Geometric Lattice Structure of Covering-Based Rough Sets through Matroids

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 (𝐸,ℐ) consisting of a finite set 𝐸 and a collection ℐ of subsets of 𝐸 satisfying the following three conditions.(I1)∅∈ℐ.(I2) If 𝐼∈ℐ and 𝐼^′⊆𝐼, then 𝐼^′∈ℐ.(I3) If 𝐼₁,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.

Let 𝑀=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 𝑟_𝑀∶2^𝐸→𝑁 defined by 𝑟_𝑀(𝑋)=max{|𝐼|∶𝐼⊆𝑋,𝐼∈ℐ}(𝑋⊆𝐸). For each 𝑋⊆𝐸, we say cl_𝑀(𝑋)={𝑎∈𝐸∶𝑟_𝑀(𝑋)=𝑟_𝑀(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]). Let 𝐸 be a set. A function 𝑟_𝑀∶2^𝐸→𝑁 is the rank function of a matroid on 𝐸 if and only if it satisfies the following conditions.(R1)For all 𝑋∈2^𝐸, 0≤𝑟_𝑀(𝑋)≤|𝑋|.(R2) If 𝑋⊆𝑌⊆𝐸, then 𝑟_𝑀(𝑋)≤𝑟_𝑀(𝑌).(R3)If 𝑋,𝑌⊆𝐸, then 𝑟_𝑀(𝑋∪𝑌)+𝑟_𝑀(𝑋∩𝑌)≤𝑟_𝑀(𝑋)+𝑟_𝑀(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]). Let 𝐸 be a set. A function cl_𝑀∶2^𝐸→2^𝐸 is the closure operator of a matroid 𝑀 on 𝐸 if and only if it satisfies the following conditions.(1)If 𝑋⊆𝐸, then 𝑋⊆cl_𝑀(𝑋).(2)If 𝑋⊆𝑌⊆𝐸, then cl_𝑀(𝑋)⊆cl_𝑀(𝑌).(3)If 𝑋⊆𝐸, cl_𝑀(cl_𝑀(𝑋))=cl_𝑀(𝑋).(4)If 𝑋⊆𝐸,𝑥∈𝐸 and 𝑦∈cl_𝑀(𝑋∪{𝑥})−cl_𝑀(𝑋), then 𝑥∈cl_𝑀(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]). Let 𝑆 be a nonempty finite set and 𝐽={1,2,…,𝑚}. ℱ={𝐹₁,𝐹₂,…,𝐹_𝑚} denotes a family of subsets of 𝑆. A transversal or system of distinct representatives of {𝐹₁,𝐹₂,…,𝐹_𝑚} is a subset {𝑒₁,𝑒₂,…,𝑒_𝑚} of 𝑆 such that 𝑒_𝑖∈𝐹_𝑖 for all 𝑖 in 𝐽. If for some subset 𝐾 of 𝐽, 𝑋 is a transversal of {𝐹_𝑖∶𝑖∈𝐾}, then 𝑋 is said to be a partial transversal of {𝐹₁,𝐹₂,…,𝐹_𝑚}.

Example 2.5. Let 𝑆={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]). Let ℱ={𝐹_𝑖∶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 𝑥_𝑛 is a subset {𝑥₀,𝑥₁,…,𝑥_𝑛} of 𝑃 such that 𝑥₀<𝑥₁<⋯<𝑥_𝑛. The length of such a chain is 𝑛, and the chain is maximal if 𝑥_𝑖 covers 𝑥_𝑖−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 ℎ_𝑃(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 ℎ_ℒ(𝑥)+ℎ_ℒ(𝑦)≥ℎ_ℒ(𝑥∨𝑦)+ℎ_ℒ(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]). Let ℒ be a lattice and 𝑎,𝑏∈ℒ. (ME) For all 𝑥,𝑧∈ℒ, 𝑥≥𝑧 implies 𝑥∧(𝑎∨𝑧)=(𝑥∧𝑎)∨𝑧, then 𝑎 is called a modular element of ℒ. (MP) For all 𝑧∈ℒ, 𝑏≥𝑧 implies 𝑏∧(𝑎∨𝑧)=(𝑏∧𝑎)∨𝑧, then (𝑎,𝑏) is called a modular pair of ℒ.

As we know, if 𝑎 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]). Let ℒ be a semimodular lattice, then (𝑎,𝑏) is a modular pair if and only if ℎ_ℒ(𝑎∨𝑏)+ℎ_ℒ(𝑎∧𝑏)=ℎ_ℒ(𝑎)+ℎ_ℒ(𝑏) for all 𝑎,𝑏∈ℒ.

2.3. Closed-Set Lattice of a Matroid

If 𝑀 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=cl_𝑀(𝑋∪𝑌) for all 𝑋,𝑌∈ℒ(𝑀). The zero of ℒ(𝑀) is cl_𝑀(∅), 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 lattice ℒ 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 𝑀 be a matroid. ℎ_ℒ(𝑀)(𝑋)=𝑟_𝑀(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). Let 𝐸 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 of 𝐸 is certainly a covering of 𝐸, so the concept of a covering is an extension of the concept of a partition.

Let 𝐸 be a finite set and 𝑅 be an equivalent relation on 𝐸. 𝑅 will generate a partition 𝐸/𝑅={𝑌₁,𝑌₂,…,𝑌_𝑚} of 𝐸, where 𝑌₁,𝑌₂,…,𝑌_𝑚 are the equivalence classes generated by 𝑅. For all 𝑋⊆𝐸, the lower and upper approximations of 𝑋, are, respectively, defined as follows: 𝑅_∗(𝑋)=𝑌_𝑖∈𝐸𝑅∶𝑌_𝑖,𝑅⊆𝑋_∗(𝑋)=𝑌_𝑖∈𝐸𝑅∶𝑌_𝑖𝑋≠∅. (2.1) 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 𝒞 be a covering of 𝐸 and 𝑥∈𝐸: 𝑀𝑑_𝒞(𝑥)={𝐾∈𝒞∶𝑥∈𝐾∧(∀𝑆∈𝒞∧𝑥∈𝑆∧𝑆⊆𝐾⇒𝐾=𝑆)}, (2.2) 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 𝐼_𝒞(𝑥). ∩{𝐾∶𝑥∈𝐾∈𝒞} is called the neighborhood of 𝑥 and denoted as 𝑁_𝒞(𝑥). When the covering is clear, we omit the lowercase 𝒞.

Definition 2.14 (A reducible covering [29]). Let 𝒞 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 covering 𝒞 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]). Let 𝒞 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]). Let 𝒞 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. Let 𝒞 be a covering of 𝐸. The covering upper approximation operators SH,XH∶2^𝐸→2^𝐸 are defined as follows: For all 𝑋∈2^𝐸, SH(𝑋)=∪{𝐾∈𝒞∶𝐾∩𝑋≠∅}=∪{𝐼(𝑥)∶𝑥∈𝑋}, XH(𝑋)={𝑥∶𝑁(𝑥)∩𝑋≠∅}.SH_𝒞 and XH_𝒞 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, if 𝑀 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.

Let 𝐸 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 covering 𝒞 with cl_𝑀(𝒞)(∅). In fact, ∅∈ℒ(𝑀(ℱ)) if and only if ℱ is a covering.

Theorem 3.1. Let 𝐸 be a set and 𝑀(ℱ) a transversal matroid induced by a family of ℱ={𝐹₁,𝐹₂,…,𝐹_𝑚} subsets of 𝐸, where 𝐹_𝑖≠∅(1≤𝑖≤𝑚). cl_ℱ(∅)=∅ 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 cl_𝑀(ℱ)(∅)=∅.
“⇒”: Since cl_𝑀(ℱ)(∅)=∅, any single-point set is an independent set, that is, for all 𝑥∈𝐸, there exists 𝑖_𝑥∈{1,2,…,𝑚} such that 𝑥∈𝐹_𝑖𝑥⊆𝐸. Hence, 𝐸=⋃_𝑥∈𝐸{𝑥}⊆⋃_𝑥∈𝐸𝐹_𝑖𝑥⊆⋃^𝑚_𝑖=1𝐹_𝑖⊆𝐸. Thus ⋃^𝑚_𝑖=1𝐹_𝑖=𝐸. For all 𝑖∈{1,2,…,𝑚}, 𝐹_𝑖≠∅ and ⋃^𝑚_𝑖=1𝐹_𝑖=𝐸, hence ℱ is a covering.

Theorem 3.1 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. Let 𝒞 be a covering of 𝐸. For all 𝑥∈𝐸, cl_𝑀(𝒞)({𝑥}) is an atom of ℒ(𝑀(𝒞)).

