#### Abstract

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.

#### 1. Introduction

Rough set theory [1] was proposed by Pawlak to deal with granularity in information systems. It is based on equivalence relations. However, the equivalence relation is rather strict, hence the applications of the classical rough set theory are quite limited. For this reason, rough set theory has been extended to generalized rough set theory based on tolerance relation [2], similarity relation [3], and arbitrary binary relation [4–8]. Through extending a partition to a covering, we generalize rough set theory to covering-based rough set theory [9–11]. Because of its high efficiency in many complicated problems such as attribute reduction and rule learning in incomplete information/decision, covering-based rough set theory has been attracting increasing research interest [12, 13].

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 [14–16], and domain theory [17].

Matroid theory [18, 19] borrows extensively from linear algebra theory and graph theory. There are dozens of equivalent ways to define a matroid. Significant definitions of a matroid include those in terms of independent sets, bases, circuits, closed sets or flats and rank functions, which provide well-established platforms to connect with other theories. In applications, matroids have been widely used in many fields such as combinatorial optimization, network flows, and algorithm design, especially greedy algorithm design [20, 21]. Some works on the connection between rough sets and matroids have been conducted [22–25].

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 2, we recall some fundamental concepts related to covering-based rough sets, lattices, and matroids. Section 3 establishes a geometric lattice structure of covering-based rough sets through the transversal matroid induced by a covering. In Section 4, we present two geometric lattice structures of covering-based rough sets through two types of upper approximation operators. Section 5 studies the relationship among these three geometric lattice structures. This paper is concluded and further work is pointed out in Section 6.

#### 2. Preliminaries

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

##### 2.1. Matroids

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.

*Definition 2.1 (Matroid [19]). *A matroid is an ordered pair consisting of a finite set and a collection of subsets of satisfying the following three conditions.(I1).(I2) If and , then .(I3) If and , then there is an element such that , where denotes the cardinality of .

Let be a matroid. The members of are the independent sets of . A set in is maximal, in the sense of inclusion, is called a base of the matroid . If , is called a dependent set of the matroid . In the sense of inclusion, a minimal dependent subset of is called a circuit of the matroid . If is a circuit, we call a loop. Moreover, if is a circuit, then and are said to be parallel. A matroid is called a simple matroid if it has no loops and no parallel elements. The rank function of a matroid is a function defined by . For each , we say is the closure of in . When there is no confusion, we use the symbol for short. is called a closure set if .

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

Proposition 2.2 (Rank axiom [19]). *Let be a set. A function is the rank function of a matroid on if and only if it satisfies the following conditions.*(R1)For all , .(R2) If , then .(R3)If , then .

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.

Proposition 2.3 (Closure axiom [19]). *Let be a set. A function is the closure operator of a matroid on if and only if it satisfies the following conditions.*(1)If , then .(2)If , then .(3)If , .(4)If and , then .

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.

*Definition 2.4 (Transversal [19]). *Let be a nonempty finite set and . denotes a family of subsets of . A transversal or system of distinct representatives of is a subset of such that for all in . If for some subset of , is a transversal of , then is said to be a partial transversal of .

*Example 2.5. *Let , and . For , is a transversal of because , and . is a partial transversal of because there exists a subset of , that is, , such that is a transversal of it.

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

Proposition 2.6 (Transversal matroid [19]). *Let be a family subsets of . is a matroid, where is the family of all partial transversals of . One calls the transversal matroid induced by .*

##### 2.2. Lattices

Let be an ordered set and . We say that is covered by (or covers ) if and there is no element in with . A chain in from to is a subset of such that . The length of such a chain is , and the chain is maximal if covers for all . If, for every pair of elements of with , all maximal chains from to have the same length, then is said to satisfy the Jordan-Dedekind chain condition. The height of an element of is the maximum length of a chain from to . A poset is a lattice if and exist for all . Suppose is a lattice with zero element . If covers , then is called an atom of . Moreover, the atoms of are precisely the elements of height one. It is not difficult to check that every finite lattice has a zero and the one. A finite lattice is called semimodular if it satisfies the Jordan-Dedekind chain condition and for every pair of elements of , the equality holds. A geometric lattice is a finite semimodular lattice in which every element is a join of atoms.

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

*Definition 2.7 (see [17]). *Let be a lattice and . (ME) For all , implies , then is called a modular element of . (MP) For all , implies , then is called a modular pair of .

As we know, if is a modular element of , then is a modular pair of for all , which roots in an important result of lattices. For a semimodular lattice, modular pair has close relation with height function.

