Soft implicative L -fuzzy interior and closure operators 1

Abstract

In this paper, we introduce soft implicative L-fuzzy interior and closure operators in a complete residuated lattice. We study the relations among soft implicative L-fuzzy interior and closure operators, soft L-fuzzy topologies and soft L-fuzzy cotopologies. In particular, we study some functorial relationships among previous spaces. We give their examples.

Keywords

Complete residuated lattice interior (closed) soft maps continuous soft maps

1 Introduction

Hájek [7] introduced a complete residuated lattice which is an algebraic structure for many valued logic. It is an important mathematical tool for algebraic structure of fuzzy contexts [2 , 26]. On the other hand, Molodtsov [18] introduced the soft set as a mathematical tool for dealing information as the uncertainty of data in engineering, physics, computer sciences and many other diverse field. Presently, the soft set theory is making progress rapidly [1 , 27]. Pawlak’s rough set [19, 20] can be viewed as a special case of soft rough sets [6]. The topological structures of soft sets have been developed by many researchers [3 , 29, 30].

Kim [13] introduced a fuzzy soft F : A → L ^U as an extension as the soft F : A → P (U) where L is a complete residuated lattice. He introduced soft L-fuzzy quasi-uniformities and soft L-fuzzy topogenous orders in complete residuated lattices.

In this paper, we introduce soft implicativeL-fuzzy interior and closure operators in a complete residuated lattice an extension as a Rodríguez’s sense [23]. We study the relations among soft implicative L-fuzzy interior and closure operators, soft L-fuzzy topologies and soft L-fuzzy cotopologies. In particular, we study some functorial relationships among previous spaces. We give their examples.

2 Preliminaries

Definition 2.1. [2 , 26] An algebra (L, ∧ , ∨ , ⊙ , → , ⊥ , ⊤) is called a complete residuated lattice if it satisfies the following conditions:

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

(L, ⊙ , ⊤) is a commutative monoid;

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

In this paper, we assume that (L, ≤ , ⊙ , → , ⊕ , ^*) is a complete residuated lattice with an order reversing involution ^* defined as x ⊕ y = (x ^* ⊙ y ^*) ^*.

Lemma 2.2. [2 , 26] For each x, y, z, x _i, y _i, w ∈ L, we have the following properties.

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

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

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

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

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

x → y ≤ (z ⊙ x) → (z ⊙ y),

If x ^* = x→ ⊥, then x → y = y ^* → x ^*.

⋁_i∈Γ x _i → ⋁ _i∈Γ y _i ≥ ⋀ _i∈Γ (x _i → y _i) and ⋀_i∈Γ x _i → ⋀ _i∈Γ y _i ≥ ⋀ _i∈Γ (x _i → y _i).

Definition 2.3 [13] Let X be an initial universe of objects and E the set of parameters (attributes) in X. A pair (F, A) is called a fuzzy soft set over X, where A ⊂ E and F : A → L ^X is a mapping. We denote S (X, A) as the family of all fuzzy soft sets under the parameter A.

Definition 2.4. [13] Let (F, A) and (G, A) be two fuzzy soft sets over a common universe X.

(F, A) is a fuzzy soft subset of (G, A), denoted by (F, A) ≤ (G, A) if F (a) ≤ G (a), for each a ∈ A.

(F, A) ∧ (G, A) = (F ∧ G, A) if (F ∧ G) (a) = F (a) ∧ G (a) for each a ∈ A.

(F, A) ∨ (G, A) = (F ∨ G, A) if (F ∨ G) (a) = F (a) ∨ G (a) for each a ∈ A.

(F, A) ⊙ (G, A) = (F ⊙ G, A) if (F ⊙ G) (a) = F (a) ⊙ G (a) for each a ∈ A.

(F, A) ^* = (F ^*, A) if F ^* (a) = (F (a)) ^* for each a ∈ A.

(F, A) ⊕ (G, A) = (F ⊕ G, A) if (F ⊕ G) (a) = (F ^* (a) ⊙ G ^* (a)) ^* for each a ∈ A.

α ⊙ (F, A) = (α ⊙ F, A) if (α ⊙ F) (a) = α ⊙ F (a) for each a ∈ A, α ∈ L.

α → (F, A) = (α → F, A) if (α → F) (a) = α → F (a) for each a ∈ A, α ∈ L.

Definition 2.5. [15] Let S (X, A) and S (Y, B) be the families of all fuzzy soft sets over X and Y, respectively. The mapping f _φ : S (X, A) → S (Y, B) is a soft mapping where f : X → Y and φ : A → B are mappings.

The image of (F, A) ∈ S (X, A) under f _φ is defined as f _φ ((F, A)) = (f _φ (F) , B) $f_{φ} (F) (b) = ⋁_{a \in φ^{- 1} ({b})} f^{\to} (F (a))$

The inverse image of (G, B) ∈ S (Y, B) under f _φ is defined by $f_{φ}^{- 1} ((G, B)) = (f_{φ}^{- 1} (G), A)$ where $f_{φ}^{- 1} (G) (a) (x) = f^{\leftarrow} (G (φ (a))) (x), \forall a \in A, x \in X .$

The soft mapping f _φ : S (X, A) → S (Y, B) is called injective (resp. surjective, bijective) if f and φ are both injective (resp. surjective, bijective).

Lemma 2.6. [15] Let f _φ : S (X, A) → S (Y, B) be a soft mapping. Then we have the following properties. For (F, A) , (F _i, A) ∈ S (X, A) and (G, B) , (G _i, B) ∈ S (Y, B),

$(G, B) \geq f_{φ} (f_{φ}^{- 1} ((G, B)))$ with equality if f is surjective,

$(F, A) \leq f_{φ}^{- 1} (f_{φ} ((F, A)))$ with equality if f is injective,

$f_{φ}^{- 1} ((G, B)^{*}) = (f_{φ}^{- 1} ((G, B)))^{*}$ ,

$f_{φ}^{- 1} (⋁_{i \in I} (G_{i}, B)) = ⋁_{i \in I} f_{φ}^{- 1} ((G_{i}, B))$ ,

$f_{φ}^{- 1} (⋀_{i \in I} (G_{i}, B)) = ⋀_{i \in I} f_{φ}^{- 1} ((G_{i}, B))$ ,

f _φ (⋁ _i∈I (F _i, A)) = ⋁ _i∈I f _φ ((F _i, A)),

f _φ (⋀ _i∈I (F _i, A)) ≤ ⋀ _i∈I f _φ ((F _i, A)) with equality if f is injective,

$f_{φ}^{- 1} ((G_{1}, B) ⊙ (G_{2}, B)) = f_{φ}^{- 1} ((G_{1}, B)) ⊙ f_{φ}^{- 1} ((G_{2}, B)),$

$f_{φ}^{- 1} ((G_{1}, B) \oplus (G_{2}, B)) = f_{φ}^{- 1} ((G_{1}, B)) \oplus f_{φ}^{- 1} ((G_{2}, B)),$

f _φ ((F ₁, A) ⊙ (F ₂, A)) ≤ f _φ ((F ₁, A)) ⊙ f _φ ((F ₂, A)) with equality if f is injective,

f _φ ((F ₁, A) ⊕ (F ₂, A)) ≤ f _φ ((F ₁, A)) ⊕ f _φ ((F ₂, A)) with equality if f is injective.

Definition 2.7. [2] 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 _X satisfies (E1) and (E2), e _X is a fuzzy preorder on X. If e _X satisfies (E1), (E2) and (E3), e _X is a fuzzy partially order on X.

3 Soft implicative L-fuzzy interior operators

Lemma 3.1. For a given set X, define a binary mapping e _X : S (X, A) × S (X, A) → L by $e_{X} ((F, A), (G, A)) = ⋀_{a \in A} ⋀_{x \in X} (F (a) (x) \to G (a) (x)) .$ Then, for each (F, A) , (G, A) , (H, A) , (K, A) ∈ S (X, A) and α ∈ L the following properties hold.

(F, A) ≤ (G, A) iff e _X ((F, A) , (G, A)) =⊤.

e _X is a fuzzy partially order on S (X, A).

If (F, A) ≤ (G, A), then $\begin{matrix} e_{X} ((H, A), (F, A)) & \leq & e_{X} ((H, A), (G, A)), \\ e_{X} ((F, A), (H, A)) & \geq & e_{X} ((G, A), (H, A)) . \end{matrix}$

e _X ((F, A) , (G, A)) ⊙ e _X ((K, A) , (H, A)) ≤ e _X ((F, A) ⊙ (K, A) , (G, A) ⊙ (H, A)) .

e _X ((F, A) , (G, A)) ⊙ e _X ((K, A) , (H, A)) ≤ e _X ((F, A) ⊕ (K, A) , (G, A) ⊕ (H, A)) .

e _X ((F, A) , α → (G, A)) = e _X (α ⊙ (F, A) , (G, A)) = α → e _X ((F, A) , (G, A)) and α ⊙ e _X ((F, A) , (G, A)) ≤ e _X ((F, A) , α ⊙ (G, A)).

(G, A) ⊙ e _X ((G, A) , (F, A)) ≤ (F, A) and (G, A) ≤ e _X ((G, A) , (F, A)) → (F, A).

e _X ((G, A) , (H, A)) ≤ e _X ((F, A) , (G, A)) → e _X ((F, A) , (H, A)).

e _X ((F, A) , (G, A)) ≤ e _X ((G, A) , (H, A)) → e _X ((F, A) , (H, A)) .

If x ^* = x → 0, then e _X ((F, A) , (G, A)) = e _X ((G, A) ^*, (F, A) ^*) .

