Matroidal Structure of Rough Sets from the Viewpoint of Graph Theory

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 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 ∅≠𝐐⊆𝐑, then ∩𝐐 is also an equivalence relation [1]. And ∩𝐐 is called an indiscernibility relation and denoted by IND(𝐐) [1]. If 𝑅∈𝐑, then 𝑈/𝑅 represents the partition of 𝑈 induced by 𝑅. That is in the partition 𝑈/𝑅={𝑇₁,𝑇₂,…,𝑇_𝑛}, for all 𝑇_𝑖∈𝑈/𝑅, 𝑇_𝑖⊆𝑈 and 𝑇_𝑖≠∅, 𝑇_𝑖∩𝑇_𝑗=∅ for 𝑖≠𝑗, 𝑖,𝑗=1,2,…,𝑛, and ∪𝑇_𝑖=𝑈. Each 𝑇_𝑖 in 𝑈/𝑅 is an equivalence class, and it can also be denoted by [𝑥]_𝑅 if 𝑥∈𝑇_𝑖.

For any subset 𝑋⊆𝑈, the lower and the upper approximations of 𝑋 with respect to 𝑅 are defined as follows [1]: 𝑅_{_}(𝑋)=∪{𝑇∈𝑈/𝑅∶𝑇⊆𝑋},𝑅(𝑋)=∪{𝑇∈𝑈/𝑅∶𝑇∩𝑋≠∅}.(2.1) Set 𝐵𝑁_𝑅(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, 𝐵𝑁_𝑅(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, 𝐵𝑁_𝑅(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 𝑋⊆𝑈, 𝑅_{_}(𝑋)=∼𝑅(∼𝑋),(P2) for all 𝑋⊆𝑈, 𝑅(𝑋)=∼𝑅_{_}(∼𝑋),where ∼𝑋 represents the set 𝑈−𝑋.

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 𝑥∈𝑈, 𝑅𝑁_𝑅(𝑥)={𝑦∈𝑈∶𝑥𝑅𝑦} 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 𝑋⊆𝑈, 𝑓^∗𝑅(𝑋)=|{𝑅𝑁_𝑅(𝑥)∶𝑥∈𝑈∧𝑅𝑁_𝑅(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 𝑢𝑣∈𝐸(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 𝑑_𝐺(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 𝑓∶𝑉(𝐺)→𝑉(𝐻) such that 𝑢𝑣∈𝐸(𝐺) if and only if 𝑓(𝑢)𝑓(𝑣)∈𝐸(𝐻). That is to say “𝐺 is isomorphic to 𝐻,” denoted by 𝐺≅𝐻 if there is an isomorphism from 𝐺 to 𝐻.

We say that 𝐺^=(𝑉^,𝐸^) is a subgraph of 𝐺=(𝑉,𝐸) if 𝑉^⊂𝑉 and 𝐸^⊂𝐸 [37]. In this case, we write 𝐺^⊂𝐺. 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,ℐ), where 𝐸 (called the ground set) is a finite set and ℐ (called the independent sets) is a family of subsets of 𝐸 satisfying the following axioms: (I1)∅∈ℐ;(I2) if 𝐼∈ℐ and 𝐼^⊆𝐼, then 𝐼^∈ℐ;(I3) if 𝐼₁,𝐼₂∈ℐ and |𝐼₁|<|𝐼₂|, then ∃𝑒∈𝐼₂−𝐼₁ such that 𝐼₁∪{𝑒}∈ℐ,where |⋅| represents the cardinality of “⋅”.

The matroid 𝑀 is generally denoted by 𝑀=𝑀(𝐸,ℐ). 𝐸(𝑀) represents the ground set of 𝑀 and ℐ(𝑀) the independent sets of 𝑀. Each element of ℐ(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)=2^𝐸−ℐ(𝑀).(2.2)

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

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 𝒜⊆2^𝐸. Then [39]: MCMOoapimnxp((((𝒜𝒜𝒜𝒜))))===={{{{𝑋𝑋𝑋𝑋⊆∈∈⊆𝐸𝐸𝒜𝒜∶∶∶∶𝐸𝑋∀∀𝑌𝑌−∉∈∈𝑋𝒜𝒜𝒜}∈,,,𝒜iiff}𝑌𝑋.⊆⊆𝑋𝑌tthheenn𝑋𝑋==𝑌𝑌}},,(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(ℐ)).

A circuit in a matroid 𝑀(𝐸,ℐ) 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,ℐ) such that 𝒞=𝒞(𝑀) if and if only 𝒞 satisfies the following properties:(𝐶1)∅∉𝒞;(𝐶2) if 𝐶₁,𝐶₂∈𝒞 and 𝐶₁⊆𝐶₂, then 𝐶₁=𝐶₂;(𝐶3) if 𝐶₁,𝐶₂∈𝒞, 𝐶₁≠𝐶₂, and ∃𝑒∈𝐶₁∩𝐶₂, then ∃𝐶₃∈𝒞 such that 𝐶₃⊆(𝐶₁∪𝐶₂)−{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 ℬ(𝑀), that is, ℬ(𝑀)=Max(ℐ).

In Example 2.4, according to Definition 2.7, we can get that ℬ(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,ℐ) be a matroid. Then the rank function 𝑟_𝑀 of 𝑀 is defined as: for all 𝑋⊆𝐸, 𝑟_𝑀(𝑋)=max||𝐼||∶𝐼∈ℐ∧𝐼⊆𝑋.(2.4)

A matroid can be determined by its base, its rank function, or its circuit. For a set, 𝐼⊆𝐸 is independent if and only if it is contained in some base, if and only if it satisfies 𝑟_𝑀(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,ℐ) be a matroid. For all 𝑋⊆𝐸, the closure operator cl_𝑀 of 𝑀 is defined as follows: cl_𝑀(𝑋)=𝑒∈𝐸∶𝑟_𝑀(𝑋)=𝑟_𝑀(𝑋∪{𝑒}).(2.5)

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

Definition 2.10 (Hyperplane [39]). Let 𝑀=(𝐸,ℐ) be a matroid. 𝐻⊆𝐸 is called a hyperplane of 𝑀 if 𝐻 is a closed set of 𝑀 and 𝑟_𝑀(𝐻)=𝑟_𝑀(𝐸)−1. And ℋ(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 𝐾_𝑛.
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,ℎ} be the universe, 𝑅 an equivalence relation, and 𝑈/𝑅={𝑇₁,𝑇₂,𝑇₃}={{𝑎},{𝑏,𝑐},{𝑑,𝑒,𝑓,𝑔,ℎ}}. 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 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 𝐸_𝑃={𝑥𝑦∶𝑥∈𝑇∈𝑃∧𝑦∈𝑇−{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 𝑥∈𝑈, [𝑥]_𝑅={𝑥}∪{𝑦∶𝑥𝑦∈𝐸}.(3.2)

Furthermore, for any subset 𝑋⊆𝑈, the lower and the upper approximations of 𝑋 can be formulated as follows: 𝑅_{_}(𝑋)=∪{𝑇∈𝑈/𝑅∶𝐺[𝑇]⊂𝐺[𝑋]},𝑅(𝑋)=∪𝑇∈𝑈/𝑅∶∃𝑌⊆Ts.t.𝐾_|𝑇|[𝑌]⊂𝐺[𝑋].(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,…,𝑒_𝑙,𝑥_𝑙 where 𝑒_𝑖=𝑥_(𝑖−1)𝑥_𝑖, 0<𝑖≤𝑙. For simplicity, the walk 𝑊 can also be denoted by 𝑥₀𝑥₁⋯𝑥_𝑙; 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.

(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 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 𝑑𝑒𝑓𝑔ℎ𝑑 and cycle 𝑑𝑓ℎ𝑒𝑔𝑑 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 𝐶_𝑇, that is, 𝐶_𝑇=(𝑇,𝐸_𝑇) is a graph (cycle) where 𝐸_𝑇 is the set of edges of 𝐶_𝑇. Then the partition 𝑈/𝑅 can be transformed to be a graph 𝐺=(𝑈,𝐸) where 𝐸=∪𝐸_𝑇∶𝑇∈𝑈/𝑅.(3.4)

One may ask which edges belong to 𝐸_𝑇 exactly? In fact, it is nonnecessary to define the edges of 𝐸_𝑇 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 𝐶_𝑇 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 𝑥∈𝑈, [𝑥]_𝑅={𝑦∈𝑈∶𝑦isconnectedto𝑥}.(3.5)

Likewise, for any subset 𝑋⊆𝑈, the lower and the upper approximations of 𝑋 can be formulated as follows: 𝑅_{_}(𝑋)=∪𝑇∈𝑈/𝑅∶𝐶_𝑇⊂𝐺[𝑋],𝑅(𝑋)=∪𝑇∈𝑈/𝑅∶∃𝑌⊆𝑇s.t.𝐶_𝑇[𝑌]⊂𝐺[𝑋].(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 𝐺_𝑃=(𝑈,𝐸_𝑃) is the graph induced by 𝑃, where 𝐸_𝑃={𝑥𝑦∶𝑥∈𝑇∈𝑃∧𝑦∈𝑇−{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 ℐ_𝑃={𝑋⊆𝑈∶𝐺_𝑃[𝑋] is an empty graph}. Then there is a matroid 𝑀 on 𝑈 such that ℐ(𝑀)=ℐ_𝑃.

