M^X and y ∈ Y, and $f_{M}^{ł} : M^{Y} \to M^{X}$ is defined by: $f_{M}^{ł} (V) (x) = V (f (x))$ for V ∈ M^Y and x ∈ X [23 , 31].

The followings are some definitions and results in convex spaces and M-fuzzifying convex spaces.

Definition 2.1. [29] The pair $(X, c)$ where $C \subseteq 2^{X}$ is called a convex space, if

$\emptyset, X \in c$ ;

$⋂_{i \in Ω} U_{i} \in c$ for all subset ${U_{i}}_{i \in Ω} \subseteq c$ ;

$C (\cup_{i \in Ω} U_{i}) \in C$ for all subset ${U_{k}}_{i \in Ω} \subseteq c$ totally ordered by inclusion.

The hull operator co : 2^X → 2^X of $(X, c)$ is defined by: $co (A) = ⋂ {B \in c : A \subseteq B}$ for A ∈ 2^X. The restriction ${Seg}_{c}$ of co on $2_{2}^{X}$ is called the segment operator of $(X, c)$ .

Definition 2.2. [29] A convex space $(X, c)$ is called a Join-Hull Commutativity (briefly, JHC) convex space, if for a ∈ X and A ∈ 2^X ∖ {∅},

(JHC)co ({a} ∪ A) = ∪ {co ({a, c}) : c ∈ co (A)}.

Definition 2.3. [29] An operator $ℐ : X \times X \to 2^{X}$ is called an interval operator on X and the pair $(X, I)$ is called an interval space, if for all x, y ∈ X,

$x, y \in I (x, y)$ ;

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

An interval space $(X, I)$ is said to has

(1) geometric property, if for all a, b, c, d ∈ X,

$I (a, a) = {a}$ ;

$c \in I (a, b)$ and $d \in I (a, c)$ imply $d \in I (a, b)$ ;

$c, d \in I (a, b)$ and $c \in I (a, d)$ imply $d \in I (c, b)$ ;

(2) Peano property, if for all a, b, c, y, z ∈ X

(Peano) $y \in I (b, c)$ and $z \in I (a, y)$ imply the existence of $x \in I (a, b)$ such that $z \in I (c, x)$ ;

(3) Pasch property, if for all $p, a, b, \bar{a}, \bar{b} \in X$ ,

$\bar{a} \in I (p, a)$ and $\bar{b} \in I (p, b)$ imply $I (\bar{a}, b) \cap I (a, \bar{b}) \neq \emptyset$ ;

(4) dense property, if for all a, b ∈ X with a ≠ b,

$I (a, b) ∖ {a, b} \neq \emptyset$ ;

(5) decomposable property, if for all a, b, x ∈ X, $x \in I (a, b)$ implies

$I (a, b) = I (a, x) \cup I (x, b)$ ;

$I (a, x) \cap I (x, b) = {x}$ ;

(6) ramification property, if for all b, c, d ∈ X,

$c \notin I (b, d)$ and $d \notin I (b, c)$ implies $I (b, d) \cap I (b, c) = {b}$ ;

(7) straight property, if for all a, b, c, d, x, y ∈ X,

$x, y \in I (a, c) \cap I (b, d)$ with x ≠ y implies the existence of u, v ∈ X such that $I (a, c) \cup I (b, d) = I (u, v)$ .

Definition 2.4. [29] A JHC convex space $(X, c)$ is called a Bryant-Webster space (briefly, BW space), if ${Seg}_{c}$ is dense, decomposable and straight.

Definition 2.5. [4, 29] An operator ∘: X × X → 2^X is called a join operator and the pair (X, ∘) is called a join space, if for all a, b, c, d ∈ X.

a, b ∉ a ∘b;

a∘b ≠ ∅

a ∘a = a/a = {a}, where /: X × X → 2^X is defined by: u/v = {x : u ∈ v ∘x};

a ∘(b ∘c) = (a ∘b) ∘c;

(a/b)∩ (c/d) ≠ ∅ implies (a∘d) ∩ (b ∘c) ≠ ∅

(a∘b) ∩ (a ∘c) ≠ ∅ implies b = c or b ∈ a ∘c or c ∈ a ∘b.

Remark 2.6. [4, 29] Let (X, ∘) be a join space.

(1) ∘satisfies (J2*) a ∘b = b ∘a for all a, b ∈ X.

(2) $(X, c_{\circ})$ is a convex space, where $C_{\circ} = {A \in 2^{X} : \forall a, b \in A, \circ \subseteq A}$ .

(3) For all A, B ∈ 2^X, we denote A ∘B = ∪ {a ∘b : a ∈ A, b ∈ B} and A/B = ∪ {a/b : a ∈ A, b ∈ B}.

Definition 2.7. [26] A mapping $C : 2^{X} \to ℳ$ is called an M-fuzzifying convex structure and the pair $(X, c)$ is called an M-fuzzifying convex structure, if

$c (\emptyset) = c (x) = ⊤$ ;

$C (\cup_{i \in I} U_{i}) \geq Λ_{i \in I} C (U_{i})$ for {U_i} _i∈I ⊆ 2^X;

$C (\cup_{i \in I} U_{i}) \geq \land_{i \in I} C (U_{i})$ for {U_i} _i∈I ⊆ 2^X totally ordered by inclusion.

Theorem 2.8. [26] The hull operator ${co}_{c} : 2^{X} \to M^{X}$ (briefly, co) of $(X, c)$ is defined by: $\begin{matrix} \forall A \in 2^{X}, \forall x \in X, co (A) (x) & = & ⋀_{x \notin B \supseteq A} (c (B))^{'} . \end{matrix}$ 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^X → M^X satisfy (MCO1)-(MCO3) and (MDF). Define $C_{c o} : 2^{X} \to M$ by: $\begin{matrix} \forall A \in 2^{X}, c_{co} (A) & = & ⋀_{x \notin A} (co (A) (x))^{'} . \end{matrix}$ Then $c_{co}$ is an M-fuzzifying convex structure with ${co}_{c_{co}} = co$ . Moreover, $c_{{co}_{c}} = c$ for any M-fuzzifying convex structure $c$ .

Definition 2.9. [26] Let $(X, C_{X})$ and $(Y, C_{Y})$ be M-fuzzifying convex spaces. f : X → Y is called an M-fuzzifying convex structure preserving (briefly, M-CP) mapping, if $C_{Y} (B) \leq C_{X} (f^{- 1} (B))$ for all B ∈ 2^Y.

Definition 2.10. [25] An operator $ℋ : 2_{f i n}^{X} \to M^{X}$ is called an M-fuzzifying restricted hull operator, if for all $F, G \in 2_{fin}^{X}$ and all x ∈ X,

$H (\emptyset) (x) = ⊥$ ;

$H (F) (x) = ⊤$ whenever x ∈ F;

$H (G) (x) \land ⋀_{y \in G} H (F) (y) \leq H (F) (x)$ .

Remark 2.11. [25 , 32] (1) $f : (X, C_{X}) \to (Y, C_{Y})$ is an M-CP mapping iff $c o_{C}_{_{X}} (f^{- 1} (B)) (x) \leq c o_{C}_{_{Y}} (B) (f (x))$ for all B ∈ 2^X and x ∈ X.

(2) For an M-fuzzifying convex space $(X, c)$ and r ∈ P (M), $(X, c_{[r^{'}]})$ is a convex space, whose hull operator is exactly co^(r) : 2^X → 2^X, defined by: co^(r) (A) = co (A) ^(r) for all A ∈ 2^X. For convenience, ${Seg}_{c_{[r^{'}]}} ({a, b})$ is simply denoted by ab^(r).

Definition 2.12. [34] An M-fuzzifying convex space $(X, c)$ is called an M-fuzzifying convex matroid, if

co (A) (p) ∨ co (A) (q) ∨ co (A ∪ {p}) (q) = co (A) (p) ∨ co (A) (q) ∨ co (A ∪ {q}) (p) for all A ∈ 2^X and all p, q ∈ X.

Definition 2.13. [33, 37] An operator $ℐ : X \times X \to M^{X}$ is called an M-fuzzifying interval operator and the pair $(X, I)$ is called an M-fuzzifying interval space, if for all x, y ∈ X,

$I (x, y) (x) = I (x, y) (y) = ⊤$ ;

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

Further, $(X, I)$ is called an M-fuzzifying geometric interval space, if for all a, b, c, d ∈ X ,

$I (a, a) = a_{⊤}$ ;

$I (a, b) (c) \land I (a, c) (d) \leq I (a, b) (d)$ .

$I (a, b) (c) \land I (a, b) (d) \land I (a, d) (c) \leq I (c, b) (d)$ .

Proposition 2.14. [33, 37] (1) An M-fuzzifying interval space $(X, I)$ satisfying (MGI1) is an M-fuzzifying geometric interval space iff for all a, b, c, d ∈ X, (MGI2&3) $I (a, b) \land I (b, c) (d) \leq I (a, b) (d) \land I (a, d) (c)$ .

(2) $I$ is an M-fuzzifying (resp. geometric) interval operator iff $ℐ^{(r)} : X \times X \to 2^{X}$ , defined by: $I^{(r)} (a, b) = I (a, b)^{(r)}$ for all a, b ∈ X, is an (resp. a geometric) interval operator for any r ∈ P (M).

