Journal of Applied Mathematics

Volume 2012 (2012), Article ID 973920, 27 pages

doi:10.1155/2012/973920

Research Article

Matroidal Structure of Rough Sets from the Viewpoint of Graph Theory

Jianguo Tang,1,2 Kun She,¹ and William Zhu³

¹School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China

²School of Computer Science and Engineering, XinJiang University of Finance and Economics, Urumqi 830012, China

³Lab of Granular Computing, Zhangzhou Normal University, Zhangzhou 363000, China

Received 4 February 2012; Revised 30 April 2012; Accepted 18 May 2012

Academic Editor: Mehmet Sezer

Copyright © 2012 Jianguo Tang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Constructing structures with other mathematical theories is an important research field of rough sets. As one mathematical theory on sets, matroids possess a sophisticated structure. This paper builds a bridge between rough sets and matroids and establishes the matroidal structure of rough sets. In order to understand intuitively the relationships between these two theories, we study this problem from the viewpoint of graph theory. Therefore, any partition of the universe can be represented by a family of complete graphs or cycles. Then two different kinds of matroids are constructed and some matroidal characteristics of them are discussed, respectively. The lower and the upper approximations are formulated with these matroidal characteristics. Some new properties, which have not been found in rough sets, are obtained. Furthermore, by defining the concept of lower approximation number, the rank function of some subset of the universe and the approximations of the subset are connected. Finally, the relationships between the two types of matroids are discussed, and the result shows that they are just dual matroids.

1. Introduction

Rough sets provide an important tool to deal with data characterized by uncertainty and vagueness. Since it was proposed by Pawlak [1, 2], rough sets have been generalized from different viewpoints such as the similarity relation [3, 4] or the tolerance relations [5] instead of the equivalence relation, and a covering over the universe instead of a partition [6–10], and the neighborhood instead of the equivalence class [11–14]. Besides, using some other mathematical theories, such as fuzzy sets [15–19], boolean algebra [20–23], topology [24–27], lattice theory [28–30], and modal logic [31], to study rough sets has became another kind of important generalizations of rough sets. Specially, matroids also have been used to study rough sets recently [32, 33].

Matroids, as a simultaneous generalization of graph theory and linear algebra, was proposed by Whitney in [34]. The original purpose of this theory is to formalize the similarities between the ideas of independence and rank in graph theory and those of linear independence and dimension in the study of vector spaces [35]. It has been found that matroids are effective to simplify various ideas in graph theory and are useful in combinatorial optimization problems.

In the existing works on the combination of rough sets and matroids, Zhu and Wang [32] constructed a matroid by defining the concepts of upper approximation number in rough sets. Then they studied the generalized rough sets with matroidal approaches. As a result, some unique properties are obtained in this way. Wang et al. [33] studied the covering-based rough sets with matroids. Two matroidal structures of covering-based rough sets are established.

In this paper, we attempt to make a further contribution to studying rough sets with matroids. As we see in Section 2.3, it is somewhat hard to understand matroids. And this will also arise in the combination of matroids and rough sets. So, in order to give an intuitive interpreting to the combination, we will study it from the viewpoint of graph theory. There are at least two kinds of graphic ways, which can be used to build relationships between matroids and rough sets. The complete graph and the cycle. More specifically, for a partition over the universe, any equivalence class of the partition can be regarded as a complete graph or a cycle. Thus a partition is transformed to a graph composed of these complete graphs or circles induced by the equivalence classes of the partition. And we can establish a matroid in terms of the graph. Afterwards, some characteristics of the matroid are formulated and some new properties, which are hard to be found via the rough sets way, are obtained. With these characteristics and properties, a matroidal structure of rough sets is constructed. Finally, the relationships between the two kinds of matroids established from the viewpoints of complete graph and cycle are discussed.

The rest of this paper is organized as follows. In Section 2, we review some basic knowledge about rough sets, matroids, and graph theory. In Section 3, we analyze the relationships between rough set theory and graph theory from the viewpoints of complete graph and cycle, respectively. In Sections 4 and 5, two kinds of matroids are established in terms of the analytical results of Section 3. And two kinds of the matroidal structures of rough sets are constructed. In Section 6, the relationships between the two kinds of matroids are discussed.

2. Preliminary

For a better understanding to this paper, in this section, some basic knowledge of rough sets, graph theory, and matroids are introduced.

2.1. Rough Sets

Let be a nonempty and finite set called universe, a family of equivalence relations over , then the relational system is called a knowledge base [1]. If , then is also an equivalence relation [1]. And is called an indiscernibility relation and denoted by [1]. If , then represents the partition of induced by . That is in the partition , for all , and , for , , and . Each in is an equivalence class, and it can also be denoted by if .

For any subset , the lower and the upper approximations of with respect to are defined as follows [1]:

Set is called the -boundary of or the boundary region of with respect to [1]. If , that is, , then is -definable, or is called a definable set with respect to ; else, if , that is, , then is rough with respect to , or is called a rough set with respect to [1]. The lower and the upper approximations satisfy duality, that is [1],

(P1) for all , ,(P2) for all , ,

where represents the set .

Neighborhood and upper approximation number are another two important concepts, which will be used in this paper. They are defined as follows.

