A natural method of constructing many-valued convex structures

Abstract

In this paper we give a natural method of constructing a bi-stratified [0, 1]-convexity by a [0, 1]-valued interval operator, where the concept of bi-stratified [0, 1]-convexities is firstly introduced. And its convex hull operators can be characterized by [0, 1]-valued interval operators. Finally, we discuss the relationship between bi-stratified [0, 1]-convexities and [0, 1]-valued interval operators from a categorical aspect.

Keywords

[0,1]-valued interval operator fuzzy convexity bi-stratified fuzzy convexity convex hull operator

1 Introduction

There exist “convex sets” in many research areas [30], such as vector spaces, ordered spaces, semi-lattice (in particular, trees), lattices [29], graphs, metric spaces [3, 4]. It is typical of an axiomatic approach not to emphasize what an object represent, but rather how it behave. In the spirit, the abstract convexity theory [30] is given, which is a description of the properties of convex sets. Explicitly, a convexity (or convex structure) on a set X is a family of subsets of X which contains the universal set X and the empty set ∅ and is stable for non-empty intersections and for unions of non-empty chains.

With the development of fuzzy mathematics, many mathematical structures have been generalized to the fuzzy setting [5–7 , 18]. Convexity has also been combined with fuzzy set theory. In 1994, Rosa generalized the concept of convexity to [0, 1]-fuzzy setting [25]. In 2009, Maruyama generalized Rosa’s definition to L-fuzzy setting [16]. In Rosa and Maruyama’s definition, an L-fuzzy convex structure is a crisp family of L-fuzzy subsets on X satisfying certain axioms. This kind of fuzzy convex structures is called L-convex structures. Up to now, many researchers studied the theory of L-convex structures from many different aspects [19–23 , 26].

In 2014, from a completely different point of view, Shi and Xiu introduced the concept of M-fuzzifying convex structures in [27]. In Shi and Xiu’s definition, an M-fuzzifying convex structure is an M-fuzzy set of 2^X satisfying certain axioms and each subset of X can be regarded as a convex set to some degree. Subsequently, many related spatial structures were introduced to characterize the M-fuzzifying convex structures, such as M-fuzzifying restricted hull operators [28], M-fuzzifying bases and M-fuzzifying bases [33] and so on. It is well known that interval operators provide a natural and frequent method of constructing convex structures. And there exist close relations between interval operators and convex structures. So they were generated in the paper [34], where Xiu introduced the notion of M-fuzzifying interval spaces and studied the relationship between M-fuzzifying interval operators and M-fuzzifying convex structures. Later, M-fuzzifying JHC convex structures and M-fuzzifying Peano (geometric) interval spaces [31, 32] were introduced and studied. By above, they only considered to construct M-fuzzifying convex structures by M-fuzzifying interval operators.

However, there exists a natural problem of how to construct many-valued convexity in the sense of Rosa by [0, 1]-valued interval operators and fuzzy inclusion order. This is one of our main aims in this paper. It is well known that category theory provides a convenient way to describe the relationship between different kinds of spatial structures [24]. So we will study the relationship between [0, 1]-valued convexity and [0, 1]-valued interval operators from a categorical aspect.

The paper is organized as follows. Section 2 places the necessary fuzzy set theory and triangular norm background, notations and basic ideas that are needed in the subsequent sections. Section 3, we introduce the concept of bi-stratified [0, 1]-convexities, which can be characterized by its convex hull. And we can construct a bi-stratified fuzzy convexity by a [0, 1]-valued interval operator. Finally, we discuss the relationship between [0, 1]-convexities and [0, 1]-valued interval operators from a categorical aspect.

2 Preliminaries

We refer to [1] for general category theory, and to [2 , 35] for fuzzy set theory. In this section, we recall briefly some basic ideas of triangular norm on the unit interval [0, 1] in [12], and fuzzy orders that will be needed.

Definition 2.1. ([12]) A triangular norm (a t-norm, in short) is a binary operation * : [0, 1] × [0, 1] → [0, 1] on [0, 1], such that for any a, b, c ∈ [0, 1] the following conditions are satisfied.

(1) (commutativity) a * b = b * a.

(2) (monotonicity) a * b ≤ a * c if b ≤ c.

(3) (associativity) (a * b) * c = a * (b * c).

(4) (boundary condition) a * 1 = a.

A t-norm * is called left continuous if for each a ∈ [0, 1], the function a * (-) : [0, 1] → [0, 1] has a right adjoint, that is a function a → (-) : [0, 1] → [0, 1] such that a * b ≤ c if and only if b ≤ a → c .

In the case, the binary function → : [0, 1] × [0, 1] → [0, 1] defined by → (a, b) = a → b, is called the residuation (or the implication) with respect to *. It is clear that the pair ([0, 1] , *) is a residuated lattice. The reader can refer to [2, 9] for more in this regard.

Some basic properties of * and → are collected in following proposition, which can be found in many places, for instance [12, 14].

Proposition 2.2. Let * be a left continuous t-norm and let → be the residuation with respect to *. Then

(I1) p → q = 1ifandonlyifp ≤ q.

(I2) 1 → q = qand0 * p = 0.

(I3) p * (⋁ _i∈Jq_i) = ⋁ _i∈J (p * q_i).

(I4) p → (q → r) = q → (p → r) = (p * q) → r.