Let f _φ : (X, A) → (Y, B) be a soft map. $\begin{matrix} e_{X} ((F, A), (G, A)) \\ \leq e_{Y} (f_{φ} ((F, A)), f_{φ} ((G, A))), \\ e_{Y} ((H, A), (K, A)) \\ \leq e_{X} (f_{φ}^{- 1} ((H, A)), f_{φ}^{- 1} ((K, A))), \end{matrix}$ and the equalities hold if f _φ is bijective.

Proof. (1) It follows from F (a) (x)→ G (a) (x) = ⊤ iff F (a) (x) ≤ G (a) (x).

(2) It follows (1) and $\begin{matrix} e_{X} ((F, A), (G, A)) ⊙ e_{X} ((G, A), (H, A)) \\ \leq (F (a) (x) \to G (a) (x)) ⊙ (G (a) (x) \to H (a) (x)) \\ \leq (F (a) (x) \to H (a) (x)) . \end{matrix}$ (5) $\begin{matrix} e_{X} ((F, A) \oplus (K, A), (G, A) \oplus (H, A)) \\ = ⋀_{a \in A} ⋀_{x \in X} ((F (a) (x) \oplus K (a) (x)) \\ \to (G (a) (x) \oplus H (a) (x)) \\ \geq ⋀_{a \in A} ⋀_{x \in X} ((F (a) (x) \to G (a) (x)) ⊙ \\ (K (a) (x) \to H (a) (x))) (by Lemma 2.2 (4)) \\ \geq (⋀_{a \in A} ⋀_{x \in X} (F (a) (x) \to G (a) (x))) ⊙ \\ (⋀_{a \in A} ⋀_{x \in X} (K (a) (x) \to H (a) (x))) \\ = e_{X} ((F, A), (G, A)) ⊙ e_{X} ((K, A), (H, A)) . \end{matrix}$ (7) Since G (a) (x) ⊙ (G (a) (x) → F (a) (x)) ≤ F (a) (x) and G (a) (x) ≤ (G (a) (x) → F (a) (x)) → F (a) (x), we have (G, A) ⊙ e _X ((G, A) , (F, A)) ≤ (F, A) and (G, A) ≤ e _X ((G, A) , (F, A)) → (F, A).

(8) and (9) are follows from e _X ((F, A) , (G, A)) ⊙ e _X ((G, A) , (H, A)) ≤ e _X ((F, A) , (H, A)) by (2).

(11) $\begin{matrix} e_{Y} (f_{φ} ((F, A)), f_{φ} ((G, A))) \\ = ⋀_{b \in B} ⋀_{y \in Y} (f_{φ} (F) (b) (y) \to f_{φ} (G) (b) (y)) \\ = ⋀_{b \in B} ⋀_{y \in Y} (⋁_{a \in φ^{- 1} ({b})} f^{\to} (F (a)) (y) \\ \to ⋁_{a \in φ^{- 1} ({b})} f^{\to} (G (a)) (y)) \\ \geq ⋀_{a \in A} ⋀_{y \in Y} (f^{\to} (F (a)) (y) \to f^{\to} (G (a)) (y)) \\ = ⋀_{a \in A} ⋀_{y \in Y} (⋁_{x \in f^{- 1} ({y})} F (a) (x) \\ \to ⋁_{x \in f^{- 1} ({y})} G (a) (x)) \\ \geq ⋀_{a \in A} ⋀_{y \in Y} (F (a) (x) \to G (a) (x)) \\ = e_{X} ((F, A), (G, A)) . \end{matrix}$ $\begin{matrix} e_{X} (f_{φ}^{- 1} ((H, B)), f_{φ}^{- 1} ((K, B))) \\ = ⋀_{a \in A} ⋀_{x \in X} (f_{φ}^{- 1} (H) (a) (x) \to f_{φ}^{- 1} (K) (a) (x)) \\ = ⋀_{a \in A} ⋀_{x \in X} (H (φ (a)) (f (x)) \to K (φ (a)) (f (x))) \\ \leq ⋀_{b \in B} ⋀_{y \in Y} (H (b) (y) \to K (b) (y)) \\ = e_{Y} ((H, B), (K, B)) . \end{matrix}$ Other cases are similarly proved.

Definition 3.2. A map $T : S (X, A) \to L$ is called a soft L-fuzzy topology on X if it satisfies:

$T ((⊥_{X}, A)) = T ((⊤_{X}, A)) = ⊤$ , where ⊥_X (a) (x) =0, ⊤ _X (a) (x) = ⊤ for all a ∈ A, x ∈ X,

$T ((F, A) ⊙ (G, A)) \geq T ((F, A)) ⊙ T ((G, A)),$

$T (⋁_{i \in I} (F_{i}, A)) \geq ⋀_{i \in I} T ((F_{i}, A))$ .

The triple $(X, A, T)$ is called a soft L-fuzzy topological space. A soft L-fuzzy topology on X is enriched if (E) $T (α ⊙ (F, A)) \geq T ((F, A))$ for each (F, A) ∈ S (X, A).

Let $(X, A, T_{X})$ and $(Y, B, T_{Y})$ be soft L-fuzzy topological spaces and f _φ : S (X, A) → S (Y, B) be a soft map. Then f _φ is called a continuous soft map if $T_{Y} ((G, B)) \leq T_{X} (f_{φ}^{- 1} ((G, B)), \forall (G, B) \in S (Y, B) .$

Definition 3.3. A mapping int : S (X, A) × L _⊥ → S (X, A) with L _⊥ = L − {⊥} is called a soft implicative L- fuzzy interior operator satisfying the following conditions;

int ((⊤ _X, A) , r) = (⊤ _X, A),

int ((F, A) , r) ≤ (F, A),

e _X (int ((F ₁, A) , r) , int ((F ₂, A) , r)) ≥ e _X ((F ₁, A) , (F ₂, A)) .

If r ₁ ≤ r ₂, then int ((F, A) , r ₁) ≥ int ((F, A) , r ₂),

int ((F ₁, A) ⊙ (F ₂, A) , r ⊙ s) ≥ int ((F ₁, A) , r) ⊙ int ((F ₂, A) , s) .

The triple (X, A, int) is called a soft implicative L-fuzzy interior space.

A soft implicative L- fuzzy interior operator is topological if

(T) int ((F, A) , r) ≤ int (int ((F, A) , r) , r) for (F, A) ∈ S (X, A) and r ∈ L _⊥.

Let (X, A, int _X) and (Y, B, int _Y) be soft implicative L-fuzzy interior spaces and f _φ : S (X, A) → S (Y, B) be a soft map. Then f _φ is called an interior soft map if, for all (G, B) ∈ S (Y, B), ${int}_{X} (f_{φ}^{- 1} ((G, B)), r) \geq f_{φ}^{- 1} ({int}_{Y} ((G, B), r)) .$

Lemma 3.4. Let int : S (X, A) × L _⊥ → S (X, A) be a map. It satisfies e _X (int ((F, A) , r) , int ((G, A) , r)) ≥ e _X ((F, A) , (G, A)) for all (F, A) , (G, A) ∈ S (X, A) iff int (α ⊙ (F, A) , r) ≥ α ⊙ int ((F, A) , r) and int ((F, A) , r) ≤ int ((G, A) , r) if (F, A) ≤ (G, A).

Proof. (⇒) If (F, A) ≤ (G, A), by Lemma 3.1(1), e _X (int ((F, A) , r) , int ((G, A) , r)) ≥ e _X ((F, A) , (G, A)) = ⊤. Then int ((F, A) , r) ≤ int ((G, A) , r). Moreover, $\begin{matrix} e_{X} (int ((F, A), r), int (α ⊙ (F, A), r)) \\ \geq e_{X} ((F, A), α ⊙ (F, A)) \geq α . \end{matrix}$ that is, α ⊙ int ((F, A) , r) ≤ int (α ⊙ (F, A) , r) .

(⇐) Put α = e _X ((F, A) , (G, A)). $\begin{matrix} e_{X} ((F, A), (G, A)) ⊙ int ((F, A), r) \\ \leq int (e_{X} ((F, A), (G, A)) ⊙ (F, A), r) \\ \leq int ((G, A), r) . (by Lemma 3.1 (7)) \end{matrix}$

Hence $\begin{matrix} e_{X} ((F, A), (G, A)) \\ \leq e_{X} (int ((F, A), r), int ((G, A), r)) . \end{matrix}$

Theorem 3.5. Let $(X, A, T)$ be a soft L-fuzzy topological space. Define the mapping ${int}_{T} : S (X, A) \times L_{⊥} \to S (X, A)$ as follows $\begin{matrix} {int}_{T} ((F, A), r) \\ = ⋁ {(G, A) ⊙ e_{X} ((G, A), (F, A)) | T ((G, A)) \geq r} . \end{matrix}$ Then we have the following properties.

$(X, A, {int}_{T})$ is a soft implicative L-fuzzy interior space,

If $(X, A, T)$ is enriched, then $(X, A, {int}_{T})$ is a topological implicative L-fuzzy interior space,

${int}_{T} ((F, A), r) \leq ⋁ {(G, A) | (G, A) \leq (F, A), T ((G, A)) \geq r}$ ,

If $(X, A, T)$ is enriched, then the equality in (3) holds.

Proof. (1) (SI1) Since, for $T ((⊤_{X}, A)) \geq r,$ $e_{X} ((⊤_{X}, A), (⊤_{X}, A)) = ⊤,$ then ${int}_{T} ((⊤_{X}, A), r) \geq (⊤_{X}, A) ⊙ ⊤ = (⊤_{X}, A)$ . Therefore, ${int}_{T} ((⊤_{X}, A), r) = (⊤_{X}, A)$ .

(SI2) By Lemma 3.1(7), we have ${int}_{T} ((F, A), r) = ⋁_{(G, A)} {(G, A) ⊙ e_{X} ((G, A), (F, A)) | T ((G, A)) \geq r} \leq (F, A)$ for all (F, A) ∈ S (X, A).