Proof. Since 𝒞 is a covering, we know any single-point set is an independent set. Thus for all 𝑥∈𝐸, 𝑟_𝑀(𝒞)(cl_𝑀(𝒞)({𝑥}))=𝑟_𝑀(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 cl_𝑀(𝒞)({𝑥}) is a closed set, we know cl_𝑀(𝒞)({𝑥}) is an atom of ℒ(𝑀(𝒞)).

Lemma 3.2 does not establish the concrete form of cl_𝑀(𝒞)({𝑥}). In order to solve that problem, we first define two sets as follows.

Definition 3.3. Let 𝒞={𝐾₁,𝐾₂,…,𝐾_𝑚} be a covering of a finite set 𝐸={𝑥₁,𝑥₂,…,𝑥_𝑛}. We define the following.(i)𝐴={𝐾_𝑖−⋃^𝑚_{𝑗=1,𝑗≠𝑖}𝐾_𝑗∶𝐾_𝑖−⋃^𝑚_{𝑗=1,𝑗≠𝑖}𝐾_𝑗≠∅,𝑖∈{1,2,…,𝑚}}={𝐴₁,𝐴₂,…,𝐴_𝑠}.(ii)𝐵=𝐸−⋃^𝑠_𝑖=1𝐴_𝑖.

Remark 3.4. For all 𝑖∈{1,2,…,𝑠} and 𝑥∈𝐴_𝑖, 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 of 𝐴 and 𝐵.

Proposition 3.5. Let 𝒞 be a covering of 𝐸. {𝐴₁,𝐴₂,…,𝐴_𝑠}∪{{𝑥}∶𝑥∈𝐵} forms a partition of 𝐸.

Proof. Let 𝑃={𝐴₁,𝐴₂,…,𝐴_𝑠}∪{{𝑥}∶𝑥∈𝐵}. According to Definition 3.3, we know (⋃^𝑠_𝑖=1𝐴_𝑖)∪(∪{{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 𝑃_𝑖 and 𝑃_𝑗 are different single-points. If 𝑃₁∈𝐴 and 𝑃₂∈{{𝑥}∶𝑥∈𝐵}, then 𝑃₁∩𝑃₂=∅ because 𝐵∩(⋃^𝑠_𝑘=1𝐴_𝑘)=∅, 𝑃₁⊆⋃^𝑠_𝑘=1𝐴_𝑘 and 𝑃₂⊆𝐵.

Proposition 3.6. 𝒞 is a partition of 𝐸 if and only if 𝐵=∅.

Proof. According to the definition of 𝐴 and 𝐵, the necessity is obvious. Now we prove the sufficiency. If 𝒞 is not a partition, then there exist 𝐾_𝑖,𝐾_𝑗∈𝒞 such that 𝐾_𝑖∩𝐾_𝑗≠∅. Thus there exists 𝑥∈𝐸 such that 𝑥∈𝐾_𝑖∩𝐾_𝑗, that is, there exist at least 𝐾_𝑖,𝐾_𝑗∈𝒞 such that 𝑥 belongs to them, hence 𝑥∈𝐵. That contradicts the assumption that 𝐵=∅.

Based on Lemma 3.2 and Definition 3.3, we can establish the concrete form of the atoms of lattice ℒ(𝑀(𝒞)).

Theorem 3.7. Let 𝒞 be a covering of 𝐸. {𝐴₁,𝐴₂,…,𝐴_𝑠}∪{{𝑥}∶𝑥∈𝐵} is the set of atoms of lattice ℒ(𝑀(𝒞)).

Proof. According to the definition of 𝐴_𝑖, we may as well suppose 𝐴_𝑖=𝐾_ℎ−⋃^𝑚_{𝑗=1,𝑗≠ℎ}𝐾_𝑗. 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∈𝐴_𝑖, {𝑥} is an independent set. For all 𝑦∈𝐴_𝑖 and 𝑦≠𝑥, we know 𝑥,𝑦∈𝐾_ℎ and 𝑥,𝑦∉𝐾_𝑗 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 𝐴_𝑖, that is, 𝑟_𝑀(𝒞)(𝐴_𝑖)=1. Next, we need to prove 𝐴_𝑖 is a closed set. Since 𝐴_𝑖⊆cl_𝑀(𝒞)(𝐴_𝑖), we need to prove cl_𝑀(𝒞)(𝐴_𝑖)⊆𝐴_𝑖, that is, 𝑦∉𝐴_𝑖 implies 𝑦∉cl_𝑀(𝒞)(𝐴_𝑖). If 𝑦∉𝐴_𝑖, based on the fact that 𝒞 is a covering and the definition of 𝐴_𝑖, then there exists 𝑗≠ℎ such that 𝑦∈𝐾_𝑗. Thus {𝑥,𝑦}(𝑥∈𝐴_𝑖) is an independent set. That implies 𝑦∉cl_𝑀(𝒞)(𝐴_𝑖), thus cl_𝑀(𝒞)(𝐴_𝑖)=𝐴_𝑖. Hence, 𝐴_𝑖∈ℒ(𝑀(𝒞)). Combining with 𝑟_𝑀(𝒞)(𝐴_𝑖)=1, we know for all 1≤𝑖≤𝑚, 𝐴_𝑖 is an atom of lattice ℒ(𝑀(𝒞)).
According to the definition of transversal matroid and the fact that 𝒞 is a covering, any single-point set is an independent set. Thus for all 𝑥∈𝐵, 𝑟_𝑀(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∈𝐾_𝑘,𝐾_𝑡 and 𝑥∈𝐾_𝑙,𝐾_𝑝, where {𝐾_𝑘,𝐾_𝑡} may be the same as {𝐾_𝑙,𝐾_𝑝}. Based on this, {𝑥,𝑦} is an independent set. This implies 𝑦∉cl_𝑀(𝒞)({𝑥}). If 𝑦∉𝐵, then we may as well suppose 𝑦∈𝐴_𝑖, thus 𝑦∈𝐾_ℎ for the definition of 𝐴_𝑖, where 𝐾_ℎ may be the same with 𝐾_𝑙 or 𝐾_𝑝. 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∉cl_𝑀(𝒞)({𝑥}). From above discussion, we have cl_𝑀(𝒞)({𝑥})={𝑥}. Hence, {𝑥}∈ℒ(𝑀(𝒞)) for all 𝑥∈𝐵. Combining with 𝑟_𝑀(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 lattice ℒ(𝑀(𝒞)) cannot be anything but {𝐴₁,𝐴₂,…,𝐴_𝑠}∪{{𝑥}∶𝑥∈𝐵}. According to Lemma 3.2, we know {cl_𝑀(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 cl_𝑀(𝒞)({𝑥})={𝑥}. If 𝑥∉𝐵, then 𝑥 belongs to one of elements in 𝐴. We may as well suppose 𝑥∈𝐴_𝑖. Combining 𝐴_𝑖 is an atom with ∅⊆cl_𝑀(𝒞)({𝑥})⊆cl_𝑀(𝒞)(𝐴_𝑖)=𝐴_𝑖, we have cl_𝑀(𝒞)({𝑥})=𝐴_𝑖. Hence, {𝐴₁,𝐴₂,…,𝐴_𝑠}∪{{𝑥}∶𝑥∈𝐵} is the set of atoms of lattice ℒ(𝑀(𝒞)).

The proposition below connects simple matroid and the cardinal number of 𝐴_𝑖. In fact, a matroid is simple if and only if |𝐴_𝑖|=1 for all 𝐴_𝑖∈𝐴.

Lemma 3.8. Let 𝒞 be a covering of 𝐸. For all 𝐴_𝑖∈𝐴, if |𝐴_𝑖|≥2, then 𝑥 and 𝑦 are parallel in 𝑀(𝒞) for all 𝑥,𝑦∈𝐴_𝑖.

Proof. According to the definition of 𝐴_𝑖, we may as well suppose 𝐴_𝑖=𝐾_ℎ−⋃^𝑚_{𝑗=1,𝑗≠ℎ}𝐾_𝑗, where 1≤ℎ≤𝑚. For all 𝑥,𝑦∈𝐴_𝑖, then 𝑥,𝑦∈𝐾_ℎ, and for all 𝑗∈{1,2,…,𝑚} and 𝑗≠ℎ, we have 𝑥∉𝐾_𝑗 and 𝑦∉𝐾_𝑗. 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. Let 𝒞 be a covering of 𝐸. 𝑀(𝒞) is a simple matroid if and only if |𝐴_𝑖|=1 for all 𝐴_𝑖∈𝐴.

