# Geometric Lattice Structure of Covering-Based Rough Sets through Matroids

Authors: Aiping Huang; William Zhu
Journal: Journal of Applied Mathematics
 (2012)

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## 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.


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## Body

## 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 Section2, 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(E,ℐ) consisting of a finite set E and a collection ℐ of subsets of E satisfying the following three conditions.(I1)
∅∈ℐ.(I2)
IfI∈ℐ and I′⊆I, then I′∈ℐ.(I3)
IfI1,I2∈ℐ and |I1|<|I2|, then there is an element e∈I2-I1 such that I1∪e∈ℐ, where |X| denotes the cardinality of X.LetM=M(E,ℐ) be a matroid. The members of ℐ are the independent sets of M. A set in ℐ is maximal, in the sense of inclusion, is called a base of the matroid M. If A∉ℐ, A is called a dependent set of the matroid M. In the sense of inclusion, a minimal dependent subset of E is called a circuit of the matroid M. If {a} is a circuit, we call {a} a loop. Moreover, if {a,b} is a circuit, then a and b 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 rM:2E→N defined by rM(X)=max{|I|:I⊆X,I∈ℐ}(X⊆E). For each X⊆E, we say clM(X)={a∈E:rM(X)=rM(X∪{a})} is the closure of X in M. When there is no confusion, we use the symbol cl(X) for short. X is called a closure set if cl(X)=X.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]).
LetE be a set. A function rM:2E→N is the rank function of a matroid on E if and only if it satisfies the following conditions.(R1)
For allX∈2E, 0≤rM(X)≤|X|.(R2)
IfX⊆Y⊆E, then rM(X)≤rM(Y).(R3)
IfX,Y⊆E, then rM(X∪Y)+rM(X∩Y)≤rM(X)+rM(Y).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]).
LetE be a set. A function clM:2E→2E is the closure operator of a matroid M on E if and only if it satisfies the following conditions.(1)
IfX⊆E, then X⊆clM(X).(2)
IfX⊆Y⊆E, then clM(X)⊆clM(Y).(3)
IfX⊆E, clM(clM(X))=clM(X).(4)
IfX⊆E,x∈E and y∈clM(X∪{x})-clM(X), then x∈clM(X∪{y}).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]).
LetS be a nonempty finite set and J={1,2,…,m}. ℱ={F1,F2,…,Fm} denotes a family of subsets of S. A transversal or system of distinct representatives of {F1,F2,…,Fm} is a subset {e1,e2,…,em} of S such that ei∈Fi for all i in J. If for some subset K of J, X is a transversal of {Fi:i∈K}, then X is said to be a partial transversal of {F1,F2,…,Fm}.Example 2.5.
LetS={1,2,3,4}, F1={2,3},F2={4} and F3={2,4}. For ℱ={F1,F2,F3}, T={2,3,4} is a transversal of ℱ because 2∈F3, 3∈F1 and 4∈F2. T′={2,4} is a partial transversal of ℱ because there exists a subset of ℱ, that is, {K1,K2}, such that T′ is a transversal of it.The following proposition shows what kind of matroids are transversal matroid.Proposition 2.6 (Transversal matroid [19]).
Letℱ={Fi:i∈J} be a family subsets of E. M(ℱ)=(E,ℐ(ℱ)) is a matroid, where ℐ(ℱ) is the family of all partial transversals of ℱ. One calls M(ℱ)=(E,ℐ(ℱ)) the transversal matroid induced by ℱ.


### 2.2. Lattices

Let(P,≤) be an ordered set and a,b∈P. We say that a is covered by b (or b covers a) if a<b and there is no element c in P with a<c<b. A chain in P from x0 to xn is a subset {x0,x1,…,xn} of P such that x0<x1<⋯<xn. The length of such a chain is n, and the chain is maximal if xi covers xi-1 for all i∈{1,2,…,n}. If, for every pair {a,b} of elements of P with a<b, all maximal chains from a to b have the same length, then P is said to satisfy the Jordan-Dedekind chain condition. The height hP(y) of an element y of P is the maximum length of a chain from 0 to y. A poset (ℒ,≤) is a lattice if a∨b and a∧b exist for all a,b∈ℒ. Suppose ℒ is a lattice with zero element 0. If a covers 0, then a∈ℒ is called an atom of ℒ. Moreover, the atoms of ℒ 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 ℒ is called semimodular if it satisfies the Jordan-Dedekind chain condition and for every pair {x,y} of elements of ℒ, the equality hℒ(x)+hℒ(y)≥hℒ(x∨y)+hℒ(x∧y) 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]).
Letℒ be a lattice and a,b∈ℒ.(ME) For allx,z∈ℒ, x≥z implies x∧(a∨z)=(x∧a)∨z, then a is called a modular element of ℒ.(MP) For allz∈ℒ, b≥z implies b∧(a∨z)=(b∧a)∨z, then (a,b) is called a modular pair of ℒ.As we know, ifa is a modular element of ℒ, then (a,b) is a modular pair of ℒ for all b∈ℒ, 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]).
Letℒ be a semimodular lattice, then (a,b) is a modular pair if and only if hℒ(a∨b)+hℒ(a∧b)=hℒ(a)+hℒ(b) for all a,b∈ℒ.


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

IfM is a matroid and ℒ(M) denotes the set of all closed sets of M ordered by inclusion, then (ℒ(M),⊆) is a lattice. In addition to that, the operations join and meet of it are, respectively, defined as X∧Y=X∩Y and X∨Y=clM(X∪Y) for all X,Y∈ℒ(M). The zero of ℒ(M) is clM(∅), while the one is E. 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 latticeℒ 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]).
LetM be a matroid. hℒ(M)(X)=rM(X) for all X∈ℒ(M).


### 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).
LetE be a universe of discourse, 𝒞 a family of subsets of E, and none of subsets in 𝒞 be empty. If ∪𝒞=E, then 𝒞 is called a covering of E. Any element of 𝒞 is called a covering block. If 𝒫 is a covering of E and it is a family of pairwise disjoint subsets of E, then P is called a partition of E.It is clear that a partition ofE is certainly a covering of E, so the concept of a covering is an extension of the concept of a partition.LetE be a finite set and R be an equivalent relation on E. R will generate a partition E/R={Y1,Y2,…,Ym} of E, where Y1,Y2,…,Ym are the equivalence classes generated by R. For all X⊆E, the lower and upper approximations of X, are, respectively, defined as follows:
(2.1)R*(X)=⋃{Yi∈ER:Yi⊆X},R*(X)=⋃{Yi∈ER:Yi⋂X≠∅}.
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]).
Let𝒞 be a covering of E and x∈E:
(2.2)Md𝒞(x)={K∈𝒞:x∈K∧(∀S∈𝒞∧x∈S∧S⊆K⇒K=S)},
is called the minimal description of x. When the covering is clear, we omit the lowercase 𝒞 in the minimal description.Definition 2.13 (Indiscernible neighborhood and neighborhood [27, 28]).
Let(E,𝒞) be a covering approximation space and x∈E.∪{K:x∈K∈𝒞} is called the indiscernible neighborhood of x and denoted as I𝒞(x).∩{K:x∈K∈𝒞} is called the neighborhood of x and denoted as N𝒞(x). When the covering is clear, we omit the lowercase 𝒞.Definition 2.14 (A reducible covering [29]).
Let𝒞 be a covering of a domain E and K∈𝒞. If K is a union of some sets in 𝒞-{K}, we say K is a reducible element in 𝒞; otherwise K is an irreducible element in 𝒞. If every element in 𝒞 is irreducible, we say 𝒞 is irreducible; otherwise 𝒞 is reducible.Definition 2.15 (Reduct [29]).
For a covering𝒞 of a universe E, when we remove all reducible elements from 𝒞, the set of remaining elements is still a covering of E, and this new irreducible covering has not reducible element. We call thus new covering a reduct of 𝒞 and it is denoted by reduct(𝒞).Definition 2.16 (Immured element [27]).
Let𝒞 be a covering of E and K an element of 𝒞. If there exists another element K′ of 𝒞 such that K⊂K′, we say that K is an immured element of covering 𝒞.Definition 2.17 (Exclusion [27]).
Let𝒞 be a covering of E. When we remove all immured elements from 𝒞, the set of all remaining elements is still a covering of E, and this new covering has no immured element. We called this new covering an exclusion of 𝒞, and it is denoted by exclusion(𝒞).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.
Let𝒞 be a covering of E. The covering upper approximation operators SH,XH:2E→2E are defined as follows: For all X∈2E,SH(X)=∪{K∈𝒞:K∩X≠∅}=∪{I(x):x∈X},XH(X)={x:N(x)∩X≠∅}.SH𝒞 and XH𝒞 are called the second and the sixth covering upper approximation operators with respect to the covering 𝒞, respectively. When there is no confusion, we omit 𝒞 at the lowercase.