Theorem 2.15. [37] TH2.16 The segment operator of an M-fuzzifying convex space is an M-fuzzifying interval operator. Conversely, if $(X, ℐ)$ is an M-fuzzifying interval space, then $C_{ℐ} : 2^{X} \to M$ , defined by: $\begin{matrix} \forall A \in 2^{X}, c_{I} (A) = ⋀_{z \notin A} ⋀_{x, y \in A} (I (x, y) (z))^{'}, \end{matrix}$ is an M-fuzzifying convex structure generated by $I$ . In addition, $I \leq {Seg}_{c_{I}}$ .

Definition 2.16. [32, 33] 2.17 (1) An M-fuzzifying convex space $(X, c)$ is called an JHC convex space, if for all a, z ∈ X and A ∈ 2^X ∖,

(MJHC)co (A) (z) = ⋁ _x∈Xax (z) ∧ co (A) (x).

(2) An M-fuzzifying interval space $(X, I)$ is called an M-fuzzifying Peano interval space, if

(MPeano) $I (b, c) (y) \land I (a, y) (z) \leq ⋁_{x \in x} I (a, b) (x) \land I (c, x) (z)$ for all a, b, c, y, z ∈ X,

(3) An M-fuzzifying interval space $(X, I)$ is called an M-fuzzifying Pasch interval space, if

(MPasch) $I (p, a) (\bar{a}) \land I (p, b) (\bar{b}) \leq ⋁_{x \in x} I (\bar{a}, b) (x) \land I (a, \bar{b}) (x)$ for all $p, a, b, \bar{a}, \bar{b} \in X$ .

3 M-fuzzifying Bryant-Webster spaces

In this section, we further define other properties, based on which we define M-fuzzifying Bryant-Webster spaces and M-fuzzifying convex geometries.

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

(1) M-fuzzifying dense interval space, if for all a, b ∈ X with a ≠ b,

$⋁_{x \in X ∖ {a, b}} I (a, b) (x) \neq ⊥$ , and for each z ∈ X ∖ {a, b}, there is w ∈ X ∖ {a, b, z} such that $I (a, b) (z) \leq I (z, b) (w)$ ;

(2) M-fuzzifying decomposable interval space, if for all a, b, x, z ∈ X

$I (a, b) (x) \land [I (a, x) \lor I (x, b)] (z) = I (a, b) (x) \land I (a, b) (z)$ ;

$I (a, b) (x) \land [I (a, x) \land I (x, b)] (z) = I (a, b) (x) \land x_{⊤} (z)$ ;

(3) M-fuzzifying ramification interval space, if for all b, c, d, z ∈ X with z ≠ b,

(MRI) $[I (b, c) \land I (b, d)] (z) \leq I (b, c) (d) \lor I (b, d) (c)$ ;

(4) M-fuzzifying straight interval space, if for all a, b, c, d, x, y ∈ X with x ≠ y,

(MSI) $t_{I} (a, c, b, d, x, y) \leq ⋀_{t_{I} (a, c, b, d, x, y) ≰^{*} r} ⋁_{u, v \in X} ⋀_{x \in A \cup B} [(r \lor (A ∖ (A \cap B))] (x)$ , where $A = I^{(r)} (a, c) \cup I^{(r)} (b, d)$ , $B = (A \cup I^{(r)} (u, v)) ∖ (A \cap I^{(r)} (u, v))$ and $t_{I} (a, c, b, d, x, y) = I (a, c) (x) \land I (a, c) (y) \land I (b, d) (x) \land I (b, d) (y)$ .

The following relations among properties defined in Definitions 2.13 and 3.1 can be easily proved.

Proposition 3.2. Let $(X, I)$ be an M-fuzzifying interval space.

(1) $I$ has M-fuzzifying decomposable (resp. ramification, straight) property iff $I^{(r)}$ has decomposable (resp. ramification, straight) property for r ∈ P (M).

(2) If $I$ has M-fuzzifying decomposable property, then (MSI) and (MSI*) are equivalent, where

$t_{I} (a, c, b, d, x, y) \leq ⋁_{u, v \in Y} ⋀_{w \in Y} I (u, v) (w)$ , where Y = {a, b, c, d} ∈2^X, x, y ∈ X (x ≠ y).

(3) (MDecI1) and (MDecI2) imply $I (a, b) (c) \land I (a, c) (b) = ⊥$ for all different points a, b, c ∈ X.

Proposition 3.3. Let $(X, I)$ be an M-fuzzifying interval space.

(1) If $I$ has M-fuzzifying decomposable property, then it also has M-fuzzifying geometric property.

(2) If $I$ has M-fuzzifying decomposable property and M-fuzzifying straight property, then it also has M-fuzzifying ramification property.

Theorem 3.4. TH3.5 The segment operator of an M-fuzzifying convex matroid satisfying (MGI1) has M-fuzzifying straight and M-fuzzifying ramification properties.

Theorem 3.5. TH3.9 Let $(X, c)$ be an M-fuzzifying convex space.

(1) $(X, c)$ is an M-fuzzifying convex geometry iff co (F ∪ {p}) (q) ∧ co (F ∪ {q}) (p) ≤ co (F) (p) ∨ co (F) (q) for all $F \in 2_{fin}^{X}$ , p, q ∈ X.

(2) If $(X, c)$ has M-fuzzifying JHC property, then it is an M-fuzzifying convex geometry iff $⋀_{i = 2}^{n - 1} a_{i - 1} a_{i + 1} (a_{i}) \leq ⋀_{i = 2}^{n - 1} a_{1} a_{n} (a_{i})$ for any finite sequence of distinct points ${a_{i}}_{i = 1}^{n} \subseteq 2^{X}$ .

(3) An M-fuzzifying BW space is an M-fuzzifying convex geometry.

Remark 3.6. In [29], if $(X, I)$ is a dense interval space, and ${Seg}_{c_{I}}$ has ramification, decomposable and Peano properties, then ${Seg}_{c_{I}}$ also has Pasch property. But, this result fails in M-fuzzifying settings.

Let M = [0, 1], and X be the set of all points surrounded by a triangle ▵pab containing its sides $\bar{p a}, \bar{p b} and \bar{a b}$ . Define an operator $ℐ : X \times X \to M^{X}$ by:

(1) ∀x ∈ X, $I (x, x) = x_{⊤}$ ;

(2) $\forall x, y \in \bar{pa}$ (resp. $x, y \in \bar{pb}$ ) with x ≠ y, ∀z ∈ X, $I (x, y) (z) = {\begin{matrix} 1, & z \in \bar{xy}; \\ 0, & z \notin \bar{xy}; \end{matrix}$

(3) $\forall {x, y} ⊈ \bar{pa}$ and ${x, y} ⊈ \bar{pb}$ with x ≠ y, ∀z ∈ X, $I (x, y) (z) = {\begin{matrix} \frac{1}{2}, & z \in \bar{xy}; \\ 0, & z \notin \bar{xy}; \end{matrix}$

Then $(X, I)$ is an M-fuzzifying interval space and ${Seg}_{c_{I}} = I$ has M-fuzzifying dense (Peano, decomposable, straight and ramification) property.

Let $c \in \bar{pa}$ and $d \in \bar{pb}$ be the middle points, and $\bar{bc} \cap \bar{ad} = {e}$ . Then $I (p, a) (c) \land I (p, b) (d) = 1$ and $⋁_{x \in X} [I (b, c) \land I (a, d)] (x) = [I (b, c) \land I (a, d)] (e) = \frac{1}{2}$ . Thus (MPasch) fails for ${Seg}_{c_{I}}$ .

Definition 3.7. (1) An M-fuzzifying convex space $(X, c)$ is called an M-fuzzifying Bryant-Webster (or, M-fuzzifying BW) space if ${Seg}_{c}$ has M-fuzzifying JHC property, M-fuzzifying decomposable property, M-fuzzifying Pasch property, M-fuzzifying straight property and M-fuzzifying dense property.

(2) An M-fuzzifying convex space $(X, c)$ is called an M-fuzzifying convex geometry if co (A ∪ {p}) ∧ co (A ∪ {q}) ≤ co (A) (p) ∨ co (A) (q) for all A ∈ L^X and p, q ∈ X.

The category of M-fuzzifying BW spaces and M-CP mappings is denoted by M-BWS.

Example 3.8. (3) Let X, M = [0, 1]. Define an operator $ℐ : X \times X \to M^{X}$ by:

(3i) $I (a, a) = a_{1}$ and $I (a, b) = I (b, a)$ ;

(3ii) if a < b, then $I (a, b) (x) = {\begin{matrix} 1, & x = a, b; \\ \frac{1}{2}, & x \in (a, b); \\ 0, & x \notin (a, b) . \end{matrix}$

Then $(X, c_{I})$ is an M-fuzzifying BW space.

(2) A BW space is a 2-fuzzifying BW space. In addition, the convex space induced by a dense and totally ordered lattice is a 2-fuzzifying BW space [29].

4 M-fuzzifying join spaces

In this section, we define an operator which can be regarded as an M-fuzzifying ‘open’ interval operator. Then we introduce M-fuzzifying join space and obtain its relations with M-fuzzifying BW spaces.

