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# Open Mathematics

### formerly Central European Journal of Mathematics

Editor-in-Chief: Vespri, Vincenzo / Marano, Salvatore Angelo

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Volume 15, Issue 1

# The characterizations of upper approximation operators based on special coverings

Pei Wang
• College of Mathematics and Econometrics, Hunan University, Hunan, 410012, China
• School of Mathematics and Information Science, Guangxi Universities Key Lab of Complex System Optimization and Big Data Processing, Yulin Normal University, Yulin 537000, China
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• Other articles by this author:
/ Qingguo Li
Published Online: 2017-03-04 | DOI: https://doi.org/10.1515/math-2017-0015

## Abstract

In this paper, we discuss the approximation operators ${\overline{apr}}_{NS}$ and ${\overline{apr}}_{S}$ which are based on NS(U) and S. We not only obtain some properties of NS(U) and S, but also give examples to show some special properties. We also study sufficient and necessary conditions when they become closure operators. In addition, we give general and topological characterizations of the covering for two types of covering-based upper approximation operators being closure operators.

MSC 2010: 54-00; 54A10; 54A99

## 1 Introduction

The concept of rough set which was first proposed by Pawlak [11] is an extension of set theory for the study of the intelligent systems characterized by insufficient and incomplete information in 1982. It is a useful and powerful tool in many fields, such as data analysis, granularity or vagueness. It has also been applied successfully in process control, economics, medical diagnosis, biochemistry, environmental science, biology, chemistry, psychology, conflict analysis, and so on. Many researchers have made some significant contributions to developing the rough theory [3,4,7-8,12,14-15,17-26]. However, a problem with Pawlak's rough set theory is that partition or equivalence relation is explicitly used in the definition of the lower and upper approximations. Such a partition or equivalence relation is too restrictive for many applications because it can only deal with complete information systems. To address this issue, generalizations of rough set theory were considered by scholars. One approach was to extend equivalence relation to tolerance [5-6,32] and others [20-21,33-34]. Another important approach was to relax the partition to a covering of the universe. In 1983, W.Zakowski generalized the classical rough set theory by using coverings of a universe instead of partitions [27]. The covering-based rough sets is one of the most important generalization of the classical Pawlak rough sets. Such generalization leads to various covering approximation operators that are both of theoretical and practical importance [28-30]. The relationships between properties of covering-based approximation and their corresponding coverings have attracted intensive research. Based on the mutual correspondence of the concepts of extension and intension, E.Bryniarski [36] and Z.Bonikowski et al. [37] gave the second type of covering-based rough sets. The third and the fourth type of covering-based rough sets were introduced in [38]. Subsequently, W.Zhu utilized the topological method to characterize covering rough sets [18]. W.Zhu and F. Wang discussed the relationship between properties of four types of covering-based upper approximation operators and their corresponding coverings [21-23]. T. Yang et al. researched attribute reduction of covering information systems [39]. G. Liu studied the two types of rough sets induced by coverings and obtained some interesting results. G. Cattaneo, D. Ciucci, and G. Liu obtained the algebraic structures of generalized rough set theory [1,2,40,44]. G. Liu also used axiomatic method to characterize covering-based rough sets [41-43]. X. Bian et al. gave characterizations of covering-based approximation spaces being closure operators [31]. Ge et al. proposed not only general but also topological characterizations of coverings for these operators being closure operators [25,29-30]. T.Lin et al. defined some approximation operators which are based on neighborhood systems and did some research on them [52]. In addition, Y. Zhang et al. discussed the relationships between generalized rough sets based on covering and reflexive neighborhood system [49] and proposed the operator ${\overline{apr}}_{S}$. Based on their works, we will investigate the properties of NS(U) and S. We also give characterizations for ${\overline{apr}}_{NS}$, ${\overline{apr}}_{S}$ being closure operators.

This paper is organized as follows. Section 2 recalls the main ideas of generalized rough set and covering approximations. Section 3 gives the properties of NS(U) and some examples. Section 4 studies the characterization of NS(U) for ${\overline{apr}}_{NS}$ being a closure operator, while Section 5 considers the properties of S and ${\overline{apr}}_{S}$ and obtains the general and topological characterizations of special covering S for covering-based upper approximation operator ${\overline{apr}}_{S}$ to be a closure operator. Finally, Section 6 concludes the paper.

## 2 Background

In this section, we introduce the fundamental concepts that are used in this paper. U is the universe of discourse and $\mathcal{P}\left(U\right)$ denotes the family of all subsets of U. If $\mathcal{C}$ is a family of subsets of U, none of sets in $\mathcal{C}$ is empty, and $\cup \mathcal{C}=U$, then $\mathcal{C}$ is called a covering of U.

