( L,M )-fuzzy internal relations and ( L,M )-fuzzy enclosed relations

Abstract

In this paper, the concepts of (L, M)-fuzzy internal relations and (L, M)-fuzzy enclosed relations between two L-fuzzy sets are introduced. They are defined respectively to be M-fuzzy subsets of L^X × L^X satisfying a set of axioms. A categorical approach is provided to present these relations. It is proved that the category of (L, M)-fuzzy internal relation spaces, the category of (L, M)-fuzzy enclosed relation spaces and the category of (L, M)-fuzzy topological spaces are isomorphic. In addition, some (L, M)-fuzzy internal relations and (L, M)-fuzzy enclosed relations are naturally constructed from (L, M)-fuzzy quasi-uniformities and (L, M)-fuzzy S-quasi-proximities.

1. Introduction

With the development of fuzzy set theory, many mathematical structures have been endowed with fuzzy sets, such as fuzzy topology [1 , 33], fuzzy convergence structures [11–13 , 17], fuzzy convex structures [14–16 , 30–32] and so on. In the framework of fuzzy topology, Chang first introduced [0, 1]-topology, which is also known as Chang’s fuzzy topology. In 1980, from a completely different direction, Höhle [4] presented the notion of a fuzzy topology being viewed as an L-subset of a powerset 2^X. Ying [33] studied Höhle’s topology from a logical point of view and called it fuzzifying topology. Kubiak [7] extendand Höhle’s fuzzy topology to M-subsets of L^X and called it (L, M)-fuzzy topology on X. $\overset{⌣}{S}$ ostak [21] independently extended Höhle’s fuzzy topology to [0, 1]-subsetsof [0, 1] ^X.

In order to characterize L-fuzzy topologies, many authors [2 , 25] have presented the notions of L-fuzzy (quasi-)neighborhood systems, L-fuzzy interior operators and L-fuzzy closure operators. Moreover, in [0, 1]-fuzzy topology the theory of topogenous orders was developed in [22]. Then a natural problem is that can (L, M)-topologies be characterized by some order relations? In this paper, we shall answer this problem.

The purpose of this paper is to introduce new definitions of (L, M)-fuzzy topogenous order spaces which are (L, M)-fuzzy internal order space and (L, M)-fuzzy enclosed order space in different from [9, 22]. They are defined to be M-fuzzifying orders on L^X satisfying a set of axioms. Two characterizations of the category (L, M)-FTOP of (L, M)-fuzzy topological spaces and their continuous mappings are presented by means of the category (L, M)-FIR of (L, M)-fuzzy internal order spaces and their continuous mappings, the category (L, M)-FER of (L, M)-fuzzy enclosed order spaces and their continuous mappings. Furthermore, some (L, M)-fuzzy internal relations and (L, M)-fuzzy enclosed relations are naturally constructed from (L, M)-fuzzy quasi-uniformities and (L, M)-fuzzy S-quasi-proximities.

2. Preliminaries

Throughout this paper, both L and M denote completely distributive De Morgan algebras. The smallest element and the largest element in L (M) are denoted by ⊥_L (⊥_M) and ⊤_L (⊤_M), respectively. For a, b ∈ L, we say that a is wedge below b in L, in symbols a ⊲ b, if for every subset D ⊆ L, ⋁D ⩾ b implies d ⩾ a for some d ∈ D. A complete lattice L is completely distributive if and only if b = ⋁ {a ∈ L ∣ a ⊲ b} for each b ∈ L. An element a in L is called co-prime if a ⩽ b ∨ c implies a ⩽ b or a ⩽ c. The set of non-zero co-prime elements in L (M) is denoted by J (L) (J (M)), respectively.

For a nonempty set X, L^X denotes the set of all L-subsets on X. The smallest element and the largest element in L^X are denoted by ⊥_{L
^X} and ⊤_{L
^X}, respectively. L^X is also a completely distributive De Morgan algebra when it inherits the structure of the lattice L in a natural way, by defining ⋁, ⋀ , ⩽ and ^′ pointwisely. 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 an L-fuzzy point. The set of non-zero co-prime elements in L^X is denoted by J (L^X). It is easy to see that J (L^X) = {x_λ ∣ x ∈ X, λ ∈ J (L)}.

A mapping f : X ⟶ Y induces a mapping $f_{L}^{\to} : L^{X} ⟶ L^{Y}$ , which is defined by ∀A ∈ L^X, y ∈ Y, $f_{L}^{\to} (A) (y) = ⋁ {A (x) ∣ f (x) = y} .$ The right adjoint of $f_{L}^{\to}$ is denoted $f_{L}^{\leftarrow}$ and given by $f_{L}^{\leftarrow} (B) = B \circ f, \forall B \in L^{Y} .$ It is known that $f_{L}^{\to}$ preserves arbitrary suprema and that $f_{L}^{\leftarrow}$ preserves arbitrary suprema, arbitrary infima and quasi-complements [20].

Definition 2.1. ([7, 21]) An (L, M)-fuzzy topology on X is a mapping τ : L^X ⟶ M which satisfies:

(LFT1)
τ (⊤ _L
^X) = τ (⊥ _L
^X) = ⊤ _M;

τ (U ∧ V) ⩾ τ (U) ∧ τ (V) for all U, V ∈ L^X;

τ (⋁ _j∈JU_j) ⩾ ⋀ _j∈Jτ (U_j) for every family {U_j ∣ j ∈ J} ⊆ L^X.
τ (U) can be interpreted as the degree to which U is an open L-set. τ^* (U) = τ (U′) will be called the degree of closedness of U, where U′ is the L-complement of U. The pair (X, τ) is called an (L, M)-fuzzy topological space.

A mapping f : (X, τ_X) ⟶ (Y, τ_Y) is called continuous with respect to (L, M)-fuzzy topologies τ_X and τ_Y if $τ_{X} (f_{L}^{\leftarrow} (U)) ⩾ τ_{Y} (U)$ for all U ∈ L^Y [20]. The category of (L, M)-fuzzy topological spaces with their continuous mappings as morphisms will be denoted by (L, M)-FTOP.

The following definitions and theorems were presented for an L-fuzzy interior operator and an L-fuzzy closure operator. They can easily be transformed to an (L, M)-fuzzy interior operator and an (L, M)-fuzzy closure operator as follows:

Definition 2.2. ([25]) An (L, M)-fuzzy interior operator on X is a mapping τ : L^X ⟶ M^{J(L^X)} satisfying the following conditions:
τ (⊤ _L
^X) (x_λ) = ⊤ _M for any x_λ ∈ J (L^X);

τ (A) (x_λ) = ⊥ _M for any x_λA;

τ (A ∧ B) = τ (A) ∧ τ (B);

τ (A) (x_λ) = ⋁ _{x_λ⩽B⩽A} ⋀ _{y_μ⊲B}τ (B) (y_μ) .

A set X equipped with an (L, M)-fuzzy interior operator τ or τ_X, denoted by (X, τ) or (X, τ_X), is called an (L, M)-fuzzy interior space.

Theorem 2.3. ([24]) topint Let τ be an (L, M)-fuzzy topology on X and let τ^τ be the (L, M)-fuzzy interior operator induced by τ. Then ∀x_λ ∈ J (L^X), ∀A ∈ L^X, $τ^{τ} (A) (x_{λ}) = \underset{x_{λ} ⩽ V ⩽ A}{⋁} τ (V) .$

Theorem 2.4. ([25]) Let τ : L^X ⟶ M^{J(L^X)} be an (L, M)-fuzzy interior operator on X. Define τ^τ : L^X ⟶ M by $τ^{τ} (A) = \underset{x_{λ} ⊲ A}{⋀} τ (A) (x_{λ}) .$ Then τ^τ is an (L, M)-fuzzy topology on X and τ^{(τ^τ)} = τ.

