Some characterizations of ( L,M )-fuzzy convex spaces

Abstract

In this paper, the notions of (L, M)-fuzzy concave spaces, (L, M)-fuzzy interior spaces, (L, M)-fuzzy interior relations and (L, M)-fuzzy hull relations are introduced. It is proved that the category of (L, M)-fuzzy concave spaces, the category of (L, M)-fuzzy interior spaces, the category of (L, M)-fuzzy interior relation spaces and the category of (L, M)-fuzzy hull relation spaces are isomorphic. Moreover, it is proved that these categories are all isomorphic to the category of (L, M)-fuzzy convex spaces when L is a completely distributive lattice with an order-reversing involution.

1 Introduction

Axiomatic convexity theory is a branch of mathematics dealing with set-theoretic structures satisfying axioms similar to that convex sets in real linear spaces fulfill. The axiomatic convexity (convex structure) exists in many mathematical structures, such as convexities in metric spaces and graphs [17, 33], convexities in lattices and in Boolean algebras [40, 41]. Also, axiomatic convexity appeared naturally in topology, especially in the theory of supercompact spaces [19]. Some more details about axiomatic convexity can be found in [40].

The notion of a fuzzy subset was introduced by Zadeh and then fuzzy subsets have been applied to various branches of mathematics, such as fuzzy topology [10, 38] and fuzzy convergence [13 , 22]. In 1994, Rosa [31, 32] first applied fuzzy subsets to convex structures and introduced a fuzzy convex structure, which was defined as a classical subset of [0, 1] ^X satisfying certain axioms. In 2009, based on a completely distributive lattice L, Maruyama [18] gave a notion of L-convex structures, which was a subset of L^X. In recent years, this kind of fuzzy convex structures were widely studied by many researchers [9 , 30].

In 2014, Shi and Xiu [36] introduced a new approach to the fuzzification of convex structures–an M-fuzzifying convex structure, which is a mapping $C : 2^{X} ⟶ M$ satisfying three axioms. As a continuation of the work, there are more and more research on this M-fuzzifying convex structures [34 , 42–45].

Recently, Shi and Xiu [37] introduced the notion of (L, M)-fuzzy convex structures, which were the generalization of M-fuzzifying convex structures and L-convex structures. In this paper, the authors gave the basic framework of (L, M)-fuzzy convex structures and discussed the categorical relationships between M-fuzzifying convex structures and (L, M)-fuzzy convex structures. Moreover, Li [15] studied the categorical properties of enriched (L, M)-fuzzy convex spaces and Pang [29] discussed hull operators, restricted hull operators, bases and subbases of (L, M)-fuzzy convex structures.

In this paper, based on the framework of (L, M)-fuzzy convex structures, the notions of (L, M)-fuzzy concave spaces, (L, M)-fuzzyinterior spaces, (L, M)-fuzzy interior relations and (L, M)-fuzzy hull relations are introduced. In some conditions, they can be used to characterize (L, M)-fuzzy convex structures. It is proved that the category of (L, M)-fuzzy concave spaces, the category of (L, M)-fuzzy interior spaces, the category of (L, M)-fuzzy interior relation spaces, the category of (L, M)-fuzzy hull relation spaces are isomorphic. Moreover, it is proved that these categories are all isomorphic to the category of (L, M)-fuzzy convex spaces whenever L is a completely distributive lattice with an order-reversing involution.

2 Preliminaries

Throughout this paper, (M, ⋁, ⋀,′) denotes a complete lattice with an order-reversing involution ′. The smallest element (or zero element) and the largest element (or unit element) in M are denoted by ⊥_M and ⊤_M, respectively. For a, b ∈ M, we say that a is wedge below b in M (in symbols, a ≺ b) if for every subset D ⩽ M, ⋁D ⩾ b implies d ⩾ a for some d ∈ M. A complete lattice M is completely distributive if and only if b = ⋁ {a ∈ M ∣ a ≺ b} for each b ∈ L.

For a nonempty set X and a completely distributive lattice L with an order-reversing involution, still denoted by, the smallest element ⊥_L and the largest element ⊤_L, L^X denotes the set of all L-subsets on X. The operators on L can be translated onto L^X in a pointwise way. The smallest element and the largest element in L^X are denoted by $⊥_{L}^{X}$ and $⊤_{L}^{X}$ , respectively. Since L is a completely distributive lattice, then L^X is also a completely distributive lattice and the wedge below relation on L^X is still denoted by “ ≺ ". For each x ∈ X and a ∈ L, the L-subset x_a, defined by x_a (y) = a if y = x, and x_a (y) = ⊥ _L if y ≠ x, is called a fuzzy point. The set of all fuzzy points in L^X is denoted by J (L^X).

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

$\forall A \in L^{X}, \forall y \in Y, f_{L}^{\to} (A) (y) = ⋁_{f (x) = y} A (x);$

$\forall B \in L^{Y}, \forall x \in X, f_{L}^{\leftarrow} (B) (x) = B (f (x)) .$

Definition 2.1. An (L, M)-fuzzy convex structure on X is a mapping $C : L^{X} ⟶ M$ which satisfies:

(LMC1) $C (⊥_{L}^{X}) = C (⊤_{L}^{X}) = ⊤_{M}$ ;

(LMC2) If {A_i} _i∈Ω ⊆ L^X is nonempty, then $C (⋀_{i \in Ω} A_{i}) ⩾ ⋀_{i \in Ω} C (A_{i})$ ;

(LMC3) If {A_k} _k∈K ⊆ L^X is nonempty and totally ordered, then $C (⋁_{k \in K} A_{k}) ⩾ ⋀_{k \in K} C (A_{k})$ . For an (L, M)-fuzzy convex structure $C$ on X, the pair $(X, C)$ is called an (L, M)-fuzzy convex space.

A mapping $f : (X, C_{X}) ⟶ (Y, C_{Y})$ between (L, M)-fuzzy convex spaces is called (L, M)-fuzzy convexity preserving ((L, M)-CP, in short) if it satisfies $\forall B \in L^{Y}, C_{X} (f_{L}^{\leftarrow} (B)) ⩾ C_{Y} (B) .$

It is easy to check that all (L, M)-fuzzy convex spaces as objects and all (L, M)-CP mappings as morphisms form a category, denoted by LMFC.

Theorem 2.2. Let $C$ be a mapping on X satisfying (LMC1) and (LMC2). Then the following two statements are equivalent.

(LMC3)^* If {A_j} _j∈J ⊆ L^X is directed, then $C (⋁_{j \in J}^{d} A_{j}) ⩾ ⋀_{j \in J} C (A_{j})$ .

(LMC3) If {A_k} _k∈K ⊆ L^X is totally ordered, then $C (⋁_{k \in K} A_{k}) ⩾ ⋀_{k \in K} C (A_{k})$ .

Remark 2.3. By Definition 2.1 and Theorem 3.6, we know that a mapping $C : L^{X} ⟶ M$ satisfying (LMC1), (LMC2) and (LMC3)^* is also an (L, M)-fuzzy convex structure on X.

Definition 2.4. A mapping $H : L^{X} ⟶ M^{J (L^{X})}$ is called an (L, M)-fuzzy hull operator on X if it satisfies:

(LMH1) $H (⊥_{L}^{X}) (x_{λ}) = ⊥_{M}$ ;

(LMH2) $H (A) (x_{λ}) = ⊤_{M}$ for each x_λ ⩽ A;

(LMH3) $H (A) (x_{λ}) = ⋀_{x_{λ} B ⩾ A} ⋁_{y_{μ} B} H (B)$ (y_μ);

(LMH4) $H (⋁_{i \in Ω}^{dir} A_{i}) (x_{λ}) = ⋀_{μ ≺ λ} ⋁_{i \in Ω} H (A_{i})$ (x_μ) for each directed subfamily {A_i} _i∈Ω ⊆ L^X.

For an (L, M)-fuzzy hull operator $H$ on X, the pair $(X, H)$ is called an (L, M)-fuzzy hull space.

