Matroidal Structure of Generalized Rough Sets Based on Tolerance Relations

Rough set theory provides an effective tool to deal with uncertain, granular, and incomplete knowledge in information systems. Matroid theory generalizes the linear independence in vector spaces and has many applications in diverse fields, such as combinatorial optimization and rough sets. In this paper, we construct a matroidal structure of the generalized rough set based on a tolerance relation. First, a family of sets are constructed through the lower approximation of a tolerance relation and they are proved to satisfy the circuit axioms of matroids. Thus we establish a matroid with the family of sets as its circuits. Second, we study the properties of the matroid including the base and the rank function. Moreover, we investigate the relationship between the upper approximation operator based on a tolerance relation and the closure operator of the matroid induced by the tolerance relation. Finally, from a tolerance relation, we can get a matroid of the generalized rough set based on the tolerance relation. The matroid can also induce a new relation. We investigate the connection between the original tolerance relation and the induced relation.


Introduction
Rough set theory was originally proposed by Pawlak [1,2] in 1982 and serves as a new mathematical approach to vague concept. It has been widely applied to many different fields, such as knowledge discovery [3], machine learning [4], knowledge acquisition [5], decision analysis [6,7], and granular computing [8]. It is well known that the classical rough set theory is based on equivalence relations. However, equivalence relations are restrictive for many applications. To address this problem, classical rough set theory has been extended from equivalence relations to some other relations, such as tolerance relation [9,10], similarity relation [11,12], and arbitrary relation [13][14][15].
Matroid theory [16,17] proposed by Whitney is a generalization of both linear algebra and graph theory. It has been successfully applied to various fields, such as combinatorial optimization, algorithm design, information coding, and cryptology. In order to enrich the theoretical system and extend the applications of rough sets, it is helpful to study rough sets with matroids. There are many works [18][19][20][21][22][23][24][25][26][27][28][29][30][31] about the connection between matroids and rough sets.
From a tolerance relation, a matroidal structure is proposed in this paper. First, we define a family of sets through the lower approximation based on a tolerance relation and prove the family to satisfy the circuit axioms of matroids. Hence, we obtain a matroid with the family of sets as its circuits. Moreover, the family of sets are proved to be a partition, so the matroid is a partition-circuit matroid. Second, we obtain that the family of circuits of this matroid is equal to the partition induced by the transitive closure of the tolerance relation. Next we investigate some characteristics of this matroid through the generalized rough set based on a tolerance relation, such as the base and the rank function. Third, we study some important relationships between the closure operator of this matroid and the upper approximation operator of the tolerance relation. Finally, we know that the matroid established by a tolerance relation can induce a relation. We prove that the original tolerance relation is contained in the induced relation. In particular, the induced relation is equal to the transitive closure of the original tolerance relation.
The rest of this paper is organized as follows. Section 2 reviews some fundamental definitions and properties of 2 The Scientific World Journal generalized rough sets and matroids. In Section 3, we propose a matroid induced by a tolerance relation and study some characteristics of this matroid through generalized rough sets. Then we investigate the relationship between the closure operator of this matroid and the upper approximation operator of a tolerance relation. We also study the relationship between a tolerance relation and the relation induced by the matroid established by the original tolerance relation. We conclude this paper in Section 4.

Basic Definitions
In this section, we recall some fundamental definitions and important conclusions of generalized rough sets and matroids.
2.1. Rough Set. Let be a universe, × the product set of and . Any subset of × is called a binary relation on . For any ( , ) ∈ × , if ( , ) ∈ , we say has relation with and denote this relationship as . In the rest of this paper, we assume is a finite and nonempty set unless otherwise stated.
In rough sets, a pair of approximation operators are used to describe an object. In the following definition, we introduce the lower and upper approximation operators of generalized rough sets through the neighborhood.
where ( ) = { ∈ : } is called the neighborhood of with respect to . , are called the lower and upper approximation operators, respectively.
The following proposition presents some properties of lower approximation operator.
The following results hold for reflexive and symmetric relations.

Matroid.
There are many equivalent ways to define a matroid. The following definition of matroid is presented from the viewpoint of independent sets.
Definition 5 (matroid [16]). A matroid is an ordered pair = ( , I) consisting of and a collection I (called independent sets) of subsets of with the following three properties: (I2) If ∈ I and ⊆ , then ∈ I; In order to make some expressions clear and brief, we introduce some symbols as follows.
Definition 7 (see [16]). Let be a universe. A ⊆ 2 is a family of subsets of ; then Base is an important concept of matroids. We give the definition of base as follows.
Definition 8 (base [16]). Let = ( , I) be a matroid. Any maximal independent set in is called a base of and the family of all bases of is denoted by B( ); that is, B( ) = Max(I).
If a subset of the universe is not an independent set of a matroid, it is called a dependent set of the matroid.
Definition 9 (circuit [16]). Let = ( , I) be a matroid. A minimal dependent set in is called a circuit of and one denotes the family of all circuits of by C( ); that is, C( ) = Min(Opp(I)).
The following proposition shows that a matroid can be defined from the viewpoint of circuits.
The Scientific World Journal 3 Proposition 10 (circuit axioms [16]). Let C be a family of subsets of . Then there exists = ( , I) such that C = C( ) if and only if C satisfies the following conditions: In matroid theory, the rank function serves as a quantitative tool. The definition of rank function is introduced as follows.
The following proposition shows the connection between the independent set and the rank function of a matroid.
The closure operator is one of the important characteristics of a matroid. We give the definition of closure operator as follows.
Definition 13 (closure operator [16]). Let = ( , I) be a matroid. For all ⊆ , is called the closure of in and cl is called the closure operator. One can omit the subscript when there is no confusion.

