Fuzzy complete lattices,Alexandrov L -topologies and fuzzy rough sets

Abstract

We introduce the concepts of fuzzy join complete lattices and Alexandrov L-pretopologies in complete residuated lattices. We show that fuzzy join complete lattices, Alexandrov L-pretopologies, fuzzy meet complete lattices and Alexandrov L-precotopologies are equivalent. Moreover, we define L-preinterior operators (resp. L-preclosure operators) as a viewpoint of fuzzy joins (resp. fuzzy meet) and fuzzy rough sets. Furthermore their properties and examples are investigated.

Keywords

Complete residuated lattices fuzzy rough sets

1 Introduction

Pawlak [13, 14] introduced the rough set theory as a formal tool to deal with imprecision and uncertainty in the data analysis. For an extension of classical rough sets, many researchers [4 , 19–23] developed L-lower and L-upper approximation operators in a complete residuated lattice which is an algebraic structure for many valued logic [1 , 18]. By using this concepts, information systems and decision rules were investigated in complete residuated lattices [1 , 23].

An interesting and natural research topic in rough set theory is the study of rough set theory and topology. The topological and lattice structures of L-fuzzy rough sets determined by lower and upper sets were investigated [11 , 20–22]. Kim [7 –9] studied the relations between L-fuzzy upper and lower approximation spaces and Alexandrov L-fuzzy topologies in complete residuated lattices. Moreover, categories of fuzzy preorders, approximating operators and Alexandrov topologies are isomorphic [8].

For a complete Heyting algebra(or a frame) as the base category [4, 26], Zhang [27, 28] introduced fuzzy complete lattices and the Dedekind-MacNeille completions for fuzzy posets in complete lattices.

In this paper, we introduce the concepts of fuzzy join(resp. meet) -complete lattice in complete residuated lattices as an extension of complete Heyting algebra(or a frame). Zhang [25] only use the complete for a fuzzy poset (L^X, e_{L
^X}). By using completeness for a fuzzy poset (τ, e_τ), we simplify Zhang’s definition. In Theorem 3.8, we show that (τ, e_τ) with ⊤_X ∈ τ is a fuzzy join complete lattice if and only if τ is an Alexandrov L-pretopology on X if and only if η_τ = {A^* ∈ L^X ∣ A ∈ τ} is an Alexandrov L-precotopology on X if and only if (η_τ, e_{η
_τ}) with ⊥_X ∈ η_τ is a fuzzy meet complete lattice.

We investigate the properties of L-preclosure (resp.L-preinterior) operators and fuzzy join(resp. meet) -complete lattices. In Theorem 3.9, let (X, e_X) be a fuzzy preordered set. We show that τ_{e
_X} = {A ∈ L^X ∣ A (x) ⊙ e_X (x, y) ≤ A (y)} = {⊓ _{τ
_{e
_X}}Θ_A = ⋁ _x∈X (A (x) ⊙ e_X (x, -)) ∣ A ∈ L^X} = {⊔ _{τ
_{e
_X}}Φ_A = ⋀ _y∈X (e_X (- , y) → A (y)) ∣ A ∈ L^X} is an Alexandrov L-topology on X and (τ_{e
_X}, e_{τ
_{e
_X}}) is a fuzzy complete lattice. In Theorems 3.16 and 3.17, we interpret L-preinterior and L-preclosure operator as join-complete and meet complete for maps Ψ_A and Θ_A on (τ, e_τ) and (η, e_η), respectively. Moreover, we show that the pair (⊔ _{τ
_{e
_X}}Φ_A, ⊓ _{τ
_{e
_X}}Θ_A) is a fuzzy rough set and give their examples.

2 Preliminaries

Definition 2.1. [1 , 24] An algebra (L, ∧ , ∨ , ⊙ , → , ⊥ , ⊤) is called a complete residuated lattice if it satisfies the following conditions:

(L1) (L, ≤ , ∨ , ∧ , ⊥ , ⊤) is a complete lattice with the greatest element ⊤ and the least element ⊥;

(L2) (L, ⊙ , ⊤) is a commutative monoid;

(L3) x ⊙ y ≤ z iff x ≤ y → z for x, y, z ∈ L.

In this paper, we always assume that (L, ≤ , ⊙ , → ^∗) is a complete residuated lattice.

For α ∈ L, A ∈ L^X, we denote (α → A) , (α ⊙ A) , α_X ∈ L^X as (α → A) (x) = α → A (x) , (α ⊙ A) (x) = α ⊙ A (x) , α_X (x) = α and x^* = x→ ⊥.

The condition x^** = x for each x ∈ L is called a double negative law.

Lemma 2.2. [1 , 24] For each x, y, z, x_i, y_i, w ∈ L, we have the following properties.

(1) ⊤ → x = x, ⊥ ⊙ x = ⊥ ,

(2) If y ≤ z, then x ⊙ y ≤ x ⊙ z, x → y ≤ x → z and z → x ≤ y → x,

(3) x ≤ y iff x→ y = ⊤.

(4) x → (⋀ _iy_i) = ⋀ _i (x → y_i),

(5) (⋁ _ix_i) → y = ⋀ _i (x_i → y),

(6) x ⊙ (⋁ _iy_i) = ⋁ _i (x ⊙ y_i),

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

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

(9) (x → y) ⊙ (y → z) ≤ x → z,

(10) ⋁_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),

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

(12) If L satisfies a double negative law, then (x ⊙ y^*) ^* = x → y and x → y = y^* → x^*.

Definition 2.3. [1 , 11] Let X be a set. A function e_X : X × X → L is called:

(E1) reflexive if e_X (x, x) =⊤ for all x ∈ X,

(E2) transitive if e_X (x, y) ⊙ e_X (y, z) ≤ e_X (x, z), for all x, y, z ∈ X,

(E3) if e_X (x, y) = e_X (y, x) =⊤, then x = y.

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

Definition 2.4. [2 , 26–28] Let (X, e_X) be a fuzzy poset and A ∈ L^X.

(1) A point x₀ is called a fuzzy join of A, denoted by x₀ = ⊔ _XA on (X, e_X), if it satisfies

(J1) A (x) ≤ e_X (x, x₀),

(J2) ⋀_x∈X (A (x) → e_X (x, y)) ≤ e_X (x₀, y).

The pair (X, e_X) is called fuzzy join complete if ⊔_XA exists for each A ∈ L^X.

A point x₁ is called a fuzzy meet of A, denoted by x₁ = ⊓ _XA on (X, e_X), if it satisfies

(M1) A (x) ≤ e_X (x₁, x),

(M2) ⋀_x∈X (A (x) → e_X (y, x)) ≤ e_X (y, x₁).

The pair (X, e_X) is called fuzzy meet complete if ⊓_XA exists for each A ∈ L^X.

The pair (X, e_X) is called fuzzy complete if ⊓_XA and ⊔_XA exists for each A ∈ L^X.

Remark 2.5. Let (X, e_X) be a fuzzy poset and A ∈ L^X.

(1) A point x₀ is x₀ = ⊔ _XA on (X, e_X) iff ⋀_x∈X (A (x) → e_X (x, y)) = e_X (x₀, y).

(2) A point x₁ is x₁ = ⊓ _XA on (X, e_X) iff ⋀_x∈X (A (x) → e_X (y, x)) = e_X (y, x₁).

Definition 2.6. [2 , 26–28] Let (L^X, e_{L
^X}) and (L^Y, e_{L
^Y}) be fuzzy posets and $F : L^{X} \to L^{Y}$ a map.

(1) $F$ is a join preserving map if $F (⊔_{L^{X}} Φ) = ⊔_{L^{Y}} F^{\to} (Φ)$ for all Φ ∈ L^{L
^X}, where $F^{\to} (Φ) (B) = ⋁_{F (A) = B} Φ (A)$ .

(2) $F$ is a meet preserving map if $F (⊓_{L^{X}} Φ) = ⊓_{L^{Y}} F^{\to} (Φ)$ for all Φ ∈ L^{L
^X}.

3 Fuzzy complete lattices, Alexandrov L-topologies and fuzzy rough sets

Definition 3.1. {(1) A subset τ ⊂ L^X is called an Alexandrov L-pretopology on X iff it satisfies the following conditions:

(O1) α_X ∈ τ.

(O2) If A_i ∈ τ for all i ∈ I, then ⋁_i∈IA_i ∈ τ.

(O3) If A ∈ τ and α ∈ L, then α ⊙ A ∈ τ.

(2) A subset η ⊂ L^X is called an Alexandrov L-precotopology on X iff it satisfies the following conditions:

(CO1) α → ⊥ _X ∈ η.

(CO2) If A_i ∈ η for all i ∈ I, then ⋀_i∈IA_i ∈ η.

(CO3) If A ∈ η and α ∈ L, then α → A ∈ η.

A subset τ ⊂ L^X is called an Alexandrov L-topology on X iff it is both Alexandrov L-pretopology and Alexandrov L-precotopology on X.

Lemma 3.2. Let τ ⊂ L^X. Define e_τ : τ × τ → L as e_τ (A, B) = ⋀ _x∈X (A (x) → B (x)). Then the following statements hold.

(1) (τ, e_τ) is a fuzzy poset.

(2) ⊔_τΦ is a fuzzy join of Φ ∈ L^τ iff ⋀_A∈τ (Φ (A) → e_τ (A, B)) = e_τ (⊔ _τΦ, B).

(3) ⊓_τΦ is a fuzzy meet of Φ ∈ L^τ iff ⋀_A∈τ (Φ (A) → e_τ (B, A)) = e_τ (B, ⊓ _τΦ).

(4) If ⊔_τΦ is a fuzzy join of Φ ∈ L^τ, then it is unique. Moreover, if ⊓_τΦ is a fuzzy meet of Φ ∈ L^τ, then it is unique.

Proof. (1) (E1) e_τ (A, A) =⋀ _x∈X (A (x) → A (x)) = ⊤ for all A ∈ τ,

(E2) By Lemma 2.2(9), e_τ (A, B) ⊙ e_τ (B, C) = ⋀ _x∈X (A (x) → B (x)) ⊙ ⋀ _x∈X (B (x) → C (x)) ≤ ⋀ _x∈X ((A (x) → B (x)) ⊙ (B (x) → C (x))) ≤ e_τ (A, C), for all A, B, C ∈ τ,

(E3) If e_τ (A, B) = e_τ (B, A) =⊤, By Lemma 2.2(3), A = B. Hence (τ, e_τ) is a fuzzy poset.

(2) Let ⊔_τΦ be a fuzzy join of Φ ∈ L^τ. By (J1), since Φ (A) ≤ e_τ (A, ⊔ _τΦ), we have Φ (A) ⊙ e_τ (⊔ _τΦ, B) ≤ e_τ (A, ⊔ _τΦ) ⊙ e_τ (⊔ _τΦ, B) ≤ e_τ (A, B) .

Hence e_τ (⊔ _τΦ, B) ≤ ⋀ _A∈τ (Φ (A) → e_τ (A, B)). By (J2), e_τ (⊔ _τΦ, B) = ⋀ _A∈τ (Φ (A) → e_τ (A, B))

(3) It is similarly proved as (2).

(4) Let A₁, A₂ be fuzzy joins of Φ ∈ L^τ. Then, for all B ∈ τ, $⋀_{A \in τ} (Φ (A) \to e_{τ} (A, B)) = e_{τ} (A_{1}, B) = e_{τ} (A_{2}, B) .$ Put B = A₁. Then ⊤ = e_τ (A₁, A₁) = e_τ (A₂, A₁) iff A₂ ≤ A₁. Put B = A₂. Then ⊤ = e_τ (A₁, A₂) = e_τ (A₂, A₂) iff A₁ ≤ A₂. Hence A₁ = A₂.

