M -fuzzifying median algebras and its induced convexities

Abstract

In the present paper we introduce the notion of M-fuzzifying median algebras, which is an extended notion of median algebras in van de Vel. It can further induce M-fuzzifying convex spaces and M-fuzzifying interval spaces, respectively. We study some mapping relations of these concepts. Some characterizations and properties of M-fuzzifying median algebras, M-fuzzifying convex spaces and M-fuzzifying interval spaces are given. Specially, an M-fuzzifying median algebra can not only induce an M-fuzzifying geometric interval operator but also obtain a special M-fuzzifying modular space.

1 Introduction

According to the literature, median algebras appear for the first time in the late 1940s. It was generated by median operation in distributive lattices. The concept of abstract median algebra was introduced by Avann [1]. Afterwards, M. Sholander [26], Bandelt [2], Bandelt and HedlíkovÁ [3], van de Vel [27] performed a detailed study of median algebras. After that, a wide range of properties of median algebras have been discovered. For example, J. Nieminen [14] and E. Evans [8] researched convex sets in median algebras. Along the way, E. Evans’s papers characterized the lattice of all convex sets. H.M. Mulder and A. Schrijver [13] studied the structure of prime convex sets in the finite case. In the paper [11], the fuzzification of convex sets in median algebras was considered, and some of their properties were investigated. Characterizations of a fuzzy number by the median value and the median interval has been discussed by Bodjanova [5]. The authors in [7] provided perfect descriptions of median-homomorphisms between conservative median algebras. E. Fioravanti [9] proved various properties of half spaces in finite rank median spaces and a duality result for locally convex median spaces.

Being a helpful mathematical tool, convexity theory has been being rapidly used in the study of extremum problems in recent years. The concept of convexity derives from thinking of some elementary geometric problems in Euclidean spaces [4]. To better examine the characteristics of convexity, a lot of specialists abstracted convexity away from Euclidean spaces and into the other mathematical structures such as posets, lattices, graphs, metric spaces, median algebras and topology. With the help of the idea of axiomatic method, scholars have achieved a description of the properties of convex sets that we regard fundamental, and the concepts of convex structures was obtained. In fact, the convex structures are studied in many aspects, such as product spaces, quotient spaces, convex invariants, separation and so on.

Since Zadeh introduced the concept of fuzzy sets, many mathematical structures have been endowed with fuzzy set theory, such as fuzzy topological structures [6, 34] and fuzzy convergence structures [17, 18]. In 1994, Rosa first proposed the notion of fuzzy convex structures [23]. Years later, Maruyama [12] further promoted generalized fuzzy convex structures, where the lattice background of the fuzzy sets was completely distributive L. One thing to point out is that each convex set was fuzzy and the convex structure is a crisp subset of L-powerset under the sense of Rosa and Maruyama. Recently, Pang et al. [15 , 19–22] provided a categorical approach to L-convex structures and present several characterizations of L-convex structures.

In 2014, Shi and Xiu [25] proposed a new approach to fuzzification of convex structures, which is called an M-fuzzifying convex structure. In this case, M-fuzzifying convex structure are actually an M-subset of the powerset 2^X. And then, the concept of M-fuzzifying restricted hull operators was introduced [24] in order to generalize the concept of restricted hull operators in classical convex structures to M-fuzzifying convex structures. Lately, Pang and Xiu [19] presented the notion of M-fuzzifying interval operators and discussed its relations with M-fuzzifying convex structures. Moreover, Xiu and Pang studied the relations between M-fuzzifying convex structures and M-fuzzifying closure systems [32] and provided the base axioms and subbase axioms [33]. Furthermore, the theoretical nature of interval operators have been considered by many scientists, such as, [29 –31].

In this paper we introduce the concept of M-fuzzifying median algebras, and then we obtain the M-fuzzifying convex spaces which is induced by the M-fuzzifying median algebras. In the following, we discuss some mapping properties of M-fuzzifying median algebras and M-fuzzifying convex spaces. Next, we find that the M-fuzzifying median algebras can be used to induced the M-fuzzifying interval operators. Fortunately, the M-fuzzifying interval operators is geometric, it is related with the M-fuzzifying modular space. Naturally, we study the relations between M-fuzzifying median algebras and M-fuzzifying modular spaces.

2 Preliminaries

In this section, we will review some basic notions about median algebras, M-fuzzifying interval operators and M-fuzzifying conver structures.

Throughout this paper, (M, ∨, ∧, ′) denotes a completely distributive lattice with an order-reversing involution ^′, i.e., M is a completely distributive De Morgan algebra. The smallest element and the largest element in M are denoted by ⊥ and ⊤, respectively. For B ⊆ M, write ⋁B for the least upper bound of B and ⋀B for the greatest lower bound of B. In particular, we adopt the convention that ⋁∅ = ⊥ and ⋀∅ = ⊤. For a nonempty set X, we denote the set of all subsets (resp., M-subsets) of X by 2^X (resp., M^X). A family {A_i: i ∈ Ω} is up-directed provided for each A₁, A₂ ∈ {A_i: i ∈ Ω} there is a third element A₃ ∈ {A_i: i ∈ Ω} such that A₁ ⊆ A₃ and A₂ ⊆ A₃, in symbols: ${A_{i} | i \in Ω} \overset{dir}{\subseteq} 2^{X}$ .

Let A ∈ M^X. For a ∈ M, we can define A_[a] = {x ∈ X ∣ A (x) ≥ a}.

Let X, Y be two nonempty sets and f: X ⟶ Y be a mapping. Define f^→: 2^X ⟶ 2^Y, f^←: 2^Y⟶2^X, $f_{M}^{\to} : 2^{X} ⟶ 2^{Y}$ and $f_{M}^{\leftarrow} : 2^{Y} ⟶ 2^{X}$ as follows:

∀A ∈ 2^X, f^→ (A) = {f (x) |x ∈ A};

∀B ∈ 2^Y, f^← (B) = {x|f (x) ∈ B};

∀U ∈ M^X, ∀y ∈ Y, $f_{M}^{\to} (U) (y) = ⋁_{f (x) = y} U (x)$ ;

∀V ∈ M^Y, ∀x ∈ X, $f_{M}^{\leftarrow} (V) (x) = V (f (x))$ .