Lemma 2.8 (see [17]). *Let be a semimodular lattice, then is a modular pair if and only if for all .*

##### 2.3. Closed-Set Lattice of a Matroid

If is a matroid and denotes the set of all closed sets of ordered by inclusion, then is a lattice. In addition to that, the operations join and meet of it are, respectively, defined as and for all . The zero of is , while the one is . The following lemma gives another definition of a geometric lattice from the viewpoint of matroid. In fact, the set of all closed sets of a matroid ordered by inclusion is a geometric lattice.

Lemma 2.9 (see [19]). *A lattice is geometric if and only if it is the lattice of closed sets of a matroid.*

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.

Lemma 2.10 (see [19]). *Let be a matroid. for all .*

##### 2.4. Covering-Based Rough Sets

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

*Definition 2.11 (Covering and partition). *Let be a universe of discourse, a family of subsets of , and none of subsets in be empty. If , then is called a covering of . Any element of is called a covering block. If is a covering of and it is a family of pairwise disjoint subsets of , then is called a partition of .

It is clear that a partition of is certainly a covering of , so the concept of a covering is an extension of the concept of a partition.

Let be a finite set and be an equivalent relation on . will generate a partition of , where are the equivalence classes generated by . For all , the lower and upper approximations of , are, respectively, defined as follows: Next, we introduce certain important concepts of covering-based rough sets, such as minimal description, indiscernible neighborhood, neighborhood, reducible element, and approximation operators.

*Definition 2.12 (Minimal description [26]). *Let be a covering of and :
is called the minimal description of . When the covering is clear, we omit the lowercase in the minimal description.

*Definition 2.13 (Indiscernible neighborhood and neighborhood [27, 28]). *Let be a covering approximation space and . is called the indiscernible neighborhood of and denoted as . is called the neighborhood of and denoted as . When the covering is clear, we omit the lowercase .

*Definition 2.14 (A reducible covering [29]). *Let be a covering of a domain and . If is a union of some sets in , we say is a reducible element in ; otherwise is an irreducible element in . If every element in is irreducible, we say is irreducible; otherwise is reducible.

*Definition 2.15 (Reduct [29]). *For a covering of a universe , when we remove all reducible elements from , the set of remaining elements is still a covering of , and this new irreducible covering has not reducible element. We call thus new covering a of and it is denoted by .

*Definition 2.16 (Immured element [27]). *Let be a covering of and an element of . If there exists another element of such that , we say that is an immured element of covering .

*Definition 2.17 (Exclusion [27]). *Let be a covering of . When we remove all immured elements from , the set of all remaining elements is still a covering of , and this new covering has no immured element. We called this new covering an of , and it is denoted by .

The second type of covering rough set model was first studied by Pomykała in [30]. While the sixth type of covering-based upper approximation operator was first defined in [31].

*Definition 2.18. *Let be a covering of . The covering upper approximation operators are defined as follows: For all , , . and are called the second and the sixth covering upper approximation operators with respect to the covering , respectively. When there is no confusion, we omit at the lowercase.

#### 3. A Geometric Lattice Structure of Covering-Based Rough Sets through Transversal Matroid

As we know, if is a matroid and denotes the set of all closed sets of ordered by inclusion, then is a geometric lattice. In this section, we study the properties such as atoms, modular elements and modular pairs of this type of geometric lattice through transversal matroid induced by a covering. We also study the structure of matroid induced by the geometric lattice. It is interesting to find that there is a one-to-one correspondence between this type of geometric lattices and transversal matroids in the context of covering-based rough sets.

Let be a nonempty finite set and a covering of . As shown in Proposition 2.6, is the transversal matroid induced by covering . is the set of all closed sets of . Especially, is the set of all closed sets of the transversal matroid induced by partition . Based on Lemma 2.9, we know and are geometric lattices.

The theorem below connects a covering with . In fact, if and only if is a covering.

Theorem 3.1. *Let be a set and a transversal matroid induced by a family of subsets of , where . if and only if is a covering of .*

*Proof. *“”: According to the definition of transversal matroid, any partial transversal is an independent set of transversal matroid. Since is a covering, any single-point set is an independent set. Based on the definition of closure operator of a matroid, we have .

“”: Since , any single-point set is an independent set, that is, for all , there exists such that . Hence, . Thus . For all , and , hence is a covering.

Theorem 3.1 indicates that the zero of is . The following lemma presents the form of the atoms of . In fact, the closure of any single-point set is an atom of the lattice.

Lemma 3.2. *Let be a covering of . For all , is an atom of .*

*Proof. *Since is a covering, we know any single-point set is an independent set. Thus for all , . As we know, the atoms of a lattice are precisely the elements of height one. Combining with Lemma 2.10 and the fact that is a closed set, we know is an atom of .

