On φ -fuzzy rough sets: Representation,special cases and induced topology

Abstract

Consider L = (L, ∗ , 1) be a complete residuated lattice. For an arbitrary function φ defined from the set of all singletons to that of all L-sets, Chen and Li introduced a type of L-fuzzy rough sets in 2007, called φ-fuzzy rough sets in this paper. The well-known R-fuzzy rough sets, where R is an L-fuzzy relation, can be regarded as a special φ-fuzzy rough sets. In this paper, we prove that φ-fuzzy rough sets can be represented by a family of R-fuzzy rough sets. Then we define some special φ of being serial, reflexive, symmetric, transitive and Euclidean, and discuss the corresponding φ-fuzzy rough sets, respectively. At last, we study the induced L-topology by φ when it is reflexive or reflexive and transitive.

Keywords

Complete residuated lattice

1 Introduction

As a technique to deal with granularity in information systems, rough set theory was proposed by Pawlak [17]. The classical rough set theory is based on equivalence relations, but in some situations, equivalence relations are not suitable for coping with granularity. Thus classical rough set theory has been extended to general relation-based rough sets [3 , 18], covering-based rough sets [23, 28] and more general L-fuzzy rough sets [4 , 24–26]. In particular, the residuated lattice-valued L-fuzzy rough sets have important meanings since residuated lattices can play the role of the set of truth-values in a many-valued logic. By generalizing crisp relations to L-fuzzy relations R, one type of L-fuzzy rough sets called R-fuzzy rough sets in this paper, were introduced in [21]. The relationships between R-fuzzy rough sets and L-topologies were studied in [5 , 24]. From a very different direction, for an arbitrary function φ from the set of all singletons to that of all L-sets, Chen and Li [2] constructed a kind of L-fuzzy rough sets called φ-fuzzy rough sets in this paper. The φ-fuzzy rough sets include both Järvinen’s rough sets [7] and R-fuzzy rough sets as special cases.

In this paper, we shall give a further study on φ-fuzzy rough sets. The contents are arranged as follows. Section 2 briefly recalls some fundamental concepts and related properties about R-fuzzy rough sets and φ-fuzzy rough sets. Section 3 gives a represented theorem for φ-fuzzy rough sets by a family of R-fuzzy rough sets, and then discusses some properties of φ-fuzzy rough sets. Section 4 investigates some special cases of φ. Section 5 discusses the L-topology induced by φ-fuzzy rough sets.

It should be pointed out that the original paper [2] also studied some special cases of φ and the induced L-topology by φ-fuzzy rough sets. But there is little overlap in the results of the present paper and the paper [2]. Indeed, except for reflexivity (which is called the extensiveness in [2]), the other φ-conditions considered in this paper are newly defined. The paper [2] discussed the induced L-topology by restricting the lattice-context, while we study the induced L-topology by restricting the function φ.

2 Preliminaries

In this paper, if there is no further statement, (L, ∗ , 1) always denotes a complete residuated lattice. That is, L is a complete lattice with a top element 1 and a bottom element 0; ∗ is a binary operation on L such that (i) (L, ∗ , 1) is a commutative monoid; and (ii) ∗ distributes over arbitrary joins. Since the binary operation * distributes over arbitrary joins, the mapping α * (-) : L → L has a right adjoint α → (-) : L → L given by α → β = ∨ {γ ∈ L : α ∗ γ ≤ β} . The binary operation → is called the residuation with respect to *. The basic properties of the complete residuated lattice are referred to [1 , 12]. A complete residuated lattice is called regular if for each a ∈ L, (a → 0) →0 = 0.

For a set X, the set L^X of functions from X to L with the pointwise order becomes a complete lattice. Each element of L^X is called an L-set of X. For any A, B ∈ L^X, {A_i} _i∈I ⊆ L^X and x ∈ X, $\begin{matrix} (A * B) (x) = A (x) * B (x), \\ (A \to B) (x) = A (x) \to B (x), \\ (\underset{i \in I}{\lor} A_{i}) (x) = \underset{i \in I}{\lor} A_{i} (x), (\underset{i \in I}{\land} A_{i}) (x) = \underset{i \in I}{\land} A_{i} (x) . \end{matrix}$

An L-set A is said to be constant if A (x) = c for all x ∈ X and some c ∈ L. Here we make no difference between a constant function and its value since no confusion will arise. For D ⊆ X, let 1_D denote the characteristic function of D.

For each x ∈ X, α ∈ L with α > 0, let x_α denote the L-set defined by $x_{α} (y) = {\begin{matrix} α, if & y = x, \\ 0, if & y \neq x . \end{matrix}$ Then x_α is called a fuzzy point, or a singleton, of L^X. The set of all singletons in L^X is denoted as M, that is, M = {{x_α} |x ∈ X, α ∈ L, α > 0}.

2.1 R-fuzzy rough sets

Definition 2.1. [21, 22] A function R : X × X ⟶ L is said to be an L-fuzzy relation on X. The pair (X, R) is called an L-fuzzy approximation space. Then two L-fuzzy rough operators $\underline{R}$ and $\bar{R}$ , called lower and upper L-fuzzy approximations, respectively, are defined as follows: for any A ∈ L^X, x ∈ X, $\begin{matrix} \underline{R} (A) (x) & = & \underset{y \in X}{\land} [R (x, y) \to A (y)], \\ \bar{R} (A) (x) & = & \underset{y \in X}{\lor} [R (x, y) * A (y)] . \end{matrix}$ The pair $(\underline{R} (A), \bar{R} (A))$ is referred as an R-fuzzy rough set in this paper.