Let ∘: X × X → M^X be an operator, a, b, c ∈ X, and U, V ∈ M^X. We define: for any z ∈ X,

(1) [a ∘(b ∘c)] (z) = ⋁ _x∈X (b ∘c) (x) ∧ (a ∘x) (z);

(2) [(a ∘b) ∘c] (z) = ⋁ _x∈X (a ∘b) (x) ∧ (x ∘c) (z);

(3) (a/b) (z) = (b ∘z) (a);

(4) (U ∘V) (z) = ⋁ _a,b∈XU (a) ∧ V (b) ∧ (a ∘b) (z);

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

For simplicity, a_⊤ ∘V (resp. a_⊤/V, U ∘b_⊤, U/b_⊤) is instead by a ∘V (resp. a/V, A ∘b, U/b). We further define some operation rules for ∘as follows:

(MJ0) if a ≠ b, then (a∘b) (a) = (a ∘b) (b) = ⊥

(MJ1) if a ≠ b, then ⋁_{x∈X∖{a,b}} (a∘b) (x) ≠ ⊥, and for each y ∉ {a, b}, there is w ∈ X ∖ {a, b, y} such that (a ∘b) (y) ≤ (y ∘b) (w);

(MJ2) a ∘a = a/a = a_⊤;

(MJ2*) a ∘b = b ∘a;

(MJ3) a ∘(b ∘c) = (a ∘b) ∘c;

(MJ4) [(a/b) ∧ (c/d)] (p) ≤ ⋁ _x∈X [(a ∘d) ∧ (b ∘c)] (x);

(MJ5) [(a ∘b) ∧ (a ∘c)] (p) ≤ b_⊤ (c) ∨ (a ∘c) (b) ∨ (a ∘b) (c).

For an operator ∘: X × X → M^X, it is clear that ∘satisfies (MJx) iff ∘^(s) : X × X → 2^X satisfies (Jx) for all s ∈ P (M), where x ∈ {0, 2, 2^*, 3, 4, 5}, and ∘^(s) is defined by: a ∘^(s)b = (a ∘b) ^(s) for all a, b ∈ X.

We define M-fuzzifying join spaces as follows.

Definition 4.1. ∘: X × X → M^X is called an M-fuzzifying join operator and the pair (X, ∘) is called an M-fuzzifying join space, if ∘satisfies (MJ0)-(MJ5).

If (X, ∘) be an M-fuzzifying join space, then $ℐ_{○} : X \times X \to M^{X}$ , defined by: $I_{\circ} (a, b) = a_{⊤} \lor b_{⊤} \lor (a \circ b)$ for all a, b ∈ X, is an M-fuzzifying interval space. For M-fuzzifying join spaces (X, ∘_X) and (Y, ∘_Y), f : X → Y is called an M-fuzzifying join preserving (briefly, M-JP) mapping, if $f : (X, I_{\circ_{X}}) \to (Y, I_{\circ_{Y}})$ is an M-fuzzifying IP mapping.

The category of M-fuzzifying join spaces and M-JP mappings is denoted by M-JS.

Example 4.2. (1) A join space [29] is a 2-fuzzifying join space.

(2) Let X, M = [0, 1]. Define an operator ∘: X × X → M^X by:

(2i) a ∘a = a₁ and a ∘b = b ∘a;

(2ii) if a < b, then $a \circ b (x) = {\begin{matrix} \frac{1}{2}, & x \in (a, b); \\ 0, & x \notin (a, b) . \end{matrix}$

Then (X, ∘) is an M-fuzzifying join space.

Lemma 4.3. Let (X, ∘) be an M-fuzzifying join space, and let a, b, c, d ∈ X. We have

(1) ∘satisfies (MJ2*).

(2) (a ∘b) (c) ∧ (b ∘c) (d) ≤ (a ∘b) (d) ∧ (a ∘d) (c);

(3) (a∘b) (c) ∧ (a ∘c) (b) = ⊥ if a, b, c are different.

Proof. If a = b (resp. z = b or z = a), then (MJ2*) is trivial. For a ≠ b and z ≠ a, b, by (MJ1), there is x ∈ X ∖ {a, b, z} with (a ∘b) (z) ≤ (z ∘b) (x). So $\begin{matrix} (a \circ b) (z) & = & [(x / z) \land (z / a)] (b) \land (z \circ b) (x) \\ \leq & ⋁_{y \in X} [(x \circ a) \land (z \circ z)] (y) \land (z \circ b) (x) \\ = & [(z \circ b) \circ a] (z) = [z \circ (b \circ a)] (z) \\ = & ⋁_{w \in X} (b \circ a) (w) \land (z / z) (w) = (b \circ a) (z) . \end{matrix}$

This shows a ∘b ≤ b ∘a. Similarly, b ∘a ≤ a ∘b.

(2) and (3) follow from Definition 4.1 and (1). □

The following result is similar to Theorem 3.6 in [37] and Lemmas 4.12 and 4.13 in [33].

Theorem 4.4. Let (X, ∘) be an M-fuzzifying join space and $C_{○} : 2^{X} \to M$ be defined by: $\begin{matrix} \forall A \in X, c_{\circ} (A) = ⋀_{x \notin A} ⋀_{a, b \in A} [(a \circ b) (x)]^{'} . \end{matrix}$ Then $c_{\circ}$ is an M-fuzzifying convexity satisfying

(1) $\circ \leq {Seg}_{c_{\circ}}$ ;

(2) $(c_{\circ})_{[s^{'}]} = c_{\circ^{(s)}}$ for all s ∈ P (M);

(3) (U ∘U) ≤ U iff $U^{(s)} \in c_{\circ^{(s)}}$ for all s ∈ P (M).

Theorem 4.5. If (X, ∘) is an M-fuzzifying join space, then $(X, c_{\circ})$ is an M-fuzzifying JHC convex space whose hull operator satisfies $co ({a_{i}}_{i = 1}^{n}) (x) = ⋁_{1 \leq k \leq n} (a_{i_{1}} \circ \cdot \cdot \cdot \circ a_{i_{k}}) (x)$ for all ${a_{i}}_{i = 1}^{n} \in 2_{fin}^{X}$ , where a_{i
₁} ∘·· · ∘a_{i
_k} = (a_{i
₁}) _⊤ whenever k = 1.

Proof. The formula of the hull operator can be proved by induction on the number k of the joined points. The M-fuzzifying JHC property of $c_{\circ}$ directly follows from this formula. □

Lemma 4.6. Let (X, ∘) be an M-fuzzifying join space and $(X, c_{\circ})$ be its induced M-fuzzifying convex space. If ${a_{i}}_{i = 1}^{n}$ is a finite sequence of distinct points, then $⋀_{i = 2}^{n - 1} a_{i - 1} a_{i + 1} (a_{i}) \leq ⋀_{i = 2}^{n - 1} a_{1} a_{n} (a_{i})$ .

Let us study relations between M-BWS and M-JS.

Theorem 4.7. If (X, ∘) is an M-fuzzifying join space, then $(X, c_{\circ})$ is an M-fuzzifying BW space.

Proof. By Theorem 4.5, $c_{\circ}$ satisfies (MJHC). By (1) of Theorem 4.4 and (MJ1), (MDenI) is trivial. Next, we prove that (MDecI1) and (MDecI2) hold for ${Seg}_{c_{\circ}}$ .

Let a, b, x, z ∈ X. (MDecI1) and (MDecI2) are trivial when a = b. If a ≠ b, we have the following cases.

(1) x = a (resp. x = b, x = z). Then (MDecI1) and (MDecI2) hold trivially.

(2) x ≠ a, b, z.

(2a) z = a. By Lemma 4.3 and (MJ2), (MDecI1) and (MDecI2) are trivial.

(2b) z = b. Similar to (2a).

(3) a, b, x, z are different. Clearly, ab (x)∧ x_⊤ (z) = ⊥. Further, by Theorem 4.5 and Lemma 4.3, $\begin{matrix} ab (x) \land [ax \land xb] (z) & \leq & (a \circ z) (x) \land (a \circ x) (z) = ⊥ . \end{matrix}$ Hence (MDecI2) holds.

Further, ab (x) ∧ [ax ∨ xb] (z) ≤ ab (x) ∧ ab (z) by (MRH3). By (MJ2), (MJ2*), (MJ4) and (MJ5), $\begin{matrix} ab (x) \land ab (z) = [(x / b) \land (z / b)] (a) \\ \leq & ⋁_{y \in X} [(b \circ x) \land (b \circ z)] (y) \land (a \circ b) (z) \\ \leq & [x_{⊤} (z) \lor (b \circ x) (z) \lor (b \circ z) (x)] \land (a \circ b) (z) \\ \leq & (b \circ x) (z) \lor [(x / z) \land (z / a)] (b) \\ \leq & (b \circ x) (z) \lor ⋁_{w \in X} [(a \circ x) \land (z \circ z)] (w) \\ = & (b \circ x) (z) \lor (a \circ x) (z) = [ax \lor xb] (z) . \end{matrix}$ Hence (MDecI1) also holds.

In conclusion, ${Seg}_{c_{\circ}}$ has M-fuzzifying decomposable property. Thus it also has M-fuzzifying geometric property by (1) of Proposition 3.3.

Finally, we prove that (MSI*) holds for ${Seg}_{c_{\circ}}$ , i.e., $[ac \land bd] (x) \land [ac \land bd] (y) \leq ⋁_{p, q \in Y} ⋀_{u \in Y} pq (u) (∇)$ for all Y = {a, b, c, d} ⊆ X and x, y ∈ X with x ≠ y.

We prove this by the following cases and subcases.

(1) a = b. Then x ≠ a or y ≠ a since x ≠ y.

(1a) x ≠ a. If c = d, then [ac ∧ bd] (x) ∧ [ac ∧ bd] (y) ≤ ⊤ = ac (b) ∧ ac (d).

If x = c ≠ d, then [ac ∧ bd] (x) ∧ [ac ∧ bd] (y) ≤ bd (c) = bd (a) ∧ bd (c).

If x = d ≠ c, then [ac ∧ bd] (x) ∧ [ac ∧ bd] (y) ≤ ac (d) = ac (b) ∧ ac (d).

