Axiomatization and conditions for neighborhoods in a covering to form a partition

doi:10.1016/j.ins.2011.01.013

Information Sciences

Volume 181, Issue 9, 1 May 2011, Pages 1735-1740

https://doi.org/10.1016/j.ins.2011.01.013 Get rights and content

Abstract

In this paper, we study the axiomatic issue of a type of covering upper approximation operations. This issue was proposed as an open problem. We also further some known results by using only a single covering approximation operator to characterize the conditions for neighborhood {N(x): x ∈ U} to form a partition of universe U.

Introduction

The classical rough set theory is built on equivalence relation [4], [10], [11], [12], [13], [14], [15], [16], [18], [23], [28]. However, equivalence relation imposes restrictions and limitations on many applications [20], [21], [30], [31]. Zakowski then established the covering-based rough set theory by exploiting coverings of a universe [24]. This brings various covering approximation operators [3], [19], [22]. Axiomatizations of these covering approximation operators are important topics in covering-based rough set theory [7], [17], [27], [28], [29], [30], [32]. As a type of special coverings, topology provides many useful techniques in covering-based rough set theory [1], [5], [6], [8], [9]. From the perspective of point-set topology, the core concept of covering approximation operators is neighborhood N(x) of a point x. Therefore, it is important to study the conditions for {N(x): x ∈ U} to form a partition of universe U [17].

In this paper, we construct a set of axioms to characterize a class of covering upper approximation operations for an arbitrary universe. In particular, when the universe is finite, our results provide answers to an open problem proposed by W. Zhu and F. Wang in paper [32]. We also further some results presented by K. Qin et al. in paper [17] by providing sufficient and necessary conditions for {N(x): x ∈ U} to form a partition on U.

Before our discussion, we introduce some basic concepts used in this paper. U always denotes a non-empty arbitrary set unless it is specially mentioned. C denotes a covering of U and P(U) denotes the family of subsets of U. We call ordered pair (U, C) a covering approximation space, and N(x) = ∩{K ∈ C: x ∈ K} neighborhood of point x for each x ∈ U.

Let (U, C) be a covering approximation space. Our discussion in this paper involves six types of covering approximation operations that are defined as follows: for X ⊆ U,

(1)
$\underset{̲}{C_{1}} (X) = ⋃ {K : K \in C ⋀ K \subseteq X}, \bar{C_{1}} (X) = ⋃ {K : K \in C ⋀ K ⋂ X \neq \emptyset}$ ;
(2)
$\underset{̲}{C_{2}} (X) = ⋃ {K : K \in C ⋀ K \subseteq X}, \bar{C_{2}} (X) = U - \underset{̲}{C_{2}} (U - X)$ ;
(3)
$\underset{̲}{C_{3}} (X) = {x \in U : N (x) \subseteq X}, \bar{C_{3}} (X) = {x \in U : N (x) ⋂ X \neq \emptyset}$ ;
(4)
$\underset{̲}{C_{4}} (X) = {x \in U : \exists u (u \in N (x) \land N (u) \subseteq X)}, \bar{C_{4}} (X) = {x \in U : \forall u (u \in N (x) \to N (u) \cap X \neq \emptyset)}$ ;
(5)
$\underset{̲}{C_{5}} (X) = {x \in U : \forall u (x \in N (u) \to N (u) \subseteq X)}, \bar{C_{5}} (X) = ⋃ {N (x) : x \in U ⋀ N (x) ⋂ X \neq \emptyset}$ ; and
(6)
$\underset{̲}{C_{6}} (X) = {x \in U : \forall u (x \in N (u) \to u \in X)}, \bar{C_{6}} (X) = ⋃ {N (x) : x \in X}$ .

Remark 1.1

We call C_n the covering lower approximation operation and $\bar{C_{n}}$ the covering upper approximation operation (n = 1, 2, 3, 4, 5, 6). C₁ and $\bar{C_{1}}$ come from [32], and for 2 ⩽ n ⩽ 6, C_n and $\bar{C_{n}}$ come from [17].

Let 1 ⩽ n ⩽ 6. If for each X ⊆ U, approximation operators C_n and $\bar{C_{n}}$ satisfy $\underset{̲}{C_{n}} (X) = - \bar{C_{n}} (- X)$ , then C_n and $\bar{C_{n}}$ are called dual approximation operators. Clearly, except C₁ and $\bar{C_{1}}$ , all other C_n and $\bar{C_{n}}$ are dual approximation operators.

The rest sections of this paper are arranged as follows. In Section 2, we construct a set of axioms that characterizes a covering upper approximation operation defined in [32] and explore the properties of this operator from the perspective of point-set topology. In Section 3, we use a single covering approximation operator to give the conditions for {N(x): x ∈ U} to form a partition of U. Section 4 concludes the paper.

Section snippets

Axiomatization and topological properties of covering approximation operation C₁

Axiomatization of covering upper approximation operation $\bar{C_{1}}$ for a non-empty finite set U was proposed as an open problem [32]. In the following, we present an axiomatization theorem of $\bar{C_{1}}$ for any nonempty arbitrary U.

Theorem 2.1

An operation L: P(U) → P(U) satisfies the following properties: for any X, Y ⊆ U,

(1)
∀x ∈ U, x ∈ L({x}),
(2)
L(∅) = ∅,
(3)
∀H ⊆ U, L(H) = ∪_x∈HL({x}), and
(4)
∀x, y ∈ U, $x \in L ({y}) \Leftrightarrow y \in L ({x})$ ,

if and only if there exists a covering C of U, such that covering upper approximation operation

\bar{C_{1}}

generated by C is equal to L.

On the conditions of neighborhoods forming a partition

We first introduce the following Theorem 3.1 that was proved in [17] for completeness.

Theorem 3.1

Let (U, C) be a covering approximation space.

(1)
{N(x): x ∈ U} forms a partition of U if and only if for each X ⊆ U, $\underset{̲}{C_{5}} (X) = \underset{̲}{C_{3}} (X)$ .
(2)
{N(x): x ∈ U} forms a partition of U if and only if for each X ⊆ U, $\bar{C_{3}} (X) = \bar{C_{4}} (X) = \bar{C_{5}} (X) = \bar{C_{6}} (X)$ .

It is then nature to raise the following question following Theorem 3.1.

Question 3.2

Can we characterize the conditions under which {N(x): x ∈ U} forms a partition of U by using only a single covering approximation

Conclusions

In this paper, we present a new method to solve axiomatization of covering upper approximation operation $\bar{C_{1}}$ , which was proposed as an open problem in [32], and explore the covering approximation space from the perspective of point-set topology. We also further the results in paper [17] by using only a single covering approximation operator to give the conditions for {N(x): x ∈ U} to form a partition of U.

There are several remaining issues in axiomatic systems of covering-based rough sets that

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^☆: This paper is supported by NSFC (Nos. 61070245, 10971185 and 11061004).

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Axiomatization and conditions for neighborhoods in a covering to form a partition☆

Abstract

Introduction

Section snippets

Axiomatization and topological properties of covering approximation operation C1

On the conditions of neighborhoods forming a partition

Conclusions

Topology and its Applications

Information Sciences

International Journal of Approximate Reasoning

Information Sciences

Information Sciences

Information Sciences

Information Sciences

Information Sciences

Information Sciences

Information Sciences

Information Sciences

Information Sciences

Information Sciences

Rings of sets

Duke Mathematical Journal

Abstract approximation spaces for rough theories

On the structure of generalized rough sets

Information Sciences

Axiomatization and topological properties of covering approximation operation C₁