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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 973920, 27 pages
http://dx.doi.org/10.1155/2012/973920
Research Article

Matroidal Structure of Rough Sets from the Viewpoint of Graph Theory

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 [610], and the neighborhood instead of the equivalence class [1114]. Besides, using some other mathematical theories, such as fuzzy sets [1519], boolean algebra [2023], topology [2427], lattice theory [2830], 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 𝑈 be a nonempty and finite set called universe, 𝐑 a family of equivalence relations over 𝑈, then the relational system 𝐾=(𝑈,𝐑) is called a knowledge base [1]. If 𝐐𝐑, then 𝐐 is also an equivalence relation [1]. And 𝐐 is called an indiscernibility relation and denoted by IND(𝐐) [1]. If 𝑅𝐑, then 𝑈/𝑅 represents the partition of 𝑈 induced by 𝑅. That is in the partition 𝑈/𝑅={𝑇1,𝑇2,,𝑇𝑛}, for all 𝑇𝑖𝑈/𝑅, 𝑇𝑖𝑈 and 𝑇𝑖, 𝑇𝑖𝑇𝑗= for 𝑖𝑗, 𝑖,𝑗=1,2,,𝑛, and 𝑇𝑖=𝑈. Each 𝑇𝑖 in 𝑈/𝑅 is an equivalence class, and it can also be denoted by [𝑥]𝑅 if 𝑥𝑇𝑖.

For any subset 𝑋𝑈, the lower and the upper approximations of 𝑋 with respect to 𝑅 are defined as follows [1]: 𝑅_(𝑋)={𝑇𝑈/𝑅𝑇𝑋},𝑅(𝑋)={𝑇𝑈/𝑅𝑇𝑋}.(2.1) Set 𝐵𝑁𝑅(𝑋)=𝑅(𝑋)𝑅_(𝑋) is called the 𝑅-boundary of 𝑋 or the boundary region of 𝑋 with respect to 𝑅 [1]. If 𝑅_(𝑋)=𝑅(𝑋), that is, 𝐵𝑁𝑅(𝑋)=, then 𝑋 is 𝑅-definable, or 𝑋 is called a definable set with respect to 𝑅; else, if 𝑅_(𝑋)𝑅(𝑋), that is, 𝐵𝑁𝑅(𝑋), then 𝑋 is rough with respect to 𝑅, or 𝑋 is called a rough set with respect to 𝑅 [1]. The lower and the upper approximations satisfy duality, that is [1],(P1) for all 𝑋𝑈, 𝑅_(𝑋)=𝑅(𝑋),(P2) for all 𝑋𝑈, 𝑅(𝑋)=𝑅_(𝑋),where 𝑋 represents the set 𝑈𝑋.

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 𝑅 be a relation on 𝑈. For all 𝑥𝑈, 𝑅𝑁𝑅(𝑥)={𝑦𝑈𝑥𝑅𝑦} is called the successor neighborhood of 𝑥 in 𝑅. When there is no confusion, we omit the subscript 𝑅.

Definition 2.2 (Upper approximation number [32]). Let 𝑅 be a relation on 𝑈. For all 𝑋𝑈, 𝑓𝑅(𝑋)=|{𝑅𝑁𝑅(𝑥)𝑥𝑈𝑅𝑁𝑅(𝑥)𝑋}| is called the upper approximation number of 𝑋 with respect to 𝑅.

2.2. Graph Theory

A graph 𝐺 is an ordered pair of disjoint sets (𝑉,𝐸) such that 𝐸 is a subset of the set 𝑉(2) of unordered pairs of 𝑉 [37]. The set 𝑉 is the set of vertices and 𝐸 is the set of edges. If 𝐺 is a graph, then 𝑉=𝑉(𝐺) is the vertex set of 𝐺, and 𝐸=𝐸(𝐺) is the edge set. An empty graph is a graph whose edge set is empty. An edge {𝑢,𝑣} is said to join the vertices 𝑢 and 𝑣 and is denoted by 𝑢𝑣. Thus 𝑢𝑣 and 𝑣𝑢 mean exactly the same edge; the vertices 𝑢 and 𝑣 are the endpoints of this edge. If 𝑢𝑣𝐸(𝐺), then 𝑢 and 𝑣 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 𝑣 in a graph 𝐺, denoted by 𝑑𝐺(𝑣) or 𝑑(𝑣), is the number of edges incident to 𝑣, except that each loop at 𝑣 counts twice.

A simple graph is a graph having no loops or parallel edges [38]. An isomorphism [38] from a simple graph 𝐺 to a simple graph 𝐻 is a bijection 𝑓𝑉(𝐺)𝑉(𝐻) such that 𝑢𝑣𝐸(𝐺) if and only if 𝑓(𝑢)𝑓(𝑣)𝐸(𝐻). That is to say “𝐺 is isomorphic to 𝐻,” denoted by 𝐺𝐻 if there is an isomorphism from 𝐺 to 𝐻.

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

2.3. Matroids

Definition 2.3 (Matroid [39]). A matroid 𝑀 is a pair (𝐸,), where 𝐸 (called the ground set) is a finite set and (called the independent sets) is a family of subsets of 𝐸 satisfying the following axioms: (I1);(I2) if 𝐼 and 𝐼𝐼, then 𝐼;(I3) if 𝐼1,𝐼2 and |𝐼1|<|𝐼2|, then 𝑒𝐼2𝐼1 such that 𝐼1{𝑒},where || represents the cardinality of “”.

The matroid 𝑀 is generally denoted by 𝑀=𝑀(𝐸,). 𝐸(𝑀) represents the ground set of 𝑀 and (𝑀) the independent sets of 𝑀. Each element of (𝑀) is called an independent set of 𝑀. If a subset 𝑋 of 𝐸 is not an independent set, then it is called a dependent set. The family of all dependent sets of 𝑀 is denoted by 𝒟(𝑀), that is, 𝒟(𝑀)=2𝐸(𝑀).(2.2)

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

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

Let 𝐸 be a set and 𝒜2𝐸. Then [39]: Max(𝒜)={𝑋𝒜𝑌𝒜,if𝑋𝑌then𝑋=𝑌},Min(𝒜)={𝑋𝒜𝑌𝒜,if𝑌𝑋then𝑋=𝑌},Opp(𝒜)={𝑋𝐸𝑋𝒜},Com(𝒜)={𝑋𝐸𝐸𝑋𝒜}.(2.3)

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

A circuit in a matroid 𝑀(𝐸,) is a set which is not independent but has the property that every proper subset of it is independent. In Example 2.4, 𝒞(𝑀)={{𝑎,𝑐}}.

Theorem 2.6 (Circuit axioms [39]). Let 𝒞 be a family of subsets of 𝐸. Then there exists 𝑀(𝐸,) such that 𝒞=𝒞(𝑀) if and if only 𝒞 satisfies the following properties:(𝐶1)𝒞;(𝐶2) if 𝐶1,𝐶2𝒞 and 𝐶1𝐶2, then 𝐶1=𝐶2;(𝐶3) if 𝐶1,𝐶2𝒞, 𝐶1𝐶2, and 𝑒𝐶1𝐶2, then 𝐶3𝒞 such that 𝐶3(𝐶1𝐶2){𝑒}.

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

In Example 2.4, according to Definition 2.7, we can get that (𝑀)={{𝑎,𝑏},{𝑏,𝑐}}.

