A new approach on covering fuzzy variable precision rough sets based on residuated lattice

Abstract

This paper constructs ℒ-covering fuzzy variable precision rough set and mainly studies some properties of type I and type II ℒ-covering fuzzy lower and upper approximation operators of ℒ-fuzzy sets on X. Topology properties of type I and type II ℒ-covering fuzzy lower and upper approximation operators are also discussed through a fuzzy interior operator and fuzzy closure operator. Moreover, we investigate the dual properties and compare between ℒ-covering fuzzy variable precision rough approximation operators.

Keywords

Residuated lattice ℒ-covering fuzzy variable precision rough approximation operators fuzzy covering fuzzy relations fuzzy topology

1 Introduction

Rough set theory was introduced by Pawlak [13] in 1982 as a mathematical approach to handle imprecision, vagueness and uncertainty in data analysis. In rough set theory, the lower and upper approximation operators are examined by many authors, which showed many interesting results and properties. Up to now, the concept of rough set had been used as a tool to model and process insufficient and incomplete information. A problem with Pawlak’s rough set theory is that an equivalence relation is explicitly used in the definition of the lower and upper approximations. However, it has been applied only in complete information systems that may limit the applications of the rough set model. Thus one of the main directions of research in rough set theory is naturally the generalization of the Pawlak rough set approximations.

To develop rough set theory, in 1990, Dubois and Prade [3, 4] put rough sets and fuzzy sets [18] together by using the triangular t-norm Min and its dual t-conorm Max, and gave out definition the pair of fuzzy rough set approximation operators with respect to a Pawlak approximation space and a fuzzy similarity relation to obtain the extended notions called rough fuzzy sets and fuzzy rough sets, respectively. In 2004, ℒ-fuzzy rough set (ℒ-FRS) was first developed by Radzikowska and Kerre [15], as a generalization of the notion of rough sets determined by a residuated lattice ℒ [6 , 14]; Estaji et al. [5] also studied a connection between rough sets and lattice theory. They generalized rough set theory to residuated lattice serving as a more general structure of truth values. In 2009, She [16] extended to the axiomatic characterization of various ℒ-FRSs. Many authors have studied topological structures, such as, Ma and Hu [12] studied the relationship between ℒ-fuzzy rough approximation operators and ℒ-fuzzy topological spaces by the lower set and upper set, Wang and Hu [17] presented fuzzy rough sets based on generalized residuated lattices. Zhao and Hu supposed fuzzy variable precision rough sets based on residuated lattices [22].

Besides, another way of extension of ordinary rough set theory is rough set based on covering. Bonikowski et al. [1] proposed covering rough set as one of extension of the concept of ordinary rough set, in which covering-based rough set was derived by replacing the partitions of a universe with its coverings. The overviews on rough sets based on covering were given in [21 , 23–26]. In 2007, Deng et al. [2] investigated fuzzy rough set based on a fuzzy covering. In 2009, Li et al. [11] developed to generalized fuzzy rough approximation operators based on a fuzzy covering, and in 2012, Zhang et al. [19] extended to generalized intuitionistic fuzzy rough sets based on intuitionistic fuzzy coverings.

Variable precision rough set (VPRS) is one of the most important generalizations of Pawlak rough set (RS), which was introduced by Ziarko in 1993 [27]. It allows partial belongingness of a set to another. Zhao et al. [20] integrated the idea of variable precision into FRSs, constructed a model of fuzzy variable precision rough sets (FVPRSs) by combining the fuzzy rough sets and variable precision rough sets, which is non-sensitive to the perturbation of the original numerical data. Recently, Hu and Wong [7, 8] studied generalized interval-valued fuzzy rough sets based on logical operators and generalized interval-valued fuzzy variable precision rough sets based on fuzzy logical operators.

This paper aims to combine the idea of variable precision rough set with ℒ-covering fuzzy rough approximation operators to obtain ℒ-covering fuzzy variable precision rough approximation operators. In what follows, we will first summarise several basic concepts and introduce some notations for convenience. Based on this, we will present ℒ-covering fuzzy variable precision rough sets to discusses some of new properties of type I/type II ℒ-covering fuzzy variable precision rough approximation operators. We will then produce ℒ-covering fuzzy variable precision rough approximation operators, investigate properties of the generalized fuzzy variable precision rough approximation operators and topology properties of type I ℒ-covering fuzzy variable precision rough approximation operators, duality of the generalized fuzzy rough approximation operators. Finally, we will discuss difference of fuzzy rough approximation operators.

2 Preliminaries

2.1 Rough set

In traditional Pawlak’s rough set theory, the pair (X, R) is called an approximation space, where X is a universe and R is an equivalence relation on X, i.e., R is reflexive, symmetrical and transitive. The relation R decomposes the set X into a disjoint class in such a way that two elements x and y are in the same class iff (x, y) ∈ R.

Suppose R to be an equivalence relation on X. With respect to R, we can define an equivalence class of an element x in X as follows: [x] _R = {y ∣ (x, y) ∈ R}.

The quotient set of U by the relation R is denoted by X/R, and $X / R = {U_{1}, U_{2}, . . ., U_{m}},$ where U_i (i = 1, 2, . . . , m) is an equivalence class of R. Elements in the same equivalence class are said to be indistinguishable, and the equivalence classes of R are called elementary sets. Every union of elementary sets is called a definable set. In particular, the empty set is also considered to be a definable set.

Given an arbitrary set U ⊆ X, it may not be possible to describe U precisely in the approximation space (X, R), the pair of lower and upper approximations of U with respect to R defined as follows $\begin{matrix} \underline{R} (U) & = & {x \in U : [x]_{R} \subseteq U}, \\ \bar{R} (U) & = & {x \in U : [x]_{R} \cap U \neq \emptyset} . \end{matrix}$

2.2 Residuated lattice

A residuated lattice ℒ is an algebra (L, ∨ , ∧ , ⊗ , → , 0, 1), where (L, ∨ , ∧ , 0, 1) is a bounded lattice with the greatest element 1 and the smallest element 0. (L, ⊗ , 1) is a commutative monoid and (⊗ , →) is an adjoint pair on ℒ satisfying a ⊗ b ≤ c ⇔ a ≤ b → c, ∀a, b, c ∈ L.

Let a ∈ L and define a unary operator, as ¬a = a → 0, referred to as the precomplement operator. If for any a ∈ L, ¬¬ a = a, then ℒ is called a regular residuated lattice.