By the above definition, we can easily verify that $f_{M}^{\leftarrow} \circ f_{M}^{\to} (A) \geq A$ and $f_{M}^{\to} \circ f_{M}^{\leftarrow} (B) \leq B$ .

We define a residual implication in M by a → b = ⋁ {c ∈ M|a ∧ c ≤ b}. Also, we denote a ↔ b = (a → b) ∧ (b → a). Some properties of the implication operation are listed in the following lemma.

Lemma 2.1. [10] Let (M, ∨, ∧) be a completely distributive lattice and let → be the implication operation corresponding to ∧. Then for all a, b, c ∈ M, {a_i} _i∈I, {b_i} _i∈I ⊆ M, the following statements hold.

⊤ → a = a.

(a → b) ≥ c ⇔ a ∧ c ≤ b.

a → b = ⊤ ⇔ a ≤ b.

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

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

(a → c) ∧ (c → b) ≤ a → b.

Definition 2.2. [28] A convex structure $C$ on X is a subset of 2^X which satisfies:

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

(C2) ${A_{i} | i \in Ω} \subseteq C$ is nonempty, then ⋂_i∈Ω $A_{i} \in C$ ;

(C3) ${A_{i} | i \in Ω} \subseteq C$ is nonempty and totally ordered by inclusion, then $⋃_{i \in Ω} A_{i} \in C$ .

For a convex structure $C$ on X, the pair $(X, C)$ is called a convex space.

A mapping $f : (X, C_{X}) ⟶ (Y, C_{Y})$ is called convexity preserving (CP, in short) provided that $B \in C_{Y}$ implies $f^{\leftarrow} (B) \in C_{X}$ , f is called convex-to-convex (CC, in short) provided that $A \in C_{X}$ implies f^→ (B) $\in C_{Y}$ .

Definition 2.3. [28] Let $C$ be a closure system on X, that is, $C$ satisfies (C1) and (C2). Then the following statements are equivalent.

(C3) ${A_{i} | i \in Ω} \subseteq C$ is nonempty and totally ordered by inclusion, then $⋃_{i \in Ω} A_{i} \in C$ .

(C3^′) ${A_{i} | i \in Ω} \overset{dir}{\subseteq} C$ implies $⋃_{i \in Ω} A_{i} \in C$ .

Definition 2.4. [28] An interval operator on X is a mapping I: X × X ⟶ 2^X which satisfies:

(I1) a, b ∈ I (a, b);

(I2) I (a, b) = I (b, a). Furthermore, I is geometric provided the following hold:

(G1) For all a ∈ X, I (a, a) = {a};

(G2) For all a, b, c, d ∈ X, if c ∈ I (a, b) and d ∈ I (c, b), then d ∈ I (a, b) and c ∈ I (a, d). For an (geometric) interval operator I on X, the pair (X, I) is called an (geometric) interval space.

A mapping f: (X, I_X) ⟶ (Y, I_Y) is called interval preserving (IP, in short) provided for all x, y ∈ X f^→ (I_X (x, y)) ⊆ I_Y (f (x), f (y)), f is called interval-to-interval (II, in short) provided that f^→ (I_X (x, y)) = I_Y (f (x), f (y)).

The concept of M-fuzzifying convex structures is proposed by Shi and Xiu as follows.

Definition 2.5. [25] A mapping $C : 2^{X} \to M$ is called an M-fuzzifying convexity on X if it satisfies the following conditions:

(MC1) $C (\emptyset) = C (X) = ⊤$ ;

(MC2) {A_i|i ∈ Ω} ⊆2^X is nonempty, then $C (⋂_{i \in Ω} A_{i}) \geq ⋀_{i \in Ω} C (A_{i})$ ;

(MC3) {A_i|i ∈ Ω} ⊆2^X is nonempty and totally ordered by inclusion, then $C (⋃_{i \in Ω} A_{i}) \geq ⋀_{i \in Ω} C (A_{i})$ .

For an M-fuzzifying convex structure $C$ on X, the pair $(X, C)$ is called an M-fuzzifying convexspace.

A mapping $f : (X, C_{X}) ⟶ (Y, C_{Y})$ is called M-fuzzifying convexity preserving (M-CP, in short) provided that $C_{X} (f^{\leftarrow} (B)) \geq C_{Y} (B)$ for each B ∈ 2^Y; f is called M-fuzzifying convex-to-convex (M-CC, in short) provided that $C_{Y} (f^{\to} (A)) \geq C_{X} (A)$ for each A ∈ 2^X.

Theorem 2.6. [25] The hull operator ${co}_{C} : 2^{X} \to M^{X}$ (briefly, co) of $(X, C)$ is defined as: $\forall A \in 2^{X}, \forall x \in X, co (A) (x) = ⋀_{x \notin B \supseteq A} (C (B))^{'} .$ Then for all A ∈ 2^X and x ∈ X,

co (∅) (x) = ⊥;

co (A) (x) =⊤ for all 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 $C_{co} : 2^{X} \to M$ as: $\forall A \in 2^{X}, C_{co} (A) = ⋀_{x \notin A} (co (A) (x))^{'} .$

Then $C_{co}$ is an M-fuzzifying convexity with ${co}_{C_{co}} = co$ . In addition, if $(X, C)$ is an M-fuzzifying convex space, then $C_{{co}_{C}} = C$ . The restriction co^seg of co on $2_{2}^{X}$ is called the segment operator of $(X, C)$ , where $2_{2}^{X} = {{x, y} | x, y \in X}$ . For convenience, co^seg is still denoted by co.

Definition 2.7. [28] A median operator on X is a mapping m: X³ ⟶ X satisfying the following properties.