A mapping $f : (X, H_{X}) ⟶ (Y, H_{Y})$ between (L, M)-fuzzy hull spaces is called (L, M)-fuzzy hull preserving ((L, M)-HP, in short) if it satisfies $\forall B \in L^{Y}, H (f_{L}^{\leftarrow} (B)) (x_{λ}) ⩽ H (B) (f (x)_{λ}) .$

It is easy to check that all (L, M)-fuzzy hull spaces as objects and all (L, M)-HP mappings as morphisms form a category, denoted by LMFH.

Definition 2.5. ([29]) The category LMFC is isomorphic to LMFH.

3 (L, M)-fuzzy concave spaces and (L, M)-fuzzy interior spaces

Definition 3.1. A mapping $A : L^{X} \to M$ is called an (L, M)-fuzzy concave structure on X if it satisfies the following three conditions:

(LMA1) $A (⊥_{L}^{X}) = A (⊤_{L}^{X}) = ⊤_{M}$ ;

(LMA2) if {A_i : i ∈ Ω} ⊆ L^X is nonempty, then $A (⋁_{i \in Ω} A_{i}) ⩾ ⋀_{i \in Ω} A (A_{i})$ ;

(LMA3) if {A_i : i ∈ Ω} ⊆ L^X is co-directed, then $A (⋀_{i \in Ω}^{cdir} A_{i}) \geq ⋀_{i \in Ω} A (A)$ .

If $A$ is an (L, M)-fuzzy concave structure, then $(X, A)$ is called an (L, M)-fuzzy concave space.

Definition 3.2. A mapping $f : (X, A_{X}) ⟶ (Y, A_{Y})$ is called (L, M)-fuzzy concavity preserving((L, M)-AP, in short) provided that $A_{Y} (B) ⩽ A_{X} (f_{L}^{\leftarrow} (B))$ for each B ∈ L^Y.

It is easy to check that all (L, M)-fuzzy concave spaces as objects and all (L, M)-AP mappings as morphisms form a category, denoted by LMFA.

Definition 3.3. An (L, M)-fuzzy interior operator on X is a mapping Int : L^X ⟶ M^{J(L^X)} satisfying the following conditions:

(LMFI1) $Int (⊤_{L}^{X}) (x_{λ}) = ⊤_{M}$ for every x_λ ∈ J (L^X);

(LMFI2) Int (A) (x_λ) = ⊥ _M for every x_λnotleqslantA;

(LMFI3) $Int (⋀_{i \in Ω}^{cdir} A_{i}) = ⋀_{i \in Ω} Int (A_{i})$ ;

(LMFI4) Int (A) (x_λ) = ⋁ _{x_λ⩽B⩽A} ⋀ _{y_μ≺B}Int (B)(y_μ).

If Int is an (L, M)-fuzzy interior operator, then (X, Int) is called an (L, M)-fuzzy interior space.

Definition 3.4. A mapping f : (X, Int_X) ⟶ (Y, Int_Y) between concave (L, M)-fuzzy interior spaces is called (L, M)-fuzzy interior preserving ((L, M)-IP, in short) provided that for each B ∈ L^Y and x_λ ∈ J (L^X), ${Int}_{Y} (B) (f (x)_{λ}) ⩽ {Int}_{X} (f_{L}^{\leftarrow} (B)) (x_{λ}) .$

It is easy to check that all concave (L, M)-fuzzy interior spaces as objects and all (L, M)-IP mappings as morphisms form a category, denoted by LMFI.

Next we will establish the relations between (L, M)-fuzzy concave spaces and (L, M)-fuzzy interior spaces.

Theorem 3.5. Let (X, Int) be an (L, M)-fuzzy interior space and defined $A^{Int} : L^{X} ⟶ M$ by

\forall A \in L^{X}, A_{Int} (A) = ⋀_{x_{λ} ≺ A} Int (A) (x_{λ}) .

Then $(X, A^{Int})$ is an (L, M)-fuzzy concave space.

Proof. (1) $A^{Int} (⊤_{L}^{X}) = ⋀_{x_{λ} ≺ ⊤_{L}^{X}} Int (⊤_{L}^{X}) (x_{λ}) = ⊤_{M}$ and $A^{Int} (⊥_{L}^{X}) = ⋀_{x_{λ} ≺ ⊥_{L}^{X}} Int (⊥_{L}^{X}) (x_{λ}) = ⊤_{M}$ .

(2) For {A_i : i ∈ Ω} ⊆ L^X, $\begin{matrix} A^{Int} (⋁_{i \in Ω} A_{i}) & = & ⋀_{x_{λ} ≺ ⋁_{i \in Ω} A_{i}} Int (⋁_{i \in Ω} A_{i}) (x_{λ}) \\ = & ⋀_{i \in Ω} ⋀_{x_{λ} ≺ A_{i}} Int (⋁_{i \in Ω} A_{i}) (x_{λ}) \\ \geq & ⋀_{i \in Ω} ⋀_{x_{λ} ≺ A_{i}} Int (A_{i}) (x_{λ}) \\ \geq & ⋀_{i \in Ω} A^{Int} (A_{i}); \end{matrix}$

(3) For a co-directed set {A_i : i ∈ Ω} ⊆ L^X, by (LMFI3), $\begin{matrix} A^{Int} (⋀_{i \in Ω}^{cdir} A_{i}) & = & ⋀_{x_{λ} ≺ ⋀_{i \in Ω}^{cdir} A_{i}} Int (⋀_{i \in Ω}^{cdir} A_{i}) (x_{λ}) \\ \geq & ⋀_{x_{λ} ≺ A_{i}} ⋀_{i \in Ω} Int (A_{i}) (x_{λ}) \\ = & ⋀_{i \in Ω} ⋀_{x_{λ} ≺ A_{i}} Int (A_{i}) (x_{λ}) \\ = & ⋀_{i \in Ω} A^{Int} (A_{i}) . □ \end{matrix}$

Theorem 3.6. If f : (X, Int_X) → (Y, Int_Y) is (L, M)-IP, then $f : (X, A^{{Int}_{X}}) \to (Y, A^{{Int}_{Y}})$ is (L, M)-AP.

Proof. For each B ∈ L^Y, $\begin{matrix} A^{{Int}_{X}} (f_{L}^{\leftarrow} (B)) & = & ⋀_{x_{λ} ≺ f_{L}^{\leftarrow} (B)} {Int}_{X} (f_{L}^{\leftarrow} (B)) (x_{λ}) \\ ⩾ & ⋀_{x_{λ} ≺ f_{L}^{\leftarrow} (B)} ({Int}_{Y} (B) (f (x)_{λ}) \\ ⩾ & ⋀_{y_{μ} ≺ B} {Int}_{Y} (B) (y_{μ}) \\ = & A_{Y} (B) . \end{matrix}$ □ By Theorems 3.5 and 3.6, we obtain a functor $A_{I} :$ LMFI ⟶ LMFA by $\begin{matrix} A_{I} : {\begin{matrix} LMFI & ⟶ & LMFA, \\ (X, Int) & ⟼ & (X, A^{Int}), \\ f & ⟼ & f . \end{matrix} \end{matrix}$

Theorem 3.7. Let $(X, A)$ be an (L, M)-fuzzy concave space and defined ${Int}^{A} : L^{X} ⟶ M^{J (L^{X})}$ by ∀A ∈ L^X and x_λ ∈ J (L^X),

{Int}^{A} (A) (x_{λ}) = ⋁_{x_{λ} ⩽ B ⩽ A} A (B) .

Then $(X, {Int}^{A})$ is an (L, M)-fuzzy interior space.

Proof. (LMFI1) For each x_λ ∈ J (L^X), ${Int}^{A} (X) (x_{λ}) = ⋁_{x_{λ} ⩽ B ⩽ ⊤_{L}^{X}} A (B) ⩾ A (⊤_{L}^{X}) = ⊤_{M}$ .

(LMFI2) For each x_λ ∈ J (L^X) and A ∈ L^X, if x_λnotleqslantA, then ${Int}^{A} (A) (x_{λ}) = ⋁_{x_{λ} ⩽ B ⩽ A} A (B) = ⊥_{M}$ .