Definition 2.1 (Neighborhood [36]). Let be a relation on . For all , is called the successor neighborhood of in . When there is no confusion, we omit the subscript .

Definition 2.2 (Upper approximation number [32]). Let be a relation on . For all , is called the upper approximation number of with respect to .

2.2. Graph Theory

A graph is an ordered pair of disjoint sets such that is a subset of the set of unordered pairs of [37]. The set is the set of vertices and is the set of edges. If is a graph, then is the vertex set of , and is the edge set. An empty graph is a graph whose edge set is empty. An edge is said to join the vertices and and is denoted by . Thus and mean exactly the same edge; the vertices and are the endpoints of this edge. If , then and are adjacent and are neighbors. A loop [38] is an edge whose endpoints are equal. Parallel edges are edges having the same pair of endpoints. The degree of vertex in a graph , denoted by or , is the number of edges incident to , except that each loop at counts twice.

A simple graph is a graph having no loops or parallel edges [38]. An isomorphism [38] from a simple graph to a simple graph is a bijection such that if and only if . That is to say “ is isomorphic to ,” denoted by if there is an isomorphism from to .

We say that is a subgraph of if and [37]. In this case, we write . If contains all edges of that join two vertices in then is said to be the subgraph induced by and is denoted by . Thus, a subgraph of is an induced subgraph if .

2.3. Matroids

Definition 2.3 (Matroid [39]). A matroid is a pair , where (called the ground set) is a finite set and (called the independent sets) is a family of subsets of satisfying the following axioms:

();() if and , then ;() if and , then such that ,

where represents the cardinality of “”.

The matroid is generally denoted by . represents the ground set of and the independent sets of . Each element of is called an independent set of . If a subset of is not an independent set, then it is called a dependent set. The family of all dependent sets of is denoted by , that is,

Example 2.4. Let , . Then is a matroid, which satisfies the axioms (I1)(I3). And each element of is an independent set. and are only two dependent sets of .

Next, we will introduce some characteristics of a matroid. For a better understanding to them, some operations will be firstly introduced as follows.

Let be a set and . Then [39]:

Definition 2.5 (Circuit [39]). Let be a matroid. A minimal dependent set is called a circuit of , and the set of all circuits of is denoted by , that is, .

A circuit in a matroid is a set which is not independent but has the property that every proper subset of it is independent. In Example 2.4, .

Theorem 2.6 (Circuit axioms [39]). Let be a family of subsets of . Then there exists such that if and if only satisfies the following properties:

();() if and , then ;() if , , and , then such that .

Definition 2.7 (Base [39]). Let be a matroid. A maximal independent set of is called a base of ; the set of all bases of is denoted by , that is, .

In Example 2.4, according to Definition 2.7, we can get that .

It is obvious that all bases of a matroid have the same cardinality, which is called the rank of the matroid.

Definition 2.8 (Rank function [39]). Let be a matroid. Then the rank function of is defined as: for all ,

A matroid can be determined by its base, its rank function, or its circuit. For a set, is independent if and only if it is contained in some base, if and only if it satisfies , or if and only if it contains no circuit. It is possible to axiomatize matroids in terms of their sets of bases, their rank functions, or their sets of circuits [40].

Definition 2.9 (Closure [39]). Let be a matroid. For all , the closure operator of is defined as follows:

If , we say that depends on . The closure of is composed of these elements of that depend on . If , then is called a closed set of .

Definition 2.10 (Hyperplane [39]). Let be a matroid. is called a hyperplane of if is a closed set of and . And represents the family of all hyperplanes of .

3. The Viewpoint of Graph Theory in Rough Sets

Graph theory provides an intuitive way to interpret and comprehend a number of practical and theoretical problems. Here, we will make use of it to interpret rough sets. There are at least two different ways to understand rough sets from the viewpoint of graph theory: the complete graph and the cycle. This will be analyzed in detail in the following subsections.

3.1. The Complete Graph

Definition 3.1 (Complete graph [38]). A complete graph is a simple undirected graph whose vertices are pairwise adjacent. A complete graph whose cardinality of vertex set is equal to is denoted by .

In rough sets, an equivalence relation can generally be regarded as an indiscernibility relation. That means any two different elements in the same equivalence class are indiscernible. In order to interpret this phenomenon from the viewpoint of graph theory, we can consider the two elements as two vertices and the indiscernibility relation between them as an edge connecting the two vertices. Then an equivalence class is represented by a complete graph. For a better understanding of it, an example is served as follows.

Example 3.2. Let be the universe, an equivalence relation, and . Then each equivalence class can be transformed to a complete graph showed in Figure 1.

Figure 1: The complete graphs of equivalence classes.

Figure 1(a) represents the complete graph of the equivalence class . Because just includes one element , there is only one vertex and no edge in the complete graph Figure 1(a). Figure 1(b) represents the complete graph of , which includes two vertices and only one edge connecting the two vertices. And Figure 1(c) represents the complete graph of . We can find that there are five vertices in Figure 1(c) and each pair of vertices are connected by an edge. Here, we denote the complete graphs Figures 1(a), 1(b), and 1(c) as , , and , respectively.