Lemma 3.2 does not establish the concrete form of . In order to solve that problem, we first define two sets as follows.

*Definition 3.3. *Let be a covering of a finite set . We define the following.(i).(ii).

*Remark 3.4. *For all and , there exists only one block such that belongs to it, and there exist at least two blocks such that belongs to them for all .

The following two propositions establish two characteristics of and .

Proposition 3.5. *Let be a covering of . forms a partition of .*

*Proof. *Let . According to Definition 3.3, we know . Now we need to prove for all , . According to the definition of , if , then . If , then because and are different single-points. If and , then because , and .

Proposition 3.6. * is a partition of if and only if .*

*Proof. *According to the definition of and , the necessity is obvious. Now we prove the sufficiency. If is not a partition, then there exist such that . Thus there exists such that , that is, there exist at least such that belongs to them, hence . That contradicts the assumption that .

Based on Lemma 3.2 and Definition 3.3, we can establish the concrete form of the atoms of lattice .

Theorem 3.7. *Let be a covering of . is the set of atoms of lattice .*

*Proof. *According to the definition of , we may as well suppose . Based on being a covering and the definition of transversal matroid, we know any single-point set is an independent set, thus for all , is an independent set. For all and , we know and for all and , thus and cannot be chosen from different blocks in the covering . That shows that is not an independent set according to the definition of transversal matroid. Hence, is a maximal independent set included in , that is, . Next, we need to prove is a closed set. Since , we need to prove , that is, implies . If , based on the fact that is a covering and the definition of , then there exists such that . Thus is an independent set. That implies , thus . Hence, . Combining with , we know for all , is an atom of lattice .

According to the definition of transversal matroid and the fact that is a covering, any single-point set is an independent set. Thus for all , . For all and , if , then there exist at least two blocks containing according to the definition of . We may as well suppose and , where may be the same as . Based on this, is an independent set. This implies . If , then we may as well suppose , thus for the definition of , where may be the same with or . Based on this, and can be chosen from different blocks in covering , thus is an independent set. That implies . From above discussion, we have . Hence, for all . Combining with , we know is an atom of lattice for all .

Next, we will prove the set of atoms of lattice cannot be anything but . According to Lemma 3.2, we know is the set of atoms of lattice . Similar to the proof of the second part, we know that if then . If , then belongs to one of elements in . We may as well suppose . Combining is an atom with , we have . Hence, is the set of atoms of lattice .

The proposition below connects simple matroid and the cardinal number of . In fact, a matroid is simple if and only if for all .

Lemma 3.8. *Let be a covering of . For all , if , then and are parallel in for all .*

*Proof. *According to the definition of , we may as well suppose , where . For all , then , and for all and , we have and . Thus is not an independent set. Based on the definition of transversal matroid and the fact that is a covering, any single-point set is an independent set. Thus or is an independent set. Hence, are parallel in .

Proposition 3.9. *Let be a covering of . is a simple matroid if and only if for all .*

*Proof. *“”: Since is a simple matroid, it does not contain parallel elements. If there exists such that , then because . According to Lemma 3.8, we know for all , are parallel which contradicts the assumption that is a simple matroid. Hence, for all , .

“”: According to the definition of parallel element, if for all , then does not contain parallel elements; otherwise, we may as well suppose are parallel, then there exists only one block which contains . Hence, there exists such that , that is, . This contradicts the fact that for all . Based on the definition of transversal matroid and the fact that is a covering, any single-point set is an independent set, thus does not contain loops. Hence, does not contain parallel elements and loops which implies that is a simple matroid.

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

Corollary 3.10. *Let be a partition of . is the set of atoms of lattice .*

Corollary 3.11. *Let be a partition of . is a simple matroid if and only if for all .*

For a geometric lattice , any closure of single-point is an atom of it. However, the closure of any two elements of may not be a element which covers some atoms of this lattice. The following proposition shows in what condition covers certain atoms of lattice .

Proposition 3.12. *Let be a covering of . For all , covers if and only if there does not exist such that .*

*Proof. *“”: For all , , then and . Now we need to prove . If , then , that is, there is only one block contains . It means that there exists such that . That contradicts the hypothesis. Hence, , that is, covers .

“”: For all , if there exists such that , then there is only one block contains , thus , hence . That implies which contradicts the assumption that covers .

The modular element and the modular pair are core concepts in lattice. As we know, if is a modular element of , then is a modular pair of for all , which roots in an important result of lattices. The following theorem shows the relationship among modular element, modular pair and rank function of a matroid in lattice .