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 𝑀=𝑀(𝐸,) be a matroid. Then the rank function 𝑟𝑀 of 𝑀 is defined as: for all 𝑋𝐸, 𝑟𝑀||𝐼||(𝑋)=max𝐼𝐼𝑋.(2.4)

A matroid can be determined by its base, its rank function, or its circuit. For a set, 𝐼𝐸 is independent if and only if it is contained in some base, if and only if it satisfies 𝑟𝑀(𝐼)=|𝐼|, 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 𝑀=𝑀(𝐸,) be a matroid. For all 𝑋𝐸, the closure operator cl𝑀 of 𝑀 is defined as follows: cl𝑀(𝑋)=𝑒𝐸𝑟𝑀(𝑋)=𝑟𝑀(𝑋{𝑒}).(2.5)

If 𝑒cl𝑀(𝑋), we say that 𝑒 depends on 𝑋. The closure of 𝑋 is composed of these elements of 𝐸 that depend on 𝑋. If cl𝑀(𝑋)=𝑋, then 𝑋 is called a closed set of 𝑀.

Definition 2.10 (Hyperplane [39]). Let 𝑀=(𝐸,) be a matroid. 𝐻𝐸 is called a hyperplane of 𝑀 if 𝐻 is a closed set of 𝑀 and 𝑟𝑀(𝐻)=𝑟𝑀(𝐸)1. And (𝑀) represents the family of all hyperplanes of 𝑀.

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 𝑛 is denoted by 𝐾𝑛.
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 𝑈={𝑎,𝑏,𝑐,𝑑,𝑒,𝑓,𝑔,} be the universe, 𝑅 an equivalence relation, and 𝑈/𝑅={𝑇1,𝑇2,𝑇3}={{𝑎},{𝑏,𝑐},{𝑑,𝑒,𝑓,𝑔,}}. Then each equivalence class can be transformed to a complete graph showed in Figure 1.

973920.fig.001
Figure 1: The complete graphs of equivalence classes.

Figure 1(a) represents the complete graph of the equivalence class 𝑇1. Because 𝑇1 just includes one element 𝑎, there is only one vertex and no edge in the complete graph Figure 1(a). Figure 1(b) represents the complete graph of 𝑇2, which includes two vertices and only one edge connecting the two vertices. And Figure 1(c) represents the complete graph of 𝑇3. 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 𝐾|𝑇1|, 𝐾|𝑇2|, and 𝐾|𝑇3|, 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 𝐺=(𝑈,𝐸), where 𝐸𝑃={𝑥𝑦𝑥𝑇𝑃𝑦𝑇{𝑥}}.(3.1)

It is obvious that if 𝑥 and 𝑦 belong to the same equivalence class, then there is an edge 𝑥𝑦 in 𝐸. So, we can formulate the equivalence class as follows: for all 𝑥𝑈, [𝑥]𝑅={𝑥}{𝑦𝑥𝑦𝐸}.(3.2)

Furthermore, for any subset 𝑋𝑈, the lower and the upper approximations of 𝑋 can be formulated as follows: 𝑅_[𝑇][𝑋](𝑋)={𝑇𝑈/𝑅𝐺𝐺},𝑅(𝑋)=𝑇𝑈/𝑅𝑌Ts.t.𝐾|𝑇|[𝑌][𝑋].𝐺(3.3)

3.2. The Cycle

A walk [37] 𝑊 in a graph is an alternating sequence of vertices and edges, say 𝑥0,𝑒1,𝑥1,𝑒2,,𝑒𝑙,𝑥𝑙 where 𝑒𝑖=𝑥(𝑖1)𝑥𝑖, 0<𝑖𝑙. For simplicity, the walk 𝑊 can also be denoted by 𝑥0𝑥1𝑥𝑙; the length of 𝑊 is 𝑙, 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 𝑈/𝑅, it can be represented by a cycle. As shown in Figure 2, the equivalence class 𝑇1, 𝑇2, and 𝑇3 are represented by Figures 2(a), 2(b), and 2(c), respectively.

fig2
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 𝑎 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 𝑑𝑒𝑓𝑔𝑑 and cycle 𝑑𝑓𝑒𝑔𝑑 can be treated as the same cycle.

For convenience, to an equivalence class 𝑇 in 𝑈/𝑅, the cycle whose vertices set is equal to 𝑇 is denoted by 𝐶𝑇, that is, 𝐶𝑇=(𝑇,𝐸𝑇) is a graph (cycle) where 𝐸𝑇 is the set of edges of 𝐶𝑇. Then the partition 𝑈/𝑅 can be transformed to be a graph 𝐺=(𝑈,𝐸) where 𝐸𝐸=𝑇𝑇𝑈/𝑅.(3.4)

One may ask which edges belong to 𝐸𝑇 exactly? In fact, it is nonnecessary to define the edges of 𝐸𝑇 exactly. Here, we just need to form a cycle with the vertex set 𝑇. That is, each pair of vertices of 𝑇 are connected and the degree of each vertex is equal to 2. In other words, we simply need to know that each vertex of 𝐶𝑇 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 𝑥𝑈, [𝑥]𝑅={𝑦𝑈𝑦isconnectedto𝑥}.(3.5)

Likewise, for any subset 𝑋𝑈, the lower and the upper approximations of 𝑋 can be formulated as follows: 𝑅_(𝑋)=𝑇𝑈/𝑅𝐶𝑇[𝑋],𝐺𝑅(𝑋)=𝑇𝑈/𝑅𝑌𝑇s.t.𝐶𝑇[𝑌][𝑋].𝐺(3.6)

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 𝑈 is the universe, 𝑅 an equivalence relation over 𝑈 and 𝑃=𝑈/𝑅 the partition. And 𝐺𝑃=(𝑈,𝐸𝑃) is the graph induced by 𝑃, where 𝐸𝑃={𝑥𝑦𝑥𝑇𝑃𝑦𝑇{𝑥}}.

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 𝑣 of a complete graph, there is not a loop whose vertex is 𝑣. 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 𝑃={𝑋𝑈𝐺𝑃[𝑋] is an empty graph}. Then there is a matroid 𝑀 on 𝑈 such that (𝑀)=𝑃.

Proof. According to Definition 2.3, we just need to prove that 𝑃 satisfies axioms (I1)(I3). It is obvious that (I1) and (I2) hold. Suppose that 𝑋,𝑌𝑃 and |𝑋|<|𝑌|. Because 𝐺𝑃[𝑋] and 𝐺𝑃[𝑌] are empty graphs, according to the definition of 𝐸𝑃, each 𝑥𝑋 belongs to a different equivalence class with the others of 𝑋 and the same to each 𝑦𝑌. Since |𝑋|<|𝑌|, thus there must be at least one element 𝑦0𝑌 such that 𝑦0 belongs to some equivalence class which does not include any element of 𝑋. Therefore, 𝑋{𝑦0}𝑃. As a result, 𝑃 satisfies (I3). That is, there exists a matroid 𝑀 on 𝑈 such that (𝑀)=𝑃.

