On L -fuzzy interior operators and L -fuzzy quasi-uniform spaces

Abstract

In this paper, we study the relationships between L-fuzzy interior spaces, L-fuzzy quasi-uniform spaces andL-fuzzy topological spaces. Also, we introduce the concept of L- fuzzy preinterior spaces and the notions of their continuities are investigated. Finally, we give their examples.

Keywords

Complete residuated lattice

1 Introduction

Rodabaugh [20] who was the first to point out the necessity for a unified approach to lattice-valued uniformities in the fuzzy context due to the diversity of reasonable approaches in the literature, e.g., Hutton [9], H $\ddot{o}$ hle [7], Katsaras [10], Lowen [14], Rodabaugh [21], as well as the explicit uniform operator redescribing of the Hutton approach by Kotz $\overset{´}{e}$ [12], the generalization of the Hutton approach by Rodabaugh [20] to include underlying complete quasi-monoidal lattices and the stratification of the Hutton approach by Zhang [23]. H $\ddot{o}$ hle [7] introduced the first step toward unification using t-norm; Kotz $\overset{´}{e}$ [12] followed that step by an additional unification using frames as the underlying lattices. On the other hand, Hájek [6] introduced a complete residuated lattice which is an algebraic structure for many valued logic. Bělohlávek [2] investigated information systems and decision rules in complete residuated lattices. Recently, Guti $\overset{´}{e}$ rrez Garc $\overset{´}{ı}$ a et al. [5] introduced L-valued Hutton quasi-uniformity where a quadruple (L, ≤ , ⊙ , ★) is defined by a GL-monoid (L, ★) dominated by ⊙, a cl-quasi-monoid (L, ≤ , ⊙). They obtained the relation between Hutton, Lowen and H $\ddot{o}$ hle categories. Kim, et al. [11] introduced the notion of fuzzy uniformities as an extension of Lowen over a complete residuated lattice.

It is well known that neighborhood systems and interior operators play an important role in topology and they are very good ways to characterize topology. Many authors [8, 22] have studied neighborhood systems and interior operators in L- fuzzy toplogical spaces. Using Lowen neighborhood system [15], Katsaras proved that every linear fuzzy neighborhood space is uniformizable in the sense of Lowen uniformity [17]. Ramadan et al. [19] investigated the relationships between L- fuzzy quasi-uniform structures and L-fuzzy topologies.

In this paper, we study the relationships betweenL-fuzzy interior spaces, L-fuzzy quasi-uniform spaces and L-fuzzy topological spaces. Also, we introduce the concept of L- fuzzy preinterior spaces and the notions of their continuities are investigated. Finally, we give their examples.

2 Preliminaries

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

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

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

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

In this paper, we assume that L = (L, ∨ , ∧ , ⊙ , → , ⊥, ⊤) be a complete residuated lattice.

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

(1) If y ≤ z, then x ⊙ y ≤ x ⊙ z.

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

(3) x→ y = ⊤ iff x ≤ y.

(4) x→ ⊤ = ⊤ and ⊤ → x = x.

(5) x ⊙ y ≤ x ∧ y .

(6) x ⊙ (⋁ _i∈Γy_i) = ⋁ _i∈Γ (x ⊙ y_i) and (⋁ _i∈Γx_i) ⊙ y = ⋁ _i∈Γ (x_i ⊙ y) .

(7) x → (⋀ _i∈Γy_i) = ⋀ _i∈Γ (x → y_i) and (⋁ _i∈Γx_i)→y = ⋀ _i∈Γ (x_i → y) .

(8) ⋁_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).

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

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

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

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

A lattice L is called s-compact if ⋁ _j∈Γc_j ≥ a for c_j, a ∈ L, there exists j_∘ ∈ Γ such that c_{j
_∘} ≥ a.

For α ∈ L, λ ∈ L^X, we denote (α → λ) , (α⊙λ) , α_X ∈ L^X as (α → λ) (x) = α → λ (x) , (α ⊙ λ) (x) = α ⊙ λ (x) , α_X (x) = α.

Definition 2.3. [2,16, 2,16] Let X be a set. A mapping R : X × X → L is called an L-partial order if it satisfies the following conditions:

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

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

(E3) antisymmetric if R (x, y) = R (y, x) =⊤, then x = y.

Lemma 2.4. [2,4, 2,4] For a given set X, define a binary mapping S : L^X × L^X → L by $S (λ, μ) = ⋀_{x \in X} (λ (x) \to μ (x)) .$

Then, for each λ, μ, ρ, ν ∈ L^X, and α ∈ L, the following properties hold.

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

(2) λ ≤ μ iff S (λ, μ)≥ ⊤,

(3) If λ ≤ μ, then S (ρ, λ) ≤ S (ρ, μ) and S (λ, ρ) ≥ S (μ, ρ),

(4) S (λ, μ) ⊙ S (ν, ρ) ≤ S (λ ⊙ ν, μ ⊙ ρ) ,

(5) S (μ, ρ) ≤ S (λ, μ) → S (λ, ρ) .