(M1) m (a, a, b) = a;

(M2) m (σ (a), σ (b), σ (c)) = m (a, b, c), where σ is any permutation of a, b and c;

(M3) m (m (a, b, c), d, c) = m (a, m (b, c, d), c).

The point m (a, b, c) is called the median of a, b, c, and the resulting pair (X, m) is called a median algebra.

A mapping f: (X, m_X) ⟶ (Y, m_Y) is called median preserving (MP, in short) provided for all a, b, c ∈ X, $f (m (a, b, c)) = m (f (a), f (b), f (c)) .$

3 M-fuzzifying median algebras

In this section, we investigate the concept of M-fuzzifying median operators, and from this we get M-fuzzifying convex structures. Meanwhile, we also obtain interval operators by M-fuzzifying median operators. Some properties of these operations are also studied.

Definition 3.1. [19 , 29–31] An M-fuzzifying interval operator on X is a mapping $I : X \times X ⟶ M^{X}$ which satisfies:

(MI1) $I (x, y) (x) = I (x, y) (y) = ⊤$ ;

(MI2) $I (x, y) = I (y, x) .$ Furthermore, $I$ is geometric provided the following hold:

(MG1) For all a ∈ X, $I (a, a) = a_{⊤}$ ;

(MG2) For all a, b, c, d ∈ X, $I (a, b) (c) \land I (c, b) (d) \leq I (a, b) (d) \land I (a, d) (c)$ .

For an M-fuzzifying (geometric) interval operator $I$ on X, the $(X, I)$ is called an M-fuzzifying (geometric) interval space.

Corollary 3.2. Let $(X, I)$ be an M-fuzzifying geometric interval space and let c, d be distinct points of X. Then for all a, b ∈ X, we have $I (a, b) (c) \land I (b, c) (d) \land I (a, c) (d) = ⊥ .$

Proof. As (MG2), we have $I (a, b) (c) \land I (b, c)$ $(d) \leq I (a, d) (c)$ and $I (a, c) (d) \land I (a, d) (c) \leq I$ (d, d) (c) =⊥, then $I (a, b) (c) \land I (b, c) (d) \land I (a,$ c) (d) =⊥. □

Definition 3.3. [19] An M-fuzzy function $f : (X, I_{X}) ⟶ (Y, I_{Y})$ is called M-fuzzifying interval preserving (M-IP, in short) provided that $\forall x, y \in X, f_{M}^{\to} (I_{X} (x, y)) \leq I_{Y} (f (x), f (y)) .$ The function is M-fuzzifying interval-to-interval (M-II, in short) provided $\forall x, y \in X, f_{M}^{\to} (I_{X} (x, y)) = I_{Y} (f (x), f (y)) .$

Theorem 3.4. [29] Let $(X, I_{X})$ and $(Y, I_{Y})$ be M-fuzzifying interval spaces and let f: X ⟶ Y be an M-II function. If $(X, I_{X})$ is an M-fuzzifying geometric interval space, then so is $(Y, I_{Y})$ .

Theorem 3.5. [29] Let $(X, C)$ be an M-fuzzifying convex structure. Define $I_{C} : X \times X ⟶ M^{X}$ 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 $C_{I} : 2^{X} \to M$ 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.

Definition 3.6. An M-fuzzifying median operator on X is a mapping $\tilde{m} : X^{3} ⟶ M^{X}$ satisfying the following properties.

(MM0) $⋁_{x \in X} \tilde{m} (a, b, c) (x) \neq ⊥$ and $\tilde{m} (a, b, c)$ $(x) \land \tilde{m} (a, b, c) (y) \neq ⊥ \Rightarrow x = y$ ;

(MM1) $\tilde{m} (a, a, b) = a_{⊤}$ ;

(MM2) $\tilde{m} (σ (a), σ (b), σ (c)) = \tilde{m} (a, b, c)$ , where σ is any permutation of a, b and c; $\begin{matrix} (MM 3) & \tilde{m} (a, b, c) (x) \land \tilde{m} (b, c, d) (y) \\ \leq & ⋀_{z \in X} (\tilde{m} (x, d, c) (z) \leftrightarrow \tilde{m} (a, y, c) (z)) . \end{matrix}$

For an M-fuzzifying median operator $\tilde{m}$ on X, the pair $(X, \tilde{m})$ is called an M-fuzzifying median algebra.

Example 3.7. Let X = {1, 2, 3, 4} and M = [0, 1]. Define

\tilde{m} : X^{3} ⟶ M^{X}

satisfies (MM1) and (MM2). The tabular representation of other cases are shown in the following table. Other values of the [0, 1]-fuzzifying operator

\tilde{m}

$\tilde{m} (a, b, c)$	(1)	(2)	(3)	(4)
(1, 2, 3)	0	0.8	0	0
(1, 2, 4)	0	0.7	0	0
(1, 3, 4)	0	0	0.5	0
(2, 3, 4)	0	0	0.5	0

Then it is easy to see that $\tilde{m}$ satisfies (MM0). Now we verify that $\tilde{m}$ satisfies (MM3). Let $s = \tilde{m} (a, b, c) (x) \land \tilde{m} (b, c, d) (y)$ and let $t = ⋀_{z \in X} (\tilde{m} (x, d, c) (z) \leftrightarrow \tilde{m} (a, y, c) (z))$ , for all a, b, c, d, x, y ∈ X. We only need to verify the following four cases:

Case1. If s = 1, ∀a, b, c, ∈ X it must be one of the following cases:

$\tilde{m} (b, b, c) (b) \land \tilde{m} (b, c, b) (b)$ ,

$\tilde{m} (b, b, c) (b) \land \tilde{m} (b, c, c) (c)$ ,

$\tilde{m} (c, b, c) (c) \land \tilde{m} (b, c, b) (b)$ ,

$\tilde{m} (c, b, c) (c) \land \tilde{m} (b, c, c) (c)$ .