Definition 2.5. ([24]) An (L, M)-fuzzy closure operator on X is a mapping Cl:L^X ⟶ M^{J(L^X)} satisfying the following conditions: (LFC1)
Cl(⊥ _L
^X) (x_λ) = ⊥ _M for any x_λ ∈ J (L^X);

Cl(A) (x_λ) = ⊤ _M for any x_λ ⩽ A;

Cl(A ∨ B) = Cl (A) ∨ Cl (B);

Cl(A) (x_λ) = ⋀ _{x_λB⩾A} ⋁ _{y
_μ
B}Cl (B) (y_μ) .

A set X equipped with an (L, M)-fuzzy closure operator Cl or Cl_X, denoted by (X, Cl) or (X, Cl_X), is called an (L, M)-fuzzy closure space.

Theorem 2.6. ([24]) Let τ be an (L, M)-fuzzy topology on X and let Cl^τ be the (L, M)-fuzzy closure operator induced by τ. Then ∀x_λ ∈ J (L^X), ∀A ∈ L^X, ${Cl}^{τ} (A) (x_{λ}) = \underset{x_{λ} D ⩾ A}{⋀} (τ (D^{'}))^{'} .$

Theorem 2.7. ([24]]) Let Cl : L^X ⟶ M^{J(L^X)} be an (L, M)-fuzzy closure operator on X. Define τ^Cl : L^X ⟶ M by $τ^{Cl} (A) = \underset{x_{λ} A^{'}}{⋀} (Cl (A^{'}) (x_{λ}))^{'} .$ Then τ^Cl is an (L, M)-fuzzy topology on X and Cl^{(τ^Cl)} = Cl.

Definition 2.8. ([23]) A mapping φ : J (L^X) ⟶ L^X is called a remote-neighborhood mapping (R-mapping for short) on L^X, if for all x_λ ∈ J (L^X), x_λφ (x_λ). φ₀ is the smallest element of $R (L^{X})$ , i.e., φ₀ (x_λ) = ⊥ _M. The set of all R-mappings on L^X is denoted by $R (L^{X}) .$

Definition 2.9. ([35]) A pointwise (L, M)-fuzzy quasi-uniformity on X is a mapping $FU : R (L^{Xitsc}) ⟶ M$ satisfying the following conditions: (FQU)
$FU (φ_{0}) = TM$ ;

$FU (φ \lor ψ) = FU (φ) \land FU (ψ)$ ;

$FU (φ) = ⋁_{ψ ⊙ ψ ⩾ φ} FU (ψ)$ for all $φ \in R (L^{X})$ , where for each x_λ ∈ J (L^X), (φ ⊙ ψ) (x_λ) = ⋀ { φ (y_μ) ∣ y_μψ (x_λ) } .

If $FU$ is a pointwise (L, M)-fuzzy quasi-uniformity on X, then we call $(X, FU)$ a pointwise (L, M)-fuzzy quasi-uniform space.

Definition 2.10. ([34]) A generalized pointwise S-quasi-proximity on L^X is a mapping δ : J (L^X) × L^X ⟶ M which satisfies the following conditions: (FSP1)
δ (x_λ, ⊥ _L
^X) = ⊥ _M for all x_λ ∈ J (L^X);

δ (x_λ, B ∨ C) = δ (x_λ, B) ∨ δ (x_λ, C);

δ (x_λ, A) ⩾ ⋀ _C∈L^X (δ (x_λ, C) ∨ ⋁ _{z
_ν
C}δ (z_ν, A));

δ (x_λ, A) < ⊤ _M ⇒ x_λA .
If δ is a generalized pointwise S-quasi-proximity on L^X, then the pair (X, δ) is called a generalized pointwise S-quasi-proximity space.
3. (L, M)-fuzzy internal relation spaces

In this section, our aim is to introduce the notion of (L, M)-fuzzy internal relations on L^X which can be used to characterize (L, M)-fuzzy topologies on X.∥Definition 3.1. A mapping $I : L^{Xitsc} \times L^{Xitsc} ⟶ M$ is called an (L, M)-fuzzy internal relation or (L, M)-fuzzy internal order on L^X if it satisfies the following conditions.

$I (⊤_{L^{Xitsc}}, ⊤_{L^{Xitsc}}) = ⊤_{Mitsc}$ ;

$I (A, B) > ⊥_{Mitsc} \Rightarrow A ⩽ B$ ;

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

$I (C, A \land B) = I (C, A) \land I (C, B)$ ;

$I (A, B) ⩽ ⋁_{C \in L^{}} (I (A, C) \land I (C, B))$ .

For an (L, M)-fuzzy internal relation

I

on L^X, the pair

(X, I)

is called an (L, M)-fuzzy internal relation space.

From (LFIO3) and (LFIO4) in Definition IO we can easily obtain the following lemma.

Lemma 3.2. inverse-order Let $I$ be an (L, M)-fuzzy internal relation on L^X. If A ⩽ B and C ⩽ D, then $I (B, C) ⩽ I (A, D)$ .

The following theorems show that we can obtain some (L, M)-fuzzy internal relations, which can be seen two examples of some (L, M)-fuzzy internal relations, from a pointwise quasi-uniformity and a pointwise S-quasi-proximity on a set.

Theorem 3.3. Let $(X, FU)$ be a pointwise (L, M)-fuzzy quasi-uniform space. For all A, B ∈ L^X, we define $I^{FU} : L^{X} \times L^{X} ⟶ M$ by $I^{FU} (A, B) = ⋀_{x_{λ} A^{'}} ⋁_{φ (x_{λ}) ⩾ B^{'}} FU (φ) .$ Then $I^{FU}$ is an (L, M)-fuzzy internal relation on L^X.

Proof. It is easy to verify that $I^{FU}$ satisfies (LFIO1). (LFIO2) can be obtained by the following fact: x_λφ (x_λ) for all $φ \in R (L^{X})$ . (LFIO3) holds by the following fact. $\begin{matrix} I^{FU} (\underset{j \in J}{⋁} A_{j}, B) & = \underset{x_{λ} ⋀_{j \in J} A_{j}^{'}}{⋀} \underset{φ (x_{λ}) ⩾ B^{'}}{⋁} FU (φ) \\ = \underset{j \in J}{⋀} \underset{x_{λ} A_{j}^{'}}{⋀} \underset{φ (x_{λ}) ⩾ B^{'}}{⋁} FU (φ) \\ = \underset{j \in J}{⋀} I^{FU} (A_{j}, B) . \end{matrix}$ (LFIO4) holds by the following fact: $\begin{matrix} I^{FU} (C, A \land B) \\ = \underset{\underset{λ}{x} C^{'}}{⋀} \underset{φ (\underset{λ}{x}) ⩾ (A^{'} \lor B^{'})}{⋁} FU (φ) \\ = \underset{\underset{λ}{x} C^{'}}{⋀} (\underset{φ (\underset{λ}{x}) ⩾ A^{'}}{⋁} FU (φ) \land \underset{φ (\underset{λ}{x}) ⩾ B^{'}}{⋁} FU (φ)) \\ = \underset{\underset{λ}{x} C^{'}}{⋀} \underset{φ (\underset{λ}{x}) ⩾ A^{'}}{⋁} FU (φ) \land \underset{\underset{λ}{x} C^{'}}{⋀} \underset{φ (\underset{λ}{x}) ⩾ B^{'}}{⋁} FU (φ) \\ = I^{FU} (C, A) \land I^{FU} (C, B) . \end{matrix}$