Proposition 2.2. [22] Let (X, R) be an L-fuzzy approximation space. Then for all A, B ∈ L^X, {A_t} _t∈T ⊆ L^X, α ∈ L we have

$\bar{R} (0) = 0$ ; (L1) $\underline{R} (1) = 1$ ;

$A \leq B \Rightarrow \bar{R} (A) \leq \bar{R} (B)$ ;

$A \leq B \Rightarrow \underline{R} (A) \leq \underline{R} (B)$ ;

$\bar{R} (\underset{t \in T}{\lor} A_{t}) = \underset{t \in T}{\lor} \bar{R} (A_{t})$ ;

$\underline{R} (\underset{t \in T}{\land} A_{t}) = \underset{t \in T}{\land} \underline{R} (A_{t})$ ;

$\bar{R} (α * A) = α * \bar{R} (A)$ ;

$\underline{R} (α * A) \geq α * \underline{R} (A)$ ;

$\bar{R} (α \to A) \leq α \to \bar{R} (A)$ ;

$\underline{R} (α \to A) = α \to \underline{R} (A)$ .

2.2 φ-fuzzy rough sets

For A, B ∈ L^X, the subsethood degree S (A, B) is defined as: $S (A, B) = \underset{x \in X}{\land} (A (x) \to B (x))$ ; the related degree of A and B is defined as: $ρ (A, B) = \underset{x \in X}{\lor} (A (x) * B (x))$ . When L = {0, 1}, S (A, B) =1 means that A ⊆ B, and ρ (A, B) =1 means that A∩ B ≠ ∅.

Definition 2.3. [2] Let φ : M ⟶ L^X be an arbitrary function, where M = {{x_α} |x ∈ X, α ∈ L, α > 0}. Then two L-fuzzy rough operators N and H are defined as follows: for all A ∈ L^X, x ∈ X, $\begin{matrix} N (A) (x) & = & \underset{x_{α} \in M}{⋁} [α * S (φ (x_{α}), A)], \\ H (A) (x) & = & \underset{x_{α} \in M}{⋁} [α * ρ (φ (x_{α}), A)] . \end{matrix}$ The pair (N (A) , H (A)) is referred to as a φ-fuzzy rough set in this paper.

In [2], Chen and Li pointed out that R-fuzzy rough sets can be regarded as special φ-fuzzy rough sets. Indeed, for an L-fuzzy relation R, we can define a function φ^R : M ⟶ L^X as φ^R (x_α) = R (x) for each x_α ∈ M, where R (x) is an L-set on X defined by R (x) (y) = R (x, y) for every y ∈ X. R (x) is usually interpreted as the successor neighborhood of x.

3 φ-fuzzy rough sets represented by a family of R-fuzzy rough sets

By using α-cut (or α-level), Yao [19] clearly demonstrated the relationships among rough sets, rough fuzzy sets, fuzzy rough sets and ordinary sets. Here by means of the idea of “level”, we define a family of L-fuzzy relations and thereby give the representation of φ-fuzzy rough sets.

Definition 3.1. Let φ : M ⟶ L^X be an arbitrary function. Then the set $Γ^{φ} = {R_{α}^{φ} | α \in L}$ , defined by $R_{α}^{φ} (x, y) = φ (x_{α}) (y)$ for all x, y ∈ X, is called the L-fuzzy relations family induced by φ. Conversely, for an L-fuzzy relations family Γ = {R_α|α ∈ L}, the function φ^Γ : M ⟶ L^X, defined by φ^Γ (x_α) = R_α (x), is called the function induced by Γ.

Remark 3.2. (1) It is easy to prove that φ^{Γ
^φ} = φ and Γ^{φ
^Γ} = Γ. Thus for a fixed set X, there exists a one-to-one correspondence between the functions φ : M ⟶ L^X and the L-fuzzy relations families Γ = {R_α|α ∈ L}. Moreover, note that $R_{α}^{φ}$ is an L-fuzzy relation and the L-set $R_{α}^{φ} (x)$ is generally called the successor neighborhood of point x. Thus, φ (x_α) can be interpreted as the the successor neighborhood of the fuzzy point x_α. In this sense, the function φ indeed assigns a successor neighborhood to each fuzzy point.

(2) In particular, let L = 2, that is, L = {0, 1}, then 2^X is just the power set of X. Since the singletons {x} (x ∈ X) of 2^X can be identified with the elements of X, the function φ : X ⟶ 2^X may be considered to be the form φ : M ⟶ L^X. Thus φ can be viewed as an extension of a neighborhood operator n in fuzzy settings.

Theorem 3.3. (φ-fuzzy rough sets represented by R-fuzzy rough sets) Let φ : M ⟶ L^X be a function. Then for all A ∈ L^X, x ∈ X, $\begin{matrix} N (A) (x) & = & \underset{x_{α} \in M}{⋁} [α * \underline{R_{α}^{φ} (} A) (x)], \\ H (A) (x) & = & \underset{x_{α} \in M}{⋁} [α * \bar{R_{α}^{φ}} (A) (x)] . \end{matrix}$

Proof. It follows immediately from the definitions of R-fuzzy rough sets and φ-fuzzy rough sets and Definition 3.1.