(LMFI3) By the definition of ${Int}^{A}$ , we know that If A ⩽ B, then ${Int}^{A} (A) ⩽ {Int}^{A} (B)$ . Hence, ${Int}^{A} (⋀_{i \in Ω}^{cdir} A_{i}) ⩽ ⋀_{i \in Ω} {Int}^{A} (A_{i})$ . Let α ∈ M and $α ≺ ⋀_{i \in Ω} {Int}^{A} (A_{i}) = ⋀_{i \in Ω} ⋁_{x_{λ} ⩽ B ⩽ A_{i}} A (B)$ . Then for each i ∈ Ω, there exists B_i such that x_λ ⩽ B_i ⩽ A_i and $α ⩽ A (B_{i})$ . Take $C_{i} = ⋁ {D \in L^{X} | D ⩽ A_{i}, α ⩽ A (D)}$ . Then {C_i|i ∈ Ω} is co-directed, x_λ ⩽ C_i and

$\begin{matrix} A (C_{i}) & = & A (⋁ {D \in L^{X} | D ⩽ A_{i}, α ⩽ A (D)}) \\ ⩾ & ⋀_{D ⩽ A_{i}, α ⩽ A (D)} A (D) \\ ⩾ & α . \end{matrix}$ Take C = ⋀ _i∈ΩC_i. Then $x_{λ} ⩽ C ⩽ ⋀_{i \in Ω}^{cdir} A_{i}$ and $A (C) = A (⋀_{i \in Ω} C_{i}) ⩾ ⋀_{i \in Ω} A (C_{i}) ⩾ α$ . So ${Int}^{A} (⋀_{i \in Ω}^{cdir} A_{i}) = ⋁_{x_{λ} ⩽ B ⩽ ⋀_{i \in Ω}^{cdir} A_{i}} A (B) ⩾ A (C) ⩾ α .$ This implies ${Int}^{A} (⋀_{i \in Ω}^{cdir} A_{i}) ⩾ ⋀_{i \in Ω} {Int}^{A} (A_{i})$ . Therefore, (LMFI3) holds.

(LMFI4) First we prove the following: ${Int}^{A} (A) (x_{λ}) = ⋀_{μ ≺ λ} {Int}^{A} (A) (x_{μ}) .$ It is obvious that ${Int}^{A} (A) (x_{λ}) ⩽ ⋀_{μ ≺ λ} {Int}^{A} (A) (x_{μ})$ . Next we prove ${Int}^{A} (A) (x_{λ}) ⩾ ⋀_{μ ≺ λ} {Int}^{A} (A) (x_{μ})$ . Let $α ≺ ⋀_{μ ≺ λ} {Int}^{A} (A) (x_{μ})$ . Then $α ≺ {Int}^{A} (A) (x_{μ})$ for every μ ≺ λ. There exists B_{x
_μ} such that x_μ ⩽ B_{x
_μ} ⩽ A and $α ⩽ A (B_{x_{μ}})$ . Take B = ⋁ _μ≺λB_{x
_μ}. Then x_λ ⩽ B ⩽ A and $α ⩽ ⋀_{μ ≺ λ} A (B_{x_{μ}}) ⩽ A (⋁_{μ ≺ λ} B_{x_{μ}}) = A (B) ⩽ ⋁_{x_{λ} ⩽ C ⩽ A} A (C) = {Int}^{A} (A) (x_{λ})$ . This means ${Int}^{A} (A) (x_{λ}) ⩾ ⋀_{μ ≺ λ} {Int}^{A} (A) (x_{μ})$ .

For every B ∈ L^X with x_λ ⩽ B ⩽ A and for every μ ≺ λ, $A (B) ⩽ ⋀_{y_{ν} ≺ B} ⋁_{y_{ν} ⩽ C ⩽ B} A (C) = ⋀_{y_{ν} ≺ B} {Int}^{A} (B) (y_{ν}) ⩽ {Int}^{A} (B) (x_{μ}) ⩽ {Int}^{A} (A) (x_{μ})$ . This shows that $\begin{matrix} {Int}^{A} (A) (x_{λ}) & = & ⋁_{x_{λ} ⩽ B ⩽ A} A (B) \\ ⩽ & ⋁_{x_{λ} ⩽ B ⩽ A} ⋀_{y_{ν} ≺ B} {Int}^{A} (B) (y_{ν}) \\ ⩽ & {Int}^{A} (A) (x_{μ}) . \end{matrix}$ Hence $\begin{matrix} {Int}^{A} (A) (x_{λ}) & ⩽ & ⋁_{x_{λ} ⩽ B ⩽ A} ⋀_{y_{ν} ≺ B} {Int}^{A} (B) (y_{ν}) \\ ⩽ & ⋀_{μ ≺ λ} {Int}^{A} (A) (x_{μ}) \\ = & {Int}^{A} (A) (x_{λ}) . \end{matrix}$ Therefore ${Int}^{A} (A) (x_{λ}) = ⋁_{x_{λ} ⩽ B ⩽ A} ⋀_{y_{μ} ≺ B} {Int}^{A} (B) (y_{μ}) .$ □

Theorem 3.8. If $f : (X, A_{X}) ⟶ (Y, A_{Y})$ is (L, M)-AP, then $f : (X, {Int}^{A_{X}}) ⟶ (Y, {Int}^{A_{Y}})$ is F-IP.

Proof. For each B ∈ L^Y, $\begin{matrix} {Int}^{A_{Y}} (B) (f (x)_{λ}) & = & ⋁_{f (x)_{λ} ⩽ C ⩽ B} A_{Y} (C) \\ ⩽ & ⋁_{f (x)_{λ} ⩽ C ⩽ B} A_{X} (f_{L}^{\leftarrow} (C)) \\ ⩽ & ⋁_{x_{λ} ⩽ D ⩽ f_{L}^{\leftarrow} (B)} A_{X} (D) \\ = & {Int}^{A_{X}} (f_{L}^{\leftarrow} (B)) (x_{λ}) . □ \end{matrix}$

By Theorems 3.7 and 3.8, we obtain a functor $I_{A} : LMFA ⟶ LMFI$ by $\begin{matrix} I_{A} : {\begin{matrix} LMFA & ⟶ & LMFI, \\ (X, A) & ⟼ & (X, {Int}^{A}), \\ f & ⟼ & f . \end{matrix} \end{matrix}$

Theorem 3.9. LMFA is isomorphic to LMFI.

Proof.

It suffices to show that $I_{A} \circ A_{I} = I_{LMFI}$ and $A_{I} \circ I_{A} = I_{LMFA}$ . That is to say, we need only verify ${Int}^{A^{Int}} = Int$ and $A^{{Int}^{A}} = A$ .

(1) For each A ∈ L^X and x_λ ∈ J (L^X), $\begin{matrix} {Int}^{A^{Int}} (A) (x_{λ}) & = & ⋁_{x_{λ} ⩽ B ⩽ A} A^{Int} (B) \\ = & ⋁_{x_{λ} ⩽ B ⩽ A} ⋀_{y_{μ} ≺ B} Int (B) (y_{μ}) \\ = & Int (A) (x_{λ}) . \end{matrix}$

(2) On one hand, for each A ∈ L^X, $\begin{matrix} A^{{Int}^{A}} (A) & = & ⋀_{x_{λ} ≺ A} {Int}^{A} (A) (x_{λ}) \\ = & ⋀_{x_{λ} ≺ A} ⋁_{x_{λ} ⩽ B ⩽ A} A (B) ⩾ A (A) . \end{matrix}$ On the other hand, let $α ≺ A^{{Int}^{A}} (A) = ⋀_{x_{λ} ≺ A} ⋁_{x_{λ} ⩽ B ⩽ A} A (B) .$ For each x_λ ≺ A, there exists B_{x
_λ} ∈ L^X such that x_λ ⩽ B_{x
_λ} ⩽ A and $α ⩽ A (B_{x_{λ}})$ . Obviously, A = ⋁ _{x_λ≺A}B_x. Then $A (A) = A (⋁_{x_{λ} ≺ A} B_{x}) ⩾ ⋀_{x_{λ} ≺ A} A (B_{x}) ⩾ α$ . This implies $A^{{Int}^{A}} ⩽ A$ . Therefore $A^{{Int}^{A}} = A$ .□

Theorem 3.10. Suppose that L is equipped with an order-reversing involution. LMFA is isomorphic to LMFC.