(LFIO5) It suffices to show that $\begin{matrix} ⋀_{x_{λ} A^{'}} ⋁_{φ (x_{λ}) ⩾ B^{'}} FU (φ) ⩽ \\ ⋁_{C \in L^{X}} (⋀_{x_{λ} A^{'}} ⋁_{ψ (x_{λ}) ⩾ C^{'}} FU (ψ) \land ⋀_{z_{ν} C^{'}} ⋁_{φ (z_{ν}) ⩾ B^{'}} FU (φ)) . \end{matrix}$ Suppose $t ⊲ ⋀_{x_{λ} A^{'}} ⋁_{φ (x_{λ}) ⩾ B^{'}} FU (φ) = ⋀_{x_{λ} A^{'}} ⋁_{φ (x_{λ}) ⩾ B^{'}} ⋁_{ψ ⊙ ψ ⩾ φ} FU (ψ) .$ Then there exist $φ, ψ \in R (L^{X})$ such that φ (x_λ) ⩾ B′, ψ ⊙ ψ ⩾ φ and $t ⩽ FU (ψ)$ for each x_λA′ . Take C = ψ (x_λ) ′. Then x_λC′ and C′ ⩾ B′. Furthermore, we get $\begin{matrix} I^{FU} (A, C) & = ⋀_{x_{λ} A^{'}} ⋁_{ψ (x_{λ}) ⩾ C^{'}} FU (ψ) \\ ⩾ ⋀_{x_{λ} A^{'}} FU (ψ) ⩾ t . \end{matrix}$

Since ⋀ {ψ (z_ν) ∣ z_νψ (x_λ)} = ψ ⊙ ψ (x_λ) ⩾ φ (x_λ) ⩾ B′, we have ψ (z_ν) ⩾ B′ for all z_νψ (x_λ) = C′ . Hence, $\begin{matrix} I^{FU} (C, B) & = ⋀_{z_{ν} C^{'}} ⋁_{φ (z_{ν}) ⩾ B^{'}} FU (φ) \\ ⩾ ⋀_{z_{ν} C^{'}} FU (ψ) ⩾ t . \end{matrix}$ Then $t ⩽ I^{FU} (A, C) \land I^{FU} (C, B)$ . Therefore $t ⩽ ⋁_{C \in L^{X}} (I^{FU} (A, C) \land I^{FU} (C, B))$ . Thus (LFIO5) is obtained from the arbitrariness of t.

Theorem 3.4. Let δ be a generalized pointwise S-quasi-proximity on L^X. For all A, B ∈ L^X, we define $I^{δ} : L^{Xitsc} \times L^{Xitsc} ⟶ M$ by $I^{δ} (A, B) = ⋀_{x_{λ} A^{'}} δ (x_{λ}, B^{'})^{'} .$ Then $I^{δ}$ is an (L, M)-fuzzy internal relation on L^X .

Proof. It is trivial to check that $I^{δ}$ satisfies (LFIO1) and (LFIO2).

(LFIO3) By the definition of $I^{δ}$ and (FSP2), we have $\begin{matrix} I^{δ} (⋁_{j \in J} A_{j}, B) & = ⋀_{x_{λ} ⋀_{j \in J} A_{j}^{'}} δ (x_{λ}, B^{'})^{'} \\ = ⋀_{j \in J} ⋀_{x_{λ} A_{j}^{'}} δ (x_{λ}, B^{'})^{'} = ⋀_{j \in J} I^{δ} (A_{j}, B) . \end{matrix}$

(LFIO4) By the definition of $I^{δ}$ and (FSP3), it follows that $\begin{matrix} I^{δ} (C, A \land B) & = ⋀_{x_{λ} C^{'}} δ (x_{λ}, A^{'} \lor B^{'})^{'} \\ = ⋀_{x_{λ} C^{'}} (δ (x_{λ}, A^{'})^{'} \land δ (x_{λ}, B^{'})^{'}) \\ = I^{δ} (C, A) \land I^{δ} (C, B) . \end{matrix}$ (LFIO5) By the definition of $I^{δ}$ and (FSP4), we have $\begin{matrix} I^{δ} (A, B) \\ = ⋀_{x_{λ} A^{'}} δ (x_{λ}, B^{'})^{'} \\ ⩽ ⋀_{x_{λ} A^{'}} ⋁_{C \in L^{X}} (δ (x_{λ}, C)^{'} \land ⋀_{z_{ν} C} δ (z_{ν}, B^{'})^{'}) \\ = ⋁_{C \in L^{X}} (I^{δ} (A, C^{'}) \land I^{δ} (C^{'}, B)) . \end{matrix}$ Thus the conclusion holds.

We will discuss that the relationships between (L, M)-fuzzy internal relations and (L, M)-fuzzy topologies in what follows. For this, the following lemma is necessary.

Lemma 3.5. Let τ : L^X ⟶ M^{J(L^X)} be a mapping. Then the following equalities are equivalent:

(LFI4)
τ (A) (x_λ) = ⋁ _{x_λ⩽B⩽A} ⋀ _{y_μ⊲B}τ (B) (y_μ) .

* τ (A) (x_λ) = ⋁ _{x_λ⩽B⩽A} (τ (B) (x_λ) ∧ ⋀ _{y_μ⊲B}τ (A) (y_μ)) .

Proof. (LFI4) ⇒ (LFI4)* is true trivially. Now suppose (LFI4) holds.

Let $\begin{matrix} t ⊲ τ (A) (x_{λ}) \\ = \underset{x λ ⩽ B ⩽ A}{⋁} (τ (B) (x_{λ}) \land ⋀_{y_{μ} ⊲ B} τ (A) (y_{μ})) . \end{matrix}$ Then there exists some B ∈ L^X such that the following statement holds. $x_{λ} ⩽ B ⩽ A, τ (B) (x_{λ}) ⊳ t (★)$ and $\forall y_{μ} ⊲ B, τ (A) (y_{μ}) ⊳ t . (★ ★)$

It is clear that the join of fuzzy sets fulfilling (★) and (★★) is still of such kind. Thus we can define B_s to be the maximal fuzzy sets fulfilling (★) and (★★), i.e., x_λ ⩽ B_s ⩽ A, τ (B_s) (x_λ) ⊳ t and τ (A) (y_μ) ⊳ t for all y_μ ⊲ B_s . Then for all y_μ ⊲ B_s, it follows from τ (A) (y_μ) ⊳ t that there exists G_μ ∈ L^X such that the following statement holds. $y_{μ} ⩽ G_{μ} ⩽ A, τ (G_{μ}) (y_{μ}) ⊳ t$ and $\forall z_{ν} ⊲ G_{μ}, τ (A) (z_{ν}) ⊳ t .$

It is easy to check that B_s ∨ G_μ satisfies (★) and (★★). Hence, by the maximality of B_s, it follows that B_s ∨ G_μ ⩽ B_s. Therefore G_μ ⩽ B_s . Then we can obtain that ∀y_μ ⊲ B, τ (B_s) (y_μ) ⩾ τ (G_μ) (y_μ) ⩾ t . Thus, ⋀_{y_μ⊲B_s}τ (B_s) (y_μ) ⩾ t . Therefore, $⋁_{x_{λ} ⩽ B ⩽ A} ⋀_{y_{μ} ⊲ B} τ (B) (y_{μ}) ⩾ t$ is obvious. From the arbitrariness of t, we have $τ (A) (x_{λ}) ⩽ ⋁_{x_{λ} ⩽ B ⩽ A} ⋀_{y_{μ} ⊲ B} τ (B) (y_{μ}) .$ Since τ (A) (x_λ) ⩾ ⋁ _{x_λ⩽B⩽A} ⋀ _{y_μ⊲B}τ (B) (y_μ) is obvious by the mapping τ is order-preserving from (LFI4), we have $τ (A) (x_{λ}) = ⋁_{x_{λ} ⩽ B ⩽ A} ⋀_{y_{μ} ⊲ B} τ (B) (y_{μ}),$ as desired.□

Theorem 3.6. Let (X, τ) be an (L, M)-fuzzy interior space. Define $I^{τ} : L^{Xitsc} \times L^{Xitsc} ⟶ M$ as follows: $\forall A, B \in L^{X}, I^{τ} (A, B) = ⋀_{x_{λ} ⊲ A} τ (B) (x_{λ}) .$ Then $I^{τ}$ is an (L, M)-fuzzy internal relation on L^X.

