Fuzzy Galois connections on Alexandrov L -topologies

Abstract

In this paper, we introduce the notion of Galois and dual Galois connections as a topological viewpoint of concept lattices in a complete residuated lattice. Under various relations, we investigate the Galois and dual Galois connections on Alexandrov L-topologies. Moreover, their properties and examples are investigated.

Keywords

Complete residuated lattices Galois and dual Galois connections

1 Introduction

Ward et al. [25] introduced a complete residuated lattice which is algebraic structure for many valued logic [3]. It is an important mathematical tool as algebraic structures for many valued logics [1–6 , 27]. Pawlak [16, 17] introduced the rough set theory as a formal tool to deal with imprecision and uncertainty in the data analysis. For an extension of Pawlak’s rough sets, many researchers [1 , 28] developed L-lower and L-upper approximation operators in complete residuated lattices.

A Galois connection is a pair of maps from a partially ordered set to another partially ordered set. Examples of two maps which form Galois connections play an important role in many areas. Some recent results on partially ordered sets can be found in Orłowska and Rewitzky [14, 15]. A context consists of a triple (U, V, R) where U is a set of objects, V is a set of attributes and R is a relation between U and V. Wille [26] introduced the formal concept lattices by allowing some uncertainty in contexts. Bělohlávek [1,2, 1,2] developed the notion of fuzzy contexts with R ∈ L^X×Y on a complete residuated lattice L and introduced the concepts of fuzzy Galois connections. Moreover, Bělohlávek [1,2, 1,2] investigated concept lattices, information systems and decision rules on complete residuated lattices.

An interesting and natural research topic in rough set theory is the study of rough set theory and topological structures. Lai [10] and Ma [11] investigated the Alexandrov L-topology and lattice structures on L-fuzzy rough sets determined by lower and upper sets. Kim [7 –9] introduced the notion of Alexandrov topologies as a topological viewpoint of fuzzy rough sets, and studied the relations among fuzzy preorders, L-lower and L-upper approximation operators, and Alexandrov topologies on complete residuated lattices.

In this paper, we introduce the notion of Galois and dual Galois connections as a topological viewpoint of Bělohlávek’s concept lattices on a complete residuated lattice where a concept lattice is β (X, Y, L) = {(A, B) ∈ L^X × L^Y ∣ F (A) = B, G (B) = A} with $\begin{matrix} F (A) (y) = ⋀_{x \in X} (A (x) \to R (x, y)), \\ G (B) (x) = ⋀_{y \in Y} (B (y) \to R (x, y)), \end{matrix}$ and (F, G) is a Galois connection on L^X × L^Y. Let τ_{e
_X} and τ_{e
_Y} be Alexandrov topologies induced by fuzzy posets (X, e_X) and (Y, e_Y). Under various relations, we investigate the Galois and dual Galois connections on τ_{e
_X} × τ_{e
_Y}. Moreover, we study their properties and give examples.

2 Preliminaries

Definition 2.1. [1 –6] 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 if and only if x ≤ y → z for all x, y, z ∈ L.

In this paper, we always assume that (L, ≤ , ⊙ , → ^∗) is complete residuated lattice with x^* = x→ ⊥ and (x^*) ^* = x for all x ∈ X.

For all α ∈ L and A ∈ L^X, we denote (α → A), (α ⊙ A) and α_X ∈ L^X by (α → A) (x) = α → A (x), (α ⊙ A) (x) = α ⊙ A (x) and α_X (x) = α, respectively.

Lemma 2.2. [1 –6] For all x, y, z, x_i, y_i, w ∈ L, the following properties hold.

⊤ → x = x and ⊥ ⊙ x = ⊥ .

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

x ≤ y if and only if x→ y = ⊤.

x → (⋀ _iy_i) = ⋀ _i (x → y_i).

(⋁ _ix_i) → y = ⋀ _i (x_i → y).

x ⊙ (⋁ _iy_i) = ⋁ _i (x ⊙ y_i).

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

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

(x → y) ⊙ (y → z) ≤ x → z.

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

(x ⊙ y^*) ^* = x → y and x → y = y^* → x^*.

Definition 2.3. [2, 10] Let X be a set. A function e_X : X × X → L is called:

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

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

if e_X (x, y) = e_X (y, x) =⊤, then x = y.

If e satisfies (E1) and (E2), then the pair (X, e_X) is call a fuzzy preorder set. If e satisfies (E1), (E2) and (E3), then the pair (X, e_X) is called a fuzzy partially order set (simply, fuzzy poset).

3 Fuzzy galois connections on Alexandrov L-topologies

Definition 3.1. Let (X, e_X) and (Y, e_Y) be fuzzy posets. Let f : X → Y and g : Y → X be maps.

A 4-tuple (e_X, f, g, e_Y) is called a Galois connection if e_Y (y, f (x)) = e_X (x, g (y)) for all x ∈ X, y ∈ Y.

A 4-tuple (e_X, f, g, e_Y) is called a dual Galois connection if e_Y (f (x) , y) = e_X (g (y) , x) for all x ∈ X, y ∈ Y.

A map f is called an isotone map if $e_{Y} (f (x_{1}), f (x_{2})) \geq e_{X} (x_{1}, x_{2}) for all x_{1}, x_{2} \in X .$

A map f is called an antitone map if $e_{Y} (f (x_{1}), f (x_{2})) \geq e_{X} (x_{2}, x_{1}) for all x_{1}, x_{2} \in X .$

Theorem 3.2. Let (X, e_X) and (Y, e_Y) be fuzzy posets. Let f : X → Y and g : Y → X be maps. Then the following hold.

(e_X, f, g, e_Y) is a Galois connection if and only if the two maps f and g are antitone, and e_Y (y, f (g (y))) = e_X (x, g (f (x))) =⊤ for all x ∈ X and y ∈ Y.

(e_X, f, g, e_Y) is a dual Galois connection if and only if the two maps f and g are antitone, and e_Y (f (g (y)) , y) = e_X (g (f (x)) , x) =⊤ for all x ∈ X and y ∈ Y.

Proof. (1) Assume that (e_X, f, g, e_Y) is a Galois connection. Since e_Y (y, f (x)) = e_X (x, g (y)), we have $⊤ = e_{Y} (f (x), f (x)) = e_{X} (x, g (f (x)))$ and $e_{Y} (y, f (g (y))) = e_{X} (g (y), g (y)) = ⊤ .$ Furthermore, $e_{Y} (f (x_{1}), f (x_{2})) = e_{X} (x_{2}, g (f (x_{1}))) \geq e_{X} (x_{2}, x_{1}) ⊙ e_{X} (x_{1}, g (f (x_{1}))) = e_{X} (x_{2}, x_{1}) .$

Conversely, assume that f and g are antitone, and e_Y (y, f (g (y))) = e_X (x, g (f (x))) =⊤ for all x ∈ X and y ∈ Y. Since e_Y (y, f (g (y)) =⊤ and f is antitone, we have $\begin{matrix} e_{Y} (y, f (x)) & \geq e_{Y} (y, f (g (y))) ⊙ e_{Y} (f (g (y)), f (x)) \\ = e_{Y} (f (g (y)), f (x)) \\ \geq e_{X} (x, g (y)) . \end{matrix}$ Similarly, e_Y (y, f (x)) ≤ e_X (x, g (y)).

(2) Assume that (e_X, f, g, e_Y) is a dual Galois connection. Since e_Y (f (x) , y) = e_X (g (y) , x), we have $⊤ = e_{Y} (f (x), f (x)) = e_{X} (g (f (x)), x)$ and $e_{Y} (f (g (y)), y) = e_{X} (g (y), g (y)) = ⊤ .$ Furthermore, $e_{Y} (f (x_{1}), f (x_{2})) = e_{X} (g (f (x_{2})), x_{1}) \geq e_{X} (x_{2}, x_{1}) ⊙ e_{X} (g (f (x_{2})), x_{2}) = e_{X} (x_{2}, x_{1}) .$

Conversely, assume that f and g are antitone, and e_Y (f (g (y)) , y) = e_X (g (f (x)) , x) =⊤ for all x ∈ X and y ∈ Y. Note that $\begin{matrix} e_{Y} (f (x), y) & \geq e_{Y} (f (x), f (g (y))) ⊙ e_{Y} (f (g (y)), y) \\ = e_{Y} (f (x), f (g (y))) \geq e_{X} (g (y), x) . \end{matrix}$ Similarly, e_Y (f (x) , y) ≤ e_X (g (y) , x).

Remark 3.3. Let (X, e_X) and (Y, e_Y) be fuzzy posets. Let f : X → Y and g : Y → X be maps.

If (e_X, f, g, e_Y) is a Galois connection, then the 4-tuple $(e_{X}, f, g, e_{Y}^{- 1})$ is an adjunction where $e_{Y}^{- 1} (y_{1}, y_{2}) = e_{Y} (y_{2}, y_{1})$ for all y₁, y₂ ∈ Y (see [27, Definition 2.7]).

If (e_X, f, g, e_Y) is a dual Galois connection, then the 4-tuple $(e_{X}^{- 1}, f, g, e_{Y})$ is an adjunction where $e_{X}^{- 1} (x_{1}, x_{2}) = e_{X} (x_{2}, x_{1})$ for all x₁, x₂ ∈ X.

(e_X, f, g, e_Y) is an adjunction if and only if f, g are isotone maps and e_Y (f (g (y)) , y) = e_X (x, g (f (x))) =⊤ for all x ∈ X, y ∈ Y (see [27, Proposition 2.9]).

Example 3.4. Define a binary operation ⊙ (called Ł ukasiewicz conjection) on L = [0, 1] by $\begin{matrix} x ⊙ y = max {0, x + y - 1}, \\ x \to y = min {1 - x + y, 1} . \end{matrix}$ Then the 5-tuple ([0, 1] , ⊙ , → , 0, 1) is a complete residuated lattice (see [2 , 24]). Let X = {a, b, c} be a set. Define the map f : X → X by f (a) = b, f (b) = a, f (c) = c.

Let (X = {a, b, c} , e₁) be a fuzzy poset where $e_{1} = (\begin{matrix} 1 & 0.6 & 0.5 \\ 0.6 & 1 & 0.7 \\ 0.7 & 0.5 & 1 \end{matrix}) .$ Since $\begin{matrix} e_{1} (x, y) & = e_{1} (f (y), f (x)), \\ e_{1} (x, f (f (x))) & = e_{1} (f (f (x)), x) = 1, \end{matrix}$ the 4-tuple (e₁, f, f, e₁) is a Galois connection and is also a dual Galois connection. Since $0.7 = e_{1} (c, a) ≰ e_{1} (f (c), f (a)) = e_{1} (c, b) = 0.5,$ f is not an isotone map.

Let (X = {a, b, c} , e₂) be a fuzzy poset where $e_{2} = (\begin{matrix} 1 & 0.6 & 0.5 \\ 0.6 & 1 & 0.5 \\ 0.7 & 0.7 & 1 \end{matrix}) .$ Since e₂ (x, y) = e₂ (f (x) , f (y)), f is an isotone map. Moreover, $e_{2} (x, f (f (x))) = e_{2} (f (f (x)), x) = 1 .$ Since $0.7 = e_{2} (c, a) ≰ e_{2} (f (a), f (c)) = e_{2} (b, c) = 0.5,$ f is not an antitone map. Hence the 4-tuple (e₂, f, f, e₂) is neither a Galois nor a dual Galois connection.

Let (X = {a, b, c} , e₃) be a fuzzy poset where $e_{3} = (\begin{matrix} 1 & 1 & 0.7 \\ 0.6 & 1 & 0.7 \\ 0.7 & 0.7 & 1 \end{matrix}) .$ Define the map g : X → X by g (a) = g (b) = b, g (c) = c. Since $\begin{matrix} e_{3} (x, y) & \leq e_{3} (g (x), g (y)), \\ e_{3} (x, y) & \leq e_{3} (g (y), g (x)), \\ g (g (a)) & = g (g (b)) = b, \\ g (g (c)) & = c, \end{matrix}$ we have $\begin{matrix} e_{X} (x, g (g (x))) & = 1, \\ e_{X} (g (g (a)), a) & = e_{X} (b, a) = 0.6 . \end{matrix}$ Hence the 4-tuple (e₃, g, g, e₃) is a Galois connection, but it is not a dual Galois connection.

Define the map h : X → X by h (a) = h (b) = a, h (c) = c. Since $\begin{matrix} e_{3} (x, y) & \leq e_{3} (h (y), h (x)), \\ h (h (a)) & = h (h (b)) = a, \\ h (h (c)) & = c, \end{matrix}$ we have $\begin{matrix} e_{X} (h (h (x)), x) & = 1, \\ e_{X} (b, h (h (b))) & = e_{X} (b, a) = 0.6 . \end{matrix}$ Hence the 4-tuple (e₃, h, h, e₃) is a dual Galois connection, but it is not a Galois connection.