Proof. “⇒”: Since 𝑀(𝒞) is a simple matroid, it does not contain parallel elements. If there exists 𝐴_𝑖∈𝐴 such that |𝐴_𝑖|≠1, then |𝐴_𝑖|≥2 because 𝐴_𝑖≠∅. According to Lemma 3.8, we know for all 𝑥,𝑦∈𝐴_𝑖, 𝑥,𝑦 are parallel which contradicts the assumption that 𝑀(𝒞) is a simple matroid. Hence, for all 𝐴_𝑖∈𝐴, |𝐴_𝑖|=1.
“⇐”: According to the definition of parallel element, if |𝐴_𝑖|=1 for all 𝐴_𝑖∈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 𝐴_𝑖∈𝐴 such that 𝑥,𝑦∈𝐴_𝑖, that is, |𝐴_𝑖|≥2. This contradicts the fact that |𝐴_𝑖|=1 for all 𝐴_𝑖∈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. Let 𝒫={𝑃₁,𝑃₂,…,𝑃_𝑚} be a partition of 𝐸. 𝒫 is the set of atoms of lattice ℒ(𝑀_𝒫).

Corollary 3.11. Let 𝒫={𝑃₁,𝑃₂,…,𝑃_𝑚} be a partition of 𝐸. 𝑀(𝒫) is a simple matroid if and only if |𝑃_𝑖|=1 for all 𝑃_𝑖∈𝑃.

For a geometric lattice ℒ(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 cl_𝑀(𝒞)({𝑥,𝑦}) covers certain atoms of lattice ℒ(𝑀(𝒞)).

Proposition 3.12. Let 𝒞 be a covering of 𝐸. For all 𝑥,𝑦∈𝐸, cl_𝑀(𝒞)({𝑥,𝑦}) covers cl_𝑀(𝒞)({𝑥}) if and only if there does not exist 𝐴_𝑖∈𝐴 such that 𝑥,𝑦∈𝐴_𝑖.

Proof. “⇐”: For all 𝑥,𝑦∈𝐸, {𝑥}⊆{𝑥,𝑦}, then cl_𝑀(𝒞)({𝑥})⊆cl_𝑀(𝒞)({𝑥,𝑦}) and 1=𝑟_𝑀(𝒞)(cl_𝑀(𝒞)({𝑥}))≤𝑟_𝑀(𝒞)(cl_𝑀(𝒞)({𝑥,𝑦}))=𝑟_𝑀(𝒞)({𝑥,𝑦})≤|{𝑥,𝑦}|=2. Now we need to prove 𝑟_𝑀(𝒞)(cl_𝑀(𝒞)({𝑥,𝑦}))=2. If 𝑟_𝑀(𝒞)(cl_𝑀(𝒞)({𝑥,𝑦}))=𝑟_𝑀(𝒞)({𝑥,𝑦})=1=𝑟_𝑀(𝒞)({𝑥}), then {𝑥,𝑦}∉ℐ(𝒞), that is, there is only one block contains 𝑥,𝑦. It means that there exists 𝐴_𝑖∈𝐴 such that 𝑥,𝑦∈𝐴_𝑖. That contradicts the hypothesis. Hence, 𝑟_𝑀(𝒞)(cl_𝑀(𝒞)({𝑥,𝑦}))=2, that is, cl_𝑀(𝒞)({𝑥,𝑦}) covers cl_𝑀(𝒞)({𝑥}).
“⇒”: For all 𝑥,𝑦∈𝐸, if there exists 𝐴_𝑖∈𝐴 such that 𝑥,𝑦∈𝐴_𝑖, then there is only one block contains 𝑥,𝑦, thus 𝑥,𝑦∉ℐ(𝒞), hence {𝑥,𝑦}⊆cl_𝑀(𝒞)({𝑥}). That implies cl_𝑀(𝒞)({𝑥,𝑦})⊆cl_𝑀(𝒞)({𝑥}) which contradicts the assumption that cl_𝑀(𝒞)({𝑥,𝑦}) covers cl_𝑀(𝒞)({𝑥}).

The modular element and the modular pair are core concepts in lattice. As we know, if 𝑎 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. Let 𝑀 be a matroid and ℒ(𝑀) the set of all closed sets of 𝑀.(1)For all 𝑋,𝑌∈ℒ(𝑀), (𝑋,𝑌) is a modular pair of ℒ(𝑀) if and only if 𝑟_𝑀(𝑋∪𝑌)+𝑟_𝑀(𝑋∩𝑌)=𝑟_𝑀(𝑋)+𝑟_𝑀(𝑌).(2)For all 𝑋∈ℒ(𝑀), 𝑋 is a modular element of ℒ(𝑀) if and only if 𝑟_𝑀(𝑋∪𝑌)+𝑟_𝑀(𝑋∩𝑌)=𝑟_𝑀(𝑋)+𝑟_𝑀(𝑌), for all 𝑌∈ℒ(𝑀).

Proof. (1) According to Lemmas 2.8 and 2.10, we know (𝑋,𝑌) is a modular pair of ℒ(𝑀) if and only if 𝑟_𝑀(𝑋)+𝑟_𝑀(𝑌)=𝑟_𝑀(𝑋∨𝑌)+𝑟_𝑀(𝑋∧𝑌)=𝑟_𝑀(cl_𝑀(𝑋∪𝑌))+𝑟_𝑀(𝑋∩𝑌)=𝑟_𝑀(𝑋∪𝑌)+𝑟_𝑀(𝑋∩𝑌).
(2) It comes from the definition of modular element and (1).

Let {𝐴_𝑖∶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. Let 𝒞 be a covering of 𝐸. For all 𝑖,𝑗∈Γ.(1)(𝐴_𝑖,𝐴_𝑗) is a modular pair of ℒ(𝑀(𝒞)).(2)𝐴_𝑖 is a modular element of ℒ(𝑀(𝒞)).

Proof. (1) Since 𝒞 is a covering, cl_𝑀(𝒞)(∅)=∅. 𝐴_𝑖 and 𝐴_𝑗 are atoms, so 𝐴_𝑖∩𝐴_𝑗=∅. According to Theorem 3.13, we need to prove 𝑟_𝑀(𝒞)(𝐴_𝑖∪𝐴_𝑗)+𝑟_𝑀(𝒞)(𝐴_𝑖∩𝐴_𝑗)=𝑟_𝑀(𝒞)(𝐴_𝑖)+𝑟_𝑀(𝒞)(𝐴_𝑗), that is, 𝑟_𝑀(𝒞)(𝐴_𝑖∪𝐴_𝑗)=2. According to the submodular inequality of 𝑟_𝑀(𝒞), we have 𝑟_𝑀(𝒞)(𝐴_𝑖∪𝐴_𝑗)+𝑟_𝑀(𝒞)(𝐴_𝑖∩𝐴_𝑗)≤𝑟_𝑀(𝒞)(𝐴_𝑖)+𝑟_𝑀(𝒞)(𝐴_𝑗), that is, 1≤𝑟_𝑀(𝒞)(𝐴_𝑖∪𝐴_𝑗)≤2. If 𝑟_𝑀(𝒞)(𝐴_𝑖∪𝐴_𝑗)=1=𝑟_𝑀(𝒞)(𝐴_𝑖), then 𝐴_𝑗⊆cl_𝑀(𝒞)(𝐴_𝑖)=𝐴_𝑖 which contradicts that 𝐴_𝑖∩𝐴_𝑗=∅.
(2) 𝐴_𝑖 is a modular element of ℒ(𝑀(𝒞)) if and only if 𝑟_𝑀(𝒞)(𝐴_𝑖∪𝐴)+𝑟_𝑀(𝒞)(𝐴_𝑖∩𝐴)=𝑟_𝑀(𝒞)(𝐴_𝑖)+𝑟_𝑀(𝒞)(𝐴) for all 𝐴∈ℒ(𝑀(𝒞)).
Case  1. If 𝐴_𝑖 and 𝐴 are comparable, that is, 𝐴_𝑖⊆𝐴, then 𝑟_𝑀(𝒞)(𝐴_𝑖∪𝐴)+𝑟_𝑀(𝒞)(𝐴_𝑖∩𝐴)=𝑟_𝑀(𝒞)(𝐴_𝑖)+𝑟_𝑀(𝒞)(𝐴).
Case  2. If 𝐴_𝑖 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∩𝐴_𝑖=∅. Hence, 𝑟_𝑀(𝒞)(𝐴)≤𝑟_𝑀(𝒞)(𝐴∪𝐴_𝑖)≤𝑟_𝑀(𝒞)(𝐴)+1. If 𝑟_𝑀(𝒞)(𝐴∪𝐴_𝑖)=𝑟_𝑀(𝒞)(𝐴), then 𝐴_𝑖⊆cl_𝑀(𝒞)(𝐴)=𝐴 which contradicts that 𝐴_𝑖∩𝐴=∅. Hence, 𝑟_𝑀(𝒞)(𝐴∪𝐴_𝑖)=𝑟_𝑀(𝒞)(𝐴)+1.
In a word, for all 𝐴∈ℒ(𝑀(𝒞)), 𝑟_𝑀(𝒞)(𝐴_𝑖∪𝐴)+𝑟_𝑀(𝒞)(𝐴_𝑖∩𝐴)=𝑟_𝑀(𝒞)(𝐴_𝑖)+𝑟_𝑀(𝒞)(𝐴), that is, 𝐴_𝑖 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. Let 𝒫 be a partition of 𝐸. For all 𝑃_𝑖,𝑃_𝑗∈𝒫:(1)(𝑃_𝑖,𝑃_𝑗) is a modular pair of ℒ(𝑀(𝒫)).(2)𝑃_𝑖 is a modular element of ℒ(𝑀(𝒫)).