Proof. Given an (L, M)-fuzzy convex space $(X, C)$ , define $A^{C}$ by $A^{C} (A) = C (A^{'})$ Then $(X, A^{C})$ is an (L, M)-fuzzy concave space.

Similarly, given an (L, M)-fuzzy concave space $(X, A)$ , define $C^{A}$ by $C^{A} (A) = A (A^{'})$ Then $(X, C^{A})$ is an (L, M)-fuzzy convex space. Since ^′ is an order-reversing involution on L, it can be easily checked that LMFC is isomorphic to LMFA.□

Corollary 3.11. Suppose that L is equipped with an order-reversing involution. LMFI is isomorphic to LMFC.

4 (L, M)-fuzzy interior relations and (L, M)-fuzzy interior operators

Definition 4.1. A (L, M)-fuzzy interior relation on L^X is a mapping $O : L^{X} \times L^{X} ⟶ M$ satisfying the following conditions:

(LMIR1) $O (⊤_{L}^{X}, ⊤_{L}^{X}) = ⊤_{M}$ ;

(LMIR2) $O (A, B) > ⊥_{M} \Rightarrow A ⩽ B$ ;

(LMIR3) $O (A, ⋀_{i \in Ω}^{cdir} B_{i}) = ⋀_{i \in Ω} O (A, B_{i})$ ;

(LMIR4) $O (⋁_{i \in Ω} A_{i}, B) = ⋀_{i \in Ω} O (A_{i}, B)$ ;

(LMIR5) $O (A, B) ⩽ ⋁_{C \in L^{X}} O (A, C) \land O (C, B)$ . If $O$ is an (L, M)-fuzzy interior relation, then $(X, O)$ is called an (L, M)-fuzzy interior relation space.

Definition 4.2. Let $(X, O_{X})$ and $(Y, O_{Y})$ be (L, M)-fuzzy interior relation spaces. A mapping f : X → Y is said to be an (L, M)-fuzzy internal relation dual preserving mapping ((L, M)-IRDP mapping for short), if for each A ∈ L^X and B ∈ L^Y, $O_{Y} (f_{L}^{\to} (A), B) ⩽ O_{X} (A, f_{L}^{\leftarrow} (B)) .$

It is easy to check that all (L, M)-fuzzy interior relation spaces as objects and all (L, M)-IRDP mappings as morphisms form a category, denoted by LMIR.

By the definition of $O$ , we can easily obtain the following lemma.

Lemma 4.3. Let $O$ be an (L, M)-fuzzy interior relation. Then the following conditions hold.

(1) If A ⩽ B, then $O (C, A) ⩽ O (C, B)$ .

(2) If A ⩽ B, then $O (A, C) ⩽ O (B, C)$ .

Theorem 4.4. Let Int be an (L, M)-fuzzy interior operator on X. Then $O^{Int} : L^{X} \times L^{X} ⟶ M$ defined by

\forall A, B \in L^{X}, O^{Int} (A, B) = ⋀_{x_{λ} ≺ A} Int (B) (x_{λ})

is an (L, M)-fuzzy interior relation.

Proof. (LMIR1) By (LMFI1), $O^{Int} (⊤_{L}^{X}, ⊤_{L}^{X}) = ⋀_{x_{λ} ≺ ⊤_{L}^{X}} Int (⊤_{L}^{X}) = ⊤_{M} .$

(LMIR2) If $O^{Int} (A, B) = ⋀_{x_{λ} ≺ A} Int (B) (x) > ⊥_{M}$ , then for each x_λ ≺ A, Int (B) (x) > ⊥ _M, that is, x_λ ⩽ B. This show that A ⩽ B.

(LMIR3) $\begin{matrix} O (A, ⋀_{i \in Ω}^{cdir} B_{i}) & = & ⋀_{x_{λ} ≺ A} Int (⋀_{i \in Ω}^{cdir} B_{i}) (x_{λ}) \\ = & ⋀_{x_{λ} ≺ A} ⋀_{i \in Ω} Int (B_{i}) (x_{λ}) \\ = & ⋀_{i \in Ω} ⋀_{x_{λ} ≺ A} Int (B_{i}) (x_{λ}) \\ = & ⋀_{i \in Ω} O (A, B_{i}) . \end{matrix}$

(LMIR4) $\begin{matrix} O (⋁_{i \in Ω} A_{i}, B) & = & ⋀_{x_{λ} ≺ ⋁_{i \in Ω} A_{i}} Int (B) (x_{λ}) \\ = & ⋀_{i \in Ω} ⋀_{x_{λ} ≺ A_{i}} Int (B) (x_{λ}) \\ = & ⋀_{i \in Ω} O (A_{i}, B) . \end{matrix}$

(LMIR5) $\begin{matrix} α ≺ O^{Int} (A, B) \\ = & ⋀_{x_{λ} ≺ A} Int (B) (x_{λ}) \\ = & ⋀_{x_{λ} ≺ A} ⋁_{x_{λ} ⩽ C ⩽ A} ⋀_{y_{μ} ≺ C} Int (C) (y_{μ}) . \end{matrix}$ For each x_λ ≺ A, there exists C_{x
_λ} such that x_λ ⩽ C_{x
_λ} ⩽ A and for each y_μ ≺ C_{x
_λ}, α ⩽ Int (C_{x
_λ}) (y_μ). Take C = ⋁ _{x_λ≺A}C_{x
_λ}. Then A ⩽ C ⩽ B,

O (A, C) = ⋀_{x_{λ} ≺ A} Int (C) (x_{λ}) ⩾ ⋀_{x_{λ} ≺ C} Int (C) (x_{λ}) ⩾ α

and $O (C, B) = ⋀_{x_{λ} ≺ C} Int (B) (x_{λ}) ⩾ ⋀_{x_{λ} ≺ C} Int (C) (x_{λ}) ⩾ α .$ By the arbitrariness of α, we obtain $O (A, B) ⩽ ⋁_{C \in L^{X}} O (A, C) \land O (C, B)$ .□

Theorem 4.5. If f : (X, Int_X) ⟶ (Y, Int_Y) is (L, M)-IP, then $f : (X, O^{{Int}_{X}}) ⟶ (Y, O^{{Int}_{Y}})$ is (L, M)-IRDP.

Proof. For each A ∈ L^X and B ∈ L^Y, $\begin{matrix} O^{{Int}_{Y}} (f (A), B) & = & ⋀_{y_{μ} ≺ f_{L}^{\to} (A)} {Int}_{Y} (B) (y_{μ}) \\ ⩽ & ⋀_{x_{λ} ≺ A} {Int}_{Y} (B) (f (x)_{λ}) \\ ⩽ & ⋀_{x_{λ} ≺ A} {Int}_{X} (f_{L}^{\leftarrow} (B)) (x_{λ}) \\ = & O^{{Int}_{X}} (A, f_{L}^{\leftarrow} (B)) . \end{matrix}$ □

By Theorems 4.4 and 4.5, we obtain a functor $O_{I} : LMFI ⟶ LMIR$ by

\begin{matrix} O_{I} : {\begin{matrix} LMFI & ⟶ & LMIR, \\ (X, Int) & ⟼ & (X, O^{Int}), \\ f & ⟼ & f . \end{matrix} \end{matrix}

Theorem 4.6. Let $O$ be an (L, M)-fuzzy interior relation on X. Then ${Int}^{O} : L^{X} ⟶ M^{J (L^{X})}$ defined by

\forall A \in L^{X} and x_{λ} \in J (L^{X}), {Int}^{O} (A) (x_{λ}) = O (x_{λ}, A)

is an (L, M)-fuzzy interior operator.