Lemma 2.5. [4] Let φ : X → Y be an ordinary mapping. Define φ^→ : L^X → L^Y and φ^← : L^Y → L^X by $φ^{\to} (λ) (y) = ⋁_{φ (x) = y} λ (x), \forall λ \in L^{X}, y \in Y,$ $φ^{\leftarrow} (μ) (x) = μ (φ (x)) = μ \circ φ (x), \forall μ \in L^{Y} .$

Then for λ, μ ∈ L^X and ρ, ν ∈ L^Y, $S (λ, μ) \leq S (φ^{\to} (λ), φ^{\to} (μ)),$ $S (ρ, ν) \leq S (φ^{\leftarrow} (ρ), φ^{\leftarrow} (ν)),$ and the equalities hold if φ is bijective.

Definition 2.6. [8,22, 8,22] A map $T : L^{X} \to L$ is called an L-fuzzy topology on X if it satisfies the following conditions:

(LO1) $T (⊥_{X}) = T (⊤_{X}) = ⊤$ ,

(LO2) $T (λ ⊙ μ) \geq T (λ) ⊙ T (μ), \forall λ, μ \in L^{X}$ ,

(LO3) $T (⋁_{i} λ_{i}) \geq ⋀_{i} T (λ_{i}), \forall {λ_{i}}_{i \in Γ} \subseteq L^{X}$ .

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

(R) $T (α ⊙ λ) \geq T (λ)$ for all λ ∈ L^X and α ∈ L.

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

Definition 2.7. [8] A map $I : L^{X} \times L_{⊥} \to L^{X}, L_{⊥} = L - {⊥}$ is called an L- fuzzy interior operator on X if $I$ satisfies the following conditions:

(LI1) $I (⊤_{X}, r) = ⊤_{X}$ , and $I (⊥_{X}, r) = ⊥_{X}$ ,

(LI2) $I (λ, r) \leq λ$ , or equivalently, $S (I (λ, r),$ λ)≥ ⊤ for all λ ∈ L^X,

(LI3) $S (λ, μ) \leq S (I (λ, r), I (μ, r))$ for all λ, μ ∈ L^X,

(LI4) If r ≤ s, then $I (λ, s) \leq I (λ, r)$ ,

(LI5) $I (λ ⊙ μ, r ⊙ s) \geq I (λ, r) ⊙ I (μ, s)$ .

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

(LI6) $I (I (λ, r), r) = I (λ, r), \forall λ \in L^{X}, r \in L_{⊥}$ .

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

(R) $I (α ⊙ λ, r) \geq α ⊙ I (λ, r)$ .

Let $(X, I_{X})$ and $(X, I_{Y})$ be two L- fuzzy interior spaces. A map φ : X → Y is called $I$ -map if $φ^{\leftarrow} (I_{Y} (μ, r)) \leq I_{X} (φ^{\leftarrow} (μ), r), \forall μ \in L^{Y}, r \in L_{⊥} .$

Definition 2.8. [11] A map $U : L^{X \times X} \to L$ is called an L-fuzzy quasi-uniformity on X iff the following conditions hold.

(LU1) There exists u ∈ L^X×X such that $U (u) = ⊤$ .

(LU2) If v ≤ u, then $U (v) \leq U (u)$ .

(LU3) For every $u, v \in L^{X \times X}, U (u ⊙ v) \geq U (u)$ $⊙ U (v)$ .

(LU4) If $U (u) \neg = ⊥$ then ⊤_▵ ≤ u where $⊤_{▵} (x, y) = {\begin{matrix} ⊤, & if x = y \\ ⊥, & if x \neq y, \end{matrix}$

(LU5) $U (u) \leq ⋁ {U (v) | v \circ v \leq u}, \forall u \in L^{X \times X},$ where $v \circ v (x, y) = ⋁_{z \in X} v (x, z) ⊙ v (z, y), \forall x, y \in X .$

An L-fuzzy quasi-uniformity $U$ on X is said to be stratified if

(R) $U (α ⊙ u) \geq α ⊙ U (u), \forall u \in L^{X \times X}$ .

The pair $(X, U)$ is called an L-fuzzy quasi-uniform space.

Let $(X, U)$ and $(Y, V)$ be L-fuzzy quasi-uniform spaces, and φ : X → Y be a mapping. Then φ is said to be L-fuzzy uniformly continuous if

$V (v) \leq U ((φ \times φ)^{\leftarrow} (v))$ , for every v ∈ L^Y×Y.

Remark 2.9. Let $(X, U)$ be an L-fuzzy quasi- uniform space. By (LU1) and (LU2), we have $U (⊤_{X \times X}) = ⊤$ because u ≤ ⊤ _X×X for all u ∈ L^X×X.

3 L- fuzzy interior space and L-fuzzy quasi- uniform space

Theorem 3.1. Let $(X, U)$ be an L-fuzzy quasi-uniform space. Define a map $I^{U} : L^{X} \times L_{⊥} \to L^{X}$ by: $I^{U} (λ, r) (x) = ⋁_{U (u) \geq r} S (u [x], λ), \forall λ \in L^{X}, x \in X,$ where u [x] (y) = u (y, x). Then the following properties hold.

(1) $(X, I^{U})$ is an L- fuzzy interior space.

(2) $I^{U} (I^{U} (λ, r_{1}), r_{1}) \geq I^{U} (λ, r)$ for each r₁ < r.