(I5) p → (⋀ _i∈Jq_i) = ⋀ _i∈J (p → q_i).

(I6) (⋁ _i∈Jp_i) → q = ⋀ _i∈J (p_i → q).

(I7) p * (a ∧ b) = (p * a) ∧ (p * b).

(I8) p * (p → q) ≤ q.

(I9) (r → p) * (p → q) ≤ r → q.

Now, we give some examples of the left continuous t-norm on [0, 1] [12], which are important for fuzzy set theory [10, 12].

Example 2.3. (1). The minimum t-norm: a * b = min {a, b} = a ∧ b . The corresponding implication is given by

$\begin{matrix} a \to b = {\begin{matrix} 1, a \leq b; \\ b, a > b . \end{matrix} \end{matrix}$

(2) The product t-norm: a * b = a · b . The corresponding implication is given by $\begin{matrix} a \to b = {\begin{matrix} 1, a \leq b; \\ \frac{b}{a}, a > b . \end{matrix} \end{matrix}$

(3) The Łukasiewicz t-norm: a * b = max {a + b - 1, 0}. The corresponding implication is given by a → b = min {1, 1 - a + b}.

[0, 1] ^X denotes the set of all fuzzy sets on X . For a p ∈ [0, 1] , the constant fuzzy set p_X is given by p_X (x) = p for all x ∈ X . The partial order and all algebraic operators on [0, 1] can be extended to [0, 1] ^X by point-wise order. For p ∈ [0, 1] and A, B ∈ [0, 1] ^X, write fuzzy sets p * A, p → A and A → B by (p * A) (x) = p * A (x) , (p → A) (x) = p → A (x) and (A → B) (x) = A (x) → B (x) for all x ∈ X .

A fuzzy relation on a set X is a mapping R : X × X → [0, 1], which is called

(R1) reflexive if R (x, x) =1 for any x ∈ X,

(R2) *-transitive if R (x, y) * R (y, z) ≤ R (x, z) for any x, y, z ∈ X,

(R3) separated if R (x, y) = R (y, x) =1 implies x = y for any x, y ∈ X .

A reflexive and *-transitive fuzzy relation is called a fuzzy preorder. A separated fuzzy preorder is called a fuzzy partial order. The pair (X, R) is called a fuzzy partial ordered set.

For instance, (1) ([0, 1] , d_[0,1]) is a fuzzy partial ordered set, where d_[0,1] : [0, 1] × [0, 1] → [0, 1] is defined by d_[0,1] (p, q) = p → q for any p, q ∈ [0, 1]. (2) The fuzzy partial order sub_X : [0, 1] ^X × [0, 1] ^X → [0, 1] is defined by

$\begin{matrix} {sub}_{X} (A, B) = ⋀_{x \in X} A (x) \to B (x) . \end{matrix}$

The value sub_X (A, B) measures the degree that A is a subset of B . So sub_X is also called the fuzzy inclusion order [2, 7] on [0, 1] ^X. For each set X, the fuzzy powerset [0, 1] ^X equipped with fuzzy inclusion order sub_X is a fuzzy partial ordered set.

Given a mapping f : X → Y, as usual, define f^→ : [0, 1] ^X → [0, 1] ^Y and f^← : [0, 1] ^Y → [0, 1] ^X by

$\begin{matrix} f^{\to} (A) (y) & = & ⋁_{f (x) = y} A (x), f^{\leftarrow} (B) (x) \\ = & B (f (x)) . \end{matrix}$

The fuzzy set f^→ (A) is called the image of A under f, and f^← (B) the preimage of B.

Definition 2.4. ([34]) A mapping $I : X \times X \to [0, 1]^{X}$ is called a [0, 1]-valued interval operator on X if it satisfies for any x, y ∈ X:

(FI-1) $I (x, y) (x) = I (x, y) (y) = 1;$

(FI-2) $I (x, y) = I (y, x) .$

If $I : X \times X \to [0, 1] X$ is a [0, 1]-valued interval operator on X, then $(X, I)$ is called a [0, 1]-valued interval space.

A mapping $f : (X, I_{1}) \to (Y, I_{2})$ is interval-preserving if $f^{\to} (I_{1} (x, y)) \leq I_{2} (f (x), f (y))$ for any x, y ∈ X . It is trivial to verify that all [0, 1]-valued interval spaces as objects and interval-preserving mappings as morphisms can form a category, denoted by MIS .

For [0, 1]-valued interval spaces $(X, I_{1})$ and $(X, I_{2})$ , we say $(X, I_{1})$ is finer than $(X, I_{2})$ , or $(X, I_{2})$ is coarser than $(X, I_{1})$ , denoted by $(X, I_{1}) \leq_{I} (X, I_{2})$ , if the identity mapping ${id}_{X} : (X, I_{1}) \to (X, I_{2})$ is interval-preserving. We also write $I_{1} \leq_{I} I_{2}$ .

Definition 2.5. ([19 , 25]) A [0, 1]-convexity (or, fuzzy convex structure) $C$ on a set X is a subset of the fuzzy power set [0, 1] ^X which satisfies the following.

(FC1) $0_{X} \in C and 1_{X} \in C$ .

(FC2) For each subset {B_i ∣ i ∈ Λ} of $C$ , ⋀_i∈ΛB_i is in $C .$

(FC3) For each directed subset {B_i ∣ i ∈ Λ} of $C$ , ⋁_i∈ΛB_i is in $C .$

If it still satisfies the condition

(WS) $p_{X} \in C$ for all p ∈ [0, 1],

then it is called weakly stratified [0, 1]-convexity in [15, 19].