Theorem 3.3. Let τ ⊂ L^X. Then the following statements are equivalent:

(1) (τ, e_τ) with ⊤_X ∈ τ is fuzzy join complete.

(2) τ is an Alexandrov L-pretopology on X.

Proof. (1) ⇒ (2) Since (τ, e_τ) is fuzzy join complete, for each Φ ∈ L^τ, we have $\begin{matrix} e_{τ} (⊔_{τ} Φ, B) = ⋀_{C \in τ} (Φ (C) \to e_{τ} (C, B)) \\ = ⋀_{C \in τ} e_{τ} (Φ (C) ⊙ C, B) (by Lemma 2.2 (5, 7)) \\ = e_{τ} (⋁_{C \in τ} Φ (C) ⊙ C, B) . \end{matrix}$ By Lemma 3.2(4), ⊔_τΦ = ⋁ _C∈τ (Φ (C) ⊙ C) ∈ τ.

(O1) Define Φ_α : τ → L as Φ_α (⊤ _X) = α for all α ∈ L and Φ_α (B) =⊥, otherwise. Then $⊔_{τ} Φ_{α} = ⋁_{A \in τ} (Φ_{α} (A) ⊙ A) = α ⊙ ⊤_{X} = α_{X} .$ So, ⊔_τΦ_α = α_X ∈ τ.

(O2) Define Φ : τ → L as Φ (A) = α for A ∈ τ and Φ (B) =⊥, otherwise. Then $⊔_{τ} Φ (x) = ⋁_{A \in τ} (Φ (A) ⊙ A (x)) = α ⊙ A (x),$ So, ⊔_τΦ = α ⊙ A ∈ τ.

(O3) Let {A_i ∈ τ ∣ i ∈ Γ} be given. Define Φ : τ → L as Φ (A_i) =⊤ for i ∈ Γ and Φ (B) =⊥, otherwise. Then $(⊔_{τ} Φ) (x) = ⋁_{A \in τ} (Φ (A) ⊙ A (x)) = ⋁_{i \in Γ} A_{i} (x) .$ So, ⊔_τΦ = ⋁ _i∈ΓA_i ∈ τ.

(2) ⇒ (1) For each Φ ∈ L^τ, by (O2) and (O3), ⋁_C∈τΦ (C) ⊙ C ∈ τ. Thus, $\begin{matrix} e_{τ} (⊔_{τ} Φ, B) \\ = ⋀_{C \in τ} (Φ (C) \to e_{τ} (C, B)) \\ = ⋀_{C \in τ} e_{τ} (Φ (C) ⊙ C, B) = e_{τ} (⋁_{C \in τ} Φ (C) ⊙ C, B) . \end{matrix}$ By Lemma 3.2 (2), ⊔_τΦ is a fuzzy join of Φ.

Remark 3.4. We can simplify Definition 4.8 in [25], A strong L-topology on a set X is a subset τ ⊂ L^X such that:

(LT1) For every function G : τ → L, ⊔_τG exists, that is, (τ, e_τ) is fuzzy join complete.

(LT2) For every function G : τ → L with a finite support, ⊓_τG exists in (τ, e_τ).

Theorem 3.5. Let η ⊂ L^X. Then the following statements are equivalent:

(1) (η, e_η) with ⊥_X ∈ η is fuzzy meet complete.

(2) η is an Alexandrov L-precotopology on X.

Proof. (1) ⇒ (2) Since (η, e_η) is fuzzy meet complete, for each Φ ∈ L^η, we have $\begin{matrix} e_{η} (B, ⊓_{η} Φ) = ⋀_{C \in η} (Φ (C) \to e_{η} (B, C)) \\ = ⋀_{C \in η} e_{η} (B, Φ (C) \to C) (by Lemma 2.2 (4, 7)) \\ = e_{η} (B, ⋀_{C \in η} (Φ (C) \to C)) . \end{matrix}$ By Lemma 3.2(4), ⊓_ηΦ = ⋀ _C∈η (Φ (C) → C) ∈ η.

(CO1) Since ⊥_X ∈ η, we define Φ_α : η → L as Φ_α (⊥ _X) = α for all α ∈ L and Φ_α (B) =⊥, otherwise. Then $⊓_{η} Φ_{α} = ⋀_{A \in η} (Φ_{α} (A) \to A) = α \to ⊥_{X} .$ So, ⊓_ηΦ_α = α → ⊥ _X ∈ η.

(CO2) Define Φ : η → L as Φ (A) = α for A ∈ η and Φ (B) =⊥, otherwise. Then $⊓_{η} Φ (x) = ⋀_{A \in η} (Φ (A) \to A (x)) = α \to A (x),$ So, ⊓_ηΦ = α → A ∈ η.

(CO3) Let {A_i ∈ η ∣ i ∈ Γ} be given. Define Φ : η → L as Φ (A_i) =⊤ for i ∈ Γ and Φ (B) =⊥, otherwise. Then $(⊓_{η} Φ) (x) = ⋀_{A \in η} (Φ (A) \to A (x)) = ⋀_{i \in Γ} A_{i} (x) .$ So, ⊓_ηΦ = ⋀ _i∈ΓA_i ∈ η.

(2) ⇒ (1) For each Φ ∈ L^η, by (CO2) and (CO3), ⋀_C∈ηΦ (C) → C ∈ η. Thus, $\begin{matrix} e_{η} (B, ⊓_{η} Φ) = ⋀_{C \in η} (Φ (C) \to e_{η} (B, C)) \\ = ⋀_{C \in η} e_{η} (B, Φ (C) \to C) \\ = e_{η} (B, ⋀_{C \in η} Φ (C) \to C) . \end{matrix}$ By Lemma 3.2 (3), ⊓_ηΦ is a fuzzy meet of Φ.

Remark 3.6. We can simplify Definition 4.11 in [25] as follows: a strong L-cotopology on a set X is a subset η ⊂ L^X such that:

(LCT1) For every function G : η → L, ⊓_ηG exists, that is, (η, e_η) is fuzzy meet complete.

(LCT2) For every function G : η → L with a finite support, ⊔_ηG exists in (η, e_η).

From Theorems 3.3 and 3.5, we can obtain the following corollary.

Corollary 3.7. Let τ ⊂ L^X. Then the following statements are equivalent:

(1) (τ, e_τ) is fuzzy complete.

(2) τ is an Alexandrov L-topology on X.

Theorem 3.8. Let L be a complete residuated lattice satisfying a double negative law and τ ⊂ L^X. Then the following statements are equivalent:

(1) (τ, e_τ) with ⊤_X ∈ τ is fuzzy join complete.

(2) η_τ = {A^* ∈ L^X ∣ A ∈ τ} is an Alexandrov L-precotopology on X.

(3) (η_τ, e_{η
_τ}) with ⊥_X ∈ η_τ is fuzzy meet complete defined as e_{η
_τ} : η_τ × η_τ → L as e_{η
_τ} (A, B) = e_τ (B^*, A^*).

Proof. (1) ⇒ (2) Since (τ, e_τ) is fuzzy join complete, τ is an Alexandrov L-pretopology on X.

(CO2) For α ∈ L and A ∈ η_τ, we have α ⊙ A^* ∈ τ. By Lemma 2.2(12), (α ⊙ A^*) ^* = α → A ∈ η_τ .

(CO1) and (CO3) are easily proved.

(2) ⇒ (3) It follows from Lemma 2.2(12) and Theorem 3.5.

(3) ⇒ (1) For Φ ∈ L^τ with Ψ (A^*) = Φ (A) and ⋁_C∈τ (Φ (C) ⊙ C) ∈ τ iff ⋀_{C^*∈η_τ} (Ψ (C^*) → C^*) ∈ η_τ from Lemma 2.2(12), we have $\begin{matrix} ⋀_{C \in τ} (Φ (C) \to e_{τ} (C, B)) \\ = ⋀_{C^{*} \in η_{τ}} (Φ (C) \to e_{η_{τ}} (B^{*}, C^{*})) \\ = e_{η_{τ}} (B^{*}, ⋀_{C^{*} \in η_{τ}} (Ψ (C^{*}) \to C^{*})) \\ = e_{τ} (⋁_{C \in τ} (Φ (C) ⊙ C), B) = e_{τ} (⊔_{τ} Φ, B) . \end{matrix}$ By Lemma 3.2 (2), ⊔_τΦ is a fuzzy join of Φ.

Theorem 3.9. 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

(1) τ_{e
_X} is an Alexandrov L-topology on X.

(2) (τ_{e
_X}, e_{τ
_{e
_X}}) is fuzzy complete.

(3) τ_{e
_X} = {⋁ _x∈X (A (x) ⊙ e_X (x, -)) ∣ A ∈ L^X}.

(4) τ_{e
_X} = {⋀ _y∈X (e_X (- , y) → A (y)) ∣ A ∈ L^X}.

(5) If $e_{X}^{- 1} (x, y) = e_{X} (y, x)$ for all x, y ∈ X, $τ_{e_{X}^{- 1}} = {A \in L^{X} ∣ A (x) ⊙ e_{X} (y, x) \leq A (y)}$ . Moreover, if L satisfies a double negative law, $τ_{e_{X}^{- 1}} = η_{τ_{e_{X}}}$ where η_{τ
_{e
_X}} = {A^* ∈ L^X ∣ A ∈ τ_{e
_X}}.

(6) If L satisfies a double negative law, then $\begin{matrix} η_{τ_{e_{X}}} = {⋁_{x \in X} (A (x) ⊙ e_{X} (-, x)) ∣ A \in L^{X}} \\ = {⋀_{y \in X} (e_{X} (y, -) \to A (y)) ∣ A \in L^{X}} . \end{matrix}$

(7) If L satisfies a double negative law and e_X (x, y) = e_X (y, x) for all x, y ∈ X, then τ_{e
_X} = η_{τ
_{e
_X}} .

Proof. (1) It follows from Theorem 4.9 in [4].

(2) By Corollary 3.7, (τ_{e
_X}, e_{τ
_{e
_X}}) is a fuzzy complete lattice.

(3) Since e_X (x, y) ⊙ e_X (y, z) ≤ e_X (x, z), we have e_X (x, -) ∈ τ_{e
_X}. By (1), ⋁_x∈X (A (x) ⊙ e_X (x, -)) ∈ τ_{e
_X} for each A ∈ L^X. Hence {⋁ _x∈X (A (x) ⊙ e_X (x, -)) ∣ A ∈ L^X} ⊂ τ_{e
_X}.

Let A ∈ τ_{e
_X}. Then ⋁_x∈X (A (x) ⊙ e_X (x, y)) ≤ A (y) = A (y) ⊙ e_X (y, y) ≤ ⋁ _x∈X (A (x) ⊙ e_X (x, y)). Hence A = ⋁ _x∈X (A (x) ⊙ e_X (x, -)).

(4) Since (e_X (x, y) → A (y)) ⊙ e_X (x, z) ⊙ e_X (z, y) ≤ (e_X (x, y) → A (y)) ⊙ e_X (x, y) ≤ A (y), we have (e_X (x, y) → A (y)) ⊙ e_X (x, z) ≤ e_X (z, y) → A (y). Hence (e_X (- , y) → A (y)) ∈ τ_{e
_X}. So, ⋀_y∈X (e_X (- , y) → A (y)) ∈ τ_{e
_X}.

