On Alexandrov L -fuzzy nearness

Abstract

This paper is dedicated to introduce the concept of a new space named "L-fuzzy nearness" and to arise a great deal of relationships among it and most of known spaces, like: L-fuzzy topological spaces (interior, closure, co-topology) and L-fuzzy pre-proximities in complete residuated lattices. We have demonstrated concrete functors between the category of Alexandrov L-fuzzy nearness, Alexandrov L-fuzzy topological structures and Alexandrov L-fuzzy pre-proximities and established the Galois correspondence between these categories.

1 Introduction

The fuzzy set theory is one of the most effective tools to understand intelligent systems with uncertain information which was proposed firstly by Zadeh [47]. In 1967, Goguen [18] introduced the concept of fuzzy set by replacing the structure of membership value [0, 1] with arbitrary set which gave more varieties to study more fuzzy structures. Ward and Dilworth [41] introduced the complete residuated lattice which is an algebraic structure for many valued logic. Through it, Bělohlávek [6] could give us the L-fuzzy sets which was capable of modeling information systems in presence of uncertainty and vagueness. The equivalence relation in Pawlak’s rough set [30] was replaced by arbitrary relation to handle more uncertainty (cf., e.g., Radzikowska and Kerre [32], Yao and Lin [46]). Yao and Lin [46] showed that upper and lower approximations of a set are noting but closure and interior of it. Hence, they could propose several models of rough sets.

The concept of fuzzy topology was first defined in 1968 by Chang [14] and later redefined in a somewhat different way by Lowen [29] and by Hutton [23]. These definitions, a fuzzy topology is a crisp subfamily of a family fuzzy sets and fuzziness in the concept of openness of a fuzzy set has not been considered, which appeared to be a drawback in the process of fuzzification of the concept of topological spaces. Therefore, Sostak introduced a new definition of fuzzy topology in 1985 [42]. Later on, he developed the theory of fuzzy topological spaces in [43, 44]. Höhle [22] toke a great step in introducing the concept of L-fuzzy topology using a different algebraic structure (cqm, quantales, MV-algebra) of truth value which allowed mathematics to play role in logic and theoretical computer science (cf., [7 , 38]).

Interior and closure operators is a very useful tools in several areas of mathematical structures with direct applications, both mathematical (e.g. topology, logic) and extramathematical (e.g. data mining, knowledge representation). In fuzzy set theory, several particular case, as well as, general theory of interior operators which operate with fuzzy sets (so called fuzzy interior operators) are studied [3 , 12]. Closure operators, however, have appeared in few studies only (Bianco and Gerla [9 –11], Bondenhfer et al [12]) and it seems that no general theory of closure operators appeared so far. Recently, Bělohlávek [8] outlined a general theory of fuzzy interior (closure [5]) operators and fuzzy interior (closure) systems using the structure of the residuated lattice over the usual structure of truth value on [0, 1]. Fang and Yue [17] studied the relationship between L-fuzzy closure systems and L-fuzzy topological spaces from a category viewpoint for a complete residuated lattice L, so as did Ramadan [34] but for L-fuzzy interior systems.

Katsaras [25] introduced fuzzy proximity in [0, 1]-fuzzy set theory. Subsequently Wang-jin Liu [37], Artico and Moresco [2] extended it into L-fuzzy set theory. F. Bayoumi [4] shows that all initial and final lifts in the category of L-proximity spaces of the internal type and hence all initial and final L-proximities of the internal type do exist.

The aim of this paper is to introduce the concept of the L-fuzzy nearness. We have arose out a great deal of relationships among it and the L-fuzzy topological spaces (interior, closure, co-topology) and the L-fuzzy pre-proximities which toke the applications to multi-attribute decision making to a whole new level and we give primitive examples on that. We have demonstrated concrete functors between the category of Alexandrov L-fuzzy nearness, Alexandrov L-fuzzy topological structures and Alexandrov L-fuzzy pre-proximities and established the Galois correspondence between these categories.

The content of this paper is organized as follows. In section 2, we recall some fundamental concepts and related definitions. In section 3, we reformulate the necessary Alexandrov topological structures studied in this paper. In sections 4-6, we investigate the relations among the given new structure Alexandrov L-fuzzy nearness, Alexandrov L-fuzzy pre-proximities and Alexandrov L-fuzzy topological structures. Several and interesting relations are introduced and finally, we conduct the Galois correspondence between the categories of Alexandrov L-fuzzy nearness, Alexandrov L-fuzzy pre-proximity spaces and that of Alexandrov L-fuzzy topological structures.

2 Preliminaries

Definition 2.1. [7 , 45] An algebra (L, ∧ , ∨ , ⊙ , → , ⊥ , ⊤) is called a complete residuated lattice if it satisfies the following conditions:

(C1) (L, ≤ , ∨ , ∧ , ⊥ , ⊤) is a complete lattice with the greatest element ⊤ and the least element ⊥;

(C2) (L, ⊙ , ⊤) is a commutative monoid;

(C3) x ⊙ y ≤ z iff x ≤ y → z for x, y, z ∈ L.

In this paper, we assume that (L, ≤ , ⊙ , ^∗) is a complete residuated lattice with an order reversing involution ^∗ which is defined by

$x \oplus y = (x^{*} ⊙ y^{*})^{*}, x^{*} = x \to ⊥ .$

For α ∈ L, f ∈ L^X, we denote (α → f) , (α ⊙ f) ,

α_X ∈ L^X as (α → f) (x) = α → f (x) ,

(α ⊙ f) (x) = α ⊙ f (x) , α_X (x) = α, $⊤_{x} (y) = {\begin{matrix} ⊤, & if y = x, \\ ⊥, & otherwise, \end{matrix} ⊤_{x}^{*} (y) = {\begin{matrix} ⊥, & if y = x, \\ ⊤, & otherwise . \end{matrix}$

Some basic properties of the binary operation ⊙ and residuated operation → are collected in the following lemma, and they can be found in many works [7 , 45].

Lemma 2.2. For each x, y, z, x_i, y_i, w ∈ L, we have the following properties.

(1) ⊤ → x = x, ⊥ ⊙ x = ⊥ ,

(2) If y ≤ z, then x ⊙ y ≤ x ⊙ z, x ⊕ y ≤ x ⊕ z,

x → y ≤ x → z and z → x ≤ y → x,

(3) x ≤ y iff x→ y = ⊤,

(4) $(⋀_{i} y_{i})^{*} = ⋁_{i} y_{i}^{*}$ , $(⋁_{i} y_{i})^{*} = ⋀_{i} y_{i}^{*}$ ,

(5) x → (⋀ _iy_i) = ⋀ _i (x → y_i),

(6) (⋁ _ix_i) → y = ⋀ _i (x_i → y),

(7) x ⊙ (⋁ _iy_i) = ⋁ _i (x ⊙ y_i),

(8) (⋀ _ix_i) ⊕ y = ⋀ _i (x_i ⊕ y),

(9) (x ⊙ y) → z = x → (y → z) = y → (x → z),

(10) x ⊙ y = (x → y^∗) ^∗, x ⊕ y = x^∗ → y and

x → y = y^* → x^*,

(11) (x → y) ⊙ (z → w) ≤ (x ⊙ z) → (y ⊙ w),

(12) x → y ≤ (x ⊙ z) → (y ⊙ z) and

(x → y) ⊙ (y → z) ≤ x → z,

(13) (x → y) ⊙ (z → w) ≤ (x ⊕ z) → (y ⊕ w),

(14) x ⊙ (x → y) ≤ y and y ≤ x → (x ⊙ y)

and x → (y ⊕ z) ≤ (x → y) ^* → z,

(15) (x ∨ y) ⊙ (z ∨ w) ≤ (x ∨ z) ∨ (y ⊙ w) ≤ (x ⊕ z) ∨ (y ⊙ w) ,

(16) ⋁_i∈Γx_i → ⋁ _i∈Γy_i ≥ ⋀ _i∈Γ (x_i → y_i) ,

⋀_i∈Γx_i → ⋀ _i∈Γy_i ≥ ⋀ _i∈Γ (x_i → y_i),

(17) (x ⊙ y) ⊙ (z ⊕ w) ≤ (x ⊙ z) ⊕ (y ⊙ w) ,

(18) z → x ≤ (x → y) → (z → y) and

y → z ≤ (x → y) → (x → z).

Definition 2.3. [7 , 35] Let X be a set. A mapping $R : X \times X \to L$ is called an L-fuzzy relation on X, then for all x, y, z ∈ X the relation $R$ is said to be

(1) reflexive if $R (x, x) = ⊤$ ,

(2) symmetric if $R (x, y) = R (y, x)$ ,

(3) transitive if $R (x, y) ⊙ R (y, z) \leq R (x, z)$ .

A L-fuzzy relation on X is called a L-fuzzy pre-order if it is reflexive and transitive and called an L-fuzzy equivalence relation if it is reflexive, symmetric and transitive.

Let $(X, R_{X})$ and $(Y, R_{Y})$ be two L-fuzzy equivalence relations, then the map $φ : (X, R_{X}) \to (Y, R_{Y})$ is called an LF-order preserving map if for each x, y ∈ X,

$R_{X} (x, y) \leq R_{Y} (φ (x), φ (y)) .$

The category of L-fuzzy equivalence relations with LF-order preserving maps as morphisms is denoted by L - FEQR.

Definition 2.4. [7 , 17] Let X be a set, define a binary map $S : L^{X} \times L^{X} \to L$ by

$S (f, g) = ⋀_{x \in X} (f (x) \to g (x)) .$

Then, for each f, g, h, k ∈ L^X, and α ∈ L, the following properties hold.

(1) $S$ is an L-partial order on L^X,

(2) f ≤ g iff $S (f, g) \geq ⊤$ ,

(3) If f ≤ g, then $S (h, f) \leq S (h, g),$ and

$S (f, h) \geq S (g, h)$ ,

(4) $S (f, g) ⊙ S (k, h) \leq S (f \oplus k, g \oplus h)$ and

$S (f, g) ⊙ S (k, h) \leq S (f ⊙ k, g ⊙ h)$ ,

(5) $⋀_{i \in Γ} S (f_{i}, g_{i}) \leq S (⋀_{i \in Γ} f_{i}, ⋀_{i \in Γ} g_{i})$ ,

(6) $S (f, h) = ⋁_{g \in L^{X}} (S (f, g) ⊙ S (g, h))$ .

If φ : X → Y is a map, then for f, g ∈ L^X

and h, k ∈ L^Y, then $\begin{matrix} S (f, g) \leq S (φ^{\to} (f), φ^{\to} (g)), \\ S (h, k) \leq S (φ^{\leftarrow} (h), φ^{\leftarrow} (k)) . \end{matrix}$ and the equalities hold if φ is bijective.

A concrete category is a pair $(C, U),$ where $C$ is a category and $U : C \to Set$ is a faithful functor (or a forgetful functor). For each $C$ -object X, U (X) is called the underlying set of X. Thus, every object in a concrete category can be regarded as a structured set. We write $C$ for $(C, U),$ if the concrete functor is obvious. All of the categories considered in this paper are concrete categories. A concrete functor between two concrete categories $(C, U),$ and $(D, V),$ is a functor $Γ : C \to D$ with U = V ∘ G, which means that G only changes the structures on the underlying sets. Hence, in order to define a concrete functor $Γ : C \to D,$ we only consider the following two requirements. First, we assign to each $C$ -object X, a $D$ -object G (X) such that V (G (X)) = U (X) . Second, we verify that if a function f : U (X) → U (Y) is a $C$ -morphism X → Y, then it is also a $D$ -morphism G (X) → G (Y) .

Theorem 2.5. [1] Suppose that $F : D \to C,$ $Γ : C \to D$ are concrete functors. Then the following conditions are equivalent:

(1) ${{id}_{Y} : F \circ Γ (Y) \to Y ∣ Y \in C}$ is a natural transformation from the functor F ∘ G to the identity functor on $C,$ and ${{id}_{X} : X \to Γ \circ F (X) ∣ X \in D}$ is a natural transformation from the identity functor $D$ to the functor G ∘ F .

(2) For each $Y \in C,$ id_Y : F ∘ G (Y) → Y is a $C$ -morphism, and for each $X \in D$ , id_X : X → G ∘ F (X) is a $D$ -morphism.

In this case, (F, G) is called a Galois correspondence between $C$ and $D .$ If (F, G) is a Galois correspondence, then it is easy to check that F is a left adjoint of G, or equivalently that G is a right adjoint of F .

3 Alexandrov topological structures

Definition 3.1. [18 , 36] A map $T : L^{X} \to L$ is called a L-fuzzy topology on X if it satisfies the following conditions:

(T1) $T (⊥_{X}) = T (⊤_{X}) = ⊤$ ,

(T2) $T (f ⊙ g) \geq T (f) ⊙ T (g) \forall f, g \in L^{X}$ ,

(T3) $T (⋁_{i} f_{i}) \geq ⋀_{i} T (f_{i}) \forall {f_{i}}_{i \in Γ} \subseteq L^{X}$ .

The pair $(X, T)$ is called a L-fuzzy topological space. A L-fuzzy topological space is said to be:

(ST) stratified if $T (α ⊙ f) \geq T (f)$ ,

(CST) co-stratified if $T (α \to f) \geq T (f)$ ,

(S) strong if it is both stratified and co-stratified,

(AL) Alexandrov if $T (⋀_{i} f_{i}) \geq ⋀_{i} T (f_{i})$

∀ {f_i} _i∈Γ ⊆ L^X,

(SE) separated if $T (⊤_{x}) = ⊤$ for all x ∈ X.

A mapping φ : X → Y between two L-fuzzy topological spaces is called LF-continuous from $(X, T_{X})$ to $(Y, T_{Y})$ if for all f ∈ L^Y, we have

$T_{Y} (f) \leq T_{X} (φ^{\leftarrow} (f)) .$

Definition 3.2. [18 , 36] A map $F : L^{X} \to L$ is called a L-fuzzy co-topology on X if it satisfies the following conditions:

(CT1) $F (⊥_{X}) = F (⊤_{X}) = ⊤$ ,

(CT2) $F (f \oplus g) \geq F (f) ⊙ F (g) \forall f, g \in L^{X}$ ,

(CT3) $F (⋀_{i} f_{i}) \geq ⋀_{i} T (f_{i}) \forall {f_{i}}_{i \in Γ} \subseteq L^{X}$ .