Proof. In order to prove that $I^{τ}$ is an (L, M)-fuzzy internal relation, we need to prove that $I^{τ}$ satisfies (LFIO1)–(LFIO5). (LFIO1) and (LFIO2) can easily obtain from (LFI1) and (LFI2), respectively.

(LFIO3) holds by the following fact. By the definition of $I^{τ}$ , we have $\begin{matrix} I^{τ} (⋁_{j \in J} A_{j}, B) & = ⋀_{x_{λ} ⊲ ⋁_{i \in Ω} A_{i}} τ (B) (x_{λ}) \\ = ⋀_{i \in Ω} ⋀_{x_{λ} ⩽ A_{i}} τ (B) (x_{λ}) \\ = ⋀_{i \in Ω} I^{τ} (A_{i}, B) . \end{matrix}$

(LFIO4) holds by the following fact. By the definition of $I^{τ}$ and (LFI3), we have $\begin{matrix} I^{τ} (C, A \land B) & = ⋀_{x_{λ} ⊲ C} τ (A \land B) (x_{λ}) \\ = ⋀_{x_{λ} ⊲ C} (τ (A) (x_{λ}) \land τ (B) (x_{λ})) \\ = I^{τ} (C, A) \land I^{τ} (C, B) . \end{matrix}$ (LFIO5) It suffices to show that $⋀_{x_{λ} ⊲ A} τ (B) (x_{λ}) ⩽ ⋁_{C \in L^{X}} (⋀_{x_{λ} ⊲ A} τ (C) (x_{λ}) \land ⋀_{y_{μ} ⊲ C} τ (B) (y_{μ})) .$ Let t ⊲ ⋀ _{x_λ⊲A}τ (B) (x_λ) . Then we have t ⊲ τ (B) (x_λ) for each x_λ ⊲ A . By (LFI4), it follows that there exists some C_λ ∈ L^X with x_λ ⩽ C_λ ⩽ B such that t ⩽ τ (C_λ) (x_λ) and t ⩽ ⋀ _{y_μ⊲C_λ}τ (B) (y_μ) for all x_λ ⊲ A. Now take C = ⋁ _{x_λ⊲A}C_λ . Then we have $⋀_{x_{λ} ⊲ A} τ (C) (x_{λ}) ⩾ ⋀_{x_{λ} ⊲ A} τ (C_{λ}) (x_{λ}) ⩾ t$ and $⋀_{y_{μ} ⊲ C} τ (B) (y_{μ}) = ⋀_{x_{λ} ⊲ A} ⋀_{y_{μ} ⊲ C_{λ}} τ (B) (y_{μ}) ⩾ t .$ Thus $t ⩽ ⋁_{C \in L^{X}} (⋀_{x_{λ} ⊲ A} τ (C) (x_{λ}) \land ⋀_{y_{μ} ⊲ C} τ (B) (y_{μ})) .$ So (LFIO5) is obtained from the arbitrariness of t.□

The following corollary gives a representation of the (L, M)-fuzzy internal relation space by means of an (L, M)-fuzzy topology. It can be seen from Theorems 2.3 and 3.6.

Corollary 3.7. Let τ : L^X ⟶ M be an (L, M)-fuzzy topology on X. Define $τ^{I} : L^{X} \times L^{X} ⟶ M$ by* $I^{τ} (A, B) = ⋀_{x_{λ} ⊲ A} ⋁_{x_{λ} ⩽ G ⩽ B} τ (G) .$ Then $I^{τ}$ is an (L, M)-fuzzy internal relation on L^X.

Theorem 3.8. Let $I$ be an (L, M)-fuzzy internal relation on L^X. For all A ∈ L^X, define $τ^{I} : L^{X} ⟶ M^{J (L^{X})}$ by $τ^{I} (A) (x_{λ}) = I (x_{λ}, A) .$ Then $τ^{I}$ is an (L, M)-fuzzy interior operator on X.

Proof. By the Lemma 3.5, we need to check (LFI1)–(LFI3) and (LFI4). First of all, (LFI1) and (LFI2) are trivial and (LFI3) is routine. Now we simply verify that it satisfies (LFI4). The key is to prove that $\begin{matrix} τ^{I} (A) (x_{λ}) \\ ⩽ ⋁_{x_{λ} ⩽ B ⩽ A} (τ^{I} (B) (x_{λ}) \land ⋀_{y_{μ} ⊲ B} τ^{I} (A) (y_{μ})) . \end{matrix}$ Let $t ⊲ τ^{I} (A) (x_{λ}) = I (x_{λ}, A) ⩽ ⋁_{C \in L^{X}} (I (x_{λ}, C) \land I (C, A)) .$

Then there exists some C ∈ L^X such that $t ⊲ (I (x_{λ}, C) \land I (C, A))$ . Then x_λ ⩽ C ⩽ A by (LFIO2). Now take B = C. Then we can immediately obtain $τ^{I} (B) (x_{λ}) = I (x_{λ}, B) = I (x_{λ}, C) ⩾ t$ and $\begin{matrix} ⋀_{y_{μ} ⊲ B} τ^{I} (A) (y_{μ}) & = ⋀_{y_{μ} ⊲ C} I (y_{μ}, A) \\ = I (⋁_{y_{μ} ⊲ C} y_{μ}, A) & (by (LFIO 3)) \\ = I (C, A) ⩾ t . \end{matrix}$ Hence $t ⩽ (τ^{I} (B) (x_{λ}) \land ⋀_{y_{μ} ⊲ B} τ^{I} (A) (y_{μ})) .$ From the arbitrariness of t, we have $τ^{I} (A) (x_{λ}) ⩽ ⋁_{x_{λ} ⩽ B ⩽ A} (τ^{I} (B) (x_{λ}) \land ⋀_{y_{μ} ⊲ B} τ^{I} (A) (y_{μ})) .$ Therefore the conclusion holds.

The following corollary gives a representation of the (L, M)-fuzzy topology by means of an (L, M)-fuzzy internal relation. It can be seen from Theorems 2.4 and 3.8.

Corollary 3.9. Let $I : L^{Xitsc} \times L^{Xitsc} ⟶ M$ be an (L, M)-fuzzy internal relation on L^X. Define $τ^{I} : L^{X} ⟶ M$ by $τ^{I} (A) = I (A, A) .$ Then $τ^{I}$ is an (L, M)-fuzzy topology on X. From Theorem 2 and Theorem 2 we obtain the following two theorems.

Theorem 3.10. For an (L, M)-fuzzy internal relation $I$ on L^X, $I^{(τ^{I})} = I$ .

Proof. This can be obtained by the following computation: $\begin{matrix} \begin{matrix} I^{(τ^{I})} (A, B) = ⋀_{x_{λ} ⊲ A} τ^{I} (B) (x_{λ}) \\ = ⋀_{x_{λ} ⊲ A} I (x_{λ}, B) = I (⋁_{x_{λ} ⊲ A} x_{λ}, B) = I (A, B) . \end{matrix} \end{matrix}$

Theorem 3.11. For an (L, M)-fuzzy interior space (X, τ), $τ^{(I^{τ})} = τ$ .

Proof. This can be obtained by the following computation: $\begin{matrix} \begin{matrix} τ^{(I^{τ})} (A) (x_{λ}) = I^{τ} (x_{λ}, A) = ⋀_{y_{μ} ⊲ x_{λ}} τ (A) (y_{μ}) \\ = ⋀_{μ ⊲ λ} τ (A) (x_{μ}) = τ (A) (x_{λ}) . \end{matrix} \end{matrix}$ □

From Theorems 2.3, 2.4, 3.10 and 3.11, Corollaries 3.7 and 3.9 we obtain the following two corollaries.

Corollary 3.12. For an (L, M)-fuzzy internal relation $I$ on L^X, $I^{(τ^{I})} = I$ .

Corollary 3.13. For an (L, M)-fuzzy topology space (X, τ), $τ^{(I^{τ})} = τ$ .

Now we consider mappings between (L, M)-fuzzy internal relation spaces.