## 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(E,ℐ) consisting of a finite set E and a collection ℐ of subsets of E satisfying the following three conditions.(I1)
∅∈ℐ.(I2)
IfI∈ℐ and I′⊆I, then I′∈ℐ.(I3)
IfI1,I2∈ℐ and |I1|<|I2|, then there is an element e∈I2-I1 such that I1∪e∈ℐ, where |X| denotes the cardinality of X.LetM=M(E,ℐ) be a matroid. The members of ℐ are the independent sets of M. A set in ℐ is maximal, in the sense of inclusion, is called a base of the matroid M. If A∉ℐ, A is called a dependent set of the matroid M. In the sense of inclusion, a minimal dependent subset of E is called a circuit of the matroid M. If {a} is a circuit, we call {a} a loop. Moreover, if {a,b} is a circuit, then a and b 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 rM:2E→N defined by rM(X)=max{|I|:I⊆X,I∈ℐ}(X⊆E). For each X⊆E, we say clM(X)={a∈E:rM(X)=rM(X∪{a})} is the closure of X in M. When there is no confusion, we use the symbol cl(X) for short. X is called a closure set if cl(X)=X.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]).
LetE be a set. A function rM:2E→N is the rank function of a matroid on E if and only if it satisfies the following conditions.(R1)
For allX∈2E, 0≤rM(X)≤|X|.(R2)
IfX⊆Y⊆E, then rM(X)≤rM(Y).(R3)
IfX,Y⊆E, then rM(X∪Y)+rM(X∩Y)≤rM(X)+rM(Y).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]).
LetE be a set. A function clM:2E→2E is the closure operator of a matroid M on E if and only if it satisfies the following conditions.(1)
IfX⊆E, then X⊆clM(X).(2)
IfX⊆Y⊆E, then clM(X)⊆clM(Y).(3)
IfX⊆E, clM(clM(X))=clM(X).(4)
IfX⊆E,x∈E and y∈clM(X∪{x})-clM(X), then x∈clM(X∪{y}).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]).
LetS be a nonempty finite set and J={1,2,…,m}. ℱ={F1,F2,…,Fm} denotes a family of subsets of S. A transversal or system of distinct representatives of {F1,F2,…,Fm} is a subset {e1,e2,…,em} of S such that ei∈Fi for all i in J. If for some subset K of J, X is a transversal of {Fi:i∈K}, then X is said to be a partial transversal of {F1,F2,…,Fm}.Example 2.5.
LetS={1,2,3,4}, F1={2,3},F2={4} and F3={2,4}. For ℱ={F1,F2,F3}, T={2,3,4} is a transversal of ℱ because 2∈F3, 3∈F1 and 4∈F2. T′={2,4} is a partial transversal of ℱ because there exists a subset of ℱ, that is, {K1,K2}, such that T′ is a transversal of it.The following proposition shows what kind of matroids are transversal matroid.Proposition 2.6 (Transversal matroid [19]).
Letℱ={Fi:i∈J} be a family subsets of E. M(ℱ)=(E,ℐ(ℱ)) is a matroid, where ℐ(ℱ) is the family of all partial transversals of ℱ. One calls M(ℱ)=(E,ℐ(ℱ)) the transversal matroid induced by ℱ.


## 2.2. Lattices

Let(P,≤) be an ordered set and a,b∈P. We say that a is covered by b (or b covers a) if a<b and there is no element c in P with a<c<b. A chain in P from x0 to xn is a subset {x0,x1,…,xn} of P such that x0<x1<⋯<xn. The length of such a chain is n, and the chain is maximal if xi covers xi-1 for all i∈{1,2,…,n}. If, for every pair {a,b} of elements of P with a<b, all maximal chains from a to b have the same length, then P is said to satisfy the Jordan-Dedekind chain condition. The height hP(y) of an element y of P is the maximum length of a chain from 0 to y. A poset (ℒ,≤) is a lattice if a∨b and a∧b exist for all a,b∈ℒ. Suppose ℒ is a lattice with zero element 0. If a covers 0, then a∈ℒ is called an atom of ℒ. Moreover, the atoms of ℒ 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 ℒ is called semimodular if it satisfies the Jordan-Dedekind chain condition and for every pair {x,y} of elements of ℒ, the equality hℒ(x)+hℒ(y)≥hℒ(x∨y)+hℒ(x∧y) 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]).
Letℒ be a lattice and a,b∈ℒ.(ME) For allx,z∈ℒ, x≥z implies x∧(a∨z)=(x∧a)∨z, then a is called a modular element of ℒ.(MP) For allz∈ℒ, b≥z implies b∧(a∨z)=(b∧a)∨z, then (a,b) is called a modular pair of ℒ.As we know, ifa is a modular element of ℒ, then (a,b) is a modular pair of ℒ for all b∈ℒ, 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]).
Letℒ be a semimodular lattice, then (a,b) is a modular pair if and only if hℒ(a∨b)+hℒ(a∧b)=hℒ(a)+hℒ(b) for all a,b∈ℒ.


## 2.3. Closed-Set Lattice of a Matroid

IfM is a matroid and ℒ(M) denotes the set of all closed sets of M ordered by inclusion, then (ℒ(M),⊆) is a lattice. In addition to that, the operations join and meet of it are, respectively, defined as X∧Y=X∩Y and X∨Y=clM(X∪Y) for all X,Y∈ℒ(M). The zero of ℒ(M) is clM(∅), while the one is E. 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 latticeℒ 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]).
LetM be a matroid. hℒ(M)(X)=rM(X) for all X∈ℒ(M).


## 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).
LetE be a universe of discourse, 𝒞 a family of subsets of E, and none of subsets in 𝒞 be empty. If ∪𝒞=E, then 𝒞 is called a covering of E. Any element of 𝒞 is called a covering block. If 𝒫 is a covering of E and it is a family of pairwise disjoint subsets of E, then P is called a partition of E.It is clear that a partition ofE is certainly a covering of E, so the concept of a covering is an extension of the concept of a partition.LetE be a finite set and R be an equivalent relation on E. R will generate a partition E/R={Y1,Y2,…,Ym} of E, where Y1,Y2,…,Ym are the equivalence classes generated by R. For all X⊆E, the lower and upper approximations of X, are, respectively, defined as follows:
(2.1)R*(X)=⋃{Yi∈ER:Yi⊆X},R*(X)=⋃{Yi∈ER:Yi⋂X≠∅}.
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]).
Let𝒞 be a covering of E and x∈E:
(2.2)Md𝒞(x)={K∈𝒞:x∈K∧(∀S∈𝒞∧x∈S∧S⊆K⇒K=S)},
is called the minimal description of x. When the covering is clear, we omit the lowercase 𝒞 in the minimal description.Definition 2.13 (Indiscernible neighborhood and neighborhood [27, 28]).
Let(E,𝒞) be a covering approximation space and x∈E.∪{K:x∈K∈𝒞} is called the indiscernible neighborhood of x and denoted as I𝒞(x).∩{K:x∈K∈𝒞} is called the neighborhood of x and denoted as N𝒞(x). When the covering is clear, we omit the lowercase 𝒞.Definition 2.14 (A reducible covering [29]).
Let𝒞 be a covering of a domain E and K∈𝒞. If K is a union of some sets in 𝒞-{K}, we say K is a reducible element in 𝒞; otherwise K is an irreducible element in 𝒞. If every element in 𝒞 is irreducible, we say 𝒞 is irreducible; otherwise 𝒞 is reducible.Definition 2.15 (Reduct [29]).
For a covering𝒞 of a universe E, when we remove all reducible elements from 𝒞, the set of remaining elements is still a covering of E, and this new irreducible covering has not reducible element. We call thus new covering a reduct of 𝒞 and it is denoted by reduct(𝒞).Definition 2.16 (Immured element [27]).
Let𝒞 be a covering of E and K an element of 𝒞. If there exists another element K′ of 𝒞 such that K⊂K′, we say that K is an immured element of covering 𝒞.Definition 2.17 (Exclusion [27]).
Let𝒞 be a covering of E. When we remove all immured elements from 𝒞, the set of all remaining elements is still a covering of E, and this new covering has no immured element. We called this new covering an exclusion of 𝒞, and it is denoted by exclusion(𝒞).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.
Let𝒞 be a covering of E. The covering upper approximation operators SH,XH:2E→2E are defined as follows: For all X∈2E,SH(X)=∪{K∈𝒞:K∩X≠∅}=∪{I(x):x∈X},XH(X)={x:N(x)∩X≠∅}.SH𝒞 and XH𝒞 are called the second and the sixth covering upper approximation operators with respect to the covering 𝒞, respectively. When there is no confusion, we omit 𝒞 at the lowercase.


