The relationships between L -fuzzy topogenous orders and topological structures

Abstract

In this paper, we introduce the notions of L-fuzzy topogenous orders, L-fuzzy topology, L-interior operators and L-closure operators in complete residuated lattices. Moreover, we investigate the relations among L-fuzzy topogenous orders, L-interior operators and L-closure operators. We show that there is a Galois correspondence between the category of separated L-fuzzy interior (resp. closure) spaces and that of separated L-fuzzy topogenous (resp. cotopogenous) spaces.

Keywords

Complete residuated lattice Galois correspondence

1 Introduction

Ward et al. [20] introduced a complete residuated lattice which is an algebraic structure for many valued logic. It is an important mathematical tool for algebraic structure. By using the concepts of topological structures, information systems and decision rules are investigated in complete residuated lattices [2 , 15–19]. Höhle [7, 8] introduced L-fuzzy topologies with algebraic structure L(cqm, quantales, MV-algebra).

Katsaras [11 –14] introduced the concepts of fuzzy topogenous order and fuzzy topogenous structures in completely distributive lattice which are a unified approach to the three spaces: Chang’s fuzzy topologies [3], Katsaras’s fuzzy proximities [10] and Hutton’s fuzzy uniformities [9]. As an extension of Katsaras’s definition, El-Dardery [5] introduced L-fuzzy topogenous order in view points of Sostak’s fuzzy topology [8] and Kim’s L-fuzzy proximities [15, 16] on strictly two-sided, commutative quantales. L-fuzzy topogenous structures [5, 8] have been developed in a slightly different sense.

In this paper, we obtain the L-interior operators and L-closure operators induced by L-fuzzy topogenous orders. We obtain the L-fuzzy topogenous order and L-fuzzy cotopogenous induced by L-interior operators and L-closure operators, respectively. Moreover, we investigate the relations among L-fuzzy topogenous orders, L-interior operators and L-closure operators. We show that there is a Galois correspondence between the category of separated L-fuzzy interior (resp. closure) spaces and that of separated L-fuzzy topogenous (resp. cotopogenous) spaces.

2 Preliminaries

Definition 2.1. [2 , 6–8] An algebra (L, ∧, ∨, ⊙, →, ⊥, ⊤) is called a complete residuated lattice if it satisfies the following conditions:

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

(L, ⊙, ⊤) is a commutative monoid;

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

In this paper, we always assume that (L, ≤, ⊙, →, ⊕, ^∗) is 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, $⊤_{x} (y) = {\begin{matrix} ⊤, & if y = x, \\ ⊥, & otherwise, \end{matrix} ⊤_{x}^{*} (y) = {\begin{matrix} ⊥, & if y = x, \\ ⊤, & otherwise . \end{matrix}$

Lemma 2.2. [2 , 6–8] For each x, y, z, x_i, y_i, w ∈ L, we have the following properties.

⊤ → x = x, ⊥ ⊙ x = ⊥,

If y ≤ z, then x ⊙ y ≤ x ⊙ z, x ⊕ y ≤ x ⊕ z, x → y ≤ x → z and z → x ≤ y → x,

x ≤ y iff x→ y = ⊤.

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

x → (⋀ _iy_i) = ⋀ _i (x → y_i),

(⋁ _ix_i) → y = ⋀ _i (x_i → y),

x ⊙ (⋁ _iy_i) = ⋁ _i (x ⊙ y_i),

(⋀ _ix_i) ⊕ y = ⋀ _i (x_i ⊕ y),

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

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

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

x → y ≤ (x ⊙ z) → (y ⊙ z) and (x → y) ⊙ (y → z) ≤ x → z,

(x → y) ⊙ (z → w) ≤ (x ⊕ z) → (y ⊕ w).

x → y = y^* → x^*.

(x ∨ y) ⊙ (z ∨ w) ≤ (x ∨ z) ∨ (y ⊙ w) ≤ (x ⊕ z) ∨ (y ⊙ w) .

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

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

z → x ≤ (x → y) → (z → y) and y → z ≤ (x → y) → (x → z).

Definition 2.3. [2] Let X be a set. A map R : X × X → L is called an L-partial order if it satisfies the following conditions:

reflexive if R (x, x) =⊤ for all x ∈ X,

transitive if R (x, y) ⊙ R (y, z) ≤ R (x, z), for all x, y, z ∈ X,

antisymmetric if R (x, y) = R (y, x) =⊤, then x = y.

Lemma 2.4. [2] For a given set X, define a binary map S : L^X × L^X → 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.

S is an L-partial order on L^X.

f ≤ g iff S (f, g)≥ ⊤,

If f ≤ g, then S (h, f) ≤ S (h, g) and S (f, h) ≥ S (g, h),

S (f, g) ⊙ S (k, h) ≤ S (f ⊕ k, g ⊕ h) and S (f, g) ⊙ S (k, h) ≤ S (f ⊙ k, g ⊙ h),

S (g, h) ≤ S (f, g) → S (f, h),

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

Definition 2.5. [8] A map $I : L^{X} \to L^{X}$ is called an L-interior operator on X if $I$ satisfies the following conditions:

$I (⊤_{X}) = ⊤_{X}$ ,

$I (f) \leq f$ , for all f ∈ L^X,

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

$I (f ⊙ g) \geq I (f) ⊙ I (g)$ .

The pair $(X, I)$ is called an L-interior space. An L-interior space is called topological if

(T) $I (I (f)) = I (f), \forall f \in L^{X}$ .

An L-interior space $(X, I)$ is said to be stratified if

(S) $I (α ⊙ f) \geq α ⊙ I (f)$ .

An L-interior space $(X, I)$ is said to be separated if

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

Let $(X, I_{X})$ and $(Y, I_{Y})$ be L-interior spaces. $φ : (X, I_{X}) \to (Y, I_{Y})$ is called an L-interior map if, for each g ∈ L^Y, $φ^{\leftarrow} (I_{Y} (g)) \leq I_{X} (φ^{\leftarrow} (g)) .$

Definition 2.6. [8] A map $C : L^{X} \to L^{X}$ is called an L-closure operator on X if $C$ satisfies the following conditions:

$C (⊥_{X}) = ⊥_{X}$ ,

$C (f) \geq f$ , for all f ∈ L^X,

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

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

The pair $(X, C)$ is called an L-closure space. An L-closure space is called topological if

(T) $C (C (f)) = C (f), \forall f \in L^{X}$ .

An L-closure space $(X, C)$ is said to be stratified if

(S) $C (α ⊙ f) \geq α ⊙ C (f)$ .

An L-closure space $(X, C)$ is said to be separated if

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

Let $(X, C_{X})$ and $(Y, C_{Y})$ be L-closure spaces. $φ : (X, C_{X}) \to (Y, C_{Y})$ is called an L-closure map if, for each f ∈ L^X, $φ^{\to} (C_{X} (f)) \leq C_{Y} (φ^{\to} (f)) .$

Definition 2.7. [8] A map $T : L^{X} \to L$ is called an L-fuzzy topology on X if it satisfies the following conditions:

$T (⊥_{X}) = T (⊤_{X}) = ⊤$ ,

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

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

The pair $(X, T)$ is called an L-fuzzy topological space.

Let $(X, T_{1})$ and $(Y, T_{2})$ be two L-fuzzy topological spaces. A map φ : X → Y is said to be L-fuzzy continuous iff for each f ∈ L^Y, $T_{2} (f) \leq T_{1} (φ^{\leftarrow} (f)) .$

Definition 2.8. [8] A map $F : L^{X} \to L$ is called an L-fuzzy cotopology on X if it satisfies the following conditions:

$F (⊥_{X}) = F (⊤_{X}) = ⊤$ ,

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

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

The pair $(X, F)$ is called an L-fuzzy cotopological space.

Let $(X, F_{1})$ and $(Y, F_{2})$ be two L-fuzzy cotopological spaces. A map φ : X → Y is said to be L-fuzzy continuous iff for each f ∈ L^Y, $F_{2} (f) \leq F_{1} (φ^{\leftarrow} (f)) .$

Definition 2.9. [4 , 17] A map ξ : L^X × L^X → L is called an L-fuzzy semi-topogenous order on X if it satisfies the following axioms.

ξ (⊤ _X, ⊤ _X) = ξ (⊥ _X, ⊥ _X) = ⊤.

If ξ (f, g) ≤ S (f, g).

If f₁ ≤ f, g ≤ g₁, then ξ (f, g) ≤ ξ (f₁, g₁).

