The reduction of covering decision systems is an important problem in data mining, and coveringbased rough sets serve as an efficient technique to process the problem. Geometric lattices have been widely used in many fields, especially greedy algorithm design which plays an important role in the reduction problems. Therefore, it is meaningful to combine coverings with geometric lattices to solve the optimization problems. In this paper, we obtain geometric lattices from coverings through matroids and then apply them to the issue of attribute reduction. First, a geometric lattice structure of a covering is constructed through transversal matroids. Then its atoms are studied and used to describe the lattice. Second, considering that all the closed sets of a finite matroid form a geometric lattice, we propose a dependence space through matroids and study the attribute reduction issues of the space, which realizes the application of geometric lattices to attribute reduction. Furthermore, a special type of information system is taken as an example to illustrate the application. In a word, this work points out an interesting view, namely, geometric lattice, to study the attribute reduction issues of information systems.
Rough set theory [
A 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. Lattices, especially geometric lattices, are important algebraic structures and are used extensively in both theoretical and applicable fields, such as rough sets [
Matroid theory [
In this paper, we pay our attention to geometric lattice structures of coverings and their applications to attribute reduction issues of information systems. First, a geometric lattice of a covering is constructed through the transversal matroid induced by the covering. Then its atoms are studied and used to characterize the lattice structure. It is interesting that any element of the lattice can be expressed as the union of all closures of singlepoint sets in the element. Second, we apply the obtained geometric lattice to attribute reduction issues in information systems. It is interesting that a subset of a finite nonempty set is a reduct of the information system if and only if it is a minimal set with respect to the property of containing an element from each nonempty complement of any coatom of the lattice.
The rest of this paper is organized as follows. In Section
In this section, we review some basic concepts of rough sets, matroids, and geometric lattices.
Rough set theory is a new mathematical tool for imprecise and incomplete data analysis. It uses equivalence relations (resp. partitions) to describe the knowledge we can master. In this subsection, we introduce some concepts of rough sets used in this paper.
Let
It is clear that a partition is certainly a covering, so the concept of a covering is an extension of the concept of a partition.
Let
Matroid theory borrows extensively from the terminology of linear algebra and graph theory, largely because it is the abstraction of various notions of central importance in these fields, such as independent sets, bases, and the rank function.
A matroid is an ordered pair
If
If
Let
Let
The following proposition is the closure axiom of a matroid. It means that an operator satisfies the following four conditions if and only if it is the closure operator of a matroid.
Let
Transversal theory is a branch of matroids. It shows how to induce a matroid, namely, transversal matroid from a family of subsets of a set. Hence, transversal matroids establish a bridge between a collection of subsets of a set and a matroid.
Let
Let
The following proposition shows what kind of matroid is a transversal matroid.
Let
A lattice is a poset
A lattice
The above proposition indicates that
As we know, a collection of all the closed sets of a matroid, in the sense of inclusion, is a geometric lattice. In this section, we convert a covering to a matroid through transversal matroids and then study the lattice of all the closed sets of the matroid. By this way, we realize the purpose to construct a geometric lattice structure from a covering.
Let
Atoms of a geometric lattice are elements that are minimal among the nonzero elements and can be used to express the lattice. Therefore, atoms play an important role in the lattices. In this subsection, we study the atoms of the geometric lattice structure induced by a covering.
A covering of universe of objects is the collection of some basic knowledge we master; therefore it is important to be studied in detail. The following theorem provides some equivalence characterizations for a covering from the viewpoint of matroids.
Let
Let
Let
“
“
“
“
Theorem
Let
For all
One example is provided to illustrate the above definition.
Let
In fact, the closure of any singleton set of universe
Let
For all
Let
According to Lemma
Let
For all
Next, we prove
The following result is the combination of Theorem
Let
Corollary
Based on Corollary
In Section
Let
In fact,
Let
It is clear that
In fact, any closed set of the matroid induced by a covering is a fixed point of the two operators induced by the covering.
Let
Utilizing Lemma
Based on the above result, any element of the geometric lattice induced by a covering can be expressed as the union of all closures of singlepoint sets in the element.
Let
It is obvious when
Suppose
The geometric lattice of
In Section
In this subsection, we apply the geometric lattices to the reduction problems of dependence spaces. First, we make certain that what is dependence space. The concept of dependence space can be found in [
Let
For a geometric lattice induced by a matroid, one can use its coatoms, namely the hyperplanes of the matroid, to induce a dependence space
Let
In fact, the issue of reduction of dependence space
Therefore, we can obtain the following result. It indicates that a subset of a finite nonempty set is a reduct of the dependence space induced by the coatoms of a geometric lattice if and only if it is a minimal set with respect to the property of containing an element from each nonempty complement of any coatom of the lattice. The symbol
According to the definition of hyperplane, we know
Suppose lattice is the one shown in Example
Considering that geometric lattices have a closed relation with matroids, we define the other dependence space from the viewpoint of matroids. It is interesting that the dependence space is equal to the one
Let
Let
Let
If
In Section
An information system is a quadruple form
In an information system,
It was noted in [
Let
According to Proposition
For all
We need to prove
When an information system satisfies the condition presented in Proposition
Let
Suppose
A relation table entirely determines an information system. The following example presents how to use above results to find all the reducts of an information system.
Let
An information system.

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In this paper, we have constructed a geometric lattice from a covering through the transversal matroid induced by the covering and have used atoms of the lattice to characterize the lattice. Furthermore, we have applied the lattice to the attribute reduction issues of information systems. Though some works have been studied in this paper, there are also many interesting topics deserving further investigation. In the future, we will study algorithm implementations of the attribute reduction issues in information systems through geometric lattices.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is supported in part by the National Natural Science Foundation of China under Grant nos. 61170128, 61379049, and 61379089, the Natural Science Foundation of Fujian Province, China, under Grant no. 2012J01294, the Science and Technology Key Project of Fujian Province, China, under Grant no. 2012H0043, and the Zhangzhou Research Fund under Grant no. Z2011001.