If x ≠ c, d and c ≠ d, then by (MJ5), $\begin{matrix} [ac \land bd] (x) \land [ac \land bd] (y) \\ \leq & [(a \circ c) \land (a \circ d)] (x) \\ \leq & c_{⊤} (d) \lor (a \circ c) (d) \lor (a \circ d) (c) \\ \leq & [ac (b) \land ac (d)] \lor [ad (c) \land ad (b)] . \end{matrix}$

(1b) y ≠ a. Similar to (1a).

(2) a = d (resp. c = b, c = d). Similar to (1).

Thus (∇) holds in either case of (1) and (2).

(3) a, c ≠ b, d. By (MDec1), $\begin{matrix} ac (x) \land ac (y) & = & [ay \lor yc] (x) \land ac (y) \\ = & [ax \lor xc] (y) \land ac (x); \\ bd (x) \land bd (y) & = & [by \lor yd] (x) \land bd (y) \\ = & [bx \lor xd] (y) \land bd (x) . \end{matrix}$ Thus we have $\begin{matrix} [ac \land bd] (x) \land [ac \land bd] (y) \\ = & {[ay \lor yc] (x) \land [ax \lor xc] (y) \land [by \lor yd] (x) \land \\ [bx \lor xd] (y)} \land [ac \land bd] (x) \land [ac \land bd] (y) \\ = & {[ay \land by \land ac \land bd] (x) \land [ax \land bx \land ac \land bd] (y)} \\ \lor \cdot \cdot \cdot \lor \\ {[yc \land yd \land ac \land bd] (x) \land [xc \land xd \land ac \land bd] (y)} . \end{matrix}$

In the last equation shown as above, there are sixteen indices, including the index φ = [ay ∧ by ∧ ac ∧ bd] (x) ∧ [cx ∧ dx ∧ ac ∧ bd] (y). To prove (∇), we prove that φ ≤ ⋁ _p,q∈Y ⋀ _u∈Ypq (u) by the have following cases. Other indices have similar property.

(3a) a = x ≠ b.

If y = d, then φ = bd (a) ∧ ac (d) ≤ bc (a) ∧ bc (d) by Lemma 4.6. Similarly, if y = c, then φ ≤ bc (a) ∧ ad (c) ≤ bd (a) ∧ bd (c). If y ≠ c, d, then by (MJ6), $\begin{matrix} φ & \leq & [cx \land dx] (y) = [(c \circ x) \land (d \circ x)] (y) \\ \leq & (d \circ x) (c) \lor (c \circ x) (d) = ad (c) \lor ac (d) . \end{matrix}$

Also, ad (c) ∧ ac (y) ≤ ad (y) by (MGI2) and ad (c) ∧ ac (y) ∧ ad (y) ≤ dy (c) by (MGI3). By Lemma 4.6, $\begin{matrix} φ & \leq & [ad (c) \lor ac (d)] \land by (a) \land ac (y) \land bd (a) \\ \leq & [by (a) \land ad (c) \land ac (y)] \lor [ac (d) \land bd (a)] \\ \leq & [by (a) \land ac (y) \land yd (c)] \lor [ac (d) \land bd (a)] \\ \leq & [bd (a) \land bd (c)] \lor [bc (a) \land bc (d)] . \end{matrix}$

(3b) b = x (resp. c = y, d = y). Similar to (3a).

(3c) x, y ≠ a, b, c, d. We have [ay ∧ by] (x) ≤ by (a) ∨ ay (b) and [xc ∧ xd] (y) ≤ xd (c) ∨ xc (d) by (MJ5), and by (a) ∧ ay (x) ≤ bx (a), ay (b) ∧ by (x) ≤ ax (b), xd (c) ∧ xc (y) ≤ dy (c) and xc (d) ∧ xd (y) ≤ cy (d) by (MGI2&3). Hence, by Lemma 4.6 again, we can prove that φ ≤ ⋁ _p,q∈Y ⋀ _u∈Ypq (u).

Therefore (∇) holds.

Finally, we prove that ${Seg}_{c_{\circ}}$ satisfies (M-Pasch). Note that ab = a_⊤ ∨ b_⊤ ∨ (a ∘b) for all a, b ∈ X by Theorem 4.5. Let $p, a, b, \bar{a}, \bar{b} \in X$ . We need to prove that $pa (\bar{a}) \land pb (\bar{b}) \leq ⋁_{x \in X} [\bar{a} b \land a \bar{b}] (x)$ .

We have the following cases.

(1) $\bar{a} = a$ . Then $pa (\bar{a}) \land pb (\bar{b}) \leq ⊤ = [\bar{a} b \land a \bar{b}] (a)$ .

(2) $\bar{a} = p$ . Then $pa (\bar{a}) \land pb (\bar{b}) = [\bar{a} b \land a \bar{b}] (\bar{b})$ .

(3) $\bar{b} = b$ . Then $pa (\bar{a}) \land pb (\bar{b}) \leq ⊤ = [\bar{a} b \land a \bar{b}] (b)$ .

(4) $\bar{b} = p$ . Then $pa (\bar{a}) \land pb (\bar{b}) = [\bar{a} b \land a \bar{b}] (\bar{a})$ .

Thus in either case of (1)–(4), (MPasch) holds.

(5) $\bar{a} \neq p, a$ and $\bar{b} \neq p, b$ . By (MJ2*) and (MJ4), $pa (\bar{a}) \land pb (\bar{b}) = [(\bar{a} / a) \land (\bar{b} / b)] (p) \leq ⋁_{x \in X} [\bar{a} b \land a \bar{b}] (x)$ . This shows that ${Seg}_{c_{\circ}}$ satisfies (MPasch). Therefore $(X, c_{\circ})$ is an M-fuzzifying BW space. □

Theorem 4.8. Let $(X, c)$ be an M-fuzzifying BW space. Define $\circ_{c} : X \times X \to M^{X}$ by: for a, b, z ∈ X, $\begin{matrix} a \circ_{c} b (z) = {\begin{matrix} a_{⊤} (z), & a = b; \\ ab (z), & a \neq b & z \in X ∖ {a, b}; \\ ⊥, & a \neq b & z \in {a, b} . \end{matrix} \end{matrix}$ Then (X, ∘) is an M-fuzzifying join space.

Proof. For convenience, we denote $\circ_{c}$ by ∘. (MJ0)–(MJ2) are clear. We prove that ∘satisfies (MJ3)-(MJ5).

(MJ3). Let a, b, c, z ∈ X. We prove [a ∘(b ∘c)] (z) = [(a ∘b) ∘c] (z) by the following cases and subcases.

(1) a = b. Then [(a ∘b) ∘c] (z) = a ∘c (z).

The case that a = c or z = a is clear. Let a ≠ c, z.

If z = c, then [(a∘b) ∘c] (c) = (a ∘c) (c) = ⊥. By (MDecI2), [a∘(b ∘c)] (c) ≤ ⋁ _{x∈X∖{a,c}} [ax ∧ xc] (c) ∧ ac (x) = ⊥. Thus [a ∘(b ∘c)] (z) = [(a ∘b) ∘c)] (z).

If z ≠ c, then by (MRH3), [a ∘(b ∘c)] (z) ≤ ac (z) = [(a ∘b) ∘c)] (z). Conversely, by (MDen1), there exists w ∈ X ∖ {a, c, z} such that ac (z) ≤ zc (w). Thus zc (w) ∧ ac (z) ≤ ac (w) by (MRH3). Hence, by (MDenI1), (MDenI2) and (3) of Proposition 3.2, $\begin{matrix} [(a \circ b) \circ c] (z) & = & [ac (z) \land ac (w)] \land zc (w) \\ = & [aw \lor wc] (z) \land ac (w) \land zc (w) \\ \leq & [aw (z) \land ac (w)] \lor [wc (z) \land zc (w)] \\ = & aw (z) \land ac (w) \\ \leq & [a \circ (b \circ c)] (z) . \end{matrix}$

(2) a = c or b = c. The desired result is easy.

(4) a, b, c are different.

We firstly prove that [a ∘(b ∘c)] (z) ≤ [(a ∘b) ∘c] (z).

Let r ∈ P (M) with ⋁_x∈X (a ∘x) (z) ∧ (b ∘c) (x) = [a ∘(b ∘c)] (z) ≰ r. So there exists x ∈ X ∖ {c, b} such that (a ∘x) (z) ∧ (b ∘c) (x) ≰ r. We next prove that $[(a \circ b) \circ c] (z) ≰ r * .$ (2)

If x = a, then x = a = z follows from a_⊤ (z) = x ∘a (z) ≰ r. By (MDenI), there is w ∈ X ∖ {b, c, x} such that r≱c ∘b (x) = cb (x) ≤ xb (w). This shows ab (w) ≰ r. Further, bc (a) ∧ ba (w) ≤ bc (w) by (MGI2) and bc (a) ∧ bc (w) ∧ ba (w) ≤ wc (a) by (MGI3). Thus [(a ∘b) ∘c] (z) ≥ (a ∘b) (w) ∧ (c ∘w) (z) ≰ r.

If x ≠ a, then z, a, x are different since (a ∘x) (z) ≰ r. Now, we have the following cases.

(4a) a ∘b (c) ≰ r. Then ab (c) = a ∘b (c) ≰ r. Thus we further have the following subcases.

(4a(i)) z = c. Then [(a ∘b) ∘c] (z) ≥ a ∘b (c) ∧ c ∘c (z) ≰ r.