For a [0, 1]-convexity $C$ on a set X, the pair $(X, C)$ is called a [0, 1]-convex space.

A mapping $f : (X, C_{1}) \to (Y, C_{2})$ is convexity-preserving (simply denoted by CP) if $f^{\leftarrow} (B) \in C_{1}$ for any $B \in C_{2} .$

3 Many-valued convex structures constructed by [0, 1]-valued interval operators

In [30], interval operators provide a natural and frequent approach of constructing convexities (or, convex structures). In fuzzy setting, it is natural to construct many-valued convex structures by [0, 1]-valued interval operators.

Firstly, we introduce the concept of bi-stratified [0, 1]-convexities, which is a kind of many-valued convex structures.

Definition 3.1. A [0, 1]-convexity $C$ on a set X is stratified if it satisfies

(S1) $p * B \in C$ for all p ∈ [0, 1] and $B \in C$ .

A fuzzy convexity $C$ on a set X is co-stratified if it satisfies

(S2) $p \to B \in C$ for all p ∈ [0, 1] and $B \in C$ .

A [0, 1]-convexity $C$ is bi-stratified if it is both stratified and co-stratified. It is clear that the condition (S1) implies the condition (WS). Then stratified [0, 1]-convexity must be weakly stratified.

It is trivial to verify that all bi-stratified [0, 1]-convex spaces as objects and convexity-preserving mappings as morphisms can form a category, denoted by BiCVS .

For bi-stratified [0, 1]-convex spaces $(X, C_{1})$ and $(X, C_{2})$ , we say $(X, C_{1})$ is finer than $(X, C_{2})$ , or $(X, C_{2})$ is coarser than $(X, C_{1})$ , denoted by $(X, C_{1}) \leq_{C} (X, C_{2})$ , if the identity mapping ${id}_{X} : (X, C_{1}) \to (X, C_{2})$ is convexity-preserving. We also write $C_{1} \leq_{C} C_{2}$ .

Now, we will give some examples of bi-stratified [0, 1]-convexities.

Example 3.2. (i) Let X = {x, y} and * be the product t-norm· on [0, 1]. We can construct a bi-stratified [0, 1]-convexity as follows: $C = {A_{1 t}, A_{s 2}, A_{r} ∣ t, s \in (0, 1], r \in [0, 1]},$

where $\begin{matrix} A_{1 t} (z) = {\begin{matrix} 0, if z = x, \\ t, if z = y; \end{matrix} \end{matrix}$ $\begin{matrix} A_{s 2} (z) = {\begin{matrix} s, if z = x, \\ 0, if z = y; \end{matrix} \end{matrix}$ $\begin{matrix} A_{r} (z) = {\begin{matrix} r, if z = x, \\ r, if z = y; \end{matrix} \end{matrix}$

for any r ∈ [0, 1] and for any s, t ∈ (0, 1] . We can simply write A_1t (resp., A_s2, A_r) by (0, t) (resp., (s, 0) , (r, r)). It is clear that $C$ satisfies the axioms (FC1)-(FC3) and (S1)-(S2).

(ii) Let E be vector space over a real number field K and let A : E → ([0, 1] , ∧) be a fuzzy set in E. A is called convex [11] if A (x) ∧ A (y) ≤ A (kx + (1 - k) y) for k ∈ [0, 1]. Then

$C = {A \in [0, 1]^{E} ∣ A is convex}$

is a bi-stratified [0, 1]-convexity.

3.1 A natural method of constructing bi-stratified [0, 1]-convex structures

In this subsection, we will give a method of constructing many-valued convex structures by [0, 1]-valued interval operators, where many-valued convex structure is a bi-stratified [0, 1]-convexity. This is why we introduce the definition of bi-stratified [0, 1]-convexities.

Theorem 3.3 Let $(X, I)$ be a [0, 1]-valued interval space and define a family $C_{I}$ by

$\begin{matrix} C_{I} = {U \in L^{X} ∣ U (x) \land U (y) \leq {sub}_{X} (I (x, y), U)} . \end{matrix}$

Then $C_{I}$ is a bi-stratified [0, 1]-convexity on X .

Proof. It suffices to show that $C_{I}$ satisfies the axioms (FC1)-(FC3) and (S1)-(S2). It is clear that 0_X and 1_X are in $C_{I}$ . Hence we only need to check the conditions (FC2)-(FC3).

(FC2): Suppose that a family {U_i ∣ i ∈ J} is included in $C_{I}$ . Then, for any x, y ∈ X,

$\begin{matrix} (⋀_{i \in J} U_{i}) (x) \land (⋀_{i \in J} U_{i}) (y) \\ \leq & ⋀_{i \in J} (U_{i} (x) \land U_{i} (y)) \\ \leq & ⋀_{i \in J} {sub}_{X} (I (x, y), U_{i}) \\ = & {sub}_{X} (I (x, y), ⋀_{i \in J} U_{i}) . \end{matrix}$