Theorem 3.13. *Let be a matroid and the set of all closed sets of .*(1)*For all , is a modular pair of if and only if .*(2)*For all , is a modular element of if and only if , for all .*

*Proof. *(1) According to Lemmas 2.8 and 2.10, we know is a modular pair of if and only if .

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

Let be the set of atoms of lattice , where denotes the index set. The following theorem shows the relationship among atoms, modular pairs, and modular elements of the lattice.

Theorem 3.14. *Let be a covering of . For all .*(1)* is a modular pair of .*(2)* is a modular element of .*

*Proof. *(1) Since is a covering, . and are atoms, so . According to Theorem 3.13, we need to prove , that is, . According to the submodular inequality of , we have , that is, . If , then which contradicts that .

(2) is a modular element of if and only if for all .*Case 1*. If and are comparable, that is, , then .*Case 2*. If and are not comparable, there are two cases. One is that is an atom of , the other is that is not an atom of . If is an atom of , then we obtain the result from . If is not an atom of , then . Hence, . If , then which contradicts that . Hence, .

In a word, for all , , that is, is a modular element of for all .

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

Corollary 3.15. *Let be a partition of . For all :*(1)* is a modular pair of .*(2)* is a modular element of .*

The following lemma shows how to induce a matroid by a lattice. In fact, if a function on a lattice is nonnegative, integer-valued, submodular and , then it can determine a matroid.

Lemma 3.16 (see [19]). *Let be a lattice of subsets of a set such that is closed under intersection, and contains and . Suppose that is a nonnegative, integer-valued, submodular function on for which . Let . is the collection of independent sets of a matroid on .*

According to the definition of , we find that is closed under intersection, and contains and . Moreover, the rank function of is a nonnegative, integer-valued, submodular function on for which . Similar to Lemma 3.16, we can obtain the following theorem.

Theorem 3.17. *Let be a covering of . We define , for all , then is a matroid.*

For any given matroid , we know that for all , is an independent set of if and only if . Moreover, based on the properties of rank function, we have . Hence, is an independent set of if and only if for all .

Lemma 3.18. *Let be a matroid. is an independent set of if and only if for all closed set of , .*

*Proof. *“”: Since for all closed set , and is an independent set, is an independent set of according to the independent set axiom of a matroid. Hence, we have .

“”: For all closed set , . Especially, for , we have . Hence, is an independent set of matroid .

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 as the rank function of on . The following theorem shows there is a one-to-one correspondence between geometric lattices and transversal matroids in the context of covering-based rough sets.

Theorem 3.19. *Let be a covering of . For all , and .*

*Proof. *According to Lemma 3.18, we know that and are equivalent, that is, and are equivalent. So does for all .

When a covering degrades into a partition, we can obtain a matroid , where and for all . As we know, for all , . If the matroid is , then .

Lemma 3.20 (see [25]). *If is a partition of and is the matroid, then for all .*

Lemma 3.21. *Let be a partition of . For all , .*

*Proof. *.

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

Proposition 3.22. *Let be a partition of . For all , and .*

*Proof. *We need to prove only . For all , then for all . Since , . Thus . Hence, . According to Lemmas 2.9 and 3.21, for all , there exists such that . Thus and . For all , . Hence, , that is, .

#### 4. Two Geometric Lattice Structures of Covering-Based Rough Sets through Approximation Operators

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 3. In this section, we study matroidal structures and the geometric lattice structures from the viewpoint of covering upper approximation operators. The conditions of two types of upper approximation operators to be matroidal closure operators are obtained, and the properties of the matroids and their geometric lattice structures induced by the operators are also established.

Pomykała first studied the second type of covering rough set model [30]. Zhu and Wang studied the axiomatization of this type of upper approximation operator and the relationship between it and the *Kuratowski* closure operator in [27]. First, we give some properties of this operator.

Proposition 4.1. *Let be a covering of . has the following properties:*(1),(2) for all ,(3) for all ,(4) for all ,(5),(6)for all , then .

*Proof. *(1)–(5) were proven in [30, 32, 33]. Here we prove only . According to , we know . If , then . According to and , we have .

We find that the idempotent of is not valid, so what is the condition that guarantees it holds for ? We have the following conclusion.

Proposition 4.2. *Let be a covering of . For all , if and only if induced by forms a partition of .*

*Proof. *“”: According to , of Proposition 4.1, we have . Now we prove . For all , there exists such that . Since , there exists such that . According to the definition of , we know , thus . For forms a partition, . Since , , that is, , thus .