The pair $(X, F)$ is called a L-fuzzy co-topological space. A L-fuzzy co-topological space is said to be:

(ST) stratified if $F (α ⊙ f) \geq F (f)$ ,

(CST) co-stratified if $F (α \to f) \geq F (f)$ ,

(S) strong if it is both stratified and co-stratified,

(SE) separated if $F (⊤_{x}^{*}) = ⊤$ for all x ∈ X,

(AL) Alexandrov if $F (⋁_{i} f_{i}) \geq ⋀_{i} F (f_{i})$

∀ {f_i} _i∈Γ ⊆ L^X.

Let $(X, T)$ be a L-fuzzy topological space. Define a map $F : L^{X} \to L$ as $F (f) = T (f^{*})$ . Then $F$ is a L-fuzzy co-topology on X. Let $(X, F)$ be a L-fuzzy co-topological space. Define a map $T : L^{X} \to L$ as $T (f) = F (f^{*})$ . Then $T$ is a L-fuzzy topology on X.

Let $(X, F_{X})$ and $(Y, F_{Y})$ be two L-fuzzy co-topological spaces. A map φ : X → Y is said to be LF-continuous iff for each f ∈ L^Y,

$F_{Y} (f) \leq F_{X} (φ^{\leftarrow} (f)) .$

Definition 3.3. A map $C : L^{X} \to L^{X}$ is called an Alexandrov L-fuzzy closure operator on X if $C$ satisfies the following conditions:

(C1) $C (⊥_{X}) = ⊥_{X}$ ,

(C2) $C (f) \geq f$ for all f ∈ L^X,

(C3) If g ≤ f, then $C (g) \leq C (f)$ for all f, g ∈ L^X,

(C4) $C (⋁_{i \in Γ} f_{i}) = ⋁_{i \in Γ} C (f_{i})$ ,

(C5) $C (α ⊙ f) = α ⊙ C (f)$ for all f ∈ L^X and α ∈ L.

The pair $(X, C)$ is called an Alexandrov L-fuzzy closure space. An Alexandrov L-fuzzy closure space is called:

(T) topological if $C (C (f)) \leq C (f) \forall f \in L^{X}$ ,

(SE) separated if $C (⊤_{x}^{*}) = ⊤_{x}^{*}$ for all x, y ∈ X,

(SY) symmetric if $C (⊤_{x}) (y) = C (⊤_{y}) (x)$ for all x, y ∈ X,

(CS) (L, ⊕)-fuzzy closure space if

$C (f \oplus g) \leq C (f) \oplus C (g) .$

Let $(X, C_{X})$ and $(Y, C_{Y})$ be Alexandrov L-fuzzy closure spaces, then the map $φ : (X, C_{X}) \to (Y, C_{Y})$ is called a LF-closure map if for each f ∈ L^Y,

$C_{X} (φ^{\leftarrow} (f)) \leq φ^{\leftarrow} (C_{Y} (f)) .$

The category of Alexandrov L-fuzzy closure spaces with LF-closure maps as morphisms is denoted by AL - FCS.

Lemma 3.4. An Alexandrov L-fuzzy closure space $(X, C_{X})$ is topological if and only if

$⋁_{z \in X} C (⊤_{y}) (z) ⊙ C (⊤_{z}) (x) \leq C (⊤_{y}) (x) .$

Proof. Since f (x) = ⋁ _y∈X ⊤ _y (x) ⊙ f (y), we have

$\begin{matrix} C (f) (x) \\ = C (⋁_{y \in X} ⊤_{y} (x) ⊙ f (y)) (x) \\ = ⋁_{y \in X} C (⊤_{y} ⊙ f (y)) (x) \\ = ⋁_{y \in X} (C (⊤_{y}) (x) ⊙ f (y)) \\ \geq ⋁_{y \in X} (⋁_{z \in X} (C (⊤_{y}) (z) ⊙ C (⊤_{z}) (x) ⊙ f (y))) \\ = ⋁_{z \in X} C (⊤_{z}) (x) ⊙ (⋁_{y \in X} (C (⊤_{y}) (z) ⊙ f (y))) \\ = ⋁_{z \in X} C (⊤_{z}) (x) ⊙ C (f) (z) = C (C (f)) (x) . \end{matrix}$

Conversely,

$\begin{matrix} C (⊤_{x}) (y) & \geq C (C (⊤_{x})) (y) \\ = ⋁_{z \in X} (C (⊤_{z}) (y) ⊙ C (⊤_{x}) (z)) . \end{matrix}$

Definition 3.5. A map $I : L^{X} \to L^{X}$ is called an Alexandrov L-fuzzy interior operator on X if $I$ satisfies the following conditions:

(I1) $I (⊤_{X}) = ⊤_{X}$ ,

(I2) $I (f) \leq f$ for all f ∈ L^X,

(I3) If g ≤ f, then $I (g) \leq I (f)$ for all f, g ∈ L^X,

(I4) $I (⋀_{i \in Γ} f_{i}) = ⋀_{i \in Γ} I (f_{i})$ ,

(I5) $I (α \to f) = α \to I (f)$ for all f ∈ L^X and α ∈ L.

The pair $(X, I)$ is called an Alexandrov L-fuzzy interior space. An Alexandrov L-fuzzy interior space is called:

(T) topological if $I (I (f)) \geq I (f) \forall f \in L^{X}$ ,

(SE) separated if $I (⊤_{x}) = ⊤_{x}$ for all x ∈ X,

(SY) symmetric if $I (⊤_{x}^{*}) (y) = I (⊤_{y}^{*}) (x)$ for all x, y ∈ X,

(IS) (L, ⊙)-fuzzy interior space if

$I (f ⊙ g) \geq I (f) ⊙ I (g) .$

Let $(X, I_{X})$ and $(Y, I_{Y})$ be Alexandrov L-fuzzy interior spaces, then the map $φ : (X, I_{X}) \to (Y, I_{Y})$ is called an LF-interior map if for each g ∈ L^Y,

$φ^{\leftarrow} (I_{Y} (g)) \leq I_{X} (φ^{\leftarrow} (g)) .$

The category of Alexandrov L-fuzzy interior spaces with LF-interior maps as morphisms is denoted by AL - FIS.

Lemma 3.6. An Alexandrov L-fuzzy interior space $(X, I_{X})$ is topological if and only if

$⋀_{z \in X} (I (⊤_{x}^{*}) (z) \oplus I (⊤_{z}^{*}) (y)) \geq I (⊤_{x}^{*}) (y) .$

Proof. Since $(⊤_{z} ⊙ f (y))^{*} = f (y) \to ⊤_{y}^{*},$ we have

$\begin{matrix} f^{*} (x) & = (⋁_{y \in X} ⊤_{y} ⊙ f (y))^{*} (x) \\ = ⋀_{y \in X} (f (y) \to ⊤_{y}^{*}) (x), \end{matrix}$

then $f (x) = ⋀_{y \in X} (f^{*} (y) \to ⊤_{y}^{*}) (x)$ .

Thus, $I (f) (x) = ⋀_{y \in X} (f^{*} (y) \to I (⊤_{y}^{*}) (x))$ , and hence

$\begin{matrix} I (f) (x) \\ \leq ⋀_{y \in X} (f^{*} (y) \to ⋀_{z \in X} (I (⊤_{y}^{*})) (z) \oplus I (⊤_{z}^{*}) (x)) \\ = ⋀_{y \in X} ⋀_{z \in X} (f^{*} (y) \to (I (⊤_{y}^{*}) (z) \oplus I (⊤_{z}^{*}) (x))) \\ \leq ⋀_{y \in X} ⋀_{z \in X} ((f^{*} (y) \to I (⊤_{y}^{*}) (z))^{*} \to I (⊤_{z}^{*}) (x)) \\ (by Lemma 2.2 (14)) \\ = ⋀_{z \in X} (⋀_{y \in X} (f^{*} (y) \to I (⊤_{y}^{*}) (z))^{*} \to I (⊤_{z}^{*}) (x)) \\ = ⋀_{z \in X} (I^{*} (f) (z) \to I (⊤_{z}^{*}) (x)) = I (I (f)) (x) . \end{matrix}$

Conversely,

$\begin{matrix} I (⊤_{x}^{*}) (y) & \leq I (I (⊤_{x}^{*}) (y)) \\ = ⋀_{z \in X} (I^{*} (⊤_{x}^{*}) (z) \to I (⊤_{z}^{*}) (y)) \\ = ⋀_{z \in X} (I (⊤_{x}^{*}) (z) \oplus I (⊤_{z}^{*}) (y)) . \end{matrix}$

4 Alexandrov L-fuzzy nearness and Alexandrov L-fuzzy Pre-proximity

Definition 4.1. A mapping $N : L^{X} \times L^{X} \to L$ is called an Alexandrov L-fuzzy nearness on X if it satisfies the following axioms:

(N1) $N (⊥_{X}, ⊤_{x}) = ⊥$ ,

(N2) $N (f, ⊤_{x}) \geq f (x)$ ,

(N3) If f ≤ g, then $N (f, ⊤_{x}) \leq N (g, ⊤_{x})$ ,

(N4) $N (⋁_{i \in Γ} f_{i}, ⊤_{x}) = ⋁_{i \in Γ} N (f_{i}, ⊤_{x})$ and $N$ $(⊤_{x}, ⋁_{i \in Γ} f_{i}) = ⋁_{i \in Γ} N (⊤_{x}, f_{i})$ ,

(N5) For all α ∈ L, f ∈ L^X we have

$N (α ⊙ f, ⊤_{x}) = α ⊙ N (f, ⊤_{x}) = N (f, α ⊙ ⊤_{x})$ .

The pair $(X, N)$ is called an Alexandrov L-fuzzy near space. An Alexandrov L-fuzzy near space $(X, N)$ is called:

(1) topological if

$⋁_{y \in X} N (⊤_{x}, ⊤_{y}) ⊙ N (⊤_{y}, ⊤_{z}) \leq N (⊤_{x}, ⊤_{z})$ ,

(2) (L, ⊕)-fuzzy near space if for every f₁, f₂ ∈ L^X, we have

$N (f_{1} \oplus f_{2}, ⊤_{x}) \leq N (f_{1}, ⊤_{x}) \oplus N (f_{2}, ⊤_{x}),$

(3) (L, ⊕)-fuzzy co-near space if for every f₁, f₂ ∈ L^X, we have

$N (⊤_{x}, f_{1} \oplus f_{2}) \leq N (⊤_{x}, f_{1}) \oplus N (⊤_{x}, f_{2}),$

(4) symmetric if $N^{s} = N$ where,

$N^{s} (f, ⊤_{x}) = N (⊤_{x}, f)$ ,

(5) separated if $N (⊤_{x}^{*}, ⊤_{x}) = N (⊤_{x}, ⊤_{x}^{*}) = ⊥$ for every x ∈ X.

Let $(X, N_{X})$ and $(Y, N_{Y})$ be two Alexandrov L-fuzzy near spaces. A mapping $φ : (X, N_{X}) \to (Y, N_{Y})$ is said to be LF-near map if for every f ∈ L^Y, we have

$N_{X} (φ^{\leftarrow} (f), φ^{\leftarrow} (⊤_{φ (x)})) \leq N_{Y} (f, ⊤_{φ (x)}),$ or

$N_{X} (φ^{\leftarrow} (⊤_{φ (x)}), φ^{\leftarrow} (f))) \leq N_{Y} (⊤_{φ (x)}, f) .$

The category of Alexandrov L-fuzzy near spaces with LF-near maps as morphisms is denoted by AL - FNS.

Definition 4.2. A mapping δ : L^X × L^X → L is called an Alexandrov L-fuzzy pre-proximity on X if it satisfies the following axioms.

(P1) δ (⊤ _X, ⊥ _X) = δ (⊥ _X, ⊤ _X) = ⊥,

(P2) δ (f, g) ≥ ⋁ _x∈X (f ⊙ g) (x),

(P3) If f₁ ≤ f₂, h₁ ≤ h₂, then

δ (f₁, h₁) ≤ δ (f₂, h₂),

(P4) For every f, f_i, g_,g_i ∈ L^X we have

δ (⋁ _i∈Γf_i, g) = ⋁ _i∈Γδ (f_i, g) ,

δ (f, ⋁ _i∈Γg_i) = ⋁ _i∈Γδ (f, g_i) .

(P5) For every α ∈ L, f ∈ L^X, we have

δ (α ⊙ f, g) = α ⊙ δ (f, g) = δ (f, α ⊙ g) .

The pair (X, δ) is called an Alexandrov L-fuzzy pre-proximity space. An Alexandrov L-fuzzy pre-proximity space (X, δ) is called:

(1) (L, ⊙ , ⊕)-fuzzy pre-proximity space if for every f₁, f₂, h₁, h₂ ∈ L^X,

δ (f₁ ⊕ f₂, h₁ ⊙ h₂) ≤ δ (f₁, h₁) ⊕ δ (f₂, h₂) ,

(2) (L, ⊙ , ⊕)-fuzzy co-pre-proximity space if

δ (f₁ ⊙ f₂, h₁ ⊕ h₂) ≤ δ (f₁, h₁) ⊕ δ (f₂, h₂) .

(3) L-fuzzy quasi-proximity on X if

δ (f, g) ≥ ⋀ _h∈L^X {δ (f, h) ⊕ δ (h^*, g)} ,

(4) L-fuzzy proximity on X if δ^s = δ where,

δ^s (f, g) = δ (g, f),

(5) topological if

⋁ _y∈Xδ (⊤ _x, ⊤ _y) ⊙ δ (⊤ _y, ⊤ _z) ≤ δ (⊤ _x, ⊤ _z),

(SE) separated if $δ (⊤_{x}, ⊤_{x}^{*}) = δ (⊤_{x}^{*}, ⊤_{x}) = ⊥$ for each x ∈ X.

Let (X, δ_X) and (Y, δ_Y) be two Alexandrov L-fuzzy pre-proximity spaces. A mapping φ : (X, δ_X) → (Y, δ_Y) is said to be LF-proximity map if for every f, g ∈ L^Y, we have

δ_X (φ^← (f) , φ^← (g)) ≤ δ_Y (f, g) .

The category of Alexandrov L-fuzzy pre-proximity spaces with LF-proximity maps as morphisms is denoted by AL - FPRXS.

Remark 4.3. Let (X, δ) be an Alexandrov L-fuzzy pre-proximity space. Then

(1) For every α ∈ L, f ∈ L^X, we have δ (α ⊙ f, α → g) ≤ δ (f, g) and δ (α → f, α ⊙ g) ≤ δ (f, g) .