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

As we know, ifM is a matroid and ℒ(M) denotes the set of all closed sets of M ordered by inclusion, then ℒ(M) 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.LetE be a nonempty finite set and 𝒞 a covering of E. As shown in Proposition 2.6, M(𝒞)=(E,ℐ(𝒞)) is the transversal matroid induced by covering 𝒞. ℒ(M(𝒞)) is the set of all closed sets of M(𝒞). Especially, ℒ(M(𝒫)) is the set of all closed sets of the transversal matroid induced by partition 𝒫. Based on Lemma 2.9, we know ℒ(M(𝒞)) and ℒ(M(𝒫)) are geometric lattices.The theorem below connects a covering𝒞 with clM(𝒞)(∅). In fact, ∅∈ℒ(M(ℱ)) if and only if ℱ is a covering.Theorem 3.1.
LetE be a set and M(ℱ) a transversal matroid induced by a family of ℱ={F1,F2,…,Fm} subsets of E, where Fi≠∅(1≤i≤m). clℱ(∅)=∅ if and only if ℱ is a covering of E.Proof.
“⇐”: According to the definition of transversal matroid, any partial transversal is an independent set of transversal matroid. Since ℱ is a covering, any single-point set is an independent set. Based on the definition of closure operator of a matroid, we have clM(ℱ)(∅)=∅.
“⇒”: Since clM(ℱ)(∅)=∅, any single-point set is an independent set, that is, for all x∈E, there exists ix∈{1,2,…,m} such that x∈Fix⊆E. Hence, E=⋃x∈E{x}⊆⋃x∈EFix⊆⋃i=1mFi⊆E. Thus ⋃i=1mFi=E. For all i∈{1,2,…,m}, Fi≠∅ and ⋃i=1mFi=E, hence ℱ is a covering.Theorem3.1 indicates that the zero of ℒ(M(𝒞)) is ∅. The following lemma presents the form of the atoms of ℒ(M(𝒞)). In fact, the closure of any single-point set is an atom of the lattice.Lemma 3.2.
Let𝒞 be a covering of E. For all x∈E, clM(𝒞)({x}) is an atom of ℒ(M(𝒞)).Proof.
Since𝒞 is a covering, we know any single-point set is an independent set. Thus for all x∈E, rM(𝒞)(clM(𝒞)({x}))=rM(𝒞)({x})=1. As we know, the atoms of a lattice are precisely the elements of height one. Combining with Lemma 2.10 and the fact that clM(𝒞)({x}) is a closed set, we know clM(𝒞)({x}) is an atom of ℒ(M(𝒞)).Lemma3.2 does not establish the concrete form of clM(𝒞)({x}). In order to solve that problem, we first define two sets as follows.Definition 3.3.
Let𝒞={K1,K2,…,Km} be a covering of a finite set E={x1,x2,…,xn}. We define the following.(i)
A={Ki-⋃j=1,j≠imKj:Ki-⋃j=1,j≠imKj≠∅,i∈{1,2,…,m}}={A1,A2,…,As}.(ii)
B=E-⋃i=1sAi.Remark 3.4.
For alli∈{1,2,…,s} and x∈Ai, there exists only one block such that x belongs to it, and there exist at least two blocks such that y belongs to them for all y∈B.The following two propositions establish two characteristics ofA and B.Proposition 3.5.
Let𝒞 be a covering of E. {A1,A2,…,As}∪{{x}:x∈B} forms a partition of E.Proof.
LetP={A1,A2,…,As}∪{{x}:x∈B}. According to Definition 3.3, we know (⋃i=1sAi)∪(∪{{x}:x∈B})=E. Now we need to prove for all P1,P2∈P, P1∩P2=∅. According to the definition of A, if P1,P2∈A, then P1∩P2=∅. If P1,P2∈{{x}:x∈B}, then P1∩P2=∅ because Pi and Pj are different single-points. If P1∈A and P2∈{{x}:x∈B}, then P1∩P2=∅ because B∩(⋃k=1sAk)=∅, P1⊆⋃k=1sAk and P2⊆B.Proposition 3.6.
𝒞 is a partition of E if and only if B=∅.Proof.
According to the definition ofA and B, the necessity is obvious. Now we prove the sufficiency. If 𝒞 is not a partition, then there exist Ki,Kj∈𝒞 such that Ki∩Kj≠∅. Thus there exists x∈E such that x∈Ki∩Kj, that is, there exist at least Ki,Kj∈𝒞 such that x belongs to them, hence x∈B. That contradicts the assumption that B=∅.Based on Lemma3.2 and Definition 3.3, we can establish the concrete form of the atoms of lattice ℒ(M(𝒞)).Theorem 3.7.
Let𝒞 be a covering of E. {A1,A2,…,As}∪{{x}:x∈B} is the set of atoms of lattice ℒ(M(𝒞)).Proof.
According to the definition ofAi, we may as well suppose Ai=Kh-⋃j=1,j≠hmKj. Based on 𝒞 being a covering and the definition of transversal matroid, we know any single-point set is an independent set, thus for all x∈Ai, {x} is an independent set. For all y∈Ai and y≠x, we know x,y∈Kh and x,y∉Kj for all 1≤j≤m and j≠h, thus x and y cannot be chosen from different blocks in the covering 𝒞. That shows that {x,y} is not an independent set according to the definition of transversal matroid. Hence, {x} is a maximal independent set included in Ai, that is, rM(𝒞)(Ai)=1. Next, we need to prove Ai is a closed set. Since Ai⊆clM(𝒞)(Ai), we need to prove clM(𝒞)(Ai)⊆Ai, that is, y∉Ai implies y∉clM(𝒞)(Ai). If y∉Ai, based on the fact that 𝒞 is a covering and the definition of Ai, then there exists j≠h such that y∈Kj. Thus {x,y}(x∈Ai) is an independent set. That implies y∉clM(𝒞)(Ai), thus clM(𝒞)(Ai)=Ai. Hence, Ai∈ℒ(M(𝒞)). Combining with rM(𝒞)(Ai)=1, we know for all 1≤i≤m, Ai is an atom of lattice ℒ(M(𝒞)).
According to the definition of transversal matroid and the fact that𝒞 is a covering, any single-point set is an independent set. Thus for all x∈B, rM(𝒞)({x})=1. For all y∈E and y≠x, if y∈B, then there exist at least two blocks containing y according to the definition of B. We may as well suppose y∈Kk,Kt and x∈Kl,Kp, where {Kk,Kt} may be the same as {Kl,Kp}. Based on this, {x,y} is an independent set. This implies y∉clM(𝒞)({x}). If y∉B, then we may as well suppose y∈Ai, thus y∈Kh for the definition of Ai, where Kh may be the same with Kl or Kp. Based on this, x and y can be chosen from different blocks in covering 𝒞, thus {x,y} is an independent set. That implies y∉clM(𝒞)({x}). From above discussion, we have clM(𝒞)({x})={x}. Hence, {x}∈ℒ(M(𝒞)) for all x∈B. Combining with rM(𝒞)({x})=1, we know {x} is an atom of lattice ℒ(M(𝒞)) for all x∈B.
Next, we will prove the set of atoms of latticeℒ(M(𝒞)) cannot be anything but {A1,A2,…,As}∪{{x}:x∈B}. According to Lemma 3.2, we know {clM(𝒞)({x}):x∈E} is the set of atoms of lattice ℒ(M(𝒞)). Similar to the proof of the second part, we know that if x∈B then clM(𝒞)({x})={x}. If x∉B, then x belongs to one of elements in A. We may as well suppose x∈Ai. Combining Ai is an atom with ∅⊆clM(𝒞)({x})⊆clM(𝒞)(Ai)=Ai, we have clM(𝒞)({x})=Ai. Hence, {A1,A2,…,As}∪{{x}:x∈B} is the set of atoms of lattice ℒ(M(𝒞)).The proposition below connects simple matroid and the cardinal number ofAi. In fact, a matroid is simple if and only if |Ai|=1 for all Ai∈A.Lemma 3.8.
Let𝒞 be a covering of E. For all Ai∈A, if |Ai|≥2, then x and y are parallel in M(𝒞) for all x,y∈Ai.Proof.
According to the definition ofAi, we may as well suppose Ai=Kh-⋃j=1,j≠hmKj, where 1≤h≤m. For all x,y∈Ai, then x,y∈Kh, and for all j∈{1,2,…,m} and j≠h, we have x∉Kj and y∉Kj. Thus {x,y} is not an independent set. Based on the definition of transversal matroid and the fact that 𝒞 is a covering, any single-point set is an independent set. Thus {x} or {y} is an independent set. Hence, x,y are parallel in M(𝒞).Proposition 3.9.
Let𝒞 be a covering of E. M(𝒞) is a simple matroid if and only if |Ai|=1 for all Ai∈A.Proof.
“⇒”: Since M(𝒞) is a simple matroid, it does not contain parallel elements. If there exists Ai∈A such that |Ai|≠1, then |Ai|≥2 because Ai≠∅. According to Lemma 3.8, we know for all x,y∈Ai, x,y are parallel which contradicts the assumption that M(𝒞) is a simple matroid. Hence, for all Ai∈A, |Ai|=1.
“⇐”: According to the definition of parallel element, if |Ai|=1 for all Ai∈A, then M(𝒞) does not contain parallel elements; otherwise, we may as well suppose x,y are parallel, then there exists only one block which contains x,y. Hence, there exists Ai∈A such that x,y∈Ai, that is, |Ai|≥2. This contradicts the fact that |Ai|=1 for all Ai∈A. Based on the definition of transversal matroid and the fact that 𝒞 is a covering, any single-point set is an independent set, thus M(𝒞) does not contain loops. Hence, M(𝒞) does not contain parallel elements and loops which implies that M(𝒞) is a simple matroid.When a covering degenerates into a partition, we also have the above results.Corollary 3.10.