If 𝐺[𝑋] is an empty graph, then it means that any two different vertices of 𝐺[𝑋] are nonadjacent. That is, each vertex of 𝐺[𝑋] comes from a different complete graph with others. For instance, in Example 3.2, we can get 𝑃 as follows: 𝑃={,{𝑎},{𝑏},{𝑐},{𝑑},{𝑒},{𝑓},{𝑔},{},{𝑎,𝑏},{𝑎,𝑐},{𝑎,𝑑},{𝑎,𝑒},{𝑎,𝑓},{𝑎,𝑔},{𝑎,},{𝑏,𝑑},{𝑏,𝑒},{𝑏,𝑓},{𝑏,𝑔},{𝑏,},{𝑐,𝑑},{𝑐,𝑒},{𝑐,𝑓},{𝑐,𝑔},{𝑐,},{𝑎,𝑏,𝑑},{𝑎,𝑏,𝑒},{𝑎,𝑏,𝑓},{𝑎,𝑏,𝑔},{𝑎,𝑏,},{𝑎,𝑐,𝑑},{𝑎,𝑐,𝑒},{𝑎,𝑐,𝑓},{𝑎,𝑐,𝑔},{𝑎,𝑐,}}.(4.1)

Definition 4.2 (The first type of matroid induced by a partition). The first type of matroid induced by a partition 𝑃 over 𝑈, denoted by 𝐼-MIP, is such a matroid whose ground set 𝐸=𝑈 and independent sets ={𝑋𝑈𝐺𝑃[𝑋] is an empty graph}.

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

Proposition 4.3. Let 𝑀𝑃=(𝑈,𝑃) be an 𝐼-MIP. Then for all 𝑋𝑈, 𝑋𝑃 if and only if for all𝑇𝑃 such that |𝑋𝑇|1.

Proof. (): If 𝑋𝑃, then 𝐺𝑃[𝑋] is an empty graph. According to the definition of 𝐸𝑃, each element of 𝑋 comes from a different equivalence class with the others of 𝑋. That is, for all 𝑇𝑃, |𝑋𝑇|1.
(): Let 𝑋𝑈. If for all 𝑇𝑃, |𝑋𝑇|1, then 𝐺𝑃[𝑋] is an empty graph. Therefore, 𝑋𝑃.

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 𝐼-𝑀𝐼𝑃 in terms of its circuit.

Theorem 4.4. Let 𝑀 be a matroid induced by 𝑃. Then 𝑀 is an 𝐼-MIP if and only if for all 𝐶𝒞(𝑀), |𝐶|=2.

Proof. According to Definition 2.3, we know that (𝑀)2𝑈. If (𝑀)=2𝑈, then 𝑀 is a 𝐼-MIP induced by 𝑃={{𝑥}𝑥𝑈}. In this case, according to (2.2), 𝒟(𝑀)= and 𝒞(𝑀)=. It indicates that there is not any circuit in 𝒞(𝑀), and, therefore, we do not need to care whether the cardinality of each circuit of 𝒞(𝑀) is equal to 2. That is, 𝒞(𝑀)= is compatible with the description that for all 𝐶𝒞(𝑀), |𝐶|=2. Similarly, if 𝒞(𝑀)=, then 𝑀 is a 𝐼-MIP induced by 𝑃={{𝑥}𝑥𝑈}. So, Theorem 4.4 is true when the set of circuits of 𝑀 is empty.
Next, we prove that Theorem 4.4 is true when the set of circuits of 𝑀 is nonempty.
(): According to (2.2), 𝒟(𝑀)=2𝑈𝑃. Therefore, in terms of Definition 4.2 and Proposition 4.1, for all 𝑋𝑈, 𝑋𝒟(𝑀) if and only if 𝐺𝑃[𝑋] is not an empty graph. That is, there is at least one edge in 𝐺𝑃[𝑋]. Obviously, the set of endpoints of each edge of 𝐺𝑃[𝑋] is a dependent set. So, for all 𝑋𝒟(𝑀), there is a set 𝑌 composed of the endpoints of some edge of 𝐺𝑃[𝑋] such that 𝑌𝑋. According to Definition 2.5, 𝑌𝒞(𝑀) and 𝑋𝒞(𝑀). That is, for all 𝐶𝒞(𝑀), |𝐶|=2.
(): 𝒟(𝑀)={𝑋𝑈𝐶𝒞(𝑀)s.t.𝐶𝑋}. According to (2.2), (𝑀)=2𝑈𝒟(𝑀). Therefore, for all 𝐼(𝑀), 𝐶𝒞(𝑀) such that 𝐶𝐼, that is, for all 𝐶𝒞(𝑀), |𝐶𝐼|1. So, for all 𝐶1,𝐶2𝒞(𝑀); if 𝐼𝐶1, then 𝐼𝐶2=𝐶1𝐶2. According to Theorem 2.6, if 𝐶1𝐶2, then 𝐶3𝒞(𝑀) such that 𝐶3=𝐶1𝐶2𝐶1𝐶2. For all 𝐶𝑖𝒞(𝑀), let 𝑇𝐶𝑖={𝐶𝒞(𝑀)𝐶𝐶𝑖}. Then |𝐼𝑇𝐶1|=1. Furthermore, if 𝐶1𝐶2=, then 𝑇𝐶1𝑇𝐶2=. If for all 𝐶𝒞(𝑀), 𝐼𝐶=, then for all 𝑦𝐼, 𝐶𝒞(𝑀) such that 𝑦𝐶. Thus, 𝑃𝒞(𝑀)={𝑇𝐶𝑖𝐶𝑖𝒞(𝑀)}{{𝑦}𝑦𝑈𝒞(𝑀)} is a partition over 𝑈. So (𝑀)={𝐼𝑈forall𝑋𝑃𝒞(𝑀),|𝐼𝑋|1}. According to Definition 4.2, 𝑀=(𝑈,) is a 𝐼-𝑀𝐼𝑃.
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 𝐼-𝑀𝐼𝑃 induced by the partition. This question will be answered by the following theorem.

Theorem 4.5. Let 𝒫 be the collection of all partitions over 𝑈, the set of all 𝐼-MIP induced by partitions of 𝒫, 𝑓𝒫, that is, for all 𝑃𝒫, 𝑓(𝑃)=𝑀𝑃, where 𝑀𝑃 is the 𝐼-MIP induced by 𝑃. Then 𝑓 satisfies the following conditions:(1)for all 𝑃1,𝑃2𝒫, and if 𝑃1𝑃2 then 𝑓(𝑃1)𝑓(𝑃2),(2)for all 𝑀, 𝑃𝑀𝒫 s.t. 𝑓(𝑃𝑀)=𝑀.

Proof. (1) Let 𝑃1,𝑃2𝒫, 𝑃1𝑃2, and 𝑀𝑃1=(𝑈,𝑃1), 𝑀𝑃2=(𝑈,𝑃2) are two 𝐼-𝑀𝐼𝑃 induced by 𝑃1 and 𝑃2, respectively. We need to prove that there is an 𝐼1𝑃1 such that 𝐼1𝑃2, or there is an 𝐼2𝑃2 such that 𝐼2𝑃1. Because 𝑃1𝑃2, there is at least one equivalence class 𝑇1𝑃1 such that 𝑇1𝑃2. If 𝑇2𝑃2 such that 𝑇1𝑇2, then 𝑋𝑈 and 𝑋𝑃1 such that 𝑋(𝑇2𝑇1). That means 𝑋𝑃2. Else, there at least two equivalence classes 𝑇2𝑖,𝑇2𝑗𝑃2 such that 𝑇2𝑖𝑇1 and 𝑇2𝑗𝑇1. That is, there is a set 𝑌𝑃2 such that 𝑌𝑇2𝑖𝑇1 and 𝑌𝑇2𝑗𝑇1. Obviously, according to Proposition 4.3, 𝑌𝑃1.
(2) Let 𝑀=(𝑈,) be a 𝐼-𝑀𝐼𝑃, for all 𝑥𝑈, 𝐶𝑥={𝑥}({𝐶𝒞(𝑀)𝐶{𝑥}}). According to Theorem 4.4, for all 𝑦𝑈 and 𝑦𝐶𝑥, then 𝐶𝑥𝐶𝑦=. Therefore, we can get a family 𝐶𝑈={𝐶𝑥𝑥𝑈}. It is obvious that 𝐶𝑈=𝑈. Therefore, 𝐶𝑈 is a partition of 𝑈. That is, 𝐶𝑈𝒫 and 𝑓(𝐶𝑈)=𝑀.