Proof. According to Definition 2.3, we just need to prove that ℐ_𝑃 satisfies axioms (I1)∼(I3). It is obvious that (I1) and (I2) hold. Suppose that 𝑋,𝑌∈ℐ_𝑃 and |𝑋|<|𝑌|. Because 𝐺_𝑃[𝑋] and 𝐺_𝑃[𝑌] are empty graphs, according to the definition of 𝐸_𝑃, each 𝑥∈𝑋 belongs to a different equivalence class with the others of 𝑋 and the same to each 𝑦∈𝑌. Since |𝑋|<|𝑌|, thus there must be at least one element 𝑦₀∈𝑌 such that 𝑦₀ belongs to some equivalence class which does not include any element of 𝑋. Therefore, 𝑋∪{𝑦₀}∈ℐ_𝑃. As a result, ℐ_𝑃 satisfies (I3). That is, there exists a matroid 𝑀 on 𝑈 such that ℐ(𝑀)=ℐ_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 ℐ_𝑃 as follows: ℐ_𝑃={∅{{𝑎𝑎,,,{ℎ𝑏𝑎,}}𝑒,,}{{,𝑏𝑏{,}𝑎𝑑,,}{𝑏,𝑐,{}𝑓𝑏,},{𝑒,𝑑}{}𝑎,,,{𝑏{𝑏,𝑒,𝑔}𝑓},}{,,{𝑓{𝑎}𝑏,,,𝑏𝑔{,}𝑔ℎ},}{,,𝑏{{,ℎ𝑎ℎ},}𝑐,,,{𝑑{𝑎𝑐},,𝑏,𝑑{}}𝑎,,,{{𝑐𝑎,𝑐,𝑒,𝑐}𝑒}},,{,{{𝑎𝑎𝑐,,,𝑐𝑑𝑓,𝑓}},},{{,𝑎{𝑐,,𝑎𝑒𝑔,}}𝑐,,,𝑔{{𝑎}𝑐,,,𝑓ℎ{𝑎}},,,𝑐{{,𝑎𝑎ℎ,,}𝑔𝑏},}𝑑.,},(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 ℐ={𝑋⊆𝑈∶𝐺_𝑃[𝑋] is an empty graph}.

Obviously, matroids proposed in Proposition 4.1 is a 𝐼-𝑀𝐼𝑃. From the above result of ℐ_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 𝑀_𝑃=(𝑈,ℐ_𝑃) be an 𝐼-MIP. Then for all 𝑋⊆𝑈, 𝑋∈ℐ_𝑃 if and only if for all𝑇∈𝑃 such that |𝑋∩𝑇|≤1.

Proof. (⇒): If 𝑋∈ℐ_𝑃, then 𝐺_𝑃[𝑋] is an empty graph. According to the definition of 𝐸_𝑃, each element of 𝑋 comes from a different equivalence class with the others of 𝑋. That is, for all 𝑇∈𝑃, |𝑋∩𝑇|≤1.
(⇐): Let 𝑋⊆𝑈. If for all 𝑇∈𝑃, |𝑋∩𝑇|≤1, then 𝐺_𝑃[𝑋] is an empty graph. Therefore, 𝑋∈ℐ_𝑃.

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 𝐶∈𝒞(𝑀), |𝐶|=2.

Proof. According to Definition 2.3, we know that ℐ(𝑀)⊆2^𝑈. If ℐ(𝑀)=2^𝑈, then 𝑀 is a 𝐼-MIP induced by 𝑃={{𝑥}∶𝑥∈𝑈}. 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 𝐶∈𝒞(𝑀), |𝐶|=2. Similarly, if 𝒞(𝑀)=∅, then 𝑀 is a 𝐼-MIP induced by 𝑃={{𝑥}∶𝑥∈𝑈}. 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)=2^𝑈−ℐ_𝑃. Therefore, in terms of Definition 4.2 and Proposition 4.1, for all 𝑋⊆𝑈, 𝑋∈𝒟(𝑀) if and only if 𝐺_𝑃[𝑋] is not an empty graph. That is, there is at least one edge in 𝐺_𝑃[𝑋]. Obviously, the set of endpoints of each edge of 𝐺_𝑃[𝑋] is a dependent set. So, for all 𝑋∈𝒟(𝑀), there is a set 𝑌 composed of the endpoints of some edge of 𝐺_𝑃[𝑋] such that 𝑌⊆𝑋. According to Definition 2.5, 𝑌∈𝒞(𝑀) and 𝑋∉𝒞(𝑀). That is, for all 𝐶∈𝒞(𝑀), |𝐶|=2.
(⇐): 𝒟(𝑀)={𝑋⊆𝑈∶∃𝐶∈𝒞(𝑀)s.t.𝐶⊆𝑋}. According to (2.2), ℐ(𝑀)=2^𝑈−𝒟(𝑀). Therefore, for all 𝐼∈ℐ(𝑀), ∄𝐶∈𝒞(𝑀) such that 𝐶⊆𝐼, that is, for all 𝐶∈𝒞(𝑀), |𝐶∩𝐼|≤1. So, for all 𝐶₁,𝐶₂∈𝒞(𝑀); if 𝐼∩𝐶₁≠∅, then 𝐼∩𝐶₂=𝐶₁∩𝐶₂. According to Theorem 2.6, if 𝐶₁∩𝐶₂≠∅, then ∃𝐶₃∈𝒞(𝑀) such that 𝐶₃=𝐶₁∪𝐶₂−𝐶₁∩𝐶₂. For all 𝐶_𝑖∈𝒞(𝑀), let 𝑇_𝐶𝑖={𝐶∈𝒞(𝑀)∶𝐶∩𝐶_𝑖≠∅}. Then |𝐼∩𝑇_𝐶1|=1. Furthermore, if 𝐶₁∩𝐶₂=∅, then 𝑇_𝐶1∩𝑇_𝐶2=∅. If for all 𝐶∈𝒞(𝑀), 𝐼∩𝐶=∅, then for all 𝑦∈𝐼, ∄𝐶∈𝒞(𝑀) such that 𝑦∈𝐶. Thus, 𝑃_𝒞(𝑀)={𝑇_𝐶𝑖∶𝐶_𝑖∈𝒞(𝑀)}∪{{𝑦}∶𝑦∈𝑈−∪𝒞(𝑀)} is a partition over 𝑈. So ℐ(𝑀)={𝐼⊆𝑈∶forall𝑋∈𝑃_𝒞(𝑀),|𝐼∩𝑋|≤1}. According to Definition 4.2, 𝑀=(𝑈,ℐ) 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, ℳ the set of all 𝐼-MIP induced by partitions of 𝒫, 𝑓∶𝒫→ℳ, that is, for all 𝑃∈𝒫, 𝑓(𝑃)=𝑀_𝑃, where 𝑀_𝑃 is the 𝐼-MIP induced by 𝑃. Then 𝑓 satisfies the following conditions:(1)for all 𝑃₁,𝑃₂∈𝒫, and if 𝑃₁≠𝑃₂ then 𝑓(𝑃₁)≠𝑓(𝑃₂),(2)for all 𝑀∈ℳ, ∃𝑃_𝑀∈𝒫 s.t. 𝑓(𝑃_𝑀)=𝑀.