The following lemma shows how to induce a matroid by a lattice. In fact, if a function 𝑓 on a lattice is nonnegative, integer-valued, submodular and 𝑓(∅)=0, then it can determine a matroid.

Lemma 3.16 (see [19]). Let ℒ_𝐸 be a lattice of subsets of a set 𝐸 such that ℒ_𝐸 is closed under intersection, and contains ∅ and 𝐸. Suppose that 𝑓 is a nonnegative, integer-valued, submodular function on ℒ_𝐸 for which 𝑓(∅)=0. Let ℐ(ℒ_𝐸,𝑓)={𝑋⊆𝐸∶𝑓(𝑋)≥|𝑋∩𝑇|,forall𝑇∈ℒ_𝐸}. ℐ(ℒ_𝐸,𝑓) is the collection of independent sets of a matroid on 𝐸.

According to the definition of ℒ(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 𝑟_𝑀(𝒞)(∅)=0. Similar to Lemma 3.16, we can obtain the following theorem.

Theorem 3.17. Let 𝒞 be a covering of 𝐸. We define ℐ(ℒ(𝑀(𝒞)),𝑟_𝑀(𝒞))={𝑋⊆𝐸∶𝑟_𝑀(𝒞)(𝑌)≥|𝑋∩𝑌|,  for  all  𝑌∈ℒ(𝑀(𝒞))}, then 𝑀(𝐸,ℐ(ℒ(𝑀(𝒞),𝑟_𝑀(𝒞))) is a matroid.

For any given matroid 𝑀, we know that for all 𝑋⊆𝐸, 𝑋 is an independent set of 𝑀 if and only if 𝑟_𝑀(𝑋)=|𝑋|. Moreover, based on the properties of rank function, we have 𝑟_𝑀(𝑋)≤|𝑋|. Hence, 𝑋 is an independent set of 𝑀 if and only if 𝑟_𝑀(𝑋)≥|𝑋| for all 𝑋⊆𝐸.

Lemma 3.18. Let 𝑀 be a matroid. 𝑋 is an independent set of 𝑀 if and only if for all closed set 𝑌 of 𝑀, 𝑟_𝑀(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 𝑟_𝑀(𝑌)≥𝑟_𝑀(𝑋∩𝑌)≥|𝑋∩𝑌|.
“⇐”: For all closed set 𝑌, 𝑟_𝑀(𝑌)≥|𝑋∩𝑌|. Especially, for 𝑌=cl_𝑀(𝑋), we have 𝑟_𝑀(𝑋)=𝑟_𝑀(cl_𝑀(𝑋))=𝑟_𝑀(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 denote 𝑟_{ℒ(𝑀(𝒞))} as the rank function of 𝑀(𝐸,ℐ(ℒ(𝑀(𝒞),𝑟_𝑀(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. Let 𝒞 be a covering of 𝐸. For all 𝑋⊆𝐸, ℐ(ℒ(𝑀(𝒞),𝑟_𝑀(𝒞))=ℐ(𝑀(𝒞)) and 𝑟_{ℒ(𝑀(𝒞))}(𝑋)=𝑟_𝑀(𝒞)(𝑋).

Proof. According to Lemma 3.18, we know that ℐ(ℒ(𝑀(𝒞),𝑟_𝑀(𝒞))={𝑋⊆𝐸∶𝑟_𝑀(𝒞)(𝑌)≥|𝑋∩𝑌|,forall𝑌∈ℒ(𝑀(𝒞))} and ℐ(𝒞)={𝑋⊆𝐸∶𝑟_𝑀(𝒞)(𝑋)=|𝑋|} are equivalent, that is, 𝑀(𝐸,ℐ(ℒ(𝑀(𝒞),𝑟_𝑀(𝒞))) and 𝑀(𝐸,ℐ(𝒞)) are equivalent. So does 𝑟_{ℒ(𝑀(𝒞))}(𝑋)=𝑟_𝑀(𝒞)(𝑋) for all 𝑋⊆𝐸.

When a covering degrades into a partition, we can obtain a matroid 𝑀(𝒫)=(𝐸,ℐ(𝑃)), where ℐ(𝒫)={𝑋⊆𝐸∶forall𝑃_𝑖∈𝒫,|𝑋∩𝑃_𝑖|≤1} and 𝑟_𝑀(𝒫)(𝑋)=max{|𝐼|∶𝐼⊆𝑋,𝐼∈ℐ(𝒫)}=|{𝑃_𝑖∈𝒫∶𝑃_𝑖∩𝑋≠∅}| for all 𝑋⊆𝐸. As we know, for all 𝑋,𝑌∈ℒ(𝑀), 𝑋∨𝑌=cl_𝑀(𝑋∪𝑌). If the matroid is 𝑀(𝒫), then 𝑋∨𝑌=𝑋∪𝑌.

Lemma 3.20 (see [25]). If 𝒫 is a partition of 𝐸 and 𝑀 is the matroid, then cl_𝑀=𝑅^∗ for all 𝑋⊆𝐸.

Lemma 3.21. Let 𝒫 be a partition of 𝐸. For all 𝑋,𝑌∈ℒ(𝑀(𝒫)), 𝑋∨𝑌=𝑋∪𝑌.

Proof. 𝑋∨𝑌=cl_𝑀(𝒫)(𝑋∪𝑌)=𝑅^∗(𝑋∪𝑌)=𝑅^∗(𝑋)∪𝑅^∗(𝑌)=cl_𝑀(𝒫)(𝑋)∪cl_𝑀(𝒫)(𝑌)=𝑋∪𝑌.

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

Proposition 3.22. Let 𝒫 be a partition of 𝐸. For all 𝑋⊆𝐸, ℐ(ℒ(𝑀(𝒫)),𝑟_𝑀(𝒫))=ℐ(𝑀(𝒫)) and 𝑟_{ℒ(𝑀(𝒫))}(𝑋)=𝑟_𝑀(𝒫)(𝑋).

Proof. We need to prove only ℐ(ℒ(𝑀(𝒫)),𝑟_𝑀(𝒫))=ℐ(𝑀(𝒫)). For all 𝑋∈ℐ(ℒ(𝑀(𝒫)),𝑟_𝑀(𝒫)), then 𝑟_𝑀(𝒫)(𝑌)≥|𝑋∩𝑌| for all 𝑌∈ℒ(𝑀(𝒫)). Since 𝑃_𝑖∈ℒ(𝑀(𝒫)), |𝑋∩𝑃_𝑖|≤𝑟_𝑀(𝒫)(𝑃_𝑖)=1. Thus 𝑋∈ℐ(𝑀(𝒫)). Hence, ℐ(ℒ(𝑀(𝒫),𝑟_𝑀(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∈𝐾𝑃_𝑖=⋃_𝑖∈𝐾𝑃_𝑖. Thus 𝑟_𝑀(𝒫)(𝑌)=|𝐾| and |𝑋∩𝑌|=|𝑋∩(⋃_𝑖∈𝐾𝑃_𝑖)|=|⋃_𝑖∈𝐾(𝑋∩𝑃_𝑖)|=∑_𝑖∈𝐾|𝑋∩𝑃_𝑖|. For all 𝑋∈ℐ(𝑀(𝒫)), |𝑋∩𝑃_𝑖|≤1. Hence, |𝑋∩𝑌|=∑_𝑖∈𝐾|𝑋∩𝑃_𝑖|≤|𝐾|=𝑟_𝑀(𝒫)(𝑌), that is, ℐ(𝑀(𝒫))⊆ℐ(ℒ(𝑀(𝒫)),𝑟_𝑀(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 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 𝒞 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 all 𝑥,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 of SH is not valid, so what is the condition that guarantees it holds for SH? We have the following conclusion.