(FC3): Let a directed family {U_i ∣ i ∈ D} be included in $C_{I}$ . Then, for any x, y ∈ X, $\begin{matrix} (⋁_{i \in D} U_{i}) (x) \land (⋁_{i \in D} U_{i}) (y) \\ = & ⋁_{i \in D} (U_{i} (x) \land U_{i} (y)) \\ \leq & ⋁_{i \in D} {sub}_{X} (I (x, y), U_{i}) \\ \leq & {sub}_{X} (I (x, y), ⋁_{i \in D} U_{i}) . \end{matrix}$

(S1): Take μ ∈ [0, 1] and $U \in C_{I}$ . Then μ * U is in $C_{I}$ . In fact, for any x, y ∈ X,

$\begin{matrix} (μ * U) (x) \land (μ * U) (y) \\ = & μ * (U (x) \land U (y)) \\ \leq & μ * {sub}_{X} (I (x, y), U) \\ \leq & {sub}_{X} (I (x, y), μ * U), \end{matrix}$

by (I7) of Proposition 2.2.

(S2): Take λ ∈ [0, 1] and $U \in C_{I}$ . Then λ → U is in $C_{I}$ . In fact, for any x, y ∈ X,

$\begin{matrix} (λ \to U) (x) \land (λ \to U) (y) \\ = & λ \to (U (x) \land U (y)) \\ \leq & λ \to {sub}_{X} (I (x, y), U) \\ = & {sub}_{X} (I (x, y), λ \to U), \end{matrix}$ by (I4) and (I5) of Proposition 2.2. □

Next, we will give some interesting examples of above method.

Example 3.4. A fuzzy metric [8, 13] on a non-empty set X is a mapping d : X × X × (0, + ∞) → ([0, 1] , *) (where * is a left-continuous t-norm) which satisfies for each x, y, z ∈ X and r, t ∈ (0, + ∞) the following conditions:

(FM1) d (x, y, t) =1⇔ x = y ;

(FM2) d (x, y, t) = d (y, x, t) ;

(FM3) d (x, z, r + t) ≥ d (x, y, r) * d (y, z, t);

(FM4) d (x, y, -) : (0, + ∞) → [0, 1] is left-continuous and $lim_{t \to + \infty} d (x, y, t) = 1$ .

From (FM3), d (x, z, t) ≥ ⋁ _r+s=td (x, y, r) * d (y, z, s). Now we can define a [0, 1]-valued (geodesic) interval operator I : X × X → [0, 1] ^X by

$I (x, z) (y) = ⋀_{t > 0} d (x, z, t) \to (⋁_{r + s = t} d (x, y, r) * d (y, z, s))$

for any x, y, z ∈ X . The value $I (x, z) (y)$ can be interpreted as the degree to which point y is (geodesically) between the points x, z.

Now we can obtain a bi-stratified [0, 1]-convexity $C$ of [0, 1] ^X, which is given by

$C = {U \in [0, 1] ∣ U (x) \land U (y) \leq {sub}_{X} (I (x, y), U)} .$

Given a bi-stratified [0, 1]-convex space $(X, C)$ , elements of $C$ are called [0, 1]-convex set and its convex hull operator ${co}_{C} : [0, 1]^{X} \to [0, 1]^{X}$ is defined by ${co}_{C} (A) = ⋀_{B \in C} {sub}_{X} (A, B) \to B$

for any A ∈ [0, 1] ^X. It follows that ${co}_{C} (A)$ is in $C$ from (FC2) and (S2). It is easy to see that ${co}_{C} (A)$ is the smallest [0, 1]-convex set of A with respect to the pointwise order on [0, 1] ^X.

From Theorem 3.3, a bi-stratified [0, 1]-convexity can be constructed by a [0, 1]-valued interval operator. So, the convex hull of a fuzzy set can also be described by [0, 1]-valued interval operator.

Theorem 3.5. Let $I$ be a [0, 1]-valued interval operator on X and let $C_{I}$ be the [0, 1]-convexity induced by $I$ . Then the convex hull of a fuzzy set A in $(X, C_{I})$ is given by $co (A) = ⋁_{k = 0}^{\infty} A_{k},$

where A₀ = A and $A_{k + 1} = ⋀ {U \in [0, 1]^{X} ∣ \forall x, y \in X, A_{k} (x) \land A_{k} (y) \leq {sub}_{X} (I (x, y), U)} .$

Proof. In order to prove $co (A) = ⋁_{k = 0}^{\infty} A_{k},$ we firstly show that the sequence ${A_{k}}_{k = 0}^{\infty}$ is increasing. In fact, suppose that a fuzzy set B is in set ${U \in [0, 1]^{X} ∣ \forall x, y \in X, A_{k} (x) \land A_{k} (y) \leq {sub}_{X} (I (x, y), U)} .$ Take y = x by the arbitrariness of y. Then, $A_{k} (x) \leq {sub}_{X} (I (x, x), B) \leq B (x)$ for every x ∈ X and whence $A_{k} \leq ⋀ {U \in [0, 1]^{X} ∣ \forall x, y \in X, A_{k} (x) \land A_{k} (y) \leq {sub}_{X} (I (x, y), U)} = A_{k + 1} .$

We prove $co (A) = ⋁_{k = 0}^{\infty} A_{k}$ in two steps.