Proof. (1) Let 𝑃₁,𝑃₂∈𝒫, 𝑃₁≠𝑃₂, and 𝑀_𝑃1=(𝑈,ℐ_𝑃1), 𝑀_𝑃2=(𝑈,ℐ_𝑃2) are two 𝐼-𝑀𝐼𝑃 induced by 𝑃₁ and 𝑃₂, respectively. We need to prove that there is an 𝐼₁∈ℐ_𝑃1 such that 𝐼₁∉ℐ_𝑃2, or there is an 𝐼₂∈ℐ_𝑃2 such that 𝐼₂∉ℐ_𝑃1. Because 𝑃₁≠𝑃₂, there is at least one equivalence class 𝑇₁∈𝑃₁ such that 𝑇₁∉𝑃₂. If ∃𝑇₂∈𝑃₂ such that 𝑇₁⊂𝑇₂, then ∃𝑋⊆𝑈 and 𝑋∈ℐ_𝑃1 such that 𝑋∩(𝑇₂−𝑇₁)≠∅. That means 𝑋∉ℐ_𝑃2. Else, there at least two equivalence classes 𝑇_2𝑖,𝑇_2𝑗∈𝑃₂ such that 𝑇_2𝑖∩𝑇₁≠∅ and 𝑇_2𝑗∩𝑇₁≠∅. That is, there is a set 𝑌∈ℐ_𝑃2 such that 𝑌∩𝑇_2𝑖∩𝑇₁≠∅ and 𝑌∩𝑇_2𝑗∩𝑇₁≠∅. Obviously, according to Proposition 4.3, 𝑌∉ℐ_𝑃1.
(2) Let 𝑀=(𝑈,ℐ) be a 𝐼-𝑀𝐼𝑃, for all 𝑥∈𝑈, 𝐶_𝑥={𝑥}∪(∪{𝐶∈𝒞(𝑀)∶𝐶∩{𝑥}≠∅}). According to Theorem 4.4, for all 𝑦∈𝑈 and 𝑦∉𝐶_𝑥, then 𝐶_𝑥∩𝐶_𝑦=∅. Therefore, we can get a family 𝐶_𝑈={𝐶_𝑥∶𝑥∈𝑈}. It is obvious that ∪𝐶_𝑈=𝑈. Therefore, 𝐶_𝑈 is a partition of 𝑈. That is, 𝐶_𝑈∈𝒫 and 𝑓(𝐶_𝑈)=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 ℐ_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 𝑀_𝑃 be the 𝐼-MIP induced by 𝑃, 𝑌⊆𝑈, and 𝐺_𝑃[𝑌]=(𝑌,𝐸_𝑌) a subgraph of 𝐺_𝑃. Then ℬ_𝑃={𝑋⊆𝑈∶|𝑋|=|𝑃|∧𝐸_𝑋=∅} is the set of bases of 𝑀_𝑃.

Proof. According to Definition 2.7, we need to prove that ℬ(𝑀_𝑃)=ℬ_𝑃, namely, Max(ℐ_𝑃)={𝑋⊆𝑈∶|𝑋|=|𝑃|∧𝐸_𝑋=∅}. In terms of Proposition 4.3, for all 𝐼∈ℐ_𝑃, |𝐼∩𝑇|≤1 for all 𝑇∈𝑃. So, for all 𝐼∈ℬ(𝑀_𝑃), |𝐼|=|𝑃|. According to Proposition 4.1 and Definition 4.2, for all 𝐼∈ℬ(𝑀_𝑃), 𝐺_𝑃[𝐼] is an empty graph, that is, 𝐼∈ℬ_𝑃. Similarly, we can prove in the same way that for all 𝑋∈ℬ_𝑃, 𝑋∈ℬ(𝑀_𝑃). That is, ℬ(𝑀_𝑃)=ℬ_𝑃.

For a base 𝐵 in ℬ_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 𝑋⊆𝑈. Then 𝑋∈ℬ_𝑃 if and only if for all𝑇∈𝑃, |𝑋∩𝑇|=1.

Proof.  According to Proposition 4.6, it is straightforward.

Corollary 4.8. ∪ℬ_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 𝑀_𝑃. 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 𝐺_𝑃[𝑋], which is adjacent. Furthermore, a minimal dependent set of 𝑀_𝑃 is the vertex set of an edge of 𝐺_𝑃. Then we can get the following proposition.

Proposition 4.9. Let 𝑀_𝑃=(𝑈,ℐ_𝑃) be the 𝐼-MIP induced by 𝑃. Then 𝒞_𝑃={{𝑥,𝑦}∶𝑥𝑦∈𝐸_𝑃} is the set of circuits of 𝑀_𝑃.

Proof. According to Definition 2.5, we need to prove 𝒞(𝑀_𝑃)=𝒞_𝑃, that is, Min(Opp(ℐ_𝑃))={{𝑥,𝑦}∶𝑥𝑦∈𝐸_𝑃}. 𝒟(𝑀)=Opp(ℐ_𝑃), for all 𝐼∈𝒟(𝑀), ∃𝑥,𝑦∈𝐼 such that 𝑥𝑦∈𝐸(𝐺_𝑃[𝐼]). Furthermore, 𝑥𝑦∈𝐸_𝑃. If {𝑥,𝑦}⊂𝐼, then {𝑥,𝑦}∈𝒞(𝑀_𝑃) and 𝐼∉𝒞(𝑀_𝑃); else, {𝑥,𝑦}=𝐼∈𝒞(𝑀_𝑃). So, for all 𝐼∈𝒞(𝑀_𝑃), 𝐼∈𝒞_𝑃. Similarly, for all {𝑥,𝑦}∈𝒞_𝑃, ∃𝐼∈𝒟(𝑀_𝑃) such that {𝑥,𝑦}⊆𝐼. Therefore, {𝑥,𝑦}∈𝒞(𝑀_𝑃). As a result, 𝒞_𝑃={{𝑥,𝑦}∶𝑥𝑦∈𝐸_𝑃}.

Likewise, Proposition 4.3 provides a necessary and sufficient condition to decide whether a set is an independent set of 𝑀_𝑃. In this way, we can get the family of dependent sets of 𝑀_𝑃 as follows: 𝒟𝑀_𝑃=𝑋⊆𝑈∶∃𝑇∈𝑃s.t.||𝑋∩𝑇||>1.(4.2)

Moreover, in terms of the Definition 2.5, we can get the set of circuits of 𝑀_𝑃 as follows: 𝒞𝑀_𝑃=Min𝒟𝑀_𝑃.(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 𝑀_𝑃, 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 𝑀_𝑃 be the 𝐼-MIP induced by 𝑃. Then 𝒞₊={𝑋⊆𝑈∶∃𝑇∈𝑃s.t.𝑋⊆𝑇∧|𝑋|=2} is the set of circuits of 𝑀_𝑃.

Proof. According to Proposition 4.3, for all 𝑋∈𝒞₊, 𝑋 is a dependent set, that is, 𝑋∈𝒟(𝑀_𝑃). In terms of Proposition 4.9, 𝑋∈𝒞(𝑀_𝑃). Similarly, for all 𝐼∈𝒞(𝑀_𝑃), according to Proposition 4.9, 𝐼∈𝒞₊. That is, 𝒞₊ is the set of circuits of 𝑀_𝑃.

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

Corollary 4.11. 𝒞_𝑃=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 𝑀_𝑃 be the 𝐼-MIP induced by 𝑃. Then for all 𝑥∈𝑈, [𝑥]_𝑅={𝑥}∪𝑦∈𝑈∶{𝑥,𝑦}∈𝒞𝑀_𝑃.(4.4)

Proof. According to Proposition 4.10, for all 𝑇∈𝑃, if 𝑥∈𝑇 then {𝑥,𝑦}∈𝒞(𝑀_𝑃) for each 𝑦∈𝑇−{𝑥}. And for any 𝑦∈𝑈 and 𝑦∉𝑇, {𝑥,𝑦}∉𝒞(𝑀_𝑃). Therefore, 𝑇=[𝑥]_𝑅={𝑥}∪{𝑦∈𝑈∶{𝑥,𝑦}∈𝒞(𝑀_𝑃)}.

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 𝐸_𝑃 whose vertex set just contains the two elements. For a subset 𝑋⊆𝑈, 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 𝐺_𝑃[𝑋] is an empty graph, that is, each pair of vertexes of 𝐺_𝑃[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 𝑀_𝑃 be the 𝐼-MIP induced by 𝑃. Then for all 𝑋⊆𝑈, 𝑟_𝑃(𝑋)=max{|𝑌|∶𝑌⊆𝑋,𝐺_𝑃[𝑌]  is  an  empty  graph} is the rank of 𝑋 in 𝑀_𝑃.

If the set of bases ℬ_𝑀𝑃 of 𝑀_𝑃 has been obtained, then for all 𝑋⊆𝑈; we can get the rank of 𝑋 as follows: 𝑟₊(𝑋)=max||𝐵∩𝑋||∶𝐵∈ℬ_𝑀𝑃.(4.5)

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

Different from the rank of 𝑋 in 𝑀_𝑃, the closure of 𝑋 is the maximal subset of 𝑈, which contains 𝑋 and its rank is equal to 𝑋. For an element 𝑦∈𝑈−𝑋, if there is an element 𝑥∈𝑋 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 𝑀_𝑃 be the 𝐼-MIP induced by 𝑃. Then for all 𝑋⊆𝑈, cl_𝑃(𝑋)=𝑋∪{𝑦∈𝑈−𝑋∶𝑥∈𝑋∧𝑥𝑦∈𝐸_𝑃} is the closure of 𝑋 in 𝑀_𝑃.