Proposition 4.2. Let 𝒞 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 for SH to be a closure operator.

Theorem 4.3. Let 𝒞 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 Propositions 4.1 and 4.2 and (2), (5), and (6) of Proposition 2.3.

For a given covering 𝒞 of 𝐸, we may as well suppose the set of indiscernible neighborhoods of 𝐸 as {𝐼(𝑥)∶𝑥∈𝐸}={𝐼(𝑥₁),𝐼(𝑥₂),…,𝐼(𝑥_𝑠)} where 𝑥₁,𝑥₂,…,𝑥_𝑠∈𝐸.

Definition 4.4. Let 𝒞 be a covering of 𝐸. We define ℐ^′={𝐼⊆𝐸∶|𝐼∩𝐼(𝑥_𝑖)|≤1,forall𝑖∈{1,2,…,𝑠}}.

As we know, if {𝐼(𝑥)∶𝑥∈𝐸}={𝐼(𝑥₁),𝐼(𝑥₂),…,𝐼(𝑥_𝑠)} 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: ℐ_SH(𝒞)={𝐼⊆𝐸∶∀𝑥∈𝐼,𝑥∉SH(𝐼−{𝑥})}. (4.1) The following proposition shows 𝑀(𝐸,ℐ_SH)=𝑀(𝐸,ℐ^′) under the condition that {𝐼(𝑥)∶𝑥∈𝐸} forms a partition of 𝐸.

Proposition 4.5. Let 𝒞 be a covering of 𝐸. If {𝐼(𝑥)∶𝑥∈𝐸} induced by 𝒞 forms a partition of 𝐸, then 𝑀(𝐸,ℐ_SH(𝒞)) is a matroid and ℐ_SH(𝒞)=ℐ^′.

Proof. Let ℐ_cl={𝐼⊆𝐸∶forall𝑥∈𝐼,𝑥∉cl(𝐼−{𝑥})}. we know that if an operator cl satisfies (1)–(4) of Proposition 2.3, 𝑀(𝐸,ℐ_cl) is a matroid. {𝐼(𝑥)∶𝑥∈𝑈} induced by 𝒞 forms a partition, hence, 𝑀(𝐸,ℐ_SH(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∈ℐ_SH(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(𝑥_𝑖)|≥2, then we may as well suppose there exist 𝑢,𝑣 such that 𝑢,𝑣∈𝐼(𝑥_𝑖) and 𝑢,𝑣∈𝐼. Since 𝑢∈𝐼(𝑢),𝑣∈𝐼(𝑣) and {𝐼(𝑥)∶𝑥∈𝐸} forms a partition, 𝐼(𝑢)=𝐼(𝑣)=𝐼(𝑥_𝑖). 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, ℐ_SH(𝒞)⊆ℐ^′. On the other hand, if 𝐼∉ℐ_SH(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^′⊆ℐ_SH(𝒞).

We denote the rank function of 𝑀(𝐸,ℐ_SH(𝒞)) by 𝑟_SH. Then some properties of 𝑀(𝐸,ℐ_SH(𝒞)) are established in the following proposition.

Proposition 4.6. Let 𝒞 be a covering of 𝐸. If {𝐼(𝑥)∶𝑥∈𝐸}={𝐼(𝑥₁),𝐼(𝑥₂),…,𝐼(𝑥_𝑠)} induced by 𝒞 forms a partition of 𝐸, then(1)𝑋 is a base of 𝑀(𝐸,ℐ_SH(𝒞)) if and only if |𝑋∩𝐼(𝑥_𝑖)|=1 for all 𝑖∈{1,2,…,𝑠}. Moreover, 𝑀(𝐸,ℐ_SH(𝒞)) has |𝐼(𝑥₁)||𝐼(𝑥₂)|⋯|𝐼(𝑥_𝑠)| bases.(2)For all 𝑋⊆𝐸, 𝑟_SH(𝑋)=|{𝐼(𝑥_𝑖)∶𝐼(𝑥_𝑖)∩𝑋≠∅,𝑖=1,2,…,𝑠}|.(3)𝑋 is a dependent set of 𝑀(𝐸,ℐ_SH(𝒞)) if and only if there exists 𝐼(𝑥_𝑖)∈{𝐼(𝑥)∶𝑥∈𝐸} such that |𝐼(𝑥_𝑖)∩𝑋|>1.(4)𝑋 is a circuit of 𝑀(𝐸,ℐ_SH(𝒞)) if and only if there exists 𝐼(𝑥_𝑖)∈{𝐼(𝑥)∶𝑥∈𝐸} such that 𝑋⊆𝐼(𝑥_𝑖) and |𝑋|=2.

Proof. (1) According to the definition of base of a matroid, we know that 𝑋 is a base of 𝑀(𝐸,ℐ_SH(𝒞))⇔𝑋 is a maximal independent set of 𝑀(𝐸,ℐ_SH(𝒞))⇔|𝑋∩𝐼(𝑥_𝑖)|=1 for all 𝑖∈{1,2,…,𝑠} because ℐ_SH(𝒞)=ℐ^′. Since 𝑋 is a base of 𝑀(𝐸,ℐ_SH(𝒞)) and 𝐼(𝑥₁),𝐼(𝑥₂),…,𝐼(𝑥_𝑠) are different, 𝑀(𝐸,ℐ_SH(𝒞)) has |𝐼(𝑥₁)||𝐼(𝑥₂)|⋯|𝐼(𝑥_𝑠)| bases.
(2) According to the definition of rank function, we know 𝑟_SH(𝑋)=|𝐵_𝑋|=|{𝐼(𝑥_𝑖)∶|𝐵_𝑋∩𝐼(𝑥_𝑖)|=1}|≤|{𝐼(𝑥_𝑖)∶𝑋∩𝐼(𝑥_𝑖)≠∅}|, where 𝐵_𝑋 is a maximal independent set included in 𝑋. Now we just need to prove the inequality |{𝐼(𝑥_𝑖)∶|𝐵_𝑋∩𝐼(𝑥_𝑖)|=1}|<|{𝐼(𝑥_𝑖)∶𝑋∩𝐼(𝑥_𝑖)≠∅}| does not hold; otherwise, there exists 1≤𝑖≤𝑠 such that 𝐼(𝑥_𝑖)∩𝑋≠∅ and 𝐼(𝑥_𝑖)∩𝐵_𝑋=∅ because 𝐵_𝑋∈ℐ_SH(𝒞). Thus there exists 𝑒_𝑖∈𝐼(𝑥_𝑖)∩𝑋 such that 𝐵_𝑋∪{𝑒_𝑖}⊆𝑋 and 𝐵_𝑋∪{𝑒_𝑖}∈ℐ_SH(𝒞). That contradicts the assumption that 𝐵_𝑋 is a maximal independent set included in 𝑋. Hence, 𝑟_SH(𝑋)=|{𝐼(𝑥_𝑖)∶𝐼(𝑥_𝑖)∩𝑋≠∅,𝑖=1,2,…,𝑠}|.
(3) According to the definition of dependent set, we know that 𝑋 is a dependent set ⇔𝑋∉ℐ_SH(𝒞)⇔ there exists 𝑖∈{1,2,…,𝑠} such that |𝑋∩𝐼(𝑥_𝑖)|>1.
(4) “⇒”: As we know, a circuit is a minimal dependent set. 𝑋 is a circuit of 𝑀(𝐸,ℐ_SH(𝒞)), then there exists 𝑖∈{1,2,…,𝑠} such that |𝐼(𝑥_𝑖)∩𝑋|=2. Now we just need to prove |𝑋|=2; otherwise, we may as well suppose 𝑋={𝑥,𝑦,𝑧} where 𝑥,𝑦∈𝐼(𝑥_𝑖)∩𝑋. Thus we can obtain |(𝑋−{𝑧})∩𝐼(𝑥_𝑖)|=2, that is, 𝑋−{𝑧}∉ℐ_SH(𝒞). That contradicts the minimality of circuit. Combining |𝑋|=2 with |𝐼(𝑥_𝑖)∩𝑋|=2, we have 𝑋⊆𝐼(𝑥_𝑖).
“⇐”: Since |𝑋|=2, we may as well suppose 𝑋={𝑥,𝑦}, and |𝑋∩𝐼(𝑥_𝑖)|=2 because there exists 𝐼(𝑥_𝑖)∈{𝐼(𝑥)∶𝑥∈𝐸} such that 𝑋⊆𝐼(𝑥_𝑖), thus 𝑋 is a dependent set. For all 𝑗∈{1,2,…,𝑠}, |{𝑥}∩𝐼(𝑥_𝑗)|≤1 and |{𝑦}∩𝐼(𝑥_𝑗)|≤1 which implies {𝑥} and {𝑦} are independent sets, hence 𝑋 is a circuit of 𝑀(𝐸,ℐ_SH(𝒞)).