Step 1. From the sequence ${A_{k}}_{k = 0}^{\infty}$ is increasing and A₀ = A, we have $A \leq ⋁_{k = 0}^{\infty} A_{k}$ . In order to prove $co (A) \leq ⋁_{k = 0}^{\infty} A_{k}$ , we only show that $⋁_{k = 0}^{\infty} A_{k}$ is in $C_{I}$ . In fact, for any x, y ∈ X, $\begin{matrix} (⋁_{k = 0}^{\infty} A_{k}) (x) \land (⋁_{k = 0}^{\infty} A_{k}) (y) \\ = & ⋁_{k = 0}^{\infty} (A_{k} (x) \land A_{k} (y)) \\ \leq & ⋁_{k = 0}^{\infty} {sub}_{X} (I (x, y), A_{k + 1}) \\ \leq & {sub}_{X} (I (x, y), ⋁_{k = 0}^{\infty} A_{k + 1}) \\ = & {sub}_{X} (I (x, y), ⋁_{k = 0}^{\infty} A_{k}) . \end{matrix}$

Step 2. We need to prove that $co (A) \geq ⋁_{k = 0}^{\infty} A_{k}$ . From $co (A) = ⋀ {U \in C_{I} ∣ A \leq U}$ , take $B \in {U \in C_{I} ∣ A \leq U}$ and hence $A (x) \land A (y) \leq B (x) \land B (y) \leq {sub}_{X} (I (x, y), B)$ for any x, y ∈ X . So A₁ ≤ B by the definition of A₁. By above analysis again, A_k ≤ B for any k and whence $⋁_{k = 0}^{\infty} A_{k} \leq {U \in C_{I} ∣ A \leq U} = co (A)$ by the arbitrariness of B.□

3.2 A characteristic of bi-stratified [0, 1]-convex spaces

In this subsection, we will show that a bi-stratified [0, 1]-convexity can be characterized by its convex hull operator.

Proposition 3.6. Let $(X, C)$ be a bi-stratified [0, 1]-convex space. Then its convex hull operator ${co}_{C} : [0, 1]^{X} \to [0, 1]^{X}$ satisfies the following axioms.

(co1) ${co}_{C} (0_{X}) = 0_{X}$ .

(co2) ${sub}_{X} (A, B) \leq {sub}_{X} ({co}_{C} (A), {co}_{C} (B))$ for any A, B ∈ [0, 1] ^X.

(co3) ${sub}_{X} (A, {co}_{C} (A)) = 1$ for all A ∈ [0, 1] ^X.

(co4) ${co}_{C} ({co}_{C} (B)) = {co}_{C} (B)$ for all B ∈ [0, 1] ^X .

(co5) ${co}_{C} (p * B) = p * {co}_{C} (B)$ for any B ∈ [0, 1] ^X and p ∈ [0, 1].

(co6) ${co}_{C} (⋁_{i \in Λ} B_{i}) = ⋁_{i \in Λ} {co}_{C} (B)$ for all directed subset {B_i ∣ i ∈ Λ}.

Conversely, if an operator co : [0, 1] ^X → [0, 1] ^X satisfies the axioms (co1)-(co6), then a family $C_{co} = {B \in [0, 1]^{X} ∣ co (B) = B}$

is a bi-stratified [0, 1]-convexity.

Proof. We prove the theorem in two steps.

Step 1. In order to complete the proof of the first part of the proposition, we only need to verify that the operator ${co}_{C}$ satisfies the axioms (co1)-(co6). It is clear that the condition (co1) holds.

(co2): For any A, B ∈ [0, 1] ^X, $\begin{matrix} {sub}_{X} ({co}_{C} (A), {co}_{C} (B)) \\ = & ⋀_{x \in X} {co}_{C} (A) (x) \to {co}_{C} (B) (x) \\ = & ⋀_{x \in X} (⋀_{C \in C} ({sub}_{X} (A, C) \to C (x))) \to \\ (⋀_{D \in C} ({sub}_{X} (B, D) \to D (x))) \\ = & ⋀_{x \in X} ⋀_{D \in C} (⋀_{C \in C} ({sub}_{X} (A, C) \to C (x))) \to \\ ({sub}_{X} (B, D) \to D (x)) \\ \geq & ⋀_{x \in X} ⋀_{D \in C} ({sub}_{X} (A, D) \to D (x)) \to \\ ({sub}_{X} (B, D) \to D (x)) \\ \geq & ⋀_{x \in X} ⋀_{D \in C} {sub}_{X} (B, D) \to {sub}_{X} (A, D) \\ \geq & {sub}_{X} (A, B) . \end{matrix}$

(co3): Since A ≤ sub_X (A, B) → B for any A, B ∈ [0, 1] ^X, then $\begin{matrix} 1 & \leq & ⋀_{x \in X} A (x) \to ⋀_{B \in C} ({sub}_{X} (A, B) \to B (x)) \\ = & {sub}_{X} (A, {co}_{C} (A)) \\ = & 1 . \end{matrix}$

(co4): It follows that $\begin{matrix} {co}_{C} ({co}_{C} (A)) & = & ⋀_{B \in C} ({sub}_{X} ({co}_{C} (A), B) \to B) \\ \leq & {sub}_{X} ({co}_{C} (A), {co}_{C} (A)) \to {co}_{C} (A) \\ = & {co}_{C} (A) \end{matrix}$ for each A ∈ [0, 1] ^X .