Proof. According to Definition 2.9, we need to prove that cl_𝑃(𝑋)=cl_𝑀𝑃(𝑋), that is, 𝑋∪{𝑦∈𝑈−𝑋∶𝑥∈𝑋∧𝑥𝑦∈𝐸_𝑃}={𝑥∈𝑈∶𝑟_𝑀𝑃(𝑋)=𝑟_𝑀𝑃(𝑋∪{𝑥})}. It is obvious that, for all 𝑋⊆𝑈, 𝑋⊆cl_𝑃(𝑋). So, we just need to prove that for all 𝑦∈𝑈−𝑋 if there is an element 𝑥∈𝑋 such that 𝑥𝑦∈𝐸_𝑃 if and if only 𝑦∈{𝑥∈𝑈∶𝑟_𝑀𝑃(𝑋)=𝑟_𝑀𝑃(𝑋∪{𝑥})}. According to (3.1), 𝑥𝑦∈𝐸_𝑃 if and if only 𝑥 and 𝑦 belong to the same equivalence. According to Definition 2.8, for all 𝑋⊆𝑈, 𝑟_𝑀𝑃(𝑋) is equal to the number of the maximal independent set contained in 𝑋. According to Proposition 4.3, for all 𝑋^⊆𝑋; if 𝑋^ is a maximal independent set contained in 𝑋, then 𝑋^∪{𝑦} is not an independent set. That is, 𝑟_𝑀𝑃(𝑋)=|𝑋^|=𝑟_𝑀𝑃(𝑋^∪{𝑦})=𝑟_𝑀𝑃(𝑋∪{𝑦}). Thus, for all 𝑥∈𝑈, 𝑥∈cl_𝑃(𝑋) if and if only 𝑥∈{𝑥∈𝑈∶𝑟_𝑀𝑃(𝑋)=𝑟_𝑀𝑃(𝑋∪{𝑥})}.

From Proposition 4.14, it can be found that, for any element 𝑦∈𝑈−𝑋, if 𝑥𝑦∈𝐸(𝐺_𝑃[𝑋]), then 𝑦∈cl_𝑃(𝑋). Therefore, the closure of 𝑋 can be equivalently represented as cl_𝑃(𝑋)=𝑋∪𝑦∈𝑈−𝑋∶𝑥𝑦∈𝐸𝐺_𝑃[𝑋].(4.6)

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

Corollary 4.15. Let 𝑥∈𝑈. For all 𝑇∈𝑃; if 𝑥∈𝑇 then cl_𝑃({𝑥})=cl_𝑃(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 𝑀_𝑃 be the 𝐼-MIP induced by 𝑃. Then ℋ_𝑃={𝑈−𝑇∶𝑇∈𝑃} is the hyperplane of 𝑀_𝑃.

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 𝑇∈𝑃, 𝑈−𝑇 is a closed set and 𝑟_𝑀𝑃(𝑈−𝑇)=|𝑃|−1. So 𝑈−𝑇∈ℋ(𝑀_𝑃), that is, for all 𝑋∈ℋ_𝑃, 𝑋∈ℋ(𝑀_𝑃). Similarly, we can get that for all 𝑋∈ℋ(𝑀_𝑃), 𝑋∈ℋ_𝑃.

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 𝑀_𝑃 be the 𝐼-MIP induced by 𝑃, ℬ(𝑀_𝑃)=ℬ_𝑃. Then for all 𝑋⊆𝑈, 𝑅(𝑋)=∪𝐵−𝐵_𝑋−𝑋∶𝐵∈ℬ_𝑃∧𝐵_𝑋−𝑋⊆𝐵,(4.7) where 𝐵_𝑋∈ℬ_𝑃 a base having the maximal intersection with 𝑋.

Proof. For all 𝐵∈ℬ_𝑃, 𝐵∩𝑋 is an independent set contained in 𝑋. Because 𝐵_𝑋 is a base having the maximal intersection with 𝑋, 𝑟_𝑀𝑃(𝑋)=|𝐵_𝑋∩𝑋|. Furthermore, for all 𝑦∈𝐵_𝑋−𝑋, [𝑦]_𝑅∩𝑋=∅. Let 𝑆₁={[𝑦]_𝑅∶𝑦∈𝐵_𝑋−𝑋} and 𝑆₂=𝑃−𝑆₁. Then 𝑅(𝑋)=𝑈−∪𝑆₁=∪𝑆₂. If 𝐵_𝑋−𝑋⊆𝐵, then 𝐵−(𝐵_𝑋−𝑋)⊆∪𝑆₂. According to Corollary 4.7, for all 𝑌⊆∪𝑆₂, if for all 𝑇∈(𝑃−𝑆₂) and |𝑌∩𝑇|=1, then 𝑌∪(𝐵_𝑋−𝑋)∈ℬ_𝑃. And ∪{𝑌⊆∪𝑆₂∶forall𝑇∈(𝑃−𝑆₂),|𝑌∩𝑇|=1}=∪𝑆₂. Therefore, ∪𝑆₂=∪{𝐵−(𝐵_𝑋−𝑋)∶𝐵∈ℬ_𝑃∧(𝐵_𝑋−𝑋)⊆𝐵}. That is, 𝑅(𝑋)=∪{𝐵−(𝐵_𝑋−𝑋)∶𝐵∈ℬ_𝑃∧(𝐵_𝑋−𝑋)⊆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 𝑀_𝑃 be the 𝐼-MIP induced by 𝑃, 𝒞(𝑀_𝑃)=𝒞_𝑃. Then for all 𝑋⊆𝑈, 𝐵𝑁_𝑅(𝑋)=∪𝐶∈𝒞_𝑃∶||𝐶∩𝑋||=1.(4.8)

Proof. According to Proposition 4.10 and Corollary 4.11, for all𝐶∈𝒞_𝑃,∃𝑇∈𝑃 such that 𝐶⊆𝑇. |𝐶∩𝑋|=1 means that each element of 𝐶 does not certainly belong either to 𝑋 or to ∼𝑋. And then ∪{𝐶∈𝒞_𝑃∶|𝐶∩𝑋|=1} is the collection of all elements, which do not certainly belong either to 𝑋 or to ∼𝑋. So 𝐵𝑁_𝑅(𝑋)=∪{𝐶∈𝒞_𝑃∶|𝐶∩𝑋|=1}.

Proposition 4.19. Let 𝑀_𝑃 be the 𝐼-MIP induced by 𝑃, 𝒞(𝑀_𝑃)=𝒞_𝑃. Then for all 𝑋⊆𝑈, 𝑅_{_}(𝑋)=𝑋−𝐵𝑁_𝑅(𝑋),𝑅(𝑋)=𝑋∪𝐵𝑁_𝑅(𝑋),(4.9) where 𝐵𝑁_𝑅(𝑋)=∪{𝐶∈𝒞_𝑃∶|𝐶∩𝑋|=1}.

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

Corollary 4.20. Let 𝑀_𝑃 be the 𝐼-MIP induced by 𝑃, 𝒞(𝑀_𝑃)=𝒞_𝑃 and 𝑋⊆𝑈. Then for all 𝐶∈𝒞_𝑃, 𝐶⊈𝑅(𝑋) if and only if for all 𝑥∈𝑋, {𝑥}∈𝑃.

Proof. (⇒): According to Proposition 4.10 and Corollary 4.11, if for all 𝐶∈𝒞_𝑃,𝐶⊈𝑅(𝑋), then for all 𝑇∈𝑃 and |𝑇|≥2, 𝑇∩𝑋=∅. That is, for all 𝑥∈𝑋, {𝑥}∈𝑃.
(⇐): It is straightforward.

Proposition 4.21. Let 𝑀_𝑃 be the 𝐼-MIP induced by 𝑃, 𝑟_𝑀𝑃=𝑟_𝑃. Then for all 𝑋⊆𝑈, the following equations hold:(1)𝑅(𝑋)={𝑥∈𝑈∶𝑟_𝑃(𝑋)=𝑟_𝑃(𝑋∪{𝑥})},(2)𝑅(𝑋)=∪{𝑇∈𝑃∶𝑟_𝑃(𝑋)=𝑟_𝑃(𝑋∪𝑇)},(3)𝑅(𝑋)=Max({𝐴⊆𝑈∶𝑟_𝑃(𝑋)=𝑟_𝑃(𝐴)}).

Proof. (1) According to Proposition 4.13, 𝑟_𝑃(𝑋)=|𝑌| where 𝑌⊆𝑋 and for all 𝑇∈𝑃, |𝑌∩𝑇|≤1. Let 𝑇∈𝑃. If |𝑌∩𝑇|=0, then 𝑇∩𝑋=∅ and for all 𝑡∈𝑇, 𝑟_𝑃(𝑋)=𝑟_𝑃(𝑋∪{𝑡})−1, that is, 𝑇⊈𝑅(𝑋) and for all 𝑡∈𝑇, 𝑡∉{𝑥∈𝑈∶𝑟_𝑃(𝑋)=𝑟_𝑃(𝑋∪{𝑥})}; else, if |𝑌∩𝑇|=1, then 𝑋∩𝑇≠∅ and for all 𝑥∈𝑇, 𝑟_𝑃(𝑋)=𝑟_𝑃(𝑋∪{𝑥}), that is, 𝑇⊆𝑅(𝑋) and for all 𝑡∈𝑇, 𝑡∈{𝑥∈𝑈∶𝑟_𝑃(𝑋)=𝑟_𝑃(𝑋∪{𝑥})}. That is, 𝑅⁻(𝑋)={𝑥∈𝑈∶𝑟_𝑃(𝑋)=𝑟_𝑃(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 𝑀_𝑃.