Let𝒫={P1,P2,…,Pm} be a partition of E. 𝒫 is the set of atoms of lattice ℒ(M𝒫).Corollary 3.11.
Let𝒫={P1,P2,…,Pm} be a partition of E. M(𝒫) is a simple matroid if and only if |Pi|=1 for all Pi∈P.For a geometric latticeℒ(M(𝒞)), any closure of single-point is an atom of it. However, the closure of any two elements of E may not be a element which covers some atoms of this lattice. The following proposition shows in what condition clM(𝒞)({x,y}) covers certain atoms of lattice ℒ(M(𝒞)).Proposition 3.12.
Let𝒞 be a covering of E. For all x,y∈E, clM(𝒞)({x,y}) covers clM(𝒞)({x}) if and only if there does not exist Ai∈A such that x,y∈Ai.Proof.
“⇐”: For all x,y∈E, {x}⊆{x,y}, then clM(𝒞)({x})⊆clM(𝒞)({x,y}) and 1=rM(𝒞)(clM(𝒞)({x}))≤rM(𝒞)(clM(𝒞)({x,y}))=rM(𝒞)({x,y})≤|{x,y}|=2. Now we need to prove rM(𝒞)(clM(𝒞)({x,y}))=2. If rM(𝒞)(clM(𝒞)({x,y}))=rM(𝒞)({x,y})=1=rM(𝒞)({x}), then {x,y}∉ℐ(𝒞), that is, there is only one block contains x,y. It means that there exists Ai∈A such that x,y∈Ai. That contradicts the hypothesis. Hence, rM(𝒞)(clM(𝒞)({x,y}))=2, that is, clM(𝒞)({x,y}) covers clM(𝒞)({x}).
“⇒”: For all x,y∈E, if there exists Ai∈A such that x,y∈Ai, then there is only one block contains x,y, thus x,y∉ℐ(𝒞), hence {x,y}⊆clM(𝒞)({x}). That implies clM(𝒞)({x,y})⊆clM(𝒞)({x}) which contradicts the assumption that clM(𝒞)({x,y}) covers clM(𝒞)({x}).The modular element and the modular pair are core concepts in lattice. As we know, ifa is a modular element of ℒ, then (a,b) is a modular pair of ℒ for all b∈ℒ, 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 ℒ(M).Theorem 3.13.
LetM be a matroid and ℒ(M) the set of all closed sets of M.(1)
For allX,Y∈ℒ(M), (X,Y) is a modular pair of ℒ(M) if and only if rM(X∪Y)+rM(X∩Y)=rM(X)+rM(Y).(2)
For allX∈ℒ(M), X is a modular element of ℒ(M) if and only if rM(X∪Y)+rM(X∩Y)=rM(X)+rM(Y), for all Y∈ℒ(M).Proof.
(1) According to Lemmas2.8 and 2.10, we know (X,Y) is a modular pair of ℒ(M) if and only if rM(X)+rM(Y)=rM(X∨Y)+rM(X∧Y)=rM(clM(X∪Y))+rM(X∩Y)=rM(X∪Y)+rM(X∩Y).
(2) It comes from the definition of modular element and(1).Let{Ai:i∈Γ} be the set of atoms of lattice ℒ(M(𝒞)), where Γ denotes the index set. The following theorem shows the relationship among atoms, modular pairs, and modular elements of the lattice.Theorem 3.14.
Let𝒞 be a covering of E. For all i,j∈Γ.(1)
(Ai,Aj) is a modular pair of ℒ(M(𝒞)).(2)
Ai is a modular element of ℒ(M(𝒞)).Proof.
(1) Since𝒞 is a covering, clM(𝒞)(∅)=∅. Ai and Aj are atoms, so Ai∩Aj=∅. According to Theorem 3.13, we need to prove rM(𝒞)(Ai∪Aj)+rM(𝒞)(Ai∩Aj)=rM(𝒞)(Ai)+rM(𝒞)(Aj), that is, rM(𝒞)(Ai∪Aj)=2. According to the submodular inequality of rM(𝒞), we have rM(𝒞)(Ai∪Aj)+rM(𝒞)(Ai∩Aj)≤rM(𝒞)(Ai)+rM(𝒞)(Aj), that is, 1≤rM(𝒞)(Ai∪Aj)≤2. If rM(𝒞)(Ai∪Aj)=1=rM(𝒞)(Ai), then Aj⊆clM(𝒞)(Ai)=Ai which contradicts that Ai∩Aj=∅.
(2)Ai is a modular element of ℒ(M(𝒞)) if and only if rM(𝒞)(Ai∪A)+rM(𝒞)(Ai∩A)=rM(𝒞)(Ai)+rM(𝒞)(A) for all A∈ℒ(M(𝒞)).
Case  1. If Ai and A are comparable, that is, Ai⊆A, then rM(𝒞)(Ai∪A)+rM(𝒞)(Ai∩A)=rM(𝒞)(Ai)+rM(𝒞)(A).
Case  2. If Ai and A are not comparable, there are two cases. One is that A is an atom of ℒ(M(𝒞)), the other is that A is not an atom of ℒ(M(𝒞)). If A is an atom of ℒ(M(𝒞)), then we obtain the result from (1). If A is not an atom of ℒ(M(𝒞)), then A∩Ai=∅. Hence, rM(𝒞)(A)≤rM(𝒞)(A∪Ai)≤rM(𝒞)(A)+1. If rM(𝒞)(A∪Ai)=rM(𝒞)(A), then Ai⊆clM(𝒞)(A)=A which contradicts that Ai∩A=∅. Hence, rM(𝒞)(A∪Ai)=rM(𝒞)(A)+1.
In a word, for allA∈ℒ(M(𝒞)), rM(𝒞)(Ai∪A)+rM(𝒞)(Ai∩A)=rM(𝒞)(Ai)+rM(𝒞)(A), that is, Ai is a modular element of ℒ(M(𝒞)) for all i∈Γ.When a covering degenerates into a partition, it is not difficult for us to obtain the following result.Corollary 3.15.
Let𝒫 be a partition of E. For all Pi,Pj∈𝒫:(1)
(Pi,Pj) is a modular pair of ℒ(M(𝒫)).(2)
Pi is a modular element of ℒ(M(𝒫)).The following lemma shows how to induce a matroid by a lattice. In fact, if a functionf on a lattice is nonnegative, integer-valued, submodular and f(∅)=0, then it can determine a matroid.Lemma 3.16 (see [19]).
LetℒE be a lattice of subsets of a set E such that ℒE is closed under intersection, and contains ∅ and E. Suppose that f is a nonnegative, integer-valued, submodular function on ℒE for which f(∅)=0. Let ℐ(ℒE,f)={X⊆E:f(X)≥|X∩T|,forallT∈ℒE}. ℐ(ℒE,f) is the collection of independent sets of a matroid on E.According to the definition ofℒ(M(𝒞)), we find that ℒ(M(𝒞)) is closed under intersection, and contains ∅ and E. Moreover, the rank function of M(𝒞) is a nonnegative, integer-valued, submodular function on ℒ(M(𝒞) for which rM(𝒞)(∅)=0. Similar to Lemma 3.16, we can obtain the following theorem.Theorem 3.17.
Let𝒞 be a covering of E. We define ℐ(ℒ(M(𝒞)),rM(𝒞))={X⊆E:rM(𝒞)(Y)≥|X∩Y|,  for  all  Y∈ℒ(M(𝒞))}, then M(E,ℐ(ℒ(M(𝒞),rM(𝒞))) is a matroid.For any given matroidM, we know that for all X⊆E, X is an independent set of M if and only if rM(X)=|X|. Moreover, based on the properties of rank function, we have rM(X)≤|X|. Hence, X is an independent set of M if and only if rM(X)≥|X| for all X⊆E.Lemma 3.18.
LetM be a matroid. X is an independent set of M if and only if for all closed set Y of M, rM(Y)≥|X∩Y|.Proof.
“⇒”: Since for all closed set Y, X∩Y⊆X and X is an independent set, X∩Y is an independent set of M according to the independent set axiom of a matroid. Hence, we have rM(Y)≥rM(X∩Y)≥|X∩Y|.
“⇐”: For all closed set Y, rM(Y)≥|X∩Y|. Especially, for Y=clM(X), we have rM(X)=rM(clM(X))=rM(Y)≥|X∩Y|=|X|. Hence, X is an independent set of matroid M.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 denoterℒ(M(𝒞)) as the rank function of M(E,ℐ(ℒ(M(𝒞),rM(𝒞))) on E. 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.
Let𝒞 be a covering of E. For all X⊆E, ℐ(ℒ(M(𝒞),rM(𝒞))=ℐ(M(𝒞)) and rℒ(M(𝒞))(X)=rM(𝒞)(X).Proof.
According to Lemma3.18, we know that ℐ(ℒ(M(𝒞),rM(𝒞))={X⊆E:rM(𝒞)(Y)≥|X∩Y|,forallY∈ℒ(M(𝒞))} and ℐ(𝒞)={X⊆E:rM(𝒞)(X)=|X|} are equivalent, that is, M(E,ℐ(ℒ(M(𝒞),rM(𝒞))) and M(E,ℐ(𝒞)) are equivalent. So does rℒ(M(𝒞))(X)=rM(𝒞)(X) for all X⊆E.When a covering degrades into a partition, we can obtain a matroidM(𝒫)=(E,ℐ(P)), where ℐ(𝒫)={X⊆E:forallPi∈𝒫,|X∩Pi|≤1} and rM(𝒫)(X)=max{|I|:I⊆X,I∈ℐ(𝒫)}=|{Pi∈𝒫:Pi∩X≠∅}| for all X⊆E. As we know, for all X,Y∈ℒ(M), X∨Y=clM(X∪Y). If the matroid is M(𝒫), then X∨Y=X∪Y.Lemma 3.20 (see [25]).
If𝒫 is a partition of E and M is the matroid, then clM=R* for all X⊆E.Lemma 3.21.
Let𝒫 be a partition of E. For all X,Y∈ℒ(M(𝒫)), X∨Y=X∪Y.Proof.
X
∨
Y
=
c
l
M
(
𝒫
)
(
X
∪
Y
)
=
R
*
(
X
∪
Y
)
=
R
*
(
X
)
∪
R
*
(
Y
)
=
c
l
M
(
𝒫
)
(
X
)
∪
c
l
M
(
𝒫
)
(
Y
)
=
X
∪
Y.Based on the above two lemmas, we can obtain the following proposition.Proposition 3.22.
Let𝒫 be a partition of E. For all X⊆E, ℐ(ℒ(M(𝒫)),rM(𝒫))=ℐ(M(𝒫)) and rℒ(M(𝒫))(X)=rM(𝒫)(X).Proof.
We need to prove onlyℐ(ℒ(M(𝒫)),rM(𝒫))=ℐ(M(𝒫)). For all X∈ℐ(ℒ(M(𝒫)),rM(𝒫)), then rM(𝒫)(Y)≥|X∩Y| for all Y∈ℒ(M(𝒫)). Since Pi∈ℒ(M(𝒫)), |X∩Pi|≤rM(𝒫)(Pi)=1. Thus X∈ℐ(M(𝒫)). Hence, ℐ(ℒ(M(𝒫),rM(𝒫))⊆ℐ(M(𝒫)). According to Lemmas 2.9 and 3.21, for all Y∈ℒ(M(𝒫)), there exists K⊆{1,2,…,m} such that Y=⋁i∈KPi=⋃i∈K‍Pi. Thus rM(𝒫)(Y)=|K| and |X∩Y|=|X∩(⋃i∈K‍Pi)|=|⋃i∈K‍(X∩Pi)|=∑i∈K|X∩Pi|. For all X∈ℐ(M(𝒫)), |X∩Pi|≤1. Hence, |X∩Y|=∑i∈K|X∩Pi|≤|K|=rM(𝒫)(Y), that is, ℐ(M(𝒫))⊆ℐ(ℒ(M(𝒫)),rM(𝒫)).