(SI3) $\begin{matrix} e_{X} ({int}_{T} ((F, A), r), {int}_{T} ((G, A), r)) \\ = ⋀_{a \in A} ⋀_{x \in X} ({int}_{T} ((F, A), r) (a) (x) \\ \to {int}_{T} ((G, A), r) (a) (x)) \\ = ⋀_{a \in A} ⋀_{x \in X} (⋁_{T ((H, A)) \geq r} H (a) (x) ⊙ e_{X} ((H, A), \\ (F, A)) \to ⋁_{T ((H, A)) \geq r} (H (a) (x) ⊙ e_{X} ((H, A), (G, A))) \\ (by Lemma 2.2 (8)) \\ \geq ⋀_{a \in A} ⋀_{x \in X} ⋀_{T ((H, A)) \geq r} (H (a) (x) \\ ⊙ e_{X} ((H, A), (F, A)) \\ \to H (a) (x) ⊙ e_{X} ((H, A), (G, A))) \\ (by Lemma 2.2 (6)) \\ \geq ⋀_{x \in X} ⋀_{T ((H, A)) \geq r} (e_{X} ((H, A), (F, A)) \\ \to e_{X} ((H, A), (G, A))) \\ \geq e_{X} ((F, A), (G, A)) . (by Lemma 3.1 (8)) \end{matrix}$

(SI4) It is easily proved.

(SI5) By Lemma 2.4(4), we have $\begin{matrix} {int}_{T} ((F, A), r) ⊙ {int}_{T} ((G, A), s) \\ = ⋁_{T ((H_{1}, A)) \geq r} (H_{1}, A) ⊙ e_{X} ((H_{1}, A), (F, A)) \\ ⊙ ⋁_{T ((H_{2}, A)) \geq s} (H_{2}, A) ⊙ e_{X} ((H_{2}, A), (G, A)) \\ = ⋁_{T ((H_{1}, A)) \geq r} ⋁_{T ((H_{2}, A)) \geq s} ((H_{1}, A) ⊙ (H_{2}, A)) \\ ⊙ e_{X} ((H_{1}, A), (F, A)) ⊙ e_{X} ((H_{2}, A), (G, A)) \\ \leq ⋁_{T ((H_{1}, A)) ⊙ T ((H_{2}, A)) \geq r ⊙ s} ((H_{1}, A) ⊙ (H_{2}, A)) \\ ⊙ e_{X} ((H_{1}, A) ⊙ (H_{2}, A), (F, A) ⊙ (G, A)) \\ = {int}_{T} ((F, A) ⊙ (G, A), r ⊙ s) . \end{matrix}$

(2) Since $T$ is enriched, $T ({int}_{T} ((F, A), r)) \geq r$ . Thus $\begin{matrix} {int}_{T} ({int}_{T} ((F, A), r), r) \\ = ⋁_{T ((G, A)) \geq r} (G, A) ⊙ e_{X} ((G, A), {int}_{T} ((F, A), r)) \\ \geq e_{X} ({int}_{T} ((F, A), r), {int}_{T} ((F, A), r)) \\ ⊙ {int}_{T} ((F, A), r) = {int}_{T} ((F, A), r) . \end{matrix}$

(3) For each $T ((G, A)) \geq r$ with (G, A) ≤ (F, A), we have $(G, A) = ⊤ ⊙ (G, A) = e_{X} ((G, A), (F, A)) ⊙ (G, A),$ it follows that $\begin{matrix} ⋁_{T ((G, A)) \geq r} {(G, A) | (G, A) \leq (F, A)} \\ \leq ⋁_{T ((G, A)) \geq r} e_{X} ((G, A), (F, A)) ⊙ (G, A) \\ = {int}_{T} ((F, A), r) . \end{matrix}$

(4) For any $T ((G, A)) \geq r, T (e_{X} ((G, A), (F, A)) ⊙ (G, A)) \geq T ((G, A)) \geq r,$ because $T$ is enriched. Thus, $\begin{matrix} {int}_{T} ((F, A), (G, A)) \\ = ⋁_{T ((G, A)) \geq r} e_{X} ((G, A), (F, A)) ⊙ (G, A) \\ \leq ⋁_{T ((G, A)) \geq r} {(G, A) | (G, A) \leq (F, A)} . \end{matrix}$

Theorem 3.6. Let (X, A, int) be a soft implicative L-fuzzy interior space. Define the mapping $T_{int} : S (X, A) \to L$ by $\begin{matrix} T_{int} ((F, A)) \\ = ⋁ {r \in L | e_{X} ((F, A), int ((F, A), r)) = ⊤} . \end{matrix}$ Then (1) $T_{int}$ is an enriched soft L-fuzzy topology on X.

(2) ${int}_{T_{int}} ((F, A), r) \leq int ((F, A), r - ∊)$ for ∊ > 0, (F, A) ∈ S (X, A) and r ∈ L _⊥.

(3) If int is topological, then ${int}_{T_{int}} \geq int$ .

(4) If $(X, T)$ is a soft L-fuzzy topological space, then $T_{{int}_{T}} = T .$

Proof. (ST1) Since (⊤ _X, A) = int ((⊤ _X, A) , r) and (⊤ _X, A) = int ((⊤ _X, A) , r) for all r ∈ L _⊥, $T_{int} ((⊤_{X}, A)) = T_{int} ((⊥_{X}, A)) = ⊤$ .

(ST2) By Lemma 3.1(4) and (SI5), we have $\begin{matrix} e_{X} ((F_{1}, A), int ((F_{1}, A), r)) \\ ⊙ e_{X} ((F_{2}, A), int ((F_{2}, A), s)) \\ \leq e_{X} ((F_{1}, A) ⊙ (F_{2}, A), int ((F_{1}, A), r) \\ ⊙ int ((F_{2}, A), s)) \\ \leq e_{X} ((F_{1}, A) ⊙ (F_{2}, A), int ((F_{1}, A) \\ ⊙ (F_{2}, A), r ⊙ s)) . \end{matrix}$

If e _X ((F ₁, A) , int ((F ₁, A) , r)) =⊤ and e _X ((F ₂, A) , int ((F ₂, A) , s)) =⊤, then $\begin{matrix} e_{X} ((F_{1}, A) ⊙ (F_{2}, A), int ((F_{1}, A) \\ ⊙ (F_{2}, A), r ⊙ s)) = ⊤ . \end{matrix}$

Thus, $T_{int} ((F_{1}, A) ⊙ (F_{2}, A)) \geq T_{int} ((F_{1}, A)) ⊙ T_{int} ((F_{2}, A)) .$

(ST3) For a family of {(F _i, A) |i ∈ I} ⊆ S (X, A), we have $\begin{matrix} T_{int} (⋁_{i \in I} (F_{i}, A)) \\ = ⋁ {r \in L | e_{X} (⋁_{i \in I} (F_{i}, A), int (⋁_{i \in I} (F_{i}, A), r)) = ⊤} \\ \geq ⋁ {r \in L | e_{X} (⋁_{i \in I} (F_{i}, A), ⋁_{i \in I} int ((F_{i}, A), r)) = ⊤} \\ \geq ⋀_{i \in I} ⋁ {r \in L | e_{X} ((F_{i}, A), int ((F_{i}, A), r)) = ⊤} \\ = ⋀_{i \in I} T_{int} ((F_{i}, A)) . \end{matrix}$

Finally, for α ∈ L _⊥ and (F, A) ∈ S (X, A), since int (α ⊙ (F, A) , r) ≥ α ⊙ int ((F, A) , r) from Lemma 3.4, we have $\begin{matrix} T_{int} (α ⊙ (F, A)) \\ = ⋁ {r \in L | e_{X} (α ⊙ (F, A), int (α ⊙ (F, A), r)) = ⊤} \\ \geq ⋁ {r \in L | e_{X} (α ⊙ (F, A), α ⊙ int ((F, A), r)) = ⊤} \\ \geq ⋁ {r \in L | e_{X} ((F, A), int ((F, A), r)) = ⊤} \\ = T_{int} ((F, A)) . \end{matrix}$

Hence, $T_{int}$ is an enriched soft L-fuzzy topology on X.

(2) Since $\begin{matrix} {int}_{T_{int}} ((F, A), r) \\ = ⋁ {(G, A) ⊙ e_{X} ((G, A), (F, A)) | T_{int} ((G, A)) \geq r} \end{matrix}$ and $T_{int}$ is enriched, then $T_{int} ({int}_{T_{int}} ((F, A), r)) \geq r$ .

By the definition of $T_{int}$ , for each ∊ > 0, $e_{X} ({int}_{T_{int}} ((F, A), r), int ({int}_{T_{int}} ((F, A), r), r - ∊)) = ⊤$ so, ${int}_{T_{int}} ((F, A), r) \leq int ({int}_{T_{int}} ((F, A), r), r - ∊)$ .

Hence $\begin{matrix} {int}_{T_{int}} ((F, A), r) \leq int ({int}_{T_{int}} ((F, A), r), r - ∊) \\ \leq int ((F, A), r - ∊) . \end{matrix}$

(3) Let int be topological. Since int ((F, A) , r) ≤ int (int ((F, A) , r) , r), we have $T_{int} (int ((F, A), r)) \geq r$ . Thus, ${int}_{T_{int}} ((F, A), r) \geq int ((F, A), r) .$

(4) Since $T_{{int}_{T}} ((F, A)) = ⋁ {r \in L | (F, A) \leq {int}_{T} ((F, A), r)}$ , ${int}_{T} ((F, A), ⋁ r) = (F, A) .$

Hence $T ({int}_{T} ((F, A), ⋁ r)) = T ((F, A)) \geq ⋁ r = T_{{int}_{T}} ((F, A))$ .