Proposition 4.22. Let 𝑀_𝑃 be the 𝐼-MIP induced by 𝑃, cl_𝑀𝑃=cl_𝑃. Then for all 𝑋⊆𝑈, 𝑅(𝑋)=cl_𝑃(𝑋).(4.10)

Proof. According to Proposition 4.14 and Corollary 4.15, we can get that cl_𝑃(𝑋)=∪{𝑇∈𝑃∶𝑥∈𝑋∧𝑥∈𝑇}. Therefore, according to the definition of the upper approximation, it is obvious that 𝑅(𝑋)=cl_𝑃(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 𝑀_𝑃 be the 𝐼-MIP induced by 𝑃, ℋ(𝑀_𝑃)=ℋ_𝑃. Then for all 𝑋⊆𝑈, 𝑅(𝑋)=∪∼𝐻∶𝐻∈ℋ_𝑃∧𝑋−𝐻≠∅.(4.11)

Proof. According to Proposition 4.16, for all 𝐻∈ℋ_𝑃, 𝑈−𝐻∈𝑃, that is, ∼𝐻∈𝑃. And if 𝑋−𝐻≠∅, then 𝑋∩(∼𝐻)≠∅. Therefore, in terms of (2.1), ∼𝐻⊆𝑅(𝑋). Thus, 𝑅(𝑋)=∪{∼𝐻∶𝐻∈ℋ_𝑃∧𝑋−𝐻≠∅}.

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 𝐺^𝑃=(𝑈,𝐸^𝑃) is the graph induced by 𝑃, where 𝐸^𝑃=∪{𝐸_𝑇𝑖∶𝑇_𝑖∈𝑃}.

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 𝑇∈𝑃 and for all 𝐾⊂𝑇, 𝐶_𝐾 is not a cycle. Obviously, any subgraph of 𝐶_𝐾 is also not a cycle. Therefore, we can get the following proposition.

Proposition 5.1. Let ℐ^𝑃={𝑋⊆𝑈∶forall𝑇∈𝑃,𝐶_𝑇⊈𝐺^𝑃[𝑋]}. Then there exists a matroid 𝑀^ on 𝑈 such that ℐ(𝑀^)=ℐ^𝑃.

Proof. According to Definition 2.3, we need to prove that ℐ^𝑃 satisfies (I1)∼(I3). In terms of Section 3.2, for all 𝑇∈𝑃, 𝐶_𝑇 is the cycle whose vertices set is 𝑇, that is, 𝐶_𝑇=(𝑇,𝐸_𝑇) where 𝐸_𝑇 is the set of edges of 𝐶_𝑇. So, it is obvious that ℐ^𝑃 satisfies (I1) and (I2). Here, we just need to prove ℐ^𝑃 satisfies (I3).
Let 𝐼₁,𝐼₂∈ℐ^𝑃, |𝐼₁|<|𝐼₂| and 𝐼₂−𝐼₁={𝑒₁,𝑒₂,…,𝑒_𝑚}(1≤𝑚≤|𝑈|). Suppose that for all 𝑒_𝑖∈𝐼₂−𝐼₁ for 1≤𝑖≤𝑚, ∃𝑇_𝑒𝑖∈𝑃 such that 𝑇_𝑒𝑖⊆𝐼₁∪{𝑒_𝑖}, that is, 𝑇_𝑒𝑖−{𝑒_𝑖}⊆𝐼₁. Because |𝑇_𝑒𝑖−{𝑒_𝑖}|≥1, |𝑇_𝑒1−{𝑒₁}|+|𝑇_𝑒2−{𝑒₂}|+⋯+|𝑇_𝑒𝑚−{𝑒_𝑚}|≥𝑚=|𝐼₂−𝐼₁|. It is obvious that 𝑇_𝑒𝑖⊈(𝐼₁∩𝐼₂)∪{𝑒_𝑖}. So |𝐼₁|=|𝑇_𝑒1−{𝑒₁}|+|𝑇_𝑒2−{𝑒₂}|+⋯+|𝑇_𝑒𝑚−{𝑒_𝑚}|+|𝐼₁∩𝐼₂|. Since |𝐼₂|=|𝐼₂−𝐼₁|+|𝐼₂∩𝐼₁|, we can get that |𝐼₁|≥|𝐼₂|. It is contradictory with the known conditions that |𝐼₁|<|𝐼₂|. So ∃𝑒_𝑖∈𝐼₂−𝐼₁ such that for all 𝑇∈𝑃, 𝑇⊈𝐼₁∪{𝑒_𝑖}, namely, 𝐶_𝑇⊈𝐺^𝑃[𝐼₁∪{𝑒_𝑖}]. That is 𝐼₁∪{𝑒_𝑖}∈ℐ^𝑃. As a result, ℐ^𝑃 satisfies (I1)∼(I3). That is, there exists a matroid 𝑀^ on 𝑈 such that ℐ(𝑀^)=ℐ^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 ℐ={𝑋⊆𝑈∶forall𝑇∈𝑃,𝐶_𝑇⊈𝐺[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 ℐ^+={⋃^𝑛𝑖=1𝑆_𝑖∶𝑆_𝑖⊂𝑇_𝑖∧𝑇_𝑖∈𝑃} where 𝑛=|𝑃|. Then 𝑀^+=(𝑈,ℐ^+) is a 𝐼𝐼-MIP.

Proof. Because 𝑆_𝑖⊂𝑇_𝑖, for all 𝑋∈ℐ^+, 𝑋∩𝑇⊂𝑇 for each 𝑇∈𝑃. That means for all 𝑇∈𝑃, 𝐶_𝑇⊈𝐺^𝑃[𝑋], that is, 𝑋∈ℐ^𝑃. Conversely, for all 𝑋∈ℐ^𝑃, since for all 𝑇∈𝑃, 𝐶_𝑇⊈𝐺^𝑃[𝑋], we can get that 𝑇⊈𝑋, that is, 𝑇∩𝑋⊂𝑇. So 𝑋∈ℐ^+. As a result, 𝑀^+=(𝑈,ℐ^+) 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,ℐ), if ℐ=ℐ^+, then 𝑀 is a 𝐼𝐼-𝑀𝐼𝑃. So, the following corollary can be obtained.

Corollary 5.4. ℐ^𝑃=ℐ^+.

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,ℐ) be a matroid. Then 𝑀 is a 𝐼𝐼-MIP if and only if for all 𝐶₁,𝐶₂∈𝒞(𝑀), 𝐶₁∩𝐶₂=∅ and ∪𝒞(𝑀)=𝑈.

Proof. (⇒): Let 𝑀=(𝑈,ℐ) be the 𝐼𝐼-𝑀𝐼𝑃 induced by 𝑃. Therefore, for all 𝑇∈𝑃, 𝑇∉ℐ, and for all 𝑋⊂𝑇, 𝑋∈ℐ. And then for all 𝑇∈𝑃, 𝑇∈𝒟(𝑀). So, according to (2.2) and Definition 2.5, 𝒞(𝑀)=𝑃. Thus, for all 𝐶₁,𝐶₂∈𝒞(𝑀), 𝐶₁∩𝐶₂=∅, and ∪𝒞(𝑀)=𝑈.
(⇐): Since for all 𝐶₁,𝐶₂∈𝒞(𝑀), 𝐶₁∩𝐶₂=∅, and ∪𝒞(𝑀)=𝑈, we can regard the 𝒞(𝑀) as a partition over 𝑈. Furthermore, 𝒟(𝑀)={𝑋⊆𝑈∶∃C∈𝒞(𝑀)s.t.𝐶⊆𝑋}. Because ℐ=2^𝑈−𝒟(𝑀), for all 𝑋∈ℐ, ∄𝐶∈𝒞(𝑀) such that 𝐶⊆𝑋. According to Proposition 5.1 and Definition 5.2, we can get that ℐ=ℐ^+=ℐ^𝑃. That is, 𝑀=(𝑈,ℐ) 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, ℳ^ the set of all 𝐼𝐼-MIP induced by partitions of 𝒫, 𝑔∶𝒫→ℳ^, that is, for all 𝑃∈𝒫, 𝑔(𝑃)=𝑀^𝑃 where 𝑀^_P is the 𝐼𝐼-MIP induced by 𝑃. Then 𝑔 satisfy the following conditions:(1)for all 𝑃₁,𝑃₂∈𝒫, if 𝑃₁≠𝑃₂ then 𝑔(𝑃₁)≠𝑔(𝑃₂),(2)for all 𝑀^∈ℳ^,∃𝑃_𝑀^′∈𝒫  s.t. (𝑃_𝑀^′)=𝑀^.