#### Definition 2.1

([46]). A mapping n : UP(U) is called a neighborhood operator If n(x) ≠ ∅ for all xU, n is called a serial neighborhood operator If xn(x) for all xU, n is called a reflexive neighborhood operator

#### Definition 2.2

([50-51]). A neighborhood system of an object xU, denoted by NS(x), is a non-empty family of neighborhoods of x. The set {NS(x) : xU} is called as a neighborhood system of U, and it is denoted by NS(U). Let NS(U) be a neighborhood system of U.

NS(U) is said to be serial if for any xU and n(x) ∈ NS(x), n(x) is non-empty (called Fré(V) Space in [32]).

NS(U) is said to be reflexive, if for any xU and n(x) ∈ NS(x), xn(x).

NS(U) is said to be symmetric, if for any x, yU, n(x) ∈ NS(x) and n(y) ∈ NS(y), xn(y) ⇒ yn(x).

NS(U) is said to be transitive, if for any x, y, zU, n(y) ∈ NS(y) and n(z) ∈ NS(z), xn(y) and yn(z) ⇒ xn(z).

#### Definition 2.3

(Covering approximation space [23]). If U is an universe and $\mathcal{C}$ is a covering of U, then we call U together with covering $\mathcal{C}$ a covering approximation space, denoted by $\left(U,\phantom{\rule{thickmathspace}{0ex}}\mathcal{C}\mathcal{\right)}$.

#### Definition 2.4

([52]). Let NS(U) be a neighborhood system of U. The lower and upper operators of X are defined as follows:

$apr_NS(X)={x∈U:∃n(x)∈NS(x),n(x)⊆X};apr¯NS(X)={x∈U:∀n(x)∈NS(x),n(x)∩X≠∅}.$

#### Definition 2.5

([49]). Let NS(U) be a neighborhood system of U.

NS(U) is referred to as weak-unary, if for any xU and n1(x), n2(x) ∈ NS(x), there exists an n3(x) ∈ NS(x) such that n3(x)⊆ n1(x)∩ n2(x);

NS(U) is referred to as weak-transitive, if for any xU and n(x) ∈ NS(x), there exists an n1(x) ∈ NS(x) satisfying thatfor any yn1(x), there exists an n(y) ∈NS(y) such that n(y)⊆ n(x);

NS(U) is referred to as a weak-S4 neighborhood system, if NS(x) is reflexive and weak-transitive.

The following topological concepts and facts are elementary and can be found in[47,51]. We list them below for the purpose of this paper being self-contained.

1. A topological space is a pair (U, τ) consisting of a set U and a family τ of subsets of U satisfying the following conditions: (a) ∅ ∈ τ and U ∈ τ (b) If U1, U2 ∈ τ, then U1U2 ∈ τ; (c)If $\mathcal{A}\subseteq \tau$, then $\cup \mathcal{A}\in \tau .\tau$ is called a topology on U and the members of τ are called open sets of (U, τ). The complementary set of an open set is called a closed set.

2. A set F is called a clopen set, if F in (U, τ) is both an open set and a closed set.

3. A family $\mathcal{B}\subseteq \tau$ is called a base for (U, τ) if for every non-empty open subset O of U and each xO, there exists a set $B\in \mathcal{B}$ such that xBO. Equivalently, a family $\mathcal{B}\subseteq \tau$ if every non-empty open subset O of U can be represented as the union of a subfamily of $\mathcal{B}$.

4. For any xU, a family $\mathcal{B}\subseteq \tau$ is called alocal base at x for (U τ) if xB for each $B\in \mathcal{B}$, and for every open subset O of U with xO, there exists a set $B\in \mathcal{B}$ such that BO.

5. If $\mathcal{P}$ is a partition of U, the topology τ = { OU O} is the union of some members of $\mathcal{P}\cup \left\{\mathrm{\varnothing }\right\}$ is called a pseudo-discrete topology in [13](also called a closed-open topology in [12]).

6. Let (U, τ) be a topological space. If for each pair of points x, yU with xy, there exist open sets O, O' such that xO, yO′ and OO′ = ∅, then (U, τ) is called a T2-space and τ is called a T2-topology.

#### Definition 2.6

(Induced topology and subspace). Let (U, τ) be a topological space and XU. It is easy to check that τ′ = {OX : O ∈ τ} is a topology on X. τ′ is called a topology induced by X, and the topology space (X, τ′) is called a subspace of (U, τ).

