^{1}

The theory of rough sets is concerned with the lower and upper approximations of objects through a binary relation on a universe. It has been applied to machine learning, knowledge discovery, and data mining. The theory of matroids is a generalization of linear independence in vector spaces. It has been used in combinatorial optimization and algorithm design. In order to take advantages of both rough sets and matroids, in this paper we propose a matroidal structure of rough sets based on a serial and transitive relation on a universe. We define the family of all minimal neighborhoods of a relation on a universe and prove it satisfies the circuit axioms of matroids when the relation is serial and transitive. In order to further study this matroidal structure, we investigate the inverse of this construction: inducing a relation by a matroid. The relationships between the upper approximation operators of rough sets based on relations and the closure operators of matroids in the above two constructions are studied. Moreover, we investigate the connections between the above two constructions.

The theory of rough sets [

Matroid theory [

Rough sets and matroids have their own application fields in the real world. In order to make use of both rough sets and matroids, researchers have combined them with each other and connected them with other theories. In this paper, we propose a matroidal structure based on a serial and transitive relation on a universe and study its relationships with the lower and upper approximations of generalized rough sets based on relations.

First, we define a family of sets of a relation on a universe, and we call it the family of all minimal neighborhoods of the relation. When the relation is serial and transitive, the family of all minimal neighborhoods satisfies the circuit axioms of matroids, then a matroid is induced. And we say the matroid is induced by the serial and transitive relation. Moreover, we study the independent sets of the matroid through the neighborhood and the lower and upper approximation operators of generalized rough sets based on relations, respectively. A sufficient and necessary condition, when different relations generate the same matroid, is investigated. And the relationships between the upper approximation operator of a relation and the closure operator of the matroid induced by the relation are studied. We employ a special type of matroids, called 2-circuit matroids, which is introduced in [

Second, the relationships between the above two constructions are studied. On the one hand, for a matroid on a universe, it can induce an equivalence relation, and the equivalence relation can generate a matroid; we prove that the circuit family of the original matroid is finer than one of the induced matroid. And the original matroid is equal to the induced matroid if and only if the circuit family of the original matroid is a partition. On the other hand, for a reflexive and transitive relation on a universe, it can generate a matroid, and the matroid can induce an equivalence relation, then the relationship between the equivalence relation and the original relation is studied. The original relation is equal to the induced equivalence relation if and only if the original relation is an equivalence relation.

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

In this section, we recall some basic definitions and related results of generalized rough sets and matroids which will be used in this paper.

Given a universe and a relation on the universe, they form a rough set. In this subsection, we introduce some concepts and properties of generalized rough sets based on relations [

Let

In the following definition, we introduce the lower and upper approximation operators of generalized rough sets based on relations through the neighborhood.

Let

We present properties of the lower and upper approximation operators in the following proposition.

Let

In [

Let

In the following definition, we use the neighborhood to describe a serial relation and a transitive relation on a universe.

Let

Matroids have many equivalent definitions. In the following definition, we will introduce one that focuses on independent sets.

A matroid is a pair

if

if

Since the above definition is from the viewpoint of independent sets to represent matroids, it is also called the independent set axioms of matroids. In order to make some expressions brief, we introduce several symbols as follows.

Let

In a matroid, a subset is a dependent set if it is not an independent set. Any circuit of a matroid is a minimal-dependent set.

Let

A matroid can be defined from the viewpoint of circuits, in other words, a matroid uniquely determines its circuits, and vice versa.

Let

if

if

The closure operator is one of the important characteristics of matroids. A matroid and its closure operator can uniquely determine each other. In the following definition, we use the circuits of matroids to represent the closure operator.

Let

In this section, we propose two constructions between a matroid and a relation. One construction is from a relation to a matroid, and the other construction is from a matroid to a relation.

In [

Let

In the following proposition, we will prove the family of all minimal neighborhoods of a relation on a universe satisfies the circuit axioms of matroids when the relation is serial and transitive.

Let

(C1): Since

(C2): According to Definitions

(C3): If

It is natural to ask the following question: “when the family of all minimal neighborhoods of a relation satisfies the circuit axioms of matroids, is the relation serial and transitive?”. In the following proposition, we will solve this issue.

Let

According to (C1) of Proposition

When the family of all minimal neighborhoods of a relation satisfies the circuit axioms of matroids, the relation is not always a transitive relation as shown in the following example.

Let

In this paper, we consider that the family of all minimal neighborhoods of a relation can generate a matroid when the relation is serial and transitive.

Let

The matroid induced by a serial and transitive relation can be illustrated by the following example.

Let

Generally speaking, a matroid is defined from the viewpoint of independent sets. In the following, we will investigate the independent sets of the matroid induced by a serial and transitive relation.

Let

According to Definition

for all

suppose

To illustrate the independent sets of a matroid induced by a serial and transitive relation, the following example is given.

Let

The lower and upper approximation operators are constructed through the neighborhood in generalized rough sets based on relations. In the following proposition, we will study the independent sets of the matroid induced by a serial and transitive relation through the lower approximation operator.

Let

According to Definition

Because of the duality of the lower and upper approximation operators, we obtain the independent sets of the matroid induced by a serial and transitive relation through the upper approximation operator.