Define the map e_[0,1] : [0, 1] × [0, 1] → [0, 1] by e_[0,1] (x, y) = x → y. Define the map f : [0, 1] → [0, 1] by f (x) = x^*. Since $\begin{matrix} e_{[0, 1]} (x, y) & = e_{[0, 1]} (f (y), f (x)), \\ e_{[0, 1]} (x, f (f (x))) & = e_{[0, 1]} (f (f (x)), x) = 1, \end{matrix}$ the 4-tuple (e_[0,1], f, f, e_[0,1]) is a Galois and is also a dual Galois connection.

Definition 3.5. [4] A subset τ_X ⊂ L^X is called an Alexandrov L-topology on X if it satisfies the following three conditions:

α_X ∈ τ_X;

If A_i ∈ τ_X for all i ∈ I, then ⋁_i∈IA_i, ⋀_i∈IA_i ∈ τ_X;

If A ∈ τ_X and α ∈ L, then α ⊙ A, α → A ∈ τ_X.

The pair (X, τ_X) is called an Alexandrov L-topological space.

Lemma 3.6. Let τ_X ⊂ L^X. Define e_{τ
_X} : τ_X × τ_X → L by e_{τ
_X} (A, B) = ⋀ _x∈X (A (x) → B (x)). Then the pair (τ_X, e_{τ
_X}) is a fuzzy poset.

Proof. (E1) Let A ∈ τ_X. Then $e_{τ_{X}} (A, A) = ⋀_{x \in X} (A (x) \to A (x)) = ⊤ .$

(E2) Let A, B, C ∈ τ_X. Then we have by Lemma 2.2(9) that $\begin{matrix} e_{τ_{X}} & (A, B) ⊙ e_{τ_{X}} (B, C) \\ = ⋀_{x \in X} (A (x) \to B (x)) ⊙ ⋀_{x \in X} (B (x) \to C (x)) \\ \leq ⋀_{x \in X} ((A (x) \to B (x)) ⊙ (B (x) \to C (x))) \\ \leq e_{τ_{X}} (A, C) . \end{matrix}$

(E3) Assume that e_{τ
_X} (A, B) = e_{τ
_X} (B, A) =⊤. By Lemma 2.2(3), we have A = B.

Hence (τ_X, e_{τ
_X}) is a fuzzy poset.

Theorem 3.7. Let e_X be a fuzzy preorder on X. Define τ_{e
_X} : = {A ∈ L^X ∣ A (x) ⊙ e_X (x, y) ≤ A (y)}. Then the following properties hold.

τ_{e
_X} = {A ∈ L^X ∣ ⋁ _x∈X (A (x) ⊙ e_X (x, y)) = A (y)} and τ_{e
_X} is an Alexandrov L-topology on X.

If τ is an Alexandrov L-topology on X, then τ = τ_e
^τ where e^τ : X × X → L is defined by e^τ (x, y) = ⋀ _B∈τ (B (x) → B (y)).

Proof. (1) and (2) follow from [4, Theorem 4.5] and [4. Theorem 5.6], respectively.

Remark 3.8.

Let (X, e_{▵_X}) be a fuzzy poset where $e_{▵_{X}} (x, y) = {\begin{matrix} ⊤ & if x = y, \\ ⊥ & if x \neq y . \end{matrix}$ Since ⋁_x∈X (A (x) ⊙ e_{▵_X} (x, y)) = A (y), we have τ_{e
_{▵_X}} = L^X where $e_{τ_{e_{▵_{X}}}} = e_{L^{X}} : L^{X} \times L^{X} \to L$ with $e_{L^{X}} (A, B) = ⋀_{x \in X} (A (x) \to B (x)) .$

Let (X, ⊤ _X×X) be a fuzzy poset where $⊤_{X \times X} (x, y) = ⊤ for all x, y \in X .$ Since $\begin{matrix} ⋁_{x \in X} (A (x) ⊙ ⊤_{X \times X} (x, y)) \\ = ⋁_{x \in X} A (x) = A (y), \end{matrix}$ we have τ_{⊤_X×X} = {α_X ∈ L^X ∣ α ∈ L} where e_{τ
_{⊤_X×X}} : τ_{⊤_X×X} × τ_{⊤_X×X} → L with e_{τ
_{⊤_X×X}} (α_X, β_X) = α → β.

Let ([0, 1] , ⊙ , 0, 1) be a left continuous t-norm where $e_{[0, 1]} (x, y) = {\begin{matrix} 1 & if x \leq y, \\ 0 & if x ≰ y . \end{matrix}$ Since $⋁_{x \leq y} (A (x) ⊙ e_{[0, 1]} (x, y)) = ⋁_{x \leq y} A (x) = A (y),$ we have $\begin{matrix} τ_{e_{X}} & = {A \in [0, 1]^{[0, 1]} ∣ A (x) ⊙ e_{X} (x, y) \leq A (y)} \\ = {A \in [0, 1]^{[0, 1]} ∣ ⋁_{x \leq y} A (x) = A (y)} \\ = {A \in [0, 1]^{[0, 1]} ∣ A (x) \leq A (y), x \leq y} . \end{matrix}$

For all R₁ ∈ L^X×Y, R₂ ∈ L^Y×Z, define $\begin{matrix} R_{1} \circ R_{2} (x, z) & = ⋁_{y} (R_{1} (x, y) ⊙ R_{2} (y, z)), \\ R^{- 1} (y, x) & = R (x, y) . \end{matrix}$

Lemma 3.9. Let (X, e_X) and (Y, e_Y) be fuzzy posets. Let R ∈ L^X×Y. Then, the following hold.

$(e_{X} \circ R)^{- 1} = R^{- 1} \circ e_{X}^{- 1}$ and ${(R \circ e_{X})}^{- 1} = e_{X}^{- 1} \circ R^{- 1}$ .

e_X ∘ R ≤ R if and only if $e_{X}^{- 1} \circ R^{*} \leq R^{*}$ .

$R \circ e_{X}^{- 1} \leq R$ if and only if R^* ∘ e_X ≤ R^*.

Proof. (1) It can be easily proved.

(2) Note that $\begin{matrix} e_{X} (x, z) ⊙ R (z, y) \leq R (x, y) \\ \Leftrightarrow R (z, y) \leq e_{X} (x, z) \to R (x, y) \\ \Leftrightarrow e_{X} (x, z) ⊙ R^{*} (x, y) \leq R^{*} (z, y) \\ \Leftrightarrow e_{X}^{- 1} (z, x) ⊙ R^{*} (x, y) \leq R^{*} (z, y) . \end{matrix}$

(3) Note that $\begin{matrix} R (w, y) ⊙ e_{X}^{- 1} (y, x) \leq R (w, x) \\ \Leftrightarrow R (w, y) ⊙ e_{X} (x, y) \leq R (w, x) \\ \Leftrightarrow e_{X} (x, y) \to R^{*} (w, y) \geq R^{*} (w, x) \\ \Leftrightarrow e_{X} (x, y) ⊙ R^{*} (w, x) \leq R^{*} (w, y) . \end{matrix}$

Theorem 3.10. Let (X, e_X) and (Y, e_Y) be fuzzy posets. Then the following three statements (1), (2) and (3) are equivalent:

(e_{τ
_{e
_X}}, F, G, e_{τ
_{e
_Y}}) is a Galois connection.

$\begin{matrix} F (⋁_{i \in Γ} A_{i}) & = ⋀_{i \in Γ} F (A_{i}), \\ F (α ⊙ A) & = α \to F (A), \\ G (⋁_{i \in Γ} B_{i}) & = ⋀_{i \in Γ} G (B_{i}), \\ G (α ⊙ B) & = α \to G (B) \end{matrix}$ where A, A_i ∈ τ_{e
_X} and B, B_i ∈ τ_{e
_Y}.

There exists a map R : X × Y → L such that $\begin{matrix} e_{X}^{- 1} \circ R \leq R, R \circ e_{Y} \leq R, \\ F (A) (y) = ⋀_{x \in X} (A (x) \to R (x, y)), \\ G (B) (x) = ⋀_{y \in Y} (B (y) \to R (x, y)) . \end{matrix}$

There exists R : X × Y → L such that $\begin{matrix} e_{X}^{- 1} \circ R \leq R, R \circ e_{Y} \leq R, \\ F ((e_{X})_{z}) (y) = R (z, y), \\ G ((e_{Y})_{w}) (x) = R (x, w) . \end{matrix}$

Proof. (1) ⇒ (2). Since $\begin{matrix} e_{τ_{e_{Y}}} (B, F (⋁_{i \in Γ} A_{i})) & = e_{τ_{e_{X}}} (⋁_{i \in Γ} A_{i}, G (B)) \\ = ⋀_{i \in Γ} e_{τ_{e_{X}}} (A_{i}, G (B)) \\ = ⋀_{i \in Γ} e_{τ_{e_{Y}}} (B, F (A_{i})) \\ = e_{τ_{e_{Y}}} (B, ⋀_{i \in Γ} F (A_{i})), \end{matrix}$ we have $F (⋁_{i \in Γ} A_{i}) = ⋀_{i \in Γ} F (A_{i}) .$ Similarly, F (α ⊙ A) = α → F (A) for all α ∈ L.

Since (e_X) _z ∈ τ_{e
_X} and F ((e_X) _z) ∈ τ_{e
_Y}, we have $\begin{matrix} F ((e_{X})_{z}) (w) & \leq e_{Y} (w, y) \to F ((e_{X})_{z}) (y), \\ F ((e_{X})_{z}) (w) & = e_{τ_{e_{Y}}} ((e_{Y})_{w}, F ((e_{X})_{z})) \\ = e_{τ_{e_{X}}} ((e_{X})_{z}, G ((e_{Y})_{w})) \\ = ⋀_{x \in X} ((e_{X})_{z} (x) \to G ((e_{Y})_{w}) (x)) \\ = G ((e_{Y})_{w}) (z) . \end{matrix}$ Let R (z, w) = F ((e_X) _z) (w) = G ((e_Y) _w) (z). Since $\begin{matrix} R (z, w) & = F ((e_{X})_{z}) (w) \\ = e_{τ_{e_{X}}} ((e_{X})_{z}, G ((e_{Y})_{w})) \\ \leq (e_{X})_{z} (x) \to G ((e_{Y})_{w}) (x) \\ = e_{X} (z, x) \to R (x, w), \end{matrix}$ we have $e_{X}^{- 1} \circ R \leq R$ . Since $\begin{matrix} R (z, w) & = F ((e_{X})_{z}) (w) \\ \leq (e_{Y})_{w} (y) \to F ((e_{X})_{z})) (y) \\ = e_{Y} (w, y) \to R (z, y), \end{matrix}$ we have R ∘ e_Y ≤ R. Moreover, $\begin{matrix} F (A) (y) & = e_{τ_{e_{Y}}} ((e_{Y})_{y}, F (A)) \\ = e_{τ_{e_{X}}} (A, G ((e_{Y})_{y})) \\ = ⋀_{x \in X} (A (x) \to G ((e_{Y})_{y}) (x)) \\ = ⋀_{x \in X} (A (x) \to R (x, y)) \end{matrix}$ and $\begin{matrix} G (B) (x) & = e_{τ_{e_{X}}} ((e_{X})_{x}, G (B)) \\ = e_{τ_{e_{Y}}} (B, F ((e_{X})_{x})) \\ = ⋀_{y \in Y} (B (y) \to F ((e_{X})_{x}) (y)) \\ = ⋀_{y \in Y} (B (y) \to R (x, y)) . \end{matrix}$

(2) ⇒ (1). Since R ∘ e_Y ≤ R and $A (x) ⊙ (A (x) \to R (x, y)) ⊙ e_{Y} (y, w) \leq R (x, y) ⊙ e_{Y} (y, w) \leq R (x, w),$ we have F (A) (y) ⊙ e_Y (y, w) ≤ F (A) (w); that is, F (A) ∈ τ_{e
_Y}.

Since $e_{X}^{- 1} \circ R \leq R$ and $B (y) ⊙ (B (y) \to R (x, y)) ⊙ e_{X} (x, z) \leq R (x, y) ⊙ e_{X} (x, z) \leq R (z, y),$ we have G (B) (x) ⊙ e_X (x, z) ≤ G (B) (z); that is, G (B) ∈ τ_{e
_X}.

Note that $\begin{matrix} e_{τ_{e_{Y}}} & (B, F (A)) \\ = ⋀_{y \in Y} (B (y) \to ⋀_{x \in X} (R (x, y) \to A^{*} (x))) \\ = ⋀_{x \in X} (A (x) \to ⋀_{y \in Y} (R (x, y) \to B^{*} (y))) \\ = e_{τ_{e_{X}}} (A, G (B)) . \end{matrix}$

Hence (e_{τ
_{e
_X}}, F, G, e_{τ
_{e
_Y}}) is a Galois connection.