Let A ∈ τ_{e
_X}. Then A (x) ⊙ e_X (x, y) ≤ A (y) iff A (x) ≤ e_X (x, y) → A (y) . Thus, $\begin{matrix} A (x) \leq ⋀_{y \in X} (e_{X} (x, y) \to A (y)) \\ \leq e_{X} (x, x) \to A (x) = A (x) . \end{matrix}$ So, ⋀_y∈X (e_X (- , y) → A (y)) ∈ τ_{e
_X}. Hence τ_{e
_X} = {⋀ _y∈X (e_X (- , y) → A (y)) ∣ A ∈ L^X}.

(5) Since η_{τ
_{e
_X}} = {A ∈ L^X ∣ A^* ∈ τ_{e
_X}}, A^* (x) ⊙ e_X (x, y) ≤ A^* (y) iff A (y) ≤ e_X (x, y) → A (x) iff A (y) ⊙ e_X (x, y) ≤ A (x) iff $A \in τ_{e_{X}^{- 1}}$ . Thus, the result holds.

(6) and (7) are easily proved from (3-5).

Definition 3.10. A map $C : L^{X} \to L^{X}$ is called an L-preclosure operator if it satisfies the following conditions:

(C1) $C (⊥_{X}) = ⊥_{X}$ ,

(C2) $A \leq C (A)$ , for A ∈ L^X,

(C3) $e_{L^{X}} (A, B) \leq e_{L^{X}} (C (A), C (B))$ for all A, B ∈ L^X.

The pair $(X, C)$ is called an L-preclosure space.

Definition 3.11. A map $I : L^{X} \to L^{X}$ is called an L-preinterior operator if it satisfies the following conditions:

(I1) $I (⊤_{X}) = ⊤_{X}$ ,

(I2) $I (A) \leq A$ , for A ∈ L^X,

(I3) $e_{L^{X}} (A, B) \leq e_{L^{X}} (I (A), I (B))$ for all A, B ∈ L^X.

The pair $(X, I)$ is called an L-preinterior space.

She and Wang [19] developed L-fuzzy rough set (I (A) , C (A)) for A ∈ L^X with L-lower approximation operator I and L-upper approximation operator C in complete residuated lattices as follows: for a fuzzy poset (X, e_X), $\begin{matrix} I (A) (x) = ⋀_{y \in X} (e_{X} (x, y) \to A (y)), \\ C (A) (y) = ⋁_{x \in X} (e_{X} (x, y) ⊙ A (x)) . \end{matrix}$

Let $(X, I)$ be an L-preinterior space and $(X, C)$ be an L-preclosure space. As a generalization of L-fuzzy rough set (I, C) as a sense in She and Wang [19], the pair $(I (A), C (A))$ is called a fuzzy rough set for A ∈ L^X.

The map α : L^X → L is an fuzzy accuracy measure defined, for A ∈ L^X $α (A) = ⋀_{x \in X} (C (A) (x) \to I (A) (x)) .$

Theorem 3.12. Let $D : L^{X} \to L^{X}$ be a map. The following statements are equivalent:

(1) $e_{L^{X}} (A, B) \leq e_{L^{X}} (D (A), D (B))$ for all A, B ∈ L^X.

(2) $α ⊙ D (A) \leq D (α ⊙ A)$ for each α ∈ L, A ∈ L^X and $D (A) \leq D (B)$ for A ≤ B.

(3) $D (α \to A) \leq α \to D (A)$ for each α ∈ L, A ∈ L^X and $D (A) \leq D (B)$ for A ≤ B.

(4) $⊔_{L^{X}} D^{\to} (Φ) \leq D (⊔_{L^{X}} Φ)$ for each Φ ∈ L^{L
^X}.

(5) $D (⊓_{L^{X}} Φ) \leq ⊓_{L^{X}} D^{\to} (Φ)$ for each Φ ∈ L^{L
^X}.

Proof. (1) ⇒ (2). If A ≤ B, then e_{L
^X} (A, B) =⊤ and $e_{L^{X}} (D (A), D (B)) = ⊤$ . Thus $D (A) \leq D (B)$ . Since $α \leq e_{L^{X}} (A, α ⊙ A) \leq e_{L^{X}} (D (A), D (α ⊙ A))$ , we have $α ⊙ D (A) \leq D (α ⊙ A)$ .

(2)⇒ (1). Let α = e_{L
^X} (A, B). Then $e_{L^{X}} (A, B) \leq e_{L^{X}} (D (A), D (B))$ from: $e_{L^{X}} (A, B) ⊙ D (A) \leq D (e_{L^{X}} (A, B) ⊙ A) \leq D (B) .$

(1) ⇒ (3). If A ≤ B, then $D (A) \leq D (B)$ . Since $α \leq e_{L^{X}} (α \to A, A) \leq e_{L^{X}} (D (α \to A), D (A))$ , we have $D (α \to A) \leq α \to D (A)$ .

(3)⇒ (1). Let α = e_{L
^X} (A, B). Then $e_{L^{X}} (A, B) \leq e_{L^{X}} (D (A), D (B))$ from: $D (A) \leq D (e_{L^{X}} (A, B) \to B) \leq e_{L^{X}} (A, B) \to D (B) .$

(4) ⇒ (2) Since $D (⊔_{L^{X}} Φ) \geq ⊔_{L^{X}} D^{\to} (Φ)$ where $D^{\to} (Φ) (B) = ⋁_{B = D (A)} Φ (A)$ for all Φ ∈ L^{L
^X}. Moreover, $\begin{matrix} e_{L^{X}} (⊔_{L^{X}} Φ, B) = ⋀_{A \in L^{X}} (Φ (A) \to e_{L^{X}} (A, B)) \\ = ⋀_{A \in L^{X}} e_{L^{X}} (Φ (A) ⊙ A, B) \\ = e_{L^{X}} (⋁_{A \in L^{X}} Φ (A) ⊙ A, B), \end{matrix}$ $\begin{matrix} e_{L^{X}} (⊔_{L^{X}} D^{\to} (Φ), B) \\ = ⋀_{C \in L^{X}} (D^{\to} (Φ) (C) \to e_{L^{X}} (C, B)) \\ = ⋀_{C \in L^{X}} e_{L^{X}} (D^{\to} (Φ) (C) ⊙ C, B) \\ = e_{L^{X}} (⋁_{C \in L^{X}} D^{\to} (Φ) (C) ⊙ C, B) . \end{matrix}$ By Lemma 3.2(4), ⊔_{L
^X}Φ = ⋁ _A∈L^X (Φ (A) ⊙ A) and $⊔_{L^{X}} D^{\to} (Φ) = ⋁_{C \in L^{X}} (D^{\to} (Φ) (C) ⊙ C)$ .

Define Φ₁ : L^X → L as Φ₁ (A) = α and Φ₁ (B) =⊥, otherwise. Then $(⊔_{L^{X}} Φ_{1}) (x) = ⋁_{D \in L^{X}} (Φ_{1} (D) ⊙ D (x)) = α ⊙ A (x) .$ Since $D^{\to} (Φ_{1}) (B) = ⋁_{B = D (A)} Φ_{1} (A)$ and $D (⊔_{L^{X}} Φ_{1}) \geq ⊔_{L^{X}} D^{\to} (Φ_{1})$ for all Φ₁ ∈ L^{L
^X}, we have $\begin{matrix} ⊔_{L^{X}} D^{\to} (Φ_{1}) (x) \\ = ⋁_{C = D (A) \in L^{X}} (D^{\to} (Φ_{1}) (C) ⊙ C (x)) \\ = Φ_{1} (A) ⊙ D (A) (x) = α ⊙ D (A) (x) \\ \leq D (⊔_{L^{X}} Φ_{1}) (x) = D (α ⊙ A) (x) . \end{matrix}$ Hence $α ⊙ D (A) \leq D (α ⊙ A)$ .

Let A ≤ B be given for A, B ∈ L^X. Define Φ₂ : L^X → L as Φ₂ (A) = Φ₂ (B) =⊤ and Φ₂ (C) =⊥, otherwise. Then $(⊔_{L^{X}} Φ_{2}) = ⋁_{C \in L^{X}} (Φ_{2} (C) ⊙ C) = A \lor B = B .$ Since $D^{\to} (Φ_{2}) (C) = ⋁_{C = D (A)} Φ_{2} (A)$ and $⊔_{L^{X}} D^{\to} (Φ_{2}) \leq D (⊔_{L^{X}} Φ_{2})$ for Φ₂ ∈ L^{L
^X}, we have $\begin{matrix} D (⊔_{L^{X}} Φ_{2}) (x) = D (A \lor B) (x) = D (B) (x), \\ ⊔_{L^{X}} D^{\to} (Φ_{2}) (x) = ⋁_{C \in L^{X}} (D^{\to} (Φ_{2}) (C) ⊙ C (x)) \\ = ⋁_{C \in L^{X}} ((⋁_{C = D (E)} Φ_{2} (E)) ⊙ C (x)) \\ = D (A) (x) \lor D (B) (x) \\ ⊔_{L^{X}} D^{\to} (Φ_{2}) = D (A) (x) \lor D (B) (x) \\ \leq D (⊔_{L^{X}} Φ_{2}) = D (B) (x) . \end{matrix}$ Hence $D (A) \leq D (B)$ .

(2) ⇒ (4) $⊔_{L^{X}} D^{\to} (Φ) \leq D (⊔_{L^{X}} Φ)$ from: $\begin{matrix} ⊔_{L^{X}} D^{\to} (Φ) (y) = ⋁_{A \in L^{X}} Φ (A) ⊙ D (A) (y) \\ \leq D (⋁_{A \in L^{X}} (Φ (A) ⊙ A)) (y) (by (2)) \\ = D (⊔_{L^{X}} Φ) (y) . \end{matrix}$

(5) ⇒ (3) Since $D (⊓_{L^{X}} Φ) \leq ⊓_{L^{X}} D^{\to} (Φ)$ for all Φ ∈ L^{L
^X}. Then $\begin{matrix} e_{L^{X}} (B, ⊓_{L^{X}} Φ) = ⋀_{A \in L^{X}} (Φ (A) \to e_{L^{X}} (B, A)) \\ = ⋀_{A \in L^{X}} e_{L^{X}} (B, Φ (A) \to A) \\ = e_{L^{X}} (B, ⋀_{A \in L^{X}} Φ (A) \to A), \end{matrix}$ $\begin{matrix} e_{L^{X}} (B, ⊓_{L^{X}} D^{\to} (Φ)) \\ = ⋀_{C \in L^{X}} (D^{\to} (Φ) (C) \to e_{L^{X}} (B, C)) \\ = ⋀_{C \in L^{X}} ((⋁_{D (A) = C} Φ (A) \to e_{L^{X}} (B, C))) \\ = ⋀_{A \in L^{X}} (Φ (A) \to e_{L^{X}} (B, D (A)) \\ = ⋀_{A \in L^{X}} e_{L^{X}} (B, Φ (A) \to D (A)) \\ = e_{L^{X}} (B, ⋀_{A \in L^{X}} Φ (A) \to D (A)) . \end{matrix}$ By Lemma 3.2(4), ⊓_{L
^X}Φ = ⋀ _A∈L^X (Φ (A) → A) and $⊓_{L^{X}} D^{\to} (Φ) = ⋀_{A \in L^{X}} (Φ (A) \to D (A)) \in L^{X}$ .