Definition 3.14. Let $(X, I_{X})$ and $(Y, I_{Y})$ be (L, M)-fuzzy internal relation spaces. A mapping f : X ⟶ Y is said to be an (L, M)-fuzzy internal relation dual-preserving mapping ((L, M)-fuzzy IRDP mapping for short), if $I_{Yitsc} (f_{Litsc}^{\to} (U), V) ⩽ I_{Xitsc} (U, f_{Litsc}^{\leftarrow} (V))$ for all U ∈ L^X, V ∈ L^Y.

The following theorems can be easily proved.

Theorem 3.15. If $f : (X, I_{X}) ⟶ (Y, I_{Y})$ and $g : (Y, I_{Y}) ⟶ (Z, I_{Z})$ are two (L, M)-fuzzy IRDP mappings, then $g \circ f : (X, I_{X}) ⟶ (Z, I_{Z})$ is also an (L, M)-fuzzy IRDP mapping.

It is easy to prove that all (L, M)-fuzzy internal relation spaces and all (L, M)-fuzzy IRDP mappings form a category, which is called the category of (L, M)-fuzzy internal relation spaces, denoted by (L, M)-FIR.

Now we consider the relationship between two categories (L, M)-TOP and (L, M)-FIR.

Theorem 3.16. If f : (X, τ_X) ⟶ (Y, τ_Y) is a continuous mapping with respect to (L, M)-fuzzy topologies τ_X and τ_Y, then $f : (X, I^{τ_{X}}) ⟶ (Y, I^{τ_{Y}})$ is an (L, M)-fuzzy IRDP mapping.

Proof. Let U ∈ L^X, V ∈ L^Y. Since f is a continuous mapping with respect to (L, M)-fuzzy topologies τ_X and τ_Y, it follows from Theorem that $\begin{matrix} I^{τ_{Y}} (f_{L}^{\to} (U), V) & = ⋀_{y_{μ} ⊲ f_{L}^{\to} (U)} ⋁_{y_{μ} ⩽ W ⩽ V} τ_{Y} (W) \\ ⩽ ⋀_{f (x)_{λ} ⊲ f_{L}^{\to} (U)} ⋁_{f (x)_{λ} ⩽ W ⩽ V} τ_{Y} (W) \\ ⩽ ⋀_{x_{λ} ⊲ U} ⋁_{x_{λ} ⩽ f_{L}^{\leftarrow} (W) ⩽ f_{L}^{\leftarrow} (V)} τ_{X} (f_{L}^{\leftarrow} (W)) \\ = I^{τ_{X}} (U, f_{L}^{\leftarrow} (V)) . \end{matrix}$ Therefore $f : (X, I^{τ_{X}}) ⟶ (Y, I^{τ_{Y}})$ is an (L, M)-fuzzy IRDP mapping.

Theorem 3.17. If $f : (X, I_{X}) ⟶ (Y, I_{Y})$ is an (L, M)-fuzzy IRDP mapping, then $f : (X, τ^{I_{Xitsc}}) ⟶ (Y, τ^{I_{Yitsc}})$ is a continuous mapping with respect to (L, M)-fuzzy topologies $τ^{I_{Xitsc}}$ and $τ^{I_{Yitsc}}$ .

Proof. Let U ∈ L^Y. Since f is an (L, M)-fuzzy IRDP mapping, it follows from Theorem 3.8 that $\begin{matrix} τ^{I_{Yitsc}} (U) & = I_{Y} (U, U) ⩽ I_{X} (f_{L}^{\leftarrow} (U), f_{L}^{\leftarrow} (U)) \\ = τ^{I_{Xitsc}} (f_{L}^{\leftarrow} (U)) . \end{matrix}$ Therefore $f : (X, I^{τ_{X}}) ⟶ (Y, I^{τ_{Y}})$ is a continuous mapping with respect to (L, M)-fuzzy topologies τ_X and τ_Y.□

Now we define a functor $S : (L, M)$ -FIR ⟶ (L, M)-FTOP such that $S (X, I) = (X, τ^{I}), S (f) = f .$ From Corollaries 3.12 and 3.13, Theorems 3.16 and 3.17 we can obtain the following theorem.

Theorem 3.18. $S : (L, M)$ -FIR ⟶ (L, M)-FTOP is an isomorphic functor, that is, the category of (L, M)-fuzzy internal relation spaces is isomorphic to the category of (L, M)-fuzzy topological spaces.
4. (L, M)-enclosed relation spaces

In this section, our aim is to introduce the notion of (L, M)-fuzzy enclosed relations which can also be used to characterize topologies.

Definition 4.1. A mapping $E : L^{Xitsc} \times L^{Xitsc} ⟶ M$ is called an (L, M)-fuzzy enclosed relation or (L, M)-fuzzy enclosed order on L^X if it satisfies the following conditions.

(LFEO1)
$E (⊥_{L^{X}}, ⊥_{L^{X}}) = ⊤_{M}$ ;

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

$E (A, ⋀_{j \in J} B_{j}) = ⋀_{j \in J} E (A, B_{j})$ ;

$E (A \lor B, C) = E (A, C) \land E (B, C)$ ;

$E (A, B) ⩽ ⋁_{C \in L^{X}} (E (A, C) \land E (C, B))$ .
For an (L, M)-fuzzy enclosed relation $E$ on L^X, the pair $(X, E)$ is called an (L, M)-fuzzy enclosed relation space.

From (LFEO3) and (LFEO4) in Definition we easily obtain the following lemma.

Lemma 4.2. Let $E$ be an (L, M)-fuzzy enclosed relation on L^X. If A ⩽ B and C ⩽ D, then $E (B, C) ⩽ E (A, D)$ .

The following theorem gives the relation between (L, M)-fuzzy internal relations and (L, M)-fuzzy enclosed relations.

Theorem 4.3. (i) For an (L, M)-fuzzy internal relation $I$ on L^X, we define a relation $E^{I}$ on L^X such that for any A, B ∈ L^X, $E^{I (A, B)} = I (B^{'}, A^{'})$ . Then $E^{I}$ is an (L, M)-fuzzy enclosed relation on L^X.

(ii) Conversely, for an (L, M)-fuzzy enclosed relation $E$ on X, we define a relation $I^{E}$ on L^X such that for any A, B ∈ L^X, $I^{E} (A, B) = E (B^{'}, A^{'})$ . Then $I^{E}$ is an (L, M)-fuzzy internal relation on L^X.

In sequel, $E$ is called the associate (L, M)-fuzzy enclosed relation of (L, M)-fuzzy internal relation $I$ .

Proof. This can easily be obtained. □

The following two theorems show that we can obtain two L-enclosed relations from a pointwise quasi-uniformity and a pointwise S-quasi-proximity on a set.

Theorem 4.4. Let $(X, FU)$ be a pointwise (L, M)-fuzzy quasi-uniform space. For all A, B ∈ L^X, we define $E^{FU} : L^{X} \times L^{X} ⟶ M$ by $E^{FU} (A, B) = ⋀_{y_{μ} B} ⋁_{φ (y_{μ}) ⩾ A} FU (φ) .$ Then $E^{FU}$ is an (L, M)-fuzzy enclosed relation on L^X.