Proof. (LMFI1) For each x_λ ∈ J (L^X), $\begin{matrix} {Int}^{O} (⊤_{L}^{X}) (x_{λ}) & = & O (x_{λ}, ⊤_{L}^{X}) \\ ⩾ & ⋀_{x_{λ} \in J (L^{X})} O (x_{λ}, ⊤_{L}^{X}) \\ = & O (⋁_{x_{λ} \in J (L^{X})} x_{λ}, ⊤_{L}^{X}) \\ = & O (⊤_{L}^{X}, ⊤_{L}^{X}) = ⊤_{M} . \end{matrix}$

(LMFI2) By (LMIR2), ${Int}^{O} (A) (x_{λ}) = O (x_{λ}, A) > ⊥_{M} \Rightarrow x_{λ} ⩽ A$ , that is, ${Int}^{O} (A) (x_{λ}) = O (x_{λ}, A) = ⊥_{M}$ for every x_λnotleqslantA.

(LMFI3) ${Int}^{O} (⋀_{i \in Ω}^{cdir} A_{i}) (x_{λ}) = O (x_{λ}, ⋀_{i \in Ω}^{cdir} A_{i}) =$ $⋀_{i \in Ω} O (x_{λ}, A_{i}) = ⋀_{i \in Ω} Int (A_{i})$ .

(LMFI4) We will prove this result in three steps.

(A) By (LMIR4), it is easy to see that ${Int}^{O} (A) (x_{λ}) = ⋀_{μ ≺ λ} {Int}^{O} (A) (x_{μ}) .$ Then we have $\begin{matrix} ⋁_{x_{λ} ⩽ B ⩽ A} ⋀_{y_{μ} ≺ B} {Int}^{O} (B) (y_{μ}) \\ ⩽ & ⋁_{x_{λ} ⩽ B ⩽ A} ⋀_{y_{μ} ≺ B} {Int}^{O} (A) (y_{μ}) \\ ⩽ & ⋁_{x_{λ} ⩽ B ⩽ A} ⋀_{ν ≺ λ ⩽ B (x)} {Int}^{O} (A) (x_{ν}) \\ = & ⋁_{x_{λ} ⩽ B ⩽ A} {Int}^{O} (A) (x_{λ}) \\ = & {Int}^{O} (A) (x_{λ}) . \end{matrix}$

(B) Let $\begin{matrix} α ≺ {Int}^{O} (A) (x_{λ}) & = & O (x_{λ}, A) \\ ⩽ & ⋁_{C \in L^{X}} O (x_{λ}, C) \land O (C, A) . \end{matrix}$ Then there exists C ∈ L^X such that $α ≺ O (x_{λ}, C) \land O (C, A)$ . Take $D = ⋁ {z_{ν} \in J (L^{X}) | α ≺ {Int}^{O} (A) (z_{ν}) = O (z_{ν}, A)} .$ By (LMFI2), D ⩽ A. For each y_μ ≺ C, $O (y_{μ}, A) ⩾ O (C, A) ≻ α$ . This implies y_μ ⩽ D and then C ⩽ D. Hence ${Int}^{O} (D) (x_{λ}) = O (x_{λ}, D) ⩾ O (x_{λ}, C) ≻ α$ . For each y_μ ≺ D, there exists z_{ν
₀} such that y_μ ⩽ z_{ν
₀} and $α ≺ O (z_{ν_{0}}, A)}$ . Hence ${Int}^{O} (A) (y_{μ}) = O (y_{μ}, A) ⩾ O (z_{ν_{0}}, A) ≻ α$ . $\begin{matrix} ⋁_{x_{λ} ⩽ B ⩽ A} ({Int}^{O} (B) (x_{λ}) \land ⋀_{y_{μ} ≺ B} {Int}^{O} (A) (y_{μ})) \\ ⩾ & {Int}^{O} (D) (x_{λ}) \land ⋀_{y_{μ} ≺ D} {Int}^{O} (A) (y_{μ}) \\ ⩾ & α . \end{matrix}$ By the arbitrariness of α, $\begin{matrix} {Int}^{O} (A) (x_{λ}) \\ ⩽ & ⋁_{x_{λ} ⩽ B ⩽ A} ({Int}^{O} (B) (x_{λ}) \land ⋀_{y_{μ} ≺ B} {Int}^{O} (A) (y_{μ})) . \end{matrix}$ (C) We only need to prove the condition (♣) as follow. $\begin{matrix} ⋁_{x_{λ} ⩽ B ⩽ A} ({Int}^{O} (B) (x_{λ}) \land ⋀_{y_{μ} ≺ B} {Int}^{O} (A) (y_{μ})) \\ ⩽ & ⋁_{x_{λ} ⩽ B ⩽ A} ⋀_{y_{μ} ≺ B} {Int}^{O} (B) (y_{μ}) . \end{matrix}$ Let $\begin{matrix} α ≺ ⋁_{x_{λ} ⩽ B ⩽ A} ({Int}^{O} (B) (x_{λ}) \land ⋀_{y_{μ} ≺ B} {Int}^{O} (A) (y_{μ})) . \end{matrix}$ Then there exists some B ∈ L^X such that (1) x_λ ⩽ B ⩽ A, $α ≺ {Int}^{O} (B) (x_{λ})$ ; (2) ∀y_μ ∈ B, $α ≺ {Int}^{O} (A) (y_{μ})$ . Let $Σ = {E | x_{λ} ⩽ E ⩽ A, α ≺ {Int}^{O} (E) (x_{λ}), \forall y_{μ} \in E, α ≺ {Int}^{O} (A) (y_{μ})}$ . Take F = ⋁ Σ. It is easy to see F ∈ Σ. Thus there exists a maximal set E₀ ∈ Σ fulfilling conditions (1) and (2). Thus, for each y_μ ≺ E₀, $α ≺ {Int}^{O} (A) (y_{μ})$ . Then it follows that there exists B_{y
_μ} ∈ L^X such that (3) y_μ ⩽ B_{y
_μ} ⩽ A, $α ≺ {Int}^{O} (B_{y_{μ}}) (y_{μ})$ ; (4) ∀z_ν ≺ B_{y
_μ}, $α ≺ {Int}^{O} (A) (z_{ν})$ . It is easy to see B_{y
_μ} ∨ E₀ satisfies conditions (1) and (2). By the maximality of E₀, B_{y
_μ} ∨ E₀ ⩽ E₀ and B_{y
_μ} ⩽ E₀. Then $α ⩽ {Int}^{O} (B_{y_{μ}}) (y_{μ}) ⩽ {Int}^{O} (E_{0}) (y_{μ})$ . Hence $α ⩽ {Int}^{O} (E_{0}) (y_{μ}) ⩽ ⋁_{x_{λ} ⩽ B ⩽ A} ⋀_{y_{μ} ≺ B} {Int}^{O} (B) (y_{μ}) .$ By the arbitrariness of α, the condition (♣) holds.

By (A),(B) and (C), (LMFI4) holds.

Theorem 4.7. If $f : (X, O_{X}) ⟶ (Y, O_{Y})$ is (L, M)-IRDP, then $f : (X, {Int}^{O_{X}}) ⟶ (Y, {Int}^{O_{Y}})$ is (L, M)-IP.

Proof. For each x_λ ∈ J (L^X) and B ∈ L^Y, $\begin{matrix} {Int}^{O_{Y}} (B) (f (x)_{λ}) & = & O_{Y} (f (x)_{λ}, B) \\ ⩽ & O_{X} (x_{λ}, f_{L}^{\leftarrow} (B)) \\ = & {Int}^{O_{X}} (f_{L}^{\leftarrow} (B)) (x_{λ}) . \end{matrix}$

By Theorems 4.6 and 4.7, we obtain a functor $A_{핆} : LMIR ⟶ LMFI$ by $\begin{matrix} A_{핆} : {\begin{matrix} LMIR & ⟶ & LMFI, \\ (X, O) & ⟼ & (X, {Int}^{O}), \\ f & ⟼ & f . \end{matrix} \end{matrix}$

Theorem 4.8. LMIR is isomorphic to LMFI.

Proof. It suffices to show that $I_{O} \circ O_{I} = I_{LMFI}$ and $O_{I} \circ I_{O} = I_{LMIR}$ . That is to say, we need only verify ${Int}^{O^{Int}} = Int$ and $O^{{Int}^{O}} = O$ .