Define Φ₁ : L^X → L as Φ₁ (A) = α and Φ₁ (B) =⊥, otherwise. Then $(⊓_{L^{X}} Φ_{1}) = ⋀_{A \in L^{X}} (Φ_{1} (A) \to A) = α \to A$ Since $D^{\to} (Φ_{1}) (B) = ⋁_{B = D (A)} Φ_{1} (A)$ and $D (⊓_{L^{X}} Φ_{1}) \leq ⊓_{L^{X}} D^{\to} (Φ_{1})$ for Φ₁ ∈ L^{L
^X}, $\begin{matrix} ⊓_{L^{X}} D^{\to} (Φ_{1}) (y) = ⋀_{B \in L^{X}} (Φ_{1} (A) \to D (A) (y)) \\ = α \to D (A) (y) \geq D (⊓_{L^{X}} Φ_{1}) (y) = D (α \to A) (y) . \end{matrix}$ Hence $D (α \to A) \leq α \to D (A) \in L^{X}$ .

Let A ≤ B be given for A, B ∈ L^X. Define Φ₂ : L^X → L as Φ₂ (A) = Φ₂ (B) =⊤ and Φ₂ (B) =⊥ otherwise. Then $⊓_{L^{X}} Φ_{2} = ⋀_{C \in L^{X}} (Φ_{2} (C) \to C) = A \land B = A .$ Since $D^{\to} (Φ_{2}) (B) = ⋁_{B = D (A)} Φ_{2} (A)$ and $D (⊓_{L^{X}} Φ_{2}) \leq ⊓_{L^{X}} D^{\to} (Φ_{2})$ for Φ₂ ∈ L^{L
^X}, we have $\begin{matrix} ⊓_{L^{X}} D^{\to} (Φ_{2}) = ⋀_{C \in L^{X}} (Φ_{2} (C) \to D (C)) \\ = D (A) \land D (B) \geq D (⊓_{L^{X}} Φ_{2}) = D (A) . \end{matrix}$ Hence $D (A) \leq D (B)$ .

(3) ⇒ (5) $D (⊓_{L^{X}} Φ) \leq ⊓_{L^{X}} D^{\to} (Φ)$ from: $\begin{matrix} ⊓_{L^{X}} D^{\to} (Φ) = ⋀_{A \in L^{X}} (Φ (A) \to D (A)) \\ \geq ⋀_{A \in L^{X}} D (Φ (A) \to A) \\ \geq D (⋀_{A \in L^{X}} (Φ (A) \to A)) \\ = D (⊓_{L^{X}} Φ) . \end{matrix}$

Theorem 3.13. Let $D : L^{X} \to L^{X}$ be a map. The following statements hold.

(1) $⊔_{L^{X}} D^{\to} (Φ) = D (⊔_{L^{X}} Φ)$ for each Φ ∈ L^{L
^X} iff $α ⊙ D (A) = D (α ⊙ A)$ for each α ∈ L, A ∈ L^X and $D (⋁_{i \in Γ} A_{i}) = ⋁_{i \in Γ} D (A_{i})$ for A_i ∈ L^X.

(2) $D (⊓_{L^{X}} Φ) = ⊓_{L^{X}} D^{\to} (Φ)$ for each Φ ∈ L^{L
^X} iff $D (α \to A) = α \to D (A)$ for each α ∈ L, A ∈ L^X and $D (⋀_{i \in Γ} A_{i}) = ⋀_{i \in Γ} D (A_{i})$ for A_i ∈ L^X.

Proof. (1) (⇒) Since $D$ is a fuzzy join preserving map, we have $D (⊔_{L^{X}} Φ) = ⊔_{L^{X}} D^{\to} (Φ)$ where ⊔_{L
^X}Φ = ⋁ _A∈L^X (Φ (A) ⊙ A) ∈ L^X and $⊔_{L^{X}} D^{\to} (Φ) = ⋁_{C \in L^{X}} (D^{\to} (Φ) (C) ⊙ C) \in L^{X}$ .

By a similar method as Theorem 3.12, $D (α ⊙ A) = α ⊙ D (A) \in L^{X}$ .

Let {A_i ∈ L^X ∣ i ∈ Γ} be given. Define Φ₂ : L^X → L as Φ₂ (A_i) =⊤ for i ∈ Γ and Φ₂ (B) =⊥, otherwise. Then $(⊔_{L^{X}} Φ_{2}) (x) = ⋁_{A \in L^{X}} (Φ_{2} (A) ⊙ A (x)) = ⋁_{i \in Γ} A_{i} (x) .$ Since $D^{\to} (Φ_{2}) (B) = ⋁_{B = D (A)} Φ_{2} (A)$ and $D (⊔_{L^{X}} Φ_{2}) = ⊔_{L^{X}} D^{\to} (Φ_{2})$ for Φ₂ ∈ L^{L
^X}, we have

$\begin{matrix} D (⊔ Φ_{2}) (y) & = D (⋁_{i \in Γ} A_{i}) (y), \\ ⊔_{L^{X}} D^{\to} (Φ_{2}) (y) & = ⋁_{B \in L^{Y}} (D^{\to} (Φ_{2}) (B) ⊙ B (y)) \\ = ⋁_{B \in L^{X}} ((⋁_{B = D (A)} Φ_{2} (A)) ⊙ B (y)) \\ = ⋁_{A \in L^{X}} (Φ_{2} (A) ⊙ D (A) (y)) \\ = ⋁_{i \in Γ} D (A_{i}) (y) . \end{matrix}$

Hence $D (⋁_{i \in Γ} A_{i}) = ⋁_{i \in Γ} D (A_{i}) \in L^{X}$ .

(⇐) Put $B_{0} = ⊔_{L^{X}} D^{\to} (Φ)$ for all Φ ∈ L^{L
^X}. Then $⋁_{A \in L^{X}} Φ (A) ⊙ D (A) \in L^{X}$ . Thus, $\begin{matrix} e_{L^{X}} (B_{0}, B) \\ = ⋀_{C \in L^{X}} (D^{\to} (Φ) (C) \to e_{L^{X}} (C, B)) \\ = ⋀_{C \in L^{X}} ((⋁_{D (A) = C} Φ (A) \to e_{L^{X}} (D (A), B)) \\ = ⋀_{A \in L^{X}} (Φ (A) \to e_{L^{X}} (D (A), B)) \\ = ⋀_{A \in L^{X}} e_{L^{X}} (Φ (A) ⊙ D (A), B) \\ = e_{L^{X}} (⋁_{A \in L^{X}} Φ (A) ⊙ D (A), B) . \end{matrix}$ Hence $D (⊔_{L^{X}} Φ) = ⊔_{L^{X}} D^{\to} (Φ)$ from: $\begin{matrix} ⊔_{L^{X}} D^{\to} (Φ) (y) = B_{0} (y) = ⋁_{A \in L^{X}} Φ (A) ⊙ D (A) (y) \\ = D (⋁_{A \in L^{X}} (Φ (A) ⊙ A)) (y) \\ = D (⊔_{L^{X}} Φ) (y) . \end{matrix}$

(2) (⇒) Since $D$ is a meet preserving map, then $D (⊓_{L^{X}} Φ) = ⊓_{L^{X}} D^{\to} (Φ)$ for all Φ ∈ L^{L
^X} where ⊓_{L
^X}Φ = ⋀ _{A∈τ_X} (Φ (A) → A) and $⊓_{L^{X}} D^{\to} (Φ) = ⋀_{A \in L^{X}} (Φ (A) \to D (A)) \in L^{X}$ .

By a similar method as Theorem 3.12, $D (α \to A) = α \to D (A) \in L^{X}$ .

Let {A_i ∈ L^X ∣ i ∈ Γ} be given. Define Φ₂ : L^X → L as Φ₂ (A_i) =⊤ for i ∈ Γ and Φ₂ (B) =⊥ otherwise. Then $⊓_{L^{X}} Φ_{2} (x) = ⋀_{A \in L^{X}} (Φ_{2} (A) \to A (x)) = ⋀_{i \in Γ} A_{i} (x) .$ Since $D^{\to} (Φ_{2}) (B) = ⋁_{B = D (A)} Φ_{2} (A)$ and $D (⊓_{L^{X}} Φ_{2}) = ⊓_{L^{X}} D^{\to} (Φ_{2})$ for Φ₂ ∈ L^{L
^X}, we have $\begin{matrix} ⊓_{L^{X}} D^{\to} (Φ_{2}) (y) = ⋀_{A \in L^{X}} (Φ_{2} (A) \to D (A) (y)) \\ = ⋀_{i \in Γ} D (A_{i}) (y) = D (⊓_{L^{X}} Φ_{2}) (y) = D (⋀_{i \in Γ} A_{i}) (y) . \end{matrix}$ Hence $D (⋀_{i \in Γ} A_{i}) = ⋀_{i \in Γ} D (A_{i}) \in L^{X}$ .

(⇐) It is similarly proved as (1).

Theorem 3.14. Let $(X, I)$ be an L-preinterior space. Define $τ_{I} \subset L^{X}$ as $τ_{I} = {A \in L^{X} ∣ A = I (A)} .$ Then the following properties hold.

(1) $τ_{I}$ is an Alexandrov L-pretopology on X.

(2) If $I (⊓_{L^{X}} Φ) = ⊓_{L^{X}} I^{\to} (Φ)$ for each Φ ∈ L^{L
^X}, then $τ_{I}$ is an Alexandrov L-topology on X.

Proof. (1) (O1) Since $α ⊙ I (⊤_{X}) \leq I (α ⊙ ⊤_{X}) \leq α_{X}$ , $I (α_{X}) = α_{X}$ . Hence $α_{X} \in τ_{I}$ .

(O2) and (O3) For $A, A_{i} \in τ_{I}$ , by Lemma 2.2(11), $⋁_{i} A_{i}, α ⊙ A \in τ_{I}$ from: $\begin{matrix} I (⋁_{i} A_{i}) \geq ⋁_{i} I (A_{i}) = ⋁_{i} A_{i}, \\ I (α ⊙ A) \geq α ⊙ I (A) = α ⊙ A . \end{matrix}$ Therefore $τ_{I}$ is an Alexandrov L-pretopology on X.

(2) Since $I (⊓_{L^{X}} Φ) = ⊓_{L^{X}} I^{\to} (Φ)$ for each Φ ∈ L^{L
^X}, by Theorem 3.13(2), $I (⋀_{i \in Γ} A_{i}) = ⋀_{i \in Γ} I (A_{i})$ and $I (α \to A) = α \to I (A)$ for all A_i, A ∈ L^X. For $A, A_{i} \in τ_{I}$ , $\begin{matrix} I (⋀_{i} A_{i}) = ⋀_{i} I (A_{i}) = ⋀_{i} A_{i} \\ I (α \to A) = α \to I (A) = α \to A . \end{matrix}$ Therefore $τ_{I}$ is an Alexandrov L-topology on X

The following corollary can be obtained by a similar method used in the proof of Theorem 3.14.

Corollary 3.15. Let $(X, C)$ be an L-preclosure space. Define $η_{C} \subset L^{X}$ as $η_{C} = {A \in L^{X} ∣ C (A) = A} .$ Then the following properties hold.

(1) $η_{C}$ is an Alexandrov L-precotopology on X.

(2) If $C (⊔_{L^{X}} Φ) = ⊔_{L^{X}} C^{\to} (Φ)$ for each Φ ∈ L^{L
^X}, then $η_{C}$ is an Alexandrov L-topology on X.

Theorem 3.16. Let (X, τ) be an Alexandrov L-pretopological space. For each A ∈ L^X and Φ_A : τ → L with Φ_A (C) = e_{L
^X} (C, A), define $I_{τ} : L^{X} \to L^{X}$ by $I_{τ} (A) = ⊔_{τ} Φ_{A}$ where ⊔_τΦ_A for (τ, e_τ). Then the following properties hold.