(2) ⇒ (3). Let A = (e_X) _z ∈ τ_{e
_X} and B = (e_Y) _w ∈ τ_{e
_X}. Since $\begin{matrix} F ((e_{X})_{z}) (y) & = ⋀_{x \in X} (e_{X} (z, x) \to R (x, y)) \\ \leq e_{X} (z, z) \to R (z, y) \\ = R (z, y), \\ e_{X} (z, x) ⊙ R (z, y) & \leq R (x, y), \end{matrix}$ we have F ((e_X) _z) (y) = R (z, y). Similarly, it holds that G ((e_Y) _w) (x) = R (x, w).

(3) ⇒ (2). Since A = ⋁ _x∈XA (x) ⊙ (e_X) _x ∈ τ_{e
_X}, we have $\begin{matrix} F (A) (y) & = F (⋁_{x \in X} A (x) ⊙ (e_{X})_{x}) (y) \\ = ⋀_{x \in X} (A (x) \to F ((e_{X})_{x}) (y)) \\ = ⋀_{z \in X} (A (z) \to R (z, y)) . \end{matrix}$ Similarly, G (B) (x) = ⋀ _y∈Y (B (y) → R (x, y)).

Remark 3.11.

Let (X, e_{▵_X}) and (Y, e_{▵_Y}) be fuzzy posets defined in Remark 3.8 (1). Then τ_{e
_{▵_X}} = L^X and τ_{e
_{▵_Y}} = L^Y. If the 4-tuple $(e_{τ_{e_{▵_{X}}}} = e_{L^{X}}, F, G, e_{τ_{e_{▵_{Y}}}} = e_{L^{Y}})$ is a Galois connection, then there exists R : X × Y → L with $\begin{matrix} F ((e_{▵_{X}})_{x}) (y) & = G ((e_{▵_{Y}})_{y}) (x) = R (x, y), \\ F (⋁_{i \in Γ} A_{i}) & = ⋀_{i \in Γ} F (A_{i}), \\ F (α ⊙ A) & = α \to F (A), \\ G (⋁_{i \in Γ} B_{i}) & = ⋀_{i \in Γ} G (B_{i}), \\ G (α ⊙ B) & = α \to G (B) \end{matrix}$ where A, A_i ∈ L^X and B, B_i ∈ L^Y. Trivially, $e_{▵_{X}}^{- 1} \circ R = R$ and R ∘ e_{▵_Y} = R.

Let (X, ⊤ _X×X) and (Y, ⊤ _Y×Y) be fuzzy posets defined in Remark 3.8 (2). If the 4-tuple $(e_{τ_{⊤_{X \times X}}}, F, G, e_{τ_{⊤_{Y \times Y}}})$ is a Galois connection with $\begin{matrix} τ_{⊤_{X \times X}} & = {γ_{X} \in L^{X} ∣ γ \in L}, \\ τ_{⊤_{Y \times Y}} & = {β_{Y} \in L^{Y} ∣ β \in L}, \end{matrix}$ then, there exists R : X × Y → L with $\begin{matrix} F ((⊤_{X \times X})_{x}) (y) & = G ((⊤_{Y \times Y})_{y}) (x) \\ = R (x, y) = α (y), \\ F (⋁_{i \in Γ} (γ_{i})_{X}) & = ⋀_{i \in Γ} F ((γ_{i})_{X}), \\ F (α ⊙ γ_{X}) & = α \to F (γ_{X}), \\ G (⋁_{i \in Γ} (β_{i})_{Y}) & = ⋀_{i \in Γ} G ((β_{i})_{Y}), \\ G (α ⊙ (β_{i})_{Y}) & = α \to G ((γ_{i})_{Y}) \end{matrix}$ where γ, (γ_i) _X ∈ τ_{⊤_X×X}, β_Y and (β_i) _Y ∈ τ_{⊤_X×X}. Trivially, $⊤_{X \times X}^{- 1} \circ R = R \circ ⊤_{X \times X} = R$ .

Theorem 3.12. Let (X, e_X) and (Y, e_Y) be fuzzy posets. Then the following three statements (1), (2) and (3) are equivalent:

(e_{τ
_{e
_X}}, F, G, e_{τ
_{e
_Y}}) is a Galois connection.

There exists R : X × Y → L such that $\begin{matrix} e_{X} \circ R \leq R, R \circ e_{Y}^{- 1} \leq R, \\ F (A) (y) = ⋀_{x \in X} (R (x, y) \to A^{*} (x)), \\ G (B) (x) = ⋀_{y \in Y} (R (x, y) \to B^{*} (y)) . \end{matrix}$

There exists R : X × Y → L such that $\begin{matrix} e_{X} \circ R \leq R, R \circ e_{Y}^{- 1} \leq R, \\ F ((e_{X})_{z}) (y) = R^{*} (z, y), \\ G ((e_{Y})_{w}) (x) = R^{*} (x, w) . \end{matrix}$

Proof. (1) ⇒ (2). Let $R^{*} (z, w) = F ((e_{X})_{z}) (w) = G ((e_{Y})_{w}) (z) .$ Since $\begin{matrix} R^{*} (z, w) & = F ((e_{X})_{z}) (w) \\ = e_{τ_{e_{X}}} ((e_{X})_{z}, G ((e_{Y})_{w})) \\ \leq (e_{X})_{z} (x) \to G ((e_{Y})_{w}) (x) \\ = e_{X} (z, x) \to R^{*} (x, w), \end{matrix}$ we have e_X ∘ R ≤ R. Since $\begin{matrix} R^{*} (z, w) & = F ((e_{X})_{z}) (w) \\ \leq (e_{Y})_{w} (y) \to F ((e_{X})_{z})) (y) \\ = e_{Y} (w, y) \to R^{*} (z, y), \end{matrix}$ we have $R \circ e_{Y}^{- 1} \leq R$ . Note that $\begin{matrix} F (A) (y) & = e_{τ_{e_{Y}}} ((e_{Y})_{y}, F (A)) \\ = e_{τ_{e_{X}}} (A, G ((e_{Y})_{y})) \\ = ⋀_{x \in X} (A (x) \to G ((e_{Y})_{y}) (x)) \\ = ⋀_{x \in X} (A (x) \to R^{*} (x, y)) \\ = ⋀_{x \in X} (R (x, y) \to A^{*} (x)) . \end{matrix}$

For all A ∈ τ_{e
_X}, B ∈ τ_{e
_Y}, we have $\begin{matrix} G (B) (x) & = e_{τ_{e_{X}}} ((e_{X})_{x}, G (B)) \\ = e_{τ_{e_{Y}}} (B, F ((e_{X})_{x})) \\ = ⋀_{y \in Y} (B (y) \to F ((e_{X})_{x}) (y)) \\ = ⋀_{y \in Y} (B (y) \to R^{*} (x, y)) \\ = ⋀_{y \in Y} (R (x, y) \to B^{*} (y)) . \end{matrix}$

The remaining can be similarly proved as in that of Theorem 3.10.

(2) ⇒ (1). Since $R \circ e_{Y}^{- 1} \leq R$ and $\begin{matrix} R (x, w) ⊙ & (R (x, y) \to A^{*}) ⊙ e_{Y} (y, w) \\ \leq R (x, y) ⊙ (R (x, y) \to A^{*}) \\ \leq A^{*} (x), \end{matrix}$ we have F (A) (y) ⊙ e_Y (y, w) ≤ F (A) (w); that is, F (A) ∈ τ_{e
_Y}. Since e_X ∘ R ≤ R and $\begin{matrix} R (z, y) ⊙ & (R (x, y) \to B^{*} (y)) ⊙ e_{X} (x, z) \\ \leq R (x, y) ⊙ (R (x, y) \to B^{*} (y)) \\ \leq B^{*} (y), \end{matrix}$ we have G (B) (x) ⊙ e_X (x, z) ≤ G (B) (z); that is, G (B) ∈ τ_{e
_X}. Note that $\begin{matrix} e_{τ_{e_{Y}}} & (B, F (A)) \\ = ⋀_{y \in Y} (B (y) \to ⋀_{x \in X} (A (x) \to R (x, y))) \\ = ⋀_{x \in X} (A (x) \to ⋀_{y \in Y} (B (y) \to R (x, y))) \\ = e_{τ_{e_{X}}} (A, G (B)) . \end{matrix}$ Hence (e_{τ
_{e
_X}}, F, G, e_{τ
_{e
_Y}}) is a Galois connection.

(2) ⇒ (3). Let A = (e_X) _x and B = (e_Y) _w. Since $e_{X} (z, x) ⊙ R (x, y) \leq R (z, y) \Leftrightarrow (R (x, y) \to e_{X}^{*} (z, x)) \geq R^{*} (z, y),$ we have F ((e_X) _z) (y) ≥ R^* (z, y). Moreover, $\begin{matrix} F ((e_{X})_{z}) (y) & = ⋀_{x \in X} (R (x, y) \to e_{X}^{*} (z, x)) \\ \leq R^{*} (z, y) . \end{matrix}$ Thus F ((e_X) _z) (y) = R^* (z, y). Similarly, one can see that G ((e_Y) _w) (x) = R^* (x, w).

(3) ⇒ (2). It can be similarly proved as in that of Theorem 3.10.

Theorem 3.13. Let (X, e_X) and (Y, e_Y) be fuzzy posets. Then the following three statements (1), (2) and (3) are equivalent:

(e_{τ
_{e
_X}}, F, G, e_{τ
_{e
_Y}}) is a dual Galois connection.

$\begin{matrix} F (⋀_{i \in Γ} A_{i}) & = ⋁_{i \in Γ} F (A_{i}), \\ F (α \to A) & = α ⊙ F (A), \\ G (⋀_{i \in Γ} B_{i}) & = ⋁_{i \in Γ} G (B_{i}), \\ G (α \to B) & = α ⊙ G (B) \end{matrix}$ where A, A_i ∈ τ_{e
_X} and B, B_i ∈ τ_{e
_Y}.

There exists R : X × Y → L such that $\begin{matrix} e_{X}^{- 1} \circ R \leq R, R \circ e_{Y} \leq R, \\ F (A) (y) = ⋁_{x \in X} (A^{*} (x) ⊙ R (x, y)), \\ G (B) (x) = ⋁_{y \in Y} (R (x, y) ⊙ B^{*} (y)) . \end{matrix}$

There exists R : X × Y → L such that $\begin{matrix} e_{X}^{- 1} \circ R \leq R, R \circ e_{Y} \leq R, \\ F ((e_{X})_{z}^{- 1 *}) (y) = R (x, y), \\ G ((e_{Y})_{w}^{- 1 *}) (x) = R (x, w) . \end{matrix}$

Proof. (1) ⇒ (2). Since $\begin{matrix} e_{τ_{e_{Y}}} (F (⋀_{i \in Γ} A_{i}), B) = e_{τ_{e_{X}}} (G (B), ⋀_{i \in Γ} A_{i}) \\ = ⋀_{i \in Γ} e_{τ_{e_{X}}} (G (B), A_{i}) = ⋀_{i \in Γ} e_{τ_{e_{Y}}} (F (A_{i}), B) \\ = e_{τ_{e_{Y}}} (⋁_{i \in Γ} F (A_{i}), B), \end{matrix}$ we have F (⋀ _i∈ΓA_i) = ⋁ _i∈ΓF (A_i). Let α ∈ L. Since $\begin{matrix} e_{τ_{e_{Y}}} (F (α \to A), B) = e_{τ_{e_{X}}} (G (B), α \to A) \\ = α \to e_{τ_{e_{X}}} (G (B), A) = α \to e_{τ_{e_{Y}}} (F (A), B) \\ = e_{τ_{e_{Y}}} (α ⊙ F (A), B), \end{matrix}$ we have F (α → A) = α ⊙ F (A).