Proposition 3.4. Let φ : M ⟶ L^X be an arbitrary function. Then for all A, B ∈ L^X,{A_t} _t∈T ⊆ L^X, α ∈ L, we have

H (0) = 0; (N1) N (1) = 1;

A ≤ B ⇒ H (A) ≤ H (B);

A ≤ B ⇒ N (A) ≤ N (B);

$H (\underset{t \in T}{\lor} A_{t}) = \underset{t \in T}{\lor} H (A_{t})$ ;

$N (\underset{t \in T}{\land} A_{t}) \leq \underset{t \in T}{\land} N (A_{t})$ ;

H (α ∗ A) = α ∗ H (A);

N (α ∗ A) ≥ α ∗ N (A);

H (α → A) ≤ α → H (A);

N (α → A) ≤ α → N (A).

Proof. (H1)-(H2) and (N1)-(N3) have been proven in [2]. Next, we prove the others. For each x ∈ X, $\begin{matrix} H (\underset{t \in T}{⋁} A_{t}) (x) \\ = \underset{x_{α} \in M}{⋁} [α * \bar{R_{α}^{φ}} (\underset{t \in T}{⋁} A_{t}) (x)] \\ \overset{(U 3)}{=} \underset{x_{α} \in M}{⋁} [α * \underset{t \in T}{⋁} \bar{R_{α}^{φ}} (A_{t}) (x)] \\ = \underset{x_{α} \in M}{⋁} \underset{t \in T}{⋁} [α * \bar{R_{α}^{φ}} (A_{t}) (x)] = \underset{t \in T}{⋁} H (A_{t}) (x), \\ H (α * A) (x) = \underset{x_{β} \in M}{⋁} [β * \bar{R_{β}^{φ}} (α * A) (x)] \\ \overset{(U 4)}{=} \underset{x_{β} \in M}{⋁} [β * α * \bar{R_{β}^{φ}} (A) (x)] = α * H (A) (x), \\ H (α \to A) (x) = \underset{x_{β} \in M}{⋁} [β * \bar{R_{β}^{φ}} (α \to A) (x)] \\ \overset{(U 5)}{\leq} \underset{x_{β} \in M}{⋁} [β * (α \to \bar{R_{β}^{φ}} (A) (x))] \\ \leq \underset{x_{β} \in M}{⋁} [α \to (β * \bar{R_{β}^{φ}} (A) (x))] \\ \leq α \to \underset{x_{β} \in M}{⋁} [β * \bar{R_{β}^{φ}} (A) (x)] \\ = α \to H (A) (x), \\ N (α * A) (x) = \underset{x_{β} \in M}{⋁} [β * \underline{R_{β}^{φ}} (α * A) (x)] \\ \overset{(L 4)}{\geq} \underset{x_{β} \in M}{⋁} [α * β * \underline{R_{β}^{φ}} (A) (x)] \\ = α * N (A) (x), \\ N (α \to A) (x) = \underset{x_{β} \in M}{⋁} [β * \underline{R_{β}^{φ}} (α \to A) (x)] \\ \overset{(L 5)}{=} \underset{x_{β} \in M}{⋁} [β * (α \to \underline{R_{β}^{φ}} (A) (x))] \\ \leq \underset{x_{β} \in M}{⋁} [α \to (β * \underline{R_{β}^{φ}} (A) (x))] \\ \leq α \to \underset{x_{β} \in M}{⋁} [β * \underline{R_{β}^{φ}} (A) (x)] \\ = α \to N (A) (x) . \end{matrix}$

4 Some special cases of φ

In the study of rough sets in the conventional settings, Yao [27] has systematically considered different types of binary relation such as the inverse serial, reflexive, symmetric and transitive relations, etc. In this section, we shall discuss similarly the corresponding φ-fuzzy rough sets.

An L-fuzzy relation R on X is called

serial if $\lor R (x) : = \underset{y \in X}{\lor} R (x, y) = 1$ for every x ∈ X;

reflexive if R (x, x) =1 for every x ∈ X;

symmetric if R (x, y) = R (y, x) for all x, y ∈ X;

transitive if R (x, y) ∗ R (y, z) ≤ R (x, z) for all x, y, z ∈ X;

Euclidean if R (x, y) ∗ R (x, z) ≤ R (y, z) for all x, y, z ∈ X.

4.1 φ is serial

Definition 4.1. A function φ : M ⟶ L^X is said to be serial if for each x_α ∈ M, $⋁ φ (x_{α}) : = \underset{y \in X}{⋁} φ (x_{α}) (y) \geq α$ .

The following proposition shows that the serial condition of φ corresponds to the serial condition of L-fuzzy relation.

Proposition 4.2. Let R be an L-fuzzy relation on X. Then φ^R is serial iff R is serial.

Proof.φ^R is serial iff for each x_α ∈ M, ⋁φ^R (x_α) ≥ α iff for each x ∈ X, ⋁R (x) ≥ ∨ α = 1 iff R is serial.

Theorem 4.3. Let φ : M ⟶ L^X be a function. Then the following conditions are equivalent:

(H6) H (1) =1; (H7) H (α) = α for all α ∈ L.

If φ is serial, then (H6) and (H7) hold.