(4a(ii) z ≠ c. By (MGI2), ab (x) ≥ ab (c) ∧ bc (x) ≰ r. Thus, by (MDecI1), ax (c) ∨ xb (c) = ab (x) ∧ (ax ∨ xb) (c) = ab (x) ∧ ab (c) ≰ r. Hence ax (c) ≰ r or xb (c) ≰ r. Note that bc (x) = b ∘c (x) ≰ r and ax (z) = a ∘x (z) ≰ r. We say xb (c) ≤ r. Otherwise, by (MDecI2), x_⊤ (c) ∧ bc (x) = bc (x) ∧ (bx ∧ xc) (c) ≰ r. This shows x = c which is a contradiction. Hence xb (c) ≤ r and ax (c) ≰ r. Further, by (MDenI1), ac (z) ∨ cx (z) ≥ ax (c) ∧ [ac ∨ cx] (z) = ax (c) ∧ ax (z) ≰ r. So ac (z) ≰ r or cx (z) ≰ r.

If cx (z) ≰ r, then c ∘x (z) = cx (z) ≰ r and thus [(a ∘b) ∘c] (z) ≥ a ∘b (x) ∧ x ∘c (z) ≰ r.

If ac (z) ≰ r, we can fix a point v ∈ X ∖ {a, c, z} such that ca (z) ≤ za (v) by (MDenI). This shows (a ∘z) (v) = az (v) ≰ r. Thus ac (v) ≥ az (v) ∧ ac (z) ≰ r by (MGI2). Further, by (MDecI1), av (z) ∨ vc (z) ≥ ac (v) ∧ [av ∨ vc] (z) = ac (v) ∧ ac (z) ≰ r.

We say that av (z) ≤ r. Otherwise, av (z) ≰ r and v_⊤ (z) ∧ az (v) = az (v) ∧ (av ∧ vz) (z) ≰ r. It implies v = z which is a contradiction. Hence av (z) ≤ r and v ∘c (z) = vc (z) ≰ r.

Note that ab (v) ≥ ab (c) ∧ ac (v) ≰ r and ab (z) ≥ ab (c) ∧ ac (z) ≰ r by (MGI2). We say that v ≠ b. Otherwise, v = b. By (MDecI2), ab (z) ∧ z_⊤ (v) = ab (z) ∧ [az ∧ zb] (v) = ab (z) ∧ az (v) ∧ zb (b) ≰ r. This shows b = v = z which is a contradiction. Hence v ≠ b and [(a ∘b) ∘c] (z) ≥ a ∘b (v) ∧ c ∘v (z) ≰ r.

(4b) a ∘c (b) ≰ r or b ∘c (a) ≰ r. Similar to (4a).

(4c) a ∘b (c) ∨ a ∘c (b) ∨ b ∘c (a) ≤ r. Note that bc (x) ∧ ax (z) ≰ r. By (MJHC) and (MRH3), $\begin{matrix} ⋁_{y \in X} ab (y) \land yc (z) = co ({a, b, c}) (z) \geq ax (z) \land bc (x) . \end{matrix}$ Thus there exists y ∈ X such that ab (y) ∧ yc (z) ≰ r.

Further, it follows from (2) of Proposition 3.4 that ${Seg}_{c}$ has M-fuzzifying ramification property. To prove (*), we check z ≠ y, c and y ≠ a, b one by one.

(4c(i)) Suppose that z = y. We further prove that ab (x) ∨ ax (b) ≤ r.

Suppose that ab (x) ≰ r. By (MRI), r ≥ bc (a) ∨ ab (c) ≥ [ba ∧ bc] (x) ≰ r which is a contradiction. Thus ab (x) ≤ r.

Suppose that ax (b) ≰ r. Note that x ≠ a, b, c and bc (x) ≰ r. Let Y = {a, b, c, x}. By (MSI*), $\begin{matrix} t_{{Seg}_{c}} (a, x, b, c, b, x) \leq ⋁_{u, v \in Y} ⋀_{w \in Y} uv (w) . \end{matrix}$ Thus there exist u, v ∈ Y with uv (w) ≰ r for all w ∈ Y. Now, we have the following cases.

If u, v ∈ {a, b} (resp. u, v ∈ {b, x}, u, v ∈ {c, x}), then ab (x) ≰ r (resp. bx (c) ≰ r, cx (b) ≰ r). Thus ⊥ = ab (x) ∧ ax (b) ≰ r (resp. ⊥ = bx (c) ∧ bc (x) ≰ r, ⊥ = cx (b) ∧ bc (x) ≰ r) by (3) of Proposition 3.2. Hence either case is a contradiction.

If u, v ∈ {a, x}, then ax (c) ≰ r. By (MDecI1), ab (c) ∨ bx (c) ≥ ax (b) ∧ [ab ∨ bx] (c) = ax (b) ∧ ax (c). Thus bx (c) ≰ r. However, by (3) of Proposition 3.2, ⊥ = bx (c) ∧ bc (x) ≰ r which is a contradiction.

If u, v ∈ {b, c} or u, v ∈ {a, c}, then bc (a) ≰ r or ac (b) ≰ r. It contradicts the assumption of (4c).

Hence u, v ∈ Y don’t exist. Therefore ax (b) ≤ r. Therefore we conclude that ab (x) ∨ ax (b) ≤ r.

Note that z ≠ a and z = y. By (MRI), r ≥ ab (x) ∨ ax (b) ≥ [ab ∧ ax] (z) = ab (y) ∧ ax (z) ≰ r which is a contradiction. Therefore z ≠ y.

(4c(ii)) Suppose that z = c. Note that x ≠ a, b, c and ax (c) = ax (z) ≰ r.

Let Y = {a, b, c, x}. By (MSI*), $⋁_{u, v \in Y} ⋀_{w \in Y} uv (w) \geq t_{{Seg}_{c}} (a, x, c, b, x, c) ≰ r$ . Thus there exist u, v ∈ Y such that uv (w) ≰ r for all w ∈ Y. Similar to (4c(i)), we can prove that u, v ∈ {a, b}. But ab (c) ≰ r is a contradiction. Hence z ≠ c. Therefore we conclude that z ∈ yc^(r) ∖ {y, c} = y ∘^(r)c.

(4c(iii)) Suppose that y = a. Note that ac (z) = yc (z) ≰ r. Since z ≠ a, ax (c) ∨ ac (x) ≥ [ax ∧ ac] (z) ≰ r by (MRI). Thus ac (x) ≰ r or ax (c) ≰ r.

If ax (c) ≰ r, then $⋁_{u, v \in Y} ⋀_{w \in Y} uv (w) \geq t_{{Seg}_{c}} (a, x, c, b, x, c) ≰ r$ by (MSI*), where Y = {a, b, c, x}. Thus there exist u, v ∈ Y such that uv (w) ≰ r for all w ∈ Y. Similar to (4c(i)), it follows that u, v ∈ {a, b}. But ab (c) ≰ r. It is a contradiction.

If ac (x) ≰ r, then ab (c) ∨ ac (b) ≥ [ac ∧ bc] (x) ≰ r by x ≠ c and (MRI). It contradicts the assumption of (4c). Therefore we conclude that y ≠ a.

(4c(iv)) Suppose that y = b. We further prove that xb (a) ∨ xa (b) ≤ r.

Suppose that xb (a) ≰ r. By (MGI2), bc (a) ≥ bc (x) ∧ bx (a) ≰ r, a contradiction. Thus xb (a) ≤ r.

Suppose that ax (b) ≰ r. By (MDecI1), ax (z) ∧ [az ∨ zx] (b) = ax (b) ∧ [ab ∨ bx] (z) = ax (z) ∧ ax (b) ≰ r. Thus [az ∨ zx] (b) ≰ r and [ab ∨ bx] (z) ≰ r.

Suppose that zx (b) ≰ r, we say bx (z) ≤ r. Otherwise, ⊥ = xz (b) ∧ xb (z) ≰ r by (3) of Proposition 3.2. Thus bx (z) ≤ r and ab (z) ≰ r. Since y = b and yc (z) ≰ r, we have bc (z) ≰ r. Note that z ≠ y = b by (4c(i)). By (MRI), ab (c) ∨ bc (a) ≥ [ab ∧ bc] (z) ≰ r. However, this contradicts the assumption of (4c). Therefore zx (b) ≤ r and az (b) ≰ r.

Note that z ≠ y = b, az (y) = az (b) ≰ r and bc (z) = yc (z) ≰ r. By (MSI*), $⋁_{u, v \in Y} ⋀_{w \in Y} uv (w) \geq t_{{Seg}_{c}} (a, z, b, c, z, b) ≰ r$ , where Y = {a, b, z, c}. Thus there exist u, v ∈ Y such that uv (w) ≰ r for all w ∈ Y. Similar to (4c(i)), we can prove u, v ∈ {a, c}. But ac (b) ≰ r is a contradiction. Hence az (b) ≰ r. Therefore we conclude that xa (b) ≤ r.

Note that z ≠ x. By (MRI), r ≥ xb (a) ∨ xa (b) ≥ [xb ∧ xa] (z). We further prove that xc (z) ≤ r.

Suppose that xc (z) ≰ r. By (MSI*), $⋁_{u, v \in Y} ⋀_{w \in Y} uv (w) \geq t_{{Seg}_{c}} (a, x, c, x, x, z) ≰ r$ , where Y = {a, z, x, c}. Thus there exist u, v ∈ Y such that uv (w) ≰ r for all w ∈ Y. Note that ax (z) ∧ cx (z) ∧ bc (x) ≰ r. Similar to (4c(i)), we can prove u, v ∈ {a, x} which shows ax (c) ≰ r. Hence by (MSI*) again, ⋁_s,t∈Z $⋀_{q \in Z} st (q) \geq t_{{Seg}_{c}} (a, x, c, b, x, c) ≰ r$ , where Z = {a, c, x, b}. Thus there exist s, t ∈ Z such that st (q) ≰ r for all q ∈ Z. However, similar to (4c(i)), s, t ∈ Z can not be find. It is a contradiction. So u, v ∉ {a, x}. That is, u, v ∈ Y can not be find. Therefore xc (z) ≤ r.