From the above example, for any two elements in the universe, if they are indiscernible then there is one edge between them. Then the partition is transformed to be a graph , where

It is obvious that if and belong to the same equivalence class, then there is an edge in . So, we can formulate the equivalence class as follows: for all ,

Furthermore, for any subset , the lower and the upper approximations of can be formulated as follows:

3.2. The Cycle

A walk [37] in a graph is an alternating sequence of vertices and edges, say where , . For simplicity, the walk can also be denoted by ; the length of is , that is, the number of its edges. A walk that starts and ends at the same vertex but otherwise has no repeated vertices is called a cycle [41]. A cycle on one vertex consists of a single vertex with a loop, and a cycle on two vertices consists of two vertices joined by a pair of parallel edges [42].

Then, how to build a bridge between rough sets and cycle? In rough sets, elements contained in the same equivalence class are indiscernible, and any proper subset of an equivalence class is no longer an equivalence class. So, we can convert an equivalence class to a cycle whose vertices set is the equivalence class. Therefore, each vertex is connected with all vertices in the cycle [42]. This reflects the indiscernible relationship among the elements of an equivalence class. Furthermore, any subgraph of the cycle does not contain a cycle. That is, any subgraph of the cycle is no longer a cycle. This can be illustrated in the following example.

Example 3.3 (Continued from Example 3.2). For any equivalence class in , it can be represented by a cycle. As shown in Figure 2, the equivalence class , , and are represented by Figures 2(a), 2(b), and 2(c), respectively.

(a)

(b)

(c)

Figure 2: The cycles of equivalence classes.

Figure 2(a) is a cycle with only one vertex and one edge. It is also called a loop. That means the vertex is connected with itself. Figure 2(b) is a cycle with two vertices and two edges. And it is generally regarded as a parallel edges. Figure 2(c) is not only a cycle but also a simple graph. Obviously, any subgraph of each cycle in Figure 2 is no longer a cycle. And, for any two different elements of the universe, they belong to the same equivalence class if and only if they are connected to each other.

It is worth noting that the sequence of vertices in a cycle is not emphasized here. We only care that the vertices, namely, elements of some equivalence class, can form a cycle. So, to Figure 2(c), the cycle and cycle can be treated as the same cycle.

For convenience, to an equivalence class in , the cycle whose vertices set is equal to is denoted by , that is, is a graph (cycle) where is the set of edges of . Then the partition can be transformed to be a graph where

One may ask which edges belong to exactly? In fact, it is nonnecessary to define the edges of exactly. Here, we just need to form a cycle with the vertex set . That is, each pair of vertices of are connected and the degree of each vertex is equal to 2. In other words, we simply need to know that each vertex of is adjacent with two other vertices (except the loop and parallel edges) and do not need to care which two vertices they are.

We can find from the Example 3.3 that, for any two elements in the universe, they belong to the same equivalence class if and only if they are connected with each other. Therefore, we can formulate the equivalence class as follows: for all ,

Likewise, for any subset , the lower and the upper approximations of can be formulated as follows:

So far, rough sets are interpreted from the viewpoints of complete graph and cycle, respectively. The above analysis shows that there are some similarities, and also some differences, between the two ways to illustrate rough sets. Because there are closed connections between graph theory and matroid theory, we will study the matroidal structure of rough sets through the two kinds of graphs.

4. Matroidal Structure of Rough Sets Constructed from the Viewpoint of Complete Graph

In Section 3, we discussed rough sets from the viewpoint of graph theory. Two graphic ways are provided to describe rough sets. In this section, we will construct two types of matroidal structures of rough sets. One of them is established by using the principle of complete graph and the other of cycle.

For convenience, in this section, we suppose that is the universe, an equivalence relation over and the partition. And is the graph induced by , where .

4.1. The First Type of Matroidal Structure of Rough Sets

We know that a complete graph is a simple graph. Then, for any vertex of a complete graph, there is not a loop whose vertex is . That is to say there is not an edge between a vertex and itself. Furthermore, for any two vertices coming from different complete graphs, there is not an edge between them as well. Because an equivalence class can be represented by a complete graph, we can construct the matroidal structure of rough sets from this perspective.

In this subsection, the first type of matroid induced by a partition will be established and defined. And then some characteristics of it such as the base, circuit, rank function, and closure are studied.

Proposition 4.1. Let is an empty graph}. Then there is a matroid on such that .

Proof. According to Definition 2.3, we just need to prove that satisfies axioms (I1)(I3). It is obvious that and hold. Suppose that and . Because and are empty graphs, according to the definition of , each belongs to a different equivalence class with the others of and the same to each . Since , thus there must be at least one element such that belongs to some equivalence class which does not include any element of . Therefore, . As a result, satisfies . That is, there exists a matroid on such that .

If is an empty graph, then it means that any two different vertices of are nonadjacent. That is, each vertex of comes from a different complete graph with others. For instance, in Example 3.2, we can get as follows:

Definition 4.2 (The first type of matroid induced by a partition). The first type of matroid induced by a partition over , denoted by -MIP, is such a matroid whose ground set and independent sets is an empty graph}.

Obviously, matroids proposed in Proposition 4.1 is a . From the above result of , we can find that any two elements of come from different equivalence class. Therefore, we can get the following proposition.

Proposition 4.3. Let be an -MIP. Then for all , if and only if for all such that .