Theorem 4.5 shows that there is one-to-one correspondence between a partition and the 𝐼-𝑀𝐼𝑃 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 𝐼-𝑀𝐼𝑃 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 𝑃 in Section 4.1, the maximal independent set is the vertex set whose cardinality is equal to the cardinality of 𝑃. Then the following proposition can be obtained.

Proposition 4.6. Let 𝑀𝑃 be the 𝐼-MIP induced by 𝑃, 𝑌𝑈, and 𝐺𝑃[𝑌]=(𝑌,𝐸𝑌) a subgraph of 𝐺𝑃. Then 𝑃={𝑋𝑈|𝑋|=|𝑃|𝐸𝑋=} is the set of bases of 𝑀𝑃.

Proof. According to Definition 2.7, we need to prove that (𝑀𝑃)=𝑃, namely, Max(𝑃)={𝑋𝑈|𝑋|=|𝑃|𝐸𝑋=}. In terms of Proposition 4.3, for all 𝐼𝑃, |𝐼𝑇|1 for all 𝑇𝑃. So, for all 𝐼(𝑀𝑃), |𝐼|=|𝑃|. According to Proposition 4.1 and Definition 4.2, for all 𝐼(𝑀𝑃), 𝐺𝑃[𝐼] is an empty graph, that is, 𝐼𝑃. Similarly, we can prove in the same way that for all 𝑋𝑃, 𝑋(𝑀𝑃). That is, (𝑀𝑃)=𝑃.

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

Corollary 4.7. Let 𝑋𝑈. Then 𝑋𝑃 if and only if for all𝑇𝑃, |𝑋𝑇|=1.

Proof.  According to Proposition 4.6, it is straightforward.

Corollary 4.8. 𝑃=𝑈.

Proof. According to Proposition 4.6, it is straightforward.

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

Proposition 4.9. Let 𝑀𝑃=(𝑈,𝑃) be the 𝐼-MIP induced by 𝑃. Then 𝒞𝑃={{𝑥,𝑦}𝑥𝑦𝐸𝑃} is the set of circuits of 𝑀𝑃.

Proof. According to Definition 2.5, we need to prove 𝒞(𝑀𝑃)=𝒞𝑃, that is, Min(Opp(𝑃))={{𝑥,𝑦}𝑥𝑦𝐸𝑃}. 𝒟(𝑀)=Opp(𝑃), for all 𝐼𝒟(𝑀), 𝑥,𝑦𝐼 such that 𝑥𝑦𝐸(𝐺𝑃[𝐼]). Furthermore, 𝑥𝑦𝐸𝑃. If {𝑥,𝑦}𝐼, then {𝑥,𝑦}𝒞(𝑀𝑃) and 𝐼𝒞(𝑀𝑃); else, {𝑥,𝑦}=𝐼𝒞(𝑀𝑃). So, for all 𝐼𝒞(𝑀𝑃), 𝐼𝒞𝑃. Similarly, for all {𝑥,𝑦}𝒞𝑃, 𝐼𝒟(𝑀𝑃) such that {𝑥,𝑦}𝐼. Therefore, {𝑥,𝑦}𝒞(𝑀𝑃). As a result, 𝒞𝑃={{𝑥,𝑦}𝑥𝑦𝐸𝑃}.

Likewise, Proposition 4.3 provides a necessary and sufficient condition to decide whether a set is an independent set of 𝑀𝑃. In this way, we can get the family of dependent sets of 𝑀𝑃 as follows: 𝒟𝑀𝑃=||||𝑋𝑈𝑇𝑃s.t.𝑋𝑇>1.(4.2)

Moreover, in terms of the Definition 2.5, we can get the set of circuits of 𝑀𝑃 as follows: 𝒞𝑀𝑃𝒟𝑀=Min𝑃.(4.3)

According to Proposition 4.1, it can be found that each subset of 𝑈 which contains exactly one element is an independent set. So, for any dependent set 𝑋 of 𝑀𝑃, if |𝑋|>2 then there must exist a subset 𝑌 of 𝑋 such that |𝑌|=2 and 𝑌 is a dependent set. Thus, we can get the following proposition.

Proposition 4.10. Let 𝑀𝑃 be the 𝐼-MIP induced by 𝑃. Then 𝒞+={𝑋𝑈𝑇𝑃s.t.𝑋𝑇|𝑋|=2} is the set of circuits of 𝑀𝑃.

Proof. According to Proposition 4.3, for all 𝑋𝒞+, 𝑋 is a dependent set, that is, 𝑋𝒟(𝑀𝑃). In terms of Proposition 4.9, 𝑋𝒞(𝑀𝑃). Similarly, for all 𝐼𝒞(𝑀𝑃), according to Proposition 4.9, 𝐼𝒞+. That is, 𝒞+ is the set of circuits of 𝑀𝑃.

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

Corollary 4.11. 𝒞𝑃=𝒞+.

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

Proposition 4.12. Let 𝑀𝑃 be the 𝐼-MIP induced by 𝑃. Then for all 𝑥𝑈, [𝑥]𝑅=𝑀{𝑥}𝑦𝑈{𝑥,𝑦}𝒞𝑃.(4.4)

Proof. According to Proposition 4.10, for all 𝑇𝑃, if 𝑥𝑇 then {𝑥,𝑦}𝒞(𝑀𝑃) for each 𝑦𝑇{𝑥}. And for any 𝑦𝑈 and 𝑦𝑇, {𝑥,𝑦}𝒞(𝑀𝑃). Therefore, 𝑇=[𝑥]𝑅={𝑥}{𝑦𝑈{𝑥,𝑦}𝒞(𝑀𝑃)}.

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 𝐸𝑃 whose vertex set just contains the two elements. For a subset 𝑋𝑈, if 𝑋 does not contain a circuit, then 𝑋 is an independent set and the rank of it is equal to |𝑋|. In other words, if 𝐺𝑃[𝑋] is an empty graph, that is, each pair of vertexes of 𝐺𝑃[𝑋] is nonadjacent, then the rank of 𝑋 is equal to |𝑋|. 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 𝑀𝑃 be the 𝐼-MIP induced by 𝑃. Then for all 𝑋𝑈, 𝑟𝑃(𝑋)=max{|𝑌|𝑌𝑋,𝐺𝑃[𝑌]  is  an  empty  graph} is the rank of 𝑋 in 𝑀𝑃.

If the set of bases 𝑀𝑃 of 𝑀𝑃 has been obtained, then for all 𝑋𝑈; we can get the rank of 𝑋 as follows: 𝑟+||||(𝑋)=max𝐵𝑋𝐵𝑀𝑃.(4.5)

It can be proved easily that 𝑟𝑃(𝑋)=𝑟+(𝑋). So, 𝑟+ is also the rank function of 𝑀𝑃.