Since y = b, bc (x) ≰ r and yc (z) ≰ r, we have bc (x) ∧ [bx ∨ xc] (z) = bc (z) ∧ bc (x) ≰ r by (MDecI1). Thus bx (z) ≰ r since xc (z) ≤ r. Since x ≠ z, $⋁_{u, v \in W} ⋀_{w \in W} uv (w) \geq t_{{Seg}_{c}} (a, x, b, z, x, z) ≰ r$ by (MSI*), where W = {a, z, x, b}. Thus there exist u, v ∈ W such that uv (w) ≰ r for all w ∈ W.

If u, v ∈ {a, z} or u, v ∈ {z, x}, then az (x) ≰ r or zx (a) ≰ r. However, ⊥ = az (x) ∧ ax (z) ≰ r or ⊥ = zx (a) ∧ xa (z) ≰ r by (3) of Proposition 3.2. Either case is a contradiction.

If u, v ∈ {z, b} or u, v ∈ {x, b}, then bz (a) ≰ r or bx (a) ≰ r. However, bc (a) ≥ bc (z) ∧ bz (a) ≰ r or bc (a) ≥ bc (z) ∧ bx (a) ≰ r by (MGI2). Either case contradicts the assumption of (4c).

If u, v ∈ {a, x}, then ax (b) ≰ r. Since x ≠ b, $⋁_{s, t \in V} ⋀_{v \in V} st (v) \geq t_{{Seg}_{c}} (a, x, b, c, b, x) ≰ r$ by (MSI*), where V = {a, b, x, c}. Thus there are s, t ∈ V such that st (v) ≰ r for all v ∈ V.

If {s, t} ∈ {{a, b} , {a, c} , {b, x} , {b, c} , {x, c}}, then it contradicts the assumption of (4c).

If s, t ∈ {a, x}, then ax (c) ≰ r. By (MDecI2), ax (b) ∧ [ab ∨ bx] (c) = ax (b) ∧ ax (c) ≰ r. Thus bx (c) ≰ r since ab (c) ≤ r. But ⊥ = bx (c) ∧ bc (x) ≰ r by (3) of Proposition 3.2. Hence s, t ∉ {a, x}. Therefore s, t ∈ V can not be found. This further implies that ax (b) ≤ r followed by u, v ∉ {a, x}. So the only case u, v ∈ {a, b} left. Hence ab (x) ≰ r and x_⊤ (z) ∧ ab (x) = ab (x) ∧ [ax ∧ xb] (z) ≰ r by (MDecI2). However, this shows x = z which is a contradiction. So u, v ∉ {a, b} either. This finally implies that u, v ∈ W can not be found.

Therefore we concluded that y ≠ b.

In conclusion, y ∈ a ∘^(r)b and z ∈ y ∘^(r)c. Thus [(a ∘b) ∘c] (z) ≥ a ∘b (y) ∧ y ∘c (z) ≰ r.

Therefore [a ∘(b ∘c)] (z) ≤ [(a ∘b) ∘c] (z).

Conversely, by exchanging a and c in the inequality [a ∘(b ∘c)] (z) ≤ [(a ∘b) ∘c] (z), we find that [(a ∘b) ∘c] (z) ≤ [a ∘(b ∘c)] (z).

Therefore [a ∘(b ∘c)] (z) = [(a ∘b) ∘c] (z) which shows (MJ3) holds.

(MJ4). Let $p, a, b, \bar{a}, \bar{b} \in X$ . We next prove that $\begin{matrix} [(\bar{a} / a) \land (\bar{b} / b)] (p) \leq ⋁_{z \in X} [(\bar{a} \circ b) \land (a \circ \bar{b})] (z) . \end{matrix}$ Let r ∈ P (M) with $(p \circ a) (\bar{a}) \land (p \circ b) (\bar{b}) = [(\bar{a} / a) \land (\bar{b} / b)] (p) ≰ r$ . We need to prove that $⋁_{z \in X} [(\bar{a} \circ b) \land (a \circ \bar{b})] (z) ≰ r . Δ$ (3) We check that ${Sec}_{c}$ satisfies (MPeano), i.e., bc (s) ∧ as (t) ≤ ⋁ _q∈Xab (q) ∧ cq (t) for all a, b, c, s, t ∈ X.

If s ∈ {b, c} or t ∈ {a, s}, then it is easy to check that the desired inequality holds. If s ≠ b, c and t ≠ a, s, then bc (s) ∧ as (t) ≤ [a ∘(b ∘c)] (t) = [(a ∘b) ∘c] (t) = ⋁ _q∈Xab (q) ∧ cq (t) by (MJ3). Thus the desired inequality holds. That is, ${Seg}_{c}$ has M-fuzzifying Peano property.

Now, we prove (Δ) by the following cases.

(1) p = a. Then $a = \bar{a}$ since $a_{⊤} (\bar{a}) = (p \circ a) (\bar{a}) ≰ r$ . Thus we have the following cases.

(1a) $\bar{b} = a$ . Then b = a since a ∘b (a) ≰ r. Thus $\begin{matrix} ⋁_{z \in X} [(\bar{a} \circ b) \land (a \circ \bar{b})] (z) \geq [(\bar{a} \circ b) \land (a \circ \bar{b})] (a) = ⊤ . \end{matrix}$

(1b) $\bar{b} \neq a$ . Thus a ≠ b (otherwise, a = b and $⊥ = a_{⊤} (\bar{b}) = (p \circ b) (\bar{b}) ≰ r$ ). By (MDenI), there is $v \in X ∖ {p, b, \bar{b}}$ such that $bp (\bar{b}) \leq \bar{b} p (v)$ . Thus $\bar{b} a (v) ≰ r$ and $ab (v) \geq a \bar{b} (v) \land ab (\bar{b}) ≰ r$ by (MRH3). Further, since a ≠ b and $(a \circ b) (\bar{b}) ≰ r$ , we have $\bar{b} \neq b$ . Further, we say v ≠ b (otherwise, $⊥ = a \bar{b} (v) \land av (\bar{b}) = a \bar{b} (v) \land ab (\bar{b}) ≰ r$ by (3) of Proposition 3.2). Thus $(\bar{a} \circ b) (v) = (a \circ b) (v) = ab (v) ≰ r$ . Hence $⋁_{z \in X} [(\bar{a} \circ b) \land (a \circ \bar{b})] (z) \geq [(\bar{a} \circ b) \land (a \circ \bar{b})] (v) ≰ r$ .

(2) p = b. Similar to (1).

(3) p ≠ a = b. Since $(p \circ a) (\bar{a}) \land (p \circ b) (\bar{b}) ≰ r$ , we have $\bar{a} \neq p, a$ and $\bar{b} \neq p, b$ . We have the following cases.

(3a) $\bar{a} = \bar{b}$ . By (MDenI), there is $v \in X ∖ {p, a, \bar{a}}$ such that $pa (\bar{a}) \leq \bar{a} a (v)$ . Thus $a \bar{a} (v) ≰ r$ and $\begin{matrix} ⋁_{z \in X} [(\bar{a} \circ b) \land (a \circ \bar{b})] (z) \geq [(\bar{a} \circ b) \land (a \circ \bar{b})] (v) ≰ r . \end{matrix}$

(3b) $\bar{a} \neq \bar{b}$ . By (MDecI1), $p \bar{b} (\bar{a}) \lor \bar{b} a (\bar{a}) \geq [p \bar{b} \lor \bar{b} a] (\bar{a}) \land pa (\bar{b}) = pa (\bar{a}) \land pa (\bar{b}) ≰ r$ . Thus $p \bar{b} (\bar{a}) \lor \bar{b} a (\bar{a}) ≰ r$ . We further have the following subcases.

(3b(i)) $\bar{b} a (\bar{a}) ≰ r$ . By (MDenI), there exists $w \in X ∖ {a, \bar{a}, \bar{b}}$ such that $\bar{b} a (\bar{a}) \leq a \bar{a} (w)$ . Thus $a \bar{a} (w) ≰ r$ . Further, $a \bar{b} (w) \geq a \bar{a} (w) \land a \bar{b} (\bar{a}) ≰ r$ by (MGI2). So $⋁_{z \in X} [(\bar{a} \circ b) \land (a \circ \bar{b})] (z) \geq [(\bar{a} \circ b) \land (a \circ \bar{b})] (w) ≰ r$ .

(3b(ii)) $p \bar{b} (\bar{a}) ≰ r$ . We have $a \bar{a} (\bar{b}) \geq pa (\bar{a}) \land pa (\bar{b}) \land p \bar{b} (\bar{a}) ≰ r$ by (MGI3). By (MDenI), there is $v \in X ∖ {a, \bar{a}, \bar{b}}$ such that $\bar{a} b (\bar{b}) \leq a \bar{b} (v)$ . Thus $a \bar{a} (v) \geq a \bar{a} (\bar{b}) \land a \bar{b} (v) ≰ r$ by (MGI2). Hence $\begin{matrix} ⋁_{z \in X} [(\bar{a} \circ b) \land (a \circ \bar{b})] (z) \geq [(\bar{a} \circ b) \land (a \circ \bar{b})] (v) ≰ r . \end{matrix}$

(4) p ≠ a, b and a ≠ b. Since $(p \circ a) (\bar{a}) \land (p \circ b) (\bar{b}) ≰ r$ , we have $\bar{a} \neq a, p$ and $\bar{b} \neq b, p$ . We have the following subcases.