Since $((e_{X})_{x}^{- 1})^{*} \in τ_{e_{X}}$ and $((e_{Y})_{y}^{- 1})^{*} \in τ_{e_{Y}}$ , we have $F (((e_{X})_{x}^{- 1})^{*}) \in τ_{e_{Y}}$ and $G (((e_{Y})_{y}^{- 1})^{*}) \in τ_{e_{X}}$ . Note that $\begin{matrix} (F (((e_{X})_{x}^{- 1})^{*}) & (y))^{*} \\ = e_{τ_{e_{Y}}} (F (((e_{X})_{x}^{- 1})^{*}), ((e_{Y})_{y}^{- 1})^{*}) \\ = e_{τ_{e_{X}}} (G (((e_{Y})_{y}^{- 1})^{*}), ((e_{X})_{x}^{- 1})^{*})) \\ = (G (((e_{Y})_{y}^{- 1})^{*})) (x))^{*} . \end{matrix}$

Let $R (x, y) = F (((e_{X})_{x}^{- 1})^{*}) (y) = G (((e_{Y})_{y}^{- 1})^{*}) (x) .$ Then R ∘ e_Y ≤ R holds from: $\begin{matrix} R (x, y)^{*} & = (F (((e_{X})_{x}^{- 1})^{*}) (y))^{*} \\ = e_{τ_{e_{Y}}} (F (((e_{X})_{x}^{- 1})^{*}), ((e_{Y})_{y}^{- 1})^{*}) \\ \leq R (x, w) \to e_{Y} (w, y)^{*} . \end{matrix}$ Since $\begin{matrix} R (x, y)^{*} & = e_{τ_{e_{X}}} (G (((e_{Y})_{y}^{- 1})^{*}), ((e_{X})_{x}^{- 1})^{*}) \\ \leq R (z, y) \to e_{X} (z, x)^{*}, \end{matrix}$ we have $e_{X}^{- 1} \circ R \leq R$ .

Let A ∈ τ_{e
_X} and B ∈ τ_{e
_Y}. Then $\begin{matrix} F (A) (y) & = F (⋀_{x \in X} ((e_{X})_{x}^{- 1} \to A (x))) (y) \\ = F (⋀_{x \in X} (A^{*} (x) \to {((e_{X})_{x}^{- 1})}^{*})) (y) \\ = ⋁_{x \in X} (A^{*} (x) ⊙ F ({((e_{X})_{x}^{- 1})}^{*}) (y)) \\ = ⋁_{x \in X} (A^{*} (x) ⊙ R (x, y)) \end{matrix}$ and $\begin{matrix} G (B) (x) & = G (⋀_{y \in Y} ((e_{Y})_{y}^{- 1} \to B (y))) (x) \\ = G (⋀_{y \in Y} (B^{*} (x) \to {((e_{Y})_{y}^{- 1})}^{*})) (x) \\ = ⋁_{y \in Y} (B^{*} (y) ⊙ G ({((e_{Y})_{y}^{- 1})}^{*})) (x)) \\ = ⋁_{y \in Y} (B^{*} (y) ⊙ R (x, y)) . \end{matrix}$

(2) ⇒ (3). Since e_X (x, z) ⊙ R (x, y) ≤ R (z, y) and $F ((e_{X})_{z}^{- 1 *}) (y) = ⋁_{x \in X} ((e_{X})_{z}^{- 1} (x) ⊙ R (x, y) \leq R (z, y),$ we have $F ((e_{X})_{z}^{- 1 *}) (y) = R (z, y)$ . Similarly, one can see that $G ((e_{Y})_{w}^{- 1 *}) (x) = R (x, w)$ .

(3) ⇒ (2). Since $((e_{X})_{x}^{- 1})^{*} \in τ_{e_{X}}$ , we have $A = ⋀_{x \in X} (A^{*} (x) \to {((e_{X})_{x}^{- 1})}^{*}) \in τ_{e_{X}} .$ Note that $\begin{matrix} F (A) (y) & = F (⋀_{x \in X} (A^{*} (x) \to {((e_{X})_{x}^{- 1})}^{*})) (y) \\ = ⋁_{x \in X} (A^{*} (x) ⊙ F ({((e_{X})_{x}^{- 1})}^{*}) (y)) \\ = ⋁_{x \in X} (A^{*} (x) ⊙ R (x, y)) . \end{matrix}$ Similarly, G (B) (x) = ⋁ _y∈Y (R (x, y) ⊙ B^* (y)) .

Theorem 3.14. Let (X, e_X) and (Y, e_Y) be fuzzy posets. Then the following three statements (1), (2) and (3) are equivalent:

(e_{τ
_{e
_X}}, F, G, e_{τ
_{e
_Y}}) is a dual Galois connection.

$\begin{matrix} F (⋀_{i \in Γ} A_{i}) & = ⋁_{i \in Γ} F (A_{i}), \\ F (α \to A) & = α ⊙ F (A), \\ G (⋀_{i \in Γ} B_{i}) & = ⋁_{i \in Γ} G (B_{i}), \\ G (α \to B) & = α ⊙ G (B) \end{matrix}$ where A, A_i ∈ τ_{e
_X} and B, B_i ∈ τ_{e
_Y}.

There exists R : X × Y → L such that $\begin{matrix} e_{X} \circ R \leq R, R \circ e_{Y}^{- 1} \leq R, \\ F (A) (y) = ⋁_{x \in X} (A^{*} (x) ⊙ R^{*} (x, y)), \\ G (B) (x) = ⋁_{y \in Y} (R^{*} (x, y) ⊙ B^{*} (y)) . \end{matrix}$

There exists R : X × Y → L such that $\begin{matrix} e_{X} \circ R \leq R, R \circ e_{Y}^{- 1} \leq R, \\ F ((e_{X})_{z}^{- 1 *}) (y) = R^{*} (z, y)), \\ G ({(e_{Y})}_{w}^{- 1 *}) (x) = R^{*} (x, w) . \end{matrix}$

Proof. (1) ⇒ (2). Let $R^{*} (x, y) = F (((e_{X})_{x}^{- 1})^{*}) (y) = G ({({(e_{Y})}_{y}^{- 1})}^{*}) (x) .$ Since $\begin{matrix} R (x, y) & = {(F ({({(e_{X})}_{x}^{- 1})}^{*}) (y))}^{*} \\ = e_{τ_{e_{Y}}} (F ({({(e_{X})}_{x}^{- 1})}^{*}), {({(e_{Y})}_{y}^{- 1})}^{*}) \\ \leq R^{*} (x, w) \to e_{Y} (w, y)^{*}, \end{matrix}$ we have R (x, y) ⊙ e_Y (w, y) ≤ R (x, w); that is, $R \circ e_{Y}^{- 1} \leq R$ . Moreover, e_X ∘ R ≤ R holds from: $\begin{matrix} R (x, y) & = e_{τ_{e_{Y}}} (G ({({(e_{Y})}_{y}^{- 1})}^{*}), ((e_{X})_{x}^{- 1})^{*}) \\ \leq R^{*} (z, y) \to e_{X} (z, x)^{*} . \end{matrix}$

Let A ∈ τ_{e
_X} and B ∈ τ_{e
_Y}. Then $\begin{matrix} F (A) (y) & = F (⋀_{x \in X} ({(e_{X})}_{x}^{- 1} \to A (x))) (y) \\ = F (⋀_{x \in X} (A^{*} (x) \to {({(e_{X})}_{x}^{- 1})}^{*})) (y) \\ = ⋁_{x \in X} (A^{*} (x) ⊙ F ({({(e_{X})}_{x}^{- 1})}^{*}) (y)) \\ = ⋁_{x \in X} (A^{*} (x) ⊙ R (x, y)) \end{matrix}$ and $\begin{matrix} G (B) (x) & = G (⋀_{y \in Y} ({(e_{Y})}_{y}^{- 1} \to B (y))) (x) \\ = G (⋀_{y \in Y} (B^{*} (x) \to {({(e_{Y})}_{y}^{- 1})}^{*})) (x) \\ = ⋁_{y \in Y} (B^{*} (y) ⊙ G ({({(e_{Y})}_{y}^{- 1})}^{*})) (x) \\ = ⋁_{y \in Y} (B^{*} (y) ⊙ R (x, y)) . \end{matrix}$

(2) ⇒ (3). The equation $F ((e_{X})_{z}^{- 1 *}) (y) = R (z, y)$ holds from: $\begin{matrix} F ({(e_{X})}_{z}^{- 1 *}) (y) & = ⋁_{x \in X} ({(e_{X})}_{z}^{- 1} (x) ⊙ R (x, y)) \\ \leq R (z, y), e_{X} (x, z) ⊙ R (x, y) \\ \leq R (z, y) . \end{matrix}$

Similarly, $G ((e_{Y})_{w}^{- 1 *}) (x) = R (x, w)$ .

Moreover, the equation $F ((e_{X})_{z}^{- 1 *}) (y) = R^{*} (z, y)$ holds from: $\begin{matrix} F ({(e_{X})}_{z}^{- 1 *}) (y) & = ⋁_{x \in X} ({(e_{X})}_{z}^{- 1} (x) ⊙ R^{*} (x, y)) \\ \geq R^{*} (x, y) \end{matrix}$ and $e_{X} (x, z) ⊙ R (z, y) \leq R (x, y) \Leftrightarrow e_{X} (x, z) ⊙ R^{*} (x, y) \leq R^{*} (z, y) .$

Similarly, $G ({(e_{Y})}_{w}^{- 1 *}) (x) = R^{*} (x, w)$ .

(3) ⇒ (2). It can be similarly proved as in that of Theorem 3.13.

Example 3.15. Let (L = [0, 1] , ⊙ , → , 0, 1) be a complete residuated lattice defined in Example 3.4. Let (X = {a, b, c} , e) be a fuzzy poset where $e = (\begin{matrix} 1.0 & 0.6 & 0.5 \\ 0.6 & 1.0 & 0.7 \\ 0.5 & 0.7 & 1.0 \end{matrix}) .$

(1) Let $R_{1} = (\begin{matrix} 0.3 & 1 & 0.2 \\ 1 & 0 & 1 \\ 0.2 & 1 & 0.5 \end{matrix}) .$ Then $e \circ R_{1} = (\begin{matrix} 0.6 & 1 & 0.6 \\ 1 & 0.7 & 1 \\ 0.7 & 1 & 0.7 \end{matrix}), R_{1} \circ e = (\begin{matrix} 0.6 & 1 & 0.7 \\ 1 & 0.7 & 1 \\ 0.6 & 1 & 0.7 \end{matrix}) .$ Thus R₁ ∘ e = R₁ ∘ e^-1 ≰ R₁. In Theorem 3.10 (3), since F (e_a) (b) ⊙ e (b, a) = R₁ (a, b) ⊙ e (b, a) =0.6 ≰ 0.3 = R₁ (a, a) = F (e_a) (a) , F (e_a) ∉ τ_e.

By Theorem 3.12 (3), we have $F (e_{b}) (b) ⊙ e (b, a) = R_{1}^{*} (b, b) ⊙ e (b, a) = 0.6 ≰ 0 = R_{1}^{*} (b, a) = F (e_{b}) (a)$ , and so F (e_b) ∉ τ_e.

By Theorem 3.13 (3), we have $F (e_{a}^{- 1 *}) (b) ⊙ e (b, a) = R_{1} (a, b) ⊙ e (b, a) = 0.6 ≰ 0.3 = R_{1} (a, a) = F (e_{a}^{- 1 *}) (a)$ , and so F (e_a) ∉ τ_e.

By Theorem 3.14 (3), we have $F (e_{b}^{- 1 *}) (b) ⊙ e (b, a) = R_{1}^{*} (b, b) ⊙ e (b, a) = 0.6 ≰ 0 = R_{1}^{*} (b, a) = F (e_{b}^{- 1 *}) (a)$ , and so $F (e_{b}^{- 1 *}) \notin τ_{e}$ .

Now e ∘ R₁ = e^-1 ∘ R₁ ≰ R₁. By Theorem 3.10, we have G (e_a) (b) ⊙ e (b, a) = R₁ (b, a) ⊙ e (b, a) =0.6 ≰ 0.2 = R₁ (a, a) = G (e_a) (a), and so G (e_a) ∉ τ_e.

By Theorem 3.12, we have $G (e_{a}) (a) ⊙ e (a, b) = R_{1}^{*} (a, a) ⊙ e (a, b) ≰ 0 = R_{1}^{*} (b, a) = G (e_{a}) (b)$ , and so G (e_a) ∉ τ_e.

By Theorem 3.13, we have $G (e_{c}^{- 1 *}) (b) ⊙ e (b, a) = R_{1} (b, c) ⊙ e (b, a) = 0.6 ≰ 0.2 = R_{1} (a, c) = G (e_{c}^{- 1 *}) (a)$ , and so $G (e_{c}^{- 1 *}) \notin τ_{e}$ .

By Theorem 3.14, we have $G (e_{a}^{- 1 *}) (a) ⊙ e (a, b) = R_{1}^{*} (a, a) ⊙ e (a, b) ≰ 0 = R_{1}^{*} (b, a) = G (e_{a}^{- 1 *}) (b)$ , and so $G (e_{a}^{- 1 *}) \notin τ_{e}$ .

(2) Let $R_{2} = (\begin{matrix} 0.7 & 0.4 & 0.3 \\ 0.6 & 0.8 & 0.5 \\ 0.3 & 0.5 & 0.8 \end{matrix}) .$ Then R₂ = e ∘ R₂ = e^-1 ∘ R₂ = R₂ ∘ e = R₂ ∘ e^-1.