An L-fuzzy semi-topogenous order ξ is called: for every f₁, f₂, g₁, g₂ ∈ L^X,

(1) L-fuzzy topogenous if

(T) ξ (f₁ ⊙ f₂, g₁ ⊙ g₂) ≥ ξ (f₁, g₁) ⊙ ξ (f₂, g₂).

(2) L-fuzzy cotopogenous if

(CT) ξ (f₁ ⊕ f₂, g₁ ⊕ g₂) ≥ ξ (f₁, g₁) ⊙ ξ (f₂, g₂),

The pair (X, ξ) is called an L-fuzzy topogenous (resp. L-fuzzy cotopogenous) space.

(3) L-fuzzy bitopogenous if ξ are L-fuzzy topogenous and L-fuzzy cotopogenous.

An L-fuzzy topogenous (resp. cotopogenuous) order ξ is called separated if ξ (⊤ _x, ⊤ _x) = ⊤ (resp. $ξ (⊤_{x}^{*}, ⊤_{x}^{*}) = ⊤$ ) for each x ∈ X.

An L-fuzzy topogenous (resp. cotopogenous) order ξ on X is said to be an L-fuzzy topogenous (resp. cotopogenous) structure if ξ ∘ ξ ≥ ξ, where

(TS) (ξ₁ ∘ ξ₂) (f, g) = ⋁ _h∈L^Xξ₁ (f, h) ⊙ ξ₂ (h, g) .

Let ξ_X and ξ_Y be two L-fuzzy topogenous (resp. cotopogenous) orders on X and Y, respectively. A map φ : (X, ξ_X) → (Y, ξ_Y) is said to be an L-topogenous (resp. L-cotopogenous) map if $ξ_{Y} (f, g) \leq ξ_{X} (φ^{\leftarrow} (f), φ^{\leftarrow} (g)), \forall f, g \in L^{Y},$

Remark 2.10. [5, 17] (1) If ξ is an L-fuzzy topogenous order on X. Define a map ξ^s : L^X × L^X → L as ξ^s (f, g) = ξ (g^*, f^*). Then ξ^s is an L-fuzzy cotopogenous order on X.

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

3 The relationships between L-fuzzy topogenous orders and topological structures

Lemma 3.1. Let $I : L^{X} \to L^{X}$ a map. The following statements are equivalent.

For all f, g ∈ L^X, $S (f, g) \leq S (I (f), I (g)) .$

If f ≤ g, then $I (f) \leq I (g)$ and $I (α ⊙ h) \geq α ⊙ I (h)$ for all f, g, h ∈ L^X and α ∈ L.

If f ≤ g, then $I (f) \leq I (g)$ and $I (α \to h) \leq α \to I (h)$ for all f, g, h ∈ L^X and α ∈ L.

Proof. (1) ⇒ (2). If f ≤ g, then $⊤ = S (f, g) \leq S (I (f), I (g)) .$ Hence $I (f) \leq I (g)$ . Put g = α ⊙ f. Then $S (f, α ⊙ f) = α \leq S (I (f), I (α ⊙ f)) .$ Hence $α ⊙ I (f) \leq I (α ⊙ f) .$

(2) ⇒ (3). Since $α ⊙ I (α \to f) \leq I (α ⊙ (α \to f)) \leq I (f),$ $I (α \to f) \leq α \to I (f)$ .

(3) ⇒ (1). Since S (f, g) ⊙ f ≤ g iff f ≤ S (f, g) → g, $I (f) \leq I (S (f, g) \to g) \leq S (f, g) \to I (g) .$ Hence $S (f, g) \leq S (I (f), I (g)) .$

Theorem 3.2. Let $(X, C)$ be an L-closure space. Define a map $F_{C} : L^{X} \to L$ by: $F_{C} (f) = S (C (f), f) .$

Then, $F_{C}$ is an L-fuzzy cotopology on X.

Proof. {(F1) $\begin{matrix} F_{C} (⊤_{X}) & = & ⋀_{x \in X} (⊤_{X} (x) \to C (⊤_{X}) (x)) \\ = & ⊤ \to ⊤ = ⊤, \\ F_{C} (⊥_{X}) & = & ⋀_{x \in X} (⊥_{X} (x) \to C (⊥_{X}) (x)) \\ = & ⊥ \to ⊥ = ⊤ . \end{matrix}$

(F2) By Lemma 2.2(13), we havre $\begin{matrix} F_{C} (f \oplus g) \\ = ⋀_{x \in X} ((f \oplus g) (x) \to C (f \oplus g) (x)) \\ \geq ⋀_{x \in X} ((f \oplus g) (x) \to (C (f) \oplus C (g)) (x)) \\ \geq ⋀_{x \in X} (f (x) \to C (f) (x)) ⊙ ⋀_{x \in X} (g (x) \to C (g) (x)) \\ = F_{C} (f) ⊙ F_{C} (g) . \end{matrix}$

(F3) By Lemma 2.2(16),we have $\begin{matrix} F_{C} (⋀_{C} f_{i}) & = & ⋀_{x \in X} (C (⋀_{i} f_{i}) (x) \to (⋀_{i} f_{i}) (x)) \\ \geq & ⋀_{x \in X} (⋀_{i} C (f_{i}) (x) \to (⋀_{i} f_{i}) (x)) \\ \geq & ⋀_{i} ⋀_{x \in X} (C (f_{i}) (x) \to f_{i} (x)) = ⋀_{i} F_{C} (f_{i}) . \end{matrix}$

The following corollary is similarly proved as Theorem 3.2.

Corollary 3.3. Let $(X, I)$ be an L-interior space. Define a map $T_{I} : L^{X} \to L$ by: $T_{I} (f) = S (f, I (f)) .$

Then $T_{I}$ is an L-fuzzy topology on X.

From the following two theorems, we obtain the L-interior operator and the L-closure operator induced by an L-fuzzy topogenous order.

Theorem 3.4. Let ξ be an L-fuzzy topogenous order on X. We define a mappings $I_{ξ} : L^{X} \to L^{X}$ as $I_{ξ} (f) = ⋁_{g \in L^{X}} (ξ (g, g) ⊙ S (g, f) ⊙ g)$

Then (1) $I_{ξ}$ is a stratified L-interior operator on X.

(2) $T_{I_{ξ}} (f) \geq ξ (f, f)$ for all f ∈ L^X.

(3) Define ξ^s (f, g) = ξ (g^*, f^*). Then ξ^s is an L-fuzzy cotopogenous order on X. Moreover, $I_{ξ^{s}}$ is a stratified L-interior operator on X defined as $I_{ξ^{s}} (f) = ⋁_{g \in L^{X}} (ξ^{s} (g^{*}, g^{*}) ⊙ S (g, f) ⊙ g)$

(4) If ξ is separated, then $I_{ξ}$ is separated.

Proof. {(1) (I1) Since ξ (⊤ _X, ⊤ _X) = ⊤, $\begin{matrix} I_{ξ} (⊤_{X}) & = & ⋁_{g \in L^{X}} (ξ (g, g) ⊙ S (g, ⊤_{X}) ⊙ g) \\ \geq & (ξ (⊤_{X}, ⊤_{X}) ⊙ S (⊤_{X}, ⊤_{X}) ⊙ ⊤_{X}) = ⊤_{X} . \end{matrix}$

(I2) Since S (g, f) ⊙ g ≤ f, then $I_{ξ} (f) \leq ⋁_{g \in L^{X}} (ξ (g, g) ⊙ f) = f .$

(I4) From Lemma 2.4, we obtain $\begin{matrix} I_{ξ} (f) ⊙ I_{ξ} (h) \\ = (⋁_{g \in L^{X}} (ξ (g, g) ⊙ S (g, f) ⊙ g)) \\ ⊙ (⋁_{k \in L^{X}} (ξ (k, k) ⊙ S (k, h) ⊙ k)) \\ = ⋁_{g, k \in L^{X}} (ξ (g, g) ⊙ ξ (k, k) \\ ⊙ S (g, f) ⊙ S (k, h) ⊙ g ⊙ k) \\ \leq ⋁_{g, k \in L^{X}} (ξ (g ⊙ k, g ⊙ k) \\ ⊙ S (g ⊙ k, f ⊙ h) ⊙ (g ⊙ k)) \\ \leq I_{ξ} (f ⊙ h) . \end{matrix}$

(I3) and by Lemma 3.1, $I_{ξ}$ is stratified from: $\begin{matrix} S (I_{ξ} (f), I_{ξ} (h)) \\ = ⋀_{x \in X} (⋁_{g \in L^{X}} (ξ (g, g) ⊙ S (g, f) ⊙ g (x))) \\ \to (⋁_{k \in L^{X}} (ξ (k, k) ⊙ S (k, h) ⊙ k (x))) \\ (by Lemma 2.2 (16)) \\ \geq ⋀_{x \in X} ⋀_{g \in L^{X}} ((ξ (g, g) ⊙ S (g, f) ⊙ g (x)) \\ \to ξ (g, g) ⊙ S (g, h) ⊙ g (x)) \\ \geq ⋀_{g \in L^{X}} (S (g, f) \to S (g, h)) = S (f, h) . \end{matrix}$

(2) $\begin{matrix} T_{I_{ξ}} (f) = S (f, I_{ξ} (f)) \\ = ⋀_{x \in X} (f (x) \to ⋁_{g \in L^{X}} (ξ (g, g) ⊙ S (g, f) ⊙ g (x))) \\ \geq ⋀_{x \in X} (f (x) \to ξ (f, f) ⊙ S (f, f) ⊙ f (x)) \\ \geq ξ (f, f) . \end{matrix}$

(3) It follows from Remark 2.10.