For a covering 𝒞 of 𝐸, we denote ℒ_SH(𝑀(𝒞)) as the set of all closed sets of 𝑀(𝐸,ℐ_SH(𝒞)). When {𝐼(𝑥)∶𝑥∈𝐸}={𝐼(𝑥₁),𝐼(𝑥₂),…,𝐼(𝑥_𝑠)} forms a partition of 𝐸, then for all 𝑋,𝑌∈ℒ_SH(𝑀(𝒞)), 𝑋∧𝑌=𝑋∩𝑌, 𝑋∨𝑌=SH(𝑋∪𝑌)=SH(𝑋)∪SH(𝑌)=𝑋∪𝑌 and SH(∅)=∅.

Proposition 4.7. Let 𝒞 be a covering of 𝐸. If {𝐼(𝑥)∶𝑥∈𝐸}={𝐼(𝑥₁),𝐼(𝑥₂),…,𝐼(𝑥_𝑠)} forms a partition of 𝐸, then(1)𝐼(𝑥₁),𝐼(𝑥₂),…,𝐼(𝑥_𝑠) are all atoms of ℒ(𝑀_SH).(2)For all 𝑥,𝑦∈𝐸 and 𝑥≠𝑦, there does not exist 𝐼(𝑧)∈{𝐼(𝑥₁),𝐼(𝑥₂),…,𝐼(𝑥_𝑠)} such that 𝑥,𝑦∈𝐼(𝑧) if and only if SH({𝑥,𝑦}) covers SH({𝑥}) or SH({𝑦}).(3)For all 𝑖,𝑗∈{1,2,…,𝑠},(𝐼(𝑥_𝑖),𝐼(𝑥_𝑗)) is a modular pair of ℒ_SH(𝑀(𝒞)).(4)For all 𝑖∈{1,2,…,𝑠},𝐼(𝑥_𝑖) is a modular element of ℒ_SH(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 Theorem 4.3, 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. Let 𝒞 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. If 𝑥∉𝐾, then 𝐼_𝒞(𝑥)=𝐼_{𝒞−{𝐾}}(𝑥). If 𝑥∈𝐾, then 𝐼_𝒞(𝑥)=⋃_{𝑥∈𝐾′}𝐾^′∪𝐾. Since 𝐾 is an immured element, there exists 𝑥∈𝐾^′ such that 𝐾⊆𝐾^′. Thus 𝐼_𝒞(𝑥)=⋃_{𝑥∈𝐾′}𝐾^′∪𝐾=⋃_{𝑥∈𝐾′}𝐾^′=𝐼_{𝒞−{𝐾}}(𝑥). Hence, 𝐼(𝑥) is the same in 𝒞 as in 𝒞−{𝐾}.

Proposition 4.9. Let 𝒞 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. Since 𝑒𝑥𝑐𝑙𝑢𝑠𝑖𝑜𝑛(𝒞) is a partition of 𝐸, {𝐼(𝑥)∶𝑥∈𝐸} induced by 𝑒𝑥𝑐𝑙𝑢𝑠𝑖𝑜𝑛(𝒞) forms a partition. Suppose {𝐾₁,𝐾₂,…,𝐾_𝑠} 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. Let 𝒞 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 for SH to be a closure operator of a matroid from the viewpoint of coverings.

Theorem 4.11. Let 𝒞 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 Theorem 4.3 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]). Let 𝒞 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 all 𝑋⊆𝑌⊆𝐸⇒XH(𝑋)⊆XH(𝑌).

From the above proposition, we find that XH 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. Let 𝒞 be a covering of 𝐸. For all 𝑥,𝑦∈𝐸 and 𝑋⊆𝐸, XH satisfies 𝑦∈XH𝑋{𝑥}−XH(𝑋)⇒𝑥∈XH𝑋𝑦 (4.2) 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 for XH to be a closure operator of a matroid.

Theorem 4.14. Let 𝒞 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 covering 𝒞 of 𝐸, we may as well suppose the set of all neighborhoods as {𝑁(𝑥)∶𝑥∈𝐸}={𝑁(𝑥₁),𝑁(𝑥₂),…,𝑁(𝑥_𝑡)} where 𝑥₁,𝑥₂,…,𝑥_𝑡∈𝐸.

Definition 4.15. Let 𝒞 be a covering of 𝐸. We define ℐ^′′={𝐼⊆𝐸∶|𝐼∩𝑁(𝑥_𝑖)|≤1,forall𝑖∈{1,2,…,𝑡}}.

Theorem 4.14 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: ℐ_XH(𝒞)={𝐼⊆𝐸∶∀𝑥∈𝐼,𝑥∉XH(𝐼−{𝑥})}. (4.3) Similar to the case of SH, we can obtain the following results.

Proposition 4.16. Let 𝒞 be a covering of 𝐸. If {𝑁(𝑥)∶𝑥∈𝐸} induced by 𝒞 forms a partition of 𝐸, then 𝑀(𝐸,ℐ_XH(𝒞)) is a matroid and ℐ_XH(𝒞)=ℐ^′′.

𝑟_XH denotes the rank function of 𝑀(𝐸,ℐ_XH(𝒞)) and ℒ_XH(𝑀(𝒞)) denotes the set of all closed sets of 𝑀(𝐸,ℐ_XH(𝒞)). Then we can obtain the following proposition.

Proposition 4.17. Let 𝒞 be a covering of 𝐸. If {𝑁(𝑥)∶𝑥∈𝐸}={𝑁(𝑥₁),𝑁(𝑥₂),…,𝑁(𝑥_𝑡)} induced by 𝒞 forms a partition of 𝐸, then(1)𝑋 is a base of 𝑀(𝐸,ℐ_XH(𝒞)) if and only if |𝑋∩𝑁(𝑥_𝑖)|=1(1≤𝑖≤𝑡), and 𝑀(𝐸,ℐ_XH(𝒞)) has |𝑁(𝑥₁)||𝑁(𝑥₂)|⋯|𝑁(𝑥_𝑠)| bases.(2)For all 𝑋⊆𝐸, 𝑟_XH(𝑋)=|{𝑁(𝑥_𝑖)∶𝑁(𝑥_𝑖)∩𝑋≠∅,1≤𝑖≤𝑡}|.(3)𝑋 is a circuit of 𝑀(𝐸,ℐ_XH(𝒞)) if and only if there exists 𝑁(𝑥_𝑖)∈{𝑁(𝑥)∶𝑥∈𝐸} such that 𝑋⊆𝑁(𝑥_𝑖) and |𝑋|=2.(4)𝑋 is a dependent set of 𝑀(𝐸,ℐ_XH(𝒞)) if and only if there exists 𝑁(𝑥_𝑖)∈{𝑁(𝑥)∶𝑥∈𝐸} such that |𝑁(𝑥_𝑖)∩𝑋|>1.(5){𝑁(𝑥₁),𝑁(𝑥₂),…,𝑁(𝑥_𝑡)} is the set of all atoms of lattice ℒ_XH(𝑀(𝒞)).(6)For all 𝑥,𝑦∈𝐸 and 𝑥≠𝑦, there does not exist 𝑁(𝑧)∈{𝑁(𝑥₁),𝑁(𝑥₂),…,𝑁(𝑥_𝑡)} such that 𝑥,𝑦∈𝑁(𝑧) if and only if XH({𝑥,𝑦}) covers XH({𝑥}) or XH({𝑦}).(7)For all 𝑖,𝑗∈{1,2,…,𝑡}, (𝑁(𝑥_𝑖),𝑁(𝑥_𝑗)) is a modular pair of lattice ℒ_XH(𝑀(𝒞)).(8)For all 𝑖∈{1,2,…,𝑡}, 𝑁(𝑥_𝑖) is a modular element of lattice ℒ_XH(𝑀(𝒞)).