(2) A map $F_{δ} : L^{X} \to L$ defined by

$F_{δ} (f) = δ^{*} (f^{*}, f)$ is a strong and Alexandrov

L-fuzzy co-topology on X .

(3) A map $T_{δ} : L^{X} \to L$ defined by

$T_{δ} (f) = δ^{*} (f, f^{*})$ is a strong and Alexandrov

L-fuzzy topology on X .

(4) A map $N_{δ} : L^{X} \times L^{X} \to L$ defined by

$N_{δ} (f, ⊤_{x}) = δ (f, ⊤_{x})$ is an Alexandrov L-fuzzy near on X .

From the following Theorem, we obtain Alexandrov L-fuzzy pre-proximity on X by an Alexandrov L-fuzzy nearness on X.

Theorem 4.4. Let $N$ be an Alexandrov L-fuzzy nearness on X. Define $δ_{N} : L^{X} \times L^{X} \to L$ as

$δ_{N} (f, g) = ⋁_{x \in X} g (x) ⊙ N (f, ⊤_{x}) \forall f, g \in L^{X} .$

Then, we have the following properties:

(1) $(X, δ_{N})$ is an Alexandrov L-fuzzy quasi-proximity space,

(2) If $(X, N)$ is (L, ⊕)-fuzzy near space, then $(X, δ_{N})$ is (L, ⊙ , ⊕)-fuzzy pre-proximity,

(3) $δ_{N} (f, g) \leq ⋀_{h \in L^{X}} δ_{N} (f, h) \oplus δ_{N} (h^{*}, g),$

(4) If $N$ is topological, then $δ_{N}$ is topological,

(5) If $N$ is separated, then $δ_{N}$ is separated.

(6) $δ_{N_{δ}} = δ, N_{δ_{N}} \geq N$ .

Proof. (1) (P1)

$\begin{matrix} δ_{N} (⊤_{X}, ⊥_{X}) = ⋁_{x \in X} N (⊤_{X}, ⊤_{x}) ⊙ ⊥_{X} (x) = ⊥, \\ δ_{N} (⊥_{X}, ⊤_{X}) = ⋁_{x \in X} N (⊥_{X}, ⊤_{x}) ⊙ ⊤_{X} (x) = ⊥, \end{matrix}$

(P2)

$\begin{matrix} δ_{N} (f, g) & = ⋁_{x \in X} N (f, ⊤_{x}) ⊙ g (x) \\ \geq ⋁_{x \in X} f (x) ⊙ g (x), \end{matrix}$

(P3) If f₁ ≤ f₂, h₁ ≤ h₂, then

$\begin{matrix} δ_{N} (f_{1}, h_{1}) & = ⋁_{x \in X} N (f_{1}, ⊤_{x}) ⊙ h_{1} (x) \\ \geq ⋁_{x \in X} N (f_{2}, ⊤_{x}) ⊙ h_{2} (x) \\ = δ_{N} (f_{2}, h_{2}), \end{matrix}$

(P4) For each f, f_i, g, g_i ∈ L^X we have

$\begin{matrix} δ_{N} (⋁_{i \in Γ} f_{i}, g) & = ⋁_{x \in X} N (⋁_{i \in Γ} f_{i}, ⊤_{x}) ⊙ g (x) \\ = ⋁_{i \in Γ} ⋁_{x \in X} N (f_{i}, ⊤_{x}) ⊙ g (x) \\ = ⋁_{i \in Γ} δ_{N} (f_{i}, g), \\ δ_{N} (f, ⋁_{i \in Γ} g_{i}) & = ⋁_{x \in X} N (f, ⊤_{x}) ⊙ ⋁_{i \in Γ} g_{i} (x) \\ = ⋁_{i \in Γ} ⋁_{x \in X} N (f, ⊤_{x}) ⊙ g_{i} (x) \\ = ⋁_{i \in Γ} δ_{N} (f, g_{i}), \end{matrix}$

(P5) For every α ∈ L, f ∈ L^X, we have

$\begin{matrix} δ_{N} (α ⊙ f, g) & = ⋁_{x \in X} N (α ⊙ f, ⊤_{x}) ⊙ g (x) \\ = α ⊙ ⋁_{x \in X} N (f, ⊤_{x}) ⊙ g (x) \\ = α ⊙ δ_{N} (f, g), \\ δ_{N} (f, α ⊙ g) & = ⋁_{x \in X} N (f, ⊤_{x}) ⊙ (α ⊙ g) (x) \\ = ⋁_{x \in X} N (f, ⊤_{x}) ⊙ α ⊙ g (x) \\ = α ⊙ ⋁_{x \in X} N (f, ⊤_{x}) ⊙ g (x) \\ = α ⊙ δ_{N} (f, g) . \end{matrix}$

(2)

$\begin{matrix} δ_{N} (f_{1} \oplus f_{2}, g_{1} ⊙ g_{2}) \\ = ⋁_{x \in X} (g_{1} ⊙ g_{2}) (x) ⊙ N (f_{1} \oplus f_{2}, ⊤_{x}) \\ \leq ⋁_{x \in X} (g_{1} (x) ⊙ g_{2} (x)) \\ ⊙ (N (f_{1}, ⊤_{x}) \oplus N (N (f_{2}, ⊤_{x}))) \\ \leq (⋁_{x \in X} g_{1} (x) ⊙ N (f_{1}, ⊤_{x})) \\ \oplus (⋁_{x \in X} g_{2} (x) ⊙ N (f_{2}, ⊤_{x})) \\ = δ_{N} (f_{1}, g_{1}) \oplus δ_{N} (f_{2}, g_{2}) . \end{matrix}$

(3)

$\begin{matrix} δ_{N}^{*} (f, h) ⊙ δ_{N}^{*} (h^{*}, g) \\ = ⋀_{x \in X} (h (x) \to N^{*} (f, ⊤_{x})) \\ ⊙ ⋀_{x \in X} (g (x) \to N^{*} (h^{*}, ⊤_{x})) \end{matrix}$

$\begin{matrix} \leq ⋀_{x \in X} (h (x) \to N^{*} (f, ⊤_{x})) ⊙ (g (x) \to h (x)) \\ \leq ⋀_{x \in X} (g (x) \to N^{*} (f, ⊤_{x})) \\ = (⋁_{x \in X} g (x) ⊙ N (f, ⊤_{x}))^{*} = δ_{N}^{*} (f, g) . \end{matrix}$

(4)

$\begin{matrix} ⋁_{y \in X} δ_{N} (⊤_{x}, ⊤_{y}) ⊙ δ_{N} (⊤_{y}, ⊤_{z}) \\ = ⋁_{y \in X} (⋁_{y \in X} ⊤_{y} (y) ⊙ N (⊤_{x}, ⊤_{y})) \\ ⊙ (⋁_{z \in X} ⊤_{z} (z) ⊙ N (⊤_{y}, ⊤_{z})) \\ = ⋁_{y, z \in X} N (⊤_{x}, ⊤_{y}) ⊙ N (⊤_{y}, ⊤_{z}) \\ \leq ⋁_{z \in X} N (⊤_{x}, ⊤_{z}) \\ = ⋁_{z \in X} ⊤_{z} (z) ⊙ N (⊤_{x}, ⊤_{z}) = δ_{N} (⊤_{x}, ⊤_{z}) . \end{matrix}$

(5)

$δ_{N} (⊤_{x}^{*}, ⊤_{x}) = ⋁_{y \in X} ⊤_{x} (y) ⊙ N (⊤_{x}^{*}, ⊤_{x}) = ⊥$ .

(6)

$\begin{matrix} δ_{N_{δ}} (f, g) & = ⋁_{x \in X} g (x) ⊙ N_{δ} (f, ⊤_{x}) \\ = ⋁_{x \in X} g (x) ⊙ δ (f, ⊤_{x}) \\ = δ (f, ⋁_{x \in X} g (x) ⊙ ⊤_{x}) = δ (f, g), \end{matrix}$

$\begin{matrix} N_{δ_{N}} (f, ⊤_{x}) & = δ_{N} (f, ⊤_{x}) \\ = ⋁_{x \in X} ⊤_{x} (x) ⊙ N (f, ⊤_{x}) \\ \geq N (f, ⊤_{x}) . \end{matrix}$

Corollary 4.5 Let $N$ be an Alexandrov L-fuzzy nearness on X. Define $δ_{N} : L^{X} \times L^{X} \to L$ as

$δ_{N} (f, g) = ⋁_{x \in X} f (x) ⊙ N (⊤_{x}, g) \forall f, g \in L^{X} .$

Then, we have the following properties:

(1) $(X, δ_{N})$ is an Alexandrov L-fuzzy pre-proximity space,

(2) $δ_{N} (f, g) \leq ⋀_{h \in L^{X}} δ_{N} (f, h) \oplus δ_{N} (h^{*}, g),$

(3) If $(X, N)$ is (L, ⊕)-fuzzy co-near space, then $(X, δ_{N})$ is (L, ⊙ , ⊕)-fuzzy co-pre-proximity,

(4) If $N$ is topological, then $δ_{N}$ is topological,

(5) If $N$ is separated, then $δ_{N}$ is separated

(6) $δ_{N_{δ}} = δ, N_{δ_{N}} \geq N$ .

Theorem 4.6. Let $(X, N_{X})$ and $(Y, N_{Y})$ be two Alexandrov L-fuzzy near spaces and $φ : (X, N_{X}) \to (Y, N_{Y})$ be a LF-near map. Then, $φ : (X, δ_{N_{X}}) \to (Y, δ_{N_{Y}})$ is a LF-proximity map.

Proof. $\begin{matrix} δ_{N_{X}} (φ^{\leftarrow} (f), φ^{\leftarrow} (g)) \\ = ⋁_{x \in X} φ^{\leftarrow} (g) (x) ⊙ N_{X} (φ^{\leftarrow} (f), ⊤_{x}) \\ = ⋁_{x \in X} φ^{\leftarrow} (g) (x) ⊙ N_{X} (φ^{\leftarrow} (f), φ^{\leftarrow} (⊤_{φ (x)})) \\ \leq ⋁_{φ (x) \in Y} g (φ (x)) ⊙ N_{Y} (f, ⊤_{φ (x)}) = δ_{N_{Y}} (f, g) . \end{matrix}$

Theorem 4.7. (1) ϒ : AL - FNS → AL - FPRXS defined as $ϒ (X, N) = (X, δ_{N})$ and ϒ (φ) = φ is a concrete functor,

(2) Φ : AL - FPRXS → AL - FNS defined as $Φ (X, δ) = (X, N_{δ})$ and Φ (φ) = φ is a concrete functor,

(3) The pair (ϒ, Φ) is a Galois correspondence between AL - FNS and AL - FPRXS.

Proof. (1) Follows from Theorems 4.4 and 4.6 and (2) Follows from Remark 4.3 and Definitions 4.1, 4.2.

(3) By Theorem 4.4 (6), if (X, δ) is an Alexandrov L-fuzzy pre-proximity space, then

$\begin{matrix} (ϒ \circ Φ) (X, δ) & = ϒ (Φ (X, δ)) \\ = ϒ (X, N_{δ}) = (X, δ_{N_{δ}}) = (X, δ) . \end{matrix}$

Hence, the identity map ϒ ∘ Φ = id_δ is a LF-proximity map.

By Theorem 4.4 (6), if $(X, N)$ is an Alexandrov L-fuzzy near space, then

$\begin{matrix} (Φ \circ ϒ) (X, N) & = Φ (ϒ (X, N)) \\ = Φ (X, δ_{N}) = (X, N_{δ_{N}}) \geq (X, N) . \end{matrix}$

Hence, the identity map $Φ \circ ϒ = {id}_{N}$ is a LF-near map. Therefore, the pair (ϒ, Φ) is a Galois correspondence.

Example 4.8. (1) Let X be a set and $R \in L^{X \times X}$ be a L-fuzzy pre-order. Define $N_{R} : L^{X} \times L^{X} \to L$ as

$N_{R} (f, ⊤_{x}) = ⋁_{x, y \in X} R (x, y) ⊙ f (y) \forall f \in L^{X} .$

(N1) and (N3) are easily proved.

(N2) For all f ∈ L^X,

$\begin{matrix} N_{R} (f, ⊤_{x}) & = ⋁_{x, y \in X} R (x, y) ⊙ f (y) \\ \geq ⋁_{x \in X} R (x, x) ⊙ f (x) \\ = ⋁_{x \in X} f (x) \geq f (x) . \end{matrix}$

(N4) and (N5) are easily proved. Hence, $N_{R}$ is Alexandrov L-fuzzy near on X.

If $R$ is an L-fuzzy pre-order on X, then

$\begin{matrix} ⋁_{y \in X} N_{R} (⊤_{x}, ⊤_{y}) ⊙ N_{R} (⊤_{y}, ⊤_{z}) \\ = ⋁_{y \in X} (⋁_{y, z \in X} R (y, z) ⊙ ⊤_{x} (y)) \\ ⊙ (⋁_{z, w \in X} R (z, w) ⊙ ⊤_{y} (z)) \\ = ⋁_{y, z, w \in X} (R (y, z) ⊙ R (z, w)) \\ ⊙ (⊤_{x} (y) ⊙ ⊤_{y} (z)) \\ \leq ⋁_{y, z, w \in X} R (y, w) ⊙ (⊤_{x} (y) ⊙ ⊤_{y} (z)) \\ = ⋁_{z, w \in X} R (z, w) ⊙ ⊤_{x} (z) = N_{R} (⊤_{x}, ⊤_{z}) . \end{matrix}$

Hence, $N_{R}$ is topological.

(2) Let $N$ be an Alexandrov L-fuzzy nearness on X. Define a map $R_{N} : X \times X \to L$ as

$R_{N} (x, y) = N (⊤_{x}, ⊤_{y}) \forall x, y \in X .$

Then, $R_{N}$ is a reflexive L-fuzzy relation on X. Moreover, If $N$ is topological, then

$\begin{matrix} R_{N} (x, z) & = N (⊤_{x}, ⊤_{z}) \\ \geq ⋁_{y \in X} N (⊤_{x}, ⊤_{y}) ⊙ N (⊤_{y}, ⊤_{z}) \\ \geq R_{N} (x, y) ⊙ R_{N} (y, z) . \end{matrix}$

Hence, $R_{N}$ is a L-fuzzy pre-order on X. Also, if $N$ is symmetric, then

$\begin{matrix} R_{N} (x, y) = N (⊤_{x}, ⊤_{y}) & = N^{s} (⊤_{y}, ⊤_{x}) \\ = R_{N^{s}} (y, x) . \end{matrix}$

Hence, $R_{N}$ is an L-fuzzy equivalence relation on X.