(4) By (I2) and $I_{ξ} (⊤_{x}) = ⋁_{g \in L^{X}} (ξ (g, g) ⊙ S (g, ⊤_{x}) ⊙ g) \geq ξ (⊤_{x}, ⊤_{x}) ⊙ S (⊤_{x}, ⊤_{x}) ⊙ ⊤_{x} = ⊤_{x}$ , we have $I_{ξ} (⊤_{x}) = ⊤_{x} .$

Theorem 3.5. Let ξ be an L-fuzzy topogenous order on X. We define a mappings $C_{ξ} : L^{X} \to L^{X}$ as $C_{ξ} (f) = ⋀_{g \in L^{X}} (S (f, g^{*}) ⊙ g \to ξ^{*} (g, g))$

Then (1) $C_{ξ}$ is a stratified L-closure operator on X.

(2) $F_{C_{ξ}} (f) \geq ξ (f^{*}, f^{*})$ for all f ∈ L^X.

(3) Define ξ^s (f, g) = ξ (g^*, f^*). Then ξ^s is an L-fuzzy cotopogenous order on X. Moreover, $C_{ξ^{s}}$ is a stratified L-closure operator on X defined as $C_{ξ^{s}} (f) = ⋀_{g \in L^{X}} (S (f, g^{*}) ⊙ g \to ξ^{s *} (g^{*}, g^{*}))$

(4) If ξ is separated, then $C_{ξ}$ is separated.

Proof. {(1) (C1) Since ξ (⊤ _X, ⊤ _X) = ⊤, $\begin{matrix} C_{ξ} (⊥_{X}) = ⋀_{g \in L^{X}} (S (⊥_{X}, g) ⊙ g \to ξ^{*} (g, g)) \\ \leq (S (⊥_{X}, ⊤_{X}) ⊙ ⊤_{X} \to ξ^{*} (⊤_{X}, ⊤_{X}) = ⊥_{X} . \end{matrix}$

(C2) Since S (f, g^*) ⊙ g = S (g, f^*) ⊙ g ≤ f^*, then $S (f, g^{*}) ⊙ g \to ξ^{*} (g, g) \geq f^{*} \to ⊥ = f .$

(C4) Since $\begin{matrix} ((a \to b) \oplus (c \to d))^{*} = (a \to b)^{*} ⊙ (c \to d)^{*} \\ = (a ⊙ b^{*}) ⊙ (c ⊙ d^{*}) = (a ⊙ c) ⊙ (b^{*} ⊙ d^{*}), \end{matrix}$ we have (a → b) ⊕ (c → d) = (a ⊙ c) → (b ⊕ d).

From Lemma 2.4, we obtain $\begin{matrix} C_{ξ} (f) \oplus C_{ξ} (h) \\ = (⋀_{g \in L^{X}} ((S (f, g^{*}) ⊙ g) \to ξ^{*} (g, g))) \\ \oplus (⋀_{k \in L^{X}} ((S (h, k^{*}) ⊙ k) \to ξ^{*} (k, k))) \\ = ⋀_{g, k \in L^{X}} ((S (f, g^{*}) ⊙ g) ⊙ (S (h, k^{*}) ⊙ k) \\ \to (ξ^{*} (g, g) \oplus ξ^{*} (k, k))) \\ \geq ⋀_{g, k \in L^{X}} ((S (f \oplus h, g^{*} \oplus k^{*}) ⊙ (g ⊙ k)) \\ \to (ξ^{*} (g ⊙ k, g ⊙ k))) \\ \geq C_{ξ} (f \oplus h) . \end{matrix}$

(C3) and by Lemma 3.1, $C_{ξ}$ is stratified from: $\begin{matrix} S (C_{ξ} (f), C_{ξ} (h)) \\ = ⋀_{x \in X} ((⋀_{g \in L^{X}} ((S (f, g^{*}) ⊙ g) \to ξ^{*} (g, g))) \\ \to (⋀_{k \in L^{X}} ((S (h, k^{*}) ⊙ k) \to ξ^{*} (k, k)))) \\ (by Lemma 2.2 (16)) \\ \geq ⋀_{x \in X} ⋀_{g \in L^{X}} (((S (f, g^{*}) ⊙ g) \to ξ^{*} (g, g)) \\ \to ((S (h, g^{*}) ⊙ g) \to ξ^{*} (g, g))) \\ \geq ⋀_{g \in L^{X}} (S (h, g^{*}) ⊙ g \to S (f, g^{*}) ⊙ g) \\ (by Lemma 2.2 (18)) \\ \geq ⋀_{g \in L^{X}} (S (h, g^{*}) \to S (f, g^{*})) = S (f, h) . \end{matrix}$

Hence $C_{ξ}$ is a stratified L-closure operator on X.

(2) $\begin{matrix} F_{C_{ξ}} (f) = S (C_{ξ} (f), f) \\ = ⋀_{x \in X} ((⋀_{g \in L^{X}} (S (f, g^{*}) ⊙ g (x)) \to ξ^{*} (g, g)) \to f (x)) \\ \geq ⋀_{x \in X} ((S (f, f) ⊙ f^{*} (x)) \to ξ^{*} (f^{*}, f^{*})) \to f (x)) \\ = ⋀_{x \in X} ((ξ (f^{*}, f^{*}) \to f (x)) \to f (x)) \\ \geq ξ (f^{*}, f^{*}) . \end{matrix}$

(3) It follows from Remark 2.10.

(4) By (C2) and $C_{ξ} (⊤_{x}^{*}) = ⋀_{g \in L^{X}} (S (⊤_{x}^{*}, g^{*}) ⊙ g \to ξ^{*} (g, g)) \leq S (⊤_{x}^{*}, ⊤_{x}^{*}) ⊙ ⊤_{x} \to ξ^{*} (⊤_{x}, ⊤_{x}) = ⊤_{x}^{*},$ we have $C_{ξ} (⊤_{x}^{*}) = ⊤_{x}^{*} .$

By similar proofs of Theorems 3.4(3) and 3.5(3), we obtain the L-interior operator and the L-closure operator induced by an L-fuzzy cotopogenous order from the following corollaries.

Corollary 3.6. Let ξ be an L-fuzzy cotopogenous order on X. We define a mappings $I_{ξ} : L^{X} \to L^{X}$ as $I_{ξ} (f) = ⋁_{g \in L^{X}} (ξ (g^{*}, g^{*}) ⊙ S (g, f) ⊙ g)$

Then (1) $I_{ξ}$ is a stratified L-interior operator on X.

(2) $T_{I_{ξ}} (f) \geq ξ (f^{*}, f^{*})$ for all f ∈ L^X.

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

Corollary 3.7. Let ξ be an L-fuzzy cotopogenous order on X. We define a mappings $C_{ξ} : L^{X} \to L^{X}$ as $C_{ξ} (f) = ⋀_{g \in L^{X}} (S (f, g^{*}) ⊙ g \to ξ^{*} (g^{*}, g^{*}))$

Then (1) $C_{ξ}$ is a stratified L-closure operator on X.

(2) $F_{C_{ξ}} (f) \geq ξ (f, f)$ for all f ∈ L^X.

(3) If ξ is separated, then $C_{ξ}$ is separated.

From the following theorem, we obtain the L-fuzzy topogenous order induced by an L-interior operator.

Theorem 3.8. Let $(X, I)$ be an L-interior space. Define a map $ξ_{I} : L^{X} \times L^{X} \to L$ by $ξ_{I} (f, g) = ⋀_{x \in X} (f (x) \to I (g) (x)) .$

Then we have the following properties.

$ξ_{I}$ is an L-fuzzy topogenous order.