Proof. For the equivalence (H6)⇔(H7), it suffices to prove (H6)⇒(H7). Indeed, for each x ∈ X, $\begin{matrix} H (α) (x) & = & \underset{x_{β} \in M}{⋁} [β * \underset{y \in X}{⋁} (φ (x_{β}) (y) * α)] \\ = & α * \underset{x_{β} \in M}{⋁} [β * \lor φ (x_{β})] = α * H (1) (x) . \end{matrix}$ This shows H (α) = α ∗ H (1). Thus H (1) =1 implies H (α) = α.

Next, we prove that (H6) holds when φ is serial. Indeed, for each x ∈ X, $\begin{matrix} H (1) (x) & = & \underset{x_{β} \in M}{⋁} [β * ⋁ φ (x_{β})] \\ \overset{serial}{\geq} \underset{x_{β} \in M}{⋁} [β * β] \geq 1 * 1 = 1 . \end{matrix}$

The following example shows that (H6) does not imply the serial condition of φ generally.

Example 4.4. Let L = {0, a, b, 1}, where 0 < a, b < 1 but a, b are incomparable. Put ∗ =∧, then it is easily seen that L = (L, ∗ , 1) is a regular residuated lattice.

Let X = {x}. Take φ (x_a) = φ (x₁) = x_a, φ (x_b) = x_b, then ⋁φ (x_a) = ⋁ φ (x₁) = a, ⋁ φ (x_b) = b. φ is not serial since ⋁φ (x₁) = a ≠ 1. While (H6) holds since H (1) = ⋁ _{x_β∈M} [β ∗ ⋁ φ (x_β)] = a ∨ b = 1.

Theorem 4.5. Let φ : M ⟶ L^X be serial. Then the operator N satisfies (N6) N (0) =0, and (N7) N (α) = α for each α ∈ L. If L is regular, then (N6)⇔(N7)⇔ φ is serial.

Proof. We prove only (N7) since (N7) implies (N6). For all α ∈ L, x ∈ X, $\begin{matrix} N (α) (x) & = & \underset{x_{β} \in M}{⋁} [β * (\lor φ (x_{β}) \to α)] \\ \geq & \underset{x_{β} \in M}{⋁} [β * α] = α * \underset{x_{β} \in M}{⋁} β = α, \\ N (α) (x) & = & \underset{x_{β} \in M}{⋁} [β * (\lor φ (x_{β}) \to α)] \\ \overset{serial}{\leq} \underset{x_{β} \in M}{⋁} [β * (β \to α)] \leq α . \end{matrix}$ Thus N (α) = α.

Let L be regular. To prove (N6)⇔(N7)⇔ φ is serial, it suffices to prove (N6)⇒ φ is serial. In fact, for each x ∈ X, $\begin{matrix} N (0) (x) & = & \underset{x_{β} \in M}{⋁} [β * (\lor φ (x_{β}) \to 0)] \overset{(N 6)}{=} 0 \\ \Leftrightarrow \forall x_{β} \in M, β * (\lor φ (x_{β}) \to 0) \leq 0 \\ \Leftrightarrow \forall x_{β} \in M, β \leq (\lor φ (x_{β}) \to 0) \\ \to 0 \overset{regular}{=} \lor φ (x_{β}) . \end{matrix}$

As we have seen that R-fuzzy rough sets can be regarded as φ-fuzzy rough sets. We observe from Example 4.1 in [22] that (N6)⇔(N7) does not hold for general complete residuated lattices.

4.2 φ is reflexive

Definition 4.6. A function φ : M ⟶ L^X is said to be reflexive (or extensive in [2]) if for each x_α ∈ M, φ (x_α) ≥ x_α.

The following proposition shows that the reflexive condition of φ corresponds to the reflexive condition of L-fuzzy relation.

Proposition 4.7. Let R be an L-fuzzy relation on X. Then φ^R is reflexive iff R is reflexive.

Proof.φ^R is reflexive iff for each x_α ∈ M, φ (x_α) (x) ≥ α iff for each x ∈ X, R (x) (x) ≥ ∨ α = 1 iff R is reflexive.

Theorem 4.8. [2] Let φ : M ⟶ L^X be reflexive. Then the following conditions hold: (N8) ∀A ∈ L^X, N (A) ≤ A; (H8) ∀A ∈ L^X, H (A) ≥ A.

Proposition 4.9. Let L be regular. Then φ is reflexive iff (N8) holds.

Proof. It is similar to Theorem 4.5.

Remark 4.10. There is no such result that (H8) ⇔ reflexivity, even if L is regular. Indeed, let X and L be defined as in Example 4.4. Take φ (x_a) = x_b, φ (x₁) = x₁, φ (x_b) = x_a. Obviously, φ is not reflexive. But we can check easily that H (A) = A for each A ∈ L^X. Thus (H8) holds.

4.3 φ is symmetric

Definition 4.11. A function φ : M ⟶ L^X is said to be symmetric if for all x_α, y_β ∈ M, α ∗ β ≤ φ (x_α) (y) → φ (y_β) (x).

The following proposition shows that the symmetry of φ corresponds to the symmetry of L-fuzzy relation.

Proposition 4.12. Let R be an L-fuzzy relation on X. Then φ^R is symmetric iff R is symmetric.