(3) If L is s-compact, then $I^{U}$ is a topological.

(4) If $U$ is stratified, then $I^{U}$ is also stratified.

Proof. (1) (LI1),(LI2) and (LI4) are easily proved.

(LI3) $\begin{matrix} S (I^{U} (λ, r), I^{U} (ρ, r)) \\ = ⋀_{x \in X} (I^{U} (λ, r) (x) \to I^{U} (ρ, r) (x)) \\ = ⋀_{x \in X} (⋁_{U (u) \geq r} S (u [x], λ) \\ \to ⋁_{U (v) \geq r} S (v [x], ρ)) \\ \geq ⋀_{x \in X} ⋀_{U (u) \geq r} (S (u [x], λ) \\ \to S (u [x], ρ)) (by Lemma 2.2 (8)) \\ \geq ⋀_{U (u) \geq r} (S (u [x], λ) \to S (u [x], ρ)) \\ \geq S (λ, ρ) . (by Lemma 2.4 (5)) \end{matrix}$

(LI5) By Lemma 2.4 (4), we have $\begin{matrix} I^{U} (λ, r) (x) ⊙ I^{U} (ρ, s) (x) \\ = (⋁_{U (u) \geq r} S (u [x], λ)) ⊙ (⋁_{U (v) \geq s} S (v [x], ρ)) \\ \leq ⋁_{U (u) ⊙ U (v) \geq r ⊙ s} (S (u [x], λ) ⊙ S (v [x], ρ)) \\ \leq ⋁_{U (u ⊙ v) \geq r ⊙ s} (S (u ⊙ v) [x], λ ⊙ ρ) \\ \leq ⋁_{U (w) \geq r ⊙ s} S (w [x], λ ⊙ ρ) = I^{U} (λ ⊙ ρ, r ⊙ s) (x) . \end{matrix}$

(2) For $U (u) \geq r$ and r > r₁, by (LU5), there exists v ∈ L^X×X such that $U (v) \geq r_{1}, v \circ v \leq u$ . $\begin{matrix} I^{U} (λ, r) (x) = ⋁_{U (u) \geq r} S (u [x], λ) \\ = ⋁_{U (u) \geq r} ⋀_{y \in X} (u (y, x) \to λ (y)) \\ \leq ⋁_{U (v) \geq r_{1}} ⋀_{y \in X} ((v \circ v) (y, x) \to λ (y)) \\ = ⋁_{U (v) \geq r_{1}} ⋀_{y \in X} ((⋁_{z \in X} v (z, x) ⊙ v (y, z)) \\ \to λ (y)) \\ = ⋁_{U (v) \geq r_{1}} ⋀_{y \in X} ⋀_{z \in X} (v (z, x) ⊙ v (y, z)) \\ \to λ (y)) \\ = ⋁_{U (v) \geq r_{1}} ⋀_{y \in X} ⋀_{z \in X} (v (z, x) \to (v (y, z) \\ \to λ (y))) (by Lemma 2.2 (12)) \\ = ⋁_{U (v) \geq r_{1}} ⋀_{z \in X} (v (z, x) \to ⋀_{y \in X} (v (y, z) \\ \to λ (y))) . \end{matrix}$ Put ρ (z) = ⋀ _y∈X (v (y, z) → λ (y)). Then $ρ (z) \leq I^{U} (λ, r_{1}) (z)$ for all z ∈ X. Thus, $\begin{matrix} I^{U} (λ, r) (x) \\ = ⋁_{U (v) \geq r_{1}} ⋀_{z \in X} (v (z, x) \to ρ (z)) \\ \leq ⋁_{U (v) \geq r_{1}} ⋀_{z \in X} (v (z, x) \to I^{U} (λ, r_{1}) (z)) \\ = ⋁_{U (v) \geq r_{1}} S (v [x], I^{U} (λ, r_{1})) \\ = I^{U} (I^{U} (λ, r_{1}), r_{1}) (x) \end{matrix}$

This implies that $(X, I^{U})$ is an L-fuzzy interior space.

(3) By (LU5), $r \leq U (u) \leq ⋁ {U (u) ∣ v \circ v \leq u}$ . Since L is s-compact, there exists v₁ ∈ L^X×X with v₁ ∘ v₁ ≤ u such that $U (v_{1}) \geq r$ . By (2), the result holds.

(4) For any λ ∈ L^X, α ∈ L and 𝒰(u) ≥ r, we have by Lemma 2.4 (4) and (LI3): $S (ℐ^{U} (λ, r), ℐ^{U} (α ⊙ λ, r)) \geq (λ, α ⊙ λ) \geq α .$ That is, $α ⊙ ℐ^{U} (λ, r) \leq ℐ^{U} (α ⊙ λ, r)$

Theorem 3.2. Let $(X, U)$ be an L-fuzzy quasi- uniform space. Define the mappings $T_{U} : L^{X} \to L$ by:

$T_{U} (λ) = ⋁ {r \in L ∣ S (λ, I^{U} (λ, r)) = ⊤} .$

Then, $T_{U}$ is L-fuzzy topology on X. If $U$ is stratified, then $T_{U}$ is enriched L-fuzzy topology on X.