(1) $I_{τ} (A) = ⊔_{τ} Φ_{A} = ⋁_{C \in τ} (e_{L^{X}} (C, A) ⊙ C) = ⋁_{i \in Γ} {A_{i} ∣ A_{i} \leq A, A_{i} \in τ} .$

(2) $I_{τ}$ is an Alexandrov L-preinterior operator on X such that $I_{τ} (I_{τ} (A)) = I_{τ} (A) .$

(3) $τ_{I_{τ}} = τ$ where $τ_{I_{τ}} = {A \in L^{X} ∣ A = I_{τ} (A)}$ .

(4) If $I$ is an L-preinterior operator on X, then $I_{τ_{I}} \leq I$ . Moreover, the equality holds if $I (I (A)) = I (A)$ for each A ∈ L^X.

(5) Define d_τ (x, y) = ⋀ _A∈τ (A (x) → A (y)). Then d_τ is a fuzzy preorder with ⊔_{τ
_{d
_τ}}Φ_A ≥ ⊔ _τΦ_A and τ ⊂ τ_{d
_τ} where τ_{d
_τ} = {B ∈ L^X ∣ B (x) ⊙ d_τ (x, y) ≤ B (y)} and ⊔_{τ
_{d
_τ}}Φ_A for (τ_{d
_τ}, e_{τ
_{d
_τ}}). Moreover, if τ is an Alexandrov L-topology on X, then ⊔_{τ
_{d
_τ}}Φ_A = ⊔ _τΦ_A and τ = τ_{d
_τ}.

Proof. {(1) Since ⋁_C∈τ (e_{L
^X} (C, A) ⊙ C) ∈ τ, $\begin{matrix} ⋀_{C \in τ} (Φ_{A} (C) \to e_{τ} (C, B) = e_{τ} (⋁_{C \in τ} Φ_{A} (C) ⊙ C, B) \\ = e_{τ} (⋁_{C \in τ} e_{L^{X}} (C, A) ⊙ C, B) = e_{τ} (⊔_{τ} Φ_{A}, B) . \end{matrix}$ Hence $I_{τ} (A) = ⊔_{τ} Φ_{A} = ⋁_{C \in τ} e_{L^{X}} (C, A) ⊙ C$ .

Put $I_{τ} (A) = ⋁_{A_{i} \in τ} (e_{L^{X}} (A_{i}, A) ⊙ A_{i})$ and I₁ (A) = ⋁ _i∈Γ {A_i ∣ A_i ≤ A, A_i ∈ τ}. Since ⋁_{A_i∈τ} (e_{L
^X} (A_i, A) ⊙ A_i) ≤ A and ⋁_{A_i∈τ} (e_{L
^X} (A_i, A) ⊙ A_i) ∈ τ, $I_{τ} (A) \leq I_{1} (A)$ .

Since I₁ (A) ∈ τ, $I_{τ} (A) \geq e_{L^{X}} (I_{1} (A), A) ⊙ I_{1} (A) = I_{1} (A) .$

(2) (I1) For all x ∈ X, we have $\begin{matrix} I_{τ} (⊤_{X}) (x) = ⋁_{C \in τ} e_{L^{X}} (C, ⊤_{X}) ⊙ C (x) \\ \geq e_{L^{X}} (⊤_{X}, ⊤_{X}) ⊙ ⊤_{X} (x) = ⊤_{X} (x) . \end{matrix}$

(I2) For each A ∈ L^X, e_{L
^X} (B, A) ⊙ B ≤ A. Hence $I_{τ} (A) \leq A$ .

(I3) For each A, C ∈ L^X, we have $\begin{matrix} e_{L^{X}} (I_{τ} (A), I_{τ} (C)) = ⋀_{x \in X} (I_{τ} (A) (x) \to I_{τ} (C) (x)) \\ = ⋀_{x \in X} (⋁_{B \in τ} (e_{L^{X}} (B, A) ⊙ B (x)) \\ \to ⋁_{D \in τ} (e_{L^{X}} (D, C) ⊙ D (x))) \\ \geq ⋀_{x \in X} ⋀_{B \in τ} ((e_{L^{X}} (B, A) ⊙ B (x)) \\ \to ((e_{L^{X}} (B, C) ⊙ B (x))) \\ \geq ⋀_{B \in τ} (e_{L^{X}} (B, A) \to e_{L^{X}} (B, C)) \geq e_{L^{X}} (A, C) . \end{matrix}$

Since $I_{τ} (A) \in τ$ $\begin{matrix} I_{τ} (I_{τ} (A)) = ⋁_{C \in τ} (e_{L^{X}} (C, I_{τ} (A)) ⊙ C) \\ \geq e_{L^{X}} (I_{τ} (A), I_{τ} (A)) ⊙ I_{τ} (A) = I_{τ} (A) . \end{matrix}$

(3) $τ_{I_{τ}} = τ$ where $τ_{I_{τ}} = {A \in L^{X} ∣ A = I_{τ} (A)}$ . $\begin{matrix} A \in τ, A = I_{τ} (A), A \in τ_{I_{τ}} \\ A \in τ_{I_{τ}}, A = I_{τ} (A) \in τ, A \in τ . \end{matrix}$

(4) For each A ∈ L^X, $\begin{matrix} I_{τ_{I}} (A) (x) = ⋁_{C \in τ_{I}} (e_{L^{X}} (C, A) ⊙ C (x)) \\ \leq ⋁_{C \in τ_{I}} (e_{L^{X}} (I (C), I (A)) ⊙ C (x)) \leq I (A) (x) . \end{matrix}$

If $I (I (A)) = I (A)$ for each A ∈ L^X, $\begin{matrix} I_{τ_{I}} (A) (x) = ⋁_{C \in L^{X}} (e_{L^{X}} (C, A) ⊙ C (x)) \\ \geq e_{L^{X}} (I (A), A) ⊙ I (A) (x) = I (A) (x) . \end{matrix}$

(5) For B ∈ τ, B (x) ⊙ ⋀ _A∈τ (A (x) → A (y)) ≤ B (x) ⊙ (B (x) → B (y)) ≤ B (y). Hence B ∈ τ_{d
_τ}. Moreover ⊔_τΦ_A = ⋁ _i∈Γ {A_i ∣ A_i ≤ A, A_i ∈ τ} ≤ ⊔ _{τ
_{d
_τ}}Φ_A .

If τ is an Alexandrov L-topology on X, for B ∈ τ_{d
_τ}, B = B (x) ⊙ ⋀ _A∈τ (A (x) → A (-)) ∈ τ. Hence B ∈ τ. Thus, τ = τ_{d
_τ} and ⊔_{τ
_{d
_τ}}Φ_A = ⊔ _τΦ_A.

Theorem 3.17. Let (X, η) be an Alexandrov L-precotopological space. For each A ∈ L^X and Θ_A : η → L with Θ_A (B) = e_{L
^X} (A, B), define $C_{η} : L^{X} \to L^{X}$ by $C_{η} (A) = ⊓_{η} Θ_{A}$ where ⊓_ηΘ_A for (η, e_η). Then the following properties hold.

(1) $C_{η} (A) = ⊓_{η} Θ_{A} = ⋀_{B \in η} (e_{L^{X}} (A, B) \to B) = ⋀ {B \in η ∣ A \leq B} .$

(2) $C_{η}$ is an L-preclosure space on X such that $C_{η} (C_{η} (A)) = C_{η} (A)$ .

(3) Define τ_η ⊂ L^X by A ∈ τ_η iff A^* ∈ η . Then τ_η is an Alexandrov L-pretopology.

(4) $η_{C_{η}} = η$ where $η_{C_{η}} = {A \in L^{X} ∣ A = C_{η} (A)}$ .

(5) $C_{η} (A) = (I_{τ_{η}} (A^{*}))^{*}$ for each A ∈ L^X. Moreover, $A \in τ_{I_{τ_{η}}}$ iff $A^{*} \in η_{C_{η}}$ .

(6) If $C$ is an L-preclosure space on X, then $C_{η_{C}} \geq C$ . Moreover, the equality holds if $C (C (A)) = C (A)$ for each A ∈ L^X.

(7) Define d_η (x, y) = ⋀ _A∈η (A (x) → A (y)). Then d_η is a fuzzy preorder with ⊓_{τ
_{d
_η}}Θ_A ≤ ⊓ _ηΘ_A and η ⊂ τ_{d
_η} where τ_{d
_η} = {B ∈ L^X ∣ B (x) ⊙ d_η (x, y) ≤ B (y)} and ⊓_{τ
_{d
_η}}Θ_A for (τ_{d
_η}, e_{τ
_{d
_η}}).

(8) If η is an Alexandrov L-topology on X, then ⊔_{τ
_{d
_η}}Φ_A = ⊔ _ηΦ_A, ⊓_{τ
_{d
_η}}Θ_A = ⊓ _ηΘ_A and η = τ_{d
_η}. Moreover, the pair (⊔ _ηΦ_A, ⊓ _ηΘ_A) is a fuzzy rough set for A.

Proof. (1) Since η is an Alexandrov L-precotopology on X, ⋀_C∈η (e_{L
^X} (A, C) → C) ∈ η. Thus $\begin{matrix} ⋀_{C \in η} (Θ_{A} (C) \to e_{η} (B, C) \\ = e_{η} (B, ⋀_{C \in η} Θ_{A} (C) \to C) \\ = e_{η} (B, ⋀_{C \in η} e_{L^{X}} (A, C) \to C) = e_{η} (B, ⊓_{η} Θ_{A}) . \end{matrix}$ Hence $C_{η} (A) = ⊓_{η} Θ_{A} = ⋀_{C \in η} (e_{L^{X}} (A, C) \to C)$ .

Put $C_{η} (A) = ⋀_{A_{i} \in η} (e_{L^{X}} (A, A_{i}) \to A_{i})$ and C₁ (A) = ⋀ _i∈Γ {A_i ∣ A ≤ A_i, A_i ∈ η}. Since A ≤ ⋀ _{A_i∈η} (e_{L
^X} (A, A_i) → A_i) and ⋀_{A_i∈η} (e_{L
^X} (A, A_i) → A_i) ∈ η, $C_{1} (A) \leq C_{η} (A)$ .

Since C₁ (A) ∈ η, $C_{η} (A) \leq e_{L^{X}} (A, C_{1} (A)) \to C_{1} (A) = C_{1} (A) .$

(2) (C1) For each x ∈ X, $\begin{matrix} C_{η} (⊥_{X}) (x) & = ⋀_{B \in L^{X}} (e_{L^{X}} (⊥_{X}, B) \to B (x)) \\ \leq e_{L^{X}} (⊥_{X}, ⊥_{X}) \to ⊥_{X} (x) = ⊥_{X} (x), \end{matrix}$

(C2) It follows A ≤ e_{L
^X} (A, B) → B.

(C3) For each A, C ∈ L^X, $\begin{matrix} e_{L^{X}} (C_{η} (A), C_{η} (C)) = ⋀_{x \in X} (C_{η} (A) (x) \to C_{η} (C) (x)) \\ = ⋀_{x \in X} (⋀_{B \in η} (e_{L^{X}} (A, B) \to B (x)) \\ \to ⋀_{D \in eta} (e_{L^{X}} (C, D) \to D))) \\ \geq ⋀_{x \in X} ⋀_{B \in η} (e_{L^{X}} (A, B) \to B (x)) \\ \to (e_{L^{X}} (C, B) \to B (x))) \\ \geq ⋀_{B \in η} (e_{L^{X}} (A, B) \to e_{L^{X}} (C, B)) \geq e_{L^{X}} (A, C), \end{matrix}$

Since $C_{η} (A) \in η$ , $C_{η} (A) = C_{η} (C_{η} (A))$ from: $\begin{matrix} C_{η} (C_{η} (A)) \leq e_{L^{X}} (C_{η} (A), C_{η} (A)) \to C_{η} (A) = C_{η} (A) . \end{matrix}$

(3) and (4) are similarly prove as Theorems 3.8(2) and 3.16 (3), respectively.