Proof. Let φ^R be symmetric. Then for all x, y ∈ X, 1 ∗ 1 ≤ φ^R (x₁) (y) → φ^R (y₁) (x) = R (x, y) → R (y, x), i.e., R (x, y) ≤ R (y, x). Dually, we obtain R (x, y) ≥ R (y, x). Thus R (x, y) = R (y, x), i.e., R is symmetric. Conversely, let R be symmetric. Then for all x_α, y_β ∈ M, we have α ∗ β ≤ 1 = φ^R (x_α) (y) → φ^R (y_β) (x) = R (x, y) → R (y, x).

Remark 4.13. In [2], a function φ : M ⟶ L^X is called symmetric if x_α ≤ φ (y_β) ⇔ y_β ≤ φ (x_α) for all x_α, y_β ∈ M. Unfortunately, under this case, we do not have the result that φ^R is symmetric iff R is symmetric.

Theorem 4.14. Let φ : M ⟶ L^X be a function.

If φ is symmetric, then HN (A) ≤ A ≤ NH (A) for every A ∈ L^X;

If L is regular, then φ is symmetric iff HN (A) ≤ A for every A ∈ L^X.

Proof. (1) Let φ be symmetric. For each x ∈ X, we have $\begin{matrix} HN (A) (x) \\ = \underset{x_{α} \in M}{⋁} α * [\underset{y \in X}{⋁} (φ (x_{α}) (y) * \\ \underset{y_{β} \in M}{⋁} [β * \underset{z \in X}{⋀} (φ (y_{β}) (z) \to A (z))])] \\ \leq \underset{x_{α} \in M}{⋁} α * [\underset{y \in X}{⋁} (φ (x_{α}) (y) * \underset{y_{β} \in M}{⋁} [β * \\ (φ (y_{β}) (x) \to A (x))])] \\ = \underset{x_{α} \in M}{⋁} \underset{y \in X}{⋁} \underset{y_{β} \in M}{⋁} [α * φ (x_{α}) (y) * β * \\ (φ (y_{β}) (x) \to A (x))] \\ \overset{symmetric}{\leq} \underset{x_{α} \in M}{⋁} \underset{y \in X}{⋁} \underset{y_{β} \in M}{⋁} [φ (y_{β}) (x) * (φ (y_{β}) (x) \\ \to A (x))] \leq A (x) . \end{matrix}$

$\begin{matrix} NH (A) (x) \\ = \underset{x_{α} \in M}{⋁} α * [\underset{y \in X}{⋀} (φ (x_{α}) (y) \to \\ \underset{y_{β} \in M}{⋁} [β * \underset{z \in X}{⋁} (φ (y_{β}) (z) * A (z))])] \\ \geq \underset{x_{α} \in M}{⋁} α * [\underset{y \in X}{⋀} (φ (x_{α}) (y) \to \underset{y_{β} \in M}{⋁} [β * \\ φ (y_{β}) (x) * A (x)])] \\ \geq \underset{x_{α} \in M}{⋁} α * [\underset{y \in X}{⋀} \underset{y_{β} \in M}{⋁} (φ (x_{α}) (y) \to [β * \\ φ (y_{β}) (x) * A (x)])] \\ \geq \underset{x_{α} \in M}{⋁} α * [\underset{y \in X}{⋀} \underset{y_{β} \in M}{⋁} (β * A (x) * [φ (x_{α}) (y) \\ \to φ (y_{β}) (x)])] \\ \overset{symmetric}{\geq} \underset{x_{α} \in M}{⋁} α * [\underset{y_{β} \in M}{⋁} (β * A (x) * α * β)] \\ = \underset{x_{α} \in M}{⋁} [α * A (x) * α] = A (x) . \end{matrix}$

(2) Let L be regular and HN (A) ≤ A for each A ∈ L^X. For each x ∈ X, take A = 1_X-{x}. Then $\begin{matrix} HN (1_{X - {x}}) (x) \\ = \underset{x_{α} \in M}{⋁} α * [\underset{y \in X}{⋁} (φ (x_{α}) (y) * \\ \underset{y_{β} \in M}{⋁} [β * (φ (y_{β}) (x) \to 0)])] \\ = \underset{x_{α} \in M}{⋁} \underset{y \in X}{⋁} \underset{y_{β} \in M}{⋁} [α * φ (x_{α}) (y) * β * \\ (φ (y_{β}) (x) \to 0)] \\ \leq 1_{X - {x}} (x) = 0 . \end{matrix}$ This means for all x_α, y_β ∈ M, we have $\begin{matrix} α * φ (x_{α}) (y) * β * (φ (y_{β}) (x) \to 0) \leq 0 \\ \Leftrightarrow α * β * φ (x_{α}) (y) \leq (φ (y_{β}) (x) \to 0) \to 0 \\ \overset{regular}{=} φ (y_{β}) (x) \\ \Leftrightarrow α * β \leq φ (x_{α}) (y) \to φ (y_{β}) (x) . \end{matrix}$

The following example shows that symmetry of φ is a sufficient but not a necessary condition such that NH (A) ≥ A for all A ∈ L^X even if L is regular.

Example 4.15. Let X and L be defined as in Example 4.4. Take $φ (x_{a}) = b, φ (x_{b}) = φ (x_{1}) = a .$ Then $1 \land a = a ≰ b = φ (x_{1}) (x) \to φ (x_{a}) (x) = a \to b .$ Thus φ is not regular.

For each A ∈ L^X, we distinguish four cases to prove NH (A) ≥ A.