For all A ∈ L^X and x_λ ∈ J (L^X), $\begin{matrix} {Int}^{O^{Int}} (A) (x_{λ}) & = & O^{Int} (x_{λ}, A) \\ = & ⋀_{y_{μ} ≺ x_{λ}} Int (A) (y_{μ}) \\ = & Int (A) (x_{λ}) . \end{matrix}$ For A, B ∈ L^X, $\begin{matrix} O^{{Int}^{O}} (A, B) & = & ⋀_{x_{λ} ≺ A} {Int}^{O} (B) (x_{λ}) \\ = & ⋀_{x_{λ} ≺ A} O (x_{λ}, A) \\ = & O (⋁_{x_{λ} ≺ A} x_{λ}, B) \\ = & O (A, B) . \end{matrix}$

Corollary 4.9. Suppose that L is equipped with an order-reversing involution ^′. LMIR is isomorphic to LMFC.

5 (L, M)-fuzzy hull relations and (L, M)-fuzzy hull operators

Definition 5.1. An (L, M)-fuzzy hull relation on L^X is a mapping Θ : L^X × L^X ⟶ M satisfying the following conditions:

(LMHR1) $Θ (⊥_{L}^{X}, ⊥_{L}^{X}) = ⊤_{M}$ ;

(LMHR2) Θ (A, B) > ⊥ _M ⇒ A ⩽ B;

(LMHR3) Θ (A, ⋀ _i∈ΩB_i) = ⋀ _i∈ΩΘ (A, B_i);

(LMHR4) $Θ (⋁_{i \in Ω}^{dir} A_{i}, B) = ⋀_{i \in Ω} Θ (A_{i}, B)$ ;

(LMHR5) Θ (A, B) ⩽ ⋁ _C∈L^XΘ (A, C) ∧ Θ(C, B).

If Θ is an (L, M)-fuzzy hull relation, then (X, Θ) is called an (L, M)-fuzzy hull relation space.

Definition 5.2. Let (X, Θ_X) and (Y, Θ_Y) be (L, M)-fuzzy hull relation spaces. A mapping f : X → Y is said to be an (L, M)-fuzzy hull relation dual preserving mapping ((L, M)-CRDP mapping for short), if for each A ∈ L^X and B ∈ L^Y, $Θ_{Y} (B, f_{L}^{\to} (A)) ⩽ Θ_{X} (f_{L}^{\leftarrow} (B), A) .$

It is easy to check that all (L, M)-fuzzy hull relation spaces as objects and all (L, M)-RRDP mappings as morphisms form a category, denoted by LMHR.

By the definition of Θ, we can easily obtain the following lemma.

Lemma 5.3. Let Θ be an (L, M)-fuzzy hull relation on L^X. Then the following conditions hold.

(1) If A ⩽ B, then Θ (C, A) ⩽ Θ (C, B).

(2) If A ⩽ B, then Θ (B, C) ⩽ Θ (A, C).

Theorem 5.4. Let $(X, H)$ be an (L, M)-fuzzy hull space. Define a mapping $Θ^{H} : L^{X} \times L^{X} ⟶ M$ by $\forall A, B \in L^{X}, Θ^{H} (A, B) = ⋀_{x_{λ} B} H (A) (x_{λ})^{'} .$ Then $Θ^{H}$ is an (L, M)-fuzzy hull relation.

Proof.

(LMHR1) Obviously, $Θ^{H} (⊥_{L}^{X}, ⊥_{L}^{X}) = ⊤_{M}$ .

(LMHR2) By (LMFH2), we know if $H (A) (x_{λ}) \neq ⊤_{M}$ , then x_λnotleqslantA. According to the definition of $Θ^{H}$ , if $Θ^{H} (A, B) = ⋀_{x_{λ} B} H (A) (x_{λ})^{'} > ⊥_{M}$ , then for each xnotleqslantB, x_λnotleqslantA. This means A ⩽ B.

(LMHR3) $\begin{matrix} Θ^{H} (A, ⋀_{i \in Ω} B_{i}) & = & ⋀_{x_{λ} ⋀_{i \in Ω} B_{i}} H (A) (x_{λ})^{'} \\ = & ⋀_{i \in Ω} ⋀_{x_{λ} B_{i}} H (A) (x_{λ})^{'} \\ = & ⋀_{i \in Ω} Θ^{H} (A, B_{i}) . \end{matrix}$

(LMHR4) $\begin{matrix} Θ^{H} (⋁_{i \in Ω}^{dir} A_{i}, B) & = & ⋀_{x_{λ} B} H (⋁_{i \in Ω}^{dir} A_{i}) (x_{λ})^{'} \\ = & ⋀_{x_{λ} B} ⋁_{μ ≺ λ} ⋀_{i \in Ω} H (A_{i}) (y_{μ})^{'} \\ = & ⋀_{x_{λ} B} ⋀_{i \in Ω} ⋁_{μ ≺ μ} H (A_{i}) (y_{μ})^{'} \\ = & ⋀_{x_{λ} B} ⋀_{i \in Ω} H (A_{i}) (x_{λ})^{'} \\ = & ⋀_{i \in Ω} Θ^{H} (A_{i}, B) . \end{matrix}$

(LMHR5) Let $α ≺ Θ^{H} (A, B)$ . By (LMFH3), $\begin{matrix} α ≺ Θ^{H} (A, B) = ⋀_{x_{λ} B} H (A) (x_{λ})^{'} \\ = ⋀_{x_{λ} B} ⋁_{x_{λ} C ⩾ A} ⋀_{y_{μ} C} H (C) (y_{μ})^{'} . \end{matrix}$ Then for each x_λnotleqslantB, there exists C_{x
_λ} ∈ L^X such that x_λnotleqslantC_{x
_λ} ⩾ A and for each y_μnotleqslantC_{x
_λ}, $α ⩽ H (C_{x_{λ}}) (y_{μ})$ . Take C = ⋀ _{x_λnotleqslantB}C_{x
_λ}. Then we have A ⩽ C ⩽ C_{x
_λ}. Thus, $\begin{matrix} Θ^{H} (A, C) & = & ⋀_{z_{ν} C} H (A) (z_{ν})^{'} \\ = & ⋀_{x_{λ} B} ⋀_{z C_{x_{λ}}} H (A) (z_{ν})^{'} \\ ⩾ & ⋀_{x B} ⋀_{z C_{x_{λ}}} H (C_{x}) (z_{ν})^{'} ⩾ α . \end{matrix}$ and $\begin{matrix} Θ^{H} (C, B) & = & ⋀_{z_{ν} B} H (C) (z_{ν})^{'} \\ ⩾ & ⋀_{z_{ν} B} H (C_{z_{ν}}) (z_{ν})^{'} \\ ⩾ & α . \end{matrix}$ By the arbitrariness of α, $Θ^{H} (A, B) ⩽ ⋁_{C \in L^{X}} Θ^{H} (A, C) \land Θ^{H} (C, B) . □$

Theorem 5.5. If $f : (X, H_{X}) \to (Y, H_{Y})$ is (L, M)-HP, then $f : (X, Θ^{H_{X}}) \to (Y, Θ^{H_{Y}})$ is (L, M)-CRDP.

Proof. For each A ∈ L^X and B ∈ L^Y, $\begin{matrix} Θ^{H_{X}} (f_{L}^{\leftarrow} (B), A) & = & ⋀_{x_{λ} A} H_{X} (f_{L}^{\leftarrow} (B)) (x_{λ})^{'} \\ ⩽ & ⋀_{x_{λ} A} H_{Y} (B) (f (x)_{λ})^{'} \\ ⩽ & ⋀_{f (x)_{λ} f (A)} H_{Y} (B) (f (x)_{λ})^{'} \\ ⩽ & Θ^{H_{Y}} (B, f (A)) . □ \end{matrix}$

By Theorems 5.4 and 5.5, we obtain a functor $ℝ_{ℍ} : LMFH ⟶ LMHR$ by $\begin{matrix} ℝ_{ℍ} : {\begin{matrix} LMFH & ⟶ & LMHR, \\ (X, H) & ⟼ & (X, Θ^{H}), \\ f & ⟼ & f . \end{matrix} \end{matrix}$

Theorem 5.6. Let (X, Θ) be an (L, M)-fuzzy hull relation space. Define a mapping $H^{Θ} : L^{X} ⟶ M^{J (L^{X})}$ by for each A ∈ L^X and x_λ ∈ J (L^X), $H^{Θ} (A) (x_{λ}) = ⋀_{x_{λ} B} Θ (A, B)^{'} .$ Then $H^{Θ}$ is an (L, M)-fuzzy hull operator.