By Theorem 3.10, (e_{τ
_e}, F, G, e_{τ
_e}) is a Galois connection where $\begin{matrix} F (A) (y) & = ⋀_{x \in X} (A (x) \to R_{2} (x, y)), \\ G (B) (x) & = ⋀_{y \in Y} (B (y) \to R_{2} (x, y)) . \end{matrix}$

By Theorem 3.12, (e_{τ
_e}, F, G, e_{τ
_e}) is a Galois connection where $\begin{matrix} F (A) (y) & = ⋀_{x \in X} (R_{2} (x, y) \to A^{*} (x)), \\ G (B) (x) & = ⋀_{y \in Y} (S_{2} (y, x) \to B^{*} (y)) . \end{matrix}$

By Theorem 3.13, (e_{τ
_e}, F, G, e_{τ
_e}) is a dual Galois connection where $\begin{matrix} F (A) (y) & = ⋁_{y \in X} (R_{2} (x, y) ⊙ A^{*} (x)), \\ G (B) (x) & = ⋁_{y \in X} (S_{2} (y, x) ⊙ B^{*} (y)) . \end{matrix}$

By Theorem 3.14, (e_{τ
_e}, F, G, e_{τ
_e}) is a dual Galois connection where $\begin{matrix} F (A) (y) & = ⋁_{x \in X} (A^{*} (x) ⊙ R^{*} (x, y)), \\ G (B) (x) & = ⋁_{y \in X} (B^{*} (y) ⊙ S^{*} (y, x)) . \end{matrix}$

(3) Let $e_{1} = (\begin{matrix} 1 & 0.7 & 0.5 \\ 0.4 & 1 & 0.3 \\ 0.3 & 0.5 & 1 \end{matrix}) .$ Then $e_{1} \circ R_{2} \circ e_{1}^{- 1} = (\begin{matrix} 0.7 & 0.5 & 0.3 \\ 0.6 & 0.8 & 0.5 \\ 0.3 & 0.5 & 0.8 \end{matrix}) .$ Since e₁ ∘ R₂ ≰ R₂ by Theorem 3.12, we have G ((e₁) _b) (a) ⊙ e₁ (a, b) = R^* (a, b) ⊙ e₁ (a, b) =0.6 ⊙ 0.7 = 0.3 ≰ 0.2 = R^* (b, b) = G ((e₁) _b) (b) and G ((e₁) _b) ∉ τ_{e
₁}. By Theorem 3.14, $G ((e_{1})_{b}^{- 1 *}) (a) ⊙ e_{1} (a, b) = R^{*} (a, b) ⊙ e_{1} (a, b) = 0.6 ⊙ 0.7 = 0.3 ≰ 0.2 = R^{*} (b, b) = G ((e_{1})_{b}^{- 1 *}) (b)$ and $G ((e_{1})_{b}^{- 1 *}) \notin τ_{e_{1}}$ .

Since $e_{1}^{- 1} \circ R_{2} = R_{2}$ and R₂ ∘ e₂ = R₂ by Theorem 3.10, we have that (e_{τ
_{e
₁}}, F, G, e_{τ
_{e
₁}}) is a Galois connection where $\begin{matrix} F (A) (y) & = ⋀_{x \in X} (A (x) \to R_{2} (x, y)), \\ G (B) (x) & = ⋀_{y \in Y} (B (y) \to R_{2} (x, y)) . \end{matrix}$

Since $e_{1}^{- 1} \circ R_{2} = R_{2}$ and R₂ ∘ e₂ = R₂ by Theorem 3.13, we have that (e_{τ
_{e
₁}}, F, G, e_{τ
_{e
₁}}) is a dual Galois connection where $\begin{matrix} F (A) (y) & = ⋁_{y \in X} (R_{2} (x, y) ⊙ A^{*} (x)), \\ G (B) (x) & = ⋁_{y \in X} (S_{2} (y, x) ⊙ B^{*} (y)) . \end{matrix}$

Theorem 3.16. Let (X, e_X) and (Y, e_Y) be fuzzy posets. Let f : X → Y and g : Y → X be maps. Then the following hold.

e_X (x, g (y)) = e_Y (y, f (x)) if and only if e_{τ
_{e
_Y}} (B, F₁ (A)) = e_{τ
_{e
_X}} (A, G₁ (B)) where $\begin{matrix} F_{1} (A) (y) & = ⋀_{x \in X} (e_{Y} (y, f (x)) \to A^{*} (x)), \\ G_{1} (B) (x) & = ⋀_{y \in X} (e_{Y} (x, g (y)) \to B^{*} (y)) . \end{matrix}$

e_X (g (y) , x) = e_Y (f (x) , y) if and only if e_{τ
_{e
_Y}} (F₂ (A) , B) = e_{τ
_{e
_X}} (G₂ (B) , A) where $\begin{matrix} F_{2} (A) (y) & = ⋁_{x \in X} (e_{Y} (f (x), y) ⊙ A^{*} (x)), \\ G_{2} (B) (x) & = ⋁_{y \in Y} (e_{X} (g (y), x) ⊙ B^{*} (y)) . \end{matrix}$

Proof. (1) Let e_X (x, g (y)) = e_Y (y, f (x)). Since $(e_{Y} (y, f (x)) \to A^{*} (x)) ⊙ e_{Y} (y, w) ⊙ e_{Y} (w, f (x)) \leq A^{*} (x),$ we have F (A) ∈ τ_{e
_Y}. Since $(e_{X} (x, g (y)) \to B^{*} (y)) ⊙ e_{X} (x, z) ⊙ e_{X} (z, g (y)) \leq B^{*} (y),$ we have G (B) ∈ τ_{e
_X}. Moreover, $\begin{matrix} e_{τ_{e_{Y}}} (B, F_{1} (A)) = ⋀_{y \in X} (B (y) \to F_{1} (A) (y)) \\ = ⋀_{y \in Y} (B (y) \to ⋀_{x \in X} (e_{Y} (y, f (x)) \to A^{*} (x))) \\ = ⋀_{y \in Y} (B (y) \to ⋀_{x \in X} (A (x) \to e_{Y}^{*} (y, f (x)))) \\ = ⋀_{y \in Y} ⋀_{y \in Y} (A (x) \to (B (y) \to e_{Y}^{*} (y, f (x)))) \\ = ⋀_{x \in X} ⋀_{y \in Y} (A (x) \to (e_{X} (x, g (y)) \to B^{*} (y))) \\ = ⋀_{x \in X} (A (x) \to ⋀_{y \in Y} (e_{X} (x, g (y)) \to B^{*} (y))) \\ = ⋀_{x \in X} (A (x) \to G_{1} (B) (x)) \\ = e_{τ_{e_{X}}} (A, G_{1} (B)) . \end{matrix}$

Conversely, $\begin{matrix} F_{1} ((e_{X})_{x}) (y) & = ⋀_{z \in X} (e_{Y} (y, f (z)) \to (e_{X})_{x}^{*} (z)) \\ \geq e_{Y}^{*} (y, f (x)) . \end{matrix}$ Moreover, $\begin{matrix} e_{Y} (y, f (z)) & ⊙ e_{X} (x, z) \\ \leq e_{Y} (y, f (z)) ⊙ e_{Y} (f (z), f (x)) \\ \leq e_{Y} (y, f (x)) \end{matrix}$ and $\begin{matrix} ⋀_{z \in X} (e_{Y} (y, f (z)) \to (e_{X})_{x}^{*} (z)) & \geq e_{Y}^{*} (y, f (x)) . \end{matrix}$ Thus $F_{1} ((e_{X})_{x}) (y) = e_{Y}^{*} (y, f (x))$ .

Similarly, $G_{1} ((e_{Y})_{y}) (x) = e_{X}^{*} (x, g (y))$ . Trivially, we have e_{τ
_{e
_Y}} ((e_Y) _y, F₁ ((e_X) _x)) ≤ F₁ ((e_X) _x) (y).

Let F₁ ((e_X) _x) ∈ τ_{e
_Y}. Since $F_{1} ((e_{X})_{x}) (y) ⊙ e_{Y} (y, w) \leq F_{1} ((e_{X})_{x}) (w) \Leftrightarrow e_{Y} (y, w) \to F_{1} ((e_{X})_{x}) (w) \geq F_{1} ((e_{X})_{x}) (y),$ we have e_{τ
_{e
_Y}} ((e_Y) _y, F₁ ((e_X) _x)) ≥ F₁ ((e_X) _x) (y). Hence e_{τ
_{e
_Y}} ((e_Y) _y, F₁ ((e_X) _x)) = F₁ ((e_X) _x) (y).

Since $G_{1} ((e_{Y})_{y}) (x) ⊙ e_{X} (x, z) \leq G_{1} ((e_{Y})_{y}) (z) \Leftrightarrow G_{1} ((e_{Y})_{y}) (x) \leq e_{X} (x, z) \to G_{1} ((e_{Y})_{y}) (z),$ we have G₁ ((e_Y) _y) (x) ≤ e_{τ
_{e
_X}} ((e_X) _x, G₁ ((e_Y) _y)). Moreover, e_{τ
_{e
_X}} ((e_X) _x, G₁ ((e_Y) _y)) ≤ G₁ ((e_Y) _y) (x). Hence e_{τ
_{e
_X}} ((e_X) _x, G₁ ((e_Y) _y)) = G₁ ((e_Y) _y) (x). Therefore $\begin{matrix} e_{τ_{e_{Y}}} ((e_{Y})_{y}, F_{1} ((e_{X})_{x})) & = F_{1} ((e_{X})_{x}) (y) \\ = e_{Y}^{*} (y, f (x)) \\ = G_{1} ((e_{Y})_{y}) (x) \\ = e_{X}^{*} (x, g (y)) \\ = e_{τ_{e_{X}}} ((e_{X})_{x}, G_{1} ((e_{Y})_{y})) . \end{matrix}$

(2) Let e_X (g (y) , x) = e_Y (f (x) , y). Since $e_{Y} (f (x), y) ⊙ e_{Y} (y, w) \leq e_{Y} (f (x), w),$ we have F (A) ∈ τ_{e
_Y}. Since $e_{X} (g (y), x) ⊙ e_{X} (x, z) \leq e_{X} (g (y), z),$ we have G (B) ∈ τ_{e
_X}. Thus $\begin{matrix} e_{τ_{e_{Y}}} (F_{1} (A), B) = ⋀_{y \in X} (F_{1} (A) (y) \to B (y)) \\ = ⋀_{y \in Y} (⋁_{x \in X} (e_{Y} (f (x), y) ⊙ A (x)) \to B (y)) \\ = ⋀_{x \in X} ⋀_{y \in Y} (B^{*} (y) ⊙ e_{X} (g (y), x) \to A (x))) \\ = e_{τ_{e_{X}}} (G_{1} (B), A) . \end{matrix}$

Conversely, $F_{1} ((e_{X})_{x}^{- 1 *}) (y) = e_{Y} (f (x), y)$ holds from: $\begin{matrix} F_{1} ((e_{X})_{x}^{- 1 *}) (y) & = ⋁_{z \in X} (e_{Y} (f (z), y) ⊙ (e_{X})_{x}^{- 1} (z)) \\ \geq e_{Y} (f (x), y) \end{matrix}$ and $e_{Y} (f (z), y) ⊙ e_{X} (z, x) \leq e_{Y} (f (z), y) ⊙ e_{Y} (f (x), f (z)) \leq e_{Y} (f (x), y) .$ Moreover, $G_{1} ((e_{Y})_{y}^{- 1 *}) (x) = e_{X} (x, g (y))$ holds from: $\begin{matrix} G_{1} ((e_{Y})_{y}^{- 1 *}) (x) \\ = ⋁_{x \in X} (e_{X} (g (w), x) ⊙ (e_{Y})_{y}^{- 1} (w)) \geq e_{X} (g (y), x), \end{matrix}$ $e_{X} (g (w), x) ⊙ e_{Y} (w, y) \leq e_{X} (g (w), x) ⊙ e_{X} (g (y), g (w)) \leq e_{X} (g (y), x)$ and $\begin{matrix} G_{1} ((e_{Y})_{y}^{- 1 *}) (x) \leq e_{X} (g (y), x) . \end{matrix}$

Since $F_{1} ((e_{X})_{x}^{- 1 *}) (w) ⊙ e_{Y} (w, y) \leq F_{1} ((e_{X})_{x}^{- 1 *}) (y) \Leftrightarrow F_{1} ((e_{X})_{x}^{- 1 *}) (w) \to e_{Y}^{*} (w, y) \geq F_{1}^{*} ((e_{X})_{x}^{- 1 *}) (y),$ we have $e_{τ_{e_{Y}}} (F_{1} ((e_{X})_{x}^{- 1 *}), (e_{Y})_{y}^{- 1 *}) = F_{1}^{*} ((e_{X})_{x}^{- 1 *}) (y) .$ Similarly, $e_{τ_{e_{X}}} (G_{1} ((e_{Y})_{y}^{- 1 *}), (e_{X})_{x}^{- 1 *}) = G_{1}^{*} ((e_{Y})_{y}^{- 1 *}) .$ Thus $e_{τ_{e_{Y}}} (F_{1} ((e_{X})_{x}^{- 1 *}), (e_{Y})_{y}^{- 1 *}) = F_{1}^{*} ((e_{X})_{x}^{- 1 *}) (y) = e_{Y}^{*} (f (x), y) = G_{1}^{*} ((e_{Y})_{y}^{- 1 *}) (x) = e_{X}^{*} (g (y), y) = e_{τ_{e_{X}}} (G_{1} ((e_{Y})_{y}^{- 1 *}), (e_{X})_{x}^{- 1 *}) .$

Theorem 3.17. Let (X, e_X) and (Y, e_Y) be fuzzy posets. Let f : X → Y and g : Y → X be antitone maps.