Conversely, let $T ((F, A)) \geq r$ . Then ${int}_{T} ((F, A), r) = (F, A)$ . Thus $T_{{int}_{T}} ((F, A)) \geq r$ . Hence $T_{{int}_{T}} \geq T .$

Theorem 3.7. Let (X, A, int _X) and (Y, B, int _Y) be two soft implicative L-fuzzy interior spaces. If f _φ : (X, A, int _X) → (Y, B, int _Y) is an interior soft map, then $f_{φ} : (X, A, T_{{int}_{X}}) \to (Y, B, T_{{int}_{Y}})$ is a continuous soft map.

Proof. From Lemma 3.1(11), we have $\begin{matrix} e_{X} (f_{φ}^{- 1} ((F, B)), {int}_{X} (f_{φ}^{- 1} ((F, B)), r)) \\ \geq e_{X} (f_{φ}^{- 1} ((F, B)), f_{φ}^{- 1} ({int}_{Y} ((F, B), r))) \\ \geq e_{X} ((F, B), {int}_{Y} ((F, B), r)) . \end{matrix}$

So, $T_{{int}_{X}} (f_{φ}^{- 1} ((F, B))) \geq T_{{int}_{Y}} ((F, B)) .$

Theorem 3.8. Let $(X, A, T_{X})$ and $(Y, B, T_{Y})$ be two soft L-fuzzy topological spaces. Then $f_{φ} : (X, A, T_{X}) \to (Y, B, T_{Y})$ is a continuous soft map iff $f_{φ} : (X, A, {int}_{T_{X}}) \to (Y, B, {int}_{T_{Y}})$ is an interior soft map.

Proof. (⇒) By Lemma 3.1(11) and a continuous soft map f _φ, we have $\begin{matrix} f_{φ}^{- 1} ({int}_{T_{Y}} ((F, B), r)) \\ = f_{φ}^{- 1} (⋁ {(G, B) ⊙ e_{X} ((G, B), (F, B)) | \\ T_{Y} ((G, B)) \geq r}) \\ = ⋁ {f_{φ}^{- 1} ((G, B)) \\ ⊙ e_{X} ((G, B), (F, B)) | T_{Y} ((G, B)) \geq r} \\ \leq ⋁ {f_{φ}^{- 1} ((G, B)) \\ ⊙ e_{X} (f_{φ}^{- 1} ((G, B)), f_{φ}^{- 1} ((F, B))) \\ | T_{X} (f_{φ}^{- 1} ((G, B))) \geq r} \\ \leq ⋁ {(H, A) ⊙ e_{X} ((H, A), f_{φ}^{- 1} ((F, B))) | \\ T_{X} ((H, A)) \geq r} \\ = {int}_{T_{X}} (f_{φ}^{- 1} ((F, B)), r) . \end{matrix}$

(⇐) It easily proved from Theorems 3.6(4) and 3.7 with $T_{{int}_{T_{X}}} = T_{X}$ and $T_{{int}_{T_{Y}}} = T_{Y}$ .

4 Soft implicative L-fuzzy closure operators

Definition 4.1. A map $F : S (X, A) \to L$ is called a soft L-fuzzy cotopology on X if it satisfies the following conditions.

$F ((⊥_{X}, A)) = F ((⊤_{X}, A)) = ⊤$ , where ⊥_X (a) (x) =0, ⊤ _X (a) (x) = ⊤ for all a ∈ A, x ∈ X,

$F ((F, A) \oplus (G, A)) \geq F ((F, A)) ⊙ F ((G, A)),$

$F (⋀_{i \in I} (F_{i}, A)) \geq ⋀_{i \in I} F ((F_{i}, A))$ .

The triple $(X, A, F)$ is called a soft L-fuzzy cotopological space. A soft L-fuzzy cotopology on X is enriched if

(E) $F (α \to (F, A)) \geq F ((F, A))$ for each (F, A) ∈ S (X, A).

Let $(X, A, F_{X})$ and $(Y, B, F_{Y})$ be soft L-fuzzy cotopological spaces and f _φ : S (X, A) → S (Y, B) be a soft map. Then f _φ is called a continuous soft map if $F_{Y} ((G, B)) \leq F_{X} (f_{φ}^{- 1} ((G, B)), \forall (G, B) \in S (Y, B) .$

Definition 4.2. A map cl : S (X, A) × L _⊥ → S (X, A) is called a soft implicative L- fuzzy closure operator if it satisfies the following conditions;

cl ((⊥ _X, A) , r) = (⊥ _X, A),

cl ((F, A) , r) ≥ (F, A),

e _X ((F ₁, A) , (F ₂, A)) ≤ e _X (cl ((F ₁, A) , r) , cl ((F ₂, A) , r)) ,

If r ₁ ≤ r ₂, then cl ((F, A) , r ₁) ≤ cl ((F, A) , r ₂),

cl ((F ₁, A) ⊕ (F ₂, A) , r ⊙ s) ≤ cl ((F ₁, A) , r) ⊕ cl ((F ₂, A) , s).

The triple (X, A, cl) is called a soft implicativeL-fuzzy closure space.

A soft implicative L- fuzzy closure operator is called topological if

(T) cl (cl ((F, A) , r) , r) ≤ cl ((F, A) , r).

Let (X, A, cl _X) and (Y, B, cl _Y) be soft implicative L-fuzzy closure spaces and f _φ : S (X, A) → S (Y, B) be a soft map. Then f _φ is called a closed soft map if, for all (F, A) ∈ S (X, A), $f_{φ} ({cl}_{X} ((F, A), r)) \leq {cl}_{Y} (f_{φ} ((F, A)), r) .$

Lemma 4.3. Let cl : S (X, A) × L _⊥ → S (X, A) be a map. It satisfies e _X (cl ((F, A) , r) , cl ((G, A) , r)) ≥ e _X ((F, A) , (G, A)) for all (F, A) , (G, A) ∈ S (X, A) iff cl (α → (F, A) , r) ≥ α → cl ((F, A) , r) and cl ((F, A) , r) ≤ cl ((G, A) , r) if (F, A) ≤ (G, A).

Proof. (⇒) If (F, A) ≤ (G, A) , ⊤ = e _X ((F, A) , (G, A)) ≤ e _X (cl ((F, A) , r) , cl ((G, A) , r)). Then cl ((F, A) , r) ≤ cl ((G, A) , r). Moreover, $\begin{matrix} e_{X} (cl (α \to (F, A), r), cl ((F, A), r)) \\ \geq e_{X} (α \to (F, A), (F, A)) \geq α \\ iff α ⊙ cl (α \to (F, A), r) \leq cl ((F, A), r) . \end{matrix}$

That is, cl (α → (F, A) , r) ≤ α → cl ((F, A) , r) .

(⇐) Put α = e _X ((F, A) , (G, A)). $\begin{matrix} e_{X} ((F, A), (G, A)) \to cl ((G, A), r) \\ \geq cl (e_{X} ((F, A), (G, A)) \to (G, A), r) \\ \geq cl ((F, A), r) \\ iff e_{X} ((F, A), (G, A)) \\ \leq cl ((F, A), r) \to cl ((G, A), r) . \end{matrix}$

Hence e _X ((F, A) , (G, A)) ≤ e _X (cl ((F, A) , r) , cl ((G, A) , r)).

Theorem 4.4. Let $(X, A, F)$ be a soft L-fuzzy co-topological space. Define the mapping ${cl}_{F} : S (X, A) \times L_{⊥} \to S (X, A)$ by $\begin{matrix} {cl}_{F} ((F, A), r) (a) (x) \\ = ⋀_{F ((G, A)) \geq r} (e_{X} ((F, A), (G, A)) \to G (a) (x)) . \end{matrix}$ Then we have the following properties.

The pair $(X, A, {cl}_{F})$ is a soft implicative L-fuzzy closure space,

If $(X, A, F)$ is enriched, then $(X, A, {cl}_{F})$ is a topological soft implicative L-fuzzy closure space.

If x ^* = x→ ⊥, then ${cl}_{F}^{*} ((F, A)^{*}, r) = {int}_{T} ((F, A), r)$ , where $T ((F, A)) = F ((F^{*}, A))$ for each (F, A) ∈ S (X, A),

${cl}_{F} ((F, A), r) \leq ⋀_{F ((G, A)) \geq r} {(G, A) | (F, A) \leq (G, A)}$ .

If $(X, F)$ is enriched, ${cl}_{F} ((F, A), r) = ⋀_{F ((G, A)) \geq r} {(G, A) | (F, A) \leq (G, A)} .$

Proof. (1) (SC1) By Lemma 3.1(2), we have $\begin{matrix} {cl}_{F} ((⊥_{X}, A), r) (a) (x) \\ = ⋀_{F ((G, A)) \geq r} (e_{X} ((⊥_{X}, A), (G, A)) \to G (a) (x)) \\ \geq ⊥_{X} (a) (x) = ⊥ . \end{matrix}$

(SC2) By Lemma 3.1(7), we have $\begin{matrix} {cl}_{F} ((F, A), r) (a) (x) \\ = ⋀_{F ((G, A)) \geq r} (e_{X} ((F, A), (G, A)) \to G (a) (x)) \\ \geq ⋀_{F ((G, A)) \geq r} F (a) (x) . \end{matrix}$

Hence ${cl}_{F} ((F, A), r) \geq (F, A)$ .