Proof. : If , then is an empty graph. According to the definition of , each element of comes from a different equivalence class with the others of . That is, for all , .

: Let . If for all , , then is an empty graph. Therefore, .

A matroid can be determined by its base, its rank function, or its circuit. So it is possible to axiomatize matroids in terms of their sets of bases, their rank functions, or their sets of circuits [40]. Here we will axiomatize the in terms of its circuit.

Theorem 4.4. Let be a matroid induced by . Then is an -MIP if and only if for all , .

Proof. According to Definition 2.3, we know that . If , then is a -MIP induced by . In this case, according to (2.2), and . It indicates that there is not any circuit in , and, therefore, we do not need to care whether the cardinality of each circuit of is equal to 2. That is, is compatible with the description that for all , . Similarly, if , then is a -MIP induced by . So, Theorem 4.4 is true when the set of circuits of is empty.

Next, we prove that Theorem 4.4 is true when the set of circuits of is nonempty.

: According to (2.2), . Therefore, in terms of Definition 4.2 and Proposition 4.1, for all , if and only if is not an empty graph. That is, there is at least one edge in . Obviously, the set of endpoints of each edge of is a dependent set. So, for all , there is a set composed of the endpoints of some edge of such that . According to Definition 2.5, and . That is, for all , .

: . According to (2.2), . Therefore, for all , such that , that is, for all , . So, for all ; if , then . According to Theorem 2.6, if , then such that . For all , let . Then . Furthermore, if , then . If for all , , then for all , such that . Thus, is a partition over . So . According to Definition 4.2, is a .

Summing up, Theorem 4.4 is true.

In terms of Proposition 4.1, we can get a matroid induced by a partition. Then one may ask whether there is a bijection between a partition and the induced by the partition. This question will be answered by the following theorem.

Theorem 4.5. Let be the collection of all partitions over , the set of all -MIP induced by partitions of , , that is, for all , , where is the -MIP induced by . Then satisfies the following conditions:

(1)for all , and if then ,(2)for all , s.t. .

Proof. Let , , and , are two induced by and , respectively. We need to prove that there is an such that , or there is an such that . Because , there is at least one equivalence class such that . If such that , then and such that . That means . Else, there at least two equivalence classes such that and . That is, there is a set such that and . Obviously, according to Proposition 4.3, .

Let be a , for all , . According to Theorem 4.4, for all and , then . Therefore, we can get a family . It is obvious that . Therefore, is a partition of . That is, and .

Theorem 4.5 shows that there is one-to-one correspondence between a partition and the induced by the partition.

4.2. Characteristics of I-MIP

The characteristics of a matroid are very important to describe the matroid from different aspects. In this subsection, we will study the characteristics of such as the base, circuit, rank function, and closure.

The set of bases of a matroid is the collection of all maximal independent sets. Observing from the result of in Section 4.1, the maximal independent set is the vertex set whose cardinality is equal to the cardinality of . Then the following proposition can be obtained.

Proposition 4.6. Let be the -MIP induced by , , and a subgraph of . Then is the set of bases of .

Proof. According to Definition 2.7, we need to prove that , namely, . In terms of Proposition 4.3, for all , for all . So, for all , . According to Proposition 4.1 and Definition 4.2, for all , is an empty graph, that is, . Similarly, we can prove in the same way that for all , . That is, .

For a base in , we can say that is such a set including one and only one element of every equivalence class of . Then we can get the following corollary.

Corollary 4.7. Let . Then if and only if for all, .

Proof.  According to Proposition 4.6, it is straightforward.

Corollary 4.8. .

Proof. According to Proposition 4.6, it is straightforward.

For a subset of , is either an independent set or a dependent set of . And so the opposition to the Proposition 4.1, is a dependent set if and only if there is at least one pair of vertices of the vertex set of , which is adjacent. Furthermore, a minimal dependent set of is the vertex set of an edge of . Then we can get the following proposition.

Proposition 4.9. Let be the -MIP induced by . Then is the set of circuits of .

Proof. According to Definition 2.5, we need to prove , that is, . , for all , such that . Furthermore, . If , then and ; else, . So, for all , . Similarly, for all , such that . Therefore, . As a result, .

Likewise, Proposition 4.3 provides a necessary and sufficient condition to decide whether a set is an independent set of . In this way, we can get the family of dependent sets of as follows:

Moreover, in terms of the Definition 2.5, we can get the set of circuits of as follows:

According to Proposition 4.1, it can be found that each subset of which contains exactly one element is an independent set. So, for any dependent set of , if then there must exist a subset of such that and is a dependent set. Thus, we can get the following proposition.

Proposition 4.10. Let be the -MIP induced by . Then is the set of circuits of .

Proof. According to Proposition 4.3, for all , is a dependent set, that is, . In terms of Proposition 4.9, . Similarly, for all , according to Proposition 4.9, . That is, is the set of circuits of .

From Propositions 4.9 and 4.10, we can find that and are the set of circuits of . Therefore, we can get the following corollary.

Corollary 4.11. .

Propositions 4.1 and 4.3 provide two ways to transform an partition to a matroid. Then, how to convert an to a partition? In the following proposition, this question is answered through the set of circuits of the .