Define the map G : τ_{e
_Y} → τ_{e
_X} by $G (B) (x) = ⋀_{y \in Y} (B (y) \to e_{X} (g (y), x)) .$ (1)Then G satisfies $\begin{matrix} G ((e_{Y})_{w}) (x) & = e_{X} (g (w), x), \\ G (α ⊙ B) & = α \to G (B), \\ G (⋁ B_{i}) & = ⋀ G (B_{i}) . \end{matrix}$ (2)Conversely, if a map G : τ_{e
_Y} → τ_{e
_X} satisfies Equation (2), then G is the map defined in Equation (1).

Furthermore, if G is the map defined in Equation (1), then there exists a unique F : τ_{e
_X} → τ_{e
_Y} with $F (A) (y) = ⋀_{x \in X} (A (x) \to e_{X} (g (y), x))$ such that (e_{tτ
_{e
_X}}, F, G, e_{τ
_{e
_Y}}) is a Galois connection.

Define the map G : τ_{e
_Y} → τ_{e
_X} by $G (B) (x) = ⋁_{y \in Y} (e_{X}^{*} (x, g (y)) ⊙ B^{*} (y)) .$ (3)Then G satisfies $\begin{matrix} G (((e_{Y})_{w}^{- 1})^{*}) (x) & = e_{X}^{*} (x, g (w)), \\ G (α \to B) & = α ⊙ G (B), \\ G (⋀ B_{i}) & = ⋁ G (B_{i}) . \end{matrix}$ (4)Conversely, if a map G : τ_{e
_Y} → τ_{e
_X} satisfies Equation (4), then G is the map defined in Equation (3).

Furthermore, if G is the map defined in Equation (3), then there exists a unique F : τ_{e
_X} → τ_{e
_Y} with $F (A) (y) = ⋁_{x \in X} (A^{*} (x) ⊙ e_{X}^{*} (x, g (y))$ such that (e_{τ
_{e
_X}}, F, G, e_{τ
_{e
_Y}}) is a dual Galois connection.

Define the map F : τ_{e
_X} → τ_{e
_Y} by $F (A) (y) = ⋁_{x \in X} (e_{Y}^{*} (y, f (x)) ⊙ A^{*} (x)) .$ (5)Then F satisfies $\begin{matrix} F (((e_{X})_{z}^{- 1})^{*}) (y) & = e_{Y}^{*} (y, f (z)), \\ F (α \to A) & = α ⊙ F (A), \\ F (⋀ A_{i}) & = ⋁ F (A_{i}) . \end{matrix}$ (6)Conversely, if a map F : τ_{e
_X} → τ_{e
_Y} satisfies Equation (6), then F is the map defined in Equation (5).

Furthermore, if F is the map defined in Equation (5), then there exists a unique G : τ_{e
_Y} → τ_{e
_X} with $G (B) (x) = ⋁_{y \in Y} (B^{*} (y) ⊙ e_{Y}^{*} (y, f (x)))$ such that (e_{τ
_{e
_X}}, F, G, e_{τ
_{e
_Y}}) is a dual Galois connection.

Define the map F : τ_{e
_X} → τ_{e
_Y} by $F (A) (y) = ⋀_{x \in X} (A (x) \to e_{Y} (f (x), y)) .$ (7)Then F satisfies $\begin{matrix} F (e_{X})_{z}) (y) & = e_{Y} (f (z), y)), \\ F (α ⊙ A) & = α \to F (A), \\ F (⋁ A_{i}) & = ⋀ G (A_{i}) . \end{matrix}$ (8)Conversely, if a map F : τ_{e
_X} → τ_{e
_Y} satisfies Equation (8), then F is the map defined in Equation (7).

Furthermore, if F is the map defined in Equation (7), then there exists a unique G : τ_{e
_Y} → τ_{e
_X} with $G (B) (x) = ⋀_{x \in X} (B (y) \to e_{Y} (f (x), y))$ such that (e_{τ
_{e
_X}}, F, G, e_{τ
_{e
_Y}}) is a Galois connection.

Proof (1) Let B ∈ τ_{e
_Y}. Then $G (B) (x) ⊙ e_{X} (x, w) = ⋀_{y \in Y} (B (y) \to e_{X} (g (y), x)) ⊙ e_{X} (x, w) \leq G (B) (w);$ that is, G (B) ∈ τ_{e
_X}. So, G is well-defined. Since g is antitone, we have $e_{X} (g (w), x) ⊙ e_{Y} (w, y) \leq e_{X} (g (w), x) ⊙ e_{X} (g (y), g (w)) \leq e_{X} (g (y), x),$ $e_{X} (g (w), x) \leq ⋀_{y \in Y} (e_{Y} (w, y) \to e_{X} (g (y), x)),$ $G ((e_{Y})_{w}) (x) = ⋀_{y \in Y} ((e_{Y})_{w} (y) \to e_{X} (g (y), x)) \leq e_{X} (g (w), x) .$ Hence G ((e_Y) _w) (x) = e_X (g (w) , x). Moreover, G (α ⊙ B) = α → G (B) and G (⋁ B_i) = ⋀ G (B_i).

Conversely, let B = ⋁ _y∈Y (B (y) ⊙ e_y (y, -)). Then $\begin{matrix} G (B) (x) & = G (⋁_{y \in Y} (B (y) ⊙ e_{Y} (y, -))) \\ = ⋀_{y \in Y} (B (y) \to G ((e_{Y})_{y}) (x)) \\ = ⋀_{y \in Y} (B (y) \to e_{X} (g (y), x)) . \end{matrix}$ Note that G (α ⊙ B) = α → G (B) and G (⋁ B_i) = ⋀ G (B_i). Define F : τ_{e
_X} → τ_{e
_Y} by $\begin{matrix} F (A) (y) = ⋁ {C (y) ∣ G (C) \geq A} \\ = ⋁ {C (y) ∣ ⋀_{y \in Y} (C (y) \to e_{X} (g (y), x)) \geq A (x)} \\ = ⋁ {C (y) ∣ C (y) \leq ⋀_{x \in X} (A (x) \to e_{X} (g (y), x))} \\ = ⋀_{x \in X} (A (x) \to e_{X} (g (y), x)) . \end{matrix}$ Since (a → b) ⊙ c ≤ a → b ⊙ c, we have F (A) ∈ τ_{e
_Y} from: $F (A) (y) ⊙ e_{Y} (y, w) \leq ⋀_{x \in X} (A (x) \to e_{X} (g (y), x)) ⊙ e_{X} (g (w), g (y)) \leq ⋀_{x \in X} (A (x) e_{X} (g (y), x) ⊙ e_{X} (g (w), g (y))) \leq F (A) (w) .$ Hence (e_{τ
_{e
_X}}, F, G, e_{τ
_{e
_Y}}) is a Galois connection.

Let F, F₁ : τ_{e
_X} → τ_{e
_Y} be two maps such that e_{τ
_{e
_X}} (A, G (B)) = e_{τ
_{e
_Y}} (B, F (A)) = e_{τ
_{e
_Y}} (B, F₁ (A)). If B = F (A), then $⊤ = e_{τ_{e_{Y}}} (F (A), F (A)) = e_{τ_{e_{Y}}} (F (A), F_{1} (A)),$ and so F ≤ F₁.

If B = F₁ (A), then $e_{τ_{e_{Y}}} (F_{1} (A), F (A)) = e_{τ_{e_{Y}}} (F_{1} (A), F_{1} (A)) = ⊤,$ and so F₁ ≤ F.

Hence F is a unique map.

(2) For all B ∈ τ_{e
_Y}, we have G (B) ∈ τ_{e
_X} from: $G (B) (x) ⊙ e_{X} (x, w) = ⋁_{y \in Y} (B^{*} (y) ⊙ e_{X}^{*} (x, g (y))) ⊙ e_{X} (x, w) \leq G (B) (w) .$ So, G is well-defined. Since g is antitone, $e_{X} (x, g (w)) ⊙ e_{Y} (y, w) \leq e_{X} (x, g (w)) ⊙ e_{X} (g (w), g (y)) \leq e_{X} (x, g (y)) .$ So, $e_{X}^{*} (x, g (y)) ⊙ e_{Y} (y, w) \leq e_{X}^{*} (x, g (w))$ and $G (((e_{Y})_{w}^{- 1})^{*}) (x) = ⋁_{y \in Y} (e_{X}^{*} (x, g (y)) ⊙ ((e_{Y})_{w}^{- 1}) (y)) \geq e_{X}^{*} (x, g (w)) .$ Hence $G (((e_{Y})_{w}^{- 1})^{*}) (x) = e_{X}^{*} (x, g (w))$ . Moreover, G (α → B) = α ⊙ G (B) and G (⋀ B_i) = ⋁ G (B_i).

Conversely, let $B = ⋀_{y \in Y} (B^{*} (y) \to e_{Y}^{*} (-, y))$ . Then $\begin{matrix} G (B) (x) & = G (⋀_{y \in Y} (B^{*} (y) \to e_{Y}^{*} (-, y))) (x) \\ = ⋁_{y \in Y} (B^{*} (y) ⊙ G (((e_{Y})_{y}^{- 1})^{*}) (x)) \\ = ⋁_{y \in Y} (B (y) ⊙ e_{X}^{*} (x, g (y))) . \end{matrix}$ Note that G (α → B) = α ⊙ G (B) and G (⋀ B_i) = ⋁ G (B_i). Define F : τ_{e
_X} → τ_{e
_Y} by $\begin{matrix} F (A) (y) = ⋀ {C (y) ∣ G (C) \leq A} \\ = ⋀ {C (y) ∣ ⋁_{y \in Y} (R^{*} (x, y) ⊙ C^{*} (y))) \leq A (x)} \\ = ⋀ {C (y) ∣ ⋀ (C^{*} (y) \to e_{X} (x, g (y)) \geq A^{*} (x)} \\ = ⋀ {C (y) ∣ C^{*} (y) \leq ⋀_{x \in X} (A^{*} (x) \to e_{X} (x, g (y)))} \\ = ⋁_{x \in X} (A^{*} (x) ⊙ e_{X}^{*} (x, g (y))) . \end{matrix}$ Then F (A) ∈ e_{τ
_{e
_Y}} because $\begin{matrix} F (A) (y) ⊙ e_{Y} (y, w) \\ = ⋁_{x \in X} (A^{*} (x) ⊙ e_{X}^{*} (x, g (y))) ⊙ e_{Y} (y, w) \\ \leq ⋁_{x \in X} (A^{*} (x) ⊙ e_{X}^{*} (x, g (y))) ⊙ e_{X} (g (w), g (y)) \\ \leq ⋁_{x \in X} (A^{*} (x) ⊙ e_{X}^{*} (x, g (w))) = F (A) (w) . \end{matrix}$ Moreover, (e_{τ
_{e
_X}}, F, G, e_{τ
_{e
_Y}}) is a dual residuated connection. By a similar method used in (1), F is unique.

(3) and (4) can be similarly proved as in (1) and (2).

Corollary 3.18. Let (X, e_X) and (Y, e_Y) be fuzzy posets. Let f : X → Y and g : Y → X be antitone maps.

Define the map G : τ_{e
_Y} → τ_{e
_X} by $G (B) (x) = ⋀_{y \in Y} (e_{X} (x, g (y)) \to B^{*} (y)) .$ (9)Then G satisfies $\begin{matrix} G (e_{Y})_{w}) (x) & = e_{X}^{*} (x, g (y)), \\ G (α ⊙ B) & = α \to G (B), \\ G (⋁ B_{i}) & = ⋀ G (B_{i}) . \end{matrix}$ (10) Conversely, if a map G : τ_{e
_Y} → τ_{e
_X} satisfies Equation (10), then G is the map defined in Equation (9).