Let

According to (4LH) of Propositions

From Examples

Let

The relationship between the two matroids induced by a serial and transitive relation and its reflexive closure is studied in the following proposition.

Let

According to Proposition

for all

for all

In fact, for a universe, any different relations generate the same matroid if and only if the families of all minimal neighborhoods are equal to each other according to Proposition

It is known that the upper approximation operator induced by a reflexive and transitive relation is exactly the closure operator of a topology [

In a matroid, the closure of any subset contains the subset itself. When a relation on a universe is reflexive, the upper approximation of any subset contains the subset itself. If a relation is reflexive, then it is serial. According to Definition

Let

From the above example, we see that the closure operator of the matroid induced by a reflexive and transitive relation dose not correspond to the upper approximation operator of the relation. A condition, when the closure operator contains the upper approximation operator, is studied in the following proposition. First, we present a remark.

Suppose

Let

According to Definition

In fact, for an equivalence relation on a universe and any subset of the universe, its closure with respect to the induced matroid can be expressed by the union of its upper approximation with respect to the relation and the family of some elements whose neighborhood is equal to itself.

Let

We need only to prove

Similarly, can the upper approximation operator of an equivalence relation contain the closure operator of the matroid induced by the equivalence relation when the cardinality of any circuit of the matroid is equal or greater than 2?

Let

According to Definition

A sufficient and necessary condition, when the upper approximation operator of an equivalence relation is equal to the closure operator of the matroid induced by the equivalence relation, is investigated in the following theorem. First, we introduce a special matroid called 2-circuit matroid.

Let

Let

According to Definition

according to Propositions

we prove this by reductio.

On the one hand, suppose there exists

On the other hand, suppose there exists

In order to further study the matroidal structure of the rough set based on a serial and transitive relation, we consider the inverse of the construction in Section

Let

The following example is to illustrate the construction of a relation from a matroid.

Let

In fact, according to Definition

Let

The following example is presented to illustrate that different matroids generate the same relation.

Let

Similarly, we will study the relationship between the closure operator of a matroid and the upper approximation operator of the equivalence relation induced by the matroid in the following proposition.

Let

Since

The above proposition can be illustrated by the following example.

Let

According to Proposition

Let

Under what condition the closure operator contains the upper approximation operator? In the following proposition, we study this issue.

Let

According to Proposition

In the following theorem, we investigate a sufficient and necessary condition when the closure operator of a matroid is equal to the upper approximation operator of the relation induced by the matroid.

Let

(1) Since

(2) According to Proposition

since

according to Propositions

In this section, we study the relationships between the two constructions in Section

Let

In the following proposition, we will represent the relationship between the circuits of a matroid and the circuits of the matroid induced by the equivalence relation which is generated by the original matroid.

Let

For all

In order to further comprehend Proposition

Let

A matroid can induce an equivalence relation, and the equivalence relation can generate a matroid, then a sufficient and necessary condition when the original matroid is equal to the induced matroid is studied in the following theorem.

Let

According to Proposition

according to Proposition

if

Similarly, a serial and transitive relation can generate a matroid, and the matroid can generate an equivalence relation, then the relationship between the original relation and the induced equivalence relation is studied as follows. First, we present a lemma about the transitivity of a relation.

Let

Suppose

If a relation is reflexive, then it is also serial. Therefore, a reflexive and transitive relation can generate a matroid according to Definition

Let

According to Definitions

The above proposition can be illustrated by the following example.

Let

A sufficient and necessary condition, when the relation is equal to the induced equivalence relation, is investigated in the following theorem.

Let

(

(

In order to broaden the theoretical and application fields of rough sets and matroids, their connections with other theories have been built. In this paper, we connected matroids and generalized rough sets based on relations. For a serial and transitive relation on a universe, we proposed a matroidal structure through the neighborhood of the relation. First, we defined the family of all minimal neighborhoods of a relation on a universe and proved it to satisfy the circuit axioms of matroids when the relation was serial and transitive. The independent sets of the matroid were studied, and the connections between the upper approximation operator of the relation and the closure operator of the matroid were investigated. In order to study the matroidal structure of the rough set based on a serial and transitive, we investigated the inverse of the above construction: inducing a relation by a matroid. Through the connectedness in a matroid, a relation was obtained and proved to be an equivalence relation. And the closure operator of the matroid was equal to the upper approximation operator of the induced equivalence relation if and only if the matroid was a 2-circuit matroid. Second, the relationships between the above two constructions were investigated. For a matroid on a universe, it induced an equivalence relation, and the equivalence relation generated a matroid, then the original matroid was equal to the induced matroid if and only if the circuit family of the original matroid was a partition on the universe. For a serial and transitive relation on a universe, it generated a matroid, and the matroid induced an equivalence relation, then the original relation was equal to the induced equivalence relation if and only if the original relation was an equivalence relation.

This work is supported in part by the National Natural Science Foundation of China under Grant no. 61170128, the Natural Science Foundation of Fujian Province, China, under Grant nos. 2011J01374 and 2012J01294, and the Science and Technology Key Project of Fujian province, China, under Grant no. 2012H0043.