It is easy to check that t = 1 under the conditions (1-1) and (1-4). As to case (1-2), $t = ⋀_{z \in X} (\tilde{m} (b, c, c) (z) \leftrightarrow \tilde{m} (b, c, c) (z)) = 1$ . Similarly, the case (1-3) is also true.

Case2. If s = 0.8 and if $\tilde{m} (σ (a), σ (b), σ (c)) = \tilde{m} (1, 2, 3)$ , then it must be one of the following cases:

$\tilde{m} (a, b, c) (2) \land \tilde{m} (b, c, a) (2)$ ,

$\tilde{m} (a, b, c) (2) \land \tilde{m} (b, c, b) (b)$ ,

$\tilde{m} (a, b, c) (2) \land \tilde{m} (b, c, c) (c)$ .

The condition (2-1) obviously hold. In case (2-2), $t = ⋀_{z \in X} (\tilde{m} (2, a, c) (z) \leftrightarrow \tilde{m} (a, 2, c) (z)) = 1$ . In case (2-3), $t = ⋀_{z \in X} (\tilde{m} (2, c, c) (z) \leftrightarrow \tilde{m} (a, c, c) (z)) = 1$ . Thus s ≤ t.

Case3. If s = 0.7 and if $\tilde{m} (σ (a), σ (b), σ (d)) = \tilde{m} (1, 2, 4)$ , then it must be one of the following cases:

$\tilde{m} (a, b, d) (2) \land \tilde{m} (b, d, a) (2)$ ,

$\tilde{m} (4, 1, 2) (2) \land \tilde{m} (1, 2, 3) (2)$ ,

$\tilde{m} (4, 2, 1) (2) \land \tilde{m} (2, 1, 3) (2)$ ,

$\tilde{m} (a, b, d) (2) \land \tilde{m} (b, d, b) (b)$ ,

$\tilde{m} (a, b, d) (2) \land \tilde{m} (b, d, d) (d)$ .

The condition (3-1) obviously hold. As to case (3-2), $t = ⋀_{z \in X} (\tilde{m} (2, 3, 2) (z) \leftrightarrow \tilde{m} (4, 2, 2) (z)) = 1$ . In case (3-3), $t = ⋀_{z \in X} (\tilde{m} (2, 3, 1) (z) \leftrightarrow \tilde{m} (4, 2, 1) (z)) = 0.7$ . In case (3-4), $t = ⋀_{z \in X} (\tilde{m} (2, b, d) (z) \leftrightarrow \tilde{m} (a, b, d) (z)) = 0.7$ . As to case (3-5), we can calculate $t = ⋀_{z \in X} (\tilde{m} (2, d, d) (z) \leftrightarrow \tilde{m} (a, d, d) (z)) = 1$ . Therefore s ≤ t.

Case4. If s = 0.5, then we only need to consider the following cases:

$\tilde{m} (1, 3, 4) (3) \land \tilde{m} (3, 4, 2) (3)$ ,

$\tilde{m} (1, 4, 3) (3) \land \tilde{m} (4, 3, 2) (3)$ ,

$\tilde{m} (3, 1, 4) (3) \land \tilde{m} (1, 4, 2) (3)$ ,

$\tilde{m} (3, 4, 1) (3) \land \tilde{m} (4, 1, 2) (2)$ ,

$\tilde{m} (3, 2, 4) (3) \land \tilde{m} (2, 4, 1) (2)$ ,

$\tilde{m} (3, 4, 2) (3) \land \tilde{m} (4, 2, 1) (2)$ ,

$\tilde{m} (4, 1, 3) (3) \land \tilde{m} (1, 3, 2) (2)$ ,

$\tilde{m} (4, 3, 1) (3) \land \tilde{m} (3, 1, 2) (2)$ ,

$\tilde{m} (4, 2, 3) (3) \land \tilde{m} (2, 3, 1) (2)$ ,

$\tilde{m} (4, 3, 2) (3) \land \tilde{m} (3, 2, 1) (2)$ .

In addition, if $\tilde{m} (σ (a), σ (c), σ (d)) = \tilde{m} (1, 3, 4)$ , we should consider that

$\tilde{m} (a, c, d) (3) \land \tilde{m} (c, d, c) (c)$ and

$\tilde{m} (a, c, d) (3) \land \tilde{m} (c, d, d) (d)$ ,

\tilde{m} (σ (b), σ (c), σ (d)) = \tilde{m} (2, 3, 4)

, we should also consider that

$\tilde{m} (b, c, d) (3) \land \tilde{m} (c, d, c) (c)$ and

$\tilde{m} (b, c, d) (3) \land \tilde{m} (c, d, d) (d)$ .

In cases like those shown above, we can obtain s ≤ t. Hence $(X, \tilde{m})$ is an M-fuzzifying medianalgebra.

Remark 3.8. Another thing we may have noticed about the above example is that $(X, {\tilde{m}}_{[α]})$ is a median algebra, for all α ≤ 0.5.

Definition 3.9. A mapping $f : (X, {\tilde{m}}_{X}) ⟶ (Y, {\tilde{m}}_{Y})$ is called M-fuzzifying median preserving (M-MP, in short) provided that $\begin{matrix} \forall a, b, c & \in & X, f_{M}^{\to} ({\tilde{m}}_{X} (a, b, c)) \\ = & {\tilde{m}}_{Y} (f (a), f (b), f (c)) . \end{matrix}$

Proposition 3.10. (1) Let $(X, \tilde{m})$ be an M-fuzzifying median algebra. Then the identity mapping ${id}_{X} : (X, {\tilde{m}}_{X}) ⟶ (X, {\tilde{m}}_{X})$ is M-MP too. (2) Let $(X, {\tilde{m}}_{X})$ , $(Y, {\tilde{m}}_{Y})$ and $(Z, {\tilde{m}}_{Z})$ be M-fuzzifying median algebras. If $f : (X, {\tilde{m}}_{X}) ⟶ (Y, {\tilde{m}}_{Y})$ and $g : (Y, {\tilde{m}}_{Y}) ⟶ (Z, {\tilde{m}}_{Z})$ are M-MP, then $g \circ f : (X, {\tilde{m}}_{X}) ⟶ (Z, {\tilde{m}}_{Z})$ is also M-MP.