(co5): Suppose that B is in [0, 1] ^X and p is in [0, 1]. Since $C$ is a bi-stratified [0, 1]-convexity, then $p * {co}_{C} (B)$ is in $C$ for any B ∈ [0, 1] ^X and p ∈ [0, 1]. Therefore, on one hand, it follows $\begin{matrix} {co}_{C} (p * B) & = & ⋀_{U \in C} ({sub}_{X} (p * B, U) \to U) \\ \leq & {sub}_{X} (p * B, p * co (B)) \to (p * co (B)) \\ = & p * co (B) . \end{matrix}$ On the other hand, $\begin{matrix} p \to {co}_{C} (p * B) \\ = & p \to ⋀_{U \in C} ({sub}_{X} (p * B, U) \to U) \\ = & p \to ⋀_{U \in C} ({sub}_{X} (B, p \to U) \to U) \\ = & ⋀_{U \in C} ({sub}_{X} (B, p \to U) \to (p \to U)) \\ \geq & ⋀_{A \in C} ({sub}_{X} (B, A) \to A) \\ = & {co}_{C} (B) \end{matrix}$ holds, i.e., ${co}_{C} (p * B) \geq p * co (B) .$

(co6): Suppose that {B_i ∣ i ∈ Λ} is directed subfamily of [0, 1] ^X. Since (co2) holds, then ${co}_{C} (⋁_{i \in Λ} B_{i}) \geq ⋁_{i \in Λ} {co}_{C} (B_{i}) .$ On the other hand, since $⋁_{i \in Λ} {co}_{C} (B_{i})$ is in $C$ , then $\begin{matrix} {co}_{C} (⋁_{i \in Λ} B_{i}) \\ = & ⋀_{D \in C} ({sub}_{X} (⋁_{i \in Λ} B_{i}, D) \to D) \\ \leq & {sub}_{X} (⋁_{i \in Λ} B_{i}, ⋁_{i \in Λ} {co}_{C} (B_{i})) \to ⋁_{i \in Λ} {co}_{C} (B_{i}) \\ = & ⋁_{i \in Λ} {co}_{C} (B_{i}) . \end{matrix}$

Step 2. Suppose that an operator co : [0, 1] ^X → [0, 1] ^X satisfies the axioms (co1)-(co6). Now let $C_{co} = {B \in [0, 1]^{X} ∣ co (B) = B}$ . In order to show that $C_{co}$ is a bi-stratified [0, 1]-convexity, we need to verify that $C_{co}$ satisfies (LC1)-(LC3) and (S1)-(S2) as follows. It is trivial that the conditions (LC1)-(LC3) hold.

(S1): Take λ ∈ [0, 1] and $U \in C_{co} .$ Then λ → U ≤ co (λ → U) by (co3). And λ → U ≥ co (λ → U) holds. In fact, by (co5), λ * co (λ → U) = co (λ * (λ → U)) ≤ co (U)

holds. Therefore, λ → U is in $C_{co}$ .

(S2): Take μ ∈ [0, 1] and $U \in C_{co} .$ Then μ * U is in $C_{co}$ , since co (μ * U) = μ * co (U) = μ * U by (co5).□

Proposition 3.7. $C_{{co}_{C}} = C$ and ${co}_{C_{co}} = co .$

Proof. (i) $A \in C$ if and only if ${co}_{C} (A) = ⋀_{B \in C} {sub}_{X} (A, B) \to B = A .$ So $C_{{co}_{C}} = C$ .

(ii) Let A ∈ [0, 1] ^X . then $\begin{matrix} {co}_{C_{co}} (A) & = & ⋀_{B \in C_{co}} {sub}_{X} (A, B) \to B \\ = & ⋀_{B \in C_{co}} {sub}_{X} (A, co (B)) \to co (B) \\ = & co (A) . \end{matrix}$ Thus ${co}_{C_{co}} = co$ by the arbitrariness of A .□

The pair (X, co) is called a convex hull space if the operator co satisfies (co1)-(co6). A mapping f : (X, co₁) → (Y, co₂) is continuous if f^→ (co₁ (A)) ≤ co₂ (f^→ (A)) for any $A \in C_{1} .$ The category of convex hull space spaces and continuous mappings is denoted by CHS.

Proposition 3.8. If f : (X, co₁) → (Y, co₂) is continuous, then $f : (X, C_{{co}_{1}}) \to (Y, C_{{co}_{2}})$ is convexity-preserving.

Proof. Suppose that f : (X, co₁) → (Y, co₂) is continuous. Then, for any $B \in C_{{co}_{2}}$ , $f^{\to} ({co}_{1} (f^{\leftarrow} (B))) \leq {co}_{2} (f^{\to} \circ f^{\leftarrow} (B)) \leq {co}_{2} (B) = B .$ It follows that co₁ (f^← (B)) ≤ f^← ∘ f^→ (co₁ (f^← (B))) ≤f^← (B). And hence co₁ (f^← (B)) = f^← (B) (i.e., $f^{\leftarrow} (B) \in C_{{co}_{2}}$ ).□

Proposition 3.9. If $f : (X, C_{1}) \to (Y, C_{2})$ is convexity-preserving, then $f : (X, {co}_{C_{1}}) \to (Y, {co}_{C_{2}})$ is continuous.