(SC3) $\begin{matrix} e_{X} ({cl}_{F} ((F, A), r), {cl}_{F} ((H, A), r)) \\ = ⋀_{a \in A} ⋀_{x \in X} ({cl}_{F} ((F, A), r) (a) (x) \\ \to {cl}_{F} ((H, A), r) (a) (x)) \\ = ⋀_{x \in X} ((⋀_{F ((G, A)) \geq r} e_{X} ((F, A), (G, A)) \to (G, A) (x)) \\ \to (⋀_{F ((G, A)) \geq r} e_{X} ((H, A), (G, A)) \to (G, A) (x))) \\ (by Lemma 2.2 (8)) \\ \geq ⋀_{x \in X} ⋀_{F ((G, A)) \geq r} ((e_{X} ((F, A), (G, A)) \\ \to (G, A) (x)) \to (e_{X} ((H, A), (G, A)) \\ \to (G, A) (x))) \\ \geq ⋀_{F ((G, A)) \geq r} (e_{X} ((H, A), (G, A)) \\ \to e_{X} ((F, A), (G, A))) \\ \geq e_{X} ((F, A), (H, A)) . (by Lemma 3.1 (9)) \end{matrix}$

(SC4) It follows from the definition of ${cl}_{F}$ .

(SC5) By Lemma 2.4(5) and Lemma 2.2(5), we have $\begin{matrix} {cl}_{F} ((F, A), r) (x) \oplus {cl}_{F} ((H, A), s) (x) \\ = (⋀_{F ((G_{1}, A)) \geq r} e_{X} ((F, A), (G_{1}, A)) \\ \to (G_{1}, A) (x)) \\ \oplus (⋀_{F ((G_{2}, A)) \geq s} e_{X} ((H, A), (G_{2}, A)) \\ \to (G_{2}, A) (x)) \\ = ⋀_{F ((G_{1}, A)) \geq r} ⋀_{F ((G_{2}, A)) \geq s} \\ ((e_{X} ((F, A), (G_{1}, A)) \to (G_{1}, A) (x)) \\ \oplus (e_{X} ((H, A), (G_{2}, A)) \to (G_{2}, A) (x))) \\ \geq ⋀_{F ((G_{1}, A) \oplus (G_{2}, A)) \geq r ⊙ s} \\ ((e_{X} ((F, A), (G_{1}, A)) ⊙ e_{X} ((H, A), (G_{2}, A))) \\ \to ((G_{1}, A) \oplus (G_{2}, A)) (x)) \\ \geq ⋀_{F ((G_{1}, A) \oplus (G_{2}, A)) \geq r ⊙ s} \\ (e_{X} ((F, A) \oplus (H, A), (G_{1}, A) \oplus (G_{2}, A)) \\ \to ((G_{1}, A) \oplus (G_{2}, A)) (x)) \\ = {cl}_{F} ((F, A) \oplus (H, A), r ⊙ s) (x) . \end{matrix}$

(2) Since $F$ is enriched, $F ({cl}_{F} ((F, A), r) \geq r$ . Then $\begin{matrix} {cl}_{F} ({cl}_{F} ((F, A), r), r) (x) \\ = ⋀_{F ((G, A)) \geq r} (e_{X} ({cl}_{F} ((F, A), r), (G, A)) \\ \to (G, A) (x)) \\ \leq ⋀_{F ({cl}_{F} ((F, A), r)) \geq r} (e_{X} ({cl}_{F} ((F, A), r), {cl}_{F} ((G, A), r)) \\ \to {cl}_{F} ((F, A), r) (x)) = {cl}_{F} ((F, A), r) (x) . \end{matrix}$

(3) Let x ^* = x→ ⊥. Then x → y = y ^* → x ^*. Hence $\begin{matrix} {cl}_{F}^{*} ((F, A)^{*}, r) \\ = {⋀_{F ((G, A)^{*}) \geq r} (e_{X} ((F, A)^{*}, (G, A)^{*}) \to (G, A)^{*})}^{*} \\ = ⋁_{F ((G, A)^{*}) \geq r} (e_{X} ((F, A)^{*}, (G, A)^{*}) ⊙ (G, A)) \\ = ⋁_{T ((G, A)) \geq r} (G, A) ⊙ e_{X} ((G, A), (F, A)) \\ = {int}_{T} ((F, A), r) . \end{matrix}$

(4) If (G, A) ≤ (F, A), then e _X ((F, A) , (G, A)) =⊤ and e _X ((F, A) , (G, A)) → (G, A) ≤ (G, A). Thus, $\begin{matrix} ⋀_{F ((G, A)) \geq r} (e_{X} ((F, A), (G, A)) \to (G, A)) \\ \leq ⋀_{F ((G, A)) \geq r} {(G, A) | (F, A) \leq (G, A)} . \end{matrix}$

(5) For any $F ((G, A)) \geq r$ , $F (e_{X} ((F, A), (G, A)) \to (G, A)) \geq F ((G, A))$ , i.e., $F (e_{X} ((F, A), (G, A)) \to (G, A)) \geq r$ , because $F$ is enriched. Thus, $\begin{matrix} ⋀_{F ((G, A)) \geq r} (e_{X} ((F, A), (G, A)) \to (G, A)) \\ \geq ⋀_{F ((G, A)) \geq r} {(G, A) | (F, A) \leq (G, A)} . \end{matrix}$

Theorem 4.5. If cl : S (X, A) × L _⊥ is a soft implicative L-fuzzy closure operator. Define the mapping $F_{cl} : e_{X} (X, A) \to L$ by $\begin{matrix} F_{cl} ((F, A)) \\ = ⋁ {r \in L | e_{X} (cl ((F, A), r), (F, A)) = ⊤} . \end{matrix}$

Then

A map $F_{cl}$ is an enriched soft L-fuzzyco-topology on X.

$cl ((F, A), r - ∊) \leq {cl}_{F_{cl}} ((F, A), r)$ for (F, A) ∈ S (X, A) and r ∈ L _⊥.

If cl is topological, ${cl}_{F_{cl}} \leq cl$ .

If $(X, A, F)$ is a soft L-fuzzy cotopological space, then $F_{{int}_{F}} = F .$

Proof. (1) (SF1) Since e _X (cl ((⊤ _X, A) , r) , (⊤ _X, A)) = e _X (cl ((⊥ _X, A) , r) , (⊥ _X, A)) = ⊤ for each r ∈ L _⊥, $F_{cl} ((⊤_{X}, A)) = F_{cl} ((⊥_{X}, A)) = ⊤$ .

(SF2) By Lemma 2.4(5) and (C4), we have $\begin{matrix} e_{X} (cl ((F_{1}, A), r), (F_{1}, A)) \\ ⊙ e_{X} (cl ((F_{2}, A), s), (F_{2}, A)) \\ \leq e_{X} (cl ((F_{1}, A), r) \\ \oplus cl ((F_{2}, A), s), (F_{1}, A) \oplus (F_{2}, A)) \\ \leq e_{X} (cl ((F_{1}, A) \\ \oplus (F_{2}, A), r ⊙ s), (F_{1}, A) \oplus (F_{2}, A)) . \end{matrix}$

If e _X (cl ((F ₁, A) , r) , (F ₁, A)) =⊤ and e _X (cl ((F ₂, A) , s) , (F ₂, A)) =⊤, then e _X (cl ((F ₁, A)⊕ (F ₂, A) , r ⊙ s) , (F ₁, A) ⊕ (F ₂, A)) = ⊤. Thus, $F_{cl} ((F_{1}, A) \oplus (F_{2}, A)) \geq F_{cl} ((F_{1}, A)) ⊙ F_{cl} ((F_{2}, A)) .$

(SF3) For a family of {(F _i, A) |i ∈ I} ⊆ S (X, A), we have $\begin{matrix} F_{cl} (⋀_{i \in I} (F_{i}, A)) \\ = ⋁ {r \in L | e_{X} (cl (⋀_{i \in I} (F_{i}, A), r), ⋀_{i \in I} (F_{i}, A)) = ⊤} \\ \leq ⋁ {r \in L | e_{X} (⋀_{i \in I} cl ((F_{i}, A), r), ⋀_{i \in I} (F_{i}, A)) = ⊤} \\ \leq ⋀_{i \in I} ⋁ {r \in L | e_{X} (cl ((F_{i}, A), r), (F_{i}, A)) = ⊤} \\ = ⋁_{i \in I} F_{cl} ((F_{i}, A)) . \end{matrix}$

Hence, $F_{cl}$ is a soft L-fuzzy co-topology on X.

By Lemma 2.4(3), (6), we have $\begin{matrix} F_{cl} (α \to (F, A)) \\ = ⋁ {r \in L | e_{X} (cl (α \to (F, A), r), α \to (F, A)) = ⊤} \\ (by Lemma 4.2) \\ \geq ⋁ {r \in L | e_{X} (α \to cl ((F, A), r), α \to (F, A)) = ⊤} \\ (by Lemma 3.1 (8)) \\ \geq ⋁ {r \in L | e_{X} (cl ((F, A), r), (F, A)) = ⊤} \\ = F_{cl} ((F, A)) . \end{matrix}$

(2) Since $\begin{matrix} {cl}_{F_{cl}} ((F, A), r) \\ = ⋀_{F_{cl} ((G, A)) \geq r} (e_{X} ((F, A), (G, A)) \to (G, A)) \end{matrix}$ and $F_{cl}$ is enriched, then $F_{cl} ({cl}_{F_{cl}} ((F, A), r)) \geq r$ . By the definition of $F_{cl}$ , for each ∊ > 0, $e_{X} (cl ({cl}_{F_{cl}} ((F, A), r), r - ∊), {cl}_{F_{cl}} ((F, A), r)) = ⊤,$ equivalently, $cl ({cl}_{F_{cl}} ((F, A), r), r - ∊) \leq {cl}_{F_{cl}} ((F, A), r)$ . Thus $\begin{matrix} cl ((F, A), r - ∊) \leq cl ({cl}_{F_{cl}} ((F, A), r), r - ∊) \\ \leq {cl}_{F_{cl}} ((F, A), r) . \end{matrix}$

(3) Let cl be topological. Since cl ((F, A) , r) ≥ cl (cl ((F, A) , r) , r), we have $F_{cl} (cl ((F, A), r)) \geq r$ . Thus, ${cl}_{F_{cl}} ((F, A), r) \leq cl ((F, A), r) .$

(4) Since $F_{{cl}_{F}} ((F, A)) = ⋁ {r \in L | {cl}_{F} ((F, A), r) \leq (F, A)}$ , ${cl}_{F} ((F, A), ⋁ r) = (F, A) .$

Hence $F ({cl}_{F} ((F, A), ⋁ r)) = F ((F, A)) \geq ⋁ r = F_{{cl}_{F}} ((F, A))$ .