Proposition 4.12. Let be the -MIP induced by . Then for all ,

Proof. According to Proposition 4.10, for all , if then for each . And for any and , . Therefore, .

Proposition 4.12 shows that if two different elements form a circuit, then they belong to the same equivalence class. In terms of (3.2), there is an edge in whose vertex set just contains the two elements. For a subset , if does not contain a circuit, then is an independent set and the rank of it is equal to . In other words, if is an empty graph, that is, each pair of vertexes of is nonadjacent, then the rank of is equal to . According to Definition 2.8, for any subset of the universe, the rank of the subset is the number of the maximal independent set contained in the subset. Therefore, we can get the following proposition.

Proposition 4.13. Let be the -MIP induced by . Then for all ,   is  an  empty  graph} is the rank of in .

If the set of bases of has been obtained, then for all ; we can get the rank of as follows:

It can be proved easily that . So, is also the rank function of .

Different from the rank of in , the closure of is the maximal subset of , which contains and its rank is equal to . For an element , if there is an element such that form a circuit, then the rank of is equal to it of . That is, belongs to the closure of . Therefore, we can get the following proposition.

Proposition 4.14. Let be the -MIP induced by . Then for all , is the closure of in .

Proof. According to Definition 2.9, we need to prove that , that is, . It is obvious that, for all , . So, we just need to prove that for all if there is an element such that if and if only . According to (3.1), if and if only and belong to the same equivalence. According to Definition 2.8, for all , is equal to the number of the maximal independent set contained in . According to Proposition 4.3, for all ; if is a maximal independent set contained in , then is not an independent set. That is, . Thus, for all , if and if only .

From Proposition 4.14, it can be found that, for any element , if , then . Therefore, the closure of can be equivalently represented as

For any element , according to Figure 1, it can be found that if there is an element such that , then and must belong to the same equivalence class. Then we can get the following corollary.

Corollary 4.15. Let . For all ; if then .

Next, we will discuss the hyperplane of the . From the Definition 2.10, we know that a hyperplane of a matroid is a closed set and the rank of it is one less than the rank of the matroid. Because the rank of the induced by is equal to the cardinality of , we can get the following proposition.

Proposition 4.16. Let be the -MIP induced by . Then is the hyperplane of .

Proof. According to (4.5) and Proposition 4.6, we know that the rank of the induced by is equal to . Furthermore, in terms of Proposition 4.14 and Corollary 4.15, for all , is a closed set and . So , that is, for all , . Similarly, we can get that for all , .

4.3. Approximations Established through I-MIP

So far, the base, circuit, rank function, closure, and hyperplane of a are established. Next, we will further study the approximations in rough sets in this subsection through these characteristics.

Proposition 4.17. Let be the -MIP induced by , . Then for all ,

where a base having the maximal intersection with .

Proof. For all , is an independent set contained in . Because is a base having the maximal intersection with , . Furthermore, for all , . Let and . Then . If , then . According to Corollary 4.7, for all , if for all and , then . And . Therefore, . That is, .

In rough sets, an element in the lower approximation certainly belongs to , while an element in the upper approximation possibly belongs to [43]. And the boundary region of is the set of elements in which each element does not certainly belong to either or . In general, we can get the boundary region of by the difference set of the lower and upper approximation of . But here, we can provide a matroidal approach to obtain the boundary region of firstly, and then the lower and the upper approximations should be established.

Proposition 4.18. Let be the -MIP induced by , . Then for all ,

Proof. According to Proposition 4.10 and Corollary 4.11, for all such that . means that each element of does not certainly belong either to or to . And then is the collection of all elements, which do not certainly belong either to or to . So .

Proposition 4.19. Let be the -MIP induced by , . Then for all ,

where .

Proof. According to the definition of the boundary region and Proposition 4.18, it is straightforward.

Corollary 4.20. Let be the -MIP induced by , and . Then for all , if and only if for all , .

Proof. : According to Proposition 4.10 and Corollary 4.11, if for all , then for all and , . That is, for all , .

: It is straightforward.

Proposition 4.21. Let be the -MIP induced by , . Then for all , the following equations hold:

(1),(2),(3).

Proof. According to Proposition 4.13, where and for all , . Let . If , then and for all , , that is, and for all , ; else, if , then and for all , , that is, and for all , . That is, .

Similarly, we can prove that and are true.

Proposition 4.21 provides three ways to get the upper approximation of with rank function. This intensifies our understanding to rank function of .

Proposition 4.22. Let be the -MIP induced by , . Then for all ,

Proof. According to Proposition 4.14 and Corollary 4.15, we can get that . Therefore, according to the definition of the upper approximation, it is obvious that .

The compact formulation of the upper approximation in Proposition 4.22 indicates that the closure is an efficient way to get the approximations in rough sets.

Proposition 4.23. Let be the -MIP induced by , . Then for all ,

Proof. According to Proposition 4.16, for all , , that is, . And if , then . Therefore, in terms of (2.1), . Thus, .

5. Matroidal Structure of Rough Sets Constructed from the Viewpoint of Cycle

In Section 3.2, the relationships between a cycle and an equivalence class are analyzed in detail. And a partition over the universe is transformed to a graph composed of some cycles. So, inspired by the cycle matroid introduced in [39], we will construct the matroidal structure of rough sets from this viewpoint. A new matroid will be established and the characteristics of it are studied. Then the approximations in rough sets are investigated via these characteristics.