## 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 Section3. 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.
Let𝒞 be a covering of E. SH has the following properties:(1)
SH(∅)=∅,(2)
X⊆SH(X) for all X⊆E,(3)
SH(X∪Y)=SH(X)∪SH(Y) for all X,Y⊆E,(4)
x∈SH({y})⇔y∈SH({x}) for all x,y∈E,(5)
X⊆Y⊆E⇒SH(X)⊆SH(Y),(6)
for allx,y∈E,y∈SH(X∪{x})-SH(X), then x∈SH(X∪{y}).Proof.
(1)–(5) were proven in [30, 32, 33]. Here we prove only (6). According to (3), we know SH(X∪{x})-SH(X)=SH(X)∪SH({x})-SH(X)=SH({x})-SH(X). If y∈SH(X∪{x})-SH(X), then y∈SH({x}). According to (4) and (5), we have x∈SH({y})⊆SH(X∪{y}).We find that the idempotent ofSH is not valid, so what is the condition that guarantees it holds for SH? We have the following conclusion.Proposition 4.2.
Let𝒞 be a covering of E. For all X⊆E, SH(SH(X))=SH(X) if and only if {I(x):x∈E} induced by 𝒞 forms a partition of E.Proof.
“⇐”: According to (2), (5) of Proposition 4.1, we have SH(X)⊆SH(SH(X)). Now we prove SH(SH(X))⊆SH(X). For all x∈SH(SH(X)), there exists y∈SH(X) such that x∈I(y). Since y∈SH(X), there exists z∈X such that y∈I(z). According to the definition of I(y), we know y∈I(y), thus I(z)∩I(y)≠∅. For {I(x):x∈E} forms a partition, I(z)=I(y). Since x∈I(y), x∈I(z), that is, x∈SH(X), thus SH(SH(X))⊆SH(X).
“⇒”: In order to prove {I(x):x∈E} forms a partition, we need to prove that for all x,y∈E, if I(x)∩I(y)≠∅, then I(x)=I(y). If I(x)∩I(y)≠∅, then there exists z∈I(x)∩I(y). For SH(SH({x}))=∪{I(u):u∈I(x)} and z∈I(x), then I(z)⊆SH(SH({x}))=SH({x})=I(x). Based on the definition of I(z) and z∈I(x), we have x∈I(z), thus I(x)⊆SH(SH({z}))=SH({z})=I(z). Hence, I(x)=I(z). Similarly, we can obtain I(y)=I(z), thus I(x)=I(z)=I(y).The following theorem establishes a necessary and sufficient condition forSH to be a closure operator.Theorem 4.3.
Let𝒞 be a covering of E. SH is a closure operator of a matroid if and only if {I(x):x∈E} induced by 𝒞 forms a partition of E.Proof.
It comes from Propositions4.1 and 4.2 and (2), (5), and (6) of Proposition 2.3.For a given covering𝒞 of E, we may as well suppose the set of indiscernible neighborhoods of E as {I(x):x∈E}={I(x1),I(x2),…,I(xs)} where x1,x2,…,xs∈E.Definition 4.4.
Let𝒞 be a covering of E. We define ℐ′={I⊆E:|I∩I(xi)|≤1,foralli∈{1,2,…,s}}.As we know, if{I(x):x∈E}={I(x1),I(x2),…,I(xs)} forms a partition of E, then M(E,ℐ′) is a matroid and SH is the closure operator of a matroid. Thus SH can determine a matroid, and the independent sets of the matroid induced by it are established as follows:
(4.1)ℐSH(𝒞)={I⊆E:∀x∈I,x∉SH(I-{x})}.
The following proposition shows M(E,ℐSH)=M(E,ℐ′) under the condition that {I(x):x∈E} forms a partition of E.Proposition 4.5.
Let𝒞 be a covering of E. If {I(x):x∈E} induced by 𝒞 forms a partition of E, then M(E,ℐSH(𝒞)) is a matroid and ℐSH(𝒞)=ℐ′.Proof.
Letℐcl={I⊆E:forallx∈I,x∉cl(I-{x})}. we know that if an operator cl satisfies (1)–(4) of Proposition 2.3, M(E,ℐcl) is a matroid. {I(x):x∈U} induced by 𝒞 forms a partition, hence, M(E,ℐSH(𝒞)) is a matroid. Since SH(I)=⋃y∈I‍I(y), SH(I-{x})=⋃y∈I-{x}‍I(y). According to the definition of I(x), we know x∈I(x). On one hand, for all I∈ℐSH(𝒞), we know that for all x∈I, x∉SH(I-{x}), that is, for all y∈I and y≠x, x∉I(y). If I∉ℐ′, that is, there exists 1≤i≤s such that |I∩I(xi)|≥2, then we may as well suppose there exist u,v such that u,v∈I(xi) and u,v∈I. Since u∈I(u),v∈I(v) and {I(x):x∈E} forms a partition, I(u)=I(v)=I(xi). Based on that, we know there exists u∈I and u≠v such that u∈I(v), that implies contradiction. Hence, I∈ℐ′, that is, ℐSH(𝒞)⊆ℐ′. On the other hand, if I∉ℐSH(𝒞), then there exists x∈I such that x∈SH(I-{x})=⋃y∈I-{x}‍I(y). That implies that there exists y∈I and y≠x such that x∈I(y). Since x∈I(x) and {I(x):x∈U} forms a partition, I(x)=I(y). Thus x,y∈I∩I(x), that implies |I∩I(x)|≥2, that is, I∉ℐ′. Hence, ℐ′⊆ℐSH(𝒞).We denote the rank function ofM(E,ℐSH(𝒞)) by rSH. Then some properties of M(E,ℐSH(𝒞)) are established in the following proposition.Proposition 4.6.
Let𝒞 be a covering of E. If {I(x):x∈E}={I(x1),I(x2),…,I(xs)} induced by 𝒞 forms a partition of E, then(1)
X is a base of M(E,ℐSH(𝒞)) if and only if |X∩I(xi)|=1 for all i∈{1,2,…,s}. Moreover, M(E,ℐSH(𝒞)) has |I(x1)||I(x2)|⋯|I(xs)| bases.(2)
For allX⊆E, rSH(X)=|{I(xi):I(xi)∩X≠∅,i=1,2,…,s}|.(3)
X is a dependent set of M(E,ℐSH(𝒞)) if and only if there exists I(xi)∈{I(x):x∈E} such that |I(xi)∩X|>1.(4)
X is a circuit of M(E,ℐSH(𝒞)) if and only if there exists I(xi)∈{I(x):x∈E} such that X⊆I(xi) and |X|=2.Proof.
(1) According to the definition of base of a matroid, we know thatX is a base of M(E,ℐSH(𝒞))⇔X is a maximal independent set of M(E,ℐSH(𝒞))⇔|X∩I(xi)|=1 for all i∈{1,2,…,s} because ℐSH(𝒞)=ℐ′. Since X is a base of M(E,ℐSH(𝒞)) and I(x1),I(x2),…,I(xs) are different, M(E,ℐSH(𝒞)) has |I(x1)||I(x2)|⋯|I(xs)| bases.
(2) According to the definition of rank function, we knowrSH(X)=|BX|=|{I(xi):|BX∩I(xi)|=1}|≤|{I(xi):X∩I(xi)≠∅}|, where BX is a maximal independent set included in X. Now we just need to prove the inequality |{I(xi):|BX∩I(xi)|=1}|<|{I(xi):X∩I(xi)≠∅}| does not hold; otherwise, there exists 1≤i≤s such that I(xi)∩X≠∅ and I(xi)∩BX=∅ because BX∈ℐSH(𝒞). Thus there exists ei∈I(xi)∩X such that BX∪{ei}⊆X and BX∪{ei}∈ℐSH(𝒞). That contradicts the assumption that BX is a maximal independent set included in X. Hence, rSH(X)=|{I(xi):I(xi)∩X≠∅,i=1,2,…,s}|.
(3) According to the definition of dependent set, we know thatX is a dependent set ⇔X∉ℐSH(𝒞)⇔ there exists i∈{1,2,…,s} such that |X∩I(xi)|>1.
(4) “⇒”: As we know, a circuit is a minimal dependent set. X is a circuit of M(E,ℐSH(𝒞)), then there exists i∈{1,2,…,s} such that |I(xi)∩X|=2. Now we just need to prove |X|=2; otherwise, we may as well suppose X={x,y,z} where x,y∈I(xi)∩X. Thus we can obtain |(X-{z})∩I(xi)|=2, that is, X-{z}∉ℐSH(𝒞). That contradicts the minimality of circuit. Combining |X|=2 with |I(xi)∩X|=2, we have X⊆I(xi).
“⇐”: Since |X|=2, we may as well suppose X={x,y}, and |X∩I(xi)|=2 because there exists I(xi)∈{I(x):x∈E} such that X⊆I(xi), thus X is a dependent set. For all j∈{1,2,…,s}, |{x}∩I(xj)|≤1 and |{y}∩I(xj)|≤1 which implies {x} and {y} are independent sets, hence X is a circuit of M(E,ℐSH(𝒞)).For a covering𝒞 of E, we denote ℒSH(M(𝒞)) as the set of all closed sets of M(E,ℐSH(𝒞)). When {I(x):x∈E}={I(x1),I(x2),…,I(xs)} forms a partition of E, then for all X,Y∈ℒSH(M(𝒞)), X∧Y=X∩Y, X∨Y=SH(X∪Y)=SH(X)∪SH(Y)=X∪Y and SH(∅)=∅.Proposition 4.7.
Let𝒞 be a covering of E. If {I(x):x∈E}={I(x1),I(x2),…,I(xs)} forms a partition of E, then(1)
I(x1),I(x2),…,I(xs) are all atoms of ℒ(MSH).(2)
For allx,y∈E and x≠y, there does not exist I(z)∈{I(x1),I(x2),…,I(xs)} such that x,y∈I(z) if and only if SH({x,y}) covers SH({x}) or SH({y}).(3)
For alli,j∈{1,2,…,s},(I(xi),I(xj)) is a modular pair of ℒSH(M(𝒞)).(4)
For alli∈{1,2,…,s},I(xi) is a modular element of ℒSH(M(𝒞)).Proof.
(
1
) comes from Corollary 3.10, Theorem 4.3 and Proposition 4.5. Based on Proposition 3.12 and Theorem 4.3, we can obtain (2). According to Corollary 3.15, Theorem 4.3 and Proposition 4.5, it is easy to obtain (3) and (4).Based on Theorem4.3, we know that a necessary and sufficient condition for SH to be a closure operator of a matroid is that {I(x):x∈E} forms a partition of E. The following two propositions show what kind of coverings can satisfy that condition.Lemma 4.8.
Let𝒞 be a covering of E and K∈𝒞. If K is an immured element, then I(x) is the same in 𝒞 as in 𝒞-{K}.Proof.
Ifx∉K, then I𝒞(x)=I𝒞-{K}(x). If x∈K, then I𝒞(x)=⋃x∈K′‍K′∪K. Since K is an immured element, there exists x∈K′ such that K⊆K′. Thus I𝒞(x)=⋃x∈K′‍K′∪K=⋃x∈K′‍K′=I𝒞-{K}(x). Hence, I(x) is the same in 𝒞 as in 𝒞-{K}.Proposition 4.9.
Let𝒞 be a covering of E. If exclusion(𝒞) is a partition of E, then {I(x):x∈E} induced by 𝒞 also forms a partition of E.Proof.
