Covering-based rough set theory is a useful tool to deal with inexact, uncertain, or vague knowledge in information systems. Geometric lattice has been widely used in diverse fields, especially search algorithm design, which plays an important role in covering reductions. In this paper, we construct three geometric lattice structures of covering-based rough sets through matroids and study the relationship among them. First, a geometric lattice structure of covering-based rough sets is established through the transversal matroid induced by a covering. Then its characteristics, such as atoms, modular elements, and modular pairs, are studied. We also construct a one-to-one correspondence between this type of geometric lattices and transversal matroids in the context of covering-based rough sets. Second, we present three sufficient and necessary conditions for two types of covering upper approximation operators to be closure operators of matroids. We also represent two types of matroids through closure axioms and then obtain two geometric lattice structures of covering-based rough sets. Third, we study the relationship among these three geometric lattice structures. Some core concepts such as reducible elements in covering-based rough sets are investigated with geometric lattices. In a word, this work points out an interesting view, namely, geometric lattice, to study covering-based rough sets.

Rough set theory [

Lattice is suggested by the form of the Hasse diagram depicting it. In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet). They encode the algebraic behavior of the entailment relation and such basic logical connectives as “and” (conjunction) and “or” (disjunction), which results in adequate algebraic semantics for a variety of logical systems. Lattice, especially geometric lattice, is one of the most important algebraic structures and is used extensively in both theoretical and applicable fields, such as data analysis, formal concept analysis [

Matroid theory [

In this paper, we pay attention to geometric lattice structures of covering based-rough sets through matroids. First, a geometric lattice structure in covering-based rough sets is generated by the transversal matroid induced by a covering. Moreover, we study the characteristics of the geometric lattice structure, such as atoms, modular elements, and modular pairs. We also point out a one-to-one correspondence between this type of geometric lattices and transversal matroids in the context of covering-based rough sets. Second, generally, covering upper approximation operators are not necessarily closure operators of matroids. Then we present three sufficient and necessary conditions for two types of covering upper approximation operators to be closure operators of matroids and exhibit representations of corresponding special matroids. We study the properties of these matroids and their closed-set lattices which are also geometric lattices. Third, we study the relationship among these three geometric lattices through corresponding matroids. Furthermore, some core concepts such as reducible and immured elements in covering-based rough sets are studied by geometric lattices.

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

In this section, we review some basic concepts of matroids, lattices, and covering-based rough sets.

Matroid theory borrows extensively from the terminology of linear algebra theory and graph theory, largely because it is the abstraction of various notions of central importance in these fields, such as independent set, base, and rank function. We introduce the concept of matroid, first.

A matroid is an ordered pair

If

If

Let

The rank function of a matroid, directly analogous to a similar theorem of linear algebra, has the following proposition.

Let

For all

If

If

The following proposition is the closure axiom of a matroid. It means that a operator satisfies the following four conditions if and only if it is the closure operator of a matroid.

Let

If

If

If

If

Transversal theory is a branch of a matroid theory. It shows how to induce a matroid, namely, transversal matroid, from a family of subsets of a set. Hence, the transversal matroid establishes a bridge between collections of subsets of a set and matroids.

Let

Let

The following proposition shows what kind of matroids are transversal matroid.

Let

Let

Next, we introduce the modular element and modular pair which are important concepts of lattices.

Let

(ME) For all

(MP) For all

As we know, if

Let

If

A lattice

The following lemma establishes the relation between the rank function of a matroid and the height function of the closed-set lattice of the matroid.

Let

In this subsection, we introduce some concepts of covering-based rough sets used in this paper.

Let

It is clear that a partition of

Let

Let

Let

Let

For a covering

Let

Let

The second type of covering rough set model was first studied by Pomykała in [

Let

As we know, if

Let

The theorem below connects a covering

Let

“

“

Theorem

Let

Since

Lemma

Let

For all

The following two propositions establish two characteristics of

Let

Let

According to the definition of

Based on Lemma

Let

According to the definition of

According to the definition of transversal matroid and the fact that

Next, we will prove the set of atoms of lattice

The proposition below connects simple matroid and the cardinal number of

Let

According to the definition of

Let

“

“

When a covering degenerates into a partition, we also have the above results.

