Rough set theory is an efficient and essential tool for dealing with vagueness and granularity in information systems. Covering-based rough set theory is proposed as a significant generalization of classical rough sets. Matroid theory is a vital structure with high applicability and borrows extensively from linear algebra and graph theory. In this paper, one type of covering-based approximations is studied from the viewpoint of Eulerian matroids. First, we explore the circuits of an Eulerian matroid from the perspective of coverings. Second, this type of covering-based approximations is represented by the circuits of Eulerian matroids. Moreover, the conditions under which the covering-based upper approximation operator is the closure operator of a matroid are presented. Finally, a matroidal structure of covering-based rough sets is constructed. These results show many potential connections between covering-based rough sets and matroids.

Various theories and methods have been proposed to deal with incomplete and insufficient information in classification, concept formation, and data analysis in data mining. For example, fuzzy set theory [

The concept of matroids was originally introduced by Whitney [

In this paper, through Eulerian matroids, one type of covering-based approximation operators is represented through the perspective of Eulerian matroids. First, since the family of circuits of an Eulerian matroid is a covering, then the properties, such as irreducible, semireduced, and minimal description, are investigated from the viewpoint of coverings. Second, the second type of covering-based upper and lower approximation operators is represented by the circuits of the Eulerian matroid. In fact, some types of covering-based lower approximation operators are equal, and they can be represented by the circuits of the restriction matroids of Eulerian matroids. Moreover, we obtain the condition under which the covering-based upper approximation operator is the closure operator of a matroid. Finally, a matroidal structure of covering-based rough sets is constructed by a type of covering with some special properties.

The remainder of this paper is organized as follows. In Section

In this section, we review some fundamental definitions and results of covering-based rough sets and matroids.

In this subsection, we recall some basic definitions and results of covering-based rough sets used in this paper. For detailed descriptions about covering-based rough sets, please refer to [

Let

It is clear that a partition of

Let

Letting

Let

Let

Let

Let

The following lemma shows that the concepts of representative and exact coverings are equivalent.

Let

Matroid theory was established as a generalization of graph theory and linear algebra. This theory was used to study abstract relations on a subset, and it used both of these areas of mathematics for its motivation, its basic examples, and its notation. With the rapid development of matroid theory in recent years, it has already become an effective mathematical tool to study other mathematic branches. In this subsection, we recall some definitions, examples, and results of matroids.

A matroid is an ordered pair

if

if

Let

A graph.

In fact, if a subset is not an independent set, then it is a dependent set of the matroid. In other words, the dependent set of a matroid generalizes the cycle in graphs. Based on the dependent set, we introduce other concepts of a matroid. For this purpose, several denotations are presented in the following definition.

Let

Let

According to Definition

A matroid and its bases are uniquely determined by each other. In other words, a matroid can also be defined from the viewpoint of the base.

Let

if

Let

According to Definition

Let

if

if

A matroid uniquely determines its circuits and vice versa. The following theorem shows that a matroid can be defined from the viewpoint of circuits.

Let

Let

Based on the rank function of a matroid, one can define the closure operator, which reflects the dependency between a set and elements.

Let

In this section, we provide equivalent formulations of some important concepts and properties of covering-based rough sets from the viewpoint of Eulerian matroids. Specifically, the covering-based upper and lower approximation operators are characterized by the circuits of matroids. For this purpose, some definitions and properties of the covering induced by an Eulerian matroid are presented.

The following definition introduces Eulerian matroids, which is a special kind of matroids, and has sound theoretical foundations and wide applications.

Let

Let

A graph.

According to the above definition, it is clear that

It is easy to check that the set family

Let

It is obvious

We have

In fact, the concepts of unary covering, representative covering, and exact covering play a vital role in covering-based rough sets. Based on the properties of the Eulerian matroid and the above proposition, the following remark is presented.

If

It is easy to know

Let

(

(

Clearly, according to Proposition

In covering-based rough sets, a pair of covering-based approximation operators are used to describe an object. In the following definition, a widely used pair of covering-based approximation operators are introduced.

Let

Let

Let

In the following proposition, a widely used pair of covering-based approximation operators are represented by the circuits of Eulerian matroids.

Let

Let

Let

Similarly, according to the above proposition, we can obtain the following proposition.

Let

It is easy to prove this proposition by Definition

As shown in Example

The above proposition shows that the covering-based lower and covering-based upper approximation operators of covering-based rough sets can be denoted by the circuits of matroids. Li and Liu [

Let

According to Proposition

Let

Let

(

(

In fact, the above proposition indicates that the independent sets of an Eulerian matroid can be judged from the viewpoint of covering-based rough sets.

Since

In fact, it is easy to find that the circuits of an Eulerian matroid contain a partition of the universe at least. Therefore, we study the relationships between the circuits of an Eulerian matroid and its subset family.

Let

It is easy to prove this proposition by Definition

It is not difficult to find that

Let

Similarly, according to the above proposition, the following proposition can be presented.

Let

Since a family of circuits determines only one matroid, then the relationship between two coverings, which are induced by two Eulerian matroids, is presented in the following proposition.

Let

In fact, the above proposition shows that different Eulerian matroids induce different covering-based approximation spaces and different covering-based approximation operators.

The above results show that many important concepts and properties of covering-based rough sets can be concisely characterized by corresponding concepts and properties in matroids. Therefore, covering-based rough sets may be efficient to study matroids.

From the above definitions and properties, we know that the circuits of an Eulerian matroid can construct only one covering-based approximation space. Now, in this section, we wonder if a covering

Let

According to the above definition, it is easy to see that the family of circuits and the family of bases of a matroid are all antichains.

Let

Let

Let

According to the above theorem, we introduce the concept of the dual matroid.

The matroid whose family of all bases is equal to

If

Now, based on the above results, a matroidal structure of covering-based rough sets will be constructed. Firstly, the concept of Eulerian covering is defined in the following definition.

An Eulerian covering

there exist

if

Let

In fact, an Eulerian covering of a universe in compliance with the above definition satisfies the circuit axiom. In other words, it determines an Eulerian matroid.

Let

It is easy to prove this proposition by Theorem

As shown in Example

Let

In fact, a partition of a universe is an Eulerian covering. Hence, a matroidal structure of classical rough sets is constructed in the following proposition.

Let

It is easy to prove that every partition of

Let

For any

Let

Let

It is easy to prove this proposition by Lemma

An Eulerian covering

In this paper, we studied one type of covering-based rough sets on Eulerian matroids. Via inducing a covering from the circuits of an Eulerian matroid, the covering-based approximation operators were represented from the viewpoint of Eulerian matroids. First, we studied the properties of the covering induced by an Eulerian matroid and the properties of the covering-based approximation operators through Eulerian matroids. Second, the relationships between two covering-based approximation spaces induced by two Eulerian matroids were explored. Finally, a matroidal structure of covering-based rough sets was constructed.

This work is in part supported by the National Science Foundation of China under Grant nos. 61170128, 61379049, and 61379089, the Natural Science Foundation of Fujian Province, China, under Grant no. 2012J01294, the Fujian Province Foundation of Higher Education under Grant no. JK2012028, and the Education Department of Fujian Province under Grant no. JA12222.