Proof. (LO1) Since $⊤_{X} = I^{U} (⊤_{X}, r)$ and $⊥_{X} = I^{U} (⊥_{X}, r)$ , we have $T_{U} (⊤_{X}) = ⊤, T_{U} (⊥_{X}) = ⊤ .$

(LO2) By Theorem 3.1(1) and Lemma 2.4, we have $\begin{matrix} S (λ_{1}, I^{U} (λ_{1}, r)) ⊙ S (λ_{2}, I^{U} (λ_{2}, s)) \\ \leq S (λ_{1} ⊙ λ_{2}, I^{U} (λ_{1}, r) ⊙ I^{U} (λ_{2}, s)) \\ \leq S (λ_{1} ⊙ λ_{2}, I^{U} (λ_{1} ⊙ λ_{2}, r ⊙ s)) \end{matrix}$

If $S (λ_{1}, I^{U} (λ_{1}, r)) = ⊤$ and $S (λ_{2}, I^{U} (λ_{2}, s)) = ⊤$ , then $S (λ_{1} ⊙ λ_{2}, I^{U} (λ_{1} ⊙ λ_{2}, r ⊙ s)) = ⊤$ . Thus, $T_{U} (λ_{1} ⊙ λ_{2}) \geq T_{U} (λ_{1}) ⊙ T_{U} (λ_{2}) .$

(LO3) For a family of {λ_i | i ∈ Γ} ⊆ L^X, we have $\begin{matrix} S (⋁_{i \in Γ} λ_{i}, I^{U} (⋁_{i \in Γ} λ_{i}, r)) \\ \geq ⋀_{i \in Γ} S (λ_{i}, I^{U} (⋁_{i \in Γ} λ_{i}, r)) \\ \geq ⋀_{i \in Γ} S (λ_{i}, I^{U} (λ_{i}, r)) = ⊤ . \end{matrix}$

Hence, $T_{U}$ is an L-fuzzy topology on X.

(R) Let $U$ be stratified. For α ∈ L_⊥ and λ ∈ L^X, we have $\begin{matrix} S (α ⊙ λ, I^{U} (α ⊙ λ, r)) \\ \geq S (α ⊙ λ, α ⊙ I^{U} (λ, r)) \\ \geq S (λ, I^{U} (λ, r)) . \end{matrix}$

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

Example 3.3. Let (L = [0, 1] , ⊙ , →) be a complete residuated lattice defined as: $x ⊙ y = (x + y - 1) \lor 0, x \to y = (1 - x + y) \land 1 .$

Let X = {x, y, z} be a set and w ∈ L^X×X such that $w = (\begin{matrix} 1 & 0.5 & 0.3 \\ 0.7 & 1 & 0.5 \\ 0.6 & 0.6 & 1 \end{matrix}) w ⊙ w = (\begin{matrix} 1 & 0 & 0 \\ 0.4 & 1 & 0 \\ 0.2 & 0.2 & 1 \end{matrix}) .$

Define $U : L^{X \times X} \to L$ as follows: $U (u) = {\begin{matrix} 1, & if u = ⊤_{X \times X}, \\ 0.6, & if w \leq u \neq ⊤_{X \times X}, \\ 0.3, & if w ⊙ w \leq u ≱ w, \\ 0, & otherwise . \end{matrix}$

(1) Since $0.3 = U (w ⊙ w) \geq U (w) ⊙ U (w) = 0.2$ and w ∘ w = w, (w ⊙ w) ∘ (w ⊙ w) = (w ⊙ w), $U$ is an L-fuzzy quasi-uniformity on X,

(2) $I^{U} (λ, r) (x) = ⋁_{U (u) \geq r} S (u [x], λ), \forall λ \in L^{X}, x \in X .$

If r > 0.6, then $\begin{matrix} I^{U} (λ, r) (x) = S (⊤_{X \times X} [x], λ) = λ (x) \land λ (y) \land λ (z), \\ I^{U} (λ, r) (y) = S (⊤_{X \times X} [y], λ) = λ (x) \land λ (y) \land λ (z), \\ I^{U} (λ, r) (z) = S (⊤_{X \times X} [z], λ) = λ (x) \land λ (y) \land λ (z) . \end{matrix}$

If 0.3 < r ≤ 0.6, then $U (w) = 0.6$ . $\begin{matrix} I^{U} (λ, r) (x) = S (w [x], λ) \\ = λ (x) \land (0.3 + λ (y)) \land (0.4 + λ (z), \\ I^{U} (λ, r) (y) = (0.5 + λ (x)) \land λ (y) \land (0.4 + λ (z), \\ I^{U} (λ, r) (z) = (0.7 + λ (x)) \land (0.5 + λ (y)) \land λ (z) . \end{matrix}$

If 0 < r ≤ 0.3, then $U (w ⊙ w) = 0.3$ . $\begin{matrix} I^{U} (λ, r) (x) = S (w [x], λ) \\ = λ (x) \land (0.6 + λ (y)) \land (0.8 + λ (z), \\ I^{U} (λ, r) (y) = λ (y) \land (0.8 + λ (z), \\ I^{U} (λ, r) (z) = λ (z) . \end{matrix}$