(4a) $\bar{a} = b$ . We have $pa (\bar{b}) \geq p \bar{a} (\bar{b}) \land pa (\bar{a}) = pb (\bar{b}) \land pa (\bar{a}) ≰ r$ by (MGI2), and $a \bar{b} (\bar{a}) \geq pa (\bar{a}) \land pa (\bar{b}) \land p \bar{a} (\bar{b}) ≰ r$ by (MGI3). Thus $(a \circ \bar{b}) (b) = a \bar{b} (b) = a \bar{b} (\bar{a}) ≰ r$ . Hence $\begin{matrix} ⋁_{z \in X} [(\bar{a} \circ b) \land (a \circ \bar{b})] (z) \geq [(b \circ b) \land (a \circ \bar{b})] (b) ≰ r . \end{matrix}$

(4b) $\bar{b} = a$ . Similar to (4a).

(4c) $a \neq \bar{b}$ and $b \neq \bar{a}$ . Since ${Sec}_{c}$ has M-fuzzifying Pasch property, we have $\begin{matrix} ⋁_{z \in X} [\bar{a} b \land \bar{b} a] (z) \geq pa (\bar{a}) \land pb (\bar{b}) ≰ r . \end{matrix}$ Hence there exists a point e ∈ X such that $[\bar{a} b \land \bar{b} a] (e) ≰ r$ . We further have the following cases.

(4c(i)) e = a. Note that $\bar{a} \neq a$ . By (MSI*), $⋁_{u, v \in Y} ⋀_{w \in Y} uv (w) \geq t_{{Seg}_{c}} (p, a, b, \bar{a}, a, \bar{a}) ≰ r$ , where $Y = {p, \bar{a}, a, b}$ . Thus there exist u, v ∈ Y such that uv (w) ≰ r for all w ∈ Y. Since ${Sec}_{c}$ has M-fuzzifying decomposable property, we can prove that u, v ∈ {p, b}. Hence $pb (a) \land pb (\bar{a}) ≰ r$ . Further, we have the following subcases.

(4c(i)a) $\bar{a} = \bar{b}$ . By (MDenI), there is $s \in X ∖ {p, a, \bar{a}}$ such that $pa (\bar{a}) \leq a \bar{a} (s)$ . Thus $a \bar{a} (s) ≰ r$ . Further, $b \bar{a} (a) \geq pb (\bar{a}) \land pb (a) \land pa (\bar{a}) ≰ r$ by (MGI3), and $b \bar{a} (s) \geq a \bar{a} (s) \land \bar{a} b (a) ≰ r$ by (MGI2). We say that s ≠ b (otherwise, $⊥ = a_{⊤} (b) \geq [\bar{a} a \land$ $ab] (b) \land \bar{a} b (a) ≰ r$ by (MDecI2)). Thus $\begin{matrix} ⋁_{z \in X} [(\bar{a} \circ b) \land (a \circ \bar{b})] (z) \geq [(\bar{a} \circ b) \land (a \circ \bar{a})] (s) ≰ r . \end{matrix}$

(4c(i)b) $\bar{a} \neq \bar{b}$ . We have $b \bar{a} (a) \geq pb (\bar{a}) \land pb (a) \land pa (\bar{a}) ≰ r$ by (MGI3). Also, by (MDecI1), $\begin{matrix} r & ≱ & [pb (\bar{b}) \land pb (a)] \land pa (\bar{a}) \\ = & [pa \lor ab] (\bar{b}) \land pb (a) \land pa (\bar{a}) \\ \leq & [pa (\bar{b}) \land pa (\bar{a})] \lor ab (\bar{b}) \\ = & {[p \bar{a} \lor \bar{a} a] (\bar{b}) \land pa (\bar{a})} \lor ab (\bar{b}) \\ \leq & p \bar{a} (\bar{b}) \lor \bar{a} a (\bar{b}) \lor ab (\bar{b}) . \end{matrix}$ Thus we further have the following three subcases.

(4c(i)b1) $ab (\bar{b}) ≰ r$ . By (MDenI), there is $t \in X ∖ {a, b, \bar{b}}$ such that $ab (\bar{b}) \leq a \bar{b} (t)$ . Thus $a \bar{b} (t) ≰ r$ . Further, $b \bar{a} (\bar{b}) \geq ab (\bar{b}) \land b \bar{a} (a) ≰ r$ by (MGI2), and $b \bar{a} (t) \geq a \bar{b} (t) \land b \bar{a} (a) \land b \bar{a} (\bar{b}) ≰ r$ by (MRH3). We say that $t \neq \bar{a}$ (otherwise, $⊥ = \bar{a} b (\bar{a}) \land b \bar{a} (a) ≰ r$ by (3) of Proposition 3.2). Therefore $\begin{matrix} ⋁_{z \in X} [(\bar{a} \circ b) \land (a \circ \bar{b})] (z) \geq [(\bar{a} \circ b) \land (a \circ \bar{b})] (t) ≰ r . \end{matrix}$

(4c(i)b2) $\bar{a} a (\bar{b}) ≰ r$ . By (MDenI), there is $x \in X ∖ {\bar{a}, a, \bar{b}}$ such that $\bar{a} a (\bar{b}) \leq a \bar{b} (x)$ . Thus $a \bar{b} (x) ≰ r$ . Similar to (4c(i)b1), we can prove that $\begin{matrix} ⋁_{z \in X} [(\bar{a} \circ b) \land (a \circ \bar{b})] (z) \geq [(\bar{a} \circ b) \land (a \circ \bar{b})] (x) ≰ r . \end{matrix}$

(4c(i)b3) $\bar{a} p (\bar{b}) ≰ r$ . Note that $a, \bar{a}, b$ are different. By (MDenI), there is $y \in X ∖ {a, \bar{a}, b}$ such that $a \bar{a} (y) \geq b \bar{a} (a) = b \bar{a} (e) ≰ r$ . Thus $a \bar{a} (y) ≰ r$ . Similar to (4c(i)b1), we can prove that $\begin{matrix} ⋁_{z \in X} [(\bar{a} \circ b) \land (a \circ \bar{b})] (z) \geq [(\bar{a} \circ b) \land (a \circ \bar{b})] (y) ≰ r . \end{matrix}$

(4c(ii)) e = b. Similar to (4c(i)).

(4c(iii)) $e = \bar{b}$ . Note that $\bar{b} \neq b$ . We have $b \bar{a} (p) \lor bp (\bar{a}) \geq [b \bar{a} \land bp] (\bar{b}) ≰ r$ by (MRI). Thus we further have the following subcases.

(4c(iii)a) $bp (\bar{a}) ≰ r$ . We have $bp (a) \lor ap (b) \geq [bp \land ap] (\bar{a}) ≰ r$ by (MRI).

If bp (a) ≰ r, then we can prove that $⋁_{z \in X} [(\bar{a} \circ b) \land (a \circ \bar{b})] (z) ≰ r$ similar to (4c(i)).

If ap (b) ≰ r, we can prove that $⋁_{z \in X} [(\bar{a} \circ b) \land (a \circ \bar{b})] (z) ≰ r$ similar to (4c(ii)).

(4c(iii)b) $b \bar{a} (p) ≰ r$ . By (MSI*), $⋁_{u, v \in Y} ⋀_{w \in Y} uv (w) \geq t_{{Seg}_{c}} (a, p, \bar{a} \bar{b}, \bar{a}, p) ≰ r$ , where $Y = {a, \bar{a}, \bar{b}, p}$ . Since ${Seg}_{c}$ has M-fuzzifying decomposable property, we can prove that $u, v \in {a, \bar{b}}$ . Thus $a \bar{b} (p) ≰ r$ . Hence $\begin{matrix} ⋁_{z \in X} [(\bar{a} \circ b) \land (a \circ \bar{b})] (z) \geq [(\bar{a} \circ b) \land (a \circ \bar{b})] (p) ≰ r . \end{matrix}$

(4c(iv)) $e = \bar{a}$ . Similar to (4c(iii)).

(4c(v)) $e \neq a, b, \bar{a}, \bar{b}$ . Then $⋁_{z \in X} [(\bar{a} \circ b) \land (a \circ \bar{b})] (z) \geq [\bar{a} b \land \bar{b} a] (e) ≰ r$ .

In conclusion, (Δ) holds. Therefore (MJ4) holds.

(MJ5). Let a, b, c, p ∈ X. We prove that $\begin{matrix} [(a \circ b) \land (a \circ c)] (p) \\ \leq b_{⊤} (c) \lor (a \circ c) (b) \lor (a \circ b) (c) \end{matrix}$ by the following cases.

(1) b = c. The desired inequality is trivial.

(2) b ≠ c. We further have the following subcases.

(2a) p = a (resp. p ≠ a = b, p ≠ a = c). The desired result follows from [(a ∘b) ∧ (a ∘c)] (p) =⊥.

(2b) p ≠ a and a ≠ b, c. By (2) of Proposition 3.3, ${Seg}_{c}$ has M-fuzzifying ramification property. Note that p ≠ a. By (MRI), $\begin{matrix} [(a \circ b) \land (a \circ c)] (p) & \leq & [ab \land ac] (p) \\ \leq & ab (c) \lor ac (b) \\ = & (a \circ b) (c) \lor (a \circ c) (b) . \end{matrix}$ Thus the desired result holds.