Proof. (1) Let 𝑃₁ and 𝑃₂ are two different partitions over 𝑈, and 𝑀^𝑃₁=(𝑈,ℐ^𝑃₁), 𝑀^𝑃₂=(𝑈,ℐ^𝑃₂) are two 𝐼𝐼-𝑀𝐼𝑃 induced by 𝑃₁ and 𝑃₂, respectively. We need to prove that there is an 𝐼₁∈ℐ^𝑃₁ such that 𝐼₁∉ℐ^𝑃₂, or there is an 𝐼₂∈ℐ^𝑃₂ such that 𝐼₂∉ℐ^𝑃₁. Because 𝑃₁≠𝑃₂, there must be an equivalence class 𝑇₁∈𝑃₁ such that 𝑇₁∉𝑃₂. Suppose that for all 𝑇∈𝑃₂, 𝑇₁⊈𝑇. Then ∃𝑋⊂𝑇₁ such that 𝑋∈{𝑆∶𝑆⊂𝑇₁} and 𝑋∉{𝑆∶𝑆⊂𝑇,forall𝑇∈𝑃₂}. According to Proposition 5.3, 𝑋∈ℐ^𝑃₁ and 𝑋∉ℐ^𝑃₂. Conversely, if ∃𝑇₂∈𝑃₂ such that 𝑇₁⊆𝑇₂, then ∃𝑋⊂𝑇₂ such that 𝑋∈{𝑆∶𝑆⊂𝑇₂} and 𝑋∉{𝑆∶𝑆⊂𝑇,forall𝑇∈𝑃₁}. Thus, according to Proposition 5.3, 𝑋∈ℐ^𝑃₂ and 𝑋∉ℐ^𝑃₁.
(2) Let 𝑀^=(𝑈,ℐ^) 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 𝑀^𝑃 be the 𝐼𝐼-MIP induced by 𝑃, 𝑛=|𝑃|. Then ℬ^𝑃={⋃^𝑛𝑖=1(𝑇_𝑖−𝑥_𝑖)∶𝑇_𝑖∈𝑃∧𝑥_𝑖∈𝑇_𝑖} is the set of bases of 𝑀^𝑃.

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 𝑇∈𝑃, |𝑇|=1, then ℬ^𝑃={∅}. Next, we can also formulate the set of bases of a 𝐼𝐼-𝑀𝐼𝑃 from the viewpoint of graph as follows.

Corollary 5.8. ℬ^𝑃=Max({𝐵⊆𝑈∶𝐺^𝑃[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 𝑇∈𝑃, 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 ℬ(𝑀^𝑃)=𝐵^𝑃. Then ∩𝐵^𝑃=∅.

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 𝑀^𝑃 be the 𝐼𝐼-MIP induced by 𝑃. Then 𝒞^𝑃=𝑃 is the set of circuits of 𝑀^𝑃.

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. 𝒞^𝑃={𝐶⊆𝑈∶𝐺^𝑃[𝐶] is a cycle}.

Proof. According to the definition of 𝐺^𝑃 and Proposition 5.10, it is straightforward.

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

Proposition 5.12. Let 𝑀^𝑃 be the 𝐼𝐼-MIP induced by 𝑃. Then for all 𝑋⊆𝑈, 𝑟^𝑃(𝑋)=max{|𝑌|∶𝑌⊆𝑋∧(forall𝑇∈𝑃,𝑇⊈𝑌)} is the rank of 𝑋 in 𝑀^𝑃.

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 𝑀^𝑃 be the 𝐼𝐼-MIP induced by 𝑃, 𝑟^𝑃=𝑟_𝑀^′𝑃, and 𝑋⊆𝑈 an 𝑅-definable set. For all 𝑌⊆𝑈, if 𝑋∩𝑌=∅, then 𝑟^𝑃(𝑋∪𝑌)=𝑟^𝑃(𝑋)+𝑟^𝑃(𝑌).

Proof. Because 𝑋 is an 𝑅-definable set and 𝑋∩𝑌=∅, according to Proposition 5.3 and Corollary 5.4, for all 𝐼₁∈ Max({𝐼∈ℐ^𝑃∶𝐼⊆𝑋}) and for all 𝐼₂∈ Max({𝐼∈ℐ^𝑃∶𝐼⊆𝑌}), 𝐼₁∩𝐼₂=∅. Furthermore, 𝐼₁∪𝐼₂∈ℐ^𝑃 and 𝐼₁∪𝐼₂∈Max({𝑍⊆𝑋∪𝑌∶forall𝑇∈𝑃,𝑇⊈𝑍}). According to Definition 2.8, 𝑟^𝑃(𝑋)=|𝐼₁| and 𝑟^𝑃(𝑌)=|𝐼₂|. Because 𝐼₁∩𝐼₂=∅, |𝐼₁∪𝐼₂|=|𝐼₁|+𝐼₂. As a result, 𝑟^𝑃(𝑋∪𝑌)=𝑟^𝑃(𝑋)+𝑟^𝑃(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 𝑋⊆𝑈, 𝑓_∗𝑅(𝑋)=|{𝑅𝑁_𝑅(𝑥)∶𝑥∈𝑈∧𝑅𝑁_𝑅(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,ℎ} be the universe, 𝑅 an equivalence relation over 𝑈, and 𝑈/𝑅={𝑇₁,𝑇₂,𝑇₃,𝑇₄}={{𝑎,𝑏},{𝑐},{𝑑,𝑒,𝑓},{𝑔,ℎ}}. Compute the lower approximation numbers of 𝑋₁, 𝑋₂, and 𝑋₃ where 𝑋₁={𝑎,𝑏,𝑐}, 𝑋₂={𝑎,𝑑,𝑔}, and 𝑋₃={𝑎,𝑐,𝑔,ℎ}.
Because 𝑅 is an equivalence relation, according to Definition 2.1, for all 𝑥∈𝑈, 𝑅𝑁_𝑅(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 𝑀^𝑃 be the 𝐼𝐼-MIP induced by 𝑃, 𝑟^𝑃=𝑟_𝑀^′𝑃. Then for all 𝑋⊆𝑈, 𝑟^𝑃(𝑋)=|𝑋|−𝑓_∗(𝑋).

Proof. Let 𝐴⊆𝑋 be the largest 𝑅-definable set contained in 𝑋. Then 𝑋=𝐴∪(𝑋−𝐴). Thus 𝑋−𝐴 is an 𝑅-indefinable set and for all 𝑇∈𝑃, 𝑇⊈𝑋−𝐴. According to Proposition 5.12 and Definition 5.14, 𝑟^𝑃(𝐴)=|𝐴|−𝑓_∗(𝑋) and 𝑟^𝑃(𝑋−𝐴)=|𝑋−𝐴|=|𝑋|−|𝐴|. Therefore, in terms of Theorem 5.13, 𝑟^𝑃(𝑋)=𝑟^𝑃(𝐴∪(𝑋−𝐴))=𝑟^𝑃(𝐴)+𝑟^𝑃(𝑋−𝐴)=|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 𝑀^𝑃 be the 𝐼𝐼-MIP induced by 𝑃, 𝑟_𝑀^′𝑃=𝑟^𝑃. Then for all 𝑋⊆𝑈, 𝑟^𝑃(𝑋)+𝑟^𝑃(∼𝑋)=|𝑈|−(𝑓_∗(𝑋)+𝑓_∗(∼𝑋)).

Proof. Let 𝐴⊆𝑋 be the largest 𝑅-definable set contained in 𝑋 and 𝐵⊆∼𝑋 the largest 𝑅-definable set contained in ∼𝑋. Then, in terms of Theorem 5.16, 𝑟^𝑃(𝑋)=𝑟^𝑃(𝐴∪(𝑋−𝐴))=|𝑋|−𝑓_∗(𝑋) and 𝑟^𝑃(∼𝑋)=𝑟^𝑃(𝐵∪(∼𝑋−𝐵))=|∼𝑋|−𝑓_∗(∼𝑋). So 𝑟^𝑃(𝑋)+𝑟^𝑃(∼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 𝑀^𝑃 be the 𝐼𝐼-MIP induced by 𝑃, 𝑟^𝑃=𝑟_𝑀^′𝑃. Then for all 𝑋⊆𝑈, 𝑋 is an 𝑅-definable set if and only if 𝑟^𝑃(𝑋)+𝑟^𝑃(∼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 𝑥∈𝑈−𝑋 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 𝑀^𝑃 be the 𝐼𝐼-MIP induced by 𝑃. Then for all 𝑋⊆𝑈, cl^𝑃(𝑋)=𝑋∪{𝑥∈𝑈−𝑋∶∃𝑌⊆𝑋s.t.𝑌∪{𝑥}∈𝑃} is the closure of 𝑋 in 𝑀^𝑃.

Proof. According to Proposition 5.12, for all 𝑥∈𝑋, 𝑟^𝑃(𝑋)=𝑟^𝑃(𝑋∪{𝑥}). Then, in terms of Definition 2.9, 𝑋⊆cl_𝑀^′𝑃(𝑋). Let 𝑌_𝑋⊆𝑋 and |𝑌_𝑋|=𝑟^𝑃(𝑋). For all 𝑥∈𝑈−𝑋, if 𝑥∈cl_𝑀^′𝑃(𝑋) then 𝑟^𝑃(𝑋)=𝑟^𝑃(𝑋∪{𝑥}). According to Proposition 5.12, for all 𝑇∈𝑃, 𝑇⊈𝑌_𝑋. That means ∃𝑇∈𝑃 such that 𝑇⊆𝑌_𝑋∪{𝑥}, that is, ∃𝑌⊆𝑌_𝑋 such that 𝑌∪{𝑥}∈𝑃. As a result cl_𝑀^′𝑃(𝑋)=cl^𝑃(𝑋).