(3) Since $T_{U} (λ) = ⋁ {r \in L ∣ S (λ, I^{U} (λ, r)) = ⊤} = ⋁ {r \in L ∣ λ = I^{U} (λ, r))}$ from Theorem 3.2 and Lemma 2.4(2), we have $\begin{matrix} T_{U} (λ) \\ = {\begin{matrix} 1, & if λ = α_{X}, \\ 0.6, & if λ (x) \leq λ (y) + 0.3, λ (x) \leq λ (z) + 0.4, \\ λ (y) \leq λ (x) + 0.5, λ (y) \leq λ (z) + 0.4, \\ λ (z) \leq λ (x) + 0.7, λ (z) \leq λ (y) + 0.5, \\ 0.3, & if λ (x) \leq λ (y) + 0.6, λ (x) \leq λ (z) + 0.8, \\ λ (y) \leq λ (z) + 0.8 \\ 0, & otherwise . \end{matrix} \end{matrix}$

Similarly, we obtain: For λ = (0.7, 0.5, 0.4) , $T_{U} (λ) = 0.6$ ,

$λ = (0.1, 0.5, 0.4), T_{U} (λ) = 0.6$ ,

$λ = (0.2, 0.9, 0.8), T_{U} (λ) = 0.3$ ,

Theorem 3.4. If $φ : (X, U) \to (Y, V)$ is L-fuzzy uniformly continuous, then

$φ : (X, I^{U}) \to (Y, I^{V})$ is I-map.

Proof. First we show that φ^← (v [φ (x)]) = (φ × φ) ^← (v) [x] from $\begin{matrix} φ^{\leftarrow} (v [φ (x)]) (z) = v [φ (x)] (φ (z)) = v (φ (z), φ (x)) \\ = (φ \times φ)^{\leftarrow} (v) (z, x) = (φ \times φ)^{\leftarrow} (v) [x] (z) . \end{matrix}$

Thus, by Lemma 2.5, we have $\begin{matrix} S (v [φ (x)], λ) & \leq & S (φ^{\leftarrow} (v [φ (x)]), φ^{\leftarrow} (λ)) \\ = & S ((φ \times φ)^{\leftarrow} (v) [x], φ^{\leftarrow} (λ)) . \end{matrix}$ $\begin{matrix} φ^{\leftarrow} (I^{V} (λ, r)) (x) = I^{V} (λ, r) (φ (x)) \\ = & ⋁_{V (v) \geq r} S (v [φ (x)], λ) \\ \leq & ⋁_{V (v) \geq r} S ((φ \times φ)^{\leftarrow} (v) [x], φ^{\leftarrow} (λ)) \\ \leq & ⋁_{U ((φ \times φ)^{\leftarrow} (v) \geq r} S ((φ \times φ)^{\leftarrow} (v) [x], φ^{\leftarrow} (λ)) \\ \leq & I^{U} (φ^{\leftarrow} (λ), r) (x) . \end{matrix}$

Theorem 3.5. If $φ : (X, I^{U}) \to (Y, I^{V})$ is I-map, then $φ : (X, T_{U}) \to (Y, T_{V})$ is L-fuzzy continuous.

Proof. By Theorem 3.4 and Lemma 2.5, we have $T_{U} (φ^{\leftarrow} (λ)) \geq T_{V} (λ)$ from: $\begin{matrix} S (φ^{\leftarrow} (λ), I^{U} (φ^{\leftarrow} (λ), r)) \\ \geq & S (φ^{\leftarrow} (λ), φ^{\leftarrow} (I^{V} (λ, r))) \\ \geq & S (λ, I^{V} (λ, r)) . \end{matrix}$

4 L-fuzzy preinterior space and L-fuzzy quasi-uniform space

Definition 4.1. A map $I : L^{X} \to L^{X}$ is called anL-fuzzy preinterior operator if it satisfies the following conditions:

(LI1) $I (⊤_{X}) = ⊤$ and $I (⊥_{X}) = ⊥$ ,

(LI2) $I (λ ⊙ μ) \geq I (λ) ⊙ I (μ)$ for each λ, μ ∈ L^X,

(LI3) If λ ≤ μ, then $I (λ) \leq I (μ)$ ,

(LI4) $I (λ) \leq λ (x)$ for all λ ∈ L^X.

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

An L-fuzzy preinterior space is called stratified if

(R) $I (α ⊙ λ) \geq α ⊙ I (λ)$ for all λ ∈ L^X and α ∈ L.

Let $(X, I_{1})$ and $(Y, I_{2})$ be two L-fuzzy preinterior spaces. A mapping φ : X → Y is said to be I-map if $φ^{\leftarrow} (I_{2} (λ)) \leq I_{1} (φ^{\leftarrow} (λ))$ for each λ ∈ L^Y .

Theorem 4.2. Let $(X, U)$ be an L-fuzzy quasi-uniform space. Define a map $I^{U} : L^{X} \to L^{X}$ by: $I^{U} (λ) (x) = ⋁_{u} U (u) ⊙ S (u [x], λ), \forall λ \in L^{X}, x \in X,$ where u [x] (y) = u (y, x). Then, $(X, I^{U})$ is an L-fuzzy preinterior space. If $U$ is stratified, then $I^{U}$ is also stratified.