$ξ_{I} (f, g) \geq ⋁_{h \in L^{X}} (ξ_{I} (f, h) ⊙ ξ_{I} (h, g))$ , the equality holds if $I$ is topological.

If $I$ is topological, then $ξ_{I}$ is an L-fuzzy topogenous structure on X.

$I \geq I_{ξ_{I}}$ , the equality holds if $I$ is topological.

$T_{I} (f) = ξ_{I} (f, f)$ ,for each f ∈ L^X.

If ξ is an L-fuzzy topogenous order on X, then $ξ_{I_{ξ}} (f, g) \geq S (f, ξ (g, g) ⊙ g)$ , for each f, g ∈ L^X.

If $I$ is separated, then $ξ_{I}$ is separated.

If ξ is a separated L-fuzzy topogenous order on X, then $ξ_{I_{ξ}} \geq ξ$ .

Proof. (1) (ST1) Since $I (⊥_{X}) = ⊥_{X}$ and $I (⊤_{X}) = ⊤_{X}$ , we have $\begin{matrix} ξ_{I} (⊤_{X}, ⊤_{X}) = ⋀_{x \in X} (⊥_{X} (x) \to I (⊤_{X}) (x)) = ⊤ . \\ ξ_{I} (⊥_{X}, ⊥_{X}) = ⋀_{x \in X} (⊥_{X} (x) \to I (⊥_{X}) (x)) = ⊤ . \end{matrix}$

(ST2) Since $I (g) \leq g$ , we have $\begin{matrix} ξ_{I} (f, g) & = & ⋀_{x \in X} (f (x) \to I (g) (x)) \\ \leq & ⋀_{x \in X} (f (x) \to g (x)) = S (f, g) . \end{matrix}$

(ST3) If g ≤ g₁, then $I (g) \leq I (g_{1})$ . Thus, $\begin{matrix} ξ_{I} (f, g) & = & ⋀_{x \in X} (f (x) \to I (g) (x)) \\ \leq & ⋀_{x \in X} (f_{1} (x) \to I (g_{1}) (x)) = ξ_{I} (f_{1}, g_{1}) . \end{matrix}$

(T) For f₁, f₂, g₁, g₂ ∈ L^X, $\begin{matrix} ξ_{I} (f_{1}, g_{1}) ⊙ ξ_{I} (f_{2}, g_{2}) \\ = ⋀_{x \in X} (f_{1} (x) \to I (g_{1}) (x)) \\ ⊙ ⋀_{x \in X} (f_{2} (x) \to I (g_{2}) (x)) \\ \leq ⋀_{x \in X} (f_{1} (x) ⊙ f_{2} (x) \to I (g_{1}) (x) ⊙ I (g_{2}) (x)) \\ (by Lemma 2.2 (11)) \\ \leq ⋀_{x \in X} (f_{1} (x) ⊙ f_{2} (x) \to I (g_{1} ⊙ g_{2}) (x)) \\ = ξ_{I} (f_{1} ⊙ f_{2}, g_{1} ⊙ g_{2}) \end{matrix}$

Hence $ξ_{I}$ is an L-fuzzy topogenous order.

(2) For f, g, h ∈ L^X, $\begin{matrix} ξ_{I} (f, h) ⊙ ξ_{I} (h, g) \\ = (⋀_{x \in X} (f (x) \to I (h) (x))) \\ ⊙ (⋀_{x \in X} (h (x) \to I (g) (x))) \\ = S (f, I (h)) ⊙ S (h, I (g)) \\ (Since I (h) \leq h,) \\ \leq S (f, h) ⊙ S (h, I (g)) \\ \leq S (f, I (g)) = ξ_{I} (f, g) . \end{matrix}$

Hence $ξ_{I} (f, g) \geq ⋁_{h \in L^{X}} (ξ_{I} (f, h) ⊙ ξ_{I} (h, g))$ .

If $I$ is topological, $\begin{matrix} ⋁_{h \in L^{X}} (ξ_{I} (f, h) ⊙ ξ_{I} (h, g)) \\ = ⋁_{h \in L^{X}} (S (f, I (h)) ⊙ S (h, I (g))) \\ (Put h = I (g),) \\ \geq S (f, I (I (g))) ⊙ S (I (g), I (g)) \\ = S (f, I (g)) = ξ_{I} (f, g) . \end{matrix}$

(3) By (2), it is trivial.

(4) Since $S (g, f) \leq S (I (g), I (f))$ from Lemma 3.1, we have $\begin{matrix} I_{ξ_{I}} (f) & = & ⋁_{g \in L^{X}} (ξ_{I} (g, g) ⊙ S (g, f) ⊙ g)) \\ = & S (g, I (g)) ⊙ S (g, f) ⊙ g \\ \leq & S (g, I (g)) ⊙ S (I (g), I (f)) ⊙ g \\ \leq & S (g, I (f)) ⊙ g \leq I (f) . \end{matrix}$

If $I$ is topological, $\begin{matrix} I_{ξ_{I}} (f) & = & ⋁_{g \in L^{X}} (ξ_{I} (g, g) ⊙ S (g, f) ⊙ g) \\ = & ⋁_{g \in L^{X}} (S (g, I (g)) ⊙ S (g, f) ⊙ g) \\ \geq & S (I (f), I (I (f))) ⊙ S (I (f), f) ⊙ I (f) \\ = & I (f) . \end{matrix}$

(5) $T_{I} (f) = ⋀_{x \in X} (f (x) \to I (f) (x)) = ξ_{I} (f, f)$ .

(6) For f, g ∈ L^X, $\begin{matrix} ξ_{I_{ξ}} (f, g) = ⋀_{x \in X} (f (x) \to I_{ξ} (g) (x)) \\ = ⋀_{x \in X} (f (x) \to ⋁_{h \in L^{X}} (ξ (h, h) ⊙ S (h, g) ⊙ h (x)) \\ \geq ⋀_{x \in X} (f (x) \to ξ (g, g) ⊙ S (g, g) ⊙ g (x)) \\ = S (f, ξ (g, g) ⊙ g) . \end{matrix}$

(7) Let $I$ be separated. Then

$ξ_{I} (⊤_{z}, ⊤_{z}) = ⋀_{x \in X} (⊤_{z} (x) \to I (⊤_{z}) (x)) = ⊤ .$

(8) For f, g ∈ L^X, $\begin{matrix} ξ_{I_{ξ}} (f, g) = ⋀_{x \in X} (f (x) \to I_{ξ} (g) (x)) \\ = ⋀_{x \in X} (f (x) \to ⋁_{h \in L^{X}} (ξ (h, h) ⊙ S (h, g) ⊙ h (x))) \\ \geq ⋀_{x \in X} (f (x) \to ξ (⊤_{x}, ⊤_{x}) ⊙ S (⊤_{x}, g) ⊙ ⊤_{x} (x)) \\ = S (f, g) \geq ξ (f, g) . \end{matrix}$

By a similar proof of Theorem 3.8, we obtain the L-fuzzy cotopogenous order induced by an L-interior operator.

Corollary 3.9. Let $(X, I)$ be an L-interior space. Define a map $ξ_{I} : L^{X} \times L^{X} \to L$ by $ξ_{I} (f, g) = ⋀_{x \in X} (g^{*} (x) \to I (f^{*}) (x)) .$

Then we have the following properties.

$ξ_{I}$ is an L-fuzzy cotopogenous order.

$ξ_{I} (f, g) \geq ⋁_{h \in L^{X}} (ξ_{I} (f, h) ⊙ ξ_{I} (h, g))$ , the equality holds if $I$ is topological.

If $I$ is topological, then $ξ_{I}$ is an L-fuzzy cotopogenous structure on X.

$I \geq I_{ξ_{I}}$ , the equality holds if $I$ is topological.

$T_{I} (f) = ξ_{I} (f^{*}, f^{*})$ ,for each f ∈ L^X.

If ξ is an L-fuzzy cotopogenous order on X, then $ξ_{I_{ξ}} (f, g) \geq S (f \to ξ^{*} (f, f), g)$ , for each f, g ∈ L^X.

If $I$ is separated, then $ξ_{I}$ is separated.

If ξ is a separated L-fuzzy cotopogenous order on X, then $ξ_{I_{ξ}} \geq ξ$ .

From the following theorem, we obtain the L-fuzzy cotopogenous order induced by an L-closure operator.

Theorem 3.10. Let $(X, C)$ be an L-closure space. Define a map $ξ_{C} : L^{X} \times L^{X} \to L$ by $ξ_{C} (f, g) = ⋀_{x \in X} (C (f) (x) \to g (x)) .$

Then we have the following properties.