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

Proposition 5.20. Let 𝑀^𝑃 be the 𝐼𝐼-MIP induced by 𝑃. Then ℋ^𝑃={𝑈−𝑋∶𝑇∈𝑃∧𝑋⊆𝑇∧|𝑋|=2} is the hyperplane of 𝑀^𝑃.

Proof. According to Definition 2.10, we need to prove that for all 𝐻∈ℋ^𝑃, 𝐻 is a close set of 𝑀^𝑃, and 𝑟_𝑀^′𝑃(𝐻)=𝑟_𝑀^′𝑃(𝑈)−1. And more, we need to prove that for all 𝑌⊆𝑈, if 𝑌∉ℋ^𝑃, then 𝑌 is not a hyperplane of 𝑀^𝑃. (1) Is a close set of 𝑀^𝑃. For all 𝐻∈ℋ^𝑃, there is an equivalence class 𝑇∈𝑃 and a subset 𝑋⊆𝑇 such that |𝑋|=2 and 𝐻=𝑈−𝑋. Therefore, for all 𝑌⊆𝐻 and for all 𝑥∈𝑋, 𝑌∪{𝑥} is not an equivalence class, that is, 𝑌∪{𝑥}∉𝑃. That is, the set {𝑥∈𝑋∶∃𝑌⊆𝐻s.t.𝑌∪{𝑥}∈𝑃}=∅. Thus, according to Proposition 5.19, cl_𝑀^′𝑃(𝐻)=𝐻. That is to say 𝐻 is a close set of 𝑀^𝑃. (2)Forall𝐻∈ℋ^𝑃, 𝑟_𝑀^′𝑃(𝐻)=𝑟_𝑀^′𝑃(𝑈)−1. For any 𝑇∈𝑃, since 𝑇 is 𝑅-definable, by Theorem 5.13𝑟_𝑀^′𝑃(𝑈)==𝑟𝑟_𝑀^′𝑃_𝑀′𝑃(𝑇∪∼𝑇)=||𝑇||−(𝑇1)++𝑟𝑟_𝑀^′𝑃_𝑀′𝑃(∼𝑇)(∼𝑇).(5.1) That is, 𝑟_𝑀^′𝑃(∼𝑇)=𝑟_𝑀^′𝑃(𝑈)−||𝑇||+1.(5.2) Furthermore, since ∼𝑇 is also 𝑅-definable, by Theorem 5.13𝑟_𝑀^′𝑃(𝐻)==𝑟𝑟_𝑀^′𝑃_𝑀′𝑃((𝑇−𝑋)∪∼𝑇)=||𝑇||−(𝑇2−+𝑋𝑟)+_𝑀′𝑃𝑟_𝑀^′𝑃(∼𝑇)(∼𝑇).(5.3) That is, 𝑟_𝑀^′𝑃(𝐻)=𝑟_𝑀^′𝑃(∼𝑇)+||𝑇||−2.(5.4) Therefore, from (5.2) and (5.4), 𝑟_𝑀^′𝑃(𝐻)=𝑟_𝑀^′𝑃(𝑈)−1. (3)Forall𝑌⊆𝑈, if 𝑌∉ℋ^𝑃, then 𝑌 is not a hyperplane of 𝑀^𝑃. If 𝑌∉ℋ^𝑃, then there are two cases. (1)|𝑈−𝑌|=2 and 𝑈−𝑌⊈𝑇, for all 𝑇∈𝑃. (2)|𝑈−𝑌|≠2. Next, we discuss these two cases, respectively.

Case 1. By Proposition 5.19, cl^𝑃(𝑌)=𝑈 so 𝑌 is not a close set.

Case 2. if |𝑈−𝑌|=1, then cl^𝑃(𝑌)=𝑈 so 𝑌 is not a close set. If |𝑈−𝑌|>2, then suppose that cl^𝑃(𝑌)=𝑌. In that case, for 𝑇∈𝑃 such that (𝑈−𝑌)∩𝑇≠∅, |(𝑈−𝑌)∩𝑇|≥2. So 𝑟_𝑀^′𝑃(𝑌)=_{𝑇∈𝑃(𝑈−𝑌)∩𝑇≠∅}||𝑇||−||(𝑈−𝑌)∩𝑇||+_{𝑇∈𝑃(𝑈−𝑌)∩𝑇=∅}||𝑇||−1.(5.5)
In that case, 𝑟_𝑀^′𝑃(𝑌) is not equal to 𝑟_𝑀^′𝑃(𝑈)−1=∑_𝑇∈𝑃(|𝑇|−1)−1.

Summing up, ℋ^𝑃={𝑈−𝑋∶𝑇∈𝑃∧𝑋⊆𝑇∧|𝑋|=2} is the hyperplane of 𝑀^𝑃.

5.3. Approximations Established through II-MIP

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

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

Proposition 5.21. Let 𝑀^𝑃 be the 𝐼𝐼-MIP induced by 𝑃, 𝒞(𝑀^𝑃)=𝒞^𝑃. Then for all𝑋⊆𝑈, 𝑅_{_}(𝑋)=∪𝐶∈𝒞^𝑃∶𝐶⊆𝑋₋,𝑅(𝑋)=∪𝐶∈𝒞^𝑃∶𝐶∩𝑋≠∅.(5.6)

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 𝑀^𝑃 be the 𝐼𝐼-MIP induced by 𝑃, 𝑟_𝑀^′𝑃=𝑟^𝑃. Then for all 𝑋⊆𝑈, 𝑅_{_}(𝑋)=Max𝑌⊆𝑋∶𝑟^𝑃(𝑌)+𝑟^𝑃(∼𝑌)=||𝑈||−||𝑃||₋,𝑅(𝑋)=Min𝑋⊆𝑌∶𝑟^𝑃(𝑌)+𝑟^𝑃(∼𝑌)=||𝑈||−||𝑃||.(5.7)

Proof. According to Lemma 5.18, for all 𝑋⊆𝑈, if 𝑟^𝑃(𝑋)+𝑟^𝑃(∼𝑋)=|𝑈|−|𝑃|, then 𝑋 is an 𝑅-definable set. So Max({𝑌⊆𝑋∶𝑟^𝑃(𝑌)+𝑟^𝑃(∼𝑌)=|𝑈|−|𝑃|}) is the largest 𝑅-definable set contained in 𝑋. And Min({𝑋⊆𝑌∶𝑟^𝑃(𝑌)+𝑟^𝑃(∼𝑌)=|𝑈|−|𝑃|}) is the smallest 𝑅-definable set contained 𝑋. And then, according to (2.1), it is straightforward.

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

Proposition 5.23. Let 𝑀^𝑃 be the 𝐼𝐼-MIP induced by 𝑃, cl_𝑀^′𝑃=cl^𝑃. Then for all 𝑋⊆𝑈, 𝑅_{_}(𝑋)=𝑥∈𝑋∶cl^𝑃(𝑋)=cl^𝑃(𝑋−{𝑥}).(5.8)

Proof. For all 𝑇∈𝑃, if 𝑇⊆𝑋, then for all 𝑥∈𝑇, 𝑇−{𝑥}⊆𝑋. According to Proposition 5.19, cl^𝑃(𝑋)=cl^𝑃(𝑋−{𝑥}). That is, 𝑥∈{𝑥∈𝑋∶cl^𝑃(𝑋)=cl^𝑃(𝑋−{𝑥})}. In terms of (2.1), 𝑅_{_}(𝑋)={𝑥∈𝑋∶cl^𝑃(𝑋)=cl^𝑃(𝑋−{𝑥})}.

In terms of (P2), we can get the upper approximation by using the duality. That is, 𝑅⁻(𝑋)=∼𝑅_{_}(∼𝑋)=∼cl^𝑃(∼𝑋−{𝑥}).

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 𝑀=(𝐸,ℐ) be a matroid, and ℬ the set of bases of 𝑀. The dual matroid 𝑀^∗ is the matroid on the set 𝐸 whose bases ℬ^∗=Com(ℬ). If ℐ(𝑀)=ℐ(𝑀^∗), then 𝑀 is called an identically self-dual matroid.

For convenience, in the following content, we suppose that 𝑈 is the universe, 𝑅 is an equivalence relation over 𝑈, 𝑃=𝑈/𝑅 is the partition on 𝑈, and 𝑀_𝑃 is the I-MIP induced by 𝑃 and 𝑀^𝑃 the II-MIP induced by 𝑃.

Proposition 6.2. Let 𝑀^∗𝑃 be the dual matroid of 𝑀_𝑃. Then 𝑀^∗𝑃=𝑀^𝑃.