Proof. (LI1) For $U (u) \neg = ⊥$ , ⊤ _▵ ≤ u. Then $\begin{matrix} I^{U} (⊥_{X}) (x) = ⋁_{u} U (u) ⊙ S (u [x], ⊥_{X}) \\ \leq ⋁_{u} (U (u) ⊙ (u (x, x) \to ⊥)) \\ = ⋁_{u} (U (u) ⊙ (⊤_{▵} (x, x) \to ⊥)) = ⊥ . \end{matrix}$

Hence, $I^{U} (⊥_{X}) = ⊥$ . Also, $I^{U} (⊤_{X}) = ⊤$ , because $\begin{matrix} I^{U} (⊤_{X}) (x) \geq U (⊤_{X \times X}) ⊙ ⋀_{y \in X} (⊤_{X \times X} (x, y) \to \\ ⊤_{X} (y)) = ⊤ . \end{matrix}$

(LI2) By Lemma 2.4 (4), we have $\begin{matrix} I^{U} (λ) (x) ⊙ I^{U} (μ) (x) \\ = (⋁_{u} U (u) ⊙ S (u [x], λ)) ⊙ (⋁_{u} U (v) ⊙ S (v [x], μ)) \\ = ⋁_{u, v} U (u) ⊙ U (v) ⊙ S (u [x], λ) ⊙ S (v [x], μ) \\ \leq ⋁_{u, v} U (u ⊙ v) ⊙ S ((u ⊙ v) [x], λ ⊙ μ) \\ \leq ⋁_{w} U (w) ⊙ S (w [x], λ ⊙ μ) = I^{U} (λ ⊙ μ) (x) . \end{matrix}$

(LI3) By Lemma 2.4 (3), we have $\begin{matrix} I^{U} (λ) (x) & = ⋁_{u} U (u) ⊙ S (u [x], λ) \\ \leq ⋁_{u} U (u) ⊙ S (u [x], μ) = I^{U} (μ) (x) . \end{matrix}$

(LI4) For $U (u) \neg = ⊥$ , ⊤ _▵ ≤ u. $\begin{matrix} I^{U} (λ) (x) & = ⋁_{u} U (u) ⊙ ⋀_{y \in X} (u (y, x) \to λ (y)) \\ \leq ⋁_{u} {U (u) ⊙ (u (x, x) \to λ (x))} \leq λ (x) . \end{matrix}$

This implies that $(X, I^{U})$ is an L-fuzzy preinterior space.

(R) $\begin{matrix} α ⊙ I^{U} (λ) (x) = α ⊙ ⋁_{u} U (u) ⊙ S (u [x], λ) \\ = ⋁_{u} α ⊙ U (u) ⊙ S (α, α) ⊙ S (u [x], λ) \\ \leq ⋁_{u} U (α ⊙ u) ⊙ S (α ⊙ u [x], α ⊙ λ) \\ \leq I^{U} (α ⊙ λ) (x) . \end{matrix}$

Theorem 4.3. (1) The L-fuzzy preinterior operator $I^{U}$ can be constructed from the cuts $U_{α}, α > ⊥$ , of the L-fuzzy quasi-uniformity by using the equality $I^{U} (λ) = ⋁_{α \in L} α ⊙ I^{U} (λ, α),$ where $I^{U} (λ, α)$ is defined from (see Theorem 3.1) $I^{U} (λ, α) (x) = ⋁_{U (u) \geq α} (S (u [x], λ) .$

(2) $I^{U} (λ) \leq I^{U} (I^{U} (λ, U (v)) .$

Proof. (1) If for some y ∈ X we have A (y) ≥ α, then we can write A (y) ⊙ B (y) ≥ α ⊙ B (y) and $⋁ {A (x) ⊙ B (x) | A (x) \geq α} \geq ⋁ {α ⊙ B (y) | A (x) \geq α} .$

Suppose $⋁ {A (x) ⊙ B (x) | x \in X} ≰ ⋁_{α \in L} ⋁ {α ⊙ B (x) | A (x) \geq α} .$

There exists x₀ ∈ X such that $A (x_{0}) ⊙ B (x_{0}) ≰ ⋁_{α \in L} ⋁ {α ⊙ B (x) | A (x) \geq α} .$

It is a contradiction. Hence $⋁ {A (x) ⊙ B (x) | x \in X} = ⋁_{α \in L} ⋁ {α ⊙ B (x) | A (x) \geq α} .$

Applying this equality to the formula giving $I^{U} (λ)$ , we obtain $\begin{matrix} I^{U} (λ) (x) = & ⋁_{α \in L} {⋁ α ⊙ S (u [x], λ) | U (u) \geq α} \\ = & ⋁_{α \in L} α ⊙ I^{U} (λ, α) (x) . \end{matrix}$