Matroidal Structure Induced by Tolerance Relation
In this section, we establish a matroidal structure of the generalized rough set based on a tolerance relation. Firstly, a family of sets are defined and they are proved to satisfy the circuit axioms of matroids.

Definition 14.
Let be a tolerance relation on . One defines a family of sets with respect to as follows: We give an example to show that C( ) is defined through the lower approximation based on a tolerance relation. In the following proposition, we will prove C( ) to satisfy the circuit axioms of matroids when the relation is a tolerance relation. (2) Let 1 , 2 ∈ C( ) and 1 ⊆ 2 . Because the elements of C( ) are minimal, we can get 1 = 2 .
(3) Let , ∈ C( ) and In order to further understand this type of matroids, we introduce a special matroid called partition-circuit matroid.
As shown in Proposition 16, we can prove that the matroid based on a tolerance relation is a partition-circuit matroid.

4
The Scientific World Journal Transitive closure of a relation is an important concept for rough sets and matroids. We give the definition of transitive closure of a relation as follows.
Definition 21 (transitive closure [35]). Let be a relation on . The smallest transitive relation on containing the relation is called the transitive closure of . One denotes the transitive closure of by ( ).
We give the properties of the corresponding transitive closure ( ) when is a tolerance relation in the following lemma.

Lemma 22 (see [36]). Let be a tolerance relation on . ( ) is an equivalence relation.
In [35], we can get ( ) = ∪ 2 ∪ ⋅ ⋅ ⋅ . We know ( ) is an equivalence relation if is a tolerance relation on , so we can get a partition / ( ) = { 1 , 2 , . . . , } on , where 1 , 2 , . . . , are the equivalence classes. Firstly, in order to show the connection between C( ) and the partition induced by the transitive closure of the tolerance relation, we give a lemma as follows.
An example can illustrate that C( ) is equal to the partition induced by the transitive closure of the tolerance relation. In [34], Liu has already shown the characteristics of independent sets about partition-circuit matroid. Combining the results of Liu about partition-circuit matroid, we can get the expression of the independent sets of the matroid induced by a tolerance relation.
( ) is the matroid induced by . Then, We present the expression of the base according to the partition induced by the transitive closure of a tolerance relation. The rank function of the matroid induced by a tolerance relation can be well expressed by the the partition induced by the transitive closure of the tolerance relation.

Proposition 28. Let be a tolerance relation on . ( ) is the matroid induced by . Then, for all ⊆ ,
Proof. According to Proposition 12, we need to prove In order to better illustrate the feature of the rank function, we give the rank for all subsets of universe. Rough set theory and matroid theory have close relationships. We study the connection between the closure of the matroid induced by a tolerance relation and the upper approximation of the tolerance relation when C( ) does not contain any single-point set. Proof. Since cl ( ) ( ) = ∪ { ∈ − : ∃ ∈ C( ) such that ∈ ⊆ ∪{ }}, we need to prove { ∈ − : ∃ ∈ C( ) such that ∈ ⊆ ∪ { }} ⊆ ( ). That is to say, for any ∈ − , there exists ∈ C( ) such that ∈ ⊆ ∪ { }. We can get ( ) ∩ ̸ = 0. If ( ) ∩ = 0, then there exist two different circuits such that ( ) and are contained in them, respectively. If there exists only one circuit ∈ C( ) such that ( ) ⊆ and ⊆ , then ∈ ( ) ⊆ ⊆ ∪ { } ⊆ ∪ { }. Because C( ) does not contain any single-point set, | | ≥ 2. It is a contradiction. So there exist two different circuits 1 ∈ C( ) and 2 ∈ C( ) such that That is to say, ∈ ( ). Hence, cl ( ) ( ) ⊆ ( ).
But if C( ) contains single-point sets, in general, cl ( ) ( ) ⊆ ( ) does not hold. We can use an example to illustrate this situation. For any ∈ C( ), we have the following conclusion about ( )( ), ( ), and cl ( ) ( ).
We have discussed how to induce a matroid from a relation. Then, how to induce a relation from a matroid is presented as follows.
Definition 34 (see [17]). Let = ( , I) be a matroid. We define a relation ( ) on as follows: for all , ∈ , We say ( ) is a relation on induced by .
An example is provided to illustrate how to induce a relation from a matroid. According to the above definition, any matroid can induce a relation. The following lemma proves that the relation induced by a matroid is an equivalence relation.
We know a tolerance relation can induce a matroid and the matroid can also induce a relation. In the following proposition, we give the relationship between the original tolerance relation and the induced relation.
In the above proposition, we give the connection between the original tolerance relation and the induced relation, while we give the relationship between the induced relation and the transitive closure of the original tolerance relation in the following proposition.

Conclusions
In this paper, we connected matroid theory and generalized rough set theory based on relations. We firstly defined a family of sets induced by a tolerance relation and proved the family to satisfy the circuit axioms of matroids. Some characteristics of this matroid, such as the base and the rank