(5) $\begin{matrix} (I_{τ_{η}} (A^{*}))^{*} = (⋁_{B \in τ_{η}} e_{L^{X}} (B, A^{*}) ⊙ B)^{*} \\ = ⋀_{B \in τ_{η}} (e_{L^{X}} (A, B^{*}) \to B^{*}) \\ = ⋀_{C \in η} (e_{L^{X}} (A, C) \to C) = C_{η} (A) . \end{matrix}$

$\begin{matrix} A \in τ_{I_{τ_{η}}} iff A = I_{τ_{η}} (A) \\ iff C_{η} (A^{*}) = A^{*} iff A^{*} \in η_{C_{η}} . \end{matrix}$

(6) $\begin{matrix} C_{η_{C}} (A) = ⋀_{B \in η_{C}} (e_{L^{X}} (A, B) \to B) \\ \geq ⋀_{B \in η_{C}} ((e_{L^{X}} (C (A), C (B))) \to B) \geq C (A) \end{matrix}$

If $C (C (A)) = C (A)$ for each A ∈ L^X, $\begin{matrix} C_{η_{C}} (A) (x) = ⋀_{B \in η_{C}} (e_{L^{X}} (A, B) \to B (x)) \\ \leq (e_{L^{X}} (A, C (A))) \to C (A) (x)) = C (A) (x) . \end{matrix}$

(7) For B ∈ η, B (x) ⊙ ⋀ _A∈η (A (x) → A (y)) ≤ B (x) ⊙ (B (x) → B (y)) ≤ B (y). Hence B ∈ τ_{d
_η} and τ_{d
_η} is an Alexandrov L-topology from Theorem 3.9. Moreover ⊓_ηΦ_A = ⋀ _i∈Γ {A_i ∣ A ≤ A_i, A_i ∈ η} ≥ ⊓ _{τ
_{d
_η}}Φ_A .

(8) If η is an Alexandrov L-topology on X, for B ∈ τ_{d
_η}, B = B (x) ⊙ ⋀ _A∈η (A (x) → A (-)) ∈ η. Hence B ∈ η. Thus, η = τ_{d
_η}, ⊓_{τ
_{d
_η}}Θ_A = ⊓ _ηΘ_A and ⊔_{τ
_{d
_η}}Φ_A = ⊔ _ηΦ_A.

Theorem 3.18. Let e_X be a fuzzy preorder on X and τ_{e
_X} = {A ∈ L^X ∣ A (x) ⊙ e_X (x, y) ≤ A (y)} be an Alexandrov L-topology on X. Then the following properties hold.

(1) If Φ_A = e_{L
^X} (- , A) : τ_{e
_X} → L, then ⊔_{τ
_{e
_X}}Φ_A = ⋁ {B ∈ τ_{e
_X} ∣ B ≤ A} = ⋀ _y∈X (e_X (- , y) → A (y)) .

(2) If Θ_A = e_{L
^X} (A, -) : τ_{e
_X} → L, then ⊓_{τ
_{e
_X}}Θ_A = ⋀ {B ∈ τ_{e
_X} ∣ A ≤ B} = ⋁ _x∈X (A (x) ⊙ e_X (x, -)) .

(3) d_{τ
_{e
_X}} = e_X where d_{τ
_{e
_X}} (x, y) = ⋀ _{A∈τ_{e
_X}} (A (x) → A (y)) for each x, y ∈ X.

(4) The pair (⊔ _{τ
_{e
_X}}Φ_A, ⊓ _{τ
_{e
_X}}Θ_A) is a fuzzy rough set for A.

Proof. (1) By Theorem 3.16(1), ⊔_{τ
_{e
_X}}Φ_A = ⋁ {B ∈ τ_{e
_X} ∣ B ≤ A} . Since ⋀_y∈X (e_X (- , y) → A (y)) ∈ τ_{e
_X} from Theorem 3.9(4) and ⋀_y∈X (e_X (- , y) → A (y)) ≤ A, ⋀_y∈X (e_X (- , y) → A (y)) ≤ ⊔ _{τ
_{e
_X}}Φ_A . Since ⊔_{τ
_{e
_X}}Φ_A ≤ A and ⊔_{τ
_{e
_X}}Φ_A ∈ τ_{e
_X}, then ⊔_{τ
_{e
_X}}Φ_A (x) ⊙ e_X (x, y) ≤ ⊔ _{τ
_{e
_X}}Φ_A (y) implies $\begin{matrix} ⊔_{τ_{e_{X}}} Φ_{A} = ⋀_{y \in X} (e_{X} (-, y) \to ⊔_{τ_{e_{X}}} Φ_{A} (y)) \\ \leq ⋀_{y \in X} (e_{X} (-, y) \to A (y)) . \end{matrix}$ Thus ⊔_{τ
_{e
_X}}Φ_A = ⋀ _y∈X (e_X (- , y) → A (y)) .

(2) By Theorem 3.17(1), ⊓_{τ
_{e
_X}}Θ_A = ⋀ {B ∈ τ_{e
_X} ∣ A ≤ B} . Since ⋁_x∈X (A (x) ⊙ e_X (x, -)) ∈ τ_{e
_X} and A ≤ ⋁ _x∈X (A (x) ⊙ e_X (x, -)), ⊓_{τ
_{e
_X}}Θ_A ≤ ⋁ _x∈X (A (x) ⊙ e_X (x, -)) . Since A ≤ ⊓ _{τ
_{e
_X}}Θ_A and ⊓_{τ
_{e
_X}}Θ_A ∈ τ_{e
_X}, then $\begin{matrix} ⋁_{x \in X} (A (x) ⊙ e_{X} (x, -)) \\ \leq ⋁_{x \in X} (⊓_{τ_{e_{X}}} Θ_{A} (x) ⊙ e_{X} (x, -)) = ⊓_{τ_{e_{X}}} Θ_{A} . \end{matrix}$ Thus, ⊓_{τ
_{e
_X}}Θ_A = ⋁ _x∈X (A (x) ⊙ e_X (x, -)) .

(3) Since e_X (z, -) ∈ τ_{e
_X} and ⋁_z∈X (A (z) ⊙ e_X (z, x)) = A (x) for A ∈ τ_{e
_X}, $\begin{matrix} d_{τ_{e_{X}}} (x, y) = ⋀_{A \in τ_{e_{X}}} (A (x) \to A (y)) \\ = ⋀_{A \in τ_{e_{X}}} (⋁_{z \in X} (A (z) ⊙ e_{X} (z, x)) \\ \to ⋁_{z \in X} (A (z) ⊙ e_{X} (z, y))) \\ \geq ⋀_{z \in X} (e_{X} (z, x) \to e_{X} (z, y)) = e_{X} (x, y) \\ d_{τ_{e_{X}}} (x, y) = ⋀_{A \in τ_{e_{X}}} (A (x) \to A (y)) \\ \leq ⋀_{z \in X} (e_{X} (z, x) \to e_{X} (z, y)) = e_{X} (x, y) . \end{matrix}$

(4) It follows from (1) and (2).

Theorem 3.19. Let $(X, I)$ be an L-preinterior space. Then the following properties hold.

(1) Define $e_{I} (x, y) = ⋀_{A \in L^{X}} (I (A) (x) \to I (A) (y))$ . Then $e_{I}$ is a fuzzy preorder with $⊔_{τ_{e_{I}}} Φ_{A} \geq I (A)$ and $τ_{I} \subset τ_{e_{I}}$ where $⊔_{τ_{e_{I}}} Φ_{A}$ for $(τ_{e_{I}}, e_{τ_{e_{I}}})$ .

(2) Let $I (I (A)) = I (A)$ for all A ∈ L^X. Define $d_{I} (x, y) = (I (⊤_{y}^{*}) (x))^{*}$ for all x, y ∈ X. Then $d_{I}$ is a fuzzy preorder with $⊔_{τ_{d_{I}}} Φ_{A} \geq I (A)$ and $τ_{I} \subset τ_{d_{I}}$ where $⊔_{τ_{d_{I}}} Φ_{A}$ for $(τ_{d_{I}}, e_{τ_{d_{I}}})$ . Moreover, if $I (⊓_{L^{X}} Φ) = ⊓_{L^{X}} I^{\to} (Φ)$ for each Φ ∈ L^{L
^X}, then $⊔_{τ_{e_{I}}} Φ_{A} = ⊔_{τ_{d_{I}}} Φ_{A} = I (A)$ and $d_{I} = e_{I}$ .

Proof. (1) Since $e_{I}$ is a fuzzy preorder, by Theorem 3.18(1), $\begin{matrix} ⊔_{τ_{e_{I}}} Φ_{A} (y) = ⋀_{x \in X} (e_{I} (y, x) \to A (x)) \\ \geq (I (A) (y) \to I (A) (x)) \to I (A) (x) \\ \geq I (A) (y) . \end{matrix}$ For $A \in τ_{I}$ , $A = I (A) \leq ⊔_{τ_{e_{I}}} Φ_{A} = I_{τ_{e_{I}}} (A)$ . Then $A \in τ_{I_{τ_{e_{I}}}} = τ_{e_{I}}$ from Theorem 3.16(4).

(2) Since $I (⊤_{x}^{*}) = ⋀_{y \in X} I (⊤_{x}^{*})^{*} (y) \to ⊤_{y}^{*}$ , $\begin{matrix} I (⊤_{x}^{*}) (z) = I (I (⊤_{x}^{*})) (z) \\ = I (⋀_{y \in X} I (⊤_{x}^{*})^{*} (y) \to ⊤_{y}^{*}) (z) \\ \leq ⋀_{y \in X} I (⊤_{x}^{*})^{*} (y) \to I (⊤_{y}^{*}) (z), \end{matrix}$ $\begin{matrix} d_{I} (z, x) = I^{*} (⊤_{x}^{*}) (z) \\ \geq ⋁_{y \in X} I (⊤_{x}^{*})^{*} (y) ⊙ I^{*} (⊤_{y}^{*}) (z) \\ = ⋁_{y \in X} (d_{I} (y, x) ⊙ d_{I} (z, y)) . \end{matrix}$ Since $α \to I (A) \geq I (α \to A)$ from Theorem 3.12(3), $\begin{matrix} ⊔_{τ_{d_{I}}} Φ_{A} (y) = ⋀_{x \in X} (d_{I} (y, x) \to A (x)) \\ = ⋀_{x \in X} (I^{*} (⊤_{x}^{*}) (y) \to A (x)) \\ = ⋀_{x \in X} (A^{*} (x) \to I (⊤_{x}^{*}) (y)) \\ \geq I (⋀_{x \in X} (A^{*} (x) \to ⊤_{x}^{*})) (y) = I (A) (y) . \end{matrix}$ If $I (⊓_{L^{X}} Φ) = ⊓_{L^{X}} I^{\to} (Φ)$ for each Φ ∈ L^{L
^X}, by Theorem 3.13(2), $I (⋀_{i \in Γ} A_{i}) = ⋀_{i \in Γ} I (A_{i})$ and $I (α \to A) = α \to I (A)$ for all A_i, A ∈ L^X. Thus, $\begin{matrix} ⊔_{τ_{d_{I}}} Φ_{A} (y) = ⋀_{x \in X} (A^{*} (x) \to I (⊤_{x}^{*}) (y)) \\ = I (⋀_{x \in X} (A^{*} (x) \to I (⊤_{x}^{*}))) (y) = I (A) (y) . \end{matrix}$ $\begin{matrix} e_{I} (x, y) = ⋀_{A \in L^{X}} (I (A) (x) \to I (A) (y)) \\ \leq I (⊤_{y}^{*}) (x) \to I (⊤_{y}^{*}) (y)) \\ \leq I (⊤_{y}^{*}) (x) \to ⊤_{y}^{*} (y) = I^{*} (⊤_{y}^{*}) (x) \\ = d_{I} (x, y) . \end{matrix}$ Since $A = ⋀_{x \in X} (A^{*} (x) \to ⊤_{x}^{*})$ , $\begin{matrix} e_{I} (x, y) = ⋀_{A \in L^{X}} (I (A) (x) \to I (A) (y)) \\ = ⋀_{A \in L^{X}} I (⋀_{z \in X} (A^{*} (z) \to ⊤_{z}^{*}) (x) \\ \to I (⋀_{z \in X} (A^{*} (z) \to ⊤_{z}^{*}) (y)) \\ = ⋀_{A \in L^{X}} (⋀_{z \in X} (A^{*} (z) \to I (⊤_{z}^{*}) (x) \\ \to ⋀_{z \in X} (A^{*} (z) \to I (⊤_{z}^{*}) (y))) \\ \geq ⋀_{A \in L^{X}} ⋀_{z \in X} ((A^{*} (z) \to I (⊤_{z}^{*}) (x) \\ \to (A^{*} (z) \to I (⊤_{z}^{*}) (y))) \\ \geq ⋀_{z \in X} (I (⊤_{z}^{*}) (x) \to I (⊤_{z}^{*}) (y)) \\ = ⋀_{z \in X} (I^{*} (⊤_{z}^{*}) (y) \to I^{*} (⊤_{z}^{*}) (x)) \\ = ⋀_{z \in X} (d_{I} (y, z) \to d_{I} (x, z)) \\ = d_{I} (x, y) . \end{matrix}$