(2) For u ∈ L^X×X and λ ∈ L^X, we have $\begin{matrix} I^{U} (λ) (x) = ⋁_{u} U (u) ⊙ S (u [x], λ) \\ = ⋁_{u} {U (u) ⊙ ⋀_{y \in X} (u (y, x) \to λ (y))} \\ \leq ⋁_{v} {U (v) ⊙ ⋀_{y \in X} ((v \circ v) (y, x) \to λ (y))} (by (LU 5)) \\ = ⋁_{v} {U (v) ⊙ ⋀_{y \in X} ((⋁_{z \in X} v (z, x) ⊙ v (y, z)) \\ \to λ (y))} \\ = ⋁_{v} {U (v) ⊙ ⋀_{y \in X} ⋀_{z \in X} ((v (z, x) ⊙ v (y, z)) \\ \to λ (y))} \\ = ⋁_{v} {U (v) ⊙ ⋀_{y \in X} ⋀_{z \in X} (v (z, x) \to (v (y, z) \\ \to λ (y))} (by Lemma 2.2 (12)) \\ = ⋁_{v} {U (v) ⊙ ⋀_{z \in X} (v (z, x) \to ⋀_{y \in X} (v (y, z) \\ \to λ (y))} . \end{matrix}$ Put ρ (z) = ⋀ _y∈X (v (y, z) → λ (y)). Then $ρ (z) \leq I^{U} (λ, U (v)) (z)$ for all z ∈ X. Thus, $\begin{matrix} I^{U} (λ) (x) \\ = & ⋁_{v} {U (v) ⊙ ⋀_{z \in X} (v (z, x) \to ρ (z)) | ρ (z) \\ \leq & I^{U} (λ, U (v)) (z)} \\ \leq & ⋁_{v} {U (v) ⊙ ⋀_{z \in X} (v (z, x) \to I^{U} (λ, U (v)) (z))} \\ \leq & I^{U} (I^{U} (λ, U (v)) (x) . \end{matrix}$

Theorem 4.4. Let $(X, U)$ be an L-fuzzy quasi-uniform space. Define a map $T_{U} : L^{X} \to L$ by: $T_{U} (λ) = S (λ, I^{U} (λ)) .$

Then, $T_{U}$ is an L-fuzzy topology on X. If $U$ is stratified, then $T_{U}$ is an enriched L-fuzzy topology.

Proof. (LO1) $T_{U} (⊤_{X}) = ⋀_{x \in X} (⊤_{X} (x) \to I^{U} (⊤_{X}) (x)) = ⊤ \to ⊤ = ⊤,$ $T_{U} (⊥_{X}) = ⋀_{x \in X} (⊥_{X} (x) \to I^{U} (⊥_{X}) (x)) = ⊥ \to ⊥ = ⊤ .$ (LO2) By Lemma 2.2(12), we havre $\begin{matrix} T_{U} (λ ⊙ μ) \\ = ⋀_{x \in X} ((λ ⊙ μ) (x) \to I^{U} (λ ⊙ μ) (x)) \\ \geq ⋀_{x \in X} ((λ ⊙ μ) (x) \to (I^{U} (λ) ⊙ I^{U} (μ)) (x)) \\ \geq ⋀_{x \in X} (λ (x) \to I^{U} (λ) (x)) ⊙ ⋀_{x \in X} (μ (x) \to I^{U} (μ) (x)) \\ = T_{U} (λ) ⊙ T_{U} (μ) . \end{matrix}$

(LO3) By Lemma 2.2 (8),we have $\begin{matrix} T_{U} (⋁_{i} λ_{i}) & = ⋀_{x \in X} ((⋁_{i} λ_{i}) (x) \to I^{U} (⋁_{i} λ_{i}) (x)) \\ \geq ⋀_{x \in X} ((⋁_{i} λ_{i}) (x) \to ⋁_{i} I^{U} (λ_{i}) (x)) \\ \geq ⋀_{i} ⋀_{x \in X} (λ_{i} (x) \to I^{U} (λ_{i}) (x)) = ⋀_{i} T_{U} (λ_{i}) . \end{matrix}$

(R) By Lemma 2.2 (12)and Theorem 4.2 (2), we have $\begin{matrix} T_{U} (α ⊙ λ) = ⋀_{x \in X} ((α ⊙ λ) (x) \to I^{U} (α ⊙ λ) (x)) \\ \geq ⋀_{x \in X} ((α ⊙ λ) (x) \to (α ⊙ I^{U} (λ)) (x)) \\ \geq ⋀_{x \in X} (λ (x) \to I^{U} (λ) (x)) = T_{U} (λ) . \end{matrix}$

Theorem 4.5. If $φ : (X, U) \to (Y, V)$ is L-fuzzy uniformly continuous, then $φ : (X, I^{U}) \to (Y, I^{V})$ is I-map.

Proof. Since v [φ (x)] = (φ × φ) ^← (v) [x], we have $\begin{matrix} φ^{\leftarrow} (I^{V} (λ)) (x) = I^{V} (λ) (φ (x)) = ⋁_{v} V (v) ⊙ S (v [φ (x)], λ) \\ \leq ⋁_{v} V (v) ⊙ S ((φ \times φ)^{\leftarrow} (v) [x], φ^{\leftarrow} (λ)) \\ \leq ⋁_{u} U ((φ \times φ)^{\leftarrow} (v)) ⊙ S ((φ \times φ)^{\leftarrow} (v) [x], φ^{\leftarrow} (λ)) \\ \leq I^{U} (φ^{\leftarrow} (λ)) (x) . \end{matrix}$

Theorem 4.6. If $φ : (X, I^{U}) \to (Y, I^{V})$ is I-map, then $φ : (X, T_{U}) \to (Y, T_{V})$ is L- fuzzy continuous.