(3) From Theorem 4.4, define $δ_{N_{R}} : L^{X} \times L^{X} \to L$ as

$\begin{matrix} δ_{N_{R}} (f, g) & = ⋁_{x \in X} g (x) ⊙ N_{R} (f, ⊤_{x}) \\ = ⋁_{x, y \in X} R (x, y) ⊙ f (y) ⊙ g (x) . \end{matrix}$

(P1) and (P3) are easily proved.

(P2) For all f, g ∈ L^X,

$\begin{matrix} δ_{N_{R}} (f, g) & = ⋁_{x, y \in X} R (x, y) ⊙ f (y) ⊙ g (x) \\ \geq ⋁_{x \in X} R (x, x) ⊙ f (x) ⊙ g (x) \\ = ⋁_{x \in X} f (x) ⊙ g (x) . \end{matrix}$

(P4) For all f₁, f₂, h₁, h₂ ∈ L^X, by Lemma 2.2 (2) and (17), we have

$\begin{matrix} δ_{N_{R}} (f_{1}, h_{1}) \oplus δ_{N_{R}} (f_{2}, h_{2}) \\ = (⋁_{x, y \in X} R (x, y) ⊙ f_{1} (y) ⊙ h_{1} (x)) \oplus \\ (⋁_{z, w \in X} R (z, w) ⊙ f_{2} (w) ⊙ h_{2} (z)) \\ \geq ⋁_{x, y, z, w \in X} (R (x, y) ⊙ R (z, w)) \\ ⊙ (f_{1} (y) ⊙ f_{2} (w)) ⊙ (h_{1} (x) \oplus h_{2} (z)) \\ \geq ⋁_{x, y, z \in X} (R (x, y) ⊙ R (y, z)) \\ ⊙ (f_{1} (z) ⊙ f_{2} (z)) ⊙ (h_{1} (x) \oplus h_{2} (x)) \\ = ⋁_{x, z \in X} (⋁_{y \in X} R (x, y) ⊙ R (y, z)) \\ ⊙ (f_{1} (z) ⊙ f_{2} (z)) ⊙ (h_{1} (x) \oplus h_{2} (x)) \\ = ⋁_{x, z \in X} R (x, z) ⊙ (f_{1} ⊙ f_{2}) (z) ⊙ \\ (h_{1} \oplus h_{2}) (x) = δ_{N_{R}} (f_{1} ⊙ f_{2}, h_{1} \oplus h_{2}) . \end{matrix}$

Hence, $δ_{N_{R}}$ is a L-fuzzy pre-proximity on X.

Since

$\begin{matrix} δ_{N_{R}} (α ⊙ f, g) \\ = ⋁_{x, y \in X} R (x, y) ⊙ (α ⊙ f) (y) ⊙ g (x) \\ = α ⊙ ⋁_{x, y \in X} R (x, y) ⊙ f (y) ⊙ g (x) \\ = α ⊙ δ_{N_{R}} (f, g) \end{matrix}$

and $δ_{N_{R}} (⋁_{i \in Γ} f_{i}, g) = ⋁_{i \in Γ} δ_{N_{R}} (f_{i}, g)$ , then $δ_{N_{R}}$ is Alexandrov. We give two special cases for the L-fuzzy pre-order $R$ .

(a) Let $R_{1} = ⊤_{X \times X}$ . Then,

$δ_{N_{R_{1}}} (f, g) = ⋁_{x, y \in X} f (x) ⊙ g (y)$ .

Since $δ_{N_{R_{1}}} (⊤_{x}, ⊤_{x}^{*}) = ⊤$ , then $δ_{N_{R_{1}}}$ is not separated.

(b) Let $R_{2} = ▵_{X \times X}$ , where

$▵_{X \times X} (x, y) = {\begin{matrix} ⊤, & if y = x, \\ ⊥, & otherwise . \end{matrix}$

Then, $δ_{N_{R_{2}}} (f, g) = ⋁_{x \in X} f (x) ⊙ g (x) .$ Hence, $δ_{N_{R_{2}}}$ is a L-fuzzy pre-proximity on X. Moreover,

(Q) For all f, g ∈ L^X,

$\begin{matrix} ⋀_{h \in L^{X}} (δ_{N_{R_{2}}} (f, h) \oplus δ_{N_{R_{2}}} (h^{*}, g)) \\ = ⋀_{h \in L^{X}} ((⋁_{x \in X} f (x) ⊙ h (x)) \\ \oplus (⋁_{x \in X} h^{*} (x) ⊙ g (x))) (Put h = g) \\ \leq ⋁_{x \in X} (f (x) ⊙ g (x)) \oplus ⋁_{x \in X} (g^{*} (x) ⊙ g (x)) \\ = ⋁_{x \in X} (f (x) ⊙ g (x)) \oplus ⊥ = δ_{N_{R_{2}}} (f, g) . \end{matrix}$

Hence, $δ_{N_{R_{2}}}$ is a L-fuzzy proximity on X. Since $δ_{N_{R_{2}}} (⊤_{x}, ⊤_{x}^{*}) = ⊥$ , then $δ_{N_{R_{2}}}$ is separated. Hence, $δ_{N_{R_{2}}}$ is separated and Alexandrov.

5 Alexandrov L-fuzzy nearness and Alexandrov L-fuzzy closure space

Theorem 5.1. Let $(X, N)$ be an Alexandrov L-fuzzy near space. Define $C_{N} : L^{X} \to L^{X}$ as

$C_{N} (f) (x) = N (f, ⊤_{x}) \forall f \in L^{X} .$

Then,

(1) $C_{N}$ is an Alexandrov L-fuzzy closure space such that $C_{N} (f) (x) = ⋁_{z \in X} f (z) ⊙ N (⊤_{z}, ⊤_{x}),$

(2) If $N$ is separated, then $C_{N}$ is separated,

(3) If $N$ is symmetric, then $C_{N}$ is symmetric,

(4) If $N$ is topological, then $C_{N}$ is topological,

(5) $C_{N} (f) (x) = ⋁_{y \in X} f (y) ⊙ C_{N} (⊤_{y}) (x)$ ,

(6) If $(X, N)$ is (L, ⊕)-fuzzy near space, then $(X, C_{N})$ is (L, ⊕)-fuzzy closure space.

Proof. (1) (C1) $C_{N} (⊥_{X}) (x) = N (⊥_{X}, ⊤_{x}) = ⊥_{X}$ ,

(C2) $C_{N} (f) (x) = N (f, ⊤_{x}) \geq f (x)$ ,

(C3) If f ≤ g, then

$C_{N} (f) (x) = N (f, ⊤_{x}) \leq N (g, ⊤_{x}) = C_{N} (g) (x)$ ,

(C4)

$\begin{matrix} C_{N} (⋁_{i \in Γ} f_{i}) (x) & = N (⋁_{i \in Γ} f_{i}, ⊤_{x}) \\ = ⋁_{i \in Γ} N (f_{i}, ⊤_{x}) \\ = ⋁_{i \in Γ} C_{N} (f_{i}) (x), \end{matrix}$

(C5) For every α ∈ L, f ∈ L^X, we have

$\begin{matrix} C_{N} (α ⊙ f) (x) & = N (α ⊙ f, ⊤_{x}) \\ = α ⊙ N (f, ⊤_{x}) = α ⊙ C_{N} (f) (x) . \end{matrix}$

$\begin{matrix} C_{N} (f) (x) & = N (f, ⊤_{x}) \\ = N (⋁_{z \in X} f (z) ⊙ ⊤_{z}, ⊤_{x}) \\ = ⋁_{z \in X} f (z) ⊙ N (⊤_{z}, ⊤_{x}) . \end{matrix}$

(2) If $N$ is separated, then

$C_{N} (⊤_{x}^{*}) (x) = N (⊤_{x}^{*}, ⊤_{x}) = ⊥ = ⊤_{x}^{*} .$

(3) if $N$ is symmetric, then

$\begin{matrix} C_{N} (⊤_{x}) (y) & = N (⊤_{x}, ⊤_{y}) \\ = N (⊤_{y}, ⊤_{x}) = C_{N} (⊤_{y}) (x) . \end{matrix}$

(4) If $N$ is topological, then

$\begin{matrix} C_{N} (f) (x) = N (f, ⊤_{x}) \\ = N (⋁_{z \in X} f (z) ⊙ ⊤_{z}, ⊤_{x}) \end{matrix}$

$\begin{matrix} = ⋁_{z \in X} f (z) ⊙ N (⊤_{z}, ⊤_{x}) \\ \geq ⋁_{z \in X} f (z) ⊙ (⋁_{y \in X} N (⊤_{z}, ⊤_{y}) ⊙ N (⊤_{y}, ⊤_{x})) \\ = ⋁_{y \in X} (⋁_{z \in X} f (z) ⊙ N (⊤_{z}, ⊤_{y})) ⊙ N (⊤_{y}, ⊤_{x}) \\ = ⋁_{y \in X} N (f, ⊤_{y}) ⊙ N (⊤_{y}, ⊤_{x}) \\ = ⋁_{y \in X} C_{N} (f) (y) ⊙ N (⊤_{y}, ⊤_{x}) \\ = N (C_{N} (f), ⊤_{x}) = C_{N} (C_{N} (f)) (x) . \end{matrix}$

Hence, $C_{N} (C_{N} (f)) \leq C_{N} (f) .$

(5) Since f (x) = ⋁ _y∈Xf (y) ⊙ ⊤ _y (x), we have

$\begin{matrix} C_{N} (f) (x) & = C_{N} (⋁_{y \in X} f (y) ⊙ ⊤_{y}) (x) \\ = ⋁_{y \in X} f (y) ⊙ C_{N} (⊤_{y}) (x) . \end{matrix}$

(6)

$\begin{matrix} C_{N} (f \oplus g) (x) & = N (f \oplus g, ⊤_{x}) \\ \leq N (f, ⊤_{x}) \oplus N (g, ⊤_{x}) \\ = C_{N} (f) (x) \oplus C_{N} (g) (x) . \end{matrix}$

Theorem 5.2. Let $(X, C)$ be an Alexandrov L-fuzzy closure space. Define a map $δ_{C} : L^{X} \times L^{X} \to L$ by

$δ_{C} (f, g) = ⋁_{x \in X} f (x) ⊙ C (g) (x) \forall f, g \in L^{X} .$

Then, we have the following properties.

(1) $δ_{C}$ is an Alexandrov L-fuzzy pre-proximity such that $δ_{C} (f, g) = ⋁_{x, y \in X} C (⊤_{y}) (x) ⊙ f (x) ⊙ g (y),$

(2) If $(X, C)$ is (L, ⊕)-fuzzy closure space, then $(X, δ_{C_{N}})$ is (L, ⊙ , ⊕)-fuzzy pre-co-proximity space,

(3) $δ_{C} (f, g) \leq ⋁_{h \in L^{X}} {δ_{C} (f, h) ⊙ δ_{C} (h^{*}, g)}$ , the equality holds if $C$ is topological,

(4) If $C$ is topological, then $δ_{C}$ is an L-fuzzy quasi-proximity on X,

(5) If $C$ is separated, then $δ_{C}$ is separated.

Proof. (1) (P1) Since $C (⊥_{X}) = ⊥_{X}$ and $C (⊤_{X}) = ⊤_{X}$ , then we have

$\begin{matrix} δ_{C} (⊤_{X}, ⊥_{X}) = ⋁_{x \in X} ⊤_{X} (x) ⊙ C (⊥_{X}) (x) = ⊥ \\ δ_{C} (⊥_{X}, ⊤_{X}) = ⋁_{x \in X} ⊥_{X} (x) ⊙ C (⊤_{X}) (x) = ⊥ . \end{matrix}$

(P2) Since $C (f) \geq f$ , we have

$\begin{matrix} δ_{C} (f, g) & = ⋁_{x \in X} f (x) ⊙ C (g) (x) \\ \geq ⋁_{x \in X} f (x) ⊙ g (x) . \end{matrix}$

(P3) If f ≤ f₁ and g ≤ g₁, then $C (g) \leq C (g_{1})$ . Thus,

$\begin{matrix} δ_{C} (f, g) & = ⋁_{x \in X} f (x) ⊙ C (g) (x) \\ \leq ⋁_{x \in X} f_{1} (x) ⊙ C (g_{1}) (x) = δ_{C} (f_{1}, g_{1}) . \end{matrix}$

(P4) It is easily proved

(P5)

$\begin{matrix} δ_{C} (f, α ⊙ g) & = ⋁_{x \in X} f (x) ⊙ C (α ⊙ g) (x) \\ = ⋁_{x \in X} f (x) ⊙ α ⊙ C (g) (x) \\ = α ⊙ ⋁_{x \in X} f (x) ⊙ C (g) (x) \\ = α ⊙ δ_{C} (f, g), \\ δ_{C} (α ⊙ f, g) & = α ⊙ δ_{C} (f, g), \end{matrix}$

$\begin{matrix} δ_{C} (α ⊙ f, α \to g) \\ = ⋁_{x \in X} (α ⊙ f) (x) ⊙ C (α \to g) (x) \\ \leq ⋁_{x \in X} α ⊙ f (x) ⊙ (α \to C (g) (x)) \\ \leq ⋁_{x \in X} f (x) ⊙ C (g) (x) = δ_{C} (f, g) . \end{matrix}$

Hence, $δ_{C}$ is an Alexandrov L-fuzzy pre-proximity on X.