Proof. The statement (1) is easy to demonstrate. As to (2). For every a, b, c ∈ X. We have $\begin{matrix} (g \circ f)_{M}^{\to} ({\tilde{m}}_{X} (a, b, c)) \\ = & g_{M}^{\to} (f_{M}^{\to} ({\tilde{m}}_{X} (a, b, c))) \\ = & g_{M}^{\to} ({\tilde{m}}_{Y} (f (a), f (b), f (c))) \\ = & {\tilde{m}}_{Z} ((g \circ f) (a), (g \circ f) (b), (g \circ f) (c)) . \end{matrix}$

This means that g ∘ f is M-MP. □ It is discovered by the research that an M-fuzzifying median algebra give rise to an M-fuzzifying convex space in the following way.

Proposition 3.11. Let $(X, \tilde{m})$ be an M-fuzzifying median algebra and define $C_{\tilde{m}} : 2^{X} \to M$ asfollows: $\forall C \in 2^{X}, C_{\tilde{m}} (C) = ⋀_{a, b \in C, x \in X} ⋀_{c \notin C} ({\tilde{m}}_{X} (a, b, x) (c))^{'} .$ Then $(X, C_{\tilde{m}})$ is an M-fuzzifying convex space.

Proof. It suffices to verify that $C_{\tilde{m}}$ satisfies (MC1)-(MC3). Actually,

$C_{\tilde{m}} (\emptyset) = C_{\tilde{m}} (X) = \land \emptyset = ⊤$ .

For every {A_i|i ∈ Ω} ⊆2^X, we have $\begin{matrix} C_{\tilde{m}} (⋂_{i \in Ω} A_{i}) \\ = & ⋀_{a, b \in ⋂_{i \in Ω} A_{i}, x \in X} ⋀_{c \notin ⋂_{i \in Ω} A_{i}} ({\tilde{m}}_{X} (a, b, x) (c))^{'} \\ = & ⋀_{a, b \in ⋂_{i \in Ω} A_{i}, x \in X} ⋀_{i \in Ω} ⋀_{c \notin A_{i}} ({\tilde{m}}_{X} (a, b, x) (c))^{'} \\ \geq & ⋀_{i \in Ω} ⋀_{a, b \in A_{i}, x \in X} ⋀_{c \notin A_{i}} ({\tilde{m}}_{X} (a, b, x) (c))^{'} \\ = & ⋀_{i \in Ω} C_{\tilde{m}} (A_{i}) . \end{matrix}$

For all ${A_{i} | i \in Ω} \overset{dir}{\subseteq} 2^{X}$ , we have $\begin{matrix} C_{\tilde{m}} (⋃_{i \in Ω} A_{i}) \\ = & ⋀_{a, b \in ⋃_{i \in Ω} A_{i}, x \in X} ⋀_{c \notin ⋃_{i \in Ω} A_{i}} ({\tilde{m}}_{X} (a, b, x) (c))^{'} \\ = & ⋀_{i_{1}, i_{2} \in Ω} ⋀_{a \in A_{i_{1}}} ⋀_{b \in A_{i_{2}}} ⋀_{x \in X} ⋀_{c \notin ⋃_{i \in Ω} A_{i}} ({\tilde{m}}_{X} (a, b, x) (c))^{'} \\ \geq & ⋀_{i_{3} \in Ω} ⋀_{a, b \in A_{i_{3}}, x \in X} ⋀_{c \notin ⋃_{i \in Ω} A_{i}} ({\tilde{m}}_{X} (a, b, x) (c))^{'} \\ \geq & ⋀_{i \in Ω} ⋀_{a, b \in A_{i}, x \in X} ⋀_{c \notin A_{i}} ({\tilde{m}}_{X} (a, b, x) (c))^{'} \\ = & ⋀_{i \in Ω} C_{\tilde{m}} (A_{i}) . \end{matrix}$

This completes the proof that $(X, C_{\tilde{m}})$ is an M-fuzzifying convex space. □

Proposition 3.12. If $f : (X, {\tilde{m}}_{X}) ⟶ (Y, {\tilde{m}}_{Y})$ is an M-MP mapping, then $f : (X, C_{{\tilde{m}}_{X}}) ⟶ (Y, C_{{\tilde{m}}_{Y}})$ is an M-CP mapping. Moreover if f: X ⟶ Y is also a surjective mapping, then $f : (X, C_{{\tilde{m}}_{X}}) ⟶ (Y, C_{{\tilde{m}}_{Y}})$ is an M-CC mapping.

Proof. Since $f : (X, {\tilde{m}}_{X}) ⟶ (Y, {\tilde{m}}_{Y})$ is an M-MP mapping, we conclude that $f_{M}^{\to} ({\tilde{m}}_{X} (a, b, c)) = {\tilde{m}}_{Y} (f (a), f (b), f (c)), \forall x, y \in X .$ Then for every B ∈ 2^Y, $\begin{array}{l} ℓ_{\tilde{m} X} (f \leftarrow (B)) \\ =_{\begin{array}{l} a, b \in f^{\leftarrow} (B) c \notin \\ x \in X \end{array}}^{\land}_{f^{\leftarrow} (B)}^{\land} (\tilde{m} X (a, b, x) (c))^{'} \\ \geq_{\begin{array}{l} a, b \in f^{\leftarrow} (B) c \notin \\ x \in X \end{array}}^{\land}_{f^{\leftarrow} (B)}^{\land} (f_{M}^{\leftarrow} \circ f_{M}^{\to} (\tilde{m} X (a, b, x) (c))^{'} \\ =_{\begin{array}{l} f (a), f (b) \in B f (c) \notin \\ x \in X \end{array}}^{\land}_{(B)}^{\land} (f_{M}^{\leftarrow} \circ f_{M}^{\to} (\tilde{m} X (a, b, x)) (c))^{'} \\ =_{\begin{array}{l} f (a), f (b) \in B f (c) \notin \\ x \in X \end{array}}^{\land}_{(B)}^{\land} (f_{M}^{\leftarrow} (\tilde{m} Y (f (a), f (b), f (x))) (c))^{'} \\ =_{\begin{array}{l} f (a), f (b) \in B f (c) \notin \\ x \in X \end{array}}^{\land}_{(B)}^{\land} (\tilde{m} Y (f (a), f (b), f (x))) (c))^{'} \\ \geq_{a_{1}, b_{1} \in B, y \in Y c_{1} \notin}^{\land}_{B}^{\land} (\tilde{m} Y (a_{1}, b_{1}, y) (c_{1}))^{'} \\ = ℓ_{\tilde{m} Y} (B) . \end{array}$