Proof. (LMFH1) For every x_λ ∈ J (L^X), $\begin{matrix} H^{Θ} (⊥_{L}^{X}) (x_{λ}) & = & ⋀_{x_{λ} B} Θ (⊥_{L}^{X}, B)^{'} \\ = & Θ (⊥_{L}^{X}, ⊥_{L}^{X})^{'} = ⊥_{M} . \end{matrix}$

(LMFH2) For every x_λ ⩽ A, if $H^{Θ} (A) (x_{λ}) = ⋀_{x_{λ} B} Θ (A, B)^{'} \neq ⊤_{M}$ , then there exists B ∈ L^X such that x_λnotleqslantB and Θ (A, B) ′ ≠ ⊤ _M, that is, Θ (A, B) > ⊥ _M. By (LMHR2), A ⩽ B. This is contradiction with x_λ ⩽ A and x_λnotleqslantB.

(LMFH3) $H^{Θ} (A) (x_{λ}) = ⋀_{x_{λ} B ⩾ A} ⋁_{y_{μ} B} H (B)^{Θ} (y_{μ});$ It is obvious to obtain $\begin{matrix} ⋀_{x_{λ} B ⩾ A} ⋁_{y_{μ} B} H (B)^{Θ} (y_{μ}) \\ ⩾ & ⋀_{x_{λ} B ⩾ A} ⋁_{y_{μ} B} H (A)^{Θ} (y_{μ}) \\ ⩾ & ⋀_{x_{λ} B ⩾ A} H (A)^{Θ} (x_{λ}) = H^{Θ} (A) (x_{λ}) . \end{matrix}$ Next we need to prove $\begin{matrix} H^{Θ} (A) (x_{λ}) \\ ⩾ ⋀_{x_{λ} B ⩾ A} (H^{Θ} (B) (x_{λ}) \lor ⋁_{y_{μ} B} H^{Θ} (A) (y_{μ})), \end{matrix}$ that is, $\begin{matrix} H^{Θ} (A) (x_{λ})^{'} \\ ⩽ & ⋁_{x_{λ} B ⩾ A} (H^{Θ} (B) (x_{λ})^{'} \land ⋀_{y_{μ} B} H^{Θ} (A) (y_{μ})^{'}) . \end{matrix}$ Let $\begin{matrix} α ≺ H^{Θ} (A) (x_{λ})^{'} & = & ⋁_{x B} Θ (A, B) \\ ⩽ ⋁_{x_{λ} B} ⋁_{C \in L^{X}} Θ (A, C) \land Θ (C, B) . \end{matrix}$ Then there exists B, C ∈ L^X such that x_λnotleqslantB, α ≺ Θ (A, C) and α ≺ Θ (C, B). Since α ≺ Θ (A, C) ⩽ ⋁ _D∈L^XΘ (A, D) ∧ Θ (D, B), we obtain that there exists D ∈ L^X such that A ⩽ D ⩽ C and α ⩽ Θ (A, D) ∧ Θ (D, B). For D ∈ L^X satisfying x_λnotleqslantD ⩾ A, $\begin{matrix} H^{Θ} (D) (x_{λ})^{'} & = & {(⋀_{x_{λ} E} Θ (D, E)^{'})}^{'} \\ = & ⋁_{x_{λ} E} Θ (D, E) \\ ⩾ & Θ (D, B) ⩾ α \end{matrix}$ and $\begin{matrix} ⋀_{y_{μ} D} H^{Θ} (A) (y_{μ})^{'} & = & ⋀_{y_{μ} D} ⋁_{y_{μ} E} Θ (A, E) \\ ⩾ & Θ (A, C) ⩾ α . \end{matrix}$ By the arbitrariness of α, $H^{Θ} (A) (x_{λ}) \leq ⋀_{x_{λ} B ⩾ A} ⋁_{y_{μ} B} H (B)^{Θ} (y_{μ}) .$ Therefore, $H^{Θ} (A) (x_{λ}) = ⋀_{x_{λ} B ⩾ A} ⋁_{y_{μ} B} H (B)^{Θ} (y_{μ}) .$

(LMFH4) $\begin{matrix} H^{Θ} (⋁_{i \in Ω}^{dir} A_{i}) (x_{λ}) & = & ⋀_{x_{λ} B} Θ (⋁_{i \in Ω}^{dir} A_{i}, B)^{'} \\ = & ⋀_{x_{λ} B} ⋁_{i \in Ω} Θ (A_{i}, B)^{'} \\ = & ⋁_{i \in Ω} ⋀_{x B} Θ (A_{i}, B)^{'} \\ = & ⋁_{i \in Ω} H^{Θ} (A_{i}) (x_{λ}) . \end{matrix}$ □

Theorem 5.7. If f : (X, Θ_X) → (Y, Θ_Y) is (L, M)-CRDP, then $f : (X, H^{Θ_{X}}) \to (Y, H^{Θ_{Y}})$ is (L, M)-HP.

Proof. For each x_λ ∈ J (L^X) and B ∈ L^Y, $\begin{matrix} H^{Θ_{X}} (f_{L}^{\leftarrow} (B)) (x_{λ}) & = & ⋀_{x_{λ} A} Θ_{X} (f_{L}^{\leftarrow} (B), A)^{'} \\ ⩽ & ⋀_{x_{λ} A} Θ_{Y} (B, f_{L}^{\to} (A))^{'} \\ ⩽ & ⋀_{f (x)_{λ} f_{L}^{\to} (A)} Θ_{Y} (B, f_{L}^{\to} (A))^{'} \\ ⩽ & H^{Θ_{Y}} (B) (f (x)_{λ}) . \end{matrix}$

By Theorems 5.6 and 5.7, we obtain a functor $ℍ_{R} : LMHR ⟶ LMFH$ by $\begin{matrix} ℍ_{R} : {\begin{matrix} LMHR & ⟶ & LMFH, \\ (X, Θ) & ⟼ & (X, H^{Θ}), \\ f & ⟼ & f . \end{matrix} \end{matrix}$

Theorem 5.8. LMHR is isomorphic to LMFH.

Proof.

It suffices to show that $ℝ_{ℍ} \circ ℍ_{R} = I_{LMHR}$ and $ℍ_{R} \circ ℝ_{ℍ} = I_{LMFH}$ . That is to say, we need only verify $H^{Θ^{H}} = H$ and $Θ^{H^{Θ}} = Θ$ .

For A, B ∈ L^X, $\begin{matrix} Θ^{H^{Θ}} (A, B) & = & ⋀_{x_{λ} B} H^{Θ} (A) (x_{λ})^{'} \\ = & ⋀_{x_{λ} B} {(⋀_{x_{λ} C} Θ (A, B)^{'})}^{'} \\ = & ⋀_{x_{λ} B} ⋁_{x_{λ} C} Θ (A, B) . \end{matrix}$ On one hand, it is easy to see $Θ^{H^{Θ}} (A, B) ⩾ Θ (A, B)$ . On the other hand, let $α ≺ Θ^{H^{Θ}} (A, B)$ . Then for each x_λnotleqslantB, there exists C_{x
_λ} ∈ L^X such that x_λnotleqslantC_{x
_λ} and α ≤ Θ (A, B). Take C = ⋀ _{x_λnotleqslantB}C_{x
_λ}. Then C ⩽ B. By (LMHR3),

$\begin{matrix} α ⩽ ⋀_{x_{λ} B} Θ (A, C_{x_{λ}}) & = & Θ (A, ⋀_{x_{λ} B} C_{x_{λ}}) \\ = & Θ (A, C) ⩽ Θ (A, B) . \end{matrix}$ This implies $Θ^{H^{Θ}} (A, B) \leq Θ (A, B)$ . Therefore $Θ^{H^{Θ}} (A, B) = Θ (A, B)$ .