Since g = ⋁ _y∈Xg (y) ⊙ ⊤ _y, then we have

$\begin{matrix} δ_{C} (f, g) = ⋁_{x \in X} f (x) ⊙ C (g) (x) \\ = ⋁_{x \in X} f (x) ⊙ C (⋁_{y \in X} g (y) ⊙ ⊤_{y}) (x) \\ = ⋁_{x, y \in X} C (⊤_{y}) (x) ⊙ f (x) ⊙ g (y) . \end{matrix}$

(2)

$\begin{matrix} δ_{C} (f_{1}, g_{1}) \oplus δ_{C} (f_{2}, g_{2}) \\ = (⋁_{x \in X} f_{1} (x) ⊙ C (g_{1}) (x)) \\ \oplus (⋁_{x \in X} f_{2} (x) ⊙ C (g_{2}) (x)) \\ \geq ⋁_{x \in X} (f_{1} (x) ⊙ C (g_{1}) (x)) \oplus (f_{2} (x) ⊙ C (g_{2}) (x)) \\ (by Lemma 2.2 (13)) \\ \geq ⋁_{x \in X} (f_{1} (x) ⊙ f_{2} (x)) ⊙ (C (g_{1}) (x) \oplus C (g_{2}) (x)) \\ \geq ⋁_{x \in X} (f_{1} (x) ⊙ f_{2} (x)) ⊙ (C (g_{1} \oplus g_{2}) (x)) \\ = δ_{C} (f_{1} ⊙ f_{2}, g_{1} \oplus g_{2}) . \end{matrix}$

(3) Since $C^{*} (h) \leq h^{*}$ , then we have

$\begin{matrix} δ_{C}^{*} (f, h) ⊙ δ_{C}^{*} (h^{*}, g) & = (⋁_{x \in X} f (x) ⊙ C (h) (x))^{*} \\ ⊙ (⋁_{x \in X} h^{*} (x) ⊙ C (g) (x))^{*} \\ = S (f, C^{*} (h)) ⊙ S (h^{*}, C^{*} (g)) \\ \leq S (f, h^{*}) ⊙ S (h^{*}, C^{*} (g)) \\ \leq S (f, C^{*} (g)) = δ_{C}^{*} (f, g) . \end{matrix}$

Hence, $δ_{C} (f, g) \leq ⋀_{h \in L^{X}} {δ_{C} (f, h) \oplus δ_{C} (h^{*}, g)}$ .

If $C$ is topological, then

$\begin{matrix} ⋁_{h \in L^{X}} (δ_{C}^{*} (f, h) ⊙ δ_{C}^{*} (h^{*}, g)) \\ = ⋁_{h \in L^{X}} S (f, C^{*} (h)) ⊙ S (h^{*}, C^{*} (g)) \\ (put C (g) = h) \\ \geq S (f, C^{*} (C (g))) ⊙ S (C^{*} (g), C (g^{*})) \\ = S (f, C^{*} (g)) = δ_{C}^{*} (f, g) . \end{matrix}$

(4) By (3), it is trivial.

(5) $\begin{matrix} δ_{C_{δ}}^{*} (⊤_{x}^{*}, ⊤_{x}^{*}) = ⋀_{x \in X} (C_{δ} (⊤_{x}^{*}) (x) \to ⊤_{x}^{*} (x)) = ⊤ \end{matrix}$ .

Theorem 5.3. Let $(X, N_{X})$ and $(Y, N_{Y})$ be two Alexandrov L-fuzzy near spaces and φ : X → Y be a map. Then,

(1)

$\begin{matrix} S (C_{N_{X}} (φ^{\leftarrow} (f)), φ^{\leftarrow} (C_{N_{Y}} (f))) \\ = ⋀_{x \in X} (N_{X} (φ^{\leftarrow} (f), φ^{\leftarrow} (⊤_{φ (x)})) \to N_{Y} (f, ⊤_{φ (x)})), \end{matrix}$

(2) If $φ : X \to Y, N_{Y})$ is a LF-near map, then, $φ : (X, C_{N_{X}}) \to (Y, C_{N_{Y}})$ is a LF-closure map.

Proof.

(1)

$\begin{matrix} S (C_{N_{X}} (φ^{\leftarrow} (f)), φ^{\leftarrow} (C_{N_{Y}} (f))) \\ = ⋀_{x \in X} (C_{N_{X}} (φ^{\leftarrow} (f) (x) \to φ^{\leftarrow} (C_{N_{Y}} (f)) (x)) \end{matrix}$

$\begin{matrix} = ⋀_{x \in X} (C_{N_{X}} (φ^{\leftarrow} (f) (x) \to C_{N_{Y}} (f) (φ (x)) \\ = ⋀_{x \in X} (N_{X} (φ^{\leftarrow} (f), φ^{\leftarrow} (⊤_{φ (x)})) \to N_{Y} (f, ⊤_{φ (x)})) . \end{matrix}$

(2) Since $N_{X} (φ^{\leftarrow} (f), φ^{\leftarrow} (⊤_{φ (x)}) \leq N_{Y} (f, ⊤_{φ (x)})$ , then by (1) we have $C_{N_{X}} (φ^{\leftarrow} (f) \leq φ^{\leftarrow} (C_{N_{Y}} (f)) .$

Theorem 5.4. Let $C$ be an Alexandrov L-fuzzy closure space. Define a map $N_{C} : L^{X} \times L^{X} \to L$ by

$N_{C} (f, ⊤_{x}) = ⋁_{x \in X} C (f) (x) \forall f \in L^{X} .$

Then, we have the following properties:

(1) $N_{C}$ is an Alexandrov L-fuzzy nearness on X such that $N_{C} (f, ⊤_{x}) = ⋁_{x, y \in X} f (y) ⊙ C (⊤_{y}) (x),$

(2) If $C$ is topological, then $N_{C}$ is topological,

(3) If $C$ is separated, then $N_{C}$ is separated,

(4) If $C$ is symmetric, then $N_{C}$ is symmetric,

(5) If $(X, C)$ is (L, ⊕)-fuzzy closure space, then $(X, N_{C})$ is (L, ⊕)-fuzzy near space,

(6) $N_{C_{N}} \geq N$ and $C_{N_{C}} \geq C .$

Proof. (1)

(N1) $N_{C} (⊥_{X}, ⊤_{x}) = ⋁_{x \in X} C (⊥_{X}) (x) = ⊥$ ,

(N2)

$N_{C} (f, ⊤_{x}) = ⋁_{x \in X} C (f) (x) \geq C (f) (x) \geq f (x),$

(N3) If f ≤ g, then

$\begin{matrix} N_{C} (f, ⊤_{x}) & = ⋁_{x \in X} C (f) (x) \\ \leq ⋁_{x \in X} C (g) (x) = N_{C} (g, ⊤_{x}), \end{matrix}$

(N4)

$\begin{matrix} N_{C} (⋁_{i \in Γ} f_{i}, ⊤_{x}) & = ⋁_{x \in X} C (⋁_{i \in Γ} f_{i}) (x) \\ = ⋁_{x \in X} ⋁_{i \in Γ} C (f_{i}) (x) \\ = ⋁_{i \in Γ} ⋁_{x \in X} C (f_{i}) (x) \\ = ⋁_{i \in Γ} N_{C} (f_{i}, ⊤_{x}), \end{matrix}$

(N5) For every α ∈ L, f ∈ L^X, we have

$\begin{matrix} N_{C} (α ⊙ f, ⊤_{x}) & = ⋁_{x \in X} C (α ⊙ f) (x) \\ = ⋁_{x \in X} α ⊙ C (f) (x) \\ = α ⊙ ⋁_{x \in X} C (f) (x) \\ = α ⊙ N_{C} (f, ⊤_{x}) . \end{matrix}$

$\begin{matrix} N_{C} (f, ⊤_{x}) & = ⋁_{x \in X} C (f) (x) \\ = ⋁_{x \in X} C (⋁_{y \in X} f (y) ⊙ ⊤_{y}) (x) \\ = ⋁_{x, y \in X} f (y) ⊙ C (⊤_{y}) (x) . \end{matrix}$

(2) Easily proved.

(3)

$\begin{matrix} N_{C} (⊤_{x}^{*}, ⊤_{x}) & = ⋁_{x \in X} C (⊤_{x}^{*}) (x) \\ = ⊥ = N_{C} (⊤_{x}, ⊤_{x}^{*}) . \end{matrix}$

(4)

$\begin{matrix} N_{C}^{s} (⊤_{y}, ⊤_{x}) & = ⋁_{x \in X} C (⊤_{y}) (x) \\ = ⋁_{x \in X} C (⊤_{x}) (y) = N_{C} (⊤_{x}, ⊤_{y}) . \end{matrix}$

(5)

$\begin{matrix} N_{C} (f \oplus g, ⊤_{x}) \\ = ⋁_{x \in X} C (f \oplus g) (x) \end{matrix}$

$\begin{matrix} \leq (⋁_{x \in X} C (f) (x)) \oplus (⋁_{x \in X} C (g) (x)) \\ = N_{C} (f, ⊤_{x}) \oplus N_{C} (g, ⊤_{x}) . \end{matrix}$

(6)

$\begin{matrix} N_{C_{N}} (f, ⊤_{x}) & = ⋁_{x \in X} C_{N} (⊤_{y}) (x) \\ = ⋁_{x \in X} N (f, ⊤_{x}) \geq N (f, ⊤_{x}), \end{matrix}$ $C_{N_{C}} (f) (x) = N_{C} (f, ⊤_{x}) = ⋁_{x \in X} C (f) (x) \geq C (f) (x) .$

Theorem 5.5. Let $(X, N)$ be an Alexandrov (L, ⊕)-fuzzy near space. Define the mapping $F_{N} : L^{X} \to L$ by: $F_{N} (f) = ⋀_{x \in X} (N (f,, ⊤_{x}) \to f (x)) \forall f \in L^{X} .$ Then,

(1) $F_{N}$ is a strong Alexandrov L-fuzzy co-topology on X with $F_{δ_{N}} (f^{*}) = F_{N} (f),$

(2) $F_{N}$ is separated if and only if $N$ is separated.

Proof. (1) (CT1)

$\begin{matrix} F_{N} (⊥_{X}) = ⋀_{x \in X} (N (⊥_{X}, ⊤_{x}) \to ⊥) = ⊤, \\ F_{N} (⊤_{X}) = ⋀_{x \in X} (N (⊤_{X}, ⊤_{x}) \to ⊤) = ⊤ . \end{matrix}$

(CT2)

$\begin{matrix} F_{N} (⋀_{i \in Γ} f_{i}) \\ = ⋀_{x \in X} (N (⋀_{i \in Γ} f_{i}, ⊤_{x}) \to ⋀_{i \in Γ} f_{i} (x)) \\ = ⋀_{i \in Γ} ⋀_{x \in X} (N (⋀_{i \in Γ} f_{i}, ⊤_{x}) \to f_{i} (x)) \\ \geq ⋀_{i \in Γ} ⋀_{x \in X} (N (f_{i}, ⊤_{x}) \to f_{i} (x)) = ⋀_{i \in Γ} F_{N} (f_{i}), \end{matrix}$

(CT3)

$\begin{matrix} F_{N} (f \oplus g) \\ = ⋀_{x \in X} (N (f \oplus g, ⊤_{x}) \to (f \oplus g) (x)) \\ \geq ⋀_{x \in X} ((N (f, ⊤_{x}) \oplus N (g, ⊤_{x})) \to (f (x) \oplus g (x))) \\ \geq ⋀_{x \in X} (N (f, ⊤_{x}) \to f (x)) \\ ⊙ ⋀_{x \in X} (N (g, ⊤_{x}) \to g (x)) \geq F_{N} (f) ⊙ F_{N} (g) . \end{matrix}$

(AL)

$\begin{matrix} F_{N} (⋁_{i \in Γ} f_{i}) \\ = ⋀_{x \in X} (N (⋁_{i \in Γ} f_{i}, ⊤_{x}) \to ⋁_{i \in Γ} f_{i} (x)) \\ = ⋀_{x \in X} (⋁_{i \in Γ} N (f_{i}, ⊤_{x}) \to ⋁_{i \in Γ} f_{i} (x)) \\ \geq ⋀_{i \in Γ} ⋀_{x \in X} (N (f_{i}, ⊤_{x}) \to f_{i} (x)) = ⋀_{i \in Γ} F_{N} (f_{i}) . \end{matrix}$

$\begin{matrix} F_{N} (α ⊙ f) \\ = ⋀_{x \in X} (N (α ⊙ f, ⊤_{x}) \to (α ⊙ f) (x)) \\ = ⋀_{x \in X} ((α ⊙ N (f, ⊤_{x})) \to (α ⊙ f (x))) \\ = ⋀_{x \in X} (N (f, ⊤_{x}) \to f (x)) = F_{N} (f) . \end{matrix}$

Since, α ⊙ (α → f) ≤ f, then

$N (α \to f, ⊤_{x}) \leq α \to N (f, ⊤_{x}),$ we have

$\begin{matrix} F_{N} (α \to f) \\ = ⋀_{x \in X} (N (α \to f, ⊤_{x}) \to (α \to f) (x)) \\ \geq ⋀_{x \in X} ((α \to N (f, ⊤_{x})) \to (α \to f (x))) \\ = ⋀_{x \in X} (N (f, ⊤_{x}) \to f (x)) = F_{N} (f), \end{matrix}$

$\begin{matrix} F_{δ_{N}} (f^{*}) & = δ_{N}^{*} (f, f^{*}) \\ = (⋁_{x \in X} N (f, ⊤_{x}) ⊙ f^{*} (x))^{*} \\ = ⋀_{x \in X} (f^{*} (x) \to N^{*} (f^{*}, ⊤_{x})) \\ = ⋀_{x \in X} (N (f, ⊤_{x}) \to f (x)) = F_{N} (f) . \end{matrix}$

(2) It is easily proved.

Theorem 5.6. Let $(X, C_{X})$ and $(Y, C_{Y})$ be two Alexandrov L-fuzzy closure spaces and φ : X → Y be a map. Then,

(1) $N_{C_{X}} (φ^{\leftarrow} (f), φ^{\leftarrow} (⊤_{φ (x)})) \to N_{C_{Y}} (f, ⊤_{φ (x)}) \geq S (C_{X} (φ^{\leftarrow} (f)), φ^{\leftarrow} (C_{Y} (f))) .$

(2) If $φ : (X, C_{X}) \to (Y, C_{Y})$ is an LF-closure map, then $φ : (X, N_{C_{X}}) \to (Y, N_{C_{Y}})$ is an LF-near map.