#### Definition 2.7

(Closure operator). An operator H: P(U) → P(U)ia called a closure operator on U if it satisfies the following conditions: for any X, YU,

$\left({H}_{1}\right)\phantom{\rule{1em}{0ex}}H\left(X\cup Y\right)=H\left(X\right)\cup H\left(Y\right)$;

$\left({H}_{2}\right)\phantom{\rule{1em}{0ex}}X\subseteq H\left(X\right)$ ;

$\left({H}_{3}\right)\phantom{\rule{1em}{0ex}}H\left(\mathrm{\varnothing }\right)=\mathrm{\varnothing }$;

$\left({H}_{4}\right)\phantom{\rule{1em}{0ex}}H\left(H\left(X\right)\right)=H\left(X\right)$.

#### Definition 2.8

(Interior operator). An operator I: P(U) → P(U)ia called a interior operator on U if it satisfies the following conditions: for any X, YU,

$\left({I}_{1}\right)\phantom{\rule{1em}{0ex}}I\left(X\cap Y\right)=I\left(X\right)\cap I\left(Y\right)$ ;

$\left({I}_{2}\right)\phantom{\rule{1em}{0ex}}I\left(X\right)\subseteq X$;

$\left({I}_{3}\right)\phantom{\rule{1em}{0ex}}I\left(U\right)=U$;

$\left({I}_{4}\right)\phantom{\rule{1em}{0ex}}I\left(I\left(X\right)\right)=I\left(X\right)$.

#### Definition 2.9

(Dual operator). Assume that H, I: P(U) → P(U) are two operators on U. If for any XU, H(X) = ∼ I(∼ X). We say that H, I are dual operators or H is the dual operator of I.

#### Proposition 2.10

([49]). Let NS(U) be a neighborhood system of U. Then the following are equivalent:

1. ${\overline{apr}}_{NS}\left(X\cup Y\right)={\overline{apr}}_{NS}\left(X\right)\cup {\overline{apr}}_{NS}\left(Y\right)for\phantom{\rule{thinmathspace}{0ex}}all\phantom{\rule{thinmathspace}{0ex}}X,Y\subseteq U$;

2. ${\underset{_}{apr}}_{NS}\left(X\cup Y\right)={\underset{_}{apr}}_{NS}\left(X\right)\cup {\underset{_}{apr}}_{NS}\left(Y\right)\phantom{\rule{thinmathspace}{0ex}}for\phantom{\rule{thinmathspace}{0ex}}all\phantom{\rule{thinmathspace}{0ex}}X,Y\subseteq U$;

3. NS(U) is weak-unary.

#### Proposition 2.11

([49]). Let NS(U) be a neighborhood system of U. Then the following are equivalent:

1. ${\overline{apr}}_{NS}\left({\overline{apr}}_{NS}\left(X\right)\right)\subseteq {\overline{apr}}_{NS}\left(X\right)\phantom{\rule{thinmathspace}{0ex}}for\phantom{\rule{thinmathspace}{0ex}}all\phantom{\rule{thinmathspace}{0ex}}X\subseteq U$

2. ${\underset{_}{apr}}_{NS}\left(X\right)\subseteq {\underset{_}{apr}}_{NS}\left({\underset{_}{apr}}_{NS}\left(Y\right)\right)\phantom{\rule{thinmathspace}{0ex}}for\phantom{\rule{thinmathspace}{0ex}}all\phantom{\rule{thinmathspace}{0ex}}X\subseteq U$

3. NS(U) is weak-transitive.

## 3 Some propositions of NS(U)

In this section, we will discuss some properties of NS(U).

#### Definition 3.1

Let NS(U) be a neighborhood system of U. NS(U) is said to be Euclidean, if for any x, yU, and n(x) ∈ NS(x), yn(x) ⇒ n(x) ⊆ n(y).

#### Proposition 3.2

Let NS(U) be a neighborhood system of U. If NS(U) is Euclidean, then ${\overline{apr}}_{NS}\left({\underset{_}{apr}}_{NS}\left(X\right)\right)\subseteq {\underset{_}{apr}}_{NS}\left(X\right)$ and ${\overline{apr}}_{NS}\left(X\right)\subseteq {\underset{_}{apr}}_{NS}\left({\overline{apr}}_{NS}\left(X\right)\right)$ for any XU.

#### Proof

For any xU, if $x\in {\overline{apr}}_{NS}\left({\underset{_}{apr}}_{