Case 1:A = 0. It is obvious.

Case 2:A = 1. Then $\begin{matrix} H (1) & = & \underset{x_{α} \in M}{⋁} (α \land φ (x_{α}) \land 1) \\ = & (1 \land a) ⋁ (a \land b) ⋁ (b \land a) = a, \\ NH (1) & = & N (a) = \underset{x_{α} \in M}{⋁} (α \land (φ (x_{α}) \to a)) \\ \geq & 1 \land (a \to a) = 1 . \end{matrix}$

Case 3:A = a. Then $\begin{matrix} H (a) & = & \underset{x_{α} \in M}{⋁} (α \land φ (x_{α}) \land a) \\ = & a \land H (1) = a, NH (a) = N (a) = 1 . \end{matrix}$

Case 4:A = b. Then H (b) = b ∧ H (1) = b ∧ a = 0, and $NH (b) = N (0) = \underset{x_{α} \in M}{⋁} (α \land (φ (x_{α}) \to 0)) \geq b \land (a \to 0) = b .$

4.4 φ is transitive

Definition 4.16. A function φ : M ⟶ L^X is said to be transitive if for all x_α, y_β, z_α ∈ M, φ (x_α) (y) ∗ φ (y_β) (z) ≤ φ (x_α) (z).

The following proposition shows that the transitivity of φ corresponds to the transitivity of L-fuzzy relation.

Proposition 4.17. Let R be an L-fuzzy relation on X. Then φ^R is transitive iff R is transitive.

Proof. Let φ^R be transitive. Then for all x, y, z ∈ X, $\begin{matrix} R (x, y) * R (y, z) & = & φ^{R} (x_{1}) (y) * φ^{R} (y_{1}) (z) \\ \leq & φ^{R} (x_{1}) (z) = R (x, z), \end{matrix}$ i.e., R is transitive. Conversely, let R be transitive. Then for all x_α, y_β, z_α ∈ M, $\begin{matrix} φ^{R} (x_{α}) (y) * φ^{R} (y_{β}) (z) = R (x, y) * R (y, z) \\ \leq R (x, z) = φ^{R} (x_{α}) (z) . \end{matrix}$

Theorem 4.18. Let φ : M ⟶ L^X be transitive. Then the following conditions hold: (N9) ∀A ∈ L^X, NN (A) ≥ N (A); (H9) ∀A ∈ L^X, HH (A) ≤ H (A).

Proof. For each x ∈ X, we have $\begin{matrix} NN (A) (x) \\ = \underset{x_{α} \in M}{⋁} α * [\underset{y \in X}{⋀} (φ (x_{α}) (y) \to \\ \underset{y_{β} \in M}{⋁} [β * \underset{z \in X}{⋀} (φ (y_{β}) (z) \to A (z))])] \\ \geq \underset{x_{α} \in M}{⋁} α * [\underset{y \in X}{⋀} \underset{y_{β} \in M}{⋁} (φ (x_{α}) (y) \to [β * \\ \underset{z \in X}{⋀} (φ (y_{β}) (z) \to A (z))])] \\ \geq \underset{x_{α} \in M}{⋁} α * [\underset{y \in X}{⋀} \underset{y_{β} \in M}{⋁} β * (φ (x_{α}) (y) \to \\ \underset{z \in X}{⋀} (φ (y_{β}) (z) \to A (z)))] \\ = \underset{x_{α} \in M}{⋁} α * [\underset{y \in X}{⋀} \underset{y_{β} \in M}{⋁} β * \underset{z \in X}{⋀} (φ (x_{α}) (y) \to \\ (φ (y_{β}) (z) \to A (z)))] \\ = \underset{x_{α} \in M}{⋁} α * [\underset{y \in X}{⋀} \underset{y_{β} \in M}{⋁} β * \underset{z \in X}{⋀} ((φ (x_{α}) (y) * \\ φ (y_{β}) (z)) \to A (z))] \\ \overset{transitive}{\geq} \underset{x_{α} \in M}{⋁} α * [\underset{y \in X}{⋀} \underset{y_{β} \in M}{⋁} β * \\ \underset{z \in X}{⋀} (φ (x_{α}) (z) \to A (z))] \\ = \underset{x_{α} \in M}{⋁} α * \underset{z \in X}{⋀} (φ (x_{α}) (z) \to A (z)) = N (A) (x) . \end{matrix}$ $\begin{matrix} HH (A) (x) & = & \underset{x_{α} \in M}{⋁} α * [\underset{y \in X}{⋁} (φ (x_{α}) (y) * \\ \underset{y_{β} \in M}{⋁} [β * \underset{z \in X}{⋁} (φ (y_{β}) (z) * A (z))])] \\ = & \underset{x_{α} \in M}{⋁} \underset{y \in X}{⋁} \underset{y_{β} \in M}{⋁} \underset{z \in X}{⋁} [α * β * φ (x_{α}) (y) \\ * φ (y_{β}) (z) * A (z)] \\ \overset{transitive}{\leq} \underset{x_{α} \in M}{⋁} \underset{y \in X}{⋁} \underset{y_{β} \in M}{⋁} \underset{z \in X}{⋁} [α * β * \\ φ (x_{α}) (z) * A (z)] \\ = & \underset{x_{α} \in M}{⋁} \underset{z \in X}{⋁} [α * φ (x_{α}) (z) * A (z)] \\ = & H (A) (x) . \end{matrix}$

Remark 4.19. We still do not know whether NN (A) ≥ N (A), or HH (A) ≤ H (A) for all A ∈ L^X implies the transitivity of φ. We tend to believe that the answer is negative.