Therefore ∘is an M-fuzzifying join operator. □

Theorem 4.9. If (X, ∘) is an M-fuzzifying join space, then $\circ_{c_{\circ}} = \circ$ . Conversely, if $(X, c)$ is an M-fuzzifying BW space, then $c_{\circ_{c}} = c$ .

Theorem 4.10. If (X, ∘_X) and (Y, ∘_Y) are M-fuzzifying join spaces and f : X → Y is an M-JP mapping, then $f : (X, c_{\circ_{X}}) \to (Y, c_{\circ_{Y}})$ is an M-CP mapping. Also, if $(X, c_{X})$ and $(Y, c_{Y})$ are M-fuzzifying BW spaces and f : X → Y is an M-CP mapping, then $f : (X, C_{\circ X}) \to (Y, C_{\circ Y})$ is an M-JP mapping.

Corollary 4.11. M-BWS is isomorphic to M-JS.

5 Conclusions

We mainly studied M-fuzzifying BW spaces and M-fuzzifying join spaces in this paper. The followings are our conclusions.

(1) Rationality of definitions.

There are non-crisp M-fuzzifying BW spaces and non-crisp M-fuzzifying join spaces listed in Example 3.7 and Example 4.2. In addition, it is showed that M-fuzzifying BW spaces and M-fuzzifying join spaces are mutually induced. Thus both of M-fuzzifying BW spaces and M-fuzzifying join spaces are reasonable.

(2) Upgrade of M-fuzzifying interval spaces.

A special kind of M-fuzzifying convex spaces, M-fuzzifying BW spaces capture almost all properties in fuzzy vector spaces. In addition, M-fuzzifying BW space makes these properties a whole. In this sense, this paper not only upgrades the theory of M-fuzzifying interval spaces but also integrates M-fuzzifying interval spaces into M-fuzzifying convex spaces.

(3) Subsequences of the previous researches.

Actually, this paper can be regarded as a subsequent article of [32 –34]. As we can see in Proposition 3.3 and Theorem 3.4, properties of M-fuzzifying interval spaces defined in Section 3 closely related to M-fuzzifying geometric property defined in [33] and M-fuzzifying convex matroids [34]. These properties behave quite similar to that in Convex Theory.

(4) Not full extensions of classic notions.

This paper is an M-fuzzifying extension of theory of Byrant-Webster spaces. However, there is a slight difference between the M-fuzzifying theory and the classic theory. As we can see in Remark 3.5, in Convex Theory, if the segment operator of a convex space induced by an interval operator has ramification, decomposable and Peano properties, then it also has Pasch property. But, this result fails in M-fuzzifying settings.

Even so, this difference doesn’t make any significant impact on the main results of this paper. That is, both M-fuzzifying BW spaces and M-fuzzifying join spaces are well defined, and most related properties of BW spaces and join spaces are still preserved. In addition, M-fuzzifying BW spaces can be successfully characterized by M-fuzzifying join spaces.

(5) Applications of M-fuzzifying BW spaces.

This paper hasn’t discuss them because many related theories haven’t been extended into M-fuzzifying setting yet, such as affine matroids, convex invariants, dimensions of M-fuzzifying convex structures and intrinsic convex spaces etc. Thus we believe that results and methods in this paper could be useful in studying of M-fuzzifying convex spaces in the further.

Footnotes

Acknowledgments

The author sincerely thank the associate editor Prof. Marek T. Malinowski and the reviewers for great helps and valuable suggestions.

This work is supported by the Youth Science Foundation of Hunan province (No. 18JJ3192).

References

Alcantud

J.C.R.

, Non-binary choice in a non-deterministic model, Econom Lett 77(1) (2002), 117–123.

Berger

, Convexity, Amer Math Monthly 97(8) (1990), 650–678.

Ben-El-Mechaiekh

, Chebbi

, Florenzano

and Llinares

J.V.

, Abstract convexity and fixed points, J Math Anal Appl 222(1) (1998), 138–150.

Bryant

V.W.

and Webster

R.J.

, Convexity spaces. I. The properties, J Math Anal Appl 37(1) (1972), 206–213 basic.

Cantwell

and Kay

D.C.

, Geometric convexity, i, Inst Math Acad Sinica 2 (1974), 289–307.

Changat

, Mulder

H.M.

and Sierksma

, Convexities related to path properties on graphs, Discrete Math 290(2) (2005), 117–131.

Coppel

W.A.

, Axioms for convexity, Bull Austral Math Soc 47(2) (1993), 179–197.

Franklin

S.P.

, Some results on order-convexity, Amer Math Monthly 69(5) (1962), 357–359.

Kubis

, Abstract convex structures in topology and set theory, PhD thesis, University of Silesia, Katowice, Poland 1999.

10.

Llinares

J.V.

, Abstract convexity, some relations and applications, Optimization 51(6) (2002), 797–818.

11.

Maruyama

, Lattice-valued fuzzy convex geometry, Optimization 1641 (2009), 22–37.

12.

Mulder

H.M.

, The structure of median graphs, Discrete Math 24(2) (1978), 197–204.

13.

Nachbin

, On the convexity of sets, Acad Sci Math 40(3) (1992), 229–234.

14.

Pang

and Shi

F.G.

, Subcategories of the category of L-convex spaces, Fuzzy Sets Syst 313 (2017), 61–74.

15.

Pang

and Zhao

, Characterizations of L-convex spaces, Iran J Fuzzy Syst 13(4) (2016), 51–61.

16.

Pang

, Zhao

and Xiu

Z.Y.

, A new definition of order relation for the introduction of algebraic fuzzy closure operators, Int J Approx Reason 92 (2018), 87–96.

17.

Prenowitz

and Jantosciak

, Geometries and join spaces, J Reine Angew Math 257 (1972), 100–128.

18.

Rapcsak

, Geodesic convexityin nonlinear optimization, J Optim Theory Appl 69(1) (1991), 169–183.

19.

Rosa

M.V.

, On fuzzy topology fuzzy convexity spaces and fuzzy local convexity, Fuzzy Sets Syst 62(1) (1994), 97–100.

20.

Ryan

M.J.

, Mixture sets on finite domains, Decis Econ Finance 33(2) (2010), 139–147.

21.

Shen

and Shi

F.G.

, L-convex systems and the categorical isomorphism to scott-hull operators, Iran J Fuzzy Syst 15(2) (2018), 23–40.

22.

Shi

F.G.

, Theory of L_β-nested sets and L_β-nested sets and its applications,65–72. (Chinese), Fuzzy Syst Math 9(4) (1995), .

23.

Shi

F.G.

, L-fuzzy interiors and L-fuzzy closures, Fuzzy Sets Syst 160 (2009), 1218–1232.

24.

Shi

F.G.

and Pang

, Categories isomorphic to the category of L-fuzzy closure system spaces, Iran J Fuzzy Syst 10(5) (2013), 127–146.

25.

Shi

F.G.

and Li

E.Q.

, The restricted hull operators of Mfuzzifying convex structures, J Intell Fuzzy Syst 30(1) (2015), 409–421.

26.

Shi

F.G.

and Xiu

Z.Y.

, A new approach to the fuzzification of convex structures,1–12, Article ID, J Appl Math (2014), 249183.

27.

Shi

F.G.

and Xiu

Z.Y.

, (L, M)-fuzzy convex structures, J Nonlinear Sci Appl 10 (2017), 3655–3669.

28.

de Vel van

M.L.J.

, Binary convexities and distributive lattices, Proc London Math Soc 48(1) (1984), 1–33.

29.

de Vel van

M.L.J.

, Noth-Holland, New-York, Theory of Convex structures 1993.

30.

Wang

G.J.

, Theory of topological molecular lattices, Fuzzy Sets Syst 47(3) (1992), 351–376.

31.

Wang

G.J.

, Shanxi Normal University Press, Xi’an, (in chinese), Theory of L-fuzzy topological spaces (1988), .

32.

X.Y.

and Bai

S.Z.

, On M-fuzzifying JHC convex structures and M-fuzzifying peano interval spaces, J Intell Fuzzy Syst 30(4) (2016), 2447–2458.

33.

X.Y.

, Li

E.Q.

and Bai

S.Z.

, Geometric properties of M-fuzzifying convex structures, J Intell Fuzzy Syst 32(4) (2017), 4273–4284.

34.

X.Y.

, Davvaz

and Bai

S.Z.

, On M-fuzzifying convex matroids and M-fuzzifying independent structures, J Intell Fuzzy Syst 33(1) (2017), 269–280.

35.

X.Y.

and Bai

S.Z.

, M-fuzzifying gated amalgamations of M-fuzzifying geometric interval spaces, J Intell Fuzzy Syst 33(6) (2017), 4017–4029.

36.

X.Y.

and Li

E.Q.

, Category and subcategories of (L,M)-fuzzy convex spaces,online, Iran J Fuzzy Syst (2018), . doi: 10.22111/ijfs.2018.3857.

37.

Xiu

Z.Y.

and Shi

F.G.

, M-fuzzifying interval spaces, Iran J Fuzzy Syst 14(1) (2017), 145–162.

38.

Xiu

Z.Y.

and Pang

, Base axioms and subbase axioms in Mfuzzifying convex spaces,online, Iran J Fuzzy Syst 15(2) (2018), 75–87.

39.

Xiu

Z.Y.

and Pang

, M-fuzzifying cotopological spaces and M-fuzzifying convex spaces as M-fuzzifying closure spaces, J Intell Fuzzy Syst 33(1) (2017), 613–620.