$ξ_{C}$ is an L-fuzzy cotopogenous order.

$ξ_{C} (f, g) \geq ⋁_{h \in L^{X}} (ξ_{C} (f, h) ⊙ ξ_{C} (h, g))$ , the equality holds if $C$ is topological.

If $C$ is topological, then $ξ_{C}$ is an L-fuzzy topogenous structure on X.

$C \leq C_{ξ_{C}}$ , the equality holds if $C$ is topological.

$F_{C} (f) = ξ_{C} (f, f)$ ,for each f ∈ L^X.

If ξ is an L-fuzzy cotopogenous order on X, then $ξ_{C_{ξ}} (f, g) \geq S ((ξ (f, f) \to f, g)$ , for each f, g ∈ L^X.

If $C$ is separated, then $ξ_{C}$ is separated.

If ξ is a separated L-fuzzy cotopogenous order on X, then $ξ_{C_{ξ}} \geq ξ$ .

Proof. {(1) (P1) Since $C (⊥_{X}) = ⊥_{X}$ and $C (⊤_{X}) = ⊤_{X}$ , we have $\begin{matrix} ξ_{C} (⊤_{X}, ⊤_{X}) = ⋀_{x \in X} (C (⊤_{X}) (x) \to ⊤_{X} (x)) = ⊤ . \\ ξ_{C} (⊥_{X}, ⊥_{X}) = ⋀_{x \in X} (C (⊥_{X}) (x) \to ⊥_{X} (x)) = ⊤ . \end{matrix}$

(P2) Since $C (f) \geq f$ , we have $\begin{matrix} ξ_{C} (f, g) & = & ⋀_{x \in X} (C (f) (x) \to g (x)) \\ \leq & ⋀_{x \in X} (f (x) \to g (x)) = S (f, g) . \end{matrix}$

(P3) If f ≥ f₁ and g ≤ g₁, then $C (f) \leq C (f_{1})$ . Thus, $\begin{matrix} ξ_{C} (f, g) & = & ⋀_{x \in X} (C (f) (x) \to g (x)) \\ \leq & ⋀_{x \in X} (C (f_{1}) (x) \to g_{1} (x)) = ξ_{C} (f_{1}, g_{1}) . \end{matrix}$

(P4) $\begin{matrix} ξ_{C} (f_{1}, g_{1}) ⊙ ξ_{C} (f_{2}, g_{2}) \\ = ⋀_{x \in X} (C (f_{1}) (x) \to g_{1} (x)) \\ ⊙ ⋀_{x \in X} (C (f_{2}) (x) \to g_{2} (x)) \\ \leq ⋀_{x \in X} (C (f_{1}) (x) \oplus C (f_{2}) (x) \to (g_{1} \oplus g_{2}) (x))) \\ (by Lemma 2.2 (13)) \\ \leq ⋀_{x \in X} (C (f_{1} \oplus f_{2}) (x) \to (g_{1} \oplus g_{2}) (x)) \\ = ξ_{C} (f_{1} \oplus f_{2}, g_{1} \oplus g_{2}) \end{matrix}$

Hence $ξ_{C}$ is an L-fuzzy cotopogenous order.

(2) $\begin{matrix} ξ_{C} (f, h) ⊙ ξ_{C} (h, g) \\ = (⋀_{x \in X} (C (f) (x) \to h (x))) \\ ⊙ (⋀_{x \in X} (C (h) (x) \to g (x))) \\ = S (C (f), h) ⊙ S (C (h), g) \\ (by C (h) \geq h,) \\ \leq S (C (f), h) ⊙ S (h, g) \\ \leq S (C (f), g) = ξ_{C} (f, g) . \end{matrix}$

Hence $ξ_{C} (f, g) \geq ⋁_{h \in L^{X}} (ξ_{C} (f, h) ⊙ ξ_{C} (h, g))$ .

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

(3) By (2), it is trivial.

(4) Since $S (g, f) \leq S (C (g), C (f))$ from Lemma 3.1, we have, by (1) and Corollary 3.7, $\begin{matrix} C_{ξ_{C}}^{*} (f) & = & ⋁_{g \in L^{X}} (ξ_{C} (g^{*}, g^{*}) ⊙ S (f, g^{*}) ⊙ g)) \\ \leq & ⋁_{g \in L^{X}} (S (C (g^{*}), g^{*}) ⊙ S (C (f), C (g^{*})) ⊙ g) \\ \leq & ⋁_{g \in L^{X}} (S (C (f), g^{*}) ⊙ g) (by Lemma 2.4 (6)) \\ = & ⋁_{g \in L^{X}} (S (g, C^{*} (f)) ⊙ g) = C^{*} (f) . \end{matrix}$

If $C$ is topological, $\begin{matrix} C_{ξ_{C}}^{*} (f) & = & ⋁_{g \in L^{X}} (ξ_{C} (g^{*}, g^{*}) ⊙ S (f, g^{*}) ⊙ g)) \\ = & ⋁_{g \in L^{X}} (S (C (g^{*}), g^{*}) ⊙ S (f, g^{*}) ⊙ g) \\ (Put g^{*} = C (f),) \\ \geq & S (C (C (f)), C (f)) ⊙ S (f, C (f)) ⊙ C^{*} (f) \\ = & C^{*} (f) . \end{matrix}$

(5) $F_{C} (f) = ⋀_{x \in X} (C (f) (x) \to f (x)) = ξ_{C} (f, f) .$

(6) $\begin{matrix} ξ_{C_{ξ}} (f, g) = ⋀_{x \in X} (C_{ξ} (f) (x) \to g (x)) \\ = ⋀_{x \in X} (⋀_{h \in L^{X}} (S (f, h^{*}) ⊙ h (x) \to ξ^{*} (h^{*}, h^{*})) \\ \to g (x)) (put h = f^{*},) \\ \geq ⋀_{x \in X} ((f^{*} (x) \to ξ^{*} (f, f)) \to g (x)) \\ = ⋀_{x \in X} ((ξ (f, f) \to f (x)) \to g (x)) \\ = S (ξ (f, f) \to f, g) . \end{matrix}$

(7) $ξ_{C_{ξ}} (⊤_{x}^{*}, ⊤_{x}^{*}) = ⋀_{x \in X} (C_{ξ} (⊤_{x}^{*}) (x) \to ⊤_{x}^{*} (x)) = ⊤$ .

(8) $\begin{matrix} ξ_{C_{ξ}} (f, g) = ⋀_{x \in X} (C_{ξ} (f) (x) \to g (x)) \\ = ⋀_{x \in X} (⋀_{h \in L^{X}} (S (f, h^{*}) ⊙ h (x) \to ξ^{*} (h^{*}, h^{*})) \\ \to g (x)) \\ \geq ⋀_{x \in X} ((S (f, ⊤_{x}^{*}) ⊙ ⊤_{x} (x) \to ξ^{*} (⊤_{x}^{*}, ⊤_{x}^{*})) \\ \to g (x)) \\ = S (f, g) \geq ξ (f, g) . \end{matrix}$

Corollary 3.11. Let $(X, C)$ be an L-closure space. Define a map $ξ_{C} : L^{X} \times L^{X} \to L$ by $ξ_{C} (f, g) = ⋀_{x \in X} (C (g^{*}) (x) \to f^{*} (x)) .$

Then we have the following properties.

$ξ_{C}$ is an L-fuzzy topogenous order.

$ξ_{C} (f, g) \geq ⋁_{h \in L^{X}} (ξ_{C} (f, h) ⊙ ξ_{C} (h, g))$ , the equality holds if $C$ is topological.

If $C$ is topological, then $ξ_{C}$ is an L-fuzzy topogenous structure on X.

$C \leq C_{ξ_{C}}$ , the equality holds if $C$ is topological.

$F_{C} (f) = ξ_{C} (f^{*}, f^{*})$ ,for each f ∈ L^X.

If ξ is an L-fuzzy topogenous order on X, then $ξ_{C_{ξ}} (f, g) \geq S ((ξ (f^{*}, f^{*}) \to f, g)$ , for each f, g ∈ L^X.

If $C$ is separated, then $ξ_{C}$ is separated.

If ξ is a separated L-fuzzy topogenous order on X, then $ξ_{C_{ξ}} \geq ξ$ .

From Theorems 3.8 and 3.10, we obtain the L-fuzzy (co)topogenous order induced by an L-topology.