Conversely, let $F ((F, A)) \geq r$ . Then ${cl}_{F} ((F, A), r) = (F, A)$ . Thus $F_{{cl}_{F}} ((F, A)) \geq r$ . Hence $F_{{cl}_{F}} \geq F .$

Theorem 4.6. Let (X, A, cl _X) and (Y, B, cl _Y) be two soft implicative L-fuzzy closure spaces. If f _φ : (X, A, cl _X) → (Y, B, cl _Y) is a closed soft map, then $f_{φ} : (X, A, F_{{cl}_{X}}) \to (Y, B, F_{{cl}_{Y}})$ is a continuous soft map.

Proof. From Theorem 4.3, we have $\begin{matrix} F_{{cl}_{X}} (f_{φ}^{- 1} ((F, A))) \\ = ⋁ {r \in L | e_{X} ({cl}_{X} (f_{φ}^{- 1} ((F, A)), r), \\ f_{φ}^{- 1} ((F, A))) = ⊤} \\ \geq ⋁ {r \in L | e_{X} (f_{φ}^{- 1} ({cl}_{Y} ((F, A), r)), \\ f_{φ}^{- 1} ((F, A))) = ⊤} \\ (by Lemma 3.1 (11)) \\ \geq ⋁ {r \in L | e_{X} ({cl}_{Y} ((F, A), r), (F, A)) = ⊤} \\ = F_{{cl}_{Y}} ((F, A)) . \end{matrix}$

Theorem 4.7. Let $(X, A, F_{X})$ and $(Y, B, F_{Y})$ be two soft L-fuzzy co-topological spaces. If $f_{φ} : (X, A, F_{X}) \to (Y, B, F_{Y})$ is a continuous soft map, then $f_{φ} : (X, A, {cl}_{F_{X}}) \to (Y, B, {cl}_{F_{Y}})$ is a closed soft map.

Proof. $\begin{matrix} f_{φ}^{- 1} ({cl}_{F_{Y}} ((F, A), r)) \\ = f_{φ}^{- 1} (⋀_{F_{Y} ((G, A)) \geq r} (e_{X} ((F, A), (G, A)) \to (G, A))) \\ = ⋀_{F_{Y} ((G, A)) \geq r} (e_{X} ((F, A), (G, A)) \to f_{φ}^{- 1} ((G, A))) \\ (by Lemma 3.1 (11)) \\ \geq ⋀_{F_{X} (f_{φ}^{- 1} ((G, A))) \geq r} (e_{X} (f_{φ}^{- 1} ((F, A)), f_{φ}^{- 1} ((G, A))) \\ \to f_{φ}^{- 1} ((G, A))) = {cl}_{F_{X}} (f_{φ}^{- 1} ((F, A)), r) . \end{matrix}$

Conversely, it easily proved from Theorems 4.5(4) and 4.6.

Example 4.8. Let X = {h _i ∣ i = {1,. . . , 6}} with h _i = house and E _X = {e, b, w, c, i} with e = expensive, b = beautiful, w = wooden, c = creative, i = in the green surroundings.

Define a binary operation ∧ on [0, 1] by $x \land y = min {x, y},$ $x \to y = {\begin{matrix} 1, & if x \leq y, \\ y, & otherwise . \end{matrix} x^{*} = 1 - x .$

Then ([0, 1] , ∧ , → , 0, 1) is a complete residuated lattice (ref. [2 , 26]). Let A = {e, b, w} ⊂ E _X, (G ₁, A) and (G ₂, A) be fuzzy soft sets as follows: $\begin{matrix} (G_{1}, A) & h^{1} & h^{2} & h^{3} & h^{4} & h^{5} & h^{6} \\ e & 0.3 & 0.6 & 0.5 & 0.0 & 0.2 & 0.6 \\ b & 0.6 & 0.3 & 0.5 & 0.3 & 0.5 & 0.6 \\ w & 0.4 & 0.4 & 0.5 & 0.6 & 0.6 & 0.5 \end{matrix}$ $\begin{matrix} (G_{2}, A) & h^{1} & h^{2} & h^{3} & h^{4} & h^{5} & h^{6} \\ e & 0.5 & 0.4 & 0.4 & 0.3 & 0.2 & 0.4 \\ b & 0.3 & 0.0 & 0.2 & 0.4 & 0.3 & 0.5 \\ w & 0.5 & 0.4 & 0.3 & 0.1 & 0.5 & 0.4 \end{matrix}$ $\begin{matrix} (G_{1} \lor G_{2}, A) & h^{1} & h^{2} & h^{3} & h^{4} & h^{5} & h^{6} \\ e & 0.5 & 0.6 & 0.5 & 0.3 & 0.2 & 0.6 \\ b & 0.6 & 0.3 & 0.5 & 0.4 & 0.5 & 0.6 \\ w & 0.5 & 0.4 & 0.5 & 0.6 & 0.6 & 0.5 \end{matrix}$ $\begin{matrix} (G_{1} \land G_{2}, A) & h^{1} & h^{2} & h^{3} & h^{4} & h^{5} & h^{6} \\ e & 0.3 & 0.4 & 0.4 & 0.0 & 0.2 & 0.4 \\ b & 0.3 & 0.0 & 0.2 & 0.3 & 0.3 & 0.5 \\ w & 0.4 & 0.4 & 0.3 & 0.1 & 0.5 & 0.4 \end{matrix}$ (1) Define a soft [0, 1]-fuzzy topology $T_{X} : S (X, A) \to [0, 1]$ as follows: $T_{X} ((H, A)) = {\begin{matrix} 1, & if (H, A) = (⊥_{X}, A) or (⊤_{X}, A), \\ 0.7, & if (H, A) = (G_{1}, A) \\ 0.4, & if (H, A) = (G_{2}, A) \\ 0.5, & if (H, A) = (G_{1} \lor G_{2}, A) \\ 0.6, & if (H, A) = (G_{1} \land G_{2}, A) \\ 0, & otherwise . \end{matrix}$

From Theorem 3.5, we obtain a soft implicative [0, 1]-fuzzy interior operator ${int}_{T_{X}} : S (X, A) \times [0, 1] \to S (X, A)$ as follows: $\begin{matrix} {int}_{T_{X}} ((F, A), r) \\ = {\begin{matrix} ⋀ (F, A) (x), if r > 0.7, \\ (⋀ (F, A) (x)) if 0.6 < r \leq 0.7, \\ \lor ((G_{1}, A) \land e_{X} ((G_{1}, A), (F, A))) \\ (⋀ (F, A) (x)) if 0.5 < r \leq 0.6, \\ \lor ((G_{1}, A) \land e_{X} ((G_{1}, A), (F, A))) \\ \lor ((G_{1} \land G_{2}, A) \land e_{X} ((G_{1} \land G_{2}, A), (F, A))) \\ (⋀ (F, A) (x)) if 0.5 < r \leq 0.6, \\ \lor ((G_{1}, A) \land e_{X} ((G_{1}, A), (F, A))) \\ \lor ((G_{1} \land G_{2}, A) \land e_{X} ((G_{1} \land G_{2}, A), (F, A))) \\ (⋀ (F, A) (x)) if 0.4 < r \leq 0.5, \\ \lor ((G_{1}, A) \land e_{X} ((G_{1}, A), (F, A))) \\ \lor ((G_{2}, A) \land e_{X} ((G_{2}, A), (F, A))) \\ \lor ((G_{1} \land G_{2}, A) \land e_{X} ((G_{1} \land G_{2}, A), (F, A))) \\ (⋀ (F, A) (x)) if r \leq 0.4, \\ \lor ((G_{1}, A) \land e_{X} ((G_{1}, A), (F, A))) \\ \lor ((G_{1}, A) \land e_{X} ((G_{1}, A), (F, A))) \\ \lor ((G_{1} \land G_{2}, A) \land e_{X} ((G_{1} \land G_{2}, A), (F, A))) \\ \lor ((G_{1} \lor G_{2}, A) \land e_{X} ((G_{1} \lor G_{2}, A), (F, A))) \end{matrix} \end{matrix}$

Put (F, A) ∈ S (X, A) as follows: $\begin{matrix} (F, A) & h^{1} & h^{2} & h^{3} & h^{4} & h^{5} & h^{6} \\ e & 0.5 & 0.5 & 0.7 & 0.4 & 0.2 & 0.4 \\ b & 0.8 & 0.6 & 0.7 & 0.5 & 0.4 & 0.6 \\ w & 0.5 & 0.5 & 0.9 & 0.8 & 0.8 & 0.4 \end{matrix}$