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

Proposition 4.14. Let 𝑀𝑃 be the 𝐼-MIP induced by 𝑃. Then for all 𝑋𝑈, cl𝑃(𝑋)=𝑋{𝑦𝑈𝑋𝑥𝑋𝑥𝑦𝐸𝑃} is the closure of 𝑋 in 𝑀𝑃.

Proof. According to Definition 2.9, we need to prove that cl𝑃(𝑋)=cl𝑀𝑃(𝑋), that is, 𝑋{𝑦𝑈𝑋𝑥𝑋𝑥𝑦𝐸𝑃}={𝑥𝑈𝑟𝑀𝑃(𝑋)=𝑟𝑀𝑃(𝑋{𝑥})}. It is obvious that, for all 𝑋𝑈, 𝑋cl𝑃(𝑋). So, we just need to prove that for all 𝑦𝑈𝑋 if there is an element 𝑥𝑋 such that 𝑥𝑦𝐸𝑃 if and if only 𝑦{𝑥𝑈𝑟𝑀𝑃(𝑋)=𝑟𝑀𝑃(𝑋{𝑥})}. According to (3.1), 𝑥𝑦𝐸𝑃 if and if only 𝑥 and 𝑦 belong to the same equivalence. According to Definition 2.8, for all 𝑋𝑈, 𝑟𝑀𝑃(𝑋) is equal to the number of the maximal independent set contained in 𝑋. According to Proposition 4.3, for all 𝑋𝑋; if 𝑋 is a maximal independent set contained in 𝑋, then 𝑋{𝑦} is not an independent set. That is, 𝑟𝑀𝑃(𝑋)=|𝑋|=𝑟𝑀𝑃(𝑋{𝑦})=𝑟𝑀𝑃(𝑋{𝑦}). Thus, for all 𝑥𝑈, 𝑥cl𝑃(𝑋) if and if only 𝑥{𝑥𝑈𝑟𝑀𝑃(𝑋)=𝑟𝑀𝑃(𝑋{𝑥})}.

From Proposition 4.14, it can be found that, for any element 𝑦𝑈𝑋, if 𝑥𝑦𝐸(𝐺𝑃[𝑋]), then 𝑦cl𝑃(𝑋). Therefore, the closure of 𝑋 can be equivalently represented as cl𝑃𝐺(𝑋)=𝑋𝑦𝑈𝑋𝑥𝑦𝐸𝑃[𝑋].(4.6)

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

Corollary 4.15. Let 𝑥𝑈. For all 𝑇𝑃; if 𝑥𝑇 then cl𝑃({𝑥})=cl𝑃(𝑇).

Next, we will discuss the hyperplane of the 𝐼-𝑀𝐼𝑃. 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 𝐼-𝑀𝐼𝑃 induced by 𝑃 is equal to the cardinality of 𝑃, we can get the following proposition.

Proposition 4.16. Let 𝑀𝑃 be the 𝐼-MIP induced by 𝑃. Then 𝑃={𝑈𝑇𝑇𝑃} is the hyperplane of 𝑀𝑃.

Proof. According to (4.5) and Proposition 4.6, we know that the rank of the 𝐼-𝑀𝐼𝑃 induced by 𝑃 is equal to |𝑃|. Furthermore, in terms of Proposition 4.14 and Corollary 4.15, for all 𝑇𝑃, 𝑈𝑇 is a closed set and 𝑟𝑀𝑃(𝑈𝑇)=|𝑃|1. So 𝑈𝑇(𝑀𝑃), that is, for all 𝑋𝑃, 𝑋(𝑀𝑃). Similarly, we can get that for all 𝑋(𝑀𝑃), 𝑋𝑃.

4.3. Approximations Established through I-MIP

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

Proposition 4.17. Let 𝑀𝑃 be the 𝐼-MIP induced by 𝑃, (𝑀𝑃)=𝑃. Then for all 𝑋𝑈, 𝐵𝑅(𝑋)=𝐵𝑋𝑋𝐵𝑃𝐵𝑋,𝑋𝐵(4.7) where 𝐵𝑋𝑃 a base having the maximal intersection with 𝑋.

Proof. For all 𝐵𝑃, 𝐵𝑋 is an independent set contained in 𝑋. Because 𝐵𝑋 is a base having the maximal intersection with 𝑋, 𝑟𝑀𝑃(𝑋)=|𝐵𝑋𝑋|. Furthermore, for all 𝑦𝐵𝑋𝑋, [𝑦]𝑅𝑋=. Let 𝑆1={[𝑦]𝑅𝑦𝐵𝑋𝑋} and 𝑆2=𝑃𝑆1. Then 𝑅(𝑋)=𝑈𝑆1=𝑆2. If 𝐵𝑋𝑋𝐵, then 𝐵(𝐵𝑋𝑋)𝑆2. According to Corollary 4.7, for all 𝑌𝑆2, if for all 𝑇(𝑃𝑆2) and |𝑌𝑇|=1, then 𝑌(𝐵𝑋𝑋)𝑃. And {𝑌𝑆2forall𝑇(𝑃𝑆2),|𝑌𝑇|=1}=𝑆2. Therefore, 𝑆2={𝐵(𝐵𝑋𝑋)𝐵𝑃(𝐵𝑋𝑋)𝐵}. That is, 𝑅(𝑋)={𝐵(𝐵𝑋𝑋)𝐵𝑃(𝐵𝑋𝑋)𝐵}.

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

Proposition 4.18. Let 𝑀𝑃 be the 𝐼-MIP induced by 𝑃, 𝒞(𝑀𝑃)=𝒞𝑃. Then for all 𝑋𝑈, 𝐵𝑁𝑅(𝑋)=𝐶𝒞𝑃||||𝐶𝑋=1.(4.8)

Proof. According to Proposition 4.10 and Corollary 4.11, for all𝐶𝒞𝑃,𝑇𝑃 such that 𝐶𝑇. |𝐶𝑋|=1 means that each element of 𝐶 does not certainly belong either to 𝑋 or to 𝑋. And then {𝐶𝒞𝑃|𝐶𝑋|=1} is the collection of all elements, which do not certainly belong either to 𝑋 or to 𝑋. So 𝐵𝑁𝑅(𝑋)={𝐶𝒞𝑃|𝐶𝑋|=1}.

Proposition 4.19. Let 𝑀𝑃 be the 𝐼-MIP induced by 𝑃, 𝒞(𝑀𝑃)=𝒞𝑃. Then for all 𝑋𝑈, 𝑅_(𝑋)=𝑋𝐵𝑁𝑅(𝑋),𝑅(𝑋)=𝑋𝐵𝑁𝑅(𝑋),(4.9) where 𝐵𝑁𝑅(𝑋)={𝐶𝒞𝑃|𝐶𝑋|=1}.

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

Corollary 4.20. Let 𝑀𝑃 be the 𝐼-MIP induced by 𝑃, 𝒞(𝑀𝑃)=𝒞𝑃 and 𝑋𝑈. Then for all 𝐶𝒞𝑃, 𝐶̸𝑅(𝑋) if and only if for all 𝑥𝑋, {𝑥}𝑃.

Proof. (): According to Proposition 4.10 and Corollary 4.11, if for all 𝐶𝒞𝑃̸,𝐶𝑅(𝑋), then for all 𝑇𝑃 and |𝑇|2, 𝑇𝑋=. That is, for all 𝑥𝑋, {𝑥}𝑃.
(): It is straightforward.