Lemma 2.1. Let (L, ∨ , ∧ , ⊗ , → , 0, 1) be a complete residuated lattice. Then the following holds: for all a, b, c, a_i, b_i ∈ L (i ∈ Λ),

a ≤ b ⇔ a → b = 1;

a → a = a → 1 =0 → a = 1, 1 → a = a;

b₁ ≤ b₂ ⇒ a ⊗ b₁ ≤ a ⊗ b₂;

b₁ ≤ b₂ ⇒ a → b₁ ≤ a → b₂, b₂ → a ≤ b₁ → a;

∨_i∈Λa_i → b = ∧ _i∈Λ (a_i → b), ∧_i∈Λa_i → b ≥ ∨ _i∈Λ (a_i → b);

b → ∧ _i∈Λa_i = ∧ _i∈Λ (b → a_i), b → ∨ _i∈Λa_i ≥ ∨ _i∈Λ (b → a_i);

b ⊗ (∨ _i∈Λa_i) = ∨ _i∈Λ (b ⊗ a_i), b ⊗ (∧ _i∈Λa_i) ≤ ∧ _i∈Λ (b ⊗ a_i);

¬ {∨ _ia_i} = ∧ _i ¬ a_i, ¬ {∧ _ia_i} ≥ ∨ _i ¬ a_i;

a → (b → c) = (a ⊗ b) → c = b → (a → c);

a ⊗ (b ∨ c) ≤ b ∨ (a ⊗ c), a ∧ (b → c) ≤ b → (a ∧ c).

The next, we will review the several typical definitions and properties of residuated lattice, fuzzy topology and fuzzy topology operators and ℒ-fuzzy variable precision rough sets. First, let us we will summarise basic definitions of classical residuated lattice.

2.3 ℒ-fuzzy sets, ℒ-relations and ℒ-topologies

The concept of ℒ-fuzzy set was first introduced by [6] when ℒ is a complete residuated lattice. And ℒ-fuzzy set was considered as a generalization of the notion of Zadeh’s fuzzy sets [18]. In what follows, specific definitions of ℒ-fuzzy sets are outlined.

Definition 2.2. Let ℒ be a complete lattice and let U be a nonempty set called the universe of discourse. A mapping $μ : U \to L$ is called an ℒ-fuzzy set in U.

Denote by $L^{X}$ the family of all ℒ-fuzzy sets on X. An ℒ-fuzzy set μ is constant if μ (x) = a, for all x ∈ X, written as $\hat{a}$ . An ℒ-fuzzy set μ is denoted by α_Y, if ∅ ≠ Y ⊆ X and $μ (x) = {\begin{matrix} α & if x \in Y \\ 0 & if x \notin Y \end{matrix}$

If Y = {y}, then α_Y is denoted by α_y.

The basic and most common operations on $L^{X}$ are as follows: for all x ∈ X and $μ \in L^{X}$ , $\begin{matrix} (μ \cup ν) (x) = μ (x) \lor ν (x); \\ (μ \cap ν) (x) = μ (x) \land ν (x); \\ (μ \otimes ν) (x) = μ (x) \otimes ν (x); \\ (μ \to ν) (x) = μ (x) \to ν (x); \\ (\neg μ) (x) = \neg μ (x) . \end{matrix}$

We also write u ⊆ v to denote u (x) ≤ v (x) for all x ∈ X.

An ℒ-fuzzy relation (take ℒ-relation for short) θ is a binary function defined on X also with truth values from ℒ, i.e., θ : X × X → L. Likewise, $L^{X \times X}$ denotes the family of all ℒ-fuzzy relations on X.

Let $θ \in L^{X \times X}$ be a relation fuzzy on X. Then θ is

serial if ∨_y∈Xθ (x, y) =1, ∀x ∈ X;

reflexive if θ (x, x) =1, ∀x ∈ X;

symmetric if θ (x, y) = θ (y, x), ∀x, y ∈ X;

transitive if θ (x, y) ⊗ θ (y, z) ≤ θ (x, z), ∀x, y, z ∈ X;

Lemma 2.3. [2] If $θ \in L^{X \times X}$ is a reflexive and transitive fuzzy relation, then $\begin{matrix} θ (x, y) & = & \lor_{z \in E} (θ (x, z) \otimes θ (z, y)) \\ = & \land_{z \in E} (θ (z, x) \to θ (z, y)) \end{matrix}$

Definition 2.4. [2] A fuzzy covering of X is a collection of fuzzy sets $Φ \subseteq L^{X}$ satisfying

every fuzzy set μ ∈ Φ is non-null, i.e., μ≠ ∅.

∀x ∈ X, ∨ _μ∈Φμ (x) >0.

Given a fuzzy relation $θ \in L^{X \times X}$ and let [x] _θ (y) = θ (x, y).

2.4 Fuzzy topology and fuzzy topology operators

Definition 2.5. $χ \subseteq L^{X}$ is a fuzzy topology on X if it satisfies the following conditions:

0, 1 ∈ χ,

μ ∩ ν ∈ χ for any μ, ν ∈ χ,

∪_i∈Iμ_i ∈ χ for any μ_i ∈ χ, i ∈ I.

Definition 2.6. [12] An operator $ψ : L^{X} \to L^{X}$ is a fuzzy interior operator if it satisfies the conditions (I1)-(I4), $\forall μ, ν \in L^{X}$ ,

ψ (1) =1,

ψ (μ) ⊆ μ,

ψ (ψ (μ)) = ψ (μ),

ψ (μ ∩ ν) = ψ (μ) ∩ ψ (ν).

An operator $ω : L^{X} \to L^{X}$ is a fuzzy closure operator if it satisfies the conditions (C1)-(C4), $\forall μ, ν \in L^{X}$ ,

ω (0) =0,

μ ⊆ ω (μ),

ω (ω (μ)) = ω (μ),

ω (μ ∪ ν) = ω (μ) ∪ ω (ν).

2.5 ℒ-fuzzy variable precision rough sets

Let X be a nonempty universe of discourse and θ be an ℒ-fuzzy relation on X. The pair (X, θ) is called an ℒ-fuzzy approximation space.

Definition 2.7. [22] Let (X, θ) be an ℒ-fuzzy approximation space and α ∈ L. Define the following two mappings ${\underline{θ}}^{α}, {\bar{θ}}^{α} : L^{X} \to L^{X}$ , called ℒ-fuzzy variable precision lower and upper approximation operators, respectively, as follows: for all x ∈ X and $μ \in L^{X}$ , $\begin{matrix} {\underline{θ}}_{α} (μ) (x) & = & \land_{y \in X} [θ (x, y) \to (α \lor μ (y))], \\ {\bar{θ}}_{α} (μ) (x) & = & \lor_{y \in X} [θ (x, y) \otimes (\neg α \land μ (y))] . \end{matrix}$ $θ^{α} (μ) = ({\underline{θ}}^{α} (μ), {\bar{θ}}^{α} (μ))$ is called an ℒ-fuzzy variable precision rough set (ℒ-FVPRS) with respect to μ.

In the following, let X be a nonempty universe of discourse and Φ be a fuzzy covering of X. The pair (X, Φ) is called an ℒ-covering fuzzy approximation space. For an arbitrary fuzzy relation θ derived from Φ, (X, θ) is also called an ℒ-covering fuzzy approximation space.

2.6 ℒ-covering fuzzy rough sets

Li [10] proposed the concept of fuzzy approximation operators based on coverings by two operators t-norm ∧ and t-conorm ∨. Deng [2] investigated fuzzy rough sets based on a fuzzy covering in the framework of residuated lattice, but they only gave type I ℒ-covering fuzzy rough set on U. Li [11] fully studied two types ℒ-covering fuzzy rough set on U by two operators t-norm $T$ and the residual implicator $I$ (R-implicator) based on t-norm $T$ .