This shows that $f : (X, C_{{\tilde{m}}_{X}}) ⟶ (Y, C_{{\tilde{m}}_{Y}})$ is an M-CP mapping. To complete the proof of the proposition, for A ∈ 2^X, we have $\begin{array}{l} ℓ_{\tilde{m} Y} (f^{\to} (A)) \\ =_{\begin{array}{l} f (a), f (b) \in f^{\to} (A) c \notin \\ f (x) \in X \end{array}}^{\land}_{f^{\to} (B)}^{\land} (\tilde{m} Y (f (a), f (b), f (x)) (f (c)))^{'} \\ =_{\begin{array}{l} a, b \in f (B) c \notin \\ x \in X \end{array}}^{\land}_{f^{\to} (A)}^{\land} (f_{M}^{\leftarrow} (\tilde{m} X (a, b, x)) (f (c)))^{'} \\ =_{\begin{array}{l} a, b \in A f \\ x \in X \end{array}}^{\land}_{(c) \notin f^{\to} (A)}^{\land} (f_{M}^{\leftarrow} (\tilde{m} X (a, b, x)) (f (c)))^{'} \\ =_{\begin{array}{l} a, b \in A f (c) \notin \\ x \in X \end{array}}^{\land}_{f^{\to} (A)}^{\land} (f_{M}^{\leftarrow} (\tilde{m} X (a, b, x)) (f (c)))^{'} \\ =_{\begin{array}{l} a, b \in A f (c) \notin \\ x \in X \end{array}}^{\land}_{f^{\to} (A)}^{\land} (f_{M}^{\to} ({\tilde{m}}_{X} (a, b, x))) (f (c)))^{'} \\ =_{a, b \in A}^{\land}_{f (c) \notin f^{\to} (A)}^{\land} (f_{M}^{\to} (\tilde{m} X (a, b, x)) (f (c))^{'} \\ =_{\begin{array}{l} a, b \in A \\ x \in X \end{array}}^{\land} {_{f (c) \notin f^{\to} (A)}^{\land}}_{f (u) = f (c)}^{\land} (\tilde{m} X (a, b, x)) (u))^{'} \\ \geq_{\begin{array}{l} a, b \in A \\ x \in X \end{array}}^{\land}_{u \notin f A}^{\land} (\tilde{m} X (a, b, x) (u))^{'} \\ = ℓ_{\tilde{m} X} (A) . \end{array}$

That is, $f : (X, C_{{\tilde{m}}_{X}}) ⟶ (Y, C_{{\tilde{m}}_{Y}})$ is an M-CCmapping. □

Note that the M-fuzzifying median algebra can be used to induce an M-fuzzifying interval operator as follows: $I_{\tilde{m}} (a, b) (x) = ⋁_{c \in X} \tilde{m} (a, b, c) (x) .$

From the above formula, we can draw some important conclusions.

Corollary 3.13. The relationship between M-fuzzifying interval operator $I_{\tilde{m}}$ and M-fuzzifying median operator $\tilde{m}$ has following characteristic: $I_{\tilde{m}} (a, b) (x) = \tilde{m} (a, b, x) (x) .$

Proof. By definition, we have $I_{\tilde{m}} (a, b) (x) = ⋁_{c \in X} \tilde{m} (a, b, c) (x) \geq \tilde{m} (a, b, x) (x)$ . Conversely, by (MM3), for all a, b, c, x ∈ X, $\begin{matrix} \tilde{m} (a, b, c) (x) & = & \tilde{m} (a, b, c) (x) \land \tilde{m} (a, a, b) (a) \\ \leq & ⋀_{z \in X} (\tilde{m} (a, x, b) (z) \leftrightarrow \tilde{m} (a, c, b) (z)) \\ \leq & ⋀_{z \in X} (\tilde{m} (a, c, b) (z) \to \tilde{m} (a, x, b) (z)) \\ \leq & \tilde{m} (a, c, b) (x) \to \tilde{m} (a, x, b) (x) . \end{matrix}$

Then $\tilde{m} (a, c, b) (x) \land \tilde{m} (a, b, c) (x) \leq \tilde{m} (a, x, b)$ (x), that is $\tilde{m} (a, b, c) (x) \leq \tilde{m} (a, b, x) (x)$ , so we have $I_{\tilde{m}} (a, b) (x) = ⋁_{c \in X} \tilde{m} (a, b, c) (x) \leq \tilde{m} (a, b, x) (x)$ . □

Proposition 3.14. If $f : (X, {\tilde{m}}_{X}) ⟶ (Y, {\tilde{m}}_{Y})$ is an M-MP mapping, then $f : (X, I_{{\tilde{m}}_{X}}) ⟶ (Y, I_{{\tilde{m}}_{Y}})$ is an M-IP mapping. Moreover if f: X ⟶ Y is also a one-one mapping, then $f : (X, I_{{\tilde{m}}_{X}}) ⟶ (Y, I_{{\tilde{m}}_{Y}})$ is an M-CC mapping.