Sinceexclusion(𝒞) is a partition of E, {I(x):x∈E} induced by exclusion(𝒞) forms a partition. Suppose {K1,K2,…,Ks} is the set of all immured elements of 𝒞. According to Lemma 4.8, we know for all x∈E, I(x) is the same in exclusion(𝒞) as in exclusion(𝒞)∪{K1}. Thus {I(x):x∈E} induced by exclusion(𝒞)∪{K1} forms a partition of E. And the rest may be deduced by analogy, we know that for all x∈E, I(x) is the same in exclusion(𝒞) as in 𝒞, thus {I(x):x∈E} induced by 𝒞 forms a partition of E.The proposition below establishes a necessary and sufficient condition for{I(x):x∈E} forms a partition of E from the viewpoint of coverings.Proposition 4.10.
Let𝒞 be a covering of E. {I(x):x∈E} induced by 𝒞 forms a partition of E if and only if 𝒞 satisfies (TRA) condition: For all x,y,z∈E,x,z∈K1∈𝒞,y,z∈K2∈𝒞, there exists K3∈𝒞 such that x,y∈K3.Proof.
“⇐”: For all x,y∈E, I(x)∩I(y)=∅ or I(x)∩I(y)≠∅. If I(x)∩I(y)≠∅, then there exists z∈I(x) and z∈I(y). According to the definition of I(x) and I(y), there exist K1,K2 such that x,z∈K1 and y,z∈K2. According to the hypothesis, we know there exists K3∈𝒞 such that x,y∈K3. Now we need to prove only I(x)=I(y). For all u∈I(x), there exists K∈𝒞 such that u,x∈K. Since x,y∈K3, there exists K′∈𝒞 such that u,y∈K′, that is, u∈I(y), thus I(x)⊆I(y). Similarly, we can prove I(y)⊆I(x). Hence, I(x)=I(y), that is, {I(x):x∈E} forms a partition of E.
“⇒”: For all x,y,z∈E,x,z∈K1∈𝒞 and y,z∈K2∈𝒞, we can obtain z∈I(x) and z∈I(y). That implies I(x)∩I(y)≠∅. Since {I(x):x∈E} forms a partition of E, I(x)=I(y). Thus there exists K3∈𝒞 such that x,y∈K3.The following theorem presents a necessary and sufficient condition forSH to be a closure operator of a matroid from the viewpoint of coverings.Theorem 4.11.
Let𝒞 be a covering of E. 𝒞 satisfies (TRA) condition if and only if SH induced by 𝒞 is a closure operator of a matroid.Proof.
It comes from Theorem4.3 and Proposition 4.10.The sixth type of covering-based upper approximation operator was first defined in [31]. Xu and Wang introduced this type of covering-based rough set model and studied the relationship between it and binary relation based rough set model. The following proposition gives some properties of this covering upper approximation operator.Proposition 4.12 (see [26]).
Let𝒞 be a covering of E. XH has the following properties:(1)
XH(E)=E,(2)
XH(∅)=∅,(3)
X⊆XH(X) for all X⊆E,(4)
XH(X∪Y)=XH(X)∪XH(Y) for all X,Y⊆E,(5)
XH(XH(X))=XH(X) for all X⊆E,(6)
for allX⊆Y⊆E⇒XH(X)⊆XH(Y).From the above proposition, we find thatXH does not satisfy the (4) of Proposition 2.3. The following proposition establishes a necessary and sufficient condition for XH to satisfy the condition.Proposition 4.13.
Let𝒞 be a covering of E. For all x,y∈E and X⊆E, XH satisfies
(4.2)y∈XH(X⋃‍{x})-XH(X)⇒x∈XH(X⋃‍{y})
if and only if {N(x):x∈E} forms a partition of E.Proof.
⇒: For all x,y∈E, if N(x)∩N(y)≠∅, then there exists z∈N(x)∩N(y). Let X=∅. According to (2) of Proposition 4.12, we know that if y∈XH({x}) then x∈XH(y), that is, if x∈N(y) then y∈N(x). Since z∈N(x), N(z)⊆N(x). According to the assumption, we also have x∈N(z), that is, N(x)⊆N(z). Thus N(x)=N(z). Similarly, z∈N(y), we have N(z)=N(y). Thus N(x)=N(z)=N(y). Hence, {N(x):x∈E} forms a partition of E.
⇐: Since XH(X∪Y)=XH(X)∪XH(Y) for all X,Y⊆U, y∈XH(X∪{x})-XH(X)=XH(X)∪XH({x})-XH(X)=XH({x})-XH(X). Now we prove x∈XH({y}). Since y∈XH({x}), x∈N(y). Because the fact that {N(x):x∈E} forms a partition of E, x∈N(x) and x∈N(y), we have N(x)=N(y), thus y∈N(x), that is, x∈XH({y}). Hence x∈XH({y})⊆XH(X∪{y}).The following theorem establishes a necessary and sufficient condition forXH to be a closure operator of a matroid.Theorem 4.14.
Let𝒞 be a covering of E. {N(x):x∈E} induced by 𝒞 forms a partition of E if and only if XH is a closure operator of a matroid.Proof.
It comes from(3), (5), and (6) of Propositions 4.12 and 4.13.For convenience, for a given covering𝒞 of E, we may as well suppose the set of all neighborhoods as {N(x):x∈E}={N(x1),N(x2),…,N(xt)} where x1,x2,…,xt∈E.Definition 4.15.
Let𝒞 be a covering of E. We define ℐ′′={I⊆E:|I∩N(xi)|≤1,foralli∈{1,2,…,t}}.Theorem4.14 indicates that if {N(x):x∈E} forms a partition of E, then XH is a closure operator of a matroid. Hence, XH can determine a matroid, and the independent sets of the matroid induced by it are established as follows:
(4.3)ℐXH(𝒞)={I⊆E:∀x∈I,x∉XH(I-{x})}.
Similar to the case of SH, we can obtain the following results.Proposition 4.16.
Let𝒞 be a covering of E. If {N(x):x∈E} induced by 𝒞 forms a partition of E, then M(E,ℐXH(𝒞)) is a matroid and ℐXH(𝒞)=ℐ′′.r
XH denotes the rank function of M(E,ℐXH(𝒞)) and ℒXH(M(𝒞)) denotes the set of all closed sets of M(E,ℐXH(𝒞)). Then we can obtain the following proposition.Proposition 4.17.
Let𝒞 be a covering of E. If {N(x):x∈E}={N(x1),N(x2),…,N(xt)} induced by 𝒞 forms a partition of E, then(1)
X is a base of M(E,ℐXH(𝒞)) if and only if |X∩N(xi)|=1(1≤i≤t), and M(E,ℐXH(𝒞)) has |N(x1)||N(x2)|⋯|N(xs)| bases.(2)
For allX⊆E, rXH(X)=|{N(xi):N(xi)∩X≠∅,1≤i≤t}|.(3)
X is a circuit of M(E,ℐXH(𝒞)) if and only if there exists N(xi)∈{N(x):x∈E} such that X⊆N(xi) and |X|=2.(4)
X is a dependent set of M(E,ℐXH(𝒞)) if and only if there exists N(xi)∈{N(x):x∈E} such that |N(xi)∩X|>1.(5)
{N(x1),N(x2),…,N(xt)} is the set of all atoms of lattice ℒXH(M(𝒞)).(6)
For allx,y∈E and x≠y, there does not exist N(z)∈{N(x1),N(x2),…,N(xt)} such that x,y∈N(z) if and only if XH({x,y}) covers XH({x}) or XH({y}).(7)
For alli,j∈{1,2,…,t}, (N(xi),N(xj)) is a modular pair of lattice ℒXH(M(𝒞)).(8)
For alli∈{1,2,…,t}, N(xi) is a modular element of lattice ℒXH(M(𝒞)).The proof of Propositions4.16 and 4.17 is similar to that of Propositions 4.5, 4.6, and 4.7, respectively. So we omit the proofs of them. Similar to the case of SH, we also study what kind of coverings can make {N(x):x∈E} form a partition of E. This paper establishes only two kinds of coverings. There are some coverings which satisfy the condition appear in [34, 35].Lemma 4.18.
Let𝒞 be a covering on E and K be reducible in 𝒞. For all x∈U, N(x) is the same in 𝒞 as in 𝒞-{K}.Proof.
For allx∈E, Md(x) is the same for covering 𝒞 and covering 𝒞-{K}, so N(x)=∩Md(x) is the same for the covering 𝒞 and covering 𝒞-{K}.Proposition 4.19.
Let𝒞 be a covering of E. If reduct(𝒞) is a partition of E, then {N(x):x∈E} induced by 𝒞 is also a partition of E.Proof.
Sincereduct(𝒞) is a partition of E, {N(x):x∈E} induced by reduct(𝒞) forms a partition of E. Suppose {K1,K2,…,Ks} is the set all reducible elements of 𝒞. According to Lemma 4.18, we know that for all x∈E, N(x) is the same in reduct(𝒞) as in reduct(𝒞)∪{K1}, thus {N(x):x∈E} induced by reduct(𝒞)∪{K1} forms a partition of E. And the rest may be deduced by analogy, then we can obtain for all x∈E, N(x) is the same in reduct(𝒞) as in 𝒞, thus {N(x):x∈E} induced by 𝒞 forms a partition of E.The following proposition establishes a sufficient condition for{N(x):x∈E} to be a partition of E from the viewpoint of coverings.Proposition 4.20.
Let𝒞 be a covering of E. If 𝒞 satisfies (EQU) condition: For all K∈𝒞, for all x,y∈K, the number of blocks which contain x is equal to that of blocks which contain y, then {N(x):x∈E} induced by 𝒞 forms a partition of E.Proof.
For allx,y∈E, if N(x)∩N(y)≠∅, then there exists z∈E such that z∈N(x) and z∈N(y), that is, the blocks which contain x also contain z and the blocks which contain y also contain z. Hence, there exist Ki,Ki′ such that x,z∈Ki and y,z∈Ki′. Without loss of generality, we suppose {K1,K2,…,Ks} is the set of all blocks which contain x and {K1′,K2′,…,Kt′} is the set of all blocks which contain y. Since the number of blocks which contain z is equal to that of blocks which contain x, {K1,K2,…,Ks} is the set of all blocks which contain z, thus {K1′,K2′,…,Kt′}⊆{K1,K2,…,Ks}. Hence, N(x)⊆N(y). Similarly, we can prove N(y)⊆N(x). Hence, N(x)=N(y), that is, {N(x):x∈E} forms a partition of E.Based on Theorem4.14, Propositions 4.19 and 4.20, we can obtain the following two corollaries.Corollary 4.21.
Let𝒞 be a covering of E. If reduct(𝒞) is a partition of E, then XH is a closure operator of a matroid.Corollary 4.22.
Let𝒞 be a covering on E. If 𝒞 satisfies (EQU) condition, then XH is a closure operator of a matroid.