Let ℒ be a complete residuated lattice. We present definitions of two types ℒ-covering fuzzy rough approximation operators on X.

Definition 2.8. [2, 11] Let (X, Φ) be an ℒ-covering fuzzy approximation space. Two pairs of approximation operators are defined as follows, for all x ∈ X and $μ \in L^{X}$ ,

$\begin{matrix} {\underline{θ}}^{I} (μ) (x) & = & \lor_{y \in X} [θ (x, y) \otimes \land_{z \in X} (θ (z, y) \\ \to μ (z))], \\ {\bar{θ}}^{I} (μ) (x) & = & \land_{y \in X} [θ (y, x) \to \lor_{z \in X} (θ (y, z) \\ \otimes μ (z))], \\ {\underline{θ}}^{II} (μ) (x) & = & \land_{y \in X} [θ (x, y) \to \land_{z \in X} (θ (z, y) \\ \to μ (z))], \\ {\bar{θ}}^{II} (μ) (x) & = & \lor_{y \in X} [θ (y, x) \otimes \lor_{z \in X} (θ (y, z) \\ \otimes μ (z))] . \end{matrix}$

The pair $({\underline{θ}}^{I} (μ), {\bar{θ}}^{I} (μ))$ is called type I ℒ-covering fuzzy rough set of μ on X and $({\underline{θ}}^{II} (μ), {\bar{θ}}^{II} (μ))$ is called type II ℒ-covering fuzzy rough set of μ on X. The $({\underline{θ}}^{I}, {\bar{θ}}^{I})$ on $L^{X}$ and $({\underline{θ}}^{II}, {\bar{θ}}^{II})$ are called the type I and type II ℒ-covering fuzzy lower and upper approximation operators on X.

3 ℒ-covering fuzzy variable precision rough approximation operators

In this subsection, we introduce new definition of ℒ-covering fuzzy variable precision rough approximation operators based on reduated lattice.

Definition 3.1. Let (X, Φ) be an ℒ-covering fuzzy approximation space. Two pairs of approximation operators are defined as follows, for all x ∈ X and $μ \in L^{X}$ , $\begin{matrix} {\underline{θ}}_{α}^{I} (μ) (x) & = & \lor_{y \in X} [θ (x, y) \otimes \land_{z \in X} (θ (z, y) \\ \to α \lor μ (z))], \\ {\bar{θ}}_{α}^{I} (μ) (x) & = & \land_{y \in X} [θ (y, x) \to \lor_{z \in X} (θ (y, z) \\ \otimes \neg α \land μ (z))], \\ {\underline{θ}}_{α}^{II} (μ) (x) & = & \land_{y \in X} [θ (x, y) \to \land_{z \in X} (θ (z, y) \\ \to α \lor μ (z))], \\ {\bar{θ}}_{α}^{II} (μ) (x) & = & \lor_{y \in X} [θ (y, x) \otimes \lor_{z \in X} (θ (y, z) \\ \otimes \neg α \land μ (z))] . \end{matrix}$

The pair $({\underline{θ}}_{α}^{I} (μ), {\bar{θ}}_{α}^{I} (μ))$ is called an α-type I ℒ-covering fuzzy rough set of μ on X and $({\underline{θ}}_{α}^{II} (μ), {\bar{θ}}_{α}^{II} (μ))$ is called α-type II ℒ-covering fuzzy rough set of μ on X. The $({\underline{θ}}_{α}^{I}, {\bar{θ}}_{α}^{I})$ and $({\underline{θ}}_{α}^{II}, {\bar{θ}}_{α}^{II})$ are called the α-type I and α-type II fuzzy variable precision lower and upper approximation operators on X, respectively.

Note 3.2. When α = 0, α-type I/α-type II ℒ-covering fuzzy variable precision lower and upper approximation operators in Definition 3.1 are degenerated into type I/type II ℒ-covering fuzzy lower and upper approximation operators in Definition 2.8, respectively.

Example 3.3. Consider the bounded lattice {0, a, b, c, 1} with the partial order ≤ defined by 0 < a < b < c < 1, in which a = ¬ c. ⊗ is defined and → is computed, respectively, as follows:

Let X = {x₁, x₂} and the ℒ-relations be defined as follows:

$θ = [\begin{matrix} 1 & b \\ a & c \end{matrix}]$ , $μ = \hat{c}$ and α = b. $\begin{matrix} {\underline{θ}}_{α}^{I} (μ) (x_{1}) \\ = \lor_{y \in X} [θ (x_{1}, y) \otimes \land_{z \in X} (θ (z, y) \to c \lor b)], \\ = \lor_{y \in X} [θ (x_{1}, y) \otimes \land_{z \in X} (θ (z, y) \to c)] \\ = θ (x_{1}, y) \otimes \land_{z \in X} (θ (z, y) \to c) \\ \lor θ (x_{2}, y) \otimes \land_{z \in X} (θ (z, y) \to c) \\ = c . \\ {\underline{θ}}_{α}^{I} (μ) (x_{1}) = c . \end{matrix}$

So, ${\underline{θ}}_{α}^{I} (μ) (x) = (c, c)$ and ${\bar{θ}}_{α}^{I} (μ) (x) = (a, a)$ .

${\underline{θ}}_{α}^{II} (μ) (x) = (c, 1)$ and ${\bar{θ}}_{α}^{II} (μ) (x) = (a, 0)$ .

Proposition 3.4. Let (X, Φ) be an ℒ-covering fuzzy approximation space. Then, for any 0 ≤ α ≤ β ≤ 1, $\forall μ \in L^{X}$ , we have the following statements.

${\underline{θ}}_{α}^{I} (μ) \subseteq {\underline{θ}}_{β}^{I} (μ)$ , ${\bar{θ}}_{β}^{I} (μ) \subseteq {\bar{θ}}_{α}^{I} (μ)$ .

${\underline{θ}}_{α}^{II} (μ) \subseteq {\underline{θ}}_{β}^{II} (μ)$ , ${\bar{θ}}_{β}^{II} (μ) \subseteq {\bar{θ}}_{α}^{II} (μ)$ .

${\underline{θ}}_{α}^{I} (μ) \subseteq {\underline{θ}}^{I} (μ)$ , ${\bar{θ}}^{I} (μ) \subseteq {\bar{θ}}_{α}^{I} (μ)$ .

${\underline{θ}}_{α}^{II} (μ) \subseteq {\underline{θ}}^{II} (μ)$ , ${\bar{θ}}^{II} (μ) \subseteq {\bar{θ}}_{α}^{II} (μ)$ .

Proof. It is easy to prove Theorem 3.4 by Lemma 2.1 and Definition 3.1. □

Proposition 3.5. Let (X, Φ) be an ℒ-covering fuzzy approximation space. If ϑ ⊆ θ, then, $\forall μ \in L^{X}$ , the following statements hold.

${\underline{ϑ}}_{α}^{I} (μ) \subseteq {\underline{θ}}_{α}^{I} (μ)$ , ${\bar{θ}}_{α}^{I} (μ) \subseteq {\bar{ϑ}}_{α}^{I} (μ)$ .