For convenience, in this section, we suppose that is a universe, an equivalence relation over , and the partition. And is the graph induced by , where .

5.1. The Second Type of Matroidal Structure of Rough Sets

In this subsection, the second type of matroid induced by a partition is defined. Similar to the discussion of , the base, circuit, rank function, and closure of the second type of matroid are investigated.

From the analysis in Section 3.2, we know that for all and for all , is not a cycle. Obviously, any subgraph of is also not a cycle. Therefore, we can get the following proposition.

Proposition 5.1. Let . Then there exists a matroid on such that .

Proof. According to Definition 2.3, we need to prove that satisfies (I1)(I3). In terms of Section 3.2, for all , is the cycle whose vertices set is , that is, where is the set of edges of . So, it is obvious that satisfies (I1) and (I2). Here, we just need to prove satisfies (I3).

Let , and . Suppose that for all for , such that , that is, . Because , . It is obvious that . So . Since , we can get that . It is contradictory with the known conditions that . So such that for all , , namely, . That is . As a result, satisfies (I1)(I3). That is, there exists a matroid on such that .

The matroid mentioned in Proposition 5.1 is established from the viewpoint of cycle. It is a new type of matroid induced by a partition and is defined as follows.

Definition 5.2 (The second type of matroid induced by a partition). The second type of matroid induced by a partition over , denoted by -MIP, is such a matroid whose ground set and independent sets .

Because of the intuition of a graph, it is easy to understand the matroid established in Definition 5.2. In fact, we can formulate a as follows.

Proposition 5.3. Let where . Then is a -MIP.

Proof. Because , for all , for each . That means for all , , that is, . Conversely, for all , since for all , , we can get that , that is, . So . As a result, is a .

In terms of Propositions 5.1 and 5.3, we can find that the two matroids established in them are equivalent. That is to say for any matroid , if , then is a . So, the following corollary can be obtained.

Corollary 5.4. .

Similar to the axiomatization of , we will axiomatize with the set of circuits of it in the following.

Theorem 5.5. Let be a matroid. Then is a -MIP if and only if for all , and .

Proof. : Let be the induced by . Therefore, for all , , and for all , . And then for all , . So, according to (2.2) and Definition 2.5, . Thus, for all , , and .

: Since for all , , and , we can regard the as a partition over . Furthermore, . Because , for all , such that . According to Proposition 5.1 and Definition 5.2, we can get that . That is, is a .

Theorem 4.5 shows there is one-to-one correspondence between a partition and the induced by the partition. In the following theorem, we will discuss the relationship between a partition and the induced by the partition.

Theorem 5.6. Let be the collection of all partitions over , the set of all -MIP induced by partitions of , , that is, for all , where is the -MIP induced by . Then satisfy the following conditions:

(1)for all , if then ,(2)for all   s.t. .

Proof. Let and are two different partitions over , and , are two induced by and , respectively. We need to prove that there is an such that , or there is an such that . Because , there must be an equivalence class such that . Suppose that for all , . Then such that and . According to Proposition 5.3, and . Conversely, if such that , then such that and . Thus, according to Proposition 5.3, and .

Let be a matroid. According to Theorem 5.5, is a partition on . That is, .

Theorem 5.6 shows that there is also a bijection between a partition and the induced by the partition.

5.2. Characteristics of II-MIP

The is established from the viewpoint of cycle. There is a big difference between the formulations of and and the same as the characteristics between them. In this subsection, we will formulate the characteristics of .

A base of a matroid is one of the maximal independent sets of the matroid. From Propositions 5.1 and 5.3, the cardinality of one of the maximal independent sets is equal to and just one element of each equivalence class does not belong to the independent set.

Proposition 5.7. Let be the -MIP induced by , . Then is the set of bases of .

Proof. According to Definition 2.7, Proposition 5.3, and Corollary 5.4, it is straightforward.

So any equivalence class is not contained in some base of a . And there will be a cycle if a new element is put in the base. Specifically, if for all , , then . Next, we can also formulate the set of bases of a from the viewpoint of graph as follows.

Corollary 5.8. does not contain a cycle.

Proof. According to Propositions 5.1 and 5.7, it is straightforward.

From Proposition 5.7 and Corollary 5.8, we can find that for all , if , then is not contained in any base of the induced by . Therefore, we can get the following corollary.

Corollary 5.9. Let . Then .

As the analysis in Section 3.2, any equivalence class of a partition can be converted to a cycle. And any proper subset of an equivalence class does not form a cycle. So we can formulate the set of circuits of a as follows.

Proposition 5.10. Let be the -MIP induced by . Then is the set of circuits of .

Proof. According to Theorem 5.5, it is straightforward.

Likewise, from the viewpoint of graph, we can get another formulation of the set of circuits of a .

Corollary 5.11. is a cycle}.

Proof. According to the definition of and Proposition 5.10, it is straightforward.

Next, we will formulate the rank function of .

Proposition 5.12. Let be the -MIP induced by . Then for all , is the rank of in .

Proof. According to Definition 2.8 and Proposition 5.3, it is straightforward.