Remark 3.12. {A subset τ ⊂ L^X is called an L-topology (ref. [8]) if (O1) ⊤_X, ⊥ _X ∈ τ, (O2) f ⊙ g ∈ τ for f, g ∈ τ, (O3) ⋁_i∈Γf_i ∈ τ for {f_i ∣ i ∈ Γ} ⊂ τ. Define $I_{τ}, C_{τ} : L^{X} \to L^{X}$ as follows: $\begin{matrix} I_{τ} (f) & = & ⋁ {g \in L^{X} ∣ g \leq f, g \in τ}, \\ C_{τ} (f) & = & ⋀ {g \in L^{X} ∣ f \leq g, g^{*} \in τ} . \end{matrix}$

$I_{τ}$ is a topological L-interior operator.

From Theorem 3.8(3), $ξ_{I_{τ}}$ is an L-fuzzy topogenous structure.

We obtain an L-fuzzy topology defined as $T_{I_{τ}} (f) = ξ_{I_{τ}} (f, f)$ , for each f ∈ L^X.

$I_{ξ_{I_{τ}}} = I_{τ}$ .

$C_{τ}$ is a topological L-fuzzy closure operator such that $C_{τ} (f) = (I_{τ} (f^{*}))^{*}$ , for each f ∈ L^X.

From Theorem 3.10(3), $ξ_{C_{τ}}$ is an L-fuzzy cotopogenous structure.

$ξ_{C_{τ}}^{s} (f, g) = ξ_{C_{τ}} (g^{*}, f^{*}) = S (C_{τ} (g^{*}), f^{*}) = S (f, C_{τ}^{*} (g^{*})) = S (f, I_{τ} (g)) = ξ_{I_{τ}} (f, g)$ for each f ∈ L^X.

We obtain an L-fuzzy cotopology defined as $F_{C_{τ}} (f) = ξ_{C_{τ}} (f, f)$ , for each f ∈ L^X.

$C_{ξ_{C_{τ}}^{s}} = C_{τ}$ .

Example 3.13. Let ([0, 1], ⊙, ⊕, →, ^*, 0, 1) be a complete residuated lattice (ref. [2 , 6–8]) as $\begin{matrix} x ⊙ y & = & max {0, x + y - 1}, x \to y \\ = & min {1 - x + y, 1} \\ x \oplus y & = & min {1, x + y}, x^{*} = 1 - x . \end{matrix}$

Let X = {x, y, z} and f ∈ [0, 1] ^X as follow: $\begin{matrix} f & = & (0.6, 0.3, 0.6), f ⊙ f = (0.2, 0, 0.2) \\ g & = & (0.5, 0.1, 0.4), f ⊙ g = (0.1, 0, 0) . \end{matrix}$

(1) Define a [0, 1]-interior operator $I : [0, 1]^{X} \to [0, 1]^{X}$ as follows: $I (h) = {\begin{matrix} 1_{X}, & if h = 1_{X}, \\ f, & if h \geq f, \\ f ⊙ f & if h \geq f ⊙ f, h ≱ f, \\ 0_{X}, & otherwise . \end{matrix}$

Since $I$ is topological, by Theorem 3.8(3), we obtain a [0.1]-fuzzy topogenous structure $ξ_{I} : [0, 1]^{X} \times [0, 1]^{X} \to [0, 1]$ as follows $ξ_{I} (h, g) = {\begin{matrix} 1, & if h = 0_{X} \\ or g = 1_{X}, \\ S (h, f), & if g \geq f, \\ S (h, f ⊙ f), & if g \geq f ⊙ f, \\ g ≱ f, \\ S (h, 0_{X}), & otherwise, \end{matrix}$

Moreover, since $I$ is topological, by Theorem 3.8(4), $I_{ξ_{I}} = I$ . Since $I (1_{x}) = 0_{X}$ , $I$ is not separated. Since $ξ_{I} (1_{x}, 1_{x}) = 0$ , $ξ_{I}$ is not separated.

(2) Define ξ : [0, 1] ^X × [0, 1] ^X → [0, 1] as $ξ (f, g) = S (f, g) .$

Then ξ is a [0, 1]-fuzzy bitopogenous structure from: $\begin{matrix} ξ (f_{1}, g_{1}) ⊙ ξ (f_{2}, g_{2}) = S (f_{1}, g_{1}) ⊙ S (f_{2}, g_{2}) \\ \leq S (f_{1} ⊙ f_{2}, g_{1} ⊙ g_{2}) = ξ_{I} (f_{1} ⊙ f_{2}, g_{1} ⊙ g_{2}) \\ ξ (f_{1}, g_{1}) ⊙ ξ (f_{2}, g_{2}) = S (f_{1}, g_{1}) ⊙ S (f_{2}, g_{2}) \\ \leq S (f_{1} \oplus f_{2}, g_{1} \oplus g_{2}) = ξ (f_{1} \oplus f_{2}, g_{1} \oplus g_{2}) \\ ⋁_{h \in L^{X}} (ξ (f, h) ⊙ ξ (h, g)) \\ = ⋁_{h \in L^{X}} (S (f, h) ⊙ S (h, g)) = S (f, g) = ξ (f, g) . \end{matrix}$

By Theorems 3.4 and 3.5, we obtain: $\begin{matrix} I_{ξ} (f) = ⋁_{g \in L^{X}} (ξ (g, g) ⊙ S (g, f) ⊙ g) \\ = ⋁_{g \in L^{X}} (S (g, g) ⊙ S (g, f) ⊙ g) = f \\ C_{ξ} (f) = ⋀_{g \in L^{X}} (S (f, g^{*}) ⊙ g \to ξ^{*} (g, g)) \\ \leq S (f, f) ⊙ f^{*} \to S^{*} (f^{*}, f^{*}) \\ = f^{*} \to 0 = f . \end{matrix}$

Thus, $ξ_{I_{ξ}} (f, g) = S (f, I_{ξ} (g)) = S (f, g) = ξ (f, g) = S (C_{ξ} (f), g) = ξ_{C_{ξ}} (f, g) .$ Moreover, $T_{I_{ξ}} (f) = S (f, I_{ξ} (f)) = S (f, f) = 1$ and $F_{C_{ξ}} (f) = S (C_{ξ} (f), f) = S (f, f) = 1 .$ Since $ξ (1_{x}, 1_{x}) = ξ (1_{x}^{*}, 1_{x}^{*}) = 1$ for each x ∈ X, ξ is separated [0, 1]-fuzzy topogenous and cotopgenous order.

(3) Define a [0, 1]-topology τ = {0_X, 1_X, f, g, f ⊙ f, f ⊙ g}. $\begin{matrix} f^{*} & = & (0.4, 0.7, 0.4), f^{*} \oplus f^{*} = (0.8, 1, 0.8) \\ g^{*} & = & (0.5, 0.9, 0.6), f^{*} \oplus g^{*} = (0.9, 1, 1) . \end{matrix}$

By Remark 3.8, we obtain topological [0, 1]-interior and closure operators $I_{τ}, C_{τ} : [0, 1]^{X} \to [0, 1]^{X}$ as follows: $\begin{matrix} I_{τ} (h) = {\begin{matrix} 1_{X}, & if h = 1_{X}, \\ f, & if h \geq f, \\ g, & if h \geq g, h ≱ f, \\ f ⊙ f & if h \geq f ⊙ f, h ≱ f, h ≱ g, \\ g ⊙ f & if h \geq g ⊙ f, h ≱ f ⊙ f, \\ 0_{X}, & otherwise . \end{matrix} \\ C_{τ} (h) \\ = {\begin{matrix} 0_{X}, & if h = 0_{X}, \\ f^{*}, & if h \leq f^{*}, \\ g^{*}, & if h \leq g^{*}, h ≰ f^{*}, \\ f^{*} \oplus f^{*} & if h \leq f^{*} \oplus f^{*}, h ≰ f^{*}, h ≰ g^{*}, \\ f^{*} \oplus g^{*} & if h \leq f^{*} \oplus g^{*}, h ≰ f^{*} \oplus f^{*}, \\ 1_{X}, & otherwise . \end{matrix} \end{matrix}$

Since $I_{τ}$ is topological, by Theorem 3.8(3), we obtain a [0, 1]-fuzzy topogenous structure $ξ_{I_{τ}} : [0, 1]^{X} \times [0, 1]^{X} \to [0, 1]$ as follows $ξ_{I_{τ}} (h, k) = {\begin{matrix} 1, & if h = 0_{X} \\ or k = 1_{X}, \\ S (h, f), & if k \geq f \\ S (h, g), & if k \geq g, k ≱ f, \\ S (h, f ⊙ f), & if k \geq f ⊙ f, \\ k ≱ f, k ≱ g, \\ S (h, f ⊙ g), & if k \geq f ⊙ g, \\ k ≱ f ⊙ f, \\ S (h, 0_{X}), & otherwise, \end{matrix}$

Since $C$ is topological, by Theorem 3.10(3), we obtain a [0, 1]-fuzzy cotopogenous structure $ξ_{C_{τ}} : [0, 1]^{X} \times [0, 1]^{X} \to [0, 1]$ as follows $ξ_{C_{τ}} (h, k) = {\begin{matrix} 1, & if h = 0_{X}, or k = 1_{X}, \\ S (f^{*}, k), & if h \leq f^{*}, \\ S (g^{*}, k), & if h \leq g^{*}, h ≰ f^{*} \\ S (f^{*} \oplus f^{*}, k), & if h \leq f^{*} \oplus f^{*}, \\ h ≰ f^{*}, h ≰ g^{*} \\ S (f^{*} \oplus g^{*}, k), & if h \leq f^{*} \oplus g^{*}, \\ h ≰ f^{*} \oplus f^{*} \\ S (1_{X}, k), & otherwise . \end{matrix}$

Moreover, since $I_{τ}$ and $C_{τ}$ are topological, by Theorems 3.8(4) and 3.10(4), $I_{ξ_{I_{τ}}} = I_{τ}$ and $C_{ξ_{C_{τ}}} = C_{τ}$ .