Proposition 4.21. Let 𝑀𝑃 be the 𝐼-MIP induced by 𝑃, 𝑟𝑀𝑃=𝑟𝑃. Then for all 𝑋𝑈, the following equations hold:(1)𝑅(𝑋)={𝑥𝑈𝑟𝑃(𝑋)=𝑟𝑃(𝑋{𝑥})},(2)𝑅(𝑋)={𝑇𝑃𝑟𝑃(𝑋)=𝑟𝑃(𝑋𝑇)},(3)𝑅(𝑋)=Max({𝐴𝑈𝑟𝑃(𝑋)=𝑟𝑃(𝐴)}).

Proof. (1) According to Proposition 4.13, 𝑟𝑃(𝑋)=|𝑌| where 𝑌𝑋 and for all 𝑇𝑃, |𝑌𝑇|1. Let 𝑇𝑃. If |𝑌𝑇|=0, then 𝑇𝑋= and for all 𝑡𝑇, 𝑟𝑃(𝑋)=𝑟𝑃(𝑋{𝑡})1, that is, 𝑇̸𝑅(𝑋) and for all 𝑡𝑇, 𝑡{𝑥𝑈𝑟𝑃(𝑋)=𝑟𝑃(𝑋{𝑥})}; else, if |𝑌𝑇|=1, then 𝑋𝑇 and for all 𝑥𝑇, 𝑟𝑃(𝑋)=𝑟𝑃(𝑋{𝑥}), that is, 𝑇𝑅(𝑋) and for all 𝑡𝑇, 𝑡{𝑥𝑈𝑟𝑃(𝑋)=𝑟𝑃(𝑋{𝑥})}. That is, 𝑅(𝑋)={𝑥𝑈𝑟𝑃(𝑋)=𝑟𝑃(𝑋{𝑥})}.
Similarly, we can prove that (2) and (3) are true.

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

Proposition 4.22. Let 𝑀𝑃 be the 𝐼-MIP induced by 𝑃, cl𝑀𝑃=cl𝑃. Then for all 𝑋𝑈, 𝑅(𝑋)=cl𝑃(𝑋).(4.10)

Proof. According to Proposition 4.14 and Corollary 4.15, we can get that cl𝑃(𝑋)={𝑇𝑃𝑥𝑋𝑥𝑇}. Therefore, according to the definition of the upper approximation, it is obvious that 𝑅(𝑋)=cl𝑃(𝑋).

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 𝑀𝑃 be the 𝐼-MIP induced by 𝑃, (𝑀𝑃)=𝑃. Then for all 𝑋𝑈, 𝑅(𝑋)=𝐻𝐻𝑃.𝑋𝐻(4.11)

Proof. According to Proposition 4.16, for all 𝐻𝑃, 𝑈𝐻𝑃, that is, 𝐻𝑃. And if 𝑋𝐻, then 𝑋(𝐻). Therefore, in terms of (2.1), 𝐻𝑅(𝑋). Thus, 𝑅(𝑋)={𝐻𝐻𝑃𝑋𝐻}.

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 𝑈 is a universe, 𝑅 an equivalence relation over 𝑈, and 𝑃=𝑈/𝑅 the partition. And 𝐺𝑃=(𝑈,𝐸𝑃) is the graph induced by 𝑃, where 𝐸𝑃={𝐸𝑇𝑖𝑇𝑖𝑃}.

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 𝐼-𝑀𝐼𝑃, 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 𝑇𝑃 and for all 𝐾𝑇, 𝐶𝐾 is not a cycle. Obviously, any subgraph of 𝐶𝐾 is also not a cycle. Therefore, we can get the following proposition.

Proposition 5.1. Let 𝑃={𝑋𝑈forall𝑇𝑃,𝐶𝑇̸𝐺𝑃[𝑋]}. Then there exists a matroid 𝑀 on 𝑈 such that (𝑀)=𝑃.

Proof. According to Definition 2.3, we need to prove that 𝑃 satisfies (I1)(I3). In terms of Section 3.2, for all 𝑇𝑃, 𝐶𝑇 is the cycle whose vertices set is 𝑇, that is, 𝐶𝑇=(𝑇,𝐸𝑇) where 𝐸𝑇 is the set of edges of 𝐶𝑇. So, it is obvious that 𝑃 satisfies (I1) and (I2). Here, we just need to prove 𝑃 satisfies (I3).
Let 𝐼1,𝐼2𝑃, |𝐼1|<|𝐼2| and 𝐼2𝐼1={𝑒1,𝑒2,,𝑒𝑚}(1𝑚|𝑈|). Suppose that for all 𝑒𝑖𝐼2𝐼1 for 1𝑖𝑚, 𝑇𝑒𝑖𝑃 such that 𝑇𝑒𝑖𝐼1{𝑒𝑖}, that is, 𝑇𝑒𝑖{𝑒𝑖}𝐼1. Because |𝑇𝑒𝑖{𝑒𝑖}|1, |𝑇𝑒1{𝑒1}|+|𝑇𝑒2{𝑒2}|++|𝑇𝑒𝑚{𝑒𝑚}|𝑚=|𝐼2𝐼1|. It is obvious that 𝑇𝑒𝑖̸(𝐼1𝐼2){𝑒𝑖}. So |𝐼1|=|𝑇𝑒1{𝑒1}|+|𝑇𝑒2{𝑒2}|++|𝑇𝑒𝑚{𝑒𝑚}|+|𝐼1𝐼2|. Since |𝐼2|=|𝐼2𝐼1|+|𝐼2𝐼1|, we can get that |𝐼1||𝐼2|. It is contradictory with the known conditions that |𝐼1|<|𝐼2|. So 𝑒𝑖𝐼2𝐼1 such that for all 𝑇𝑃, 𝑇̸𝐼1{𝑒𝑖}, namely, 𝐶𝑇̸𝐺𝑃[𝐼1{𝑒𝑖}]. That is 𝐼1{𝑒𝑖}𝑃. As a result, 𝑃 satisfies (I1)(I3). That is, there exists a matroid 𝑀 on 𝑈 such that (𝑀)=𝑃.

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 𝑃 over 𝑈, denoted by 𝐼𝐼-MIP, is such a matroid whose ground set 𝐸=𝑈 and independent sets ={𝑋𝑈forall𝑇𝑃,𝐶𝑇̸𝐺[𝑋]}.

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 𝐼𝐼-𝑀𝐼𝑃 as follows.

Proposition 5.3. Let +={𝑛𝑖=1𝑆𝑖𝑆𝑖𝑇𝑖𝑇𝑖𝑃} where 𝑛=|𝑃|. Then 𝑀+=(𝑈,+) is a 𝐼𝐼-MIP.

Proof. Because 𝑆𝑖𝑇𝑖, for all 𝑋+, 𝑋𝑇𝑇 for each 𝑇𝑃. That means for all 𝑇𝑃, 𝐶𝑇̸𝐺𝑃[𝑋], that is, 𝑋𝑃. Conversely, for all 𝑋𝑃, since for all 𝑇𝑃, 𝐶𝑇̸𝐺𝑃[𝑋], we can get that 𝑇̸𝑋, that is, 𝑇𝑋𝑇. So 𝑋+. As a result, 𝑀+=(𝑈,+) is a 𝐼𝐼-𝑀𝐼𝑃.