${\underline{ϑ}}_{α}^{II} (μ) \subseteq {\underline{θ}}_{α}^{II} (μ)$ , ${\bar{θ}}_{α}^{II} (μ) \subseteq {\bar{ϑ}}_{α}^{II} (μ)$ .

Proof. It is easy to prove Theorem 3.5 by Lemma 2.1 and Definition 3.1. □

3.1 Properties of α-type I ℒ-covering fuzzy variable precision rough approximation operators

Proposition 3.6. Let (X, Φ) be an ℒ-covering fuzzy approximation space. Then we have: for all α, β ∈ L and $μ, ν, μ_{i} \in L^{X}$ (i ∈ Λ),

${\underline{θ}}_{α}^{I} (\hat{0}) = \hat{0}, {\bar{θ}}_{α}^{I} (\hat{1}) = \hat{1}$

μ ⊆ ν implies ${\underline{θ}}_{α}^{I} (μ) \subseteq {\underline{θ}}_{α}^{I} (ν)$ , ${\bar{θ}}_{α}^{I} (μ) \subseteq {\bar{θ}}_{α}^{I} (ν)$ .

${\underline{θ}}_{α}^{I} (μ \cap ν) \subseteq {\underline{θ}}_{α}^{I} (μ) \cap {\underline{θ}}_{α}^{I} (ν)$ , ${\underline{θ}}_{α}^{I} (μ \cup ν) \subseteq {\underline{θ}}_{α}^{I} (μ) \cup {\underline{θ}}_{α}^{I} (ν)$ .

${\bar{θ}}_{α}^{I} (μ) \cap {\bar{θ}}_{α}^{I} (ν) \subseteq {\bar{θ}}_{α}^{I} (μ \cap ν)$ , ${\bar{θ}}_{α}^{I} (μ) \cup {\bar{θ}}_{α}^{I} (ν) \subseteq {\bar{θ}}_{α}^{I} (μ \cup ν)$ .

${\underline{θ}}_{α}^{I} (\hat{α} \cup μ) = {\underline{θ}}_{α}^{I} (μ)$ , ${\bar{θ}}_{α}^{I} (\hat{\neg α} \cap μ) = {\bar{θ}}_{α}^{I} (μ)$ ;

If θ₁ ⊆ θ₂, then ${\underline{θ_{1}}}_{α}^{I} (μ) \subseteq {\underline{θ_{2}}}_{α}^{I} (μ)$ and ${\bar{θ_{1}}}_{α}^{I} (μ) \subseteq {\bar{θ_{2}}}_{α}^{I} (μ)$ .

${\underline{θ}}^{I} (\hat{a}) \subseteq \hat{α \cup a}$ , $\hat{\neg α \cap a} \subseteq {\bar{θ}}^{I} (\hat{a})$ .

${\underline{θ}}_{α}^{I} ({\underline{θ}}_{α}^{I} (μ)) = \hat{α} \cup {\underline{θ}}_{α}^{I} (μ)$ , ${\bar{θ}}_{α}^{I} ({\underline{θ}}_{α}^{I} (μ)) = \hat{\neg α} \cap {\underline{θ}}_{α}^{I} (μ)$ .

Proof. It is easy to prove 3.6(1)-3.6(7) by Lemma 2.1 and Definition 3.1.

(8) For all $μ \in L^{X}, x \in X$ , $\begin{matrix} {\underline{θ}}_{α}^{I} ({\underline{θ}}_{α}^{I} (μ)) (x) \\ = \lor_{y \in X} [θ (x, y) \otimes \land_{z \in X} (θ (z, y) \to α \lor {\underline{θ}}_{α}^{I} (μ (z)))] \\ \geq \lor_{y \in X} {θ (x, y) \otimes [\land_{z \in X} (θ (z, y) \to α)] \\ \lor [\land_{z \in X} (θ (z, y) \to {\underline{θ}}_{α}^{I} (μ (z)))]} \\ = \lor_{y \in X} [θ (x, y) \otimes \land_{z \in X} (θ (z, y) \to α)] \\ \lor \lor_{y \in X} [θ (x, y) \otimes \land_{z \in X} (θ (z, y) \to {\underline{θ}}_{α}^{I} (μ (z)))] \\ = {\underline{θ}}_{α}^{I} (\hat{α}) \lor \lor_{y \in X} [θ (x, y) \otimes \land_{z \in X} (θ (z, y) \to \\ {\underline{θ}}_{α}^{I} (μ (z)))] . \end{matrix}$

And, $\begin{matrix} \lor_{y \in X} [θ (x, y) \otimes \land_{z \in X} (θ (z, y) \to {\underline{θ}}_{α}^{I} (μ (z)))] \\ = \lor_{y \in X} [θ (x, y) \otimes \land_{z \in X} (θ (z, y) \to \\ \lor_{y^{'} \in X} [θ (z, y^{'}) \otimes \land_{z^{'} \in X} (θ (z^{'}, y^{'}) \to α \lor μ (z^{'}))] \\ \geq \lor_{y \in X} [θ (x, y) \otimes \land_{z \in X} (θ (z, y) \to \\ [θ (z, y) \otimes \land_{z^{'} \in X} (θ (z^{'}, y) \to α \lor μ (z^{'}))] \\ \geq \lor_{y \in X} [θ (x, y) \otimes \\ \land_{z \in X} \land_{z^{'} \in X} (θ (z^{'}, y) \to α \lor μ (z^{'})) \\ = \lor_{y \in X} [θ (x, y) \otimes \land_{z^{'} \in X} (θ (z^{'}, y) \to α \lor μ (z^{'})) \\ = {\underline{θ}}_{α}^{I} (μ (z)) (x) . \end{matrix}$

So, $α \lor {\underline{θ}}_{α}^{I} (μ (z)) \leq {\underline{θ}}_{α}^{I} (\hat{α}) \lor {\underline{θ}}_{α}^{I} (μ (z)) \leq {\underline{θ}}_{α}^{I} ({\underline{θ}}_{α}^{I} (μ))$ . By applying 3.6(7), we have ${\underline{θ}}_{α}^{I} ({\underline{θ}}_{α}^{I} (μ)) = α \lor {\underline{θ}}_{α}^{I} (μ (z))$ . □

In general, ${\bar{θ}}_{α}^{I}$ and ${\underline{θ}}_{α}^{I}$ may not satisfy the following properties. ${\bar{θ}}_{α}^{I} (μ \cup ν) = {\bar{θ}}_{α}^{I} (μ) \cup {\bar{θ}}_{α}^{I} (ν)$ (1) ${\underline{θ}}_{α}^{I} (μ \cap ν) = {\underline{θ}}_{α}^{I} (μ) \cup {\underline{θ}}_{α}^{I} (ν)$ (2)

In the next subsection we will find the conditions under which ${\bar{θ}}_{α}^{I}$ satisfies (1) and ${\underline{θ}}_{α}^{I}$ satisfies (2). Besides, we consider relation between topology structure of α-type I ℒ-covering fuzzy variable precision rough approximation operators and fuzzy interior/closure operator.