4 Galois correspondences

Theorem 4.1. Let (X, ξ_X) and (Y, ξ_Y) be L-fuzzy topogenous spaces and φ : X → Y a map. Then the following hold:

For each f ∈ L^Y, $\begin{matrix} ⋀_{g \in L^{Y}} (ξ_{Y} (g, g) \to ξ_{X} (φ^{\leftarrow} (g), φ^{\leftarrow} (g))) \\ \leq S (φ^{\leftarrow} (I_{ξ_{Y}} (f)), I_{ξ_{X}} (φ^{\leftarrow} (f))) . \end{matrix}$

For each f ∈ L^Y, $\begin{matrix} ⋀_{g \in L^{Y}} (ξ_{Y} (g, g) \to ξ_{X} (φ^{\leftarrow} (g), φ^{\leftarrow} (g))) \\ \leq S (C_{ξ_{X}} (φ^{\leftarrow} (f)), φ^{\leftarrow} (C_{ξ_{Y}} (f))) . \end{matrix}$

If a map φ : (X, ξ_X) → (Y, ξ_Y) is L-topogenous map, then $φ : (X, I_{ξ_{X}}) \to (Y, I_{ξ_{Y}})$ is an L-interior map.

If a map φ : (X, ξ_X) → (Y, ξ_Y) is L-topogenous map, then $φ : (X, C_{ξ_{X}}) \to (Y, C_{ξ_{Y}})$ is an L-closed map.

Proof. (1) For each f ∈ L^Y, $\begin{matrix} S (φ^{\leftarrow} (I_{ξ_{Y}} (f)), I_{ξ_{X}} (φ^{\leftarrow} (f))) \\ = ⋀_{x \in X} (φ^{\leftarrow} (⋁_{g \in L^{Y}} ξ_{Y} (g, g) ⊙ S (g, f) ⊙ g) (x) \\ \to ⋁_{h \in L^{X}} (ξ_{X} (h) ⊙ S (h, φ^{\leftarrow} (f)) ⊙ h (x))) \\ \geq ⋀_{x \in X} ((⋁_{g \in L^{Y}} ξ_{Y} (g, g) ⊙ S (g, f) ⊙ g (φ (x))) \to \\ ⋁_{g \in L^{Y}} (ξ_{X} (φ^{\leftarrow} (g)) ⊙ S (φ^{\leftarrow} (g), φ^{\leftarrow} (f)) \\ ⊙ g (φ (x)))) \\ \geq ⋀_{x \in X} ⋀_{g \in L^{Y}} (ξ_{Y} (g, g) ⊙ S (φ^{\leftarrow} (g), φ^{\leftarrow} (f)) \\ ⊙ g (φ (x)) \to ξ_{X} (φ^{\leftarrow} (g), φ^{\leftarrow} (g)) \\ ⊙ S (φ^{\leftarrow} (g), φ^{\leftarrow} (f)) ⊙ g (φ (x))) \\ \geq ⋀_{g \in L^{Y}} (ξ_{Y} (g, g) \to ξ_{X} (φ^{\leftarrow} (g), φ^{\leftarrow} (g))) . \end{matrix}$

(2) $\begin{matrix} S (C_{ξ_{X}} (φ^{\leftarrow} (f)), φ^{\leftarrow} (C_{ξ_{Y}} (f))) \\ = ⋀_{x \in X} (⋀_{h \in L^{X}} (ξ_{X} (h) ⊙ S (φ^{\leftarrow} (f), h^{*}) ⊙ h (x) \\ \to ξ_{X}^{*} (h, h))) \\ \to (⋀_{y \in Y} φ^{\leftarrow} (⋀_{g \in L^{Y}} (S (f, g^{*}) ⊙ g (y) \to ξ_{Y}^{*} (g, g))) \\ \geq ⋀_{x \in X} (⋀_{g \in L^{Y}} (S (φ^{\leftarrow} (f), φ^{\leftarrow} (g)^{*}) ⊙ φ^{\leftarrow} (g) (x) \\ \to ξ_{X}^{*} (φ^{\leftarrow} (g), φ^{\leftarrow} (g)))) \\ \to (⋀_{x \in X} (⋀_{g \in L^{Y}} (S (f, g^{*}) ⊙ g (φ (x)) \\ \to ξ_{Y}^{*} (g, g))) \\ \geq ⋀_{x \in X} (⋀_{g \in L^{Y}} (S (φ^{\leftarrow} (f), φ^{\leftarrow} (g)^{*}) ⊙ φ^{\leftarrow} (g) (x) \\ \to ξ_{X}^{*} (φ^{\leftarrow} (g), φ^{\leftarrow} (g)))) \\ \to (⋀_{x \in X} (⋀_{g \in L^{Y}} (S (φ^{\leftarrow} (f), φ^{\leftarrow} (g)^{*}) ⊙ φ^{\leftarrow} (g) (x) \\ \to ξ_{Y}^{*} (g, g))) \\ \geq ⋀_{x \in X} ⋀_{g \in L^{Y}} (S (φ^{\leftarrow} (f), φ^{\leftarrow} (g)^{*}) ⊙ φ^{\leftarrow} (g) (x) \\ \to ξ_{X}^{*} (φ^{\leftarrow} (g), φ^{\leftarrow} (g))) \\ \to (S (φ^{\leftarrow} (f), φ^{\leftarrow} (g)^{*}) ⊙ φ^{\leftarrow} (g) (x) \to ξ_{Y}^{*} (g, g))) \\ \geq ⋀_{g \in L^{Y}} (ξ_{X}^{*} (φ^{\leftarrow} (g), φ^{\leftarrow} (g)) \to ξ_{Y}^{*} (g, g)) \\ = ⋀_{g \in L^{Y}} (ξ_{Y} (g, g) \to ξ_{X} (φ^{\leftarrow} (g), φ^{\leftarrow} (g))) \end{matrix}$

(3) By (1), if ξ_Y (g, h) ≤ ξ_X (φ^← (g), φ^← (h)) for each g ∈ L^Y, then $φ^{\leftarrow} (I_{ξ_{Y}} (f)) \leq I_{ξ_{X}} (φ^{\leftarrow} (f))$ .

(4) By (2), if ξ_Y (g, h) ≤ ξ_X (φ^← (g), φ^← (h)) for each g ∈ L^Y, then $C_{ξ_{X}} (φ^{\leftarrow} (f)) \leq φ^{\leftarrow} (C_{ξ_{Y}} (f)))$ .

Corollary 4.2. Let (X, ξ_X) and (Y, ξ_Y) be L-fuzzy cotopogenous spaces and φ : X → Y a map. Then

If a map φ : (X, ξ_X) → (Y, ξ_Y) is L-cotopogenous map, then $φ : (X, I_{ξ_{X}}) \to (Y, I_{ξ_{Y}})$ is an L-interior map.

If a map φ : (X, ξ_X) → (Y, ξ_Y) is L-cotopogenous map, then $φ : (X, C_{ξ_{X}}) \to (Y, C_{ξ_{Y}})$ is an L-closed map.

Theorem 4.3. Let $(X, I_{X})$ and $(Y, I_{Y})$ be L-interior spaces and φ : X → Y a map. Then $\begin{matrix} S (φ^{\leftarrow} (I_{Y} (f)), I_{X} (φ^{\leftarrow} (f))) \\ \leq ξ_{I_{Y}} (f, g) \to ξ_{I_{X}} (φ^{\leftarrow} (f), φ^{\leftarrow} (g))) . \end{matrix}$

Moreover, if a map $φ : (X, I_{X}) \to (Y, I_{Y})$ is an L-interior map, then $φ : (X, ξ_{I_{X}}) \to (Y, ξ_{I_{Y}})$ is L-fuzzy topogenous map.