Furthermore, if G is the map defined in Equation (9), then there exists a unique F : τ_{e
_X} → τ_{e
_Y} with $F (A) (y) = ⋀_{x \in X} (e_{X} (x, g (y)) \to A^{*} (x))$ such that (e_{τ
_{e
_X}}, F, G, e_{τ
_{e
_Y}}) is a Galois connection.

Define the map G : τ_{e
_Y} → τ_{e
_X} by $G (B) (x) = ⋁_{y \in Y} (e_{X} (g (y), x) ⊙ B^{*} (y)) .$ (11)Then G satisfies $\begin{matrix} G (((e_{Y})_{w}^{- 1})^{*}) (x) & = e_{X} (g (w), x), \\ G (α \to B) & = α ⊙ G (B), \\ G (⋀ B_{i}) & = ⋁ G (B_{i}) . \end{matrix}$ (12)Conversely, if a map G : τ_{e
_Y} → τ_{e
_X} satisfies Equation (12), then G is the map defined in Equation (11).

Furthermore, if G is the map defined in Equation (11), then there exists a unique F : τ_{e
_X} → τ_{e
_Y} with $F (A) (y) = ⋁_{x \in X} (A^{*} (x) ⊙ e_{X} (g (y), x)$ such that (e_{τ
_{e
_X}}, F, G, e_{τ
_{e
_Y}}) is a dual Galois connection.

Define the map F : τ_{e
_X} → τ_{e
_Y} by $F (A) (y) = ⋁_{x \in X} (e_{Y} (f (x), y) ⊙ A^{*} (x)) .$ (13)Then F satisfies $\begin{matrix} F (((e_{X})_{z}^{- 1})^{*}) (y) & = e_{Y} (f (z), y), \\ F (α \to A) & = α ⊙ F (A), \\ F (⋀ A_{i}) & = ⋁ F (A_{i}) . \end{matrix}$ (14)Conversely, if a map F : τ_{e
_X} → τ_{e
_Y} satisfies Equation (14), then F is the map defined in Equation (13).

Furthermore, if F is the map defined in Equation (13), then there exists a unique G : τ_{e
_Y} → τ_{e
_X} with $G (B) (x) = ⋁_{y \in Y} (B^{*} (y) ⊙ e_{Y} (f (x), y))$ such that (e_{τ
_{e
_X}}, F, G, e_{τ
_{e
_Y}}) is a dual Galois connection.

Define the map F : τ_{e
_X} → τ_{e
_Y} by $F (A) (y) = ⋀_{x \in X} (e_{Y} (y, f (x)) \to A^{*} (x)) .$ (15)Then F satisfies $\begin{matrix} F (e_{X})_{z}) (y) & = e_{Y}^{*} (y, f (z)), \\ F (α ⊙ A) & = α \to F (A), \\ F (⋁ A_{i}) & = ⋀ G (A_{i}) . \end{matrix}$ (16) Conversely, if a map F : τ_{e
_X} → τ_{e
_Y} satisfies Equation (16), then F is the map defined in Equation (15).

Furthermore, if F is the map defined in Equation (15), then there exists a unique G : τ_{e
_Y} → τ_{e
_X} with $G (B) (x) = ⋀_{x \in X} (e_{Y} (y, f (x)) \to B^{*} (y))$ such that (e_{τ
_{e
_X}}, F, G, e_{τ
_{e
_Y}}) is a Galois connection.

Example 3.19. Let (L = [0, 1] , ⊙ , → , 0, 1) be a complete residuated lattice defined in Example 3.4. Let X = {a, b, c} be a set. Let f : X → X be a map by f (a) = b, f (b) = a, f (c) = c.

(1) Let (X = {a, b, c} , e₁, e₂) be a fuzzy poset where $e_{1} = (\begin{matrix} 1 & 0.6 & 0.5 \\ 0.6 & 1 & 0.6 \\ 0.7 & 0.5 & 1 \end{matrix}), e_{2} = (\begin{matrix} 1 & 0.7 & 0.5 \\ 0.7 & 1 & 0.7 \\ 0.6 & 0.6 & 1 \end{matrix}) .$ Since e₁ (x, y) ≤ e₂ (f (y) , f (x)) and e₁ (x, f (f (x))) = e₂ (y, f (f (y))) =1, the 4-tuple (e₁, f, f, e₂) is a Galois connection but it is not a dual Galois connection because 0.7 = e₂ (a, b) ≰ e₁ (f (b) , f (a)) = e₁ (a, b) =0.6.

Let R₁ (x, y) = e₂ (y, f (x)) and R₂ (x, y) = e₂ (f (x) , y) for all x, y ∈ X where $R_{1} = (\begin{matrix} 0.7 & 1 & 0.6 \\ 1 & 0.7 & 0.6 \\ 0.5 & 0.7 & 1 \end{matrix}), R_{2} = (\begin{matrix} 0.7 & 1 & 0.7 \\ 1 & 0.7 & 0.5 \\ 0.6 & 0.6 & 1 \end{matrix}) .$

(2) By Theorem 3.17 (3), (e_{τ
_{e
_X}}, F, G, e_{τ
_{e
_Y}}) is a dual Galois connection with two maps: F : τ_{e
_X} → τ_{e
_Y} and G : τ_{e
_Y} → τ_{e
_X} where $\begin{matrix} F (A) (y) & = ⋁_{x \in X} (e_{2}^{*} (y, f (x)) ⊙ A^{*} (x)) \\ = (\begin{matrix} (0.3 ⊙ A^{*} (a)) \lor (0.5 ⊙ A^{*} (c)) \\ (0.3 ⊙ A^{*} (b)) \lor (0.3 ⊙ A^{*} (c)) \\ (0.4 ⊙ A^{*} (a)) \lor (0.4 ⊙ A^{*} (b)) \end{matrix}) \end{matrix}$ and $\begin{matrix} G (B) (x) & = ⋁_{x \in X} (e_{2}^{*} (y, f (x)) ⊙ B^{*} (y)) \\ = (\begin{matrix} (0.3 ⊙ B^{*} (a)) \lor (0.4 ⊙ B^{*} (c)) \\ (0.3 ⊙ B^{*} (b)) \lor (0.4 ⊙ B^{*} (c)) \\ (0.5 ⊙ B^{*} (a)) \lor (0.3 ⊙ B^{*} (b)) \end{matrix}) . \end{matrix}$

(3) By Corollary 3.18 (4), (e_{τ
_{e
_X}}, F, G, e_{τ
_{e
_Y}}) is a dual Galois connection with two maps: F : τ_{e
_X} → τ_{e
_Y} and G : τ_{e
_Y} → τ_{e
_X} where $\begin{matrix} F (A) (y) = ⋀_{x \in X} (e_{2} (y, f (x)) \to A^{*} (x)) \\ = (\begin{matrix} (0.7 \to A^{*} (a)) \land A^{*} (b) \land (0.5 \to A^{*} (c)) \\ A^{*} (a) \land (0.7 \to A^{*} (b)) \land (0.7 \to A^{*} (c)) \\ (0.6 \to A^{*} (a)) \land (0.6 \to A^{*} (b) \land A^{*} (c) \end{matrix}) \end{matrix}$ and $\begin{matrix} G (B) (x) = ⋀_{x \in X} (e_{2} (y, f (x)) \to B^{*} (y)) \\ = (\begin{matrix} (0.7 \to B^{*} (a)) \land B^{*} (b) \land (0.6 \to B^{*} (c)) \\ B^{*} (a) \land (0.7 \to B^{*} (b)) \land (0.6 \to B^{*} (c)) \\ (0.5 \to B^{*} (a)) \land (0.7 \to B^{*} (b)) \land B^{*} (c) \end{matrix}) . \end{matrix}$

(4) By Theorem 3.17 (4), (e_{τ
_{e
_X}}, F, G, e_{τ
_{e
_Y}}) is a Galois connection with two maps: F : τ_{e
_X} → τ_{e
_Y} and G : τ_{e
_Y} → τ_{e
_X} where $\begin{matrix} F (A) (y) = ⋀_{x \in X} (A (x) \to e_{2} (f (x), y)) \\ = (\begin{matrix} (A (a) \to 0.7) \land (A (c) \to 0.6) \\ (A (b) \to 0.7) \land (A (c) \to 0.6) \\ (A (a) \to 0.7) \land (A (b) \to 0.5) \end{matrix}) \end{matrix}$ and $\begin{matrix} G (B) (x) = ⋀_{y \in X} (B (y) \to e_{2} (f (x), y)) \\ = (\begin{matrix} (B (a) \to 0.7) \land (B (c) \to 0.7) \\ (B (b) \to 0.7) \land (B (c) \to 0.5) \\ (B (a) \to 0.6) \land (B (b) \to 0.6) \end{matrix}) . \end{matrix}$

(5) By Corollary 3.18 (3), (e_{τ
_{e
_X}}, F, G, e_{τ
_{e
_Y}}) is a dual Galois connection with two maps: F : τ_{e
_X} → τ_{e
_Y} and G : τ_{e
_Y} → τ_{e
_X} where $\begin{matrix} F (A) (y) & = ⋁_{x \in X} (e_{2} (f (x), y) ⊙ A^{*} (x)) \\ = (\begin{matrix} (0.7 ⊙ A^{*} (a)) \lor (0.6 ⊙ A^{*} (c)) \\ (0.7 ⊙ A^{*} (b)) \lor (0.6 ⊙ A^{*} (c)) \\ (0.7 ⊙ A^{*} (a)) \lor (0.5 ⊙ A^{*} (b)) \end{matrix}) \end{matrix}$ and $\begin{matrix} G (B) (x) & = ⋁_{x \in X} (e_{2}^{*} (y, f (x)) ⊙ B^{*} (y)) \\ = (\begin{matrix} (0.7 ⊙ B^{*} (a)) \lor (0.7 ⊙ B^{*} (c)) \\ (0.7 ⊙ B^{*} (b)) \lor (0.5 ⊙ B^{*} (c)) \\ (0.6 ⊙ B^{*} (a)) \lor (0.6 ⊙ B^{*} (b)) \end{matrix}) . \end{matrix}$

Example 3.20. Let (L = [0, 1] , ⊙ , → , 0, 1) be a complete residuated lattice defined in Example 3.4. Let X = {h_i ∣ i = {1, . . . , 3}} with h_i= “house". Let Y = {e, b, w, c, i} with e= “expensive", b= “beautiful", w= “wooden", c= “creative", i= “in the green surroundings". Let R ∈ [0, 1] ^X×Y be a fuzzy information where $\begin{matrix} R & e & b & w & c & i \\ h_{1} & 0.7 & 0.6 & 0.5 & 0.9 & 0.2 \\ h_{2} & 0.6 & 0.8 & 0.4 & 0.3 & 0.5 \\ h_{3} & 0.4 & 0.9 & 0.8 & 0.6 & 0.6 \end{matrix}$ Define [0, 1]-fuzzy preorders $e_{X}^{Y}, e_{X}^{{b, w}} \in [0, 1]^{X \times X}$ by $\begin{matrix} e_{X}^{Y} (h_{i}, h_{j}) & = ⋀_{y \in Y} (R (h_{i}, y) \to R (h_{j}, y)), \\ e_{X}^{{b, w}} (h_{i}, h_{j}) & = ⋀_{y \in {b, w}} (R (h_{i}, y) \to R (h_{j}, y)) . \end{matrix}$ Then $e_{X}^{Y} = (\begin{matrix} 1 & 0.4 & 0.7 \\ 0.7 & 1 & 0.8 \\ 0.6 & 0.6 & 1 \end{matrix}), e_{X}^{{b, w}} = (\begin{matrix} 1 & 0.4 & 0.7 \\ 0.8 & 1 & 1 \\ 0.7 & 0.6 & 1 \end{matrix}) .$

Define [0, 1]-fuzzy preorders e_Y ∈ [0, 1] ^Y×Y and e_{b,w,c} ∈ [0, 1] ^{{b,w,c}×{b,w,c}} by $\begin{matrix} e_{Y} (y_{i}, y_{j}) & = ⋀_{x \in X} (R (x, y_{i}) \to R (x, y_{j})), \\ e_{{b, w, c}} (y_{i}, y_{j}) & = ⋀_{x \in X} (R (x, y_{i}) \to R (x, y_{j})) . \end{matrix}$ Then $\begin{matrix} e_{Y} & = (\begin{matrix} 1 & 0.9 & 0.8 & 0.7 & 0.5 \\ 0.5 & 1 & 0.6 & 0.5 & 0.5 \\ 0.6 & 1 & 1 & 0.8 & 0.7 \\ 0.8 & 0.7 & 0.6 & 1 & 0.3 \\ 0.8 & 1 & 0.9 & 0.8 & 1 \end{matrix}), \\ e_{{b, w, c}} & = (\begin{matrix} 1 & 0.6 & 0.5 \\ 1 & 1 & 0.8 \\ 0.7 & 0.6 & 1 \end{matrix}), \\ R_{{b, w, c}} & = (\begin{matrix} 0.6 & 0.5 & 0.9 \\ 0.8 & 0.4 & 0.3 \\ 0.9 & 0.8 & 0.6 \end{matrix}) . \end{matrix}$