Proof. (1)

$\begin{matrix} N_{C_{X}} (φ^{\leftarrow} (f), φ^{\leftarrow} (⊤_{φ (x)})) \to N_{C_{Y}} (f, ⊤_{φ (x)}) \\ = ⋁_{x \in X} C_{X} (φ^{\leftarrow} (f)) (x) \to ⋁_{φ (x) \in Y} C_{Y} (f) (φ (x)) \\ = ⋁_{x \in X} C_{X} (φ^{\leftarrow} (f)) (x) \to ⋁_{φ (x) \in Y} φ^{\leftarrow} (C_{Y} (f)) (x) \\ \geq ⋀_{x \in X} (C_{X} (φ^{\leftarrow} (f) (x) \to φ^{\leftarrow} (C_{Y} (f)) (x)) \\ = S (C_{X} (φ^{\leftarrow} (f), φ^{\leftarrow} (C_{Y} (f))) . \end{matrix}$

(2) Since $C_{X} (φ^{\leftarrow} (f)) \leq φ^{\leftarrow} (C_{Y} (f)),$ we have $N_{C_{X}} (φ^{\leftarrow} (f), φ^{\leftarrow} (⊤_{φ (x)}) \leq N_{C_{Y}} (f, ⊤_{φ (x)}) .$

Theorem 5.7. Let $(X, N_{X})$ and $(Y, N_{Y})$ be Alexandrov L-fuzzy near spaces and φ : X → Y be a map. Then,

(1)

$\begin{matrix} F_{N_{Y}} (f) \to F_{N_{X}} (φ^{\leftarrow} (f)) \\ \geq ⋀_{x \in X} (N_{X} (φ^{\leftarrow} (f), φ^{\leftarrow} (⊤_{φ (x)})) \to N_{Y} (f, ⊤_{φ (x)})) . \end{matrix}$

(2) If $φ : (X, N_{X}) \to (Y, N_{Y})$ is an LF-near map, then $φ : (X, F_{N_{X}}) \to (Y, F_{N_{Y}})$ is a LF-continuous map.

Proof. (1)

$\begin{matrix} F_{N_{Y}} (f) \to F_{N_{X}} (φ^{\leftarrow} (f)) \\ = ⋀_{y \in Y} (N_{Y} (f, ⊤_{y}) \to f (y)) \\ \to ⋀_{x \in X} (N_{X} (φ^{\leftarrow} (f), φ^{\leftarrow} (⊤_{φ (x)})) \to φ^{\leftarrow} (f) (x)) \\ \geq ⋀_{x \in X} (N_{Y} (f, ⊤_{φ (x)}) \to f (φ (x))) \\ \to ⋀_{x \in X} (N_{X} (φ^{\leftarrow} (f), φ^{\leftarrow} (⊤_{φ (x)})) \to φ^{\leftarrow} (f) (x)) \\ \geq ⋀_{x \in X} (N_{X} (φ^{\leftarrow} (f), φ^{\leftarrow} (⊤_{φ (x)})) \to N_{Y} (f, ⊤_{φ (x)})) . \end{matrix}$

(2) Since, $N_{X} (φ^{\leftarrow} (f), φ^{\leftarrow} (⊤_{φ (x)})) \leq N_{Y} (f, ⊤_{φ (x)}),$ by (1), we have $F_{N_{Y}} (f) \leq F_{N_{X}} (φ^{\leftarrow} (f)) .$

Theorem 5.8. (1)ϒ : AL - FNS → AL - FCS defined as $ϒ (X, N) = (X, C_{N})$ and ϒ (φ) = φ is a concrete functor,

(2) Φ : AL - FCS → AL - FNS defined as $Φ (X, C) = (X, N_{C})$ and Φ (φ) = φ is a concrete functor,

(3) The pair (ϒ, Φ) is a Galois correspondence between AL - FNS and AL - FCS.

Proof.(1) Follows from Theorems 5.1 and 5.3 and (2) Follows from Theorems 5.4 and 5.6.

(3) By Theorem 5.4 (6), if $(X, C)$ is an Alexandrov L-fuzzy closure space, then

$\begin{matrix} (ϒ \circ Φ) (X, C) & = ϒ (Φ (X, C)) \\ = ϒ (X, N_{C}) = (X, C_{N_{C}}) \geq (X, C) . \end{matrix}$

Hence, the identity map $ϒ \circ Φ = {id}_{C}$ is a LF-closure map.

By Theorem 5.4 (6), if $(X, N)$ is an Alexandrov L-fuzzy near space, then

$\begin{matrix} (Φ \circ ϒ) (X, N) & = Φ (ϒ (X, N)) \\ = Φ (X, C_{N}) = (X, N_{C_{N}}) \geq (X, N) . \end{matrix}$

Hence, the identity map $Φ \circ ϒ = {id}_{N}$ is a LF-near map. Therefore, the pair (ϒ, Φ) is a Galois correspondence.

6 Alexandrov L-fuzzy nearness and Alexandrov L-fuzzy interior space

Theorem 6.1. Let $(X, N)$ be an Alexandrov L-fuzzy near space. Define $I_{N} : L^{X} \to L^{X}$ as

$I_{N} (f) (x) = N^{*} (⊤_{x}, f^{*}) \forall f \in L^{X} .$

Then,

(1) $I_{N} (f) (x) = ⋀_{y \in X} (N (⊤_{x}, ⊤_{y}) \to f (y))$ ,

(2) $I_{N}$ is an Alexandrov L-fuzzy interior space,

(3) If $N$ is separated, then $I_{N}$ is separated,

(4) If $N$ is symmetric, then $I_{N}$ is symmetric,

(5) If $N$ is topological, then $I_{N}$ is topological,

(6) If $(X, N)$ is (L, ⊕)-fuzzy co-near space, then $(X, I_{N})$ is (L, ⊙)-fuzzy interior space.

Proof. (1)

$\begin{matrix} I_{N} (f) (x) & = N^{*} (⊤_{x}, f^{*}) \\ = N^{*} (⊤_{x}, ⋁_{y \in X} f^{*} (y) ⊙ ⊤_{y}) \\ = ⋀_{y \in X} (f^{*} (y) \to N^{*} (⊤_{x}, ⊤_{y})) \\ = ⋀_{y \in X} (N (⊤_{x}, ⊤_{y}) \to f (y)) . \end{matrix}$

(2)(I1)

$\begin{matrix} I_{N} (⊤_{X}) (x) & = N^{*} (⊤_{x}, ⊤_{X}^{*}) \\ = N^{*} (⊤_{x}, ⊥_{X}) = ⊤_{X}, \end{matrix}$

(I2) $I_{N} (f) (x) = N^{*} (⊤_{x}, f^{*}) \leq f (x)$ ,

(I3) If g ≤ f, then

$\begin{matrix} I_{N} (g) (x) & = N^{*} (⊤_{x}, g^{*}) \\ \leq N^{*} (⊤_{x}, f^{*}) = I_{N} (f) (x), \end{matrix}$

(I4)

$\begin{matrix} I_{N} (⋀_{i \in Γ} f_{i}) (x) & = N^{*} (⊤_{x}, (⋀_{i \in Γ} f_{i})^{*}) \\ = ⋀_{i \in Γ} N^{*} (⊤_{x}, f_{i}^{*}) \\ = ⋀_{i \in Γ} I_{N} (f_{i}) (x), \end{matrix}$

(I5)

$\begin{matrix} I_{N} (α \to f) (x) & = N^{*} (⊤_{x}, (α \to f)^{*}) \\ = N^{*} (⊤_{x}, α ⊙ f^{*}) \\ = α ⊙ N^{*} (⊤_{x}, f^{*}) \\ = α ⊙ I_{N} (f) (x) . \end{matrix}$

(3)

$\begin{matrix} I_{N} (⊤_{x}) (x) & = N^{*} (⊤_{x}, ⊤_{x}^{*}) \\ = N^{*} (⊤_{x}, ⊥_{x}) = ⊤_{x} . \end{matrix}$

(4)

$\begin{matrix} I_{N} (⊤_{x}^{*}) (y) & = N^{*} (⊤_{y}, (⊤_{x}^{*})^{*}) \\ = N^{*} (⊤_{y}, ⊤_{x}) = I_{N} (⊤_{y}^{*}) (x) . \end{matrix}$

(5)

$\begin{matrix} I_{N} (f) (x) \\ = ⋀_{y \in X} (N (⊤_{x}, ⊤_{y}) \to f (y)) \\ \leq ⋀_{y \in X} ((⋁_{z \in X} N (⊤_{x}, ⊤_{z}) \\ ⊙ N (⊤_{z}, ⊤_{y})) \to f (y)) \\ = ⋀_{y \in X} ⋁_{z \in X} ((N (⊤_{x}, ⊤_{z}) \\ ⊙ N (⊤_{z}, ⊤_{y})) \to f (y)) \\ = ⋁_{z \in X} (N (⊤_{x}, ⊤_{z}) \to \\ ⋀_{y \in X} (N (⊤_{z}, ⊤_{y})) \to f (y))) \\ = ⋁_{z \in X} (N (⊤_{x}, ⊤_{z}) \to I_{N} (f) (z)) \\ = I_{N} (I_{N} (f)) (x) . \end{matrix}$

(6)

$\begin{matrix} I_{N} (f ⊙ g) (x) & = N^{*} (⊤_{x}, (f ⊙ g)^{*}) \\ = N^{*} (⊤_{x}, f^{*} \oplus g^{*}) \\ \geq N^{*} (⊤_{x}, f^{*}) ⊙ N^{*} (⊤_{x}, g^{*}) \\ = I_{N} (f) (x) ⊙ I_{N} (g) (x) . \end{matrix}$

Theorem 6.2. Let $(X, I)$ be an Alexandrov L-fuzzy interior space. Define a map $δ_{I} : L^{X} \times L^{X} \to L$ by

$δ_{I} (f, g) = ⋁_{x \in X} f (x) ⊙ I^{*} (g^{*}) (x) \forall f, g \in L^{X} .$

Then, we have the following properties.

(1) $δ_{I}$ is an Alexandrov L-fuzzy pre-proximity such that $δ_{I} (f, g) = ⋁_{x, y \in X} I^{*} (⊤_{y}^{*}) (x) ⊙ f (x) ⊙ g (y),$

(2) If $(X, I)$ is (L, ⊙)-fuzzy interior space, then $(X, δ_{I_{N}})$ is (L, ⊙ , ⊕)-fuzzy pre-co-proximity space,

(3) $δ_{I} (f, g) \leq ⋁_{h \in L^{X}} {δ_{I} (f, h) ⊙ δ_{I} (h^{*}, g)$ }, the equality holds if $I$ is topological,

(4) If $I$ is topological, then $δ_{I}$ is an L-fuzzy quasi-proximity on X,

(5) If $I$ is separated, then $δ_{I}$ is separated.

Proof. (1) (P1) Since $I (⊥_{X}) = ⊥_{X}$ and $I (⊤_{X}) = ⊤_{X}$ , we have

$\begin{matrix} δ_{I} (⊤_{X}, ⊥_{X}) = ⋁_{x \in X} ⊤_{X} (x) ⊙ I^{*} (⊥_{X}^{*}) (x) = ⊥, \\ δ_{I} (⊥_{X}, ⊤_{X}) = ⋁_{x \in X} ⊥_{X} (x) ⊙ I^{*} (⊤_{X}^{*}) (x) = ⊥ . \end{matrix}$

(P2) Since $I (g) \leq g$ , then we have

$\begin{matrix} δ_{I} (f, g) & = ⋁_{x \in X} f (x) ⊙ I^{*} (g^{*}) (x) \\ \geq ⋁_{x \in X} f (x) ⊙ g (x) = ⋁_{x \in X} (f ⊙ g) (x) . \end{matrix}$

(P3) If g ≤ g₁, f ≤ f₁ and by (I3), then

$I^{*} (g^{*}) \leq I^{*} (g_{1}^{*})$ . Thus,

$\begin{matrix} δ_{I} (f, g) & = ⋁_{x \in X} f (x) ⊙ I^{*} (g^{*}) (x) \\ \leq ⋁_{x \in X} f_{1} (x) ⊙ I^{*} (g_{1}^{*}) (x) = δ_{I} (f_{1}, g_{1}) . \end{matrix}$

(P4) Easily proved.

(P5)

$\begin{matrix} δ_{I} (α ⊙ f, α \to g) \\ = ⋁_{x \in X} (α ⊙ f) (x) ⊙ I^{*} (α ⊙ g^{*}) (x) \\ \leq ⋁_{x \in X} (α ⊙ f (x)) ⊙ (α \to I^{*} (g^{*}) (x)) \\ \leq ⋁_{x \in X} f (x) ⊙ I^{*} (g^{*}) (x) = δ_{I} (f, g) . \end{matrix}$

Hence, $δ_{I}$ is a L-fuzzy pre-proximity on X.

Since g = ⋁ _y∈Xg (y) ⊙ ⊤ _y, then by lemma 2.2 (10) we have

$\begin{matrix} δ_{I} (f, g) = ⋁_{x \in X} f (x) ⊙ I^{*} (g^{*}) (x) \\ = ⋁_{x \in X} f (x) ⊙ I^{*} (⋀_{y \in X} g^{*} (y) \oplus ⊤_{y}^{*}) (x) \\ = ⋁_{x \in X} f (x) ⊙ (⋀_{y \in X} I (g (y) \to ⊤_{y}^{*}))^{*} (x) \\ = ⋁_{x \in X} f (x) ⊙ (⋀_{y \in X} g (y) \to I (⊤_{y}^{*}))^{*} (x) \\ = ⋁_{x, y \in X} f (x) ⊙ g (y) ⊙ I^{*} (⊤_{y}^{*}) (x) . \end{matrix}$

(2) For f₁, f₂, g₁, g₂ ∈ L^X, we have

$\begin{matrix} δ_{I} (f_{1}, g_{1}) \oplus δ_{I} (f_{2}, g_{2}) \\ = (⋁_{x \in X} f_{1} (x) ⊙ I^{*} (g_{1}^{*}) (x)) \\ \oplus (⋁_{x \in X} f_{2} (x) ⊙ I^{*} (g_{2}^{*}) (x)) \\ \geq ⋁_{x \in X} (f_{1} (x) ⊙ f_{2} (x)) ⊙ (I^{*} (g_{1}^{*}) (x) \oplus I^{*} (g_{2}^{*}) (x)) \\ (by Lemma 2.2 (11)) \\ \geq ⋁_{x \in X} (f_{1} (x) ⊙ f_{2} (x)) ⊙ I^{*} ((g_{1} \oplus g_{2})^{*}) (x) \\ = δ_{I} (f_{1} ⊙ f_{2}, g_{1} \oplus g_{2}) . \end{matrix}$

(3) For f, g, h ∈ L^X, we have

$\begin{matrix} δ_{I}^{*} (f, h) ⊙ δ_{I}^{*} (h^{*}, g) \\ = (⋁_{x \in X} f (x) ⊙ I^{*} (h^{*}) (x))^{*} \\ ⊙ (⋁_{x \in X} h^{*} (x) ⊙ I^{*} (g^{*}) (x))^{*} \\ = ⋀_{x \in X} (f (x) \to I (h^{*}) (x)) \\ ⊙ ⋀_{x \in X} (h^{*} (x) \to I (g^{*}) (x)) (Since I (h^{*}) \leq h^{*}) \\ \leq ⋀_{x \in X} (f (x) \to h^{*} (x)) ⊙ (h^{*} (x) \to I (g^{*}) (x)) \\ \leq ⋀_{x \in X} (f (x) \to I (g^{*}) (x)) = δ_{I}^{*} (f, g) . \end{matrix}$

Hence, $δ_{I} (f, g) \leq ⋀_{h \in L^{X}} (δ_{I} (f, h) \oplus δ_{I} (h^{*}, g))$ .