For all A ∈ L^X and x_λ ∈ J (L^X), $\begin{matrix} H^{Θ^{H}} (A) (x_{λ}) & = & ⋀_{x_{λ} B} Θ^{H} (A, B)^{'} \\ = & ⋀_{x_{λ} B} {(⋀_{y_{μ} B} H (A) (y_{μ})^{'})}^{'} \\ = & ⋀_{x_{λ} B} ⋁_{y_{μ} B} H (A) (y_{μ}) . \end{matrix}$ On one hand, it is easy to see $\begin{matrix} H^{Θ^{H}} (A) (x_{λ}) & = & ⋀_{x_{λ} B} ⋁_{y_{μ} B} H (A) (y_{μ}) \\ ⩾ & ⋀_{x_{λ} B} H (A) (x_{λ}) \\ = & H (A) (x_{λ}) . \end{matrix}$ On the other hand, let α ≺ ⋀ _{x_λnotleqslantB} $⋁_{y_{μ} B} H (A) (y_{μ})$ . Then for each B ∈ L^X and x_λnotleqslantB, there exists y_μ ∈ J (L^X) such that y_μnotleqslantB and $α ⩽ H (A) (y_{μ})$ . For each C ∈ L^X satisfying x_λnotleqslantC ⩾ A, there exists z_ν ∈ J (L^X) such that z_νnotleqslantC and $α ⩽ H (A) (z_{ν}) ⩽ H (C) (z_{ν})$ . Hence $H (A) (x_{λ}) = ⋀_{x_{λ} C ⩾ A} ⋁_{y_{μ} C} H (C) (y_{μ}) ⩾ α$ . This implies $⋀_{x_{λ} B} ⋁_{y_{μ} B} H (A) (y_{μ}) ⩽ H (A) (x_{λ}) .$ Therefore $H^{Θ^{H}} = H$ .□

Corollary 5.9. Suppose that L is equipped with an order-reversing involution. LMFH is isomorphic to LMFC.

6 Conclusions

In this paper, some characterizations of (L, M)-fuzzy convex spaces are given. The notions of (L, M)-fuzzy concave spaces, (L, M)-fuzzy interior spaces, (L, M)-fuzzy interior relations and (L, M)-fuzzy hull relations are introduced. It is proved that the category of (L, M)-fuzzy concave spaces, the category of (L, M)-fuzzy interior spaces, the category of (L, M)-fuzzy interior relation spaces, the category of (L, M)-fuzzy hull relation spaces are isomorphic. When L is a completely distributive lattice with an order-reversing involution, these categories are all isomorphic to the category of (L, M)-fuzzy convex spaces.

Footnotes

Acknowledgments

The authors would like to express their sincere thanks to the anonymous reviewers and the editor for their careful reading and constructive comments. This work is supported by the Project funded by the Natural Science Foundation of China (11771134,11871097), the Project funded by China Postdoctoral Science Foundation (2017M622563) and the Project (2017Z056) Supported by the Scientific Research Foundation of CUIT.

References

Berger , Convexity, American Mathematical Monthly 97(8) (1990), 650–678.

Dwinger , Characterizations of the complete homomorphic images of a completely distributive complete lattice I, Indagationes Mathematicae (Proceedings) 85 (1982), 403–414.

J.M.

Fang , I-FTOP is isomorphic to I-FQN and I-AITOP, Fuzzy Sets Syst 147 (2004), 317–325.

J.M.

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

J.M.

Fang and

Y.L.

Yue , L-fuzzy closure systems, Fuzzy Sets Syst 161 (2010), 1242–1252.

Höhle , Upper semicontinuous fuzzy sets and applications, J Math Anal Appl 78 (1980), 659–673.

Höhle and

S.E.

Rodabaugh , (Eds.), Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, The Handbooks of Fuzzy Sets Series, Vol. 3, Kluwer Academic Publishers, Boston, Dordrecht, London, 1999.

Höhle and

Šostak , Axiomatic foundations of variable-basis fuzzy topology, in:

Höhle and

S.E.

Rodabaugh (Eds.), Mathematics of Fuzzy Sets: Logic Topology, and Measure Theory, The Handbooks of Fuzzy Sets Series, Vol. 3, Kluwer Academic Publishers, Boston, Dordrecht, London, 1999, pp. 123–272.

Jin and

L.Q.

Li , On the embedding of L-convex spaces in stratified L-convex spaces, Springer Plus 5 (2016), 1610.

10.

Kubiak , On fuzzy topologies, Ph.D. Thesis, Adam Mickiewicz, Poznan, Poland, 1985.

11.

Kubiak and

Šostak , Lower set-valued fuzzy topologies, Quaestiones Mathematicae 20(3) (1997), 423–429.

12.

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

13.

L.Q.

Li and

Jin , p-Topologicalness and p-Regularity for latticevalued convergence spaces, Fuzzy Sets Syst 238 (2014), 26–45.

14.

L.Q.

Li ,

Jin and

Hu , On stratified L-convergence spaces: Fischer's diagonal axiom, Fuzzy Sets Syst 267 (2015), 31–40.

15.

L.Q.

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

16.

E.Q.

Li and

F.G.

Shi , Some properties of M-fuzzifying convexities induced by M-orders, Fuzzy Sets Syst (2018). DOI: 10.1016/j.fss.2018.05.008

17.

Lassak , On metric B-convexity for which diameters of any set and its hull are equal, Bull Acad Polon Sci 25 (1977), 969–975.

18.

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

19.

van Mill , Supercompactness and Wallman Spaces, Math Centre Tracts 85, Amsterdam, 1977.

20.

C.V.

Negoita and

D.A.

Ralescu , Applications of fuzzy sets to systems analysis, Interdisciplinary Systems Research Series, vol. 11, Birkhauser, Basel, Stuttgart and Halsted Press, New York, 1975.

21.

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

22.

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

23.

Pang and

F.G.

Shi , Subcategories of the category of Lconvex spaces, Fuzzy Sets Syst 313 (2017), 61–74.

24.

Pang and

F.G.

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

25.

Pang and and

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

26.

Pang ,

Zhao and

Z.Y.

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

27.

Pang and

Z.Y.

Xiu , Lattice-valued interval operators and its induced lattice-valued convex structures, IEEE T Fuzzy Syst 26(3) (2018), 1525–1534.

28.

Pang and

Z.Y.

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

29.

Pang , (L, M)-fuzzy convex spaces: (restricted) hull operators, bases and subbases, Fuzzy Sets Syst, submitted.

30.

Shen and

F.G.

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

31.

M.V.

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

32.

M.V.

Rosa , A study of fuzzy convexity with special reference to separation properties, PhD thesis, Cochin University of Science and Technology, 1994.

33.

V.P.

Soltan , d-convexity in graphs, Soviet Math Dokl 28 (1983), 419–421.

34.

X.Y.

Wu and

S.Z.

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

35.

F.G.

Shi and

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

36.

F.G.

Shi and

Z.Y.

Xiu , A new approach to the fuzzification of convex structures, Journal of Applied Mathematics 2014. Article ID 249183.

37.

F.G.

Shi and

Z.Y.

Xiu , (L, M)-fuzzy convex structures, J-Nolinear Sci Appl 10(7) (2017), 3655–3669.

38.

A.P.

Sostak , On a fuzzy topological structure, Rend Circ Mat Palermo (Suppl Ser II) 11 (1985), 89–103.

39.

A.P.

Sostak , Two decades of fuzzy topology: Basic ideas notions and results, Russian Math Surveys 44 (1989), 125–186.

40.

M.L.J.

Van de Vel , Theory of Convex Structures, North Holland, NY, 1993.

41.

J.C.

Varlet , Remarks on distributive lattices, Bull Acad Polon Sci 23 (1975), 1143–1147.

42.

Z.Y.

Xiu and

Pang , A degree approach to special mappings between L-fuzzifying convex spaces, J Intell Fuzzy Syst 35(1) (2018), 705–716.

43.

Z.Y.

Xiu and

F.G.

Shi , M-fuzzifying interval spaces, Iran J Fuzzy Syst 147(1) (2017), 145–162.

44.

Z.Y.

Xiu and

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

45.

Z.Y.

Xiu and

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