Let

Let

For a geometric lattice

Let

“

“

The modular element and the modular pair are core concepts in lattice. As we know, if

Let

For all

For all

(1) According to Lemmas

(2) It comes from the definition of modular element and

Let

Let

(1) Since

(2)

In a word, for all

When a covering degenerates into a partition, it is not difficult for us to obtain the following result.

Let

The following lemma shows how to induce a matroid by a lattice. In fact, if a function

Let

According to the definition of

Let

For any given matroid

Let

“

“

What is the relation between the two matroids induced by a covering and a geometric lattice, respectively? In order to establish the relation between them, we first denote

Let

According to Lemma

When a covering degrades into a partition, we can obtain a matroid

If

Let

Based on the above two lemmas, we can obtain the following proposition.

Let

We need to prove only

A geometric lattice structure of covering-based rough sets is established through the transversal matroid induced by a covering, and its characteristics including atoms, modular elements, and modular pairs are studied in Section

Pomykała first studied the second type of covering rough set model [

Let

for all

(1)–(5) were proven in [

We find that the idempotent of

Let

“

“

The following theorem establishes a necessary and sufficient condition for

Let

It comes from Propositions

For a given covering

Let

As we know, if

Let

Let

We denote the rank function of

Let

For all

(1) According to the definition of base of a matroid, we know that

(2) According to the definition of rank function, we know

(3) According to the definition of dependent set, we know that

(4) “

“

For a covering

Let

For all

For all

For all

Based on Theorem

Let

If

Let

Since

The proposition below establishes a necessary and sufficient condition for

Let

“

“

The following theorem presents a necessary and sufficient condition for

Let

It comes from Theorem

The sixth type of covering-based upper approximation operator was first defined in [

Let

for all

From the above proposition, we find that

Let

The following theorem establishes a necessary and sufficient condition for

Let

It comes from

For convenience, for a given covering

Let

Theorem

Let

Let

For all

For all

For all

For all

The proof of Propositions

Let

For all

Let

Since

The following proposition establishes a sufficient condition for

Let

For all

Based on Theorem

Let

Let

In Section

The following proposition shows the relationship between

Let

Since

For all

The following proposition illustrates that in what condition the indiscernible neighborhoods are included in the geometric lattice induced by

Let

Since

We give an example to help understand the relationship between

Let

The lattice of

Let

The following example illustrates the above statements.

Let

The following proposition shows the relationship between

Let

If

For all

From Example

When a covering degenerates into a partition, we can obtain the following result.

If

Since

Next, we discuss the reducible element and immured element’s influence on matroidal structures and geometric lattice structures. First, we study the reducible element and immured element’s influence on

Let

For all

The following example illustrates

Let

Let

Let

Let

Let

Let

The following theorem shows the reducible element and immured element’s influence on geometric lattice structure

Let

First, we prove

Second, we need to prove that any atom of

Third, we need to prove

The following example shows that

Based on Example

Similarly, when

Let

Let

Let

Let

Let

Let

The following theorem presents an immured element’s influence on

Let

It comes from Lemma

As we know, if

Let

Now we consider the reducible element’s influence on

Let

Since

As we know, if

Let

Let

Suppose

This paper has studied the geometric lattice structures of covering based-rough sets through matroids. The important contribution of this paper is that we have established a geometric lattice structure of covering-based rough sets through the transversal matroid induced by a covering and have presented two geometric lattice structures of covering-based rough sets through two types of covering upper approximation operators. Moreover, we have discussed the relationship among the three geometric lattice structures. To study other properties of the geometric lattice structure induced by a covering and to study other geometric lattices from the viewpoint of other upper approximation operators are our future work.

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