## 5. Relationships among Three Geometric Lattice Structures of Covering-Based Rough Sets

In Section3, the properties of the geometric lattice induced by a covering have been studied by the matroid M(E,ℐ(𝒞)). Section 4 presents three sufficient and necessary conditions for two types of covering upper approximation operators to be closure operators of matroids. Moreover, we exhibit two types of matroidal structures through closure axioms, and then obtain two geometric lattice structures of covering-based rough sets. In this section, we study the relationship among above three types of geometric lattices through corresponding matroids. We also discuss the reducible element and the immured element’s influence on the relationship among this three types of matroidal structures and geometric lattice structures.The following proposition shows the relationship betweenℐSH(𝒞) and ℐ(𝒞), and the relationship between ℒSH(M(𝒞)) and ℒ(M(𝒞)).Proposition 5.1.
Let𝒞 be a covering of E. If SH induced by 𝒞 is a closure operator, then ℐSH(𝒞)⊆ℐ(𝒞) and ℒSH(M(𝒞))⊆ℒ(M(𝒞)).Proof.
SinceSH induced by 𝒞 is a closure operator, {I(x):x∈E}={I(x1),I(x2),…,I(xs)} forms a partition of E. For all I∈ℐSH(𝒞), suppose I={i1,i2,…,iα}(α≤s) such that i1∈I(xi1),i2∈I(xi2),…,iα∈I(xiα) and {I(xi1),I(xi2),…,I(xiα)}⊆{I(x1),I(x2),…,I(xs)}. According to the definition of I(x), there exists {Ki1,Ki2,…,Kiα}⊆𝒞 such that i1∈Ki1,i2∈Ki2,…,iα∈Kiα. Since {I(x1),I(x2),…,I(xs)} forms a partition of E, thus {Ki1,Ki2,…,Kiα} are different blocks. According to the definition of transversal matroid, we have I∈ℐ. Hence, ℐSH(𝒞)⊆ℐ(𝒞).
For allX∈ℒSH(M(𝒞)), X=SH(X)=⋃x∈X‍I(x). Now we need to prove X∈ℒ(M(𝒞)), that is, clM(𝒞)(X)=X={x∣rM(𝒞)(X)=rM(𝒞)(X∪{x})}. Since X⊆clM(𝒞)(X), if clM(𝒞)(X)≠X, then clM(𝒞)(X)⊈X, that is, there exists y∉X such that rM(𝒞)(X)=rM(𝒞)(X∪{y}). Suppose T={t1,t2,…,tt}(t≤s) is a maximal independent set included in X, then {t1,t2,…,tt}⊆X=⋃x∈X‍I(x) and there exist different K1,K2,…,Kt such that for all i∈{1,2,…,t},ti∈Ki. Since y∉X, I(x)∩I(y)=∅ for all x∈X. Based on {I(x1),I(x2),…,I(xs)} forms a partition of E, there exists K⊆I(y) such that K1,K2,…,Kt,K are different blocks and y∈K, thus T∪{y} is a maximal independent set included in X∪{y}. Hence, we have rM(𝒞)(X∪{y})=rM(𝒞)(X)+1 which contradicts rM(𝒞)(X)=rM(𝒞)(X∪{y}). Thus we can obtain clM(𝒞)(X)=X.The following proposition illustrates that in what condition the indiscernible neighborhoods are included in the geometric lattice induced by𝒞.Proposition 5.2.
Let𝒞 be a covering of E. If SH induced by 𝒞 is a closure operator, then for all x∈E,I(x)∈ℒ(M(𝒞)).Proof.
SinceSH induced by 𝒞 is a closure operator, {I(x):x∈E} forms a partition of E. Thus, for all x∈E, SH(I(x))=⋃y∈I(x)‍I(y)=I(x), that is, I(x)∈ℒSH(M(𝒞)). According to Proposition 5.1, I(x)∈ℒ(M(𝒞)).We give an example to help understand the relationship betweenℒSH(M(𝒞)) and ℒ(M(𝒞)) better.Example 5.3.
Let𝒞={{1,2},{1,3},{2,3},{4,5}}. I(1)=I(2)=I(3)={1,2,3}, I(4)=I(5)={4,5}. Let T={1,2}. T∈ℐ(𝒞) but T∉ℐSH(𝒞) because |T∩I(1)|=2. Hence, ℐ(𝒞)⊈ℐSH(𝒞). ℒSH(M(𝒞))={∅,{1,2,3},{4,5},{1,2,3,4,5}}. ℒ(M(𝒞))={∅, {1}, {2}, {3}, {4,5}, {1,2}, {1,3}, {2,3}, {1,4,5}, {2,4,5}, {3,4,5}, {1,2,3}, {1,2,4,5}, {1,3,4,5}, {2,3,4,5}, {1,2,3,4,5}}. It is obvious that ℒ(M(𝒞))⊈ℒSH(M(𝒞)). The structures of ℒSH(M(𝒞)) and ℒ(M(𝒞)) are showed in Figure 1.Figure 1
The lattice ofℒ(M(𝒞)) (resp., ℒXH(M(𝒞))) and ℒSH(M(𝒞)).Remark 5.4.
Let𝒞 be a covering of E. Although XH induced by 𝒞 is a closure operator, it has no relationship between ℐXH(𝒞) and ℐ(𝒞), and has no relationship between ℒXH(M(𝒞)) and ℒ(M(𝒞)).The following example illustrates the above statements.Example 5.5.
Let𝒞={K1,K2,K3,K4} be a covering of E={a,b,c,d,e,f,g,h,i}, where K1={a,b,i}, K2={a,b,c,d,e,f}, K3={f,g,h}, K4={c,d,e,g,h,i}. Then N(a)=N(b)={a,b},N(c)=N(d)=N(e)={c,d,e}, N(f)={f}, N(g)=N(h)={g,h} and N(i)={i}. Let T={a,c,f,g,i}. It is clear that T∈ℐXH(𝒞), but T∉ℐ(𝒞) because |T∩K2|=2, thus ℐXH(𝒞)⊈ℐ(𝒞). Let T′={a,c,d}. It is clear that T′∈ℐ(𝒞), but T′∉ℐXH because |T′∩N(c)|=2, thus ℐ(𝒞)⊈ℐXH(𝒞). Let X={a,b,i}. X∈ℒXH(M(𝒞)) for XH(X)=X. However, X∉ℒ(M(𝒞)) for clM(𝒞)(X)={a,b,c,d,e,i}. Let X={a}. X∈ℒ(M(𝒞)) for clM(𝒞)(X)=X. However, X∉ℒXH(M(𝒞)) for XH(X)={a,b}≠X.The following proposition shows the relationship betweenℐSH(𝒞) and ℐXH(𝒞), and the relationship between ℒSH(M(𝒞)) and ℒXH(M(𝒞)).Proposition 5.6.
Let𝒞 be a covering of E. If XH and SH induced by 𝒞 are closure operators, then ℐSH(𝒞)⊆ℐXH(𝒞) and ℒSH(M(𝒞))⊆ℒXH(M(𝒞)).Proof.
IfXH and SH induced by 𝒞 are closure operators, then {N(x):x∈E} and {I(x):x∈E} form a partition of E, respectively. For all x∈X, N(x)=⋂x∈K‍K⊆⋃i∈K‍K=I(x), thus {N(x):x∈E} is finer than {I(x):x∈E}. Based on this, we can obtain ℐSH(𝒞)⊆ℐXH(𝒞).
For allX∈ℒSH(M(𝒞)), X=SH(X)=⋃x∈X‍I(x). Since x∈N(x)⊆I(x), X=⋃x∈X‍{x}⊆⋃x∈X‍N(x)⊆⋃x∈X‍I(x)=X, thus, X=⋃x∈X‍N(x), that is, X∈ℒXH(M(𝒞)). Hence, ℒSH(M(𝒞))⊆ℒXH(M(𝒞)).Example 5.7.
From Example5.3, we know N(1)={1}, N(2)={2}, N(3)={3}, N(4)=N(5)={4,5} and I(1)=I(2)=I(3)={1,2,3}, I(4)=I(5)={4,5}. Let I={1,2}. It is clear that I∈ℐXH(M(𝒞)) but I∉ℐSH(M(𝒞)) because |I∩I(1)|=2. Hence ℐXH(M(𝒞))⊈ℐSH(M(𝒞)). ℒSH(M(𝒞))={∅,{1,2,3},{4,5},{1,2,3,4,5}} and ℒXH(M(𝒞))={∅, {1}, {2}, {3}, {4,5}, {1,2}, {1,3}, {2,3}, {1,4,5}, {2,4,5}, {3,4,5}, {1,2,3}, {1,2,4,5}, {1,3,4,5}, {2,3,4,5}, {1,2,3,4,5}}. It is obvious that ℒXH(M(𝒞))⊈ℒSH(M(𝒞)). The structures of them are showed in Figure 1.When a covering degenerates into a partition, we can obtain the following result.Theorem 5.8.
If𝒞 is a partition of E, then ℐSH(𝒞)=ℐXH(𝒞)=ℐ(𝒞) and ℒSH(M(𝒞))=ℒXH(M(𝒞))=ℒ(M(𝒞)).Proof.
Since𝒞 is a partition of E, I(x)=N(x)=K(x∈K) for all x∈E, and for all X⊆E, SH(X)=XH(X)=R*(X). Thus ℐSH(𝒞)=ℐXH(𝒞)=ℐ(𝒞) and ℒSH(M(𝒞))=ℒXH(M(𝒞))=ℒ(M(𝒞)).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 onℐ(𝒞).Theorem 5.9.