In all the characteristics of a matroid introduced in this paper, the rank function of a matroid is the one and only one numeric characteristic. In the following content, we will further study some properties of the rank function of .

Theorem 5.13. Let be the -MIP induced by , , and an -definable set. For all , if , then .

Proof. Because is an -definable set and , according to Proposition 5.3 and Corollary 5.4, for all Max and for all Max, . Furthermore, and . According to Definition 2.8, and . Because , . As a result, .

Theorem 5.13 provides a way to separate the rank of a subset into the sum of ranks of two disjoint sets contained in the subset. And one of the two disjoint sets is a definable set. It is obvious that any subset of the universe can be represented as the union of a definable set, and an indefinable set which are disjoint and contained in the subset. This will be very helpful to some proofs in the following. In order to study the properties of the rank function of conveniently, the concept of lower approximation number will be defined as follows.

Definition 5.14 (Lower approximation number). Let be a relation on . For all , is called the lower approximation number of with respect to . When there is no confusion, we omit the subscript .

In classical rough sets, usually refers to the equivalence relation. For a better understanding to the lower approximation number, an example is served.

Example 5.15. Let be the universe, an equivalence relation over , and . Compute the lower approximation numbers of , , and where , , and .

Because is an equivalence relation, according to Definition 2.1, for all , . Therefore, we can get that , , and .

Similarly, according to Definition 2.2, we can get that , , and .

Theorem 5.16. Let be the -MIP induced by , . Then for all , .

Proof. Let be the largest -definable set contained in . Then . Thus is an -indefinable set and for all , . According to Proposition 5.12 and Definition 5.14, and . Therefore, in terms of Theorem 5.13, .

Theorem 5.16 combines the lower approximation number and the rank function of closely. And the formulation is very simple. This is useful to study rough sets with matroidal approaches and vice verse.

Lemma 5.17. Let be the -MIP induced by , . Then for all , .

Proof. Let be the largest -definable set contained in and the largest -definable set contained in . Then, in terms of Theorem 5.16, and . So .

The last lemma discusses the relationship between the ranks of a subset and its complementary set. For a subset of , if is a definable set then is also a definable set. So, we can get the following lemma.

Lemma 5.18. Let be the -MIP induced by , . Then for all , is an -definable set if and only if .

Proof. : According to Theorems 5.13 and 5.16 and Lemma 5.17, it is straightforward.

: According to Lemma 5.17, . Then, according to Definition 5.14, is an -definable set.

From Definition 2.9, we know that, for any subset of , if such that the rank of is equal to the rank of , then belongs to the closure of . In , we can say that if contains one cycle more than , then belongs to the closure of .

Proposition 5.19. Let be the -MIP induced by . Then for all , is the closure of in .

Proof. According to Proposition 5.12, for all , . Then, in terms of Definition 2.9, . Let and . For all , if then . According to Proposition 5.12, for all , . That means such that , that is, such that . As a result .

The hyperplane of can be formulated as follows.

Proposition 5.20. Let be the -MIP induced by . Then is the hyperplane of .

Proof. According to Definition 2.10, we need to prove that for all , is a close set of , and . And more, we need to prove that for all , if , then is not a hyperplane of .

  Is a close set of . For all , there is an equivalence class and a subset such that and . Therefore, for all and for all , is not an equivalence class, that is, . That is, the set . Thus, according to Proposition 5.19, . That is to say is a close set of . , . For any , since is -definable, by Theorem 5.13

 That is,

 Furthermore, since is also -definable, by Theorem 5.13

 That is,

 Therefore, from (5.2) and (5.4), . , if , then is not a hyperplane of . If , then there are two cases.  and , for all . . Next, we discuss these two cases, respectively.

Case 1. By Proposition 5.19, so is not a close set.

Case 2. if , then so is not a close set. If , then suppose that . In that case, for such that , . So

In that case, is not equal to .

Summing up, is the hyperplane of .

5.3. Approximations Established through II-MIP

In this subsection, we will redefine the lower and the upper approximations with some characteristics of . Three pairs of approximations are established.

Proposition 5.10 shows that the set of circuits of a is equal to the partition which induces the . So we can get the following proposition.

Proposition 5.21. Let be the -MIP induced by , . Then for all,

Proof. According to Proposition 5.10, (2.1), it is straightforward.

We know that the lower and the upper approximations of a subset are all definable sets. The lower approximation is the largest definable set contained in the subset. And the upper approximation is the smallest definable set, which contains the subset. Since Lemma 5.18 provides a way to decide whether a subset is a definable set, we can define the lower and the upper approximations as follows.

Proposition 5.22. Let be the -MIP induced by , . Then for all ,

Proof. According to Lemma 5.18, for all , if , then is an -definable set. So is the largest -definable set contained in . And is the smallest -definable set contained . And then, according to (2.1), it is straightforward.

Next, we will establish the lower approximation via the closure of .

Proposition 5.23. Let be the -MIP induced by , . Then for all ,

Proof. For all , if , then for all , . According to Proposition 5.19, . That is, . In terms of (2.1), .

In terms of (), we can get the upper approximation by using the duality. That is, .