3.2 Topology structure of α-type I ℒ-covering fuzzy variable precision rough approximation operators

Definition 3.7. Let (X, Φ) be an ℒ-covering fuzzy approximation space. We define

${\underline{H}}^{α} = {μ \in L ∣ {\underline{θ}}_{α}^{I} (μ) = \hat{α} \cup μ}$ .

${\bar{H}}^{α} = {μ \in L ∣ {\bar{θ}}_{α}^{I} (μ) = \neg \hat{α} \cap μ}$ .

Theorem 3.8. Let (X, Φ) be an $L$ -covering fuzzy approximation space. For any $μ \in L^{X}, α \in L$ , then the following hold.

$0, 1 \in {\underline{H}}^{α}$ .

For all i, $μ_{i} \in {\underline{H}}^{α}, i = 1, . ., n$ , $\cup_{i \in I} μ_{i} \in {\underline{H}}^{α}$ .

$μ \in {\underline{H}}^{α}$ implies ${\underline{θ}}_{α}^{I} ({\underline{θ}}_{α}^{I} (μ)) \in {\underline{H}}^{α}$ .

Proof.

Obviously, $0, 1 \in {\underline{H}}^{α}$ .

$\forall μ_{i} \in {\underline{H}}^{α}, i \in I$ , we have $\cup_{i \in I} (\hat{α} \cup μ_{i}) = \cup_{i \in I} {\underline{θ}}_{α}^{I} (μ) \subseteq {\underline{θ}}_{α}^{I} (\cup_{i \in I} μ) \subseteq \hat{α} \cup (\cup_{i \in I} μ_{i})$ . So, ${\underline{θ}}_{α}^{I} (\cup_{i \in I} μ_{i}) = \hat{α} \cup (\cup_{i \in I} μ_{i})$ , i.e., $\cup_{i \in I} μ_{i} \in {\underline{H}}^{α}$ .

It follows immediately from Theorem 3.6(8).

□

Remark 3.9. ${\underline{H}}^{α}$ is not closed under finite intersection of sets.

Theorem 3.10. Let (X, Φ) be an ℒ-covering fuzzy approximation space. For any $μ \in L^{X}, α \in L$ , then the following hold.

$0, 1 \in {\bar{H}}^{α}$ .

For all i, $μ_{i} \in {\bar{H}}^{α}, i = 1, . ., n$ , $\cap_{i \in I} μ_{i} \in {\bar{H}}^{α}$ .

$μ \in {\bar{H}}^{α}$ implies ${\bar{θ}}_{α}^{I} (μ) \in {\bar{H}}^{α}$ .

$μ \in {\bar{H}}^{α}$ , $μ \in {\underline{H}}^{α}$ iff $\neg μ \in {\bar{H}}^{α}$ .

Proof.

Obviously, $0, 1 \in {\underline{H}}^{α}$ .

$\forall μ \in {\bar{H}}^{α}, i \in I$ , we have $\cap_{i \in I} (\neg \hat{α} \cap μ_{i}) = \cap_{i \in I} {\bar{θ}}_{α}^{I} μ \supseteq {\bar{θ}}_{α}^{I} (\cap_{i \in I} μ) \supseteq \neg \hat{α} \cap_{i \in I} μ_{i}$ . So, $\neg \hat{α} \cap (\cap_{i \in I} μ_{i}) = {\bar{θ}}_{α}^{I} (\cap_{i \in I} μ)$ , i.e., $\cap_{i \in I} μ_{i} \in {\bar{H}}^{α}$ .

It follows immediately from Theorem 3.6(8).

It can be proven by the duality of ⊗ and → w.r.t to ¬. □

Remark 3.11. ${\bar{H}}^{α}$ is not closed under finite union of sets.

Theorem 3.12. Let (X, Φ) be an $L$ -covering fuzzy approximation space. For $μ, ν \in L^{X}$ and $α \in L$ , then the following hold.

${\underline{θ}}_{α}^{I}$ satisfies ${\underline{θ}}_{α}^{I} (μ \cap ν) = {\underline{θ}}_{α}^{I} (μ) \cap {\underline{θ}}_{α}^{I} (ν)$ if and only if the family ${\underline{H}}^{α}$ is a fuzzy topology on X.

${\bar{θ}}_{α}^{I}$ satisfies ${\bar{θ}}_{α}^{I} (μ \cup ν) = {\bar{θ}}_{α}^{I} (μ) \cup {\bar{θ}}_{α}^{I} (ν)$ if and only if the family ${\bar{H}}^{α}$ is a fuzzy topology on X.

Proof.

(⇒) For all $μ, ν \in {\underline{H}}^{α}$ , ${\underline{θ}}_{α}^{I} (μ \cap ν) = {\underline{θ}}_{α}^{I} (μ) \cap {\underline{θ}}_{α}^{I} (ν) = \hat{α} \cap μ \cap ν$ , so $μ \cap ν \in {\underline{H}}^{α}$ . By combining with Theorem 3.10(1)(2), we obtain ${\underline{H}}^{α}$ is a fuzzy topology on X.

((⇐) Let ${\underline{H}}^{α}$ be a fuzzy topology on X. For any $μ, ν \in {\underline{H}}^{α}$ , by Theorem 3.6(3), we have ${\underline{θ}}_{α}^{I} (μ \cap ν) \subseteq {\underline{θ}}_{α}^{I} (μ) \cap {\underline{θ}}_{α}^{I} (ν)$ . Applying Theorem 3.6(7), ${\underline{θ}}_{α}^{I} (μ) \cap {\underline{θ}}_{α}^{I} (ν) \subseteq \hat{α} \cap μ \cap ν$ . Again, by Theorem 3.6(7), ${\underline{θ}}_{α}^{I} ({\underline{θ}}_{α}^{I} (μ) \cap {\underline{θ}}_{α}^{I} (ν)) \subseteq \hat{α} \cap {\underline{θ}}_{α}^{I} (μ \cap ν))$ . On the other hand, ${\underline{H}}^{α}$ is a fuzzy topology, we have ${\underline{θ}}_{α}^{I} (μ) \cap {\underline{θ}}_{α}^{I} (ν) \in {\underline{H}}^{α}$ , so ${\underline{θ}}_{α}^{I} ({\underline{θ}}_{α}^{I} (μ) \cap {\underline{θ}}_{α}^{I} (ν)) = \hat{α} \cap {\underline{θ}}_{α}^{I} (μ) \cap {\underline{θ}}_{α}^{I} (ν)$ . Then, ${\underline{θ}}_{α}^{I} (μ) \cap {\underline{θ}}_{α}^{I} (ν) \subseteq {\underline{θ}}_{α}^{I} (μ \cap ν))$ . Thus, we can conclude ${\underline{θ}}_{α}^{I} (μ) \cap {\underline{θ}}_{α}^{I} (ν) = {\underline{θ}}_{α}^{I} (μ \cap ν)$ .

It can be proved in a similar way of item (1).

□

Theorem 3.13. Let (X, Φ) be an ℒ-covering fuzzy approximation space. Then the following hold.

Operator ${\underline{θ}}_{α}^{I}$ is a fuzzy interior operator if and only if the family ${\underline{H}}^{α}$ is a fuzzy topology on X.