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 𝑀=(𝑈,), if =+, then 𝑀 is a 𝐼𝐼-𝑀𝐼𝑃. So, the following corollary can be obtained.

Corollary 5.4. 𝑃=+.

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

Theorem 5.5. Let 𝑀=(𝑈,) be a matroid. Then 𝑀 is a 𝐼𝐼-MIP if and only if for all 𝐶1,𝐶2𝒞(𝑀), 𝐶1𝐶2= and 𝒞(𝑀)=𝑈.

Proof. (): Let 𝑀=(𝑈,) be the 𝐼𝐼-𝑀𝐼𝑃 induced by 𝑃. Therefore, for all 𝑇𝑃, 𝑇, and for all 𝑋𝑇, 𝑋. And then for all 𝑇𝑃, 𝑇𝒟(𝑀). So, according to (2.2) and Definition 2.5, 𝒞(𝑀)=𝑃. Thus, for all 𝐶1,𝐶2𝒞(𝑀), 𝐶1𝐶2=, and 𝒞(𝑀)=𝑈.
(): Since for all 𝐶1,𝐶2𝒞(𝑀), 𝐶1𝐶2=, and 𝒞(𝑀)=𝑈, we can regard the 𝒞(𝑀) as a partition over 𝑈. Furthermore, 𝒟(𝑀)={𝑋𝑈C𝒞(𝑀)s.t.𝐶𝑋}. Because =2𝑈𝒟(𝑀), for all 𝑋, 𝐶𝒞(𝑀) such that 𝐶𝑋. According to Proposition 5.1 and Definition 5.2, we can get that =+=𝑃. That is, 𝑀=(𝑈,) is a 𝐼𝐼-𝑀𝐼𝑃.

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

Theorem 5.6. Let 𝒫 be the collection of all partitions over 𝑈, the set of all 𝐼𝐼-MIP induced by partitions of 𝒫, 𝑔𝒫, that is, for all 𝑃𝒫, 𝑔(𝑃)=𝑀𝑃 where 𝑀P is the 𝐼𝐼-MIP induced by 𝑃. Then 𝑔 satisfy the following conditions:(1)for all 𝑃1,𝑃2𝒫, if 𝑃1𝑃2 then 𝑔(𝑃1)𝑔(𝑃2),(2)for all 𝑀,𝑃𝑀𝒫  s.t. (𝑃𝑀)=𝑀.

Proof. (1) Let 𝑃1 and 𝑃2 are two different partitions over 𝑈, and 𝑀𝑃1=(𝑈,𝑃1), 𝑀𝑃2=(𝑈,𝑃2) are two 𝐼𝐼-𝑀𝐼𝑃 induced by 𝑃1 and 𝑃2, respectively. We need to prove that there is an 𝐼1𝑃1 such that 𝐼1𝑃2, or there is an 𝐼2𝑃2 such that 𝐼2𝑃1. Because 𝑃1𝑃2, there must be an equivalence class 𝑇1𝑃1 such that 𝑇1𝑃2. Suppose that for all 𝑇𝑃2, 𝑇1̸𝑇. Then 𝑋𝑇1 such that 𝑋{𝑆𝑆𝑇1} and 𝑋{𝑆𝑆𝑇,forall𝑇𝑃2}. According to Proposition 5.3, 𝑋𝑃1 and 𝑋𝑃2. Conversely, if 𝑇2𝑃2 such that 𝑇1𝑇2, then 𝑋𝑇2 such that 𝑋{𝑆𝑆𝑇2} and 𝑋{𝑆𝑆𝑇,forall𝑇𝑃1}. Thus, according to Proposition 5.3, 𝑋𝑃2 and 𝑋𝑃1.
(2) Let 𝑀=(𝑈,) be a 𝐼𝐼-𝑀𝐼𝑃 matroid. According to Theorem 5.5, 𝒞(𝑀) is a partition on 𝑈. That is, 𝑔(𝒞(𝑀))=𝑀.

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

5.2. Characteristics of II-MIP

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

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 |𝑈||𝑃| and just one element of each equivalence class does not belong to the independent set.

Proposition 5.7. Let 𝑀𝑃 be the 𝐼𝐼-MIP induced by 𝑃, 𝑛=|𝑃|. Then 𝑃={𝑛𝑖=1(𝑇𝑖𝑥𝑖)𝑇𝑖𝑃𝑥𝑖𝑇𝑖} is the set of bases of 𝑀𝑃.

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 𝐼𝐼-𝑀𝐼𝑃. And there will be a cycle if a new element is put in the base. Specifically, if for all 𝑇𝑃, |𝑇|=1, then 𝑃={}. Next, we can also formulate the set of bases of a 𝐼𝐼-𝑀𝐼𝑃 from the viewpoint of graph as follows.

Corollary 5.8. 𝑃=Max({𝐵𝑈𝐺𝑃[𝐵] 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 𝑇𝑃, if |𝑇|=1, then 𝑇 is not contained in any base of the 𝐼𝐼-𝑀𝐼𝑃 induced by 𝑃. Therefore, we can get the following corollary.

Corollary 5.9. Let (𝑀𝑃)=𝐵𝑃. Then 𝐵𝑃=.

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 𝐼𝐼-𝑀𝐼𝑃 as follows.

Proposition 5.10. Let 𝑀𝑃 be the 𝐼𝐼-MIP induced by 𝑃. Then 𝒞𝑃=𝑃 is the set of circuits of 𝑀𝑃.

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 𝐼𝐼-𝑀𝐼𝑃.

Corollary 5.11. 𝒞𝑃={𝐶𝑈𝐺𝑃[𝐶] is a cycle}.

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

Next, we will formulate the rank function of 𝐼𝐼-𝑀𝐼𝑃.

Proposition 5.12. Let 𝑀𝑃 be the 𝐼𝐼-MIP induced by 𝑃. Then for all 𝑋𝑈, 𝑟𝑃̸(𝑋)=max{|𝑌|𝑌𝑋(forall𝑇𝑃,𝑇𝑌)} is the rank of 𝑋 in 𝑀𝑃.

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 𝐼𝐼-𝑀𝐼𝑃.

Theorem 5.13. Let 𝑀𝑃 be the 𝐼𝐼-MIP induced by 𝑃, 𝑟𝑃=𝑟𝑀𝑃, and 𝑋𝑈 an 𝑅-definable set. For all 𝑌𝑈, if 𝑋𝑌=, then 𝑟𝑃(𝑋𝑌)=𝑟𝑃(𝑋)+𝑟𝑃(𝑌).

Proof. Because 𝑋 is an 𝑅-definable set and 𝑋𝑌=, according to Proposition 5.3 and Corollary 5.4, for all 𝐼1 Max({𝐼𝑃𝐼𝑋}) and for all 𝐼2 Max({𝐼𝑃𝐼𝑌}), 𝐼1𝐼2=. Furthermore, 𝐼1𝐼2𝑃 and 𝐼1𝐼2̸Max({𝑍𝑋𝑌forall𝑇𝑃,𝑇𝑍}). According to Definition 2.8, 𝑟𝑃(𝑋)=|𝐼1| and 𝑟𝑃(𝑌)=|𝐼2|. Because 𝐼1𝐼2=, |𝐼1𝐼2|=|𝐼1|+𝐼2. As a result, 𝑟𝑃(𝑋𝑌)=𝑟𝑃(𝑋)+𝑟𝑃(𝑌).

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 𝐼𝐼-𝑀𝐼𝑃 conveniently, the concept of lower approximation number will be defined as follows.