Theorem 3.20. Let $(X, C)$ be an L-preclosure space. Then the following properties hold.

(1) Define $e_{C} (x, y) = ⋀_{A \in L^{X}} (C (A) (x) \to C (A) (y))$ . Then $e_{C}$ is a fuzzy preorder with $⊓_{τ_{e_{C}}} Θ_{A} \leq C (A)$ and $η_{C} \subset τ_{e_{C}}$ where $⊓_{τ_{e_{C}}} Θ_{A}$ for $(τ_{e_{C}}, e_{τ_{e_{C}}})$ .

(2) Let $C (C (A)) = C (A)$ for all A ∈ L^X. Define $d_{C} (x, y) = C (⊤_{y}) (x)$ for all x, y ∈ X. Then $d_{C}$ is a fuzzy preorder with $⊓_{τ_{d_{C}}} Θ_{A} \leq C (A)$ where $⊓_{τ_{d_{C}}} Θ_{A}$ for $(τ_{d_{C}}, e_{τ_{d_{C}}})$ . Moreover, if $C (⊔_{L^{X}} Φ) = ⊔_{L^{X}} C^{\to} (Φ)$ for each Φ ∈ L^{L
^X}, then $e_{C} = d_{C}$ and $⊓_{τ_{d_{C}}} Θ_{A} = ⊓_{τ_{e_{C}}} Θ_{A} = C (A)$ .

Proof. (1) Since $e_{C}$ is a fuzzy preorder, by Theorem 3.18(2), $\begin{matrix} ⊓_{τ_{e_{C}}} Θ_{A} (y) = ⋁_{x \in X} (e_{I} (x, y) ⊙ A (x)) \\ \leq ⋁_{x \in X} (C (A) (x) \to C (A) (y)) ⊙ C (A) (y) \\ \leq C (A) (y) . \end{matrix}$

(2) Since $α ⊙ C (A) \leq C (α ⊙ A)$ from Theorem 3.12(2) and $C (⊤_{x}) = ⋁_{y \in X} C (⊤_{x}) (y) ⊙ ⊤_{y})$ , we have $\begin{matrix} C (⊤_{x}) (z) = C (C (⊤_{x})) (z) \\ = C (⋁_{y \in X} C (⊤_{x}) (y) ⊙ ⊤_{y}) (z) \\ \geq ⋁_{y \in X} C (⊤_{x}) (y) ⊙ C (⊤_{y}) (z), \end{matrix}$ $\begin{matrix} d_{C} (x, z) = C (⊤_{x}) (z) \\ \geq ⋁_{y \in X} C (⊤_{x}) (y) ⊙ C (⊤_{y}) (z) \\ = ⋁_{y \in X} (d_{C} (x, y) ⊙ d_{C} (y, z)) . \end{matrix}$

$\begin{matrix} ⊓_{τ_{d_{C}}} Θ_{A} (y) = ⋁_{x \in X} (d_{C} (x, y) ⊙ A (x)) \\ = ⋁_{x \in X} (C (⊤_{x}) (y) ⊙ A (x)) \\ \leq C (⋁_{x \in X} (A (x) ⊙ ⊤_{x})) (y) = C (A) (y) . \end{matrix}$

If $C (⊔_{L^{X}} Φ) = ⊔_{L^{X}} C^{\to} (Φ)$ for each Φ ∈ L^{L
^X}, by Theorem 3.13(2), $C (⋁_{i \in Γ} A_{i}) = ⋁_{i \in Γ} C (A_{i})$ and $C (α ⊙ A) = α ⊙ C (A)$ for all A_i, A ∈ L^X. Thus, $\begin{matrix} ⊓_{τ_{d_{C}}} Θ_{A} (y) = ⋁_{x \in X} (d_{C} (x, y) ⊙ A (x)) \\ = ⋁_{x \in X} (C (⊤_{x}) (y) ⊙ A (x)) \\ = C (⋁_{x \in X} (A (x) ⊙ ⊤_{x})) (y) = C (A) (y) . \end{matrix}$

$\begin{matrix} e_{C} (x, y) = ⋀_{A \in L^{X}} (C (A) (x) \to C (A) (y)) \\ \leq C (⊤_{x}) (x) \to C (⊤_{x}) (y) = d_{C} (x, y) . \end{matrix}$

Since A = ⋁ _x∈X (A (x) ⊙ ⊤ _x), $\begin{matrix} e_{C} (x, y) = ⋀_{A \in L^{X}} (C (A) (x) \to C (A) (y)) \\ = ⋀_{A \in L^{X}} C (⋁_{z \in X} (A (z) ⊙ ⊤_{z}) (x) \\ \to C (⋁_{z \in X} (A (z) ⊙ ⊤_{z}) (y)) \\ = ⋀_{A \in L^{X}} (⋁_{z \in X} (A (z) ⊙ C (⊤_{z}) (x) \\ \to ⋁_{z \in X} (A (z) ⊙ C (⊤_{z}) (y))) \\ \geq ⋀_{A \in L^{X}} ⋀_{z \in X} ((A (z) ⊙ C (⊤_{z}) (x) \\ \to (A (z) ⊙ C (⊤_{z}) (y))) \\ \geq ⋀_{z \in X} (C (⊤_{z}) (x) \to C (⊤_{z}) (y)) \\ = ⋀_{z \in X} (d_{C} (z, x) \to d_{C} (z, y)) \\ = d_{C} (x, y) . \end{matrix}$

Example 3.21. Let ([0, 1] , ⊙ , → , 0, 1) be a complete residuated lattice (ref.[1 , 24]) as $\begin{matrix} x ⊙ y & = max {0, x + y - 1}, \\ x \to y = min {1 - x + y, 1} . \end{matrix}$ Define x^* = x → 0 =1 - x. Then (x^*) ^* = x. Let X = {x, y, z} and A ∈ [0, 1] ^X with A (x) =0.6, A (y) =0.7, A (z) =0.4.

(1) Define an Alexandrov [0, 1]-pretopology $τ_{X} = {(α ⊙ A) \lor β_{X}) ∣ α, β \in [0, 1]} .$ By Theorem 3.16 (3), $τ_{I_{τ}} = τ$ where $τ_{I_{τ}} = {A \in L^{X} ∣ A = I_{τ} (A)}$ . Since $0.8 \to A = (0.8, 0.9, 0.6) \notin τ_{X} = τ_{I_{τ}}$ , $τ_{X} = τ_{I_{τ}}$ is not an Alexandrov [0, 1]-topology. By Lemma 3.2, (τ_X, e_{τ
_X}) is a fuzzy poset. For each Φ : τ_X → [0, 1], since ⋁_{C∈τ_X} (Φ (C) ⊙ C) ∈ τ_X for C ∈ τ_X, it follows that $\begin{matrix} e_{τ} (⊔_{τ_{X}} Φ, B) & = ⋀_{C \in τ_{X}} (Φ (C) \to e_{τ_{X}} (C, B)) \\ = ⋀_{C \in τ_{X}} e_{τ_{X}} (Φ (C) ⊙ C, B) \\ = e_{τ_{X}} (⋁_{C \in τ_{X}} (Φ (C) ⊙ C), B) \end{matrix}$ By Lemma 3.2(2), (τ_X, e_{τ
_X}) is a fuzzy join complete lattice.

For B = (0.3, 0.4, 0.5) ∈ [0, 1] ^X and Φ_B (A_i) = e_{L
^X} (A_i, B), $\begin{matrix} ⊔_{τ_{X}} Φ_{B} = ⋁ {A_{i} \in τ_{X} ∣ A_{i} \leq B} \\ = (0.7 ⊙ A) \lor 0 . 3_{X} = (0.3, 0.4, 0.3) . \end{matrix}$ Since ((α ⊙ A (x)) ∨ β_X) ⊙ (A (x) → A (y)) ≤ (α ⊙ A (x)) ∨ β_X, we have d_{τ
_X} (x, y) = ⋀ _{C∈τ_X} (C (x) → C (y)) = A (x) → A (y) as $d_{τ_{X}} = (\begin{matrix} 1 & 1 & 0.8 \\ 0.9 & 1 & 0.7 \\ 1 & 1 & 1 \end{matrix})$ By Theorem 3.9(3), we obtain an Alexandrov [0, 1]-topology τ_{d
_{τ
_X}} = {((α ⊙ A) ∨ β_X) , (α → A) ∧ β_X ∣ α, β ∈ [0, 1]} = {⋁ _x∈X (C (x) ⊙ d_{τ
_X} (x, -)) ∣ C ∈ L^X} = {⋀ _x∈X (d_{τ
_X} (- , x) → D (x)) ∣ D ∈ L^X} where $\begin{matrix} ⊓_{τ_{d_{τ_{X}}}} Θ_{C} = ⋁_{x \in X} (C (x) ⊙ d_{τ_{X}} (x, -)) \\ = (\begin{matrix} C (x) \lor (C (y) - 0.1) \lor C (z) \\ C (x) \lor C (y) \lor C (z) \\ (C (x) - 0.2) \lor (C (y) - 0.3) \lor C (z) \end{matrix}) \end{matrix}$ $\begin{matrix} ⊔_{τ_{d_{τ_{X}}}} Φ_{D} = ⋀_{x \in X} (d_{τ_{X}} (-, x) \to D (x)) \\ = (\begin{matrix} D (x) \land D (y) \land (0.2 + D (z)) \\ (0.1 + D (x)) \land D (y) \land (0.3 + D (z)) \\ D (x) \land D (y) \land D (z) \end{matrix}) \end{matrix}$ The pair (⊔ _{τ
_{d
_{τ
_X}}}Φ_B = (0.3, 0.4, 0.3) , ⊓ _{τ
_{d
_{τ
_X}}}Θ_B = (0.5, 0.5, 0.5)) is a fuzzy rough set of B = (0.3, 0.4, 0.5). The fuzzy accuracy measure α (B) of B is $α (B) = ⋀_{x \in X} (⊓_{τ_{d_{τ_{X}}}} Θ_{B} (x) \to ⊔_{τ_{d_{τ_{X}}}} Φ_{B} (x)) = 0.8 .$