Operator ${\bar{θ}}_{α}^{I}$ is a fuzzy closure operator if and only if the family ${\underline{H}}^{α}$ is a fuzzy topology on X.

Proof. It can be proved by Theorem 3.12 and Definition 2.6. □

Theorem 3.14. Let (X, Φ) be an ℒ-covering fuzzy approximation space and X be a finite partially ordered set. Then the following hold.

${\underline{H}}^{α}$ is a fuzzy topology on X if only if ${\underline{H}}^{α}$ is a lattice on X.

${\underline{H}}^{α}$ is a fuzzy topology on X if and only if ${\bar{H}}^{α}$ is a lattice on X.

Proof. It follows immediately from Theorems 3.8, 3.10 and 3.12. □

Theorem 3.15. Let (X, Φ) be an ℒ-covering fuzzy approximation space and X be a finite partially ordered set. If ${\underline{H}}^{α}$ and ${\bar{H}}^{α}$ are fuzzy topology on X, then ${\bar{H}}^{α}$ is dually isomorphic to ${\underline{H}}^{α}$ .

Proof. Let $f : {\bar{H}}^{α} \to {\underline{H}}^{α}$ be a mapping such that $\forall μ \in {\underline{H}}^{α}$ , $f (μ) = \neg (\hat{α} \cup μ)$ and f be definable on Theorem 3.10. Then f is bijective. For $μ, ν \in {\bar{H}}^{α}$ , $f (μ \cup ν) = \neg (\hat{α} \cup μ \cup ν) = \neg \hat{α} \cap \neg μ \cap \neg ν = f (μ) \cap f (ν)$ and $f (μ \cap ν) = f ({\underline{θ}}_{α}^{I} (μ \cap ν) = \neg ({\underline{θ}}_{α}^{I} (μ \cap ν)) = {\underline{θ}}_{α}^{I} (\neg (μ \cap ν)) = \neg (\hat{α}) \land (μ \cup ν) = (\neg \hat{α} \cap μ) \cup (\neg \hat{α} \cap ν) = f (μ) \cup f (ν)$ . □

Theorem 3.16. Let (X, Φ) be an ℒ-covering fuzzy approximation space. For any $μ \in L^{X}$ ,

${\underline{θ}}_{α}^{I} (μ) = \cup {ξ \in {\underline{H}}^{α} : \hat{α} \cup ξ \subseteq μ}$ ,

${\bar{θ}}_{α}^{I} (μ) = \cap {ξ \in {\underline{H}}^{α} : \neg \hat{α} \cap ξ \subseteq μ}$ .

Proof. $\forall ξ \in {ξ \in {\underline{H}}^{α} : \hat{α} \cup ξ \subseteq μ}$ , $\hat{α} \cup ξ = {\underline{θ}}_{α}^{I} (ξ) \subseteq {\underline{θ}}_{α}^{I} (μ)$ . Thus, $\cup {ξ \in {\underline{H}}^{α} : \hat{α} \cup ξ \subseteq μ} \subseteq {\underline{θ}}_{α}^{I} (μ)$ . Meanwhile, by Theorem 3.6(7)(8), we have $\hat{α} \cup {\underline{θ}}_{α}^{I} (μ) = {\underline{θ}}_{α}^{I} ({\underline{θ}}_{α}^{I} (μ)) \subseteq \hat{α} \cup μ$ . So ${\underline{θ}}_{α}^{I} (μ) \subseteq μ$ . Thus, ${\underline{θ}}_{α}^{I} (μ) \in \cup {ξ \in {\underline{H}}^{α} : \hat{α} \cup ξ \subseteq μ}$ . □

3.3 Properties of α-type II ℒ-covering fuzzy variable precision rough approximation operators

Let (X, Φ) be an ℒ-covering fuzzy approximation space. We can define a fuzzy tolerance relation (i.e., it satisfies reflexive and symmetric) on X: θ^Φ (x, z) = ∨ _y∈X (θ (x, y) ⊗ θ (z, y)) , ∀ x, y, z ∈ X.

Theorem 3.17. Let (X, Φ) be an ℒ-covering fuzzy approximation space. Then, we have ${\underline{θ}}_{α}^{II} = {\underline{θ}}_{α}^{Φ}$ , ${\bar{θ}}_{α}^{II} = {\bar{θ}}_{α}^{Φ}$ .

Proof. For all $x \in X, μ \in L^{X}$ and α ∈ L, $\begin{matrix} {\underline{θ}}_{α}^{II} (μ) (x) \\ = \land_{y \in X} [θ (x, y) \to \land_{z \in X} (θ (z, y) \to α \lor μ (z))] \\ = \land_{y \in X} \land_{z \in X} [θ (x, y) \to (θ (z, y) \to α \lor μ (z))] \\ = \land_{y \in X} \land_{z \in X} [(θ (x, y) \otimes θ (z, y)) \to α \lor μ (z)] \\ = \land_{y \in X} \land_{z \in X} [(θ (x, y) \otimes θ (z, y)) \to α \lor μ (z)] \\ = \land_{z \in X} [\lor_{y \in X} (θ (x, y) \otimes θ (z, y)) \to α \lor μ (z)] \\ = \land_{z \in X} [θ^{Φ} (x, z) \to α \lor μ (z)] \\ = {\underline{θ}}_{α}^{Φ} (μ) . \end{matrix}$

Hence, ${\underline{θ}}_{α}^{II} = {\underline{θ}}_{α}^{Φ}$ .

The others can be proved similarly. □

Lemma 3.18. [22] Let (X, θ) be an ℒ-fuzzy approximation space. Then, we have the following statements.

${\underline{θ}}_{α}^{Φ} (\hat{1}) = \hat{1}$ , ${\bar{θ}}_{α}^{Φ} (\hat{0}) = \hat{0}$ .

$\hat{α \cup a} \subseteq {\underline{θ}}_{α}^{Φ} (\hat{a})$ , ${\bar{θ}}_{α}^{Φ} (\hat{a}) \subseteq \hat{\neg α \cap a}$ .

${\underline{θ}}_{α}^{Φ} (\cap_{i \in λ} μ_{i}) = \cap_{i \in λ} {\underline{θ}}_{α}^{Φ} (μ_{i})$ , ${\bar{θ}}_{α}^{Φ} (\cup_{i \in λ} μ_{i}) = \cup_{i \in λ} {\underline{θ}}_{α}^{Φ} (μ_{i})$ .

$\forall μ, ν \in L^{X}$ , $μ \subseteq ν \Rightarrow {\underline{θ}}_{α}^{Φ} (μ) \subseteq {\underline{θ}}_{α}^{Φ} (ν)$ , ${\bar{θ}}_{α}^{Φ} (ν) \subseteq {\bar{θ}}_{α}^{Φ} (ν)$ .

$\forall μ \in L^{X}, {\underline{θ}}_{α}^{Φ} (μ) \subseteq \hat{α} \cup μ, \neg \hat{α} \cap μ \subseteq {\underline{θ}}_{α}^{Φ} (μ)$ .