Definition 5.14 (Lower approximation number). Let 𝑅 be a relation on 𝑈. For all 𝑋𝑈, 𝑓𝑅(𝑋)=|{𝑅𝑁𝑅(𝑥)𝑥𝑈𝑅𝑁𝑅(𝑥)𝑋}| is called the lower approximation number of 𝑋 with respect to 𝑅. When there is no confusion, we omit the subscript 𝑅.

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

Example 5.15. Let 𝑈={𝑎,𝑏,𝑐,𝑑,𝑒,𝑓,𝑔,} be the universe, 𝑅 an equivalence relation over 𝑈, and 𝑈/𝑅={𝑇1,𝑇2,𝑇3,𝑇4}={{𝑎,𝑏},{𝑐},{𝑑,𝑒,𝑓},{𝑔,}}. Compute the lower approximation numbers of 𝑋1, 𝑋2, and 𝑋3 where 𝑋1={𝑎,𝑏,𝑐}, 𝑋2={𝑎,𝑑,𝑔}, and 𝑋3={𝑎,𝑐,𝑔,}.
Because 𝑅 is an equivalence relation, according to Definition 2.1, for all 𝑥𝑈, 𝑅𝑁𝑅(𝑥)=[𝑥]𝑅. Therefore, we can get that 𝑓(𝑋1)=|{𝑇1,𝑇2}|=2, 𝑓(𝑋2)=||=0, and 𝑓(𝑋3)=|{𝑇2,𝑇4}|=2.
Similarly, according to Definition 2.2, we can get that 𝑓(𝑋1)=|{𝑇1,𝑇2}|=2, 𝑓(𝑋2)=|{𝑇1,𝑇3,𝑇4}|=3, and 𝑓(𝑋3)=|{𝑇1,𝑇2,𝑇4}|=3.

Theorem 5.16. Let 𝑀𝑃 be the 𝐼𝐼-MIP induced by 𝑃, 𝑟𝑃=𝑟𝑀𝑃. Then for all 𝑋𝑈, 𝑟𝑃(𝑋)=|𝑋|𝑓(𝑋).

Proof. Let 𝐴𝑋 be the largest 𝑅-definable set contained in 𝑋. Then 𝑋=𝐴(𝑋𝐴). Thus 𝑋𝐴 is an 𝑅-indefinable set and for all 𝑇𝑃, 𝑇̸𝑋𝐴. According to Proposition 5.12 and Definition 5.14, 𝑟𝑃(𝐴)=|𝐴|𝑓(𝑋) and 𝑟𝑃(𝑋𝐴)=|𝑋𝐴|=|𝑋||𝐴|. Therefore, in terms of Theorem 5.13, 𝑟𝑃(𝑋)=𝑟𝑃(𝐴(𝑋𝐴))=𝑟𝑃(𝐴)+𝑟𝑃(𝑋𝐴)=|𝑋|𝑓(𝑋).

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

Lemma 5.17. Let 𝑀𝑃 be the 𝐼𝐼-MIP induced by 𝑃, 𝑟𝑀𝑃=𝑟𝑃. Then for all 𝑋𝑈, 𝑟𝑃(𝑋)+𝑟𝑃(𝑋)=|𝑈|(𝑓(𝑋)+𝑓(𝑋)).

Proof. Let 𝐴𝑋 be the largest 𝑅-definable set contained in 𝑋 and 𝐵𝑋 the largest 𝑅-definable set contained in 𝑋. Then, in terms of Theorem 5.16, 𝑟𝑃(𝑋)=𝑟𝑃(𝐴(𝑋𝐴))=|𝑋|𝑓(𝑋) and 𝑟𝑃(𝑋)=𝑟𝑃(𝐵(𝑋𝐵))=|𝑋|𝑓(𝑋). So 𝑟𝑃(𝑋)+𝑟𝑃(𝑋)=|𝑋|+|𝑋|𝑓(𝑋)𝑓(𝑋)=|𝑈|(𝑓(𝑋)+𝑓(𝑋)).

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

Lemma 5.18. Let 𝑀𝑃 be the 𝐼𝐼-MIP induced by 𝑃, 𝑟𝑃=𝑟𝑀𝑃. Then for all 𝑋𝑈, 𝑋 is an 𝑅-definable set if and only if 𝑟𝑃(𝑋)+𝑟𝑃(𝑋)=|𝑈||𝑃|.

Proof. (): According to Theorems 5.13 and 5.16 and Lemma 5.17, it is straightforward.
(): According to Lemma 5.17, |𝑃|=𝑓(𝑋)+𝑓(𝑋). Then, according to Definition 5.14, 𝑋 is an 𝑅-definable set.

From Definition 2.9, we know that, for any subset 𝑋 of 𝑈, if 𝑥𝑈𝑋 such that the rank of 𝑋{𝑥} is equal to the rank of 𝑋, then 𝑥 belongs to the closure of 𝑋. In 𝐼𝐼-𝑀𝐼𝑃, we can say that if 𝑋{𝑥} contains one cycle more than 𝑋, then 𝑥 belongs to the closure of 𝑋.

Proposition 5.19. Let 𝑀𝑃 be the 𝐼𝐼-MIP induced by 𝑃. Then for all 𝑋𝑈, cl𝑃(𝑋)=𝑋{𝑥𝑈𝑋𝑌𝑋s.t.𝑌{𝑥}𝑃} is the closure of 𝑋 in 𝑀𝑃.

Proof. According to Proposition 5.12, for all 𝑥𝑋, 𝑟𝑃(𝑋)=𝑟𝑃(𝑋{𝑥}). Then, in terms of Definition 2.9, 𝑋cl𝑀𝑃(𝑋). Let 𝑌𝑋𝑋 and |𝑌𝑋|=𝑟𝑃(𝑋). For all 𝑥𝑈𝑋, if 𝑥cl𝑀𝑃(𝑋) then 𝑟𝑃(𝑋)=𝑟𝑃(𝑋{𝑥}). According to Proposition 5.12, for all 𝑇𝑃, 𝑇̸𝑌𝑋. That means 𝑇𝑃 such that 𝑇𝑌𝑋{𝑥}, that is, 𝑌𝑌𝑋 such that 𝑌{𝑥}𝑃. As a result cl𝑀𝑃(𝑋)=cl𝑃(𝑋).

The hyperplane of 𝐼𝐼-𝑀𝐼𝑃 can be formulated as follows.

Proposition 5.20. Let 𝑀𝑃 be the 𝐼𝐼-MIP induced by 𝑃. Then 𝑃={𝑈𝑋𝑇𝑃𝑋𝑇|𝑋|=2} is the hyperplane of 𝑀𝑃.

Proof. According to Definition 2.10, we need to prove that for all 𝐻𝑃, 𝐻 is a close set of 𝑀𝑃, and 𝑟𝑀𝑃(𝐻)=𝑟𝑀𝑃(𝑈)1. And more, we need to prove that for all 𝑌𝑈, if 𝑌𝑃, then 𝑌 is not a hyperplane of 𝑀𝑃.(1) Is a close set of 𝑀𝑃.For all 𝐻𝑃, there is an equivalence class 𝑇𝑃 and a subset 𝑋𝑇 such that |𝑋|=2 and 𝐻=𝑈𝑋. Therefore, for all 𝑌𝐻 and for all 𝑥𝑋, 𝑌{𝑥} is not an equivalence class, that