Note that $(e_{X}^{{b, w, c}})^{- 1} \circ R_{{b, w, c}} = R_{{b, w, c}}$ but $e_{X}^{{b, w, c}} \circ R_{{b, w, c}} ≰ R_{{b, w, c}}$ because $e_{X}^{{b, w, c}} \circ R_{{b, w, c}} = (\begin{matrix} 0.6 & 0.5 & 0.9 \\ 0.9 & 0.8 & 0.7 \\ 0.9 & 0.8 & 0.6 \end{matrix}) .$

Note that R_{b,w,c} ∘ e_{b,w,c} = R_{b,w,c} but $R_{{b, w, c}} \circ e_{{b, w, c}}^{- 1} ≰ R_{{b, w, c}}$ because $R_{{b, w, c}} \circ e_{{b, w, c}}^{- 1} = (\begin{matrix} 0.6 & 0.7 & 0.9 \\ 0.8 & 0.8 & 0.5 \\ 0.9 & 0.9 & 0.6 \end{matrix}) .$

Note $R ≱ e_{X}^{Y} \circ R = (\begin{matrix} 0.7 & 0.6 & 0.5 & 0.9 & 0.4 \\ 0.6 & 0.8 & 0.6 & 0.6 & 0.5 \\ 0.4 & 0.9 & 0.8 & 0.6 & 0.6 \end{matrix}) .$

Note that $(e_{X}^{Y})^{- 1} \circ R = R$ and R ∘ e_Y = R, but $R ≱ R \circ e_{Y}^{- 1} = (\begin{matrix} 0.7 & 0.6 & 0.7 & 0.9 & 0.7 \\ 0.7 & 0.8 & 0.8 & 0.5 & 0.8 \\ 0.8 & 0.9 & 0.9 & 0.6 & 0.9 \end{matrix}) .$

(1) Since $(e_{X}^{Y})^{- 1} \circ R = R$ and R ∘ e_Y = R, Theorems 3.10 and 3.13 hold.

(2) Since $e_{X}^{Y} (h_{2}, h_{3}) ⊙ R (h_{3}, w) = 0.8 ⊙ 0.8 = 0.6 ≰ R (h_{2}, w) = 0.4$ by Theorem 3.12, we have $G ((e_{Y})_{w}) (h_{2}) \notin τ_{e_{X}^{Y}}$ from: $G ((e_{Y})_{w}) (h_{2}) ⊙ e_{X}^{Y} (h_{2}, h_{3}) = R^{*} (h_{2}, w) ⊙ e_{X}^{Y} (h_{2}, h_{3}) ≰ R^{*} (h_{3}, w) = G ((e_{Y})_{w}) (h_{3}) .$ Since $R (h_{2}, b) ⊙ e_{Y}^{- 1} (b, e) = 0.8 ⊙ 0.9 = 0.7 ≰ R (h_{2}, e) = 0.6$ by Theorem 3.12, we have F ((e_X) _{h
₂}) ∉ τ_{e
_Y} from: $F ((e_{X})_{h_{2}}) (e) ⊙ e_{Y} (e, b) = R^{*} (h_{2}, e) ⊙ e_{Y} (e, b) = 0.4 ⊙ 0.9 ≰ 0.3 = R^{*} (h_{2}, b) = F ((e_{X})_{h_{2}}) (b) .$

By Theorem 3.14, we have $F ((e_{X})_{h_{2}}^{- 1 *}) \notin τ_{e_{Y}}$ from: $F ((e_{X})_{h_{2}}^{- 1 *}) (e) ⊙ e_{Y} (e, b) = R^{*} (h_{2}, e) ⊙ e_{Y} (e, b) = 0.4 ⊙ 0.9 ≰ 0.3 = R^{*} (h_{2}, b) = F ((e_{X})_{h_{2}}^{- 1 *}) (b) .$ Also we have $G ((e_{Y})_{w}^{- 1 *}) \notin τ_{e_{X}^{Y}}$ from: $G ((e_{Y})_{w}^{- 1 *}) (h_{2}) ⊙ e_{X}^{Y} (h_{2}, h_{3}) = R^{*} (h_{2}, w) ⊙ e_{X}^{Y} (h_{2}, h_{3}) ≰ R^{*} (h_{3}, w) = G ((e_{Y})_{w}^{- 1 *}) (h_{3}) .$

(3) Since $e_{X}^{{b, w, c}} (h_{2}, h_{3}) ⊙ R_{{b, w, c}} (h_{3}, b) = 1 ⊙ 0.9 = 0.9 ≰ R_{{b, w, c}} (h_{2}, b) = 0.8$ by Theorem 3.12, we have $G ((e_{{b, w, c}})_{b}) \notin τ_{e_{X}^{{b, w, c}}}$ from: $G ((e_{{b, w, c}})_{b}) (h_{2}) ⊙ e_{X}^{{b, w, c}} (h_{2}, h_{3}) = R_{{b, w, c}}^{*} (h_{2}, b) ⊙ e_{X}^{{b, w, c}} (h_{2}, h_{3}) ≰ R_{{b, w, c}}^{*} (h_{3}, w) = G ((e_{{b, w, c}})_{b}) (h_{3}) .$

Since $R_{{b, w, c}} (h_{3}, b) ⊙ e_{{b, w, c}}^{- 1} (b, w) = 0.9 ⊙ 1 = 0.9 ≰ R_{{b, w, c}} (h_{3}, w) = 0.8$ by Theorem 3.12, we have $F ((e_{X}^{{b, w, c}})_{h_{3}}) \notin τ_{e_{{b, w, c}}}$ from: $F ((e_{X}^{{b, w, c}})_{h_{3}}) (w) ⊙ e_{{b, w, c}} (w, b) = R_{{b, w, c}}^{*} (h_{3}, w) ⊙ e_{{b, w, c}} (w, b) = 0.2 ⊙ 1 ≰ 0.1 = R_{{b, w, c}}^{*} (h_{3}, b) = F ((e_{X}^{{b, w, c}})_{h_{3}}) (b) .$

By Theorem 3.14, we have $F ((e_{X}^{{b, w, c}})_{h_{3}}^{- 1 *}) \notin τ_{e_{{b, w, c}}}$ from: $F ((e_{X}^{{b, w, c}})_{h_{3}}^{- 1 *}) (w) ⊙ e_{{b, w, c}} (w, b) = R_{{b, w, c}}^{*} (h_{3}, w) ⊙ e_{{b, w, c}} (w, b) = 0.2 ⊙ 1 ≰ 0.1 = R_{{b, w, c}}^{*} (h_{3}, b) = F ((e_{X}^{{b, w, c}})_{h_{3}}^{- 1 *}) (b) .$ Also we have $G ((e_{{b, w, c}})_{b}^{- 1 *}) \notin τ_{e_{X}^{{b, w, c}}}$ from: $G ((e_{{b, w, c}})_{b}^{- 1 *}) (h_{2}) ⊙ e_{X}^{{b, w, c}} (h_{2}, h_{3}) = R_{{b, w, c}}^{*} (h_{2}, b) ⊙ e_{X}^{{b, w, c}} (h_{2}, h_{3}) ≰ R_{{b, w, c}}^{*} (h_{3}, w) = G ((e_{{b, w, c}})_{b}^{- 1 *}) (h_{3}) .$

4 Conclusion

As an extension of Bělohlávek’s Galois connections, we have introduced the notion of Galois and dual Galois connections on Alexandrov topologies. Under various relations, we have investigated the Galois and dual Galois connections on τ_{e
_X} × τ_{e
_Y}. We have studied their properties and have given examples.

In the future, we consider whether β (X, Y, L) = {(A, B) ∈ τ_{e
_X} × τ_{e
_X} ∣ F (A) = B, G (B) = A} is a complete lattice or not. Using the concepts of Galois connections, we investigate information systems and decision rules on residuated lattices in terms of applications to multi-attribute decision-making.

References

Bělohlávek

, Fuzzy Galois connections, Math Logic Quart 45(4) (1999), 497–504.

Bělohlávek

, Fuzzy Relational Systems, Kluwer Academic Publishers, New York, (2002).

Hájek

, Metamathematices of Fuzzy Logic, Kluwer Academic Publishers, Dordrecht, (1998).

Hao

and Li

, The relationship between-fuzzy rough set and-topology, Fuzzy Sets and Systems 178 (2011), 74–83.

Höhle

and Klement

E.P.

, Non-classical logic and their applications to fuzzy subsets, Kluwer Academic Publishers, Boston, (1995).

Höhle

and Rodabaugh

S.E.

, Mathematics of Fuzzy Sets, Logic, Topology and Measure Theory, The Handbooks of Fuzzy Sets Series, Kluwer Academic Publishers, Dordrecht, (1999).

Kim

Y.C.

, Join-meet preserving maps and fuzzy preorders, Journal of Intelligent and Fuzzy Systems 28 (2015), 1089–1097.

Kim

Y.C.

, Categories of fuzzy preorders, approximation operators and Alexandrov topologies, Journal of Intelligent and Fuzzy Systems 31 (2016), 1787–1793.

J.M.

and Kim

Y.C.

, Fuzzy complete lattices, Alexandrov-topologies and fuzzy rough sets, Journal of Intelligent and Fuzzy Systems 38 (2020), 3253–3266.

10.

Lai

, Zhang

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

11.

Z.M.

and Hu

B.Q.

, Topological and lattice structures of Lfuzzy rough set determined by lower and upper sets, Information Sciences 218 (2013), 194–204.

12.

J.M.

and Kim

Y.C.

, Fuzzy transformations and fuzzy residuated connections, Journal of Intelligent and Fuzzy Systems 36(4) (2019), 3555–3566.

13.

J.M.

and Kim

Y.C.

, The relations between residuated frames and residuated connections, Mathematics 8(295) (2020), 1–24.

14.

Orłowska

and Rewitzky

, Context algebras, context frames and their discrete duality, Transactions on Rough Sets IX, Springer, Berlin, (2008), 212–229.

15.

Orłowska

and Rewitzky

, Algebras for Galois-style connections and their discrete duality, Fuzzy Sets and Systems 161 (2010), 1325–1342.

16.

Pawlak

, Rough sets, Internat J Comput Inform Sci 11 (1982), 341–356.

17.

Pawlak

, Rough sets: Theoretical Aspects of Reasoning about Data, System Theory, Knowledge Engineering and Problem Solving, Kluwer Academic Publishers, Dordrecht, The Netherlands, (1991).

18.

Radzikowska

A.M.

and Kerre

E.E.

, A comparative study of fuzy rough sets, Fuzzy Sets and Systems 126 (2002), 137–155.

19.

Radzikowska

A.M.

and Kerre

E.E.

, Characterisation of main classes of fuzzy relations using fuzzy modal operators, Fuzzy Sets and Systems 152 (2005), 223–247.

20.

Ramadan

and Li

, Categories of lattice-valued closure (interior) operators and Alexandroff L-fuzzy topologies, Iranian Journal of Fuzzy Systems 16(3) (2019), 73–84.

21.

Rodabaugh

S.E.

and Klement

E.P.

, Topological and Algebraic Structures In Fuzzy Sets, The Handbook of Recent Developments in the Mathematics of Fuzzy Sets, Kluwer Academic Publishers, Boston, Dordrecht, London, (2003).

22.

She

Y.H.

and Wang

G.J.

, An axiomatic approach of fuzzy rough sets based on residuated lattices, Computers and Mathematics with Applications 58 (2009), 189–201.

23.

Tiwari

S.P.

and Srivastava

A.K.

, Fuzzy rough sets, fuzzy preorders and fuzzy topologies, Fuzzy Sets and Systems 210 (2013), 63–68.

24.

Turunen

, Mathematics Behind Fuzzy Logic, A Springer-Verlag Co., (1999).

25.

Ward

and Dilworth

R.P.

, Residuated lattices, Trans Amer Math Soc 45 (1939), 335–354.

26.

Wille

, Restructuring lattice theory; an approach based on hierarchies of concept, in: 1. Rival(Ed.), Ordered Sets, Reidel, Dordrecht, Boston, (1982), 445–470.

27.

Zhang

, An enrched category approach to many valed topology, Fuzzy Sets and Systems 158 (2007), 349–366.

28.

Zhao

F.F.

, Li

L.Q.

, Sun

S.B.

and Jin

, Rough approximation operators based on quantale-valued fuzzy generalized neighborhood systems, Iranian Journal of Fuzzy Systems 16 (6) (2019), 53–63.