$\forall μ_{i} \in L^{X}, {\underline{θ}}_{α}^{Φ} (\cap_{i \in λ} μ_{i}) = \cap_{i \in λ} {\underline{θ}}_{α}^{Φ} (μ_{i})$ , ${\bar{θ}}_{α}^{Φ} (\cup_{i \in λ} μ_{i}) = \cup_{i \in λ} {\bar{θ}}_{α}^{Φ} (μ_{i})$ .

Lemma 3.19. [22] Let (X, θ) be an ℒ-fuzzy approxi- mation space. Then the following statements are equivalent.

θ is symmetric.

${\bar{θ}}_{α} (1_{y}) (x) = {\bar{θ}}_{α} (1_{x}) (y)$ .

${\underline{θ}}_{α} (1_{X - {x}}) (y) = {\underline{θ}}_{α} (1_{X - {y}}) (x)$ .

Lemma 3.20. [22] Let (X, θ) be an ℒ-fuzzy approximation space. Then the following statements are equivalent.

θ is reflexive.

${\underline{θ}}_{α} (μ) \subseteq \hat{α} \cup μ$ .

$\neg \hat{α} \cap μ \subseteq {\bar{θ}}_{α} (μ)$ .

Theorem 3.21. Let (X, Φ) be an ℒ-covering fuzzy approximation space. The operators ${\underline{θ}}_{α}^{II}$ and ${\bar{θ}}_{α}^{II}$ satisfy the following properties:

${\underline{θ}}_{α}^{II} (\hat{1}) = \hat{1}$ and ${\bar{θ}}_{α}^{II} (\hat{0}) = \hat{0}$ .

For any a ∈ L, ${\underline{θ}}_{α}^{II} (\hat{α \lor a}), {\bar{θ}}_{α}^{II} (\hat{a}) = \hat{\neg α \land a}$ .

For any $μ_{i} \in L^{X}, i \in I$ , ${\underline{θ}}_{α}^{II} (\cap_{i \in I} μ_{i}) = \cap_{i \in I} {\underline{θ}}_{α}^{II} (μ_{i})$ , ${\bar{θ}}_{α}^{II} (\cup_{i \in I} μ_{i}) = \cup_{i \in I} {\bar{θ}}_{α}^{II} (μ_{i})$ .

$\forall μ, ν \in L^{X}, μ \subseteq ν \Rightarrow {\underline{θ}}_{α}^{II} (μ) \subseteq {\underline{θ}}_{α}^{II} (ν), {\bar{θ}}_{α}^{II} (μ) \subseteq {\bar{θ}}_{α}^{II} (ν)$ .

For any $μ \in L^{X}$ , $\neg \hat{α} \cap μ \subseteq {\underline{θ}}_{α}^{II} (μ), {\bar{θ}}_{α}^{II} (μ) \subseteq \hat{α} \cup μ$ .

For any $μ_{i} \in L^{X}, i \in I$ , ${\underline{θ}}_{α}^{II} (\cup_{i \in I} μ_{i}) \supseteq \cup_{i \in I} {\underline{θ}}_{α}^{II} (μ_{i})$ , ${\bar{θ}}_{α}^{II} (\cap_{i \in I} μ_{i}) \subseteq \cap_{i \in I} {\bar{θ}}_{α}^{II} (μ_{i})$ .

For any $μ \in L^{X}$ , ${\bar{θ}}_{α}^{II} (\neg (\hat{a}) \cap μ) = {\bar{θ}}_{α}^{II} (μ), {\underline{θ}}_{α}^{II} (\hat{a} \cup μ) = {\underline{θ}}_{α}^{II} (μ)$ .

For any $μ \in L^{X}$ , ${\bar{θ}}_{α}^{II} ({\underline{θ}}_{α}^{II} (μ)) \subseteq \neg \hat{α} \cap \hat{α} \cup μ$ and $\hat{α} \cup \neg \hat{α} \cap μ \subseteq {\underline{θ}}_{α}^{II} ({\bar{θ}}_{α}^{II} (μ))$ .

Proof. By Theorem 3.18, Lemmas 3.19 and 3.20, it can immediately prove Theorem 3.21(1)–(6).

We only prove the first inclusion relation in item (8).

For any $μ \in L^{X}, α \in L$ , we have $\begin{matrix} {\bar{θ}}_{α}^{II} ({\underline{θ}}_{α}^{II} (μ)) \\ = \lor_{y \in X} {θ (y, x) \to \lor_{z \in X} (θ (y, z) \otimes \neg α \land \\ \land_{y^{'} \in X} [θ (z, y^{'}) \to \land_{z^{'} \in X} (θ (z^{'}, y^{'}) \\ \to α \lor μ (z^{'}))])} \\ = \lor_{y \in X} {θ (y, x) \to \lor_{z \in X} (θ (y, z) \\ \otimes \land_{y^{'} \in X} (\neg α \land [θ (z, y^{'}) \to \land_{z^{'} \in X} (θ (z^{'}, y^{'}) \\ \to α \lor μ (z^{'}))]))} \\ \leq \lor_{y \in X} {θ (y, x) \to \lor_{z \in X} (θ (y, z) \\ \otimes \land_{y^{'} \in X} [θ (z, y^{'}) \to \neg α \land \land_{z^{'} \in X} (θ (z^{'}, y^{'}) \\ \to α \lor μ (z^{'}))])} \\ \leq \lor_{y \in X} {θ (y, x) \to \lor_{z \in X} (θ (y, z) \\ \otimes \land_{y^{'} \in X} [θ (z, y^{'}) \to \land_{z^{'} \in X} (θ (z^{'}, y^{'}) \\ \to \neg α \land α \lor μ (z^{'}))])} \\ \leq \lor_{y \in X} {θ (y, x) \to \lor_{z \in X} (θ (y, z) \\ \otimes [θ (z, y) \to \land_{z^{'} \in X} (θ (x, y) \to \neg α \\ \land α \lor μ (x))])} \leq \neg α \land α \lor μ (x) . □ \end{matrix}$

3.4 Duality of the ℒ-covering fuzzy rough approximation operators

The duality between rough approximation operators is an important property in rough set theory and fuzzy rough set theory.

Theorem 3.22. Let (X, Φ) be an ℒ-covering fuzzy approximation space. Suppose that ℒ is a complete regular residuated lattice, then, for all $μ \in L^{X}, α \in L$ , we have

${\underline{θ}}_{α}^{I} (μ) = \neg ({\bar{θ}}_{α}^{I} (\neg μ)$ and ${\bar{θ}}_{α}^{I} (μ) = \neg ({\underline{θ}}_{α}^{I} (\neg μ)$ .

${\bar{θ}}_{α}^{II} (μ) \subseteq {\underline{θ}}_{α}^{II} (μ) \subseteq \hat{α} \cup μ$ .

$\neg \hat{α} \cap μ \subseteq {\underline{θ}}_{α}^{II} (μ) \subseteq {\bar{θ}}_{α}^{II} (μ)$ .