If $I$ is topological, then

$\begin{matrix} ⋁_{h \in L^{X}} δ_{I}^{*} (f, h) ⊙ δ_{I}^{*} (h^{*}, g) \\ = ⋁_{h \in L^{X}} (⋀_{x \in X} (f (x) \to I (h^{*}) (x))) \\ ⊙ (⋀_{x \in X} (h^{*} (x) \to I (g^{*}) (x))) (Put h^{*} = I (g^{*})) \\ \geq ⋀_{x \in X} (f (x) \to I (I (g^{*}) (x))) \\ ⊙ ⋀_{x \in X} (I (g^{*}) (x) \to I (g^{*}) (x)) \\ = ⋀_{x \in X} (f (x) \to I (g^{*}) (x)) = δ_{I}^{*} (f, g) . \end{matrix}$

(4) By (3), it is trivial.

(5) Let $I$ be separated, then

$δ_{I} (⊤_{z}, ⊤_{z}^{*}) = ⋁_{x \in X} ⊤_{z} (x) ⊙ I^{*} (⊤_{z}) (x) = ⊥ .$

Theorem 6.3 Let $(X, N_{X})$ and $(Y, N_{Y})$ be two Alexandrov L-fuzzy near spaces and φ : X → Y be a map. Then,

(1)

$\begin{matrix} S (φ^{\leftarrow} (I_{Y}) (f), I_{X} (φ^{\leftarrow} (f))) \\ = ⋀_{x \in X} (N_{X} (φ^{\leftarrow} (⊤_{φ (x)}), φ^{\leftarrow} (f^{*})) \\ \to N_{Y} (⊤_{φ (x)}, f^{*})) . \end{matrix}$

(2) If $φ : (X, N_{X}) \to (Y, N_{Y})$ is an LF-near map, then $φ : (X, I_{N_{X}}) \to (Y, I_{N_{Y}})$ is an LF-interior map.

Proof. (1)

$\begin{matrix} S (φ^{\leftarrow} (I_{Y} (f), I_{X} (φ^{\leftarrow} (f))) \\ = ⋀_{x \in X} (φ^{\leftarrow} (I_{Y} (f)) (x) \to I_{X} (φ^{\leftarrow} (f)) (x)) \\ = ⋀_{x \in X} (I_{Y} (f) (φ (x)) \to I_{X} (φ^{\leftarrow} (f)) (x)) \\ = ⋀_{x \in X} (N_{Y}^{*} (⊤_{φ (x)}, f^{*}) \\ \to N_{X}^{*} (φ^{\leftarrow} (⊤_{φ (x)}), φ^{\leftarrow} (f^{*}))) \\ = ⋀_{x \in X} (N_{X} (φ^{\leftarrow} (⊤_{φ (x)}), φ^{\leftarrow} (f^{*})) \\ \to N_{Y} (⊤_{φ (x)}, f^{*})), \end{matrix}$

(2) By (1), if

$N_{X} (φ^{\leftarrow} (⊤_{φ (x)}), φ^{\leftarrow} (h)) \leq N_{Y} (⊤_{φ (x)}, h)$

for each h ∈ L^Y, then $φ^{\leftarrow} (I_{Y} (f)) \leq I_{X} (φ^{\leftarrow} (f)) .$

Theorem 6.4. Let $I$ be an Alexandrov L-fuzzy interior space. Define a map $N_{I} : L^{X} \times L^{X} \to L$ by

$N_{I} (⊤_{x}, f) = ⋁_{x \in X} I^{*} (f^{*}) (x) \forall f \in L^{X} .$

Then, we have the following properties.

(1) $N_{I}$ is an Alexandrov L-fuzzy nearness on X,

(2) If $I$ is separated, then $N_{I}$ is separated,

(3) If $I$ is symmetric, then $N_{I}$ is symmetric,

(4) If $I$ is topological, then $N_{I}$ is topological,

(5) If $(X, I)$ is (L, ⊙)-fuzzy interior space, then $(X, N_{I})$ is (L, ⊕)-fuzzy co-near space,

(6) $N_{I_{N}} \geq N$ and $I_{N_{I}} \leq I .$

Proof. (1)

(N1) $N_{I} (⊥_{X}, ⊤_{x}) = ⋁_{x \in X} I^{*} (⊥_{X}^{*}) (x) = ⊥$ ,

(N2) $N_{I} (⊤_{x}, f) = ⋁_{x \in X} I^{*} (f^{*}) (x) \geq (f) (x)$ ,

(N3) If f ≤ g, then

$\begin{matrix} N_{I} (⊤_{x}, f) & = ⋁_{x \in X} I^{*} (f^{*}) (x) \\ \leq ⋁_{x \in X} I^{*} (g^{*}) (x) = N_{I} (⊤_{x}, g), \end{matrix}$

(N4)

$\begin{matrix} N_{I} (⊤_{x}, ⋁_{i \in Γ} f_{i}) & = ⋁_{x \in X} I^{*} (⋀_{i \in Γ} f_{i}^{*}) (x) \\ = ⋁_{i \in Γ} ⋁_{x \in X} I^{*} (f_{i}^{*}) (x) \\ = ⋁_{i \in Γ} N_{I} (⊤_{x}, f_{i}^{*}), \end{matrix}$

(N5) For every α ∈ L, f ∈ L^X, we have

$\begin{matrix} N_{I} (⊤_{x}, α ⊙ f) \\ = ⋁_{x \in X} I^{*} ((α ⊙ f)^{*}) (x) \\ = ⋁_{x \in X} I^{*} (α \to f^{*}) (x) \\ = α ⊙ ⋁_{x \in X} I^{*} (f^{*}) (x) \\ = α ⊙ N_{I} (⊤_{x}, f) = N_{I} (α ⊙ f, ⊤_{x}) . \end{matrix}$

(2) $N_{I} (⊤_{x}, ⊤_{x}^{*}) = ⋁_{x \in X} I^{*} (⊤_{x}) (x) = ⊥,$

$N_{I} (⊤_{x}^{*}, ⊤_{x}) = ⋁_{x \in X} I^{*} (⊤_{x}^{*})^{*} (x) = ⊥ .$

(3)

$\begin{matrix} N_{I}^{s} (⊤_{x}, ⊤_{y}) & = ⋁_{x \in X} I^{*} (⊤_{y}^{*}) (x) \\ = ⋁_{x \in X} I^{*} (⊤_{x}^{*}) (y) = N_{I} (⊤_{y}, ⊤_{x}) . \end{matrix}$

(4)

$\begin{matrix} ⋁_{y \in X} N_{I} (⊤_{x}, ⊤_{y}) ⊙ N_{I} (⊤_{y}, ⊤_{z}) \\ = ⋁_{y \in X} (⋁_{x \in X} I^{*} (⊤_{y}^{*}) (x)) \\ ⊙ (⋁_{y \in X} I^{*} (⊤_{z}^{*}) (y)) \\ = ⋁_{x, y \in X} I^{*} (⊤_{z}^{*}) (y) ⊙ I^{*} (⊤_{y}^{*}) (x) \\ \leq ⋁_{x \in X} I^{*} (⊤_{z}^{*}) (x) = N_{I} (⊤_{x}, ⊤_{z}) . \end{matrix}$

(5)

$\begin{matrix} N_{I} (⊤_{x}, f \oplus g) \\ = ⋁_{x \in X} I^{*} ((f \oplus g)^{*}) (x) \\ = ⋁_{x \in X} I^{*} (f^{*} ⊙ g^{*}) (x) \\ \leq (⋁_{x \in X} I^{*} (f^{*}) (x)) \oplus (⋁_{x \in X} I^{*} (g^{*}) (x)) \\ = N_{I} (⊤_{x}, f) \oplus N_{I} (⊤_{x}, g) . \end{matrix}$

(6)

$\begin{matrix} N_{I_{N}} (⊤_{x}, f) & = ⋁_{x \in X} I_{N}^{*} (f^{*}) (x) \\ = ⋁_{x \in X} (N^{*} (⊤_{x}, f))^{*} (x) \geq N (f, ⊤_{x}), \\ I_{N_{I}} (f) (x) & = N_{I}^{*} (⊤_{x}, f^{*}) \\ = (⋁_{x \in X} I^{*} (f) (x))^{*} = ⋀_{x \in X} I (f) (x) . \end{matrix}$

Example 6.5. (1) Define $C : L^{X} \to L^{X}$ as $C (f) = f$ . Hence, $C$ is a topological, separated and Alexandrov L-fuzzy closure operator on X.

By Theorem 5.4, we have $N_{C} (f, ⊤_{x}) = ⋁_{x \in X} f (x) .$

By Theorem 5.1, we have

$\begin{matrix} C_{N_{C}} (f) (x) & = N_{C} (f, ⊤_{x}) \\ = ⋁_{x \in X} f (x) \geq f (x) = C (f) (x) . \end{matrix}$

By Theorem 5.4, we have

$\begin{matrix} N_{C_{N}} (f, ⊤_{x}) & = ⋁_{x \in X} C_{N} (f) (x) \\ = ⋁_{x \in X} N (f, ⊤_{x}) \geq N (f, ⊤_{x}) . \end{matrix}$

(2) Define $I : L^{X} \to L^{X}$ as $I (f) = f$ . Hence, $I$ is a topological, separated and Alexandrov L-fuzzy interior operator on X.

By Theorem 6.4, we have

$N_{I} (⊤_{x}, f) = ⋁_{x \in X} I^{*} (f^{*}) (x) = ⋁_{x \in X} f (x) .$

By Theorem 6.1, we have

$\begin{matrix} I_{N_{I}} (f) (x) & = N_{I}^{*} (⊤_{x}, f^{*}) \\ = ⋀_{x \in X} f (x) \leq f (x) = I (f) (x) . \end{matrix}$

By Theorem 6.4, we have

$\begin{matrix} N_{I_{N}} (⊤_{x}, f) & = ⋁_{x \in X} I_{N}^{*} (f^{*}) (x) \\ = ⋁_{x \in X} N (⊤_{x}, f) \geq N (⊤_{x}, f) . \end{matrix}$

Theorem 6.6. Let $(X, N)$ be an Alexandrov (L, ⊕)-fuzzy co-near space. Define a mapping $T_{N} : L^{X} \to L$ by:

$T_{N} (f) = ⋀_{x \in X} (f (x) \to N^{*} (⊤_{x}, f^{*})) \forall f \in L^{X} .$

Then,

(1) $T_{N}$ is a strong Alexandrov (L, ⊙)- fuzzy topology on X with $T_{δ_{N}} = T_{N}$ ,

(2) If $T_{N}$ is separated, then $N$ is separated.

Proof. (1) (T1)

$\begin{matrix} T_{N} (⊥_{X}) = ⋀_{x \in X} (⊥_{X} (x) \to N^{*} (⊤_{x}, ⊤_{X})) = ⊤, \\ T_{N} (⊤_{X}) = ⋀_{x \in X} (⊤_{X} (x) \to N^{*} (⊤_{x}, ⊥_{X})) = ⊤ . \end{matrix}$

(T2)

$\begin{matrix} T_{N} (f ⊙ g) \\ = ⋀_{x \in X} ((f ⊙ g) (x) \to N^{*} (⊤_{x}, f^{*} \oplus g^{*})) \\ \geq ⋀_{x \in X} ((f ⊙ g) (x) \to (N^{*} (⊤_{x}, f^{*}) ⊙ N^{*} (⊤_{x}, g^{*}))) \\ \geq ⋀_{x \in X} ((f (x) \to N^{*} (⊤_{x}, f^{*})) \\ ⊙ (g (x) \to N^{*} (⊤_{x}, g^{*}))) \geq T_{N} (f) ⊙ T_{N} (g) . \end{matrix}$

(T3)

$\begin{matrix} T_{N} (⋁_{i \in Γ} f_{i}) \\ = ⋀_{x \in X} (⋁_{i \in Γ} f_{i} (x) \to N^{*} (⊤_{x}, ⋀_{i \in Γ} f_{i}^{*})) \\ \geq ⋀_{x \in X} (⋁_{i \in Γ} f_{i} (x) \to ⋁_{i \in Γ} N^{*} (⊤_{x}, f_{i}^{*})) \\ \geq ⋀_{i \in Γ} ⋀_{x \in X} (f_{i} (x) \to N^{*} (⊤_{x}, f_{i}^{*})) \\ = ⋀_{i \in Γ} T_{N} (f_{i}) . \end{matrix}$

(AL)

$\begin{matrix} T_{N} (⋀_{i \in Γ} f_{i}) \\ = ⋀_{x \in X} (⋀_{i \in Γ} f_{i} (x) \to N^{*} (⊤_{x}, ⋁_{i \in Γ} f_{i}^{*})) \\ = ⋀_{x \in X} (⋀_{i \in Γ} f_{i} (x) \to ⋀_{i \in Γ} N^{*} (⊤_{x}, f_{i}^{*})) \\ \geq ⋀_{i \in Γ} ⋀_{x \in X} (f_{i} (x) \to N^{*} (⊤_{x}, f_{i}^{*})) \\ = ⋀_{i \in Γ} T_{N} (f_{i}) . \end{matrix}$

Since α ⊙ (α → f) ≤ f, then

$\begin{matrix} α ⊙ N (⊤_{x}, α \to f) & = N (⊤_{x}, α ⊙ (α \to f)) \\ \leq N (⊤_{x}, f) . \end{matrix}$

Thus, $N (⊤_{x}, α \to f) \leq α \to N (⊤_{x}, f),$ we have

$\begin{matrix} T_{N} (α ⊙ f) \\ = ⋀_{x \in X} ((α ⊙ f (x)) \to N^{*} (⊤_{x}, α \to f^{*})) \\ \geq ⋀_{x \in X} ((α ⊙ f (x)) \to (α ⊙ N^{*} (⊤_{x}, f^{*}))) \\ \geq ⋀_{x \in X} (f (x) \to N^{*} (⊤_{x}, f^{*})) = T_{N} (f), \end{matrix}$

$\begin{matrix} T_{N} (α \to f) \\ = ⋀_{x \in X} ((α \to f (x)) \to N^{*} (⊤_{x}, α ⊙ f^{*})) \\ = ⋀_{x \in X} ((α \to f (x)) \to (α \to N^{*} (⊤_{x}, f^{*}))) \\ \geq ⋀_{x \in X} (f (x) \to N^{*} (⊤_{x}, f^{*})) = T_{N} (f) . \\ T_{N} (f) & = ⋀_{x \in X} (f (x) \to N^{*} (⊤_{x}, f^{*})) \\ = (⋁_{x \in X} f (x) ⊙ N (⊤_{x}, f^{*}))^{*} \\ = δ_{N}^{*} (f, f^{*}) = T_{δ_{N}} (f) . \end{matrix}$

(2) It is easily proved.