Letℱ be a family of subset of E and K∈ℱ. ℐ(ℱ-{K})⊆ℐ(ℱ).Proof.
For allI∈ℐ(ℱ-{K}), we may as well suppose I={i1,i2,…,it} where i1,i2,…,it∈E. According to the definition of transversal matroid, there exist different blocks K1,K2,…,Kt∈ℱ satisfy Ki≠K and ij∈Kj for all 1≤i, j≤t. Thus I∈ℐ(ℱ).The following example illustratesℐ(ℱ)⊈ℐ(ℱ-{K}).Example 5.10.
Letℱ={K1,K2,K3} be a family of subset of E={1,2,3,4}, where K1={1,2}, K2={1,3}, K3={3}. ℐ(ℱ)=2E, ℐ(ℱ-{K3})={∅,{1},{2},{3},{1,3},{1,2}{2,3}}. Hence, ℐ(ℱ)⊈ℐ(ℱ-{K}).Let𝒞 be a covering of E and ℱ=𝒞. As we know, reducible elements and immured elements are members of 𝒞. Based on Theorem 5.9, it is not difficult for us to obtain the following four corollaries.Corollary 5.11.
Let𝒞 be a covering of E and K∈𝒞. If K is reducible, then ℐ(𝒞-{K})⊆ℐ(𝒞).Corollary 5.12.
Let𝒞 be a covering of E. ℐ(reduct(𝒞))⊆ℐ(𝒞).Corollary 5.13.
Let𝒞 be a covering of E and K∈𝒞. If K is an immured element, then ℐ(𝒞-{K})⊆ℐ(𝒞).Corollary 5.14.
Let𝒞 be a covering of E. ℐ(exclusion(𝒞))⊆ℐ(𝒞).The following theorem shows the reducible element and immured element’s influence on geometric lattice structureℒ(M(𝒞)).Theorem 5.15.
Letℱ be a family of subset of E and for all K∈ℱ. ℒ(M(ℱ-{K}))⊆ℒ(M(ℱ)).Proof.
First, we proveclℱ({x})⊆clℱ-{K}({x}) for all x∈E. For all y∉clℱ-{K}({x}), {x,y}∈ℐℱ-{K}⊆ℐℱ. Thus y∉clℱ({x}) which implies clℱ({x})⊆clℱ-{K}({x}).
Second, we need to prove that any atom ofℒ(M(ℱ-{K})) is a closed set of ℒ(M(ℱ)), that is, clℱ(clℱ-{K}({x}))=clℱ-{K}({x}). Since clℱ-{K}({x})⊆clℱ(clℱ-{K}({x}))⊆clℱ-{K}(clℱ-{K}({x}))=clℱ-{K}({x}), clℱ(clℱ-{K}({x}))=clℱ-{K}({x}).
Third, we need to proveℒ(M(ℱ-{K}))⊆ℒ(M(ℱ)). For all X⊆ℒ(M(ℱ-{K})), X=⋁i=1mclℱ-{K}({x})=⋁i=1m⋁j=1siclℱ({xj}) because ℒ(M(ℱ-{K})) is a atomic lattice, thus X∈ℒ(M(ℱ)).The following example shows thatℒ(M(ℱ))⊈ℒ(M(ℱ-{K})), where K∈ℱ.Example 5.16.
Based on Example5.10, we have ℒ(M(ℱ))=2E and ℒ(M(ℱ-{K3}))={∅,{1},{2},{3},{1,2,3}}. It is clear that ℒ(M(ℱ))⊈ℒ(M(ℱ-{K3})).Similarly, whenℱ is equal to 𝒞, we can obtain the following four corollaries.Corollary 5.17.
Let𝒞 be a covering of E and K∈𝒞. If K is reducible, then ℒ(M(𝒞-{K}))⊆ℒ(M(𝒞)).Corollary 5.18.
Let𝒞 be a covering of E. ℒ(M(reduct(𝒞)))⊆ℒ(M(𝒞)).Corollary 5.19.
Let𝒞 be a covering of E and K∈𝒞. If K is an immured element, then ℒ(M(𝒞-{K}))⊆ℒ(M(𝒞)).Corollary 5.20.
Let𝒞 be a covering of E. ℒ(M(exclusion(𝒞)))⊆ℒ(M(𝒞)).Let𝒞 be a covering of E and SH induced by 𝒞 a closure operator. If a reducible element K of 𝒞 is removed from the covering 𝒞, then covering 𝒞-{K} may not still be a covering which makes SH be a closure operator. Hence, we omit the discussion of the relationship between ℐSH(𝒞) and ℐSH(𝒞-{K}).Example 5.21.
LetE={1,2,3} and 𝒞={K1,K2,K3} where K1={1,2}, K2={1,3}, K3={1,2,3}. I(1)=I(2)=I(3)={1,2,3}, thus {I(x):x∈E} forms a partition of E. Hence, SH is a closure operator induced by 𝒞. It is clear that K3 is a reducible element, 𝒞-{K3}={K1,K2}. Then the indiscernible neighborhoods induced by 𝒞-{K3} are I(1)={1,2,3}, I(2)={1,2}, I(3)={1,3}. we find that {I(x):x∈E} cannot form a partition of E. Hence, SH is not a closure operator induced by 𝒞-{K3}.The following theorem presents an immured element’s influence onℐSH(𝒞) and ℒSH(M(𝒞)).Theorem 5.22.
Let𝒞 be a covering of E and K an immured element of 𝒞. If SH induced by 𝒞 is a closure operator, then SH induced by covering 𝒞-{K} is also a closure operator. Moreover, ℐSH(𝒞)=ℐSH(𝒞-{K}) and ℒSH(M(𝒞))=ℒSH(M(𝒞-{K})).Proof.
It comes from Lemma4.8 and Theorem 4.3.As we know, if𝒞 is a covering of E, K1 and K2 are two elements of 𝒞, and K1 is an immured element of 𝒞, then K2 is an immured element of 𝒞 if and only if K2 is an immured element of the covering 𝒞-{K1}. Combining this with Theorem 5.22, we can obtain the following corollary.Corollary 5.23.
Let𝒞 be a covering of E. If SH induced by 𝒞 is a closure operator, then SH induced by exclusion(𝒞) is also a closure operator. Moreover, ℐSH(𝒞)=ℐSH(exclusion(𝒞)) and ℒSH(M(𝒞))=ℒSH(M(exclusion(𝒞))).Now we consider the reducible element’s influence onℐXH(𝒞) and ℒXH(M(𝒞)).Theorem 5.24.
Let𝒞 be a covering of E and K be a reducible element. If XH induced by 𝒞 is a closure operator, then XH induced by covering 𝒞-{K} is also a closure operator. Moreover, ℐXH(𝒞)=ℐXH(𝒞-{K}) and ℒXH(M(𝒞))=ℒXH(M(𝒞-{K})).Proof.
SinceXH induced by 𝒞 is a closure operator, {N(x):x∈E} induced by 𝒞 forms a partition. Based on the definition of ℐXH(𝒞) and Lemma 4.18, XH induced by 𝒞-{K} is also a closure operator and ℐXH(𝒞)=ℐXH(𝒞-{K}).As we know, if𝒞 is a covering of E, K∈𝒞, K is reducible in 𝒞 and K1∈𝒞-{K}, then K1 is reducible in covering 𝒞 if and only if it is reducible in covering 𝒞-{K}. Based on this and Theorem 5.24, we can obtain the following result.Corollary 5.25.
Let𝒞 be a covering of E. If XH induced by 𝒞 is a closure operator, then XH induced by reduct(𝒞) is a closure operator. Moreover, ℐXH(𝒞)=ℐXH(reduct(𝒞)) and ℒXH(M(𝒞))=ℒXH(M(reduct(𝒞))).Let𝒞 be a covering of E and XH induced by 𝒞 a closure operator. If an immured element K is removed from the covering 𝒞, then 𝒞-{K} may not still be a covering which makes XH be a closure operator. So we omit the discussion of the relationship between ℐXH(𝒞) and ℐXH(𝒞-{K}).Example 5.26.
SupposeK1={1}, K2={1,2}, K3={2,3}, K4={3}, K5={1,2,3} and 𝒞1={K1,K2,K3,K4}. Then N(1)={1}, N(2)={2} and N(3)={3}, thus {N(x):x∈E} forms a partition of E. Hence, XH is a closure operator. It is clear that K1 is an immured element, and the neighborhoods induced by 𝒞-{K1} are N(1)={1,2}, N(2)={2} and N(3)={3}, thus {N(x):x∈E} cannot form a partition of E. Hence, 𝒞-{K1} is not a covering which makes XH be a closure operator.


## 6. Conclusions

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.



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