4.5 φ is Euclidean

Definition 4.20. A function φ : M ⟶ L^X is said to be Euclidean if for all x, y, z ∈ X, α, β ∈ L - {0}, φ (x_α) (y) ∗ φ (x_α) (z) ≤ φ (y_β) (z).

Proposition 4.21. Let R be an L-fuzzy relation on X. Then φ^R is Euclidean iff R is Euclidean.

Proof. It follows from the following equivalence: for all x, y, z ∈ X, α, β ∈ L - {0}, $\begin{matrix} φ^{R} (x_{α}) (y) * φ^{R} (x_{α}) (z) \leq φ^{R} (y_{β}) (z) \\ \Leftrightarrow R (x, y) * R (x, z) \leq R (y, z) . \end{matrix}$

Proposition 4.22. Let φ : M ⟶ L^X be Euclidean. Then NH (A) ≥ H (A) and HN (A) ≤ N (A) for each A ∈ L^X.

Proof. For each x ∈ X, we have $\begin{matrix} NH (A) (x) \\ = \underset{x_{α} \in M}{⋁} α * [\underset{y \in X}{⋀} (φ (x_{α}) (y) \to \\ \underset{y_{β} \in M}{⋁} [β * \underset{z \in X}{⋁} (φ (y_{β}) (z) * A (z))])] \\ \geq \underset{x_{α} \in M}{⋁} α * [\underset{y \in X}{⋀} \underset{y_{β} \in M}{⋁} \underset{z \in X}{⋁} (φ (x_{α}) (y) \to \\ [β * φ (y_{β}) (z) * A (z)])] \\ \geq \underset{x_{α} \in M}{⋁} α * [\underset{y \in X}{⋀} \underset{y_{β} \in M}{⋁} \underset{z \in X}{⋁} (β * A (z) * \\ [φ (x_{α}) (y) \to φ (y_{β}) (z)])] \\ \overset{Euclidean}{\geq} \underset{x_{α} \in M}{⋁} α * [\underset{y \in X}{⋀} \underset{y_{β} \in M}{⋁} \underset{z \in X}{⋁} (β * \\ A (z) * φ (x_{α}) (z))] \\ = \underset{x_{α} \in M}{⋁} α * [\underset{z \in X}{⋁} (A (z) * φ (x_{α}) (z))] = H (A) (x), \end{matrix}$ $\begin{matrix} HN (A) (x) \\ = \underset{x_{α} \in M}{⋁} α * [\underset{y \in X}{⋁} (φ (x_{α}) (y) * \\ \underset{y_{β} \in M}{⋁} [β * \underset{z \in X}{⋀} (φ (y_{β}) (z) \to A (z))])] \\ \leq \underset{x_{α} \in M}{⋁} α * \underset{y \in X}{⋁} \underset{y_{β} \in M}{⋁} \underset{z \in X}{⋀} [β * φ (x_{α}) (y) * \\ (φ (y_{β}) (z) \to A (z))] \\ \leq \underset{x_{α} \in M}{⋁} α * \underset{y \in X}{⋁} \underset{y_{β} \in M}{⋁} \underset{z \in X}{⋀} [β * ([φ (x_{α}) (y) \to \\ φ (y_{β}) (z)] \to A (z))] \\ \overset{Euclidean}{\leq} \underset{x_{α} \in M}{⋁} α * \underset{y \in X}{⋁} \underset{y_{β} \in M}{⋁} \underset{z \in X}{⋀} [β * \\ (φ (x_{α}) (z) \to A (z))] \\ = \underset{x_{α} \in M}{⋁} α * \underset{z \in X}{⋀} [(φ (x_{α}) (z) \to A (z))] \\ = N (A) (x) . \end{matrix}$

In general, for each A ∈ L^X, NH (A) ≥ H (A), or HN (A) ≤ N (A) does not imply that φ is Euclidean.

Example 4.23. Let X and L be defined as in Example 4.4. Take $φ (x_{a}) = a, φ (x_{b}) = b, φ (x_{1}) = 1 .$ Then $φ (x_{a}) (x) \land φ (x_{a}) (x) = a \land a = a ≰ b = φ (x_{b}) (x) .$ Thus φ is not Euclidean. But it is easily seen that N (A) = H (A) = A for each A ∈ L^X.

5 The L-topology induced by φ-fuzzy rough sets

In this section, when φ is reflexive, or reflexive and transitive, we shall study the L-topology induced by φ-fuzzy rough sets.

Let X be a non-empty set. Then a subset $T \subseteq L^{Xitsc}$ is called an Alexandrov L-topology if it satisfies (LT1) $0, 1 \in T$ ; (LT2) ${A_{j} | j \in J} \subseteq T \Rightarrow ⋀_{j \in J} A_{j} \in T$ ; (LT3) ${A_{j} | j \in J} \subseteq T \Rightarrow ⋁_{j \in J} A_{j} \in T$ ; (LT4) $A \in T, α \in L \Rightarrow α * A \in T$ ; and (LT5) $A \in T, α \in L \Rightarrow α \to A \in T$ .