Theorem 6.7. Let $(X, I_{X})$ and $(Y, I_{Y})$ be two L-fuzzy interior spaces and φ : X → Y be a map. Then,

(1)

$\begin{matrix} S (φ^{\leftarrow} (I_{Y} (f)), I_{X} (φ^{\leftarrow} (f))) \\ = ⋀_{x \in X} (N_{I_{X}} (φ^{\leftarrow} (⊤_{φ (x)}), φ^{\leftarrow} (f^{*})) \\ \to N_{I_{Y}} (⊤_{φ (x)}, f^{*}) . \end{matrix}$

(2) If $φ : (X, I_{X}) \to (Y, I_{Y})$ is an LF-interior map, then $φ : (X, N_{I_{X}}) \to (Y, N_{I_{Y}})$ is an LF-near map.

Proof. (1)

$\begin{matrix} S (φ^{\leftarrow} (I_{Y} (f)), I_{X} (φ^{\leftarrow} (f))) \\ = ⋀_{x \in X} (φ^{\leftarrow} (I_{Y} (f)) (x) \to I_{X} (φ^{\leftarrow} (f)) (x)) \\ = ⋀_{x \in X} (I_{Y} (f) (φ (x)) \to I_{X} (φ^{\leftarrow} (f)) (x)) \\ = ⋀_{x \in X} (N_{I_{Y}}^{*} (⊤_{φ (x)}, f^{*}) \\ \to N_{I_{X}}^{*} (φ^{\leftarrow} (⊤_{φ (x)}), φ^{\leftarrow} (f^{*})) \\ = ⋀_{x \in X} (N_{I_{X}} (φ^{\leftarrow} (⊤_{φ (x)}), φ^{\leftarrow} (f^{*}) \\ \to N_{I_{Y}} (⊤_{φ (x)}, f^{*})) . \end{matrix}$

(2) By (1), if $φ^{\leftarrow} (I_{Y} (f)) \leq I_{X} (φ^{\leftarrow} (f)),$ then

$N_{I_{X}} (φ^{\leftarrow} (⊤_{φ (x)}), φ^{\leftarrow} (h)) \leq N_{I_{Y}} (⊤_{φ (x)}, h)$

for every h ∈ L^Y .

Theorem 6.8. Let $(X, N_{X})$ and $(Y, N_{Y})$ be Alexandrov L-fuzzy near spaces and $φ : (X, N_{X}) \to (Y, N_{Y})$ be a LF-near map. Then, $φ : (X, T_{N_{X}}) \to (Y, T_{N_{Y}})$ is a LF-continuous map.

Proof. $\begin{matrix} T_{N_{X}} (φ^{\leftarrow} (f)) \\ = S (φ^{\leftarrow} (f), I_{N_{X}} (φ^{\leftarrow} (f))) \\ = ⋀_{x \in X} (φ^{\leftarrow} (f) (x) \to I_{N_{X}} (φ^{\leftarrow} (f)) (x)) \\ = ⋀_{x \in X} (φ^{\leftarrow} (f) (x) \to N_{X}^{*} (⊤_{x}, (φ^{\leftarrow} (f))^{*})) \end{matrix}$ $\begin{matrix} = ⋀_{x \in X} (f (φ (x)) \to N_{X}^{*} (φ^{\leftarrow} (⊤_{φ (x)}), φ^{\leftarrow} (f^{*}))) \\ \geq ⋀_{φ (x) \in Y} (f (φ (x)) \to N_{Y}^{*} ((⊤_{φ (x)}), f^{*})) \\ = T_{N_{Y}} (f) . \end{matrix}$

Theorem 6.9. (1) ϒ : AL - FNS → AL - FIS defined as $ϒ (X, N) = (X, I_{N})$ and ϒ (φ) = φ is a concrete functor,

(2) Φ : AL - FIS → AL - FNS defined as $Φ (X, I) = (X, N_{I})$ and Φ (φ) = φ is a concrete functor,

(3) The pair (ϒ, Φ) is a Galois correspondence between AL - FNS and AL - FIS.

Proof.{(1) follows from Theorems 6.1 and 6.3 and (2) follows from Theorems 6.4 and 6.7.

(3) By Theorem 6.4 (6), if $(X, I)$ is an Alexandrov L-fuzzy interior space, then

$\begin{matrix} (ϒ \circ Φ) (X, I) & = ϒ (Φ (X, I)) \\ = ϒ (X, N_{I}) = (X, I_{N_{I}}) \leq (X, I) . \end{matrix}$

Hence, the identity map $ϒ \circ Φ = {id}_{I}$ is an LF-interior map.

By Theorem 6.4 (6), if $(X, N)$ is an Alexandrov L-fuzzy near space, then

$\begin{matrix} (Φ \circ ϒ) (X, N) & = Φ (ϒ (X, N)) \\ = Φ (X, I_{N}) = (X, N_{I_{N}}) \geq (X, N) . \end{matrix}$

Hence, the identity map $Φ \circ ϒ = {id}_{N}$ is a LF-near map. Therefore, the pair (ϒ, Φ) is a Galois correspondence.

Example 6.10. Let X = {h_i ∣ i = {1, . . . , 3}} with h_i=house and Y = {e, b, w, c, i} with e=expensive, b= beautiful, w=wooden, c= creative, i=in the green surroundings. Let ([0, 1] , ⊙ , → , ^*, 0, 1) be a complete residuated lattice (ref.[2,6-8, 2,6-8]) as

$\begin{matrix} x ⊙ y = max {0, x + y - 1}, \\ x \to y = min {1 - x + y, 1}, x^{*} = 1 - x . \end{matrix}$ Let I ∈ [0, 1] ^X×Y be a fuzzy information as follows:

$\begin{matrix} I & e & b & w & c & i \\ h_{1} & 0.7 & 0.6 & 0.5 & 0.9 & 0.2 \\ h_{2} & 0.6 & 0.8 & 0.4 & 0.3 & 0.5 \\ h_{3} & 0.4 & 0.9 & 0.8 & 0.6 & 0.6 \end{matrix}$

Define [0, 1]-fuzzy pre-orders $R_{X}^{Y}, R_{X}^{{b, w}} \in [0, 1]^{X \times X}$ by

$\begin{matrix} R_{X}^{Y} (h_{i}, h_{j}) & = ⋀_{y \in Y} (I (h_{i}, y) \to I (h_{j}, y)), \\ R_{X}^{{b, w}} (h_{i}, h_{j}) & = ⋀_{y \in {b, w}} (R (h_{i}, y) \to R (h_{j}, y)) . \end{matrix}$

$R_{X}^{Y} = (\begin{matrix} 1 & 0.4 & 0.7 \\ 0.7 & 1 & 0.8 \\ 0.6 & 0.6 & 1 \end{matrix}), R_{X}^{{b, w}} = (\begin{matrix} 1 & 0.9 & 1 \\ 0.8 & 1 & 1 \\ 0.7 & 0.6 & 1 \end{matrix}) .$

(1) For each $R \in {R_{X}^{Y}, R_{X}^{{b, w}}}$ , by Example 4.8, we obtain Alexandrov [0, 1]-fuzzy pre-proximity

$δ_{R} : [0, 1]^{X} \times [0, 1]^{X} \to [0, 1]$ as $δ_{R} (f, g) = ⋁_{h_{i}, h_{j} \in X} R_{X}^{Y} (h_{i}, h_{j}) ⊙ f (h_{i}) ⊙ g (h_{j}) .$

We obtain Alexandrov [0, 1]-fuzzy nearness

$N_{R} : [0, 1]^{X} \times [0, 1]^{X} \to [0, 1]$ as

$\begin{matrix} N_{R} (f, ⊤_{h_{j}}) = ⋁_{h_{i} \in X} R (h_{i}, h_{j}) ⊙ f (h_{i}) . \end{matrix}$

By Remark 4.3 (2), we obtain a strong [0, 1]-fuzzy co-topology $F_{δ_{R}} : [0, 1]^{X} \to [0, 1]$ as

$\begin{matrix} F_{δ_{R}} (f) = δ_{R}^{*} (f^{*}, f) \\ = (⋁_{h_{i}, h_{j} \in X} (R_{X}^{Y} (h_{i}, h_{j}) ⊙ f^{*} (h_{i}) ⊙ f (h_{j})))^{*} \\ = ⋀_{h_{i}, h_{j} \in X} (R_{X}^{Y} (h_{i}, h_{j}) ⊙ f (h_{j}) \to f (h_{i})) . \end{matrix}$

By Theorem 5.5, we obtain [0, 1]-fuzzy co-topology $F_{N_{R}} : [0, 1]^{X} \to [0, 1]$ as

$\begin{matrix} F_{N_{R}} (f) = ⋀_{h_{j} \in X} (N_{R} (f, ⊤_{h_{j}}) \to f (h_{j})) \\ = ⋀_{h_{j} \in X} (⋁_{h_{i} \in X} (R (h_{i}, h_{j}) ⊙ f (h_{i}) \to f (h_{j}))) \\ = ⋀_{h_{i}, h_{j} \in X} ((R (h_{i}, h_{j}) ⊙ f (h_{i})) \to f (h_{j})) . \end{matrix}$

(2) For each $R \in {R_{X}^{Y}, R_{X}^{{b, w}}}$ , we obtain a strong topological and Alexandrov [0, 1]-fuzzy closure operator $C_{R} : [0, 1]^{X} \to [0, 1]^{X}$ as

$\begin{matrix} C_{R} (f) (h_{j}) = ⋁_{h_{i} \in X} R (h_{i}, h_{j}) ⊙ f (h_{i}) . \end{matrix}$

By Theorem 5.2, we obtain a topological and Alexandrov [0, 1]-fuzzy quasi-proximity $δ_{C_{R}}$ as

$\begin{matrix} δ_{C_{R}} (f, g) = ⋁_{h_{i} \in X} f (h_{i}) ⊙ C_{R} (g) (h_{i}) \\ = ⋁_{h_{i} \in X} (f (h_{i}) ⊙ ⋁_{h_{j} \in X} (R (h_{j}, h_{i}) ⊙ g (h_{j}))) \\ = ⋁_{h_{i}, h_{j} \in X} (R (h_{j}, h_{i}) ⊙ f (h_{i}) ⊙ g (h_{j})) . \end{matrix}$

By Remark 4.3 (2) and Theorem 5.5, we obtain [0, 1]-fuzzy co-topologies $F_{δ_{C_{R}}}$ and $F_{N_{R}}$ as follows $\begin{matrix} F_{δ_{C_{R}}} (f) = δ_{C_{R}}^{*} (f^{*}, f) \\ = (⋁_{h_{i}, h_{j} \in X} (R (h_{i}, h_{j}) ⊙ f^{*} (h_{i}) ⊙ f (h_{j})))^{*} \\ = ⋀_{h_{i}, h_{j} \in X} (R_{X}^{Y} (h_{i}, h_{j}) ⊙ f (h_{j}) \to f (h_{i})) . \end{matrix}$

Since $N_{R} (f, ⊤_{h_{j}}) = ⋁_{h_{i} \in X} R (h_{i}, h_{j}) ⊙ f (h_{i})$ ,

(3) For each $R \in {R_{X}^{Y}, R_{X}^{{b, w}}}$ and by Example 4.8, we obtain Alexandrov [0, 1]-fuzzy near $N_{R} : [0, 1]^{X} \times [0, 1]^{X} \to [0, 1]$ as

$\begin{matrix} N_{R} (⊤_{h_{i}}, f) = ⋁_{h_{j} \in X} R (h_{i}, h_{j}) ⊙ f (h_{j}) . \end{matrix}$

From Theorem 6.6, we obtain [0, 1]-fuzzy topology $T_{N_{R}}$ as follows

$\begin{matrix} T_{N_{R}} (f) = ⋀_{h_{j} \in X} (f (h_{j}) \to N_{R}^{*} (⊤_{h_{j}}, f^{*})) \\ = ⋀_{h_{j} \in X} (f (h_{j}) \to ⋀_{h_{i} \in X} (R (h_{j}, h_{i}) \to f (h_{i}))) \\ = ⋀_{h_{i}, h_{j} \in X} ((R (h_{j}, h_{i}) ⊙ f (h_{j})) \to f (h_{i})) . \end{matrix}$

7 Conclusion

As an unified structure as an extension of Pawlak’s rough set [30, 31], we introduce a new structure Alexandrov L-fuzzy nearness and investigate the relations among it and Alexandrov L-fuzzy pre-proximities, Alexandrov L-fuzzy topological structures. Several and interesting relations are introduced. Finally, we have shown the existence of concrete functors one from the category of Alexandrov L-fuzzy nearness AL - FNS, Alexandrov L-fuzzy pre-proximity spaces AL - FPRXS and Alexandrov L-fuzzy topological structures (AL - FCS, AL - FIS) and demonstrate the Galois correspondence between these categories. In future we will discuss concepts like fibers, concrete isomorphism and duality concepts of such concrete categories. We give Example 6.10 as a viewpoint of the topological structure for fuzzy information and fuzzy rough sets in a complete residuated lattice.

Using the concepts of L-fuzzy nearness, information systems and decision rules with a view point of applications to multi-attribute decision-making will be investigated in complete residuated lattices [24, 48] soon.

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