Combining Corollary 3.13, we conclude that for every a, b, x ∈ X, $\begin{matrix} f_{M}^{\leftarrow} (I_{{\tilde{m}}_{Y}} (f (a), f (b))) (x) \\ = & I_{{\tilde{m}}_{Y}} (f (a), f (b)) (f (x)) \\ = & {\tilde{m}}_{Y} (f (a), f (b), f (x)) (f (x)) \\ = & f_{M}^{\to} ({\tilde{m}}_{X} (a, b, x)) (f (x)) \\ = & ⋁_{f (z) = f (x)} {\tilde{m}}_{X} (a, b, x) (z) \\ \geq & {\tilde{m}}_{X} (a, b, x) (x) \\ = & I_{{\tilde{m}}_{X}} (a, b) (x) . \end{matrix}$

Then $f_{M}^{\to} (I_{{\tilde{m}}_{X}} (a, b)) \leq f_{M}^{\to} \circ f_{M}^{\leftarrow} (I_{{\tilde{m}}_{Y}} (f (a),$ $f (b))) \leq I_{{\tilde{m}}_{Y}} (f (a), f (b))$ , showing that f: (X, $I_{{\tilde{m}}_{X}}) ⟶ (Y, I_{{\tilde{m}}_{Y}})$ is an M-IP mapping. The rest of the proposition is obviously valid. □

Proposition 3.15. If the M-fuzzifying intervaloperator $I_{\tilde{m}}$ is induced by an M-fuzzifying median algebra $(X, \tilde{m})$ , then $I_{\tilde{m}}$ is an M-fuzzifying geometric interval operator.

Proof. According to Corollary 3.13 (MG1) is easy to obtain. As to the axiom (MG2), by (MM3) and Corollary 3.13 ∀a, b, c, d ∈ X, $\begin{matrix} I_{\tilde{m}} (a, b) (c) \land I_{\tilde{m}} (b, c) (d) \\ = & \tilde{m} (a, b, c) (c) \land \tilde{m} (b, c, d) (d) \\ \leq & ⋀_{z \in X} (\tilde{m} (c, d, c) (z) \leftrightarrow \tilde{m} (a, d, c) (z)) \\ \leq & ⊤ \leftrightarrow \tilde{m} (a, d, c) (c) \\ = & \tilde{m} (a, d, c) (c) \\ = & I_{\tilde{m}} (a, d) (c) . \end{matrix}$ Meanwhile, $\begin{matrix} I_{\tilde{m}} (b, c) (d) \land I_{\tilde{m}} (a, b) (c) \\ = & \tilde{m} (b, c, d) (d) \land \tilde{m} (a, b, c) (c) \\ \leq & ⋀_{z \in X} (\tilde{m} (a, d, b) (z) \leftrightarrow \tilde{m} (c, d, b) (z)) \\ \leq & \tilde{m} (c, d, b) (d) \to \tilde{m} (a, d, b) (d) \\ \leq & I_{\tilde{m}} (b, c) (d) \to I_{\tilde{m}} (a, b) (d), \end{matrix}$ that is $I_{\tilde{m}} (b, c) (d) \land I_{\tilde{m}} (a, b) (c) \leq I_{\tilde{m}} (a, b) (d)$ . Hence $I_{\tilde{m}} (a, b) (c) \land I_{\tilde{m}} (b, c) (d) \leq I_{\tilde{m}} (a, b) (d) \land I_{\tilde{m}} (a, d) (c)$ . □

4 M-fuzzifying modular spaces

In this section, we will introduce the concept of M-fuzzifying modular operators, after that we will study the relations between M-fuzzifying modular operators and M-fuzzifying median operators. What is really needed to demonstrate that ⊥ in this section is prime in M.

Definition 4.1. Let $(X, I)$ be an M-fuzzifying geometric interval space and ∀a, b, c, x ∈ X, we define $M (a, b, c) (x) = I (a, b) (x) \land I (b, c) (x) \land I (c, a) (x),$ and we call that M-fuzzifying geometric interval operator is M-fuzzifying modular operator provided $⋁_{x \in X} M (a, b, c) (x) \neq ⊥$ for each a, b, c ∈ X.

For an M-fuzzifying modular operator $M$ on X, the pair $(X, I)$ is called an M-fuzzifying modular space.

Lemma 4.2. In an M-fuzzifying geometric interval space $(X, I)$ the following are true.

$M (a, b, c) (c) = I (a, b) (c)$ ,

If $I (a, b) (c) = λ \neq ⊥$ , then $M (a, b, c) = c_{λ}$ .

Proof. (1) It is straightforward. As to (2), note that $M (a, b, c) (c) = I (a, b) (c) = λ$ . And according to Corollary 3.2, we have $I (a, b) (c) \land I (b, c) (d) \land I (a, c) (d) = ⊥$ for each d ∈ X \ c, combined with prime ⊥ and $I (a, b) (c) \neq ⊥$ , yields $I (b, c) (d) \land I (a, c) (d) = ⊥$ . This gives $M (a, b, c) = c_{λ}$ . □

Proposition 4.3. Let $(X, \tilde{m})$ be an M-fuzzifying median algebra and define $I_{\tilde{m}} : X \times X \to M^{X}$ as follows: $I_{\tilde{m}} (a, b) (x) = ⋁_{c \in X} \tilde{m} (a, b, c) (x) .$

Then $(X, I_{\tilde{m}})$ is an M-fuzzifying modular space with $\tilde{m} (a, b, c) \leq M_{\tilde{m}} (a, b, c)$ and $M_{\tilde{m}} (a, b, c)$ is an fuzzy point for all a, b, c ∈ X.

Proof. We reviewed Proposition 3.15 that an M-fuzzifying median algebra can induces an M-fuzzifying geometric interval operator $I_{\tilde{m}}$ . By Definition of M-fuzzifying interval operator, we have $⊥ \neq ⋁_{x \in X} \tilde{m} (a, b, c) (x) \leq ⋁_{x \in X} (I_{\tilde{m}} (a, b) (x) \land I_{\tilde{m}} (b, c) (x) \land I_{\tilde{m}} (c, a) (x)) = ⋁_{x \in X} M_{\tilde{m}} (a, b, c) (x)$ for each a, b, c ∈ X, this showing that $(X, I_{\tilde{m}})$ is an M-fuzzifying modular interval space and $\tilde{m} (a, b, c) \leq M_{\tilde{m}} (a, b, c)$ .