Proof. For all 𝐵∈ℬ(𝑀_𝑃), according to Definition 6.1, 𝑈−𝐵∈ℬ(𝑀^∗𝑃). For all 𝑇∈𝑃 such that |𝑇∩𝐵|=1, ∃𝑥∈𝑇 such that 𝑇−{𝑥}⊆𝑈−𝐵. Therefore, ℬ(𝑀^∗𝑃)={⋃^𝑛𝑖=1(𝑇_𝑖−𝑥_𝑖)∶𝑇_𝑖∈𝑃∧𝑥_𝑖∈𝑇_𝑖} where 𝑛=|𝑃|. In terms of Proposition 5.7, ℬ(𝑀^∗𝑃)=ℬ(𝑀^𝑃). As a result, 𝑀^∗𝑃=𝑀^𝑃.

Proposition 6.2 shows that 𝑀_𝑃 and 𝑀^𝑃 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 𝑀_𝑃 and 𝑀^𝑃 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 𝑃₁ and 𝑃₂ be two partitions over 𝑈, 𝑀_𝑃1 the I-MIP induced by 𝑃₁, and 𝑀_𝑃2 the 𝐼𝐼-𝑀𝐼𝑃 induced by 𝑃₂. Then 𝑀^∗𝑃₁=𝑀_𝑃2 if and only if 𝑃₁=𝑃₂.

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

Proposition 6.3 shows that a 𝐼-𝑀𝐼𝑃 and a 𝐼𝐼-𝑀𝐼𝑃 induced by different partitions are not dual matroids.

Proposition 6.4. Let 𝑃₁ and 𝑃₂ be two different partitions over 𝑈 and 𝑀_𝑃1 and 𝑀_𝑃2 are the 𝐼-𝑀𝐼𝑃 induced by 𝑃₁ and 𝑃₂, respectively. Then 𝑀^∗𝑃₁≠𝑀_𝑃2.

Proof. Suppose that 𝑀^∗𝑃₁=𝑀_𝑃2. From Proposition 6.2, 𝑀^∗𝑃₁ is a II-MIP. In terms of Proposition 5.10, we can get that 𝒞(𝑀^∗𝑃₁)=𝑃₁=𝒞(𝑀_𝑃2). Therefore, according to Theorem 4.4, for all 𝑇∈𝑃₁, |𝑇|=2. Thus 𝑃₁=𝑃₂. This is contrary to the known condition that 𝑃₁ and 𝑃₂ are two different partitions over 𝑈. As a result, 𝑀^∗𝑃₁≠𝑀_𝑃2.

Proposition 6.5. Let 𝑃₁ and 𝑃₂ be two different partitions over 𝑈 and 𝑀_𝑃1 and 𝑀_𝑃2 are the II-MIP induced by 𝑃₁ and 𝑃₂, respectively. Then 𝑀^∗𝑃₁≠𝑀_𝑃2.

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 𝑃₁ and 𝑃₂ are two different partitions over 𝑈. So, in Propositions 6.4 and 6.5, could 𝑀^∗𝑃₁ be equal to 𝑀_𝑃2 when 𝑃₁=𝑃₂? We will discuss this problem in the following proposition.

Proposition 6.6. 𝑀_𝑃=𝑀^𝑃 if and only if for all𝑇∈𝑃, |𝑇|=2.

Proof. (⇒): According to Definition 6.1, if 𝑀_𝑃=𝑀^𝑃, then ℐ(𝑀_𝑃)=ℐ(𝑀^𝑃). According to Propositions 4.1, 4.3, and 5.1, for all 𝑇∈𝑃, |𝑇|=2.
(⇐): It is straightforward.

Proposition 6.6 provides a necessary and sufficient condition to decide whether a 𝐼-𝑀𝐼𝑃 or a 𝐼𝐼-𝑀𝐼𝑃 is a self-dual matroid. This gives a good answer to the previous question that whether 𝑀^∗𝑃₁ could be equal to 𝑀_𝑃2 when 𝑃₁=𝑃₂.

The set of circuits of a matroid is complementary to the set of hyperplanes of its dual matroid [39]. That is, ℋℋ𝑀𝑀_𝑃𝑃=𝑈−𝐶∶𝐶∈𝒞^𝑃,=𝑈−𝐶∶𝐶∈𝒞_𝑃.(6.1)

Next, we will study when a hyperplane of 𝑀_𝑃 is also a hyperplane of 𝑀^𝑃.

Proposition 6.7. Let 𝐻∈ℋ(𝑀_𝑃). Then 𝐻∈ℋ(𝑀^𝑃) if and only if 𝑈−𝐻∈𝒞(𝑀_𝑃).

Proof. (⇒): According to Proposition 4.16, 𝑈−𝐻∈𝑃, that is, ∃𝑇∈𝑃 such that 𝑈−𝐻=𝑇. If 𝐻∈ℋ(𝑀^𝑃), according to Proposition 5.20, then |𝑈−𝐻|=|𝑇|=2. According to Proposition 4.10 and Corollary 4.11, 𝑇∈𝒞(𝑀_𝑃), that is, 𝑈−𝐻∈𝒞(𝑀_𝑃).
(⇐): If 𝑈−𝐻∈𝒞(𝑀_𝑃), then |𝑈−𝐻|=2 and ∃𝑇∈𝑃 such that 𝑇=𝑈−𝐻. Therefore, according to Propositions 4.16 and 5.20, 𝐻∈ℋ(𝑀^𝑃).

From the Propositions 4.14 and 5.19, we can find, for any subset 𝑋⊆𝑈, that the cl^𝑃(𝑋) is generally the subset of cl_𝑃(𝑋). Now, we will study under what conditions that the cl^𝑃(𝑋) is certain the subset of cl_𝑃(𝑋).

Proposition 6.8. Let 𝑋⊆𝑈, cl_𝑀𝑃(𝑋)=cl_𝑃(𝑋), and cl_𝑀^′𝑃(𝑋)=cl^𝑃(𝑋). Then cl^𝑃(𝑋)⊆cl_𝑃(𝑋) if and only if for all𝑥∈𝑈−𝑋, 𝑟^𝑃({𝑥})≠0.

Proof. (⇒): Let cl^𝑃(𝑋)⊆cl_𝑃(𝑋). According to Propositions 4.14 and 5.19, we can get that {𝑦∈𝑈−𝑋∶∃𝑌⊆𝑋s.t.𝑌∪{𝑦}∈𝑃}⊆{𝑦∈𝑈−𝑋∶𝑥∈𝑋∧𝑥𝑦∈𝐸_𝑃}. That is, for all 𝑦∈{𝑦∈𝑈−𝑋∶∃𝑌⊆𝑋s.t.𝑌∪{𝑦}∈𝑃}, 𝑦∈{𝑦∈𝑈−𝑋∶𝑥∈𝑋∧𝑥𝑦∈𝐸_𝑃}. In terms of Section 4, we know that 𝐸_𝑃={𝑥𝑦∶𝑥≠𝑦∧(∃𝑇∈𝑃s.t.{𝑥,𝑦}⊆𝑇)}. Therefore, for all 𝑦∈{𝑦∈𝑈−𝑋∶𝑥∈𝑋∧𝑥𝑦∈𝐸_𝑃}, {𝑦}∉𝑃. And for all 𝑦∈{𝑦∈𝑈−𝑋∶∃𝑌⊆𝑋s.t.𝑌∪{𝑦}∈𝑃}, {𝑦}∉𝑃. That is, for all 𝑥∈𝑈−𝑋, 𝑟^𝑃({𝑥})≠0.
(⇐): According to Propositions 4.14 and 5.19, it is straightforward.

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

Theorem 6.9. Let 𝑋⊆𝑈, 𝑟_𝑀𝑃=𝑟_𝑃, and 𝑟_𝑀^′𝑃=𝑟^𝑃. Then 𝑟_𝑃(𝑋)=𝑟^𝑃(𝑋) if and only if |𝑋|=𝑓^∗(𝑋)+𝑓_∗(𝑋).

Proof. (⇒): If 𝑟_𝑃(𝑋)=𝑟^𝑃(𝑋), according to Theorem 5.16, 𝑟_𝑃(𝑋)=|𝑋|−𝑓_∗(𝑋). Then, in terms of Definition 2.2 and Proposition 4.13, we can get that 𝑟_𝑃(𝑋)=𝑓^∗(𝑋). Therefore, 𝑓^∗(𝑋)=|𝑋|−𝑓_∗(𝑋). And then |𝑋|=𝑓_∗𝑋+𝑓^∗(𝑋).
(⇐): 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 𝑀_𝑃 and 𝑀^𝑃. 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 𝑋⊆𝑈, the rank of 𝑋 within the context of 𝐼𝐼-𝑀𝐼𝑃 is equal to the difference of the cardinality and the lower approximation number of 𝑋. And Theorem 6.9 indicates that the rank of 𝑋 within the context of 𝐼-𝑀𝐼𝑃 is equal to it within the context of 𝐼𝐼-𝑀𝐼𝑃 if and only if the cardinality of 𝑋 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.