Proof. If ℒ is a complete regular residuated lattice, then the theorem can be easily proved by property ¬¬ a = a. □

Theorem 3.23. Let (X, Φ) be an ℒ-covering fuzzy approximation space. Then, for all $μ \in L^{X}, α \in L$ , we have

${\underline{θ}}_{α}^{I} (μ) \subseteq \neg ({\underline{θ}}_{α}^{I} (\neg μ))$ and ${\bar{θ}}_{α}^{I} (μ) \subseteq \neg ({\underline{θ}}_{α}^{I} (\neg μ))$ .

${\underline{θ}}_{α}^{II} (μ) \subseteq \neg ({\bar{θ}}_{α}^{II} (\neg μ))$ and ${\bar{θ}}_{α}^{II} (μ) \subseteq \neg ({\underline{θ}}_{α}^{II} (\neg μ))$ .

Proof. If ℒ is a complete residuated lattice, then the theorem can be easily proved by property a ≤ ¬¬ a. □

4 Comparison of ℒ-covering fuzzy variable precision rough approximation operators

Theorem 4.1. Let (X, Φ) be an ℒ-covering fuzzy approximation space. If θ is a reflexive, then for all $μ \in L^{X}, α \in L$ , we have

${\underline{θ}}_{α}^{II} (μ) \subseteq {\underline{θ}}_{α} (μ) \subseteq {\underline{θ}}_{α}^{I} (μ) \subseteq \hat{α} \cup μ$ ,

$\neg \hat{α} \cap μ \subseteq {\bar{θ}}_{α}^{I} (μ) \subseteq {\bar{θ}}_{α} (μ) \subseteq {\bar{θ}}_{α}^{I} (μ)$ .

Proof. For all $μ \in L^{X}, x \in X, α \in L$ , we have $\begin{matrix} {\underline{θ}}_{α}^{I} (μ) (x) \\ = \lor_{y \in X} [θ (x, y) \otimes \land_{z \in X} (θ (z, y) \to α \lor μ (z))] \\ \geq θ (x, x) \otimes \land_{z \in X} (θ (z, x) \to α \lor μ (z)) \\ = 1 \otimes \land_{z \in X} (θ (z, x) \to α \lor μ (z)) \\ = \land_{z \in X} (θ (z, x) \to α \lor μ (z)) \\ = {\underline{θ}}_{α} (μ) (x) . \end{matrix}$

And, $\begin{matrix} {\underline{θ}}_{α}^{II} (μ) (x) \\ = \land_{y \in X} [θ (x, y) \to \land_{z \in X} (θ (z, y) \to α \lor μ (z))] \\ \leq θ (x, x) \to \land_{z \in X} (θ (z, x) \to α \lor μ (z)) \\ = 1 \to \land_{z \in X} (θ (z, x) \to α \lor μ (z)) \\ = \land_{z \in X} (θ (z, x) \to α \lor μ (z)) \\ = {\underline{θ}}_{α} (μ) (x) . \end{matrix}$

Thus, for $μ \in L^{X}$ , we have ${\underline{θ}}_{α}^{II} (μ) \subseteq {\underline{θ}}_{α} (μ) \subseteq {\underline{θ}}_{α}^{I} (μ)$ . By applying Proposition 3.6(7), we can conclude that ${\underline{θ}}_{α}^{II} (μ) \subseteq {\underline{θ}}_{α} (μ) \subseteq {\underline{θ}}_{α}^{I} (μ) \subseteq \hat{α} \cup μ$ . □

Corollary 4.2. If θ is a reflexive and transitive fuzzy relation on X, then ${\underline{θ}}_{α}^{I} = {\underline{θ}}_{α} (μ)$ and ${\bar{θ}}_{α}^{I} = {\bar{θ}}_{α} (μ)$ .

Proof. For all $μ \in L^{X}, x \in X, α \in L$ , we have $\begin{matrix} {\underline{θ}}_{α}^{I} (μ) (x) \\ = \lor_{y \in X} [θ (x, y) \otimes \land_{z \in X} (θ (z, y) \to α \lor μ (z))], \\ \leq \lor_{y \in X} \land_{z \in X} [θ (x, y) \otimes (\lor_{u \in X} (θ (z, u) \\ \otimes θ (u, y)) \to α \lor μ (z))] \\ \leq \lor_{y \in X} \land_{z \in X} \land_{u \in X} [θ (x, y) \otimes ((θ (z, u) \\ \otimes θ (u, y)) \to α \lor μ (z))] \\ \leq \lor_{y \in X} \land_{z \in X} [θ (x, y) \otimes ((θ (z, x) \otimes θ (x, y)) \\ \to α \lor μ (z))] \\ \leq \lor_{y \in X} \land_{z \in X} (θ (z, x) \to α \lor μ (z)) \\ = {\underline{θ}}_{α} (μ) . \end{matrix}$

So, ${\underline{θ}}_{α}^{I} (μ) \subseteq {\underline{θ}}_{α} (μ)$ , it follows from Theorem 4.1 that ${\underline{θ}}_{α}^{I} (μ) = {\underline{θ}}_{α} (μ)$ . □

Theorem 4.3. If θ is a symmetric fuzzy relation on X, then for any $μ \in L^{X}$ and x ∈ X,

${\underline{θ}}_{α}^{I} (μ) (x) = \lor_{y \in C_{x}} \land_{z \in C_{y}} α \lor μ (z)$ and ${\bar{θ}}_{α}^{I} (μ) (x) = \land_{y \in C_{x}} \lor_{z \in C_{y}} \neg α \land μ (z)$ .

${\underline{θ}}_{α}^{II} (μ) (x) = \land_{y \in C_{x}} \land_{z \in C_{y}} α \lor μ (z)$ and ${\bar{θ}}_{α}^{II} (μ) (x) = \lor_{y \in C_{x}} \lor_{z \in C_{y}} \neg α \land μ (z)$ .

Proof. Let x ∈ X. Then for each y ∈ X, y ∈ C_x, θ (x, y) =1. $\begin{matrix} {\underline{θ}}_{α}^{I} (μ) (x) \\ = \lor_{y \in X} [θ (x, y) \otimes \land_{z \in X} (θ (z, y) \\ \to α \lor μ (z))] \\ = \lor_{y \in C_{x}} [1 \otimes \land_{z \in X} (θ (z, y) \to α \lor μ (z))] \\ = \lor_{y \in C_{x}} \land_{z \in X} (θ (z, y) \to α \lor μ (z)) \\ = \lor_{y \in C_{x}} \land_{z \in C_{x}} (1 \to α \lor μ (z)) \\ = \lor_{y \in C_{x}} \land_{z \in C_{x}} (α \lor μ (z)) . □ \end{matrix}$

5 Conclusion

This paper mainly proposes two types of ℒ-covering fuzzy variable precision rough approximation operators, α-type I and α-type II ℒ-covering fuzzy variable precision rough set. Some basic properties of the new approximation operators are discussed. Besides, we also construct topology structure of ℒ-covering fuzzy variable precision rough approximation operators and consider their relationship and fuzzy topology operators. The duality and comparison between α-type I and α-type II ℒ-covering fuzzy variable precision rough set under certain conditions were also studied in this paper.

Footnotes

Acknowledgments

This research was supported by the National Nature Science Foundation of China (Grand No. 61179038).

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