Proof. Suppose that $f : (X, C_{1}) \to (Y, C_{2})$ is continuous. Then $A \leq f^{\leftarrow} \circ f^{\to} (A) \leq f^{\leftarrow} ({co}_{C_{2}} (f^{\to} (A))),$ and hence ${co}_{C_{1}} (A) \leq f^{\leftarrow} ({co}_{C_{2}} (f^{\to} (A))) .$ Thus we can obtain that $\begin{matrix} f^{\to} ({co}_{C_{1}} (A)) & \leq & f^{\to} (f^{\leftarrow} ({co}_{C_{2}} (f^{\to} (A)))) \\ \leq & {co}_{C_{2}} (f^{\to} (A)) . □ \end{matrix}$

Remark 3.10. Now, the category of bi-stratified [0, 1]-convex space and convexity-preserving mappings is denoted by BiCVS. From Proposition 3.7–3.9, we obtain that BiCVS is isomorphic to CHS.

By above, we know that a bi-stratified [0, 1]-convexity can be characterized by its convex hull operator.

3.3 Relationship between BiCVS and MIS

In this subsection, we will discuss the relation between bi-stratified [0, 1]-convexities and [0, 1]-valued interval operators.

Now, we shall construct a [0, 1]-valued interval operator by a bi-stratified [0, 1]-convex space.

Let $(X, C)$ be a bi-stratified [0, 1]-convex space and define a mapping $I_{C} : [0, 1]^{X} \to [0, 1]^{X}$ by $I_{C} (x, y) = ⋀ {U \in C ∣ U (x) = U (y) = 1} .$

Then $I_{C}$ is a [0, 1]-valued interval operator on X. In fact, from Definition 2.4, we only need to verify that $I_{C}$ satisfies the axioms (FI-1)-(FI-2). It is clear that $I_{C} (x, y) (x) = I_{C} (x, y) (y) = 1$ and $I_{C} (x, y) = I_{C} (y, x)$ .

From Theorem 3.3, we can construct a bi-stratified [0, 1]-convexity $C_{I}$ by a [0, 1]-valued interval operator $I .$ Connecting above, we have the following.

Proposition 3.11. $I \leq_{I} I_{C_{I}}$ and $C \geq_{C} C_{I_{C}}$ .

(1) $I \leq I_{C_{I}}$ . In fact, for any x, y ∈ X, $\begin{matrix} I_{C_{I}} (x, y) \\ = & ⋀ {U \in C_{I} ∣ U (y) = U (x) = 1} \\ \geq & I (x, y) \end{matrix}$ holds by the definition of $C_{I}$ . That is to say, $I \leq_{I} I_{C_{I}}$ holds.

(2) $C \subseteq C_{I_{C}}$ . In fact, let U be in $C$ . Take x, y ∈ X. We denote U (x) ∧ U (y) by t. Then t → U is in $C$ from the condition (S2), and hence $I_{C} (x, y) (z) = ⋀ {B \in C_{I} ∣ B (y) = B (x) = 1} \leq (t \to U) (z)$ for each z ∈ X. So, $t * I_{C} (x, y) (z) \leq t * (t \to U) (z) \leq U (z)$

for each z ∈ X. Therefore, $t \leq {sub}_{X} (I_{C} (x, y), U)$ . By the arbitrariness of x and y, we obtain that U is in $C_{I_{C}}$ . That is to say, $C \leq_{C} C_{I_{C}}$ holds.□

Proposition 3.12. If $f : (X, C_{1}) \to (Y, C_{2})$ is convexity-preserving, then $f : (X, I_{C_{1}}) \to (Y, I_{C_{2}})$ is interval-preserving.

Proof. We need to show that $f^{\to} (I_{C_{1}} (x, y)) \leq I_{C_{2}} (f (x), f (y))$ for any x, y ∈ X . It is equivalent to $I_{C_{1}} (x, y) \leq f^{\leftarrow} (I_{C_{2}} (f (x), f (y)))$ . So we only check that one of them holds. In fact, for any x, y ∈ X, $\begin{matrix} f^{\leftarrow} (I_{C_{2}} (f (x), f (y))) (z) \\ = & I_{C_{2}} (f (x), f (y)) (f (z)) \\ = & ⋀ {V (f (z)) ∣ V \in C_{2}, V (f (x)) = V (f (y)) = 1} \\ = & ⋀ {f^{\leftarrow} (V) (z) ∣ f^{\leftarrow} (V) (x) = f^{\leftarrow} (V) (y) = 1} \\ \geq & ⋀ {U (z) ∣ V \in C_{1}, U (x) = U (y) = 1} \\ \geq & I_{C_{1}} (x, y) \end{matrix}$ holds.□

Proposition 3.13. If $f : (X, I_{1}) \to (Y, I_{2})$ is interval-preserving, then $f : (X, C_{I_{1}}) \to (Y, C_{I_{2}})$ is convexity-preserving.

Proof. Let U be in $C_{I_{2}}$ . We only need to show that f^← (U) is in $C_{I_{1}}$ . Since $f : (X, I_{1}) \to (Y, I_{2})$ is interval-preserving, it follows that $f^{\to} (I_{1} (x, y)) \leq I_{2} (f (x), f (y))$ . Therefore, $\begin{matrix} f^{\leftarrow} (U) (x) \land f^{\leftarrow} (U) (y) \\ = & U (f (x)) \land U (f (y)) \\ \leq & {sub}_{Y} (I_{2} (f (x), f (y)), U) \\ \leq & {sub}_{Y} (f^{\to} (I_{1} (x, y)), U) \\ = & {sub}_{X} (I_{1} (x, y), f^{\leftarrow} (U)) \end{matrix}$ holds for any x, y ∈ X. By the arbitrariness of x and y, f^← (U) is in $C_{I_{1}}$ .□

Remark 3.14 From Theorem 3.3 and above analysis, there exist two functors F : BiCVS → MIS and G : MIS → BiCVS such that (F, G) is a Galois correspondence. It is shown that a [0, 1]-valued interval space contains not less information than the induced [0, 1]-convex space.