“”: In order to prove forms a partition, we need to prove that for all , if , then . If , then there exists . For and , then . Based on the definition of and , we have , thus . Hence, . Similarly, we can obtain , thus .

The following theorem establishes a necessary and sufficient condition for to be a closure operator.

Theorem 4.3. *Let be a covering of . is a closure operator of a matroid if and only if induced by forms a partition of .*

*Proof. *It comes from Propositions 4.1 and 4.2 and , , and of Proposition 2.3.

For a given covering of , we may as well suppose the set of indiscernible neighborhoods of as where .

*Definition 4.4. *Let be a covering of . We define .

As we know, if forms a partition of , then is a matroid and is the closure operator of a matroid. Thus can determine a matroid, and the independent sets of the matroid induced by it are established as follows: The following proposition shows under the condition that forms a partition of .

Proposition 4.5. *Let be a covering of . If induced by forms a partition of , then is a matroid and .*

*Proof. *Let . we know that if an operator satisfies (1)–(4) of Proposition 2.3, is a matroid. induced by forms a partition, hence, is a matroid. Since , . According to the definition of , we know . On one hand, for all , we know that for all , , that is, for all and , . If , that is, there exists such that , then we may as well suppose there exist such that and . Since and forms a partition, . Based on that, we know there exists and such that , that implies contradiction. Hence, , that is, . On the other hand, if , then there exists such that . That implies that there exists and such that . Since and forms a partition, . Thus , that implies , that is, . Hence, .

We denote the rank function of by . Then some properties of are established in the following proposition.

Proposition 4.6. *Let be a covering of . If induced by forms a partition of , then*(1) is a base of if and only if for all . Moreover, has bases.(2)For all , .(3) is a dependent set of if and only if there exists such that .(4) is a circuit of if and only if there exists such that and .

*Proof. *(1) According to the definition of base of a matroid, we know that is a base of is a maximal independent set of for all because . Since is a base of and are different, has bases.

(2) According to the definition of rank function, we know , where is a maximal independent set included in . Now we just need to prove the inequality does not hold; otherwise, there exists such that and because . Thus there exists such that and . That contradicts the assumption that is a maximal independent set included in . Hence, .

(3) According to the definition of dependent set, we know that is a dependent set there exists such that .

(4) “”: As we know, a circuit is a minimal dependent set. is a circuit of , then there exists such that . Now we just need to prove ; otherwise, we may as well suppose where . Thus we can obtain , that is, . That contradicts the minimality of circuit. Combining with , we have .

“”: Since , we may as well suppose , and because there exists such that , thus is a dependent set. For all , and which implies and are independent sets, hence is a circuit of .

For a covering of , we denote as the set of all closed sets of . When forms a partition of , then for all , , and .

Proposition 4.7. *Let be a covering of . If forms a partition of , then*(1) are all atoms of .(2)For all and , there does not exist such that if and only if covers or .(3)For all is a modular pair of .(4)For all is a modular element of .

*Proof. * comes from Corollary 3.10, Theorem 4.3 and Proposition 4.5. Based on Proposition 3.12 and Theorem 4.3, we can obtain . According to Corollary 3.15, Theorem 4.3 and Proposition 4.5, it is easy to obtain and .

Based on Theorem 4.3, we know that a necessary and sufficient condition for to be a closure operator of a matroid is that forms a partition of . The following two propositions show what kind of coverings can satisfy that condition.

Lemma 4.8. *Let be a covering of and . If is an immured element, then is the same in as in .*

*Proof. *If , then . If , then . Since is an immured element, there exists such that . Thus . Hence, is the same in as in .

Proposition 4.9. *Let be a covering of . If is a partition of , then induced by also forms a partition of .*

*Proof. *Since is a partition of , induced by forms a partition. Suppose is the set of all immured elements of . According to Lemma 4.8, we know for all , is the same in as in . Thus induced by forms a partition of . And the rest may be deduced by analogy, we know that for all , is the same in as in , thus induced by forms a partition of .

The proposition below establishes a necessary and sufficient condition for forms a partition of from the viewpoint of coverings.

Proposition 4.10. *Let be a covering of . induced by forms a partition of if and only if satisfies condition: For all , there exists such that .*

*Proof. *“”: For all , or . If , then there exists and . According to the definition of and , there exist such that and . According to the hypothesis, we know there exists such that . Now we need to prove only . For all , there exists such that . Since , there exists such that , that is, , thus . Similarly, we can prove . Hence, , that is, forms a partition of .

“”: For all and , we can obtain and . That implies . Since forms a partition of , . Thus there exists such that