The proof of Propositions 4.16 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. Let 𝒞 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 all 𝑥∈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. Let 𝒞 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. Since 𝑟𝑒𝑑𝑢𝑐𝑡(𝒞) is a partition of 𝐸, {𝑁(𝑥)∶𝑥∈𝐸} induced by 𝑟𝑒𝑑𝑢𝑐𝑡(𝒞) forms a partition of 𝐸. Suppose {𝐾₁,𝐾₂,…,𝐾_𝑠} 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. Let 𝒞 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 all 𝑥,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 𝐾_𝑖,𝐾^′_𝑖 such that 𝑥,𝑧∈𝐾_𝑖 and 𝑦,𝑧∈𝐾^′_𝑖. Without loss of generality, we suppose {𝐾₁,𝐾₂,…,𝐾_𝑠} is the set of all blocks which contain 𝑥 and {𝐾^′₁,𝐾^′₂,…,𝐾^′_𝑡} 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,…,𝐾_𝑠} is the set of all blocks which contain 𝑧, thus {𝐾^′₁,𝐾^′₂,…,𝐾^′_𝑡}⊆{𝐾₁,𝐾₂,…,𝐾_𝑠}. 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 Theorem 4.14, Propositions 4.19 and 4.20, we can obtain the following two corollaries.

Corollary 4.21. Let 𝒞 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. Let 𝒞 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 Section 3, 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 between ℐ_SH(𝒞) and ℐ(𝒞), and the relationship between ℒ_SH(𝑀(𝒞)) and ℒ(𝑀(𝒞)).

Proposition 5.1. Let 𝒞 be a covering of 𝐸. If SH induced by 𝒞 is a closure operator, then ℐ_SH(𝒞)⊆ℐ(𝒞) and ℒ_SH(𝑀(𝒞))⊆ℒ(𝑀(𝒞)).

Proof. Since SH induced by 𝒞 is a closure operator, {𝐼(𝑥)∶𝑥∈𝐸}={𝐼(𝑥₁),𝐼(𝑥₂),…,𝐼(𝑥_𝑠)} forms a partition of 𝐸. For all 𝐼∈ℐ_SH(𝒞), suppose 𝐼={𝑖₁,𝑖₂,…,𝑖_𝛼}(𝛼≤𝑠) such that 𝑖₁∈𝐼(𝑥_𝑖1),𝑖₂∈𝐼(𝑥_𝑖2),…,𝑖_𝛼∈𝐼(𝑥_𝑖𝛼) and {𝐼(𝑥_𝑖1),𝐼(𝑥_𝑖2),…,𝐼(𝑥_𝑖𝛼)}⊆{𝐼(𝑥₁),𝐼(𝑥₂),…,𝐼(𝑥_𝑠)}. According to the definition of 𝐼(𝑥), there exists {𝐾_𝑖1,𝐾_𝑖2,…,𝐾_𝑖𝛼}⊆𝒞 such that 𝑖₁∈𝐾_𝑖1,𝑖₂∈𝐾_𝑖2,…,𝑖_𝛼∈𝐾_𝑖𝛼. Since {𝐼(𝑥₁),𝐼(𝑥₂),…,𝐼(𝑥_𝑠)} forms a partition of 𝐸, thus {𝐾_𝑖1,𝐾_𝑖2,…,𝐾_𝑖𝛼} are different blocks. According to the definition of transversal matroid, we have 𝐼∈ℐ. Hence, ℐ_SH(𝒞)⊆ℐ(𝒞).
For all 𝑋∈ℒ_SH(𝑀(𝒞)), 𝑋=SH(𝑋)=⋃_𝑥∈𝑋𝐼(𝑥). Now we need to prove 𝑋∈ℒ(𝑀(𝒞)), that is, cl_𝑀(𝒞)(𝑋)=𝑋={𝑥∣𝑟_𝑀(𝒞)(𝑋)=𝑟_𝑀(𝒞)(𝑋∪{𝑥})}. Since 𝑋⊆cl_𝑀(𝒞)(𝑋), if cl_𝑀(𝒞)(𝑋)≠𝑋, then cl_𝑀(𝒞)(𝑋)⊈𝑋, that is, there exists 𝑦∉𝑋 such that 𝑟_𝑀(𝒞)(𝑋)=𝑟_𝑀(𝒞)(𝑋∪{𝑦}). Suppose 𝑇={𝑡₁,𝑡₂,…,𝑡_𝑡}(𝑡≤𝑠) is a maximal independent set included in 𝑋, then {𝑡₁,𝑡₂,…,𝑡_𝑡}⊆𝑋=⋃_𝑥∈𝑋𝐼(𝑥) and there exist different 𝐾₁,𝐾₂,…,𝐾_𝑡 such that for all 𝑖∈{1,2,…,𝑡},𝑡_𝑖∈𝐾_𝑖. Since 𝑦∉𝑋, 𝐼(𝑥)∩𝐼(𝑦)=∅ for all 𝑥∈𝑋. Based on {𝐼(𝑥₁),𝐼(𝑥₂),…,𝐼(𝑥_𝑠)} forms a partition of 𝐸, there exists 𝐾⊆𝐼(𝑦) such that 𝐾₁,𝐾₂,…,𝐾_𝑡,𝐾 are different blocks and 𝑦∈𝐾, thus 𝑇∪{𝑦} is a maximal independent set included in 𝑋∪{𝑦}. Hence, we have 𝑟_𝑀(𝒞)(𝑋∪{𝑦})=𝑟_𝑀(𝒞)(𝑋)+1 which contradicts 𝑟_𝑀(𝒞)(𝑋)=𝑟_𝑀(𝒞)(𝑋∪{𝑦}). Thus we can obtain cl_𝑀(𝒞)(𝑋)=𝑋.

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

Proposition 5.2. Let 𝒞 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. Since SH 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)∈ℒ_SH(𝑀(𝒞)). According to Proposition 5.1, 𝐼(𝑥)∈ℒ(𝑀(𝒞)).

We give an example to help understand the relationship between ℒ_SH(𝑀(𝒞)) and ℒ(𝑀(𝒞)) better.

Example 5.3. Let 𝒞={{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∉ℐ_SH(𝒞) because |𝑇∩𝐼(1)|=2. Hence, ℐ(𝒞)⊈ℐ_SH(𝒞). ℒ_SH(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))⊈ℒ_SH(𝑀(𝒞)). The structures of ℒ_SH(𝑀(𝒞)) and ℒ(𝑀(𝒞)) are showed in Figure 1.

Figure 1 

The lattice of ℒ(𝑀(𝒞)) (resp., ℒXH(𝑀(𝒞))) and ℒSH(𝑀(𝒞)).

Remark 5.4. Let 𝒞 be a covering of 𝐸. Although XH induced by 𝒞 is a closure operator, it has no relationship between ℐ_XH(𝒞) and ℐ(𝒞), and has no relationship between ℒ_XH(𝑀(𝒞)) and ℒ(𝑀(𝒞)).

The following example illustrates the above statements.

Example 5.5. Let 𝒞={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∈ℐ_XH(𝒞), but 𝑇∉ℐ(𝒞) because |𝑇∩𝐾₂|=2, thus ℐ_XH(𝒞)⊈ℐ(𝒞). Let 𝑇^′={𝑎,𝑐,𝑑}. It is clear that 𝑇^′∈ℐ(𝒞), but 𝑇^′∉ℐ_XH because |𝑇^′∩𝑁(𝑐)|=2, thus ℐ(𝒞)⊈ℐ_XH(𝒞). Let 𝑋={𝑎,𝑏,𝑖}. 𝑋∈ℒ_XH(𝑀(𝒞)) for XH(𝑋)=𝑋. However, 𝑋∉ℒ(𝑀(𝒞)) for cl_𝑀(𝒞)(𝑋)={𝑎,𝑏,𝑐,𝑑,𝑒,𝑖}. Let 𝑋={𝑎}. 𝑋∈ℒ(𝑀(𝒞)) for cl_𝑀(𝒞)(𝑋)=𝑋. However, 𝑋∉ℒ_XH(𝑀(𝒞)) for XH(𝑋)={𝑎,𝑏}≠𝑋.

The following proposition shows the relationship between ℐ_SH(𝒞) and ℐ_XH(𝒞), and the relationship between ℒ_SH(𝑀(𝒞)) and ℒ_XH(𝑀(𝒞)).

Proposition 5.6. Let 𝒞 be a covering of 𝐸. If XH and SH induced by 𝒞 are closure operators, then ℐ_SH(𝒞)⊆ℐ_XH(𝒞) and ℒ_SH(𝑀(𝒞))⊆ℒ_XH(𝑀(𝒞)).