6. Relationship between I-MIP and II-MIP

In the previous two sections, we have got two types of matroids induced by a partition from the viewpoints of complete graph and cycle, respectively. And it is can be found that the matroidal characteristics of them are very different. But, if the two types of matroid are induced by the same partition, what are the relationships between them? In this section, we will study this issue.

Definition 6.1 (Dual matroid see [39]). Let be a matroid, and the set of bases of . The dual matroid is the matroid on the set whose bases . If , then is called an identically self-dual matroid.

For convenience, in the following content, we suppose that is the universe, is an equivalence relation over , is the partition on , and is the I-MIP induced by and the II-MIP induced by .

Proposition 6.2. Let be the dual matroid of . Then .

Proof. For all , according to Definition 6.1, . For all such that , such that . Therefore, where . In terms of Proposition 5.7, . As a result, .

Proposition 6.2 shows that and are dual matroids. This result is very interesting and helpful to study rough sets. Maybe people want to ask some questions about the relationships between and as follows: whether a I-MIP and a II-MIP which induced by different partitions could be dual matroids? And whether two different I-MIP (or II-MIP) could be dual matroids? Next, we will answer them.

Proposition 6.3. Let and be two partitions over , the I-MIP induced by , and the induced by . Then if and only if .

Proof. According to Theorems 4.5 and 5.6 and Proposition 6.2, it is straightforward.

Proposition 6.3 shows that a and a induced by different partitions are not dual matroids.

Proposition 6.4. Let and be two different partitions over and and are the induced by and , respectively. Then .

Proof. Suppose that . From Proposition 6.2, is a II-MIP. In terms of Proposition 5.10, we can get that . Therefore, according to Theorem 4.4, for all , . Thus . This is contrary to the known condition that and are two different partitions over . As a result, .

Proposition 6.5. Let and be two different partitions over and and are the II-MIP induced by and , respectively. Then .

Proof. Similar to the proof of Proposition 6.4, it is straightforward.

Propositions 6.4 and 6.5 show that two different I-MIP are not dual matroids. And the same as two different II-MIP. One can find that we emphasize in Propositions 6.4 and 6.5 that and are two different partitions over . So, in Propositions 6.4 and 6.5, could be equal to when ? We will discuss this problem in the following proposition.

Proposition 6.6. if and only if for all, .

Proof. : According to Definition 6.1, if , then . According to Propositions 4.1, 4.3, and 5.1, for all , .

: It is straightforward.

Proposition 6.6 provides a necessary and sufficient condition to decide whether a or a is a self-dual matroid. This gives a good answer to the previous question that whether could be equal to when .

The set of circuits of a matroid is complementary to the set of hyperplanes of its dual matroid [39]. That is,

Next, we will study when a hyperplane of is also a hyperplane of .

Proposition 6.7. Let . Then if and only if .

Proof. : According to Proposition 4.16, , that is, such that . If , according to Proposition 5.20, then . According to Proposition 4.10 and Corollary 4.11, , that is, .

: If , then and such that . Therefore, according to Propositions 4.16 and 5.20, .

From the Propositions 4.14 and 5.19, we can find, for any subset , that the is generally the subset of . Now, we will study under what conditions that the is certain the subset of .

Proposition 6.8. Let , , and . Then if and only if for all, .

Proof. : Let . According to Propositions 4.14 and 5.19, we can get that . That is, for all , . In terms of Section 4, we know that . Therefore, for all , . And for all , . That is, for all , .

: According to Propositions 4.14 and 5.19, it is straightforward.

The formulation of rank functions of and are very different. In consideration of that and are dual matroids, it is meaningful to discuss the relationship between the rank functions of them.

Theorem 6.9. Let , , and . Then if and only if .

Proof. : If , according to Theorem 5.16, . Then, in terms of Definition 2.2 and Proposition 4.13, we can get that . Therefore, . And then .

: With the same way used in the above proof, it is straightforward.

In Theorem 6.9, we make use of the lower and the upper approximations numbers to discuss the relationship between the rank functions of and . It adequately reflects the close relation between rough sets and matroids.

7. Conclusions

We make a further study to the combination of rough sets and matroids. Two graphical ways are provided to establish and understand the matroidal structure of rough sets intuitively. For a better research on the relationships between rough sets and matroids, the concept of lower approximation number is proposed. And then, some meaningful results are obtained. For example, Theorem 5.16 shows that, for any subset , the rank of within the context of is equal to the difference of the cardinality and the lower approximation number of . And Theorem 6.9 indicates that the rank of within the context of is equal to it within the context of if and only if the cardinality of is equal to the difference of its lower and upper approximation numbers. Furthermore, the relationships between the two kinds of matroids established in Sections 4 and 5 are discussed. It is so exciting that the two kinds of matroids are dual matroids. And this is meaningful to the study of the combination of rough sets and matroids.

Matroids possess a sophisticated mathematical structure. And it has been widely used in real world. So we hope our work in this paper could be contributive to the theoretical development and applications of rough sets. In future works, we will study the axiomatization of rough sets with matroidal approaches and explore the wider applications of rough sets with matroids.

Acknowledgments

This work is in part supported by National Science Foundation of China under Grant nos. 60873077, 61170128 and the Natural Science Foundation of Fujian Province, China under Grant no. 2011J01374.

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