Proof. It is easy to verify that $E^{FU}$ satisfies (LFEO1) and (LFEO2). (LFEO3) holds by the following fact. $\begin{matrix} E^{FU} (A, ⋁_{j \in J} B_{j}) & = ⋀_{y_{μ} ⋁_{j \in J} B_{j}} ⋁_{φ (y_{μ}) ⩾ A} FU (φ) \\ = ⋀_{j \in J} ⋀_{y_{μ} B_{j}} ⋁_{φ (y_{μ}) ⩾ A} FU (φ) \\ = ⋀_{j \in J} E^{FU} (A, B_{j}) . \end{matrix}$ (LFEO4) holds by the following fact: $\begin{matrix} E^{FU} (A \lor B, C) & = ⋀_{y_{μ} C} ⋁_{φ (y_{μ}) ⩾ (A \lor B)} FU (φ) \\ = ⋀_{y_{μ} C} (⋁_{φ (y_{μ}) ⩾ A} FU (φ) \land ⋁_{φ (y_{μ}) ⩾ B} FU (φ)) \\ = E^{FU} (A, C) \land E^{FU} (B, C) . \end{matrix}$ (LFEO5) It suffices to show that $\begin{matrix} ⋀_{y_{μ} B} ⋁_{φ (y_{μ}) ⩾ A} FU (φ) \\ = ⋁_{C \in L^{X}} (⋀_{z_{ν} C} ⋁_{ψ (z_{ν}) ⩾ A} FU (ψ) \land ⋀_{y_{μ} B} ⋁_{φ (y_{μ}) ⩾ C} FU (φ)) . \end{matrix}$ Suppose $t ⊲ ⋀_{y_{μ} B} ⋁_{φ (y_{μ}) ⩾ A} FU (φ) = ⋀_{y_{μ} B} ⋁_{φ (y_{μ}) ⩾ A} ⋁_{φ ⊙ φ ⩾ φ} FU (φ) .$

Then there exist $φ, φ \in R (L^{X})$ such that φ (y_μ) ⩾ A, φ ⊙ φ ⩾ φ and $t ⩽ FU (φ)$ for each y_μB . Let C = φ (y_μ). Then y_μC and A ⩽ C. Furthermore, we get∥ $E^{FU} (C, B) = ⋀_{y_{μ} B} ⋁_{φ (y_{μ}) ⩾ C} FU (φ) ⩾ ⋀_{y_{μ} B} FU (φ) ⩾ t .$ ∥Since ⋀ {φ (z_ν) ∣ z_νφ (y_μ)} = φ ⊙ φ (y_μ) ⩾ φ (y_μ) ⩾ A, we have φ (z_ν) ⩾ A for all z_νC . Hence, ∥ $E^{FU} (A, C) = ⋀_{z_{ν} C} ⋁_{ψ (z_{ν}) ⩾ A} FU (ψ) ⩾ ⋀_{z_{ν} C} FU (φ) ⩾ t .$ ∥Then $t ⩽ E^{FU} (A, C) \land E^{FU} (C, B)$ . Therefore $t ⩽ ⋁_{C \in L^{X}} (E^{FU} (A, C) \land E^{FU} (C, B))$ . Thus (LFEO5) is obtained from the arbitrariness of t. □

Theorem 4.5. Let δ be a generalized pointwise S-quasi-proximity on L^X. For all A, B ∈ L^X, we define $E^{δ} : L^{X} \times L^{X} ⟶ M$ by
$E^{δ (A, B)} = ⋀_{y_{μ} B} δ (y_{μ}, A)^{'} .$
Then $E^{δ}$ is an (L, M)-fuzzy enclosed relation on L^X .

Proof. It is trivial to check that $E^{δ}$ satisfies (LFEO1) and (LFEO2).

(LFEO3) By the definition of $E^{δ}$ and (FSP2), we have $\begin{matrix} E^{δ} (A, ⋀_{j \in J} B_{j}) & = ⋀_{y_{μ} ⋀_{j \in J} B_{j}} δ (y_{μ}, A)^{'} \\ = ⋀_{j \in J} ⋀_{y_{μ} B_{j}} δ (y_{μ}, A)^{'} = ⋀_{j \in J} E^{δ} (A, B_{j}) . \end{matrix}$

(LFEO4) holds by the following fact. By the definition of $E^{δ}$ and (FSP3), it follows that $\begin{matrix} E^{δ} (A \lor B, C) & = ⋀_{y_{μ} C} δ (y_{μ}, A \lor B)^{'} \\ = ⋀_{y_{μ} C} (δ (y_{μ}, A)^{'} \land δ (y_{μ}, B)^{'}) \\ = E^{δ} (A, C) \land E^{δ} (B, C) . \end{matrix}$ (LFEO5) By the definition of $E^{δ}$ and (FSP4), we have $\begin{matrix} E^{δ} (A, B) & = ⋀_{y_{μ} B} δ (y_{μ}, A)^{'} \\ ⩽ ⋀_{y_{μ} B} ⋁_{C \in L^{X}} (δ (y_{μ}, C)^{'} \land ⋀_{z_{ν} C} δ (z_{ν}, A)^{'}) \\ = ⋁_{C \in L^{X}} (E^{δ} (C, B) \land E^{δ} (A, C)) . \end{matrix}$ Thus the conclusion holds.

We will discuss that the relationships between (L, M)-fuzzy enclosed relations and (L, M)-fuzzy topologies in what follows. For this, the following lemma is necessary.

Lemma 4.6. Let Cl : L^X ⟶ M^{J(L^X)} be an mapping. Then the following equalities are equivalent: (LFC4)
Cl (A) (x_λ) = ⋀_{x_λB⩾A} ⋁ _{y
_μ
B}Cl (B) (y_μ) .

* Cl (A) (x_λ) =⋀_{x_λ !¬! ⩽B⩾A} (Cl (B) (x_λ) ∨ ⋁ _{y
_μ
B}Cl (A) (y_μ)) .

Proof. (LFC4) ⇒ (LFC4) is trivial. Now suppose (LFC4) holds. It is obvious that Cl (A) (x_λ) ⩽ ⋀ _{x_λB⩾A} ⋁ _{y
_μ
B}Cl (B) (y_μ) by a mapping Cl is order-preserving from (LFC4). In order to prove Cl (A) (x_λ) ⩾ ⋀ _{x_λB⩾A} ⋁ _{y
_μ
B}Cl (B) (y_μ), it suffices to show that $(Cl (A) (x_{λ}))^{'} ⩽ ⋁_{x_{λ} B ⩾ A} ⋀_{y_{μ} B} (Cl (B) (y_{μ}))^{'} .$

Let $\begin{matrix} t ⊲ (Cl (A) (x_{λ}))^{'} \\ = ⋁_{x_{λ} B ⩾ A} ((Cl (B) (x_{λ}))^{'} \land ⋀_{y_{μ} B} (Cl (A) (y_{μ}))^{'}) . \end{matrix}$ Then there exists some B ∈ L^X such that the following statement holds. $x_{λ} B ⩾ A, (Cl (B) (x_{λ}))^{'} ⊳ t ()$ and $\forall y_{μ} B, (Cl (A) (y_{μ}))^{'} ⊳ t . (* )$ It is clear that the meet of fuzzy sets fulfilling (∗) and (∗∗) is still of such kind. Thus we can define B_t to be the minimal fuzzy sets fulfilling (∗) and (∗∗), i.e., x_λB_t ⩾ A, (Cl (B_t) (x_λ)) ′ ⊳ t and (Cl (A) (y_μ)) ′ ⊳ t for all y_μ ⊲ B_t . Then for all y_μB_t, it follows from (Cl (A) (y_μ)) ′ ⊳ t that there exists G_μ ∈ L^X such that the following statement holds. $y_{μ} G_{μ} ⩾ A, (Cl (G_{μ}) (y_{μ}))^{'} ⊳ t$ and $\forall z_{ν} G_{μ}, (Cl (A) (z_{ν}))^{'} ⊳ t .$ It is easy to check that B_t ∧ G_μ satisfies (∗) and (∗∗). Hence, by the minimality of B_t, it follows that B_t ⩽ B_t ∧ G_μ. Therefore B_t ⩽ G_μ . Then we can obtain that ∀y_μB_t, (Cl (B_t) (y_μ)) ′ ⩾ (Cl (G_μ) (y_μ)) ′ ⊳ t . Thus, ⋀_{y
_μ
B
_t} (Cl (B_t) (y_μ)) ′ ⩾ t . Therefore, ⋁_{x_λB⩽A} ⋀ _{y
_μ
B} (Cl (B) (y_μ)) ′ ⩾ t is obvious. From the arbitrariness of t, we have $(Cl (A) (x_{λ}))^{'} ⩽ ⋁_{x_{λ} B ⩽ A} ⋀_{y_{μ} B} (Cl (B) (y_{μ}))^{'} .$ □

The following theorem shows that we can obtain an (L, M)-fuzzy enclosed relation from an (L, M)-fuzzy topology.