Proof. By Lemma 2.2, we have $\begin{matrix} ξ_{I_{Y}} (f, g) \to ξ_{I_{X}} (φ^{\leftarrow} (f), φ^{\leftarrow} (g)) \\ = ⋀_{y \in Y} (f (y) \to I_{Y} (g) (y)) \\ \to ⋀_{x \in X} (φ^{\leftarrow} (f) (x) \to I_{X} (φ^{\leftarrow} (g)) (x)) \\ \geq ⋀_{x \in X} (φ (f (x)) \to I_{Y} (g) (φ (f (x)))) \\ \to ⋀_{x \in X} (φ^{\leftarrow} (f) (x) \to I_{X} (φ^{\leftarrow} (g)) (x)) \\ \geq ⋀_{x \in X} (φ^{\leftarrow} (f) (x) \to φ^{\leftarrow} (I_{X} (g) (x)) \\ \to (φ^{\leftarrow} (f) (x) \to I_{X} (φ^{\leftarrow} (g)) (x)) \\ \geq ⋀_{x \in X} (φ^{\leftarrow} (I_{Y} (g)) (x) \to I_{X} (φ^{\leftarrow} (g)) (x)) . \end{matrix}$

Moreover, if $φ^{\leftarrow} (I_{Y} (f)) \leq I_{X} (φ^{\leftarrow} (f))$ , then $ξ_{I_{Y}} (f) \leq ξ_{I_{X}} (φ^{\leftarrow} (f))$ .

Theorem 4.4. Let $(X, C_{X})$ and $(Y, C_{Y})$ be L-closure spaces and φ : X → Y a map. Then $\begin{matrix} S (C_{ξ_{X}} (φ^{\leftarrow} (f)), φ^{\leftarrow} (C_{ξ_{Y}} (f))) \\ \leq ξ_{C_{Y}} (f, g) \to ξ_{C_{X}} (φ^{\leftarrow} (f), φ^{\leftarrow} (g))) . \end{matrix}$

Moreover, if a map $φ : (X, C_{X}) \to (Y, C_{Y})$ is an L-closed map, then $φ : (X, ξ_{C_{X}}) \to (Y, ξ_{C_{Y}})$ is L-fuzzy topogenous map.

Proof. By Lemma 2.2, we have $\begin{matrix} ξ_{C_{Y}} (f, g) \to ξ_{C_{X}} (φ^{\leftarrow} (f), φ^{\leftarrow} (g)) \\ = ⋀_{y \in Y} (C_{Y} (f) (y) \to g (y)) \\ \to ⋀_{x \in X} (C_{Y} (φ^{\leftarrow} (f)) (x) \to φ^{\leftarrow} (g) (x)) \\ \geq ⋀_{x \in X} (C_{Y} (f) (φ (x)) \to g (φ (x))) \\ \to ⋀_{x \in X} (C_{Y} (φ^{\leftarrow} (f)) (x) \to φ^{\leftarrow} (g) (x)) \\ = ⋀_{x \in X} (φ^{\leftarrow} (C_{Y} (f)) (x) \to φ^{\leftarrow} (g) (x)) \\ \to ⋀_{x \in X} (C_{Y} (φ^{\leftarrow} (f)) (x) \to φ^{\leftarrow} (g) (x)) \\ = ⋀_{x \in X} (φ^{\leftarrow} (C_{Y} (f)) (x) \to φ^{\leftarrow} (g) (x)) \\ \to (C_{Y} (φ^{\leftarrow} (f)) (x) \to φ^{\leftarrow} (g) (x))) \\ \geq ⋀_{x \in X} (C_{Y} (φ^{\leftarrow} (f)) (x) \to φ^{\leftarrow} (C_{Y} (f) (x)) . \end{matrix}$

Moreover, if $C_{Y} (φ^{\leftarrow} (f)) \leq φ^{\leftarrow} (C_{Y} (f)$ , then $ξ_{C_{Y}} (f, g) \leq ξ_{C_{X}} (φ^{\leftarrow} (f), φ^{\leftarrow} (g))$ .

Corollary 4.5. (1) Let $(X, I_{X})$ and $(Y, I_{Y})$ be L-interior spaces. If a map $φ : (X, I_{X}) \to (Y, I_{Y})$ is an L-interior map, then $φ : (X, ξ_{I_{X}}) \to (Y, ξ_{I_{Y}})$ is L-fuzzy cotopogenous map.

(2) Let $(X, C_{X})$ and $(Y, C_{Y})$ be L-closure spaces. If a map $φ : (X, C_{X}) \to (Y, C_{Y})$ is an L-closed map, then $φ : (X, ξ_{C_{X}}) \to (Y, ξ_{C_{Y}})$ is L-fuzzy cotopogenous map.

Definition 4.6. [1] Suppose that $F : D \to C,$ $G : C \to D$ are concrete functors. The pair (F, G) is called a Galois correspondence between $C$ and $D$ if 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.

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 .

The category of separated L-fuzzy topogenous spaces (resp. separated L-fuzzy cotopogenous spaces) with L-topogenous mappings (resp. L-fuzzy cotopogenous maps) as morphisms is denoted by SL-FTOP (resp. SL-FCTOP).

The category of separated L-fuzzy interior spaces (resp. separated L-fuzzy closure spaces) with L-interior mappings (resp. L-closure maps) as morphisms is denoted by SL-FI (resp. SL-FC).

From Theorems 3.4 and 4.1, we obtain a concrete functor ϒ : SL - FTOP → SL - FI defined as $ϒ (X, ξ) = (X, I_{ξ}), ϒ (φ) = φ .$

From Theorems 3.8 and 4.3, we obtain a concrete functor Ω : SL - FI → SL - FTOP defined as $Ω (X, I) = (X, ξ_{I}), Ω (φ) = φ .$

Theorem 4.7. Ω : SL - FI → SL - FTOP is a left adjoint of ϒ : SL - FTOP → SL - FI, i.e., (ϒ, Ω) is a Galois connection.

Proof. By Theorem 3.8(4), if $I_{X}$ is an separated L-fuzzy interior operator on a set X, then $ϒ (Ω (I_{X})) = I_{ξ_{I_{X}}} \leq I_{X}$ . Hence, the identity map ${id}_{X} : (X, I_{X}) \to (X, I_{T_{I_{X}}}) = (X, ϒ (Ω (I_{X})))$ is an L-interior map. Moreover, if ξ_Y is a separated L-fuzzy topogeous order on a set Y, by Theorem 3.8(8), $Ω (ϒ (ξ_{Y})) = ξ_{I_{ξ_{Y}}} \geq ξ_{Y}$ . Hence the identity map ${id}_{Y} : (Y, ξ_{I_{ξ_{Y}}}) \to (Y, ξ_{Y})$ is L-topogenous map. Therefore (ϒ, Ω) is a Galois correspondence.

From Corollaries 3.7 and 4.2, we obtain a concrete functor Θ : SL - FCTOP → SL - FC defined as $Θ (X, ξ) = (X, C_{ξ}), Θ (φ) = φ .$

From Theorem 3.10 and Corollary 4.5, we obtain a concrete functor Γ : SL - FC → SL - FCTOP defined as $Γ (X, C) = (X, ξ_{C}), Γ (φ) = φ .$

Theorem 4.8. Γ : SL - FC → SL - FCTOP is a left adjoint of Θ : SL - FCTOP → SL - FC, i.e., (Θ, Γ) is a Galois connection.

Proof. By Theorem 3.10(4), if $C_{X}$ is a separated L-fuzzy closure operator on a set X, then $Θ (Γ (C_{X})) = C_{ξ_{C_{X}}} \geq C_{X}$ . Hence, the identity map ${id}_{X} : (X, C_{X}) \to (X, C_{T_{C_{X}}}) = (X, Θ (Γ (C_{X})))$ is an L-closed map. Moreover, if ξ_Y is a separated L-fuzzy cotopogeous order on a set Y, by Theorem 3.10(8), $Γ (Θ (ξ_{Y})) = ξ_{C_{ξ_{Y}}} \geq ξ_{Y}$ . Hence the identity map id_Y : (Y, Γ (Θ (ξ_Y))) → (Y, ξ_Y) is L-cotopogenous map. Therefore (Θ, Γ) is a Galois correspondence.

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