Proof. If XH 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 ℐ_SH(𝒞)⊆ℐ_XH(𝒞).
For all 𝑋∈ℒ_SH(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∈ℒ_XH(𝑀(𝒞)). Hence, ℒ_SH(𝑀(𝒞))⊆ℒ_XH(𝑀(𝒞)).

Example 5.7. From Example 5.3, 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∈ℐ_XH(𝑀(𝒞)) but 𝐼∉ℐ_SH(𝑀(𝒞)) because |𝐼∩𝐼(1)|=2. Hence ℐ_XH(𝑀(𝒞))⊈ℐ_SH(𝑀(𝒞)). ℒ_SH(𝑀(𝒞))={∅,{1,2,3},{4,5},{1,2,3,4,5}} and ℒ_XH(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_XH(𝑀(𝒞))⊈ℒ_SH(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. If 𝒞 is a partition of 𝐸, then ℐ_SH(𝒞)=ℐ_XH(𝒞)=ℐ(𝒞) and ℒ_SH(𝑀(𝒞))=ℒ_XH(𝑀(𝒞))=ℒ(𝑀(𝒞)).

Proof. Since 𝒞 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 ℐ_SH(𝒞)=ℐ_XH(𝒞)=ℐ(𝒞) and ℒ_SH(𝑀(𝒞))=ℒ_XH(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 on ℐ(𝒞).

Theorem 5.9. Let ℱ be a family of subset of 𝐸 and 𝐾∈ℱ. ℐ(ℱ−{𝐾})⊆ℐ(ℱ).

Proof. For all 𝐼∈ℐ(ℱ−{𝐾}), we may as well suppose 𝐼={𝑖₁,𝑖₂,…,𝑖_𝑡} where 𝑖₁,𝑖₂,…,𝑖_𝑡∈𝐸. According to the definition of transversal matroid, there exist different blocks 𝐾₁,𝐾₂,…,𝐾_𝑡∈ℱ satisfy 𝐾_𝑖≠𝐾 and 𝑖_𝑗∈𝐾_𝑗 for all 1≤𝑖, 𝑗≤𝑡. Thus 𝐼∈ℐ(ℱ).

The following example illustrates ℐ(ℱ)⊈ℐ(ℱ−{𝐾}).

Example 5.10. Let ℱ={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)=2^𝐸, ℐ(ℱ−{𝐾₃})={∅,{1},{2},{3},{1,3},{1,2}{2,3}}. Hence, ℐ(ℱ)⊈ℐ(ℱ−{𝐾}).

Let 𝒞 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. Let 𝒞 be a covering of 𝐸 and 𝐾∈𝒞. If 𝐾 is reducible, then ℐ(𝒞−{𝐾})⊆ℐ(𝒞).

Corollary 5.12. Let 𝒞 be a covering of 𝐸. ℐ(𝑟𝑒𝑑𝑢𝑐𝑡(𝒞))⊆ℐ(𝒞).

Corollary 5.13. Let 𝒞 be a covering of 𝐸 and 𝐾∈𝒞. If 𝐾 is an immured element, then ℐ(𝒞−{𝐾})⊆ℐ(𝒞).

Corollary 5.14. Let 𝒞 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 structure ℒ(𝑀(𝒞)).

Theorem 5.15. Let ℱ be a family of subset of 𝐸 and for all 𝐾∈ℱ. ℒ(𝑀(ℱ−{𝐾}))⊆ℒ(𝑀(ℱ)).

Proof. First, we prove cl_ℱ({𝑥})⊆cl_ℱ−{𝐾}({𝑥}) for all 𝑥∈𝐸. For all 𝑦∉cl_ℱ−{𝐾}({𝑥}), {𝑥,𝑦}∈ℐ_ℱ−{𝐾}⊆ℐ_ℱ. Thus 𝑦∉cl_ℱ({𝑥}) which implies cl_ℱ({𝑥})⊆cl_ℱ−{𝐾}({𝑥}).
Second, we need to prove that any atom of ℒ(𝑀(ℱ−{𝐾})) is a closed set of ℒ(𝑀(ℱ)), that is, cl_ℱ(cl_ℱ−{𝐾}({𝑥}))=cl_ℱ−{𝐾}({𝑥}). Since cl_ℱ−{𝐾}({𝑥})⊆cl_ℱ(cl_ℱ−{𝐾}({𝑥}))⊆cl_ℱ−{𝐾}(cl_ℱ−{𝐾}({𝑥}))=cl_ℱ−{𝐾}({𝑥}), cl_ℱ(cl_ℱ−{𝐾}({𝑥}))=cl_ℱ−{𝐾}({𝑥}).
Third, we need to prove ℒ(𝑀(ℱ−{𝐾}))⊆ℒ(𝑀(ℱ)). For all 𝑋⊆ℒ(𝑀(ℱ−{𝐾})), 𝑋=⋁^𝑚_𝑖=1cl_ℱ−{𝐾}({𝑥})=⋁^𝑚_𝑖=1⋁^𝑠𝑖_𝑗=1cl_ℱ({𝑥_𝑗}) because ℒ(𝑀(ℱ−{𝐾})) is a atomic lattice, thus 𝑋∈ℒ(𝑀(ℱ)).

The following example shows that ℒ(𝑀(ℱ))⊈ℒ(𝑀(ℱ−{𝐾})), where 𝐾∈ℱ.

Example 5.16. Based on Example 5.10, we have ℒ(𝑀(ℱ))=2^𝐸 and ℒ(𝑀(ℱ−{𝐾₃}))={∅,{1},{2},{3},{1,2,3}}. It is clear that ℒ(𝑀(ℱ))⊈ℒ(𝑀(ℱ−{𝐾₃})).

Similarly, when ℱ is equal to 𝒞, we can obtain the following four corollaries.

Corollary 5.17. Let 𝒞 be a covering of 𝐸 and 𝐾∈𝒞. If 𝐾 is reducible, then ℒ(𝑀(𝒞−{𝐾}))⊆ℒ(𝑀(𝒞)).

Corollary 5.18. Let 𝒞 be a covering of 𝐸. ℒ(𝑀(𝑟𝑒𝑑𝑢𝑐𝑡(𝒞)))⊆ℒ(𝑀(𝒞)).

Corollary 5.19. Let 𝒞 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. Let 𝒞 be a covering of 𝐸. ℒ(𝑀(𝑒𝑥𝑐𝑙𝑢𝑠𝑖𝑜𝑛(𝒞)))⊆ℒ(𝑀(𝒞)).

Let 𝒞 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 ℐ_SH(𝒞) and ℐ_SH(𝒞−{𝐾}).

Example 5.21. Let 𝐸={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 on ℐ_SH(𝒞) and ℒ_SH(𝑀(𝒞)).

Theorem 5.22. Let 𝒞 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, ℐ_SH(𝒞)=ℐ_SH(𝒞−{𝐾}) and ℒ_SH(𝑀(𝒞))=ℒ_SH(𝑀(𝒞−{𝐾})).

Proof. It comes from Lemma 4.8 and Theorem 4.3.

As we know, if 𝒞 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. Let 𝒞 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, ℐ_SH(𝒞)=ℐ_SH(𝑒𝑥𝑐𝑙𝑢𝑠𝑖𝑜𝑛(𝒞)) and ℒ_SH(𝑀(𝒞))=ℒ_SH(𝑀(𝑒𝑥𝑐𝑙𝑢𝑠𝑖𝑜𝑛(𝒞))).

Now we consider the reducible element’s influence on ℐ_XH(𝒞) and ℒ_XH(𝑀(𝒞)).

Theorem 5.24. Let 𝒞 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, ℐ_XH(𝒞)=ℐ_XH(𝒞−{𝐾}) and ℒ_XH(𝑀(𝒞))=ℒ_XH(𝑀(𝒞−{𝐾})).

Proof. Since XH induced by 𝒞 is a closure operator, {𝑁(𝑥)∶𝑥∈𝐸} induced by 𝒞 forms a partition. Based on the definition of ℐ_XH(𝒞) and Lemma 4.18, XH induced by 𝒞−{𝐾} is also a closure operator and ℐ_XH(𝒞)=ℐ_XH(𝒞−{𝐾}).

As we know, if 𝒞 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. Let 𝒞 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, ℐ_XH(𝒞)=ℐ_XH(𝑟𝑒𝑑𝑢𝑐𝑡(𝒞)) and ℒ_XH(𝑀(𝒞))=ℒ_XH(𝑀(𝑟𝑒𝑑𝑢𝑐𝑡(𝒞))).

Let 𝒞 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 ℐ_XH(𝒞) and ℐ_XH(𝒞−{𝐾}).

Example 5.26. Suppose 𝐾₁={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.

Acknowledgments

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