Let $M_{\tilde{m}} (a, b, c) (x) \neq ⊥$ , ∀x ∈ X. Suppose that $\tilde{m} (a, b, c) = y_{λ}$ , λ≠ ⊥, and by (MM3) $\tilde{m} (x,$ $a, b) (x) \land \tilde{m} (a, b, c) (y) \leq \tilde{m} (x, c, b) (x) \leftrightarrow \tilde{m} (x, y, b)$ $(x) \leq (\tilde{m} (x, c, b) (x) \to \tilde{m} (x, y, b) (x))$ , that is $I_{\tilde{m}} (a,$ $b) (x) \land I_{\tilde{m}} (b, c) (x) \land \tilde{m} (a, b, c) (y) \leq I_{\tilde{m}} (b, y) (x)$ . In the same way, $I_{\tilde{m}} (a, b) (x) \land I_{\tilde{m}} (a, c) (x) \land \tilde{m} (a, b, c) (y) \leq I_{\tilde{m}} (a, y) (x)$ . Then we have $I_{\tilde{m}} (a,$ $b) (x) \land I_{\tilde{m}} (b, c) (x) \land I_{\tilde{m}} (c, a) (x) \land \tilde{m} (a, b, c) (y) \leq$ $I_{\tilde{m}} (a, y) (x) \land I_{\tilde{m}} (b, y) (x)$ , this yields $M_{\tilde{m}} (a,$ $b, c) (x) \land \tilde{m} (a, b, c) (y) \leq I_{\tilde{m}} (a, y) (x) \land I_{\tilde{m}} (b, y) (x)$ $\land I_{\tilde{m}} (a, b) (y) .$ If x ≠ y, then by Corollary 3.2 yields $M_{\tilde{m}} (a, b, c) (x) \land \tilde{m} (a, b, c) (y) = ⊥$ . Then we have $M_{\tilde{m}} (a, b, c) (x) = ⊥$ or $\tilde{m} (a, b, c) (y) = ⊥$ since ⊥ is prime, contradiction. Hence x = y and $\tilde{m} (a, b, c) = x_{λ}$ . Thus $M_{\tilde{m}} (a, b, c) = x_{μ}$ andλ ≤ μ. □

From the above Proposition, we conclude that M-fuzzifying medina algebra is a special kind of M-fuzzifying modular space in which $\tilde{m} \leq M_{\tilde{m}}$ .

Proposition 4.4. If $(X, I_{X})$ is an M-fuzzifying modular space and if f: X ⟶ Y is a surjective M-II function, then so is $(Y, I_{Y})$ .

Proof. $(Y, I_{Y})$ is an M-fuzzifying geometric interval space by Theorem 1. Since f: X ⟶ Y is an M-II function, then for all f (a), f (b), f (c) ∈ X, $\begin{matrix} ⋁_{f (x) \in Y} M (f (a), f (b), f (c)) (f (x)) \\ = ⋁_{f (x) \in Y} (I_{Y} (f (a), f (b)) \land I_{Y} (f (b), f (c)) \land I_{Y} (f (c), f (a))) (f (x)) \\ = ⋁_{x \in X} f_{M}^{\leftarrow} (I_{Y} (f (a), f (b)) \land I_{Y} (f (b), f (c)) \land I_{Y} (f (c), f (a))) (x) \\ = ⋁_{x \in X} f_{M}^{\leftarrow} (f_{M}^{\to} (I_{X} (a, b)) \land f_{M}^{\to} (I_{X} (b, c)) \land f_{M}^{\to} (I_{X} (c, a))) (x) \\ \geq ⋁_{x \in X} (I_{X} (a, b) \land I_{X} (b, c) \land I_{X} (c, a)) (x) \\ \neq ⊥ . \end{matrix}$ This shows that $(Y, I_{Y})$ is an M-fuzzifying modular space. □

Proposition 4.5. Let $I_{1}$ and $I_{2}$ be two M-fuzzifying geometric interval operators on X with corresponding M-fuzzifying modular operators $M_{1}$ and $M_{2}$ . If for each a, b, c ∈ X there is a point z in X with $M_{1} (a, b, c) (z) \land M_{2} (a, b, c) (z) \neq ⊥$ ,then $\forall a, b, c, x \in X, I_{1} (a, b) (x) \neq ⊥$ ( $M_{1} (a, b, c) (x)$ ≠⊥) if and only if $I_{2} (a, b) (x) \neq ⊥$ ( $M_{2} (a, b, c)$ (x)≠ ⊥).

Proof. For all a, b, x ∈ X, if $I_{1} (a, b) (x) = λ \neq ⊥$ , by Remark 1 we have $M_{1} (a, b, x) = x_{λ}$ . It follows that $M_{2} (a, b, x) = x_{μ}$ and μ≠ ⊥ since $M_{1} (a, b, x) (x) \land M_{2} (a, b, x) (x) \neq ⊥$ . Thus $I_{2} (a, b)$ $(x) = M_{2} (a, b, x) (x) = μ \neq ⊥$ . After that change the role of the indices, we can complete theproof. □

5 Conclusion

In order to obtain M-fuzzifying interval spaces and M-fuzzifying convex spaces spaces, this paper presents an M-fuzzifying median algebras in a new concept and method. And then, we discuss some mapping properties of M-fuzzifying median algebras and M-fuzzifying convex spaces. It is also shown that induced the M-fuzzifying interval operators is geometric. Afterwards, we study the relations between M-fuzzifying median algebras and M-fuzzifying modular spaces.

Footnotes

Acknowledgments

The project is supported by the National Natural Science Foundation of China (11371002) and the Specialized Research Fund for the Doctoral Program of Higher Education (20131101110048).

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