4 Conclusions

In this paper, it is shown that a [0, 1]-valued convexity can by constructed by a [0, 1]-valued interval operator, where the [0, 1]-valued convexity is just a bi-stratified [0, 1]-convexity. And the bi-stratified [0, 1]-convexity can be characterized by its convex hull. Finally, we discuss the relation between fuzzy convexities and [0, 1]-valued interval operators from a categorical aspect.

Footnotes

Acknowledgement

The authors would like to express their sincere thanks to the anonymous reviewers for their careful reading and constructive comments. This work is supported by the National Natural Science Foundation of China (11871097, 11371002) and Specialized Research Fund for the Doctoral Program of Higher Education (20131101110048).

References

Adámek

, Herrlich

, Strecker

G.E.

, Abstract and Concrete Categories, Wiley, New York, 1990.

Běelohlávek

Foundations and Principles, Fuzzy Relational Systems, Kluwer Academic /Plenum Publishers, New York, Boston, Dordrecht, London, Moscow, 2002.

Boltyanskii

V.G.

, Soltan

P.S.

, Combinatorial geometry and convexity class, Russian Mathematical Surveys 33 (1978), 1–45.

Busemann

, The geometry of geodesics, Academic Press, New York, 1955.

Chang

C.L.

, Fuzzy topological spaces, J Math Anal Appl 24 (1968), 190–201.

Fang

, I-fuzzy Alexandrov topologies and specialization orders, Fuzzy Sets Syst 158 (2007), 2359–2374.

Fang

, Relationships between L-ordered convergence structures and strong L-topologies, Fuzzy Sets Syst 161 (2010), 292–2944.

George

, Veeramani

, Some results in fuzzy metric spaces, Fuzzy Sets Syst 64 (1994), 395–399.

Gierz

, Hofmann

K.H.

, Keimel

, Lawson

J.D.

, Mislove

M.W.

, Scott

D.S.

, Continuous Lattices and Domains, Cambridge University Press, Cambridge, 2003.

10.

Hájek

, Metamathematics of Fuzzy Logic, Kluwer Academic Publishers, Dordrecht, 1998.

11.

Katsaras

, Fuzzy vector spaces and fuzzy topological vector spaces, J Math Anal Appl 68 (1977), 135–146.

12.

Klement

E.P.

, Mesiar

, Pap

, Triangular Norms, Trends in Logic, Vol. 8, Kluwer Academic Publishers, Dordrecht, 2000.

13.

Kramosil

, Michalek

, Fuzzy metrics and statistical metric spaces, Kybernetika 11 (1975), 336–344.

14.

Lai

, Zhang

, Fuzzy preorder and fuzzy topology, Fuzzy Sets Syst 157 (2006), 1865–1885.

15.

L.-Q.

, On the category of enriched (L,M)-convex spaces, J Intell Fuzzy Syst 33 (2017), 3209–3216.

16.

Maruyama

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

17.

Pang

, Stratified L-ordered filter spaces, Quaest Math 40(5) (2017), 661–678.

18.

Pang

, Degrees of separation properties in stratified L-generalized convergence spaces using residual implication, Filomat 31(20) (2017), 6293–6305.

19.

Pang

, Shi

F.-G.

, Fuzzy counterparts of hull operators and interval operators in the framework of L-convex spaces, Fuzzy Sets Syst (2018). DOI: 10.1016/j.fss.2018.05.012

20.

Pang

, Shi

F.-G.

, Strong inclusion orders between L-subsets and its applications in L-convex spaces, Quaest Math 41(8) (2018), 1021–1043.

21.

Pang

, Xiu

Z.-Y.

, An axiomatic approach to bases and subbases in L-convex spaces and their applications, Fuzzy Sets Syst (2018). DOI: 10.1016/j.fss.2018.08.002

22.

Pang

, Zhao

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

23.

Pang

, Zhao

, 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.

24.

Preuss

, Foundations of Topology–An Approach to Convenient Topology, Kluwer Academic Publisher, Dordrecht, Boston, London, 2002.

25.

Rosa

M.V.

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

26.

Shen

, Shi

F.-G.

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

27.

Shi

F.-G.

, Xiu

Z.-Y.

, A new approach to the fuzzification of convex spaces, J Appl Math (2014), Article ID 249183. https://doi.org/10.1155/2014-249183

28.

Shi

F.-G.

, Li

E.-Q.

, The restricted hull operator of M-fuzzifying convex structures, J Intell Fuzzy Syst 30 (2016), 409–421.

29.

van de Vel

M.L.J.

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

30.

van de Vel

M.L.J.

, Theory of convex spaces, North Holland, New York, 1993.

31.

X-Y

and Bai

S.-Z.

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

32.

X.-Y.

, Bai

S.-Z.

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

33.

Xiu

Z.-Y.

, Pang

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

34.

Xiu

Z.-Y.

, Shi

F.-G.

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

35.

Zadeh

L.A.

, Fuzzy sets, Inf Control 8 (1965), 338–353.