Since e _X ((G ₁, A) , (F, A)) = e _X ((G ₁ ∨ G ₂, A) , (F, A)) =0.4 and e _X ((G ₂, A) , (F, A)) = e _X ((G ₁ ∧ G₂, A) , (F, A)) =0.4, we have for r > 0.7, $\begin{matrix} {int}_{T_{X}} ((F, A), r) (e) = 0 . 2_{X}, \\ {int}_{T_{X}} ((F, A), r) (b) = 0 . 4_{X}, \\ {int}_{T_{X}} ((F, A), r) (e) = 0 . 4_{X} \end{matrix}$ for 0.5 < r ≤ 0.7, $\begin{matrix} {int}_{T_{X}} ((F, A)) & h^{1} & h^{2} & h^{3} & h^{4} & h^{5} & h^{6} \\ e & 0.3 & 0.4 & 0.4 & 0.2 & 0.2 & 0.4 \\ b & 0.4 & 0.4 & 0.4 & 0.4 & 0.4 & 0.4 \\ w & 0.4 & 0.4 & 0.4 & 0.4 & 0.4 & 0.4 \end{matrix}$ 0.4 < r ≤ 0.5, $\begin{matrix} {int}_{T_{X}} ((F, A)) & h^{1} & h^{2} & h^{3} & h^{4} & h^{5} & h^{6} \\ e & 0.4 & 0.4 & 0.4 & 0.3 & 0.2 & 0.4 \\ b & 0.4 & 0.4 & 0.4 & 0.4 & 0.4 & 0.4 \\ w & 0.4 & 0.4 & 0.4 & 0.4 & 0.4 & 0.4 \end{matrix}$ for r ≤ 0.5, $\begin{matrix} {int}_{T_{X}} ((F, A)) & h^{1} & h^{2} & h^{3} & h^{4} & h^{5} & h^{6} \\ e & 0.5 & 0.4 & 0.4 & 0.3 & 0.2 & 0.4 \\ b & 0.4 & 0.4 & 0.4 & 0.4 & 0.4 & 0.5 \\ w & 0.5 & 0.4 & 0.4 & 0.4 & 0.5 & 0.4 \end{matrix}$

(2) From Theorem 4.4(3), we obtain a soft [0, 1]-fuzzy cotopology $F_{X} : S (X, A) \to [0, 1]$ as follows: $\begin{matrix} F_{X} ((H, A)) \\ = {\begin{matrix} 1, & if (H, A) = (⊥_{X}, A) or (⊤_{X}, A), \\ 0.7, & if (H, A) = (G_{1}^{*}, A) \\ 0.4, & if (H, A) = (G_{2}^{*}, A) \\ 0.5, & if (H, A) = (G_{1}^{*} \land G_{2}^{*}, A) \\ 0.6, & if (H, A) = (G_{1}^{*} \lor G_{2}^{*}, A) \\ 0, & otherwise . \end{matrix} \end{matrix}$

From Theorem 4.4, we obtain a soft implicative [0, 1]-fuzzy closure operator ${cl}_{F_{X}} : S (X, A) \times [0, 1] \to S (X, A)$ as follows: $\begin{matrix} {cl}_{F_{X}} ((F, A), r) \\ = {\begin{matrix} ⋁ (F, A) (x), if r > 0.7, \\ (⋁ (F, A) (x)) if 0.6 < r \leq 0.7, \\ \land (e_{X} (F (A), (G_{1}^{*}, A)) \to (G_{1}^{*}, A)) \\ (⋁ (F, A) (x)) if 0.5 < r \leq 0.6, \\ \land (e_{X} (F (A), (G_{1}^{*}, A)) \to (G_{1}^{*}, A)) \\ \land (e_{X} (F (A), (G_{1}^{*} \lor G_{2}^{*}, A)) \to (G_{1}^{*} \lor G_{2}^{*}, A)) \\ (⋁ (F, A) (x)) if 0.4 < r \leq 0.5, \\ \land (e_{X} (F (A), (G_{1}^{*}, A)) \to (G_{1}^{*}, A)) \\ \land (e_{X} (F (A), (G_{2}^{*}, A)) \to (G_{2}^{*}, A)) \\ \land (e_{X} (F (A), (G_{1}^{*} \lor G_{2}^{*}, A)) \to (G_{1}^{*} \lor G_{2}^{*}, A)) \\ (⋁ (F, A) (x)) if r \leq 0.4, \\ \land (e_{X} (F (A), (G_{1}^{*}, A)) \to (G_{1}^{*}, A)) \\ \land (e_{X} (F (A), (G_{2}^{*}, A)) \to (G_{2}^{*}, A)) \\ \land (e_{X} (F (A), (G_{1}^{*} \lor G_{2}^{*}, A)) \to (G_{1}^{*} \lor G_{2}^{*}, A)) \\ \land (e_{X} (F (A), (G_{1}^{*} \land G_{2}^{*}, A)) \to (G_{1}^{*} \land G_{2}^{*}, A)) \end{matrix} \end{matrix}$

Since (F ^*, A) ∈ S (X, A) as follows: $\begin{matrix} (F^{*}, A) & h^{1} & h^{2} & h^{3} & h^{4} & h^{5} & h^{6} \\ e & 0.5 & 0.5 & 0.3 & 0.6 & 0.8 & 0.6 \\ b & 0.2 & 0.4 & 0.3 & 0.5 & 0.6 & 0.4 \\ w & 0.5 & 0.5 & 0.1 & 0.2 & 0.2 & 0.6 \end{matrix}$

Since $e_{X} ((F^{*}, A), (G_{1}^{*}, A)) = e_{X} ((F^{*}, A), (G_{1}^{*} \land G_{2}^{*}, A)) = 0.4$ and $e_{X} ((F^{*}, A), (G_{2}^{*}, A)) = e_{X} ((F^{*}, A), (G_{1}^{*} \lor G_{2}^{*}, A)) = 1$ , we have for r > 0.6, $\begin{matrix} {cl}_{F_{X}} ((F^{*}, A), r) (e) = 0 . 8_{X}, \\ {cl}_{F_{X}} ((F^{*}, A), r) (b) = 0 . 6_{X}, \\ {cl}_{F_{X}} ((F^{*}, A), r) (e) = 0 . 6_{X} \end{matrix}$ for 0.5 < r ≤ 0.6, $\begin{matrix} {cl}_{T_{X}} ((F^{*}, A)) & h^{1} & h^{2} & h^{3} & h^{4} & h^{5} & h^{6} \\ e & 0.5 & 0.6 & 0.6 & 0.7 & 0.8 & 0.6 \\ b & 0.6 & 0.6 & 0.6 & 0.6 & 0.6 & 0.5 \\ w & 0.6 & 0.6 & 0.6 & 0.6 & 0.5 & 0.6 \end{matrix}$ for r ≤ 0.5, $\begin{matrix} {cl}_{T_{X}} ((F^{*}, A)) & h^{1} & h^{2} & h^{3} & h^{4} & h^{5} & h^{6} \\ e & 0.5 & 0.6 & 0.6 & 0.7 & 0.8 & 0.6 \\ b & 0.6 & 0.6 & 0.6 & 0.6 & 0.6 & 0.5 \\ w & 0.5 & 0.6 & 0.6 & 0.6 & 0.5 & 0.6 \end{matrix}$

Since a ^* ≠ a → 0, we have $\begin{matrix} 0.2 & = {cl}_{F} ((F^{*}, A), 0.7)^{*} (e) (h^{2}) \\ \neq {int}_{T} ((F, A), 0.7) (e) (h^{2}) = 0.4 . \end{matrix}$

Hence, in general, ${int}_{T} ((F, A), r) \neq ({cl}_{F} ((F, A)^{*}, r))^{*}$ .

Example 4.9. Let X and E _X be given as Example 4.8. Define a binary operation ⊙ on [0, 1] by $\begin{matrix} x ⊙ y = max {0, x + y - 1}, \\ x \to y = min {1 - x + y, 1} \\ x \oplus y = min {1, x + y}, x^{*} = 1 - x \end{matrix}$

Then ([0, 1] , ∧ , → , 0, 1) is a complete residuated lattice (ref.[2,8,26 , 2,8,26]). Let B = {b, c, i} ⊂ E _X and Y = {h ¹, h ⁴, h ⁵, h ⁶} ⊂ X. Put (H, B) be a fuzzy soft set as follow: $\begin{matrix} (H, B) & h^{1} & h^{4} & h^{5} & h^{6} \\ b & 0.5 & 0.6 & 0.2 & 0.6 \\ c & 0.1 & 0.5 & 0.5 & 0.6 \\ i & 0.4 & 0.6 & 0.6 & 0.5 \end{matrix}$ $\begin{matrix} (H, B) ⊙ (H, B) & h^{1} & h^{4} & h^{5} & h^{6} \\ b & 0.0 & 0.2 & 0.0 & 0.2 \\ c & 0.0 & 0.0 & 0.0 & 0.2 \\ i & 0.0 & 0.2 & 0.2 & 0.0 \end{matrix}$ $\begin{matrix} (H^{*}, B) & h^{1} & h^{4} & h^{5} & h^{6} \\ b & 0.5 & 0.4 & 0.8 & 0.4 \\ c & 0.9 & 0.5 & 0.5 & 0.4 \\ i & 0.6 & 0.4 & 0.4 & 0.5 \end{matrix}$ $\begin{matrix} (H^{*}, B) \oplus (H^{*}, B) & h^{1} & h^{4} & h^{5} & h^{6} \\ b & 1.0 & 0.8 & 1.0 & 0.8 \\ c & 1.0 & 1.0 & 1.0 & 0.8 \\ i & 1.0 & 0.8 & 0.8 & 1.0 \end{matrix}$ (1) Define a soft L-fuzzy topology $T : S (Y, B) \to L$ as follows $\begin{matrix} T ((F, B)) \\ = {\begin{matrix} 1, & if (F, B) = (⊤_{X}, B), (F, B) = (⊥_{X}, B), \\ 0.6, & if (F, B) = (H, B), \\ 0.3, & if (F, B) = (H, B) ⊙ (H, B), \\ 0, & otherwise . \end{matrix} \end{matrix}$