Theorem 5.1. Let φ : M ⟶ L^X be reflexive. Then the set $T_{H} = {H (A) = A | A \in L^{Xitsc}}$ forms an Alexandrov L-topology on X.

Proof. (1) From (H1) and (H8), we know that $0, 1 \in T_{H}$ .

(2) Let ${A_{t}}_{t \in T} \subseteq T_{H} .$ Then for every t ∈ T we have H (A_t) = A_t. By (H2) we have $H (\underset{t \in T}{\land} A_{t}) \leq \underset{t \in T}{\land} H (A_{t}) = \underset{t \in T}{\land} A_{t}$ , then by (H8) we obtain $\underset{t \in T}{\land} A_{t} \leq H (\underset{t \in T}{\land} A_{t})$ . Thus $H (\underset{t \in T}{\land} A_{t}) = \underset{t \in T}{\land} A_{t}$ , i.e., $\underset{t \in T}{\land} A_{t} \in T_{H} .$

(3) Let ${A_{t}}_{t \in T} \subseteq T_{H}$ . Then for every t ∈ T we have H (A_t) = A_t. Next, we prove $\underset{i \in I}{\lor} A_{i} \in T_{H}$ . In fact, by (H3) we have $H (\underset{t \in T}{\lor} A_{t}) = \underset{t \in T}{\lor} H (A_{t}) = \underset{t \in T}{\lor} A_{t}$ , i.e., $\underset{t \in T}{\lor} A_{t} \in T_{H}$ .

(4) Let $A \in T_{H}$ , by (H4) we have H (α ∗ A) = α ∗ H (A) = α ∗ A, i.e., $α * A \in T_{H}$ .

(5) Let $A \in T_{H}$ . Then by (H5) we have H (α → A) ≤ α → H (A) = α → A. On the other hand, by (H8) we have H (α → A) ≥ α → A. So, H (α → A) = α → A, i.e., $α \to A \in T_{H}$ .

Theorem 5.2. Let φ : M ⟶ L^X be reflexive and transitive. Then the set ${\tilde{T}}_{H} = {H (A) | A \in L^{X}}$ forms an Alexandrov L-topology on X.

Proof. (1) By (H1) and (H8), we have $0, 1 \in {\tilde{T}}_{H}$ .

(2) For all ${B_{t}}_{t \in T} \subseteq {\tilde{T}}_{H}$ , we have {A_t} _t∈T ⊆ L^X such that H (A_t) = B_t. From (H8), we obtain $H (\underset{t \in T}{\land} H (A_{t})) \geq \underset{t \in T}{\land} H (A_{t})$ . On the other hand, $H (\underset{t \in T}{\land} H (A_{t})) \overset{(H 2)}{\leq} \underset{t \in T}{\land} HH (A_{t}) \overset{(H 9)}{=} \underset{t \in T}{\land} H (A_{t}) .$ So, $\underset{t \in T}{\land} B_{t} = \underset{t \in T}{\land} H (A_{t}) = H (\underset{t \in T}{\land} H (A_{t})) \in {\tilde{T}}_{H}$ .

(3) For all ${B_{t}}_{t \in T} \subseteq {\tilde{T}}_{H}$ , we have {A_t} _t∈T ⊆ L^X such that H (A_t) = B_t. Then $\underset{t \in T}{\lor} B_{t} = \underset{t \in T}{\lor} H (A_{t}) \overset{(H 3)}{=} H (\underset{t \in T}{\lor} A_{t}) \in {\tilde{T}}_{H} .$

(4) For each $B \in {\tilde{T}}_{H}$ , we have A ∈ L^X such that H (A) = B. Then $α * B = α * H (A) \overset{(H 4)}{=} H (α * A) \in {\tilde{T}}_{H} .$

(5) For each $B \in {\tilde{T}}_{H}$ , we have A ∈ L^X such that H (A) = B. Then $H (α \to H (A)) \overset{(H 5)}{\leq} α \to HH (A) \overset{(H 9)}{=} α \to H (A) .$

On the other hand, by (H8), we have H (α → H (A)) ≥ α → H (A). Therefore, $α \to B = α \to H (A) = H (α \to H (A)) \in {\tilde{T}}_{H} .$

Remark 5.3. (1) Even if φ = φ^R, the reflexivity in Theorem 5.1, and the transitivity in Theorem 5.2, can not be removed generally, see [13].

(2) It is easily seen that ${\tilde{T}}_{H} = T_{H}$ when φ is reflexive and transitive.

(3) The operator N does not induce someL-topologies since it is not meet-preserving generally.

6 Conclusion

This paper has examined several issues related to φ-fuzzy rough sets. To some extent, the results are generalizations of the corresponding notions in existing studies to a setting of L-fuzzy sets. Particularly, we show that φ-fuzzy rough sets can be represented by a family of R-fuzzy rough sets. This result further exhibits the close relationships between R-fuzzy rough sets and φ-fuzzy rough sets.

In addition, we have made a systematic study of some special cases of φ-fuzzy rough sets. Since the main properties of these special cases are related to some axioms of fuzzy modal logics, we may formulate the characteristic axioms for the systems of fuzzy modal logics.

Footnotes

Acknowledgments

The authors thank the reviewers and the Associate Editor for their valuable comments and suggestions. This work is supported by National Natural Science Foundation of China (11371130, 11501278).

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