(2) From (1) and Theorem 3.8(2), we obtain an Alexandrov [0, 1]-precotopology $\begin{matrix} η_{τ_{X}} & = {A^{*} ∣ A \in τ_{X}} \\ = {(α \to A^{*}) \land β_{X} ∣ α, β \in [0, 1]} . \end{matrix}$ Then (η_{τ
_X}, e_{η
_{τ
_X}}) is a fuzzy poset. For each Ψ : η_{τ
_X} → [0, 1] such that Ψ (A) = Φ (A^*), since ⋁_{C^*∈τ_X} (Φ (C^*) ⊙ C^*) ∈ τ_X, we have $\begin{matrix} e_{η_{τ_{X}}} (B, ⊓_{η_{τ_{X}}} Ψ) & = ⋀_{C \in η_{τ_{X}}} (Ψ (C) \to e_{η_{τ_{X}}} (B, C)) \\ = ⋀_{C^{*} \in τ_{X}} (Ψ (C) \to e_{τ_{X}} (C^{*}, B^{*})) \\ = ⋀_{C^{*} \in τ_{X}} e_{τ_{X}} (Ψ (C) ⊙ C^{*}, B^{*}) \\ = e_{τ_{X}} (⋁_{C^{*} \in τ_{X}} (Φ (C^{*}) ⊙ C^{*}), B^{*}) \\ = e_{η_{τ_{X}}} (B, ⋀_{C^{*} \in τ_{X}} (Φ (C^{*}) \to C)) \\ = e_{η_{τ_{X}}} (B, ⋀_{C \in τ_{X}^{*}} (Ψ (C) \to C)) \end{matrix}$ By Lemma 3.2(3), (η_{τ
_X}, e_{η
_{τ
_X}}) is a fuzzy meet complete lattice.

For B = (0.3, 0.4, 0.5) ∈ [0, 1] ^X and Θ_B (A_i) = e_{L
^X} (B, A_i), $\begin{matrix} ⊓_{η_{τ_{X}}} Θ_{B} = ⋀ {A_{i} \in η_{τ_{X}} ∣ B \leq A_{i}} \\ = (0.7 \to A^{*}) \land 0 . 5_{X} = (0.5, 0.4, 0.5) . \end{matrix}$ Since ((α → A^* (x)) ∧ β_X) ⊙ (A^* (x) → A^* (y)) ≤ (α → A^* (y)) ∧ β_X, we obtain a fuzzy preorder d_{η
_{τ
_X}} with d_{η
_{τ
_X}} (x, y) = ⋀ _{B∈η_{τ
_X}} (B (x) → B (y)) = A^* (x) → A^* (y) = A (y) → A (x) = d_{τ
_X} (y, x) = ⋀ _{B∈τ_X} (B (x) → B (y)) as $d_{η_{τ_{X}}} = (\begin{matrix} 1 & 0.9 & 1 \\ 1 & 1 & 1 \\ 0.8 & 0.7 & 1 \end{matrix})$ By Theorem 3.9(3), we obtain an Alexandrov [0, 1]-topology τ_{d
_{η
_{τ
_X}}} = {((α ⊙ A^*) ∨ β_X) , (α → A^*) ∧ β_X ∣ α, β ∈ [0, 1]} = {⋁ _x∈X (C (x) ⊙ d_{η
_{τ
_X}} (x, -)) ∣ C ∈ L^X} = {⋀ _x∈X (d_{η
_{τ
_X}} (- , x) → D (x)) ∣ D ∈ L^X} where $\begin{matrix} ⊓_{τ_{d_{η_{τ_{X}}}}} Θ_{C} = ⋁_{x \in X} (C (x) ⊙ d_{η_{τ_{X}}} (x, -)) \\ = (\begin{matrix} C (x) \lor C (y) \lor (C (z) - 0.2) \\ (C (x) - 0.1) \lor C (y) \lor (C (z) - 0.3) \\ C (x) \lor C (y) \lor C (z) \end{matrix}) \end{matrix}$ $\begin{matrix} ⊔_{τ_{d_{η_{τ_{X}}}}} Φ_{D} = ⋀_{x \in X} (d_{η_{τ_{X}}} (-, x) \to D (x)) \\ = (\begin{matrix} D (x) \land (D (y) + 0.1) \land D (z) \\ D (x) \land D (y) \land D (z) \\ (D (x) + 0.2) \land (D (x) + 0.3) \land D (z) \end{matrix}) \end{matrix}$ The pair (⊔ _{τ
_{d
_{η
_{τ
_X}}}}Φ_B = (0.3, 0.3, 0.5) , ⊓ _{τ
_{d
_{η
_{τ
_X}}}}Θ_B = (0.4, 0.4, 0.5)) is a fuzzy rough set of B = (0.3, 0.4, 0.5). The fuzzy accuracy measure α (B) of B is $α (B) = ⋀_{x \in X} (⊓_{τ_{d_{η_{τ_{X}}}}} Θ_{B} (x) \to ⊔_{τ_{d_{η_{τ_{X}}}}} Φ_{B} (x)) = 0.9 .$

Example 3.22. Let X = {h_i ∣ i = {1, . . . , 3}} and Y = {e, b, w, c, i} be sets with h_i=house and e=expensive,b= beautiful, w=wooden, c= creative, i=in the green surroundings. Let ([0, 1] , ⊙ , → , ^*, 0, 1) be a complete residuated lattice as in Example 3.21. Let R ∈ [0, 1] ^X×Y be a fuzzy information as follows: $\begin{matrix} R & e & b & w & c & i \\ h_{1} & 0.7 & 0.6 & 0.5 & 0.9 & 0.2 \\ h_{2} & 0.6 & 0.8 & 0.4 & 0.3 & 0.5 \\ h_{3} & 0.4 & 0.9 & 0.8 & 0.6 & 0.6 \end{matrix}$ Define L-fuzzy preordered relations $e_{X}^{Y}, e_{X}^{{b, w}} \in [0, 1]^{X \times X}$ by $\begin{matrix} e_{X}^{Y} (h_{i}, h_{j}) = ⋀_{y \in Y} (R (h_{i}, y) \to R (h_{j}, y)), \\ e_{X}^{{b, w}} (h_{i}, h_{j}) = ⋀_{y \in {b, w}} (R (h_{i}, y) \to R (h_{j}, y)) . \end{matrix}$ $e_{X}^{Y} = (\begin{matrix} 1 & 0.4 & 0.7 \\ 0.7 & 1 & 0.8 \\ 0.6 & 0.6 & 1 \end{matrix}) e_{X}^{{b, w}} = (\begin{matrix} 1 & 0.9 & 1 \\ 0.8 & 1 & 1 \\ 0.7 & 0.6 & 1 \end{matrix})$ Then $e_{X}^{Y} (h_{1}, -) = (1, 0.4, 0.7) \in τ_{e_{X}^{Y}}, e_{X}^{Y} (h_{1}, -) \notin τ_{e_{X}^{{b, w}}}$ because $\begin{matrix} e_{X}^{Y} (h_{1}, h_{1}) ⊙ e_{X}^{{b, w}} (h_{1}, h_{2}) \\ = 0.9 ≰ e_{X}^{Y} (h_{1}, h_{2}) = 0.4 . \end{matrix}$ By Theorem 3.9(3,4), we obtain an Alexandrov [0, 1]-topology $τ_{e_{X}^{Y}} = {⋁_{x \in X} (A (x) ⊙ e_{X}^{Y} (x, -)) ∣ A \in L^{X}} = {⋀_{x \in X} (e_{X}^{Y} (-, x) \to A (x)) ∣ A \in L^{X}}$ where $\begin{matrix} ⊓_{τ_{e_{X}^{Y}}} Φ_{A} = C_{τ_{e_{X}^{Y}}} (A) = ⋁_{x \in X} (A (x) ⊙ e_{X}^{Y} (x, -)) \\ = (\begin{matrix} A (x) \lor (A (y) - 0.3) \lor (A (z) - 0.4) \\ (A (x) - 0.6) \lor A (y) \lor (A (z) - 0.4) \\ (A (x) - 0.3) \lor (A (y) - 0.2) \lor A (z) \end{matrix}) \end{matrix}$ $\begin{matrix} ⊔_{τ_{e_{X}^{Y}}} Φ_{A} = I_{τ_{e_{X}^{Y}}} (A) = ⋀_{x \in X} (e_{X}^{Y} (-, x) \to A (x)) \\ = (\begin{matrix} A (x) \land (0.6 + A (y)) \land (0.3 + A (z)) \\ (0.3 + A (x)) \land A (y) \land (0.2 + A (z)) \\ (0.4 + A (x)) \land (0.4 + A (y)) \land A (z) \end{matrix}) \end{matrix}$ The pair $(⊔_{τ_{e_{X}^{Y}}} Φ_{A}, ⊓_{τ_{e_{X}^{Y}}} Φ_{A})$ is a fuzzy rough set for A.

Moreover, we obtain an Alexandrov [0, 1]-topology $τ_{e_{X}^{{b, w}}} = {⋁_{x \in X} (A (x) ⊙ e_{X}^{{b, w}} (x, -)) ∣ A \in L^{X}} = {⋀_{x \in X} (e_{X}^{{b, w}} (-, x) \to A (x)) ∣ A \in L^{X}}$ where $\begin{matrix} ⊓_{τ_{e_{X}^{{b, w}}}} Φ_{A} = C_{τ_{e_{X}^{{b, w}}}} (A) \\ = ⋁_{x \in X} (A (x) ⊙ e_{X}^{{b, w}} (x, -)) \\ = (\begin{matrix} A (x) \lor (A (y) - 0.2) \lor (A (z) - 0.3) \\ (A (x) - 0.1) \lor A (y) \lor (A (z) - 0.4) \\ A (x) \lor A (y) \lor A (z) \end{matrix}) \end{matrix}$ $\begin{matrix} ⊔_{τ_{e_{X}^{{b, w}}}} Φ_{A} = I_{τ_{e_{X}^{{b, w}}}} (A) \\ = ⋀_{x \in X} (e_{X}^{{b, w}} (-, x) \to A (x)) \\ = (\begin{matrix} A (x) \land (0.1 + A (y)) \land A (z) \\ (0.2 + A (x)) \land A (y) \land A (z) \\ (0.3 + A (x)) \land (0.4 + A (y)) \land A (z) \end{matrix}) \end{matrix}$ The pair $(⊔_{τ_{e_{X}^{{b, w}}}} Φ_{A}, ⊓_{τ_{e_{X}^{{b, w}}}} Φ_{A})$ is a fuzzy rough set for A.

4 Conclusion

As an extension of Zhang’s fuzzy complete lattices for frames [27, 28], we define Alexandrov L-pretopology, Alexandrov L-precotopology, Alexandrov L-preinterior (L-preclosure) operators and fuzzy rough sets as a view point of fuzzy join and meet complete lattices in a complete residuated lattice. Moreover, we investigate their relations and Example 3.22 as a viewpoint of the topological structure for fuzzy information and fuzzy rough sets in a complete residuated lattice.

In the future, by using the concepts of fuzzy join and meet complete lattices, information systems and decision rules with a view point of applications to multi-attribute decision-making are investigated in complete residuated lattices.

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