Theorem 4.7. Let (X, Cl) be an (L, M)-fuzzy closure space. For any A, B ∈ L^X. Define $E^{Cl} : L^{X} \times L^{X} ⟶ M$ as follows:*
$E^{Cl} (A, B) = ⋀_{x_{λ} B} (Cl (A) (x_{λ}))^{'} .$
Then $E^{Cl}$ is an (L, M)-fuzzy enclosed relation.∥Proof. In order to prove that $E^{Cl}$ is an (L, M)-fuzzy enclosed relation, we need to prove that $E^{Cl}$ satisfies the conditions (LFEO1)–(LFEO5). (LFEO1) and (LFEO2) can easily obtain from (LFC1) and (LFC2), respectively.

(LFEO3) holds by the following fact. By the definition of $E^{Cl}$ , we have $\begin{matrix} E^{Cl} (A, ⋀_{j \in J} B_{j}) & = ⋀_{x_{λ} ⋀_{j \in J} B_{j}} (Cl (A) (x_{λ}))^{'} \\ = ⋀_{j \in J} ⋀_{x_{λ} B_{j}} (Cl (A) (x_{λ}))^{'} \\ = ⋀_{j \in J} E^{Cl} (A, B_{j}) . \end{matrix}$ (LFEO4) holds by the following fact. By the definition of $E^{Cl}$ and (LFC3), we have $\begin{matrix} E^{Cl} (A \lor B, C) & = ⋀_{x_{λ} C} (Cl (A \lor B) (x_{λ}))^{'} \\ = ⋀_{x_{λ} C} ((Cl (A) (x_{λ}))^{'} \land (Cl (B) (x_{λ}))^{'}) \\ = E^{Cl} (A, C) \land E^{Cl} (B, C) . \end{matrix}$ (LFEO5) It suffices to show that $\begin{matrix} ⋀_{x_{λ} B} (Cl (A) (x_{λ}))^{'} \\ ⩽ ⋁_{C \in L^{X}} (⋀_{z_{ν} C} (Cl (A) (z_{ν}))^{'} \land ⋀_{x_{λ} B} (Cl (C) (x_{λ}))^{'}) . \end{matrix}$ Let t ⊲ ⋀ _{x
_λ
B} (Cl (A) (x_λ)) ′ . Then we have t ⊲ (Cl (A) (x_λ)) ′ for each x_λB . By (LFC4), it follows that there exists some C_λ ∈ L^X with x_λC_λ ⩾ A such that t ⩽ (Cl (C_λ) (x_λ)) ′ and t ⩽ ⋀ _{y
_μ
C
_λ} (Cl (A) (y_μ)) ′ for all x_λB. Now take C = ⋀ _{x
_λ
B}C_λ . Then we have C ⩾ A and B ⩽ C. Therefore, we have $⋀_{x_{λ} B} (Cl (C) (x_{λ}))^{'} ⩾ ⋀_{x_{λ} B} (Cl (C_{λ}) (x_{λ}))^{'} ⩾ t$ and $⋀_{z_{ν} C} (Cl (A) (z_{ν}))^{'} = ⋀_{x_{λ} B} ⋀_{z_{ν} C_{λ}} (Cl (A) (z_{ν}))^{'} ⩾ t .$ Thus $t ⩽ ⋁_{C \in L^{X}} (⋀_{z_{ν} C} (Cl (A) (z_{ν}))^{'} \land ⋀_{x_{λ} B} (Cl (C) (x_{λ}))^{'}) .$ So (LFEO5) is obtained from the arbitrariness of t.

Corollary 4.8. Let τ : L^X ⟶ M be an (L, M)-fuzzy topology on X. Define $E^{τ} : L^{X} \times L^{X} ⟶ M$ by* $E^{τ} (A, B) = ⋀_{x_{λ} B} ⋁_{x_{λ} D ⩾ A} τ (D^{'}) .$ Then $E^{τ}$ is an (L, M)-fuzzy enclosed relation on L^X.

From an (L, M)-fuzzy enclosed relation on L^X we can obtain an (L, M)-fuzzy topology on X as follows.

Theorem 4.9. Let $E$ be an (L, M)-fuzzy enclosed relation on L^X. For all A, B ∈ L^X, we define ${Cl}^{E} : L^{X} ⟶ M^{J (L^{X})}$ by
${Cl}^{E} (A) (x_{λ}) = ⋀_{x_{λ} B} E (A, B)^{'} .$
Then ${Cl}^{E}$ is an (L, M)-fuzzy closure operator on X.

Proof. By the Lemma 2, we need to check (LFC1)–(LFC3) and (LFC4). First of all, (LFC1) and (LFC2) are trivial and (LFC3) is routine. Now we simply verify that it satisfies (LFC4). The key is to prove that∥ $\begin{matrix} ({Cl}^{E} (A) (x_{λ}))^{'} \\ ⩽ ⋁_{x_{λ} B ⩾ A} (({Cl}^{E} (B) (x_{λ}))^{'} \land ⋀_{y_{μ} B} ({Cl}^{E} (A) (y_{μ}))^{'}) . \end{matrix}$ Let $\begin{matrix} t ⊲ ({Cl}^{E} (A) (x_{λ}))^{'} & = ⋁_{x_{λ} B} E (A, B) \\ ⩽ ⋁_{x_{λ} B} ⋁_{C \in L^{X}} (E (A, C) \land E (C, B)) . \end{matrix}$ Then there exist B, C ∈ L^X such that x_λB and $t ⩽ E (A, C) \land E (C, B)$ . Then A ⩽ C ⩽ B by (LFEO2). It follows from $t ⩽ E (A, C)$ that there exists some D ∈ L^X such that $t ⩽ (E (A, D) \land E (D, C))$ by (LFEO5), and thus A ⩽ D ⩽ C . Therefore, $\begin{matrix} ({Cl}^{E} (D) (x_{λ}))^{'} & = ⋁_{x_{λ} G} E (D, G) ⩾ E (D, B) \\ ⩾ E (D, C) ⩾ t \end{matrix}$ and $\begin{matrix} ⋀_{y_{μ} D} ({Cl}^{E} (A) (y_{μ}))^{'} & = ⋀_{y_{μ} D} ⋁_{y_{μ} G} E (A, G) \\ ⩾ ⋀_{y_{μ} D} E (A, C) = E (A, C) ⩾ t . \end{matrix}$ Hence $t ⩽ ({Cl}^{E} (D) (x_{λ}))^{'} \land ⋀_{y_{μ} D} ({Cl}^{E} (A) (y_{μ}))^{'} .$ From the arbitrariness of t, we have $\begin{matrix} ({Cl}^{E} (A) (x_{λ}))^{'} \\ ⩽ ⋁_{x_{λ} B ⩾ A} (({Cl}^{E} (B) (x_{λ}))^{'} \land ⋀_{y_{μ} B} ({Cl}^{E} (A) (y_{μ}))^{'}) . \end{matrix}$ Therefore the conclusion holds. □

Corollary 4.10. Let $E : L^{X} \times L^{X} ⟶ M$ be an (L, M)-fuzzy enclosed relation on L^X. Define $τ^{E} : L^{X} ⟶ M$ by
$τ^{E} (A) = ⋀_{x_{λ} A^{'}} ⋁_{x_{λ} B} E (A^{'}, B) .$
Then $τ^{E}$ is an (L, M)-fuzzy topology on X.

From Theorems 4.7 and 4.9 we obtain the following theorems.

Theorem 4.11. For an (L, M)-fuzzy enclosed relation $I$ on L^X, $E^{({Cl}^{E})} = E$ .