Proof. By Lemma 2.2, we have $\begin{matrix} T_{V} (λ) \to T_{U} (φ^{\leftarrow} (λ)) \\ = ⋀_{y \in Y} (λ (y) \to I^{V} (λ) (y)) \\ \to ⋀_{x \in X} (φ^{\leftarrow} (λ) (x) \to I^{U} (φ^{\leftarrow} (λ)) (x)) \\ \geq ⋀_{x \in X} (φ^{\leftarrow} (λ) (x) \to φ^{\leftarrow} (I^{V} (λ)) (x)) \\ \to ⋀_{x \in X} (φ^{\leftarrow} (λ) (x) \to I^{U} (φ^{\leftarrow} (λ)) (x)) \\ \geq ⋀_{x \in X} (φ^{\leftarrow} (λ) (x) \to φ^{\leftarrow} (I^{V} (λ) (x)) \\ \to (φ^{\leftarrow} (λ) (x) \to I^{U} (φ^{\leftarrow} (λ)) (x)) \\ \geq ⋀_{x \in X} (φ^{\leftarrow} (I^{V} (λ)) (x) \to I^{U} (φ^{\leftarrow} (λ)) (x)) . \end{matrix}$

Thus, if $φ^{\leftarrow} (I^{V} (λ)) \leq I^{U} (φ^{\leftarrow} (λ))$ , then $T_{V} (λ) \leq T_{U} (φ^{\leftarrow} (λ))$ .

Example 4.7. Let (L = [0, 1] , ⊙ , →) be a complete residuated lattice defined by: $x ⊙ y = (x + y - 1) \lor 0, x \to y = (1 - x + y) \land 1 .$

Let X = {x, y, z} be a set and w, w ⊙ w ∈ L^X×X such that $w = (\begin{matrix} 1 & 0.7 & 0.6 \\ 0.7 & 1 & 0.3 \\ 0.6 & 0.3 & 1 \end{matrix}), w ⊙ w = (\begin{matrix} 1 & 0.4 & 0.2 \\ 0.4 & 1 & 0 \\ 0.2 & 0 & 1 \end{matrix}) .$

Define $U : L^{X \times X} \to L$ as follows: $U (u) = {\begin{matrix} 1, & if u = 1_{X \times X}, \\ 0.6, & if w \leq u \neq 1_{X \times X}, \\ 0.3, & if w ⊙ w \leq u ≱ w, \\ 0, & otherwise . \end{matrix}$

Since $0.3 = U (w ⊙ w) \geq U (w) ⊙ U (w) = 0.2$ and w ∘ w = w, (w ⊙ w) ∘ (w ⊙ w) = (w ⊙ w), $U$ is an L-fuzzy quasi-uniformity on X.

$I^{U} (λ) (x) = ⋁_{u} U (u) ⊙ S (u [x], λ), \forall λ \in L^{X}, x \in X .$ $\begin{matrix} I^{U} (λ) (x) = ⋁_{u} U (u) ⊙ S (u [x], λ) \\ = (λ (x) \land λ (y) \land λ (z)) \lor \\ (0.6 ⊙ (λ (x) \land (0.3 + λ (y)) \land (0.4 + λ (z))) \\ \lor (0.3 ⊙ (λ (x) \land (0.6 + λ (y)) \land (0.8 + λ (z))), \end{matrix}$

$\begin{matrix} I^{U} (λ) (y) = (λ (x) \land λ (y) \land λ (z)) \lor \\ (0.6 ⊙ ((0.3 + λ (x)) \land λ (y) \land (0.7 + λ (z))) \\ \lor (0.3 ⊙ ((0.6 + λ (x)) \land λ (y))), \end{matrix}$ $\begin{matrix} I^{U} (λ) (z) = (λ (x) \land λ (y) \land λ (z)) \lor \\ (0.6 ⊙ ((0.4 + λ (x)) \land (0.7 + λ (y)) \land λ (z))) \\ \lor (0.3 ⊙ ((0.8 + λ (x)) \land λ (z))) . \end{matrix}$

For λ = (0.4, 0.6, 0.8), $I^{U} (λ) (x) = 0.4, I^{U} (λ) (y) = 0.4, I^{U} (λ) (z) = 0.4$ $T_{U} (λ) = ⋀_{x \in X} (λ (x) \to I^{U} (λ) (x)) = 0.7 .$

For ρ = (0.8, 0.2, 0.7), $I^{U} (λ) (x) = 0.2, I^{U} (λ) (y) = 0.2, I^{U} (λ) (z) = 0.3$ $T_{U} (λ) = ⋀_{x \in X} (λ (x) \to I^{U} (λ) (x)) = 0.4 .$

5 Conclusion

The main purpose of this paper is to study the relationships between L-fuzzy interior spaces, L-fuzzy quasi-uniform spaces and L-fuzzy topological spaces. Also, we introduce the concept ofL- fuzzy preinterior spaces and the notions of their continuities are investigated.

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