From Theorem 3.5, we obtain a soft implicative L-fuzzy interior operator ${int}_{T} : S (Y, B) \times L_{⊥} \to L$ as follows $\begin{matrix} {int}_{T} ((F, B), r) \\ = {\begin{matrix} ⋀ (F, B) (x) & if r > 0.6 \\ (⋀ (F, B) (x)), & if r \in (0.3, 0.6] \\ \lor (H, B) ⊙ R ((H, B), (F, A)), \\ (⋀ (F, B) (x)) & if r \in (0, 0.3] \\ \lor (H, B) ⊙ R ((H, B), (F, A)) \\ \lor (H, B) ⊙ (H, B) \\ ⊙ e_{Y} ((H, B) ⊙ (H, B), (F, A)), \end{matrix} \end{matrix}$

Put (F, B) be a fuzzy soft set as follow: $\begin{matrix} (F, B) & h^{1} & h^{4} & h^{5} & h^{6} \\ b & 0.7 & 0.5 & 0.8 & 0.5 \\ c & 0.3 & 0.9 & 0.3 & 0.6 \\ i & 0.5 & 0.6 & 0.4 & 0.8 \end{matrix}$

We obtain $\begin{matrix} e_{Y} ((H, B), (F, B)) = 0.8, \\ e_{Y} ((H ⊙ H, B), (F, B)) = 1 . \end{matrix}$ for r > 0.6, $\begin{matrix} {int}_{T_{Y}} ((F, B), r) (b) = 0 . 5_{Y}, \\ {int}_{T_{Y}} ((F, B), r) (c) = 0 . 3_{Y}, \\ {int}_{T_{Y}} ((F, B), r) (i) = 0 . 4_{Y} \end{matrix}$ $\begin{matrix} (H, B) ⊙ e_{Y} ((H, B), (F, B)) & h^{1} & h^{4} & h^{5} & h^{6} \\ b & 0.3 & 0.4 & 0 & 0.4 \\ c & 0 & 0.3 & 0.3 & 0.4 \\ i & 0.2 & 0.4 & 0.4 & 0.3 \end{matrix}$ for r ≤ 0.6, $\begin{matrix} {int}_{T_{Y}} ((F, B), r) & h^{1} & h^{4} & h^{5} & h^{6} \\ b & 0.5 & 0.5 & 0.5 & 0.5 \\ c & 0.3 & 0.3 & 0.3 & 0.4 \\ i & 0.4 & 0.4 & 0.4 & 0.4 \end{matrix}$

(2) We obtain a soft L-fuzzy cotopology $F : S (Y, B) \to L$ as follows $\begin{matrix} F ((F, B)) \\ = {\begin{matrix} 1, & if (F, B) = (⊤_{X}, B), (F, B) = (⊥_{X}, B), \\ 0.6, & if (F, B) = (H^{*}, B), \\ 0.3, & if (F, B) = (H^{*} ⊙ H^{*}, B), \\ 0, & otherwise . \end{matrix} \end{matrix}$

From Theorem 4.4, we obtain a soft [0, 1]-fuzzy closure operator ${cl}_{F_{Y}} : S (Y, B) \times [0, 1] \to S (Y, A)$ as follows: $\begin{matrix} {cl}_{F_{Y}} ((F, B), r) \\ = {\begin{matrix} ⋁ (F, B) (x), if r > 0.6, \\ (⋁ (F, B) (x)) if 0.3 < r \leq 0.6, \\ \land (e_{Y} ((F, B), (H^{*}, B)) \to (H^{*}, B)) \\ (⋁ (F, B) (x)) if r \leq 0.3, \\ \land (e_{Y} ((F, B), (H^{*}, B)) \to (H^{*}, B)) \\ \land (e_{Y} ((F, B), (H^{*} \oplus H^{*}, B)) \to (H^{*} \oplus H^{*}, B)) \end{matrix} \end{matrix}$

We obtain (F ^*, B) be a fuzzy soft set as follow: $\begin{matrix} (F^{*}, B) & h^{1} & h^{4} & h^{5} & h^{6} \\ b & 0.3 & 0.5 & 0.2 & 0.5 \\ c & 0.7 & 0.1 & 0.7 & 0.4 \\ i & 0.5 & 0.4 & 0.6 & 0.2 \end{matrix}$

We obtain $\begin{matrix} e_{Y} ((F^{*}, B), (H^{*}, B)) = 0.8, \\ e_{Y} ((F^{*}, B), (H^{*} \oplus H^{*}, B)) = 1 . \end{matrix}$ for r > 0.6, $\begin{matrix} {cl}_{F_{Y}} ((F^{*}, B), r) (b) = 0 . 5_{Y}, \\ {cl}_{F_{Y}} ((F^{*}, B), r) (c) = 0 . 7_{Y}, \\ {cl}_{F_{Y}} ((F^{*}, B), r) (i) = 0 . 6_{Y} . \end{matrix}$ $\begin{matrix} e_{Y} ((F^{*}, B), (H^{*}, B)) \to (H^{*}, B) & h^{1} & h^{4} & h^{5} & h^{6} \\ b & 0.7 & 0.6 & 1 & 0.6 \\ c & 1 & 0.7 & 0.7 & 0.6 \\ i & 0.8 & 0.6 & 0.6 & 0.7 \end{matrix}$ for r ≤ 0.6, $\begin{matrix} {cl}_{F_{Y}} ((F^{*}, B), r) & h^{1} & h^{4} & h^{5} & h^{6} \\ b & 0.5 & 0.5 & 0.5 & 0.5 \\ c & 0.7 & 0.7 & 0.7 & 0.6 \\ i & 0.6 & 0.6 & 0.6 & 0.6 \end{matrix}$

Since a ^* = a → 0, we have ${int}_{T} ((F, A), r) = ({cl}_{F} ((F, A)^{*}, r))^{*}$ .

References

Babitha

K.V.

, and Sunil

J.J.

, Soft set relations and functions, Computers and Mathematics with Applications 60 (2010), 1840–1849.

Bělohlávek

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

Cagman

, Karatas

, and Enginoglu

, Soft topology, Comput Math Appl 62(1) (2011), 351–358.

Čimoka

, and Šostak

, L-fuzzy syntopogenous structures, Fuzzy Sets and Systems 232 (2013), 74–97.

Dubois

, and Prade

, Rough fuzzy sets and fuzzy rough sets, Internat J Gen Systems 17(2-3) (1990), 191–209.

Feng

, Liu

, Fotea

V.L.

, and Jun

Y.B.

, Soft sets and soft rough sets, Information Sciences 181 (2011), 1125–1137.

Hájek

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

Höhle

, and Rodabaugh

S.E.

, Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, The Handbooks of Fuzzy Sets Series 3, Kluwer Academic Publishers, Boston, 1999.

Katsaras

A.K.

, Fuzzy proximity spaces, J Math Anal Appl 68 (1979), 100–110.

10.

Katsaras

A.K.

, Fuzzy quasi-proximities and fuzzy quasiuniformities, Fuzzy Sets and Systems 27 (1988), 335–343.

11.

Kharal

, and Ahmad

, Mappings on soft classes, New Math Nat Comput 7(3) (2011), 471–481.

12.

Kim

Y.C.

, and Min

K.C.

, L-fuzzy preproximities andL-fuzzy topologies, Inform Sci 173 (2005), 93–113.

13.

Kim

Y.C.

, and Ko

J.M.

, Soft L-fuzzy quasi-uniformities and soft L-fuzzy topogenous orders, (submit to) J Intelligent and Fuzzy Systems.

14.

Maji

P.K.

, Biswas

, and Roy

, Soft set theory, Computers and Mathematics with Applications 45 (2003), 555–562.

15.

Maji

P.K.

, Biswas

, and Roy

, An application of soft sets in decision making problems, Computers and Mathematics with Applications 44 (2002), 1077–1083.

16.

Maji

P.K.

, and Roy

, A fuzzy set theoretic approach to decision making problems, J Computational and Applied Mathematics 203 (2007), 412–418.

17.

Majumdar

, and Samanta

S.K.

, Generalised fuzzy soft sets, Computaters and Mathematics with Applications 59 (2010), 1425–1432.

18.

Molodtsov

, Soft set theory, Computers and Mathematics with Applications 37 (1999), 19–31.

19.

Pawlak

, Rough sets, Int J Comput Inf Sci 11 (1982), 341–356.

20.

Pawlak

, Rough probability, Bull Pol Acad Sci Math 32 (1984), 607–615.

21.

Qin

, and Pei

, On topological properties of fuzzy rough sets, Fuzzy Sets and Systems 151 (2005), 601–613.

22.

Radzikowska

A.M.

, and Kerre

E.E.

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

23.

Rodríguez

R.O.

, Esteva

, Garcia

, and Godo

, On implicative closure operators in approximate reasing, Internat J Approx Reason 33(2) (2003), 159–184.

24.

Shabir

, and Naz

, On soft topological spaces, Comput Math Appl 61 (2011), 1786–1799.

25.

Tanay

, and Kandemir

M.B.

, Topological structure of fuzzy soft sets, Comput Math Appl 61(10) (2011), 2952–2957.

26.

Turunen

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

27.

Zhan

J.M.

, and Jun

Y.B.

, Soft BL-algebras based on fuzzy sets, Comput Math Appl 59(6) (2010), 2037–2046.

28.

W.Z.

, and Zhang

W.X.

, Neighborhood operator systems and approximations, Information Sciences 144 (2002), 201–217.

29.

Zhao

, and Li

S.-G.

, L-fuzzifying soft topological spaces and L-fuzzifying soft interior operators, Ann Fuzzy Math Inform 5(3) (2013), 493–503.

30.

Zorlutuna

Í.

, Akdag

, Min

W.K.

, and Atmaca

, Remarks on soft topological spaces, Ann Fuzzy Math Inform 3(2) (2012), 171–185.