Proof. $E^{({Cl}^{E})} ⩾ E$ is trivial. The key is to prove that $E^{({Cl}^{E})} ⩽ E .$

This can be obtained by the following computation: $\begin{matrix} E^{({Cl}^{E})} (A, B) \\ = ⋀_{x_{λ} B} {({Cl}^{E} (A) (x_{λ}))}^{'} \\ = ⋀_{x_{λ} B} ⋁_{x_{λ} D} E (A, D) \\ = ⋁_{f \in \prod_{x_{λ} B} B_{x_{λ}}} ⋀_{x_{λ} B} E (A, f (x_{λ})) (by CD - law) \\ = ⋁_{f \in \prod_{x_{λ} B} B_{x_{λ}}} E (A, ⋀_{x_{λ} B} f (x_{λ})) (by (LFEO 3)) \\ ⩽ ⋁_{f \in \prod_{x_{λ} B} B_{x_{λ}}} E (A, B) \\ = E (A, B), \end{matrix}$ where $B_{λ} = {D \in L^{X} ∣ x_{λ D}}$ . The last inequality is obtained by ⋀_{x
_λ
B}f (x_λ) ⩽ B. In fact, for any y_{μ
B}, y_μf (y_μ) ⩾ ⋀ _{x
_λ
B}f (x_λ). So for any y_μB, y_μ ⋀ _{x
_λ
B}f (x_λ) , and hence ⋀_{x
_λ
B}f (x_λ) ⩽ B . □

Theorem 4.12. For an (L, M)-fuzzy closure space (X, Cl), ${Cl}^{(E^{Cl})} = Cl$ .

Proof. ${Cl}^{(E^{Cl})} ⩾ E$ is obvious. The key is to prove that ${Cl}^{(E^{Cl})} ⩽ E$ . This can be obtained by the following computation: $\begin{matrix} {Cl}^{(E^{Cl})} (A) (x_{λ}) & = ⋀_{x_{λ} B} E^{Cl} (A, B)^{'} \\ = ⋀_{x_{λ} B ⩾ A} ⋁_{y_{μ} B} Cl (A) (y_{μ}) \\ ⩽ ⋀_{x_{λ} B ⩾ A} ⋁_{y_{μ} B} Cl (B) (y_{μ}) \\ = Cl (A) (x_{λ}) . \end{matrix}$ □∥From Theorems 2.7, 2.6, 4.11 and 4.12, Corollaries 4.8 and 4.10 we obtain the following corollaries.

Corollary 4.13. For an (L, M)-fuzzy enclosed relation $E$ on L^X, $E^{(τ^{E})} = E$ .

Corollary 4.14. For an (L, M)-fuzzy topology space (X, τ), $τ^{(E^{τ})} = τ$ .

In what follows, we shall consider mappings between two (L, M)-fuzzy enclosed relation spaces.

Definition 4.15. Let $(X, E_{X})$ and $(Y, E_{Y})$ be two (L, M)-fuzzy enclosed relation spaces. A mapping f : X ⟶ Y is said to be an (L, M)-fuzzy enclosed relation dual-preserving mapping, or (L, M)-fuzzy ERDP mapping for short, if $E_{Y} (U, f_{L}^{\to} (V)) ⩽ E_{X} (f_{L}^{\leftarrow} (U), V)$ for all U ∈ L^Y, V ∈ L^X,

The following theorem is obvious.

Theorem 4.16. If $f : (X, E_{X}) ⟶ (Y, E_{Y})$ and $g : (Y, E_{Y}) ⟶ (Z, E_{Z})$ are two (L, M)-fuzzy ERDP mapping, then $g \circ f : (X, E_{X}) ⟶ (Z, E_{Z})$ is also an (L, M)-fuzzy ERDP mapping.

It is easy to prove that all (L, M)-fuzzy enclosed relation spaces and all (L, M)-fuzzy ERDP mappings form a category, which is called the category of (L, M)-fuzzy enclosed relation spaces, denoted by (L, M)-FER.

Now we consider the relationship between two categories (L, M)-TOP and (L, M)-FER.

Theorem 4.17. ERP-con If $f : (X, E_{X}) ⟶ (Y, E_{Y})$ is an (L, M)-fuzzy ERDP mapping, then $f : (X, τ^{E_{X}}) ⟶ (Y, τ^{E_{Y}})$ is a continuous mapping with respect to (L, M)-fuzzy topologies $τ^{E_{X}}$ and $τ^{E_{Y}}$ .

Proof. Let U ∈ L^Y, V ∈ L^X. Since f is an (L, M)-fuzzy ERDP mapping, it follows from Corollary 4.10 that $\begin{matrix} τ^{E_{X}} (f_{L}^{\leftarrow} (U)) & = ⋀_{x_{λ} f_{L}^{\leftarrow} (U)^{'}} ⋁_{x_{λ} V} E_{X} (f_{L}^{\leftarrow} (U)^{'}, V) \\ ⩾ ⋀_{f (x)_{λ} U^{'}} ⋁_{f (x)_{λ} f_{L}^{\to} (V)} E_{Y} (U^{'}, f_{L}^{\to} (V)) \\ ⩾ ⋀_{y_{μ} U^{'}} ⋁_{y_{μ} f_{L}^{\to} (V)} E_{Y} (U^{'}, f_{L}^{\to} (V)) \\ = τ^{E_{Y}} (U) . \end{matrix}$ Therefore f is a continuous mapping with respect to (L, M)-fuzzy topologies τ_X and τ_Y. □

Theorem 4.18. If f : (X, τ_X) ⟶ (Y, τ_Y) is a continuous mapping with respect to (L, M)-fuzzy topologies τ_X and τ_Y, then $f : (X, E^{τ_{X}}) ⟶ (Y, E^{τ_{Y}})$ is an (L, M)-fuzzy ERDP mapping.

Proof. Let U ∈ L^Y, V ∈ L^X. Since f is a continuous mapping with respect to (L, M)-fuzzy topologies τ_X and τ_Y, it follows from Corollary 4.8 that $\begin{matrix} E^{τ_{X}} (f_{L}^{\leftarrow} (U), V) \\ = ⋀_{x_{λ} V} ⋁_{x_{λ} f_{L}^{\leftarrow} (W) ⩾ f_{L}^{\leftarrow} (U)} τ_{X} (f_{L}^{\leftarrow} (W)^{'}) \\ ⩾ ⋀_{f (x)_{λ} f_{L}^{\to} (V)} ⋁_{f (x)_{λ} W ⩾ U} τ_{Y} (W^{'}) \\ ⩾ ⋀_{y_{μ} f_{L}^{\to} (V)} ⋁_{y_{μ} W ⩾ U} τ_{Y} (W^{'}) \\ = E^{τ_{Y}} (U, f_{L}^{\to} (V)) . \end{matrix}$ Therefore f is an (L, M)-fuzzy ERDP mapping. □

Now we define a functor $T : (L, M)$ -TOP ⟶ (L, M)-FER such that $T (X, τ) = (X, E^{τ}), T (f) = f .$

By Corollaries 4.13 and 4.14, Theorems 4.17 and 4.18 we can obtain the following theorem.

Theorem 4.19. $T$ is an isomorphic functor, that is, the category of (L, M)-fuzzy enclosed relation spaces is isomorphic to the category of topological spaces.∥
5. Conclusions

∥n this paper, we presented the notions of (L, M)-fuzzy internal relations and (L, M)-fuzzy enclosed relations. The category of (L, M)-fuzzy internal relation spaces, the category of (L, M)-fuzzy enclosed relation spaces and the category of topological spaces are isomorphism. Based on these conclusions, we may study the separation axioms and compactness by using (L, M)-fuzzy internal relations and (L, M)-fuzzy enclosed relations.

Footnotes

Acknowledgements

This work is supported by the National Natural Science Foundation of China (11871097) and Specialized Research Fund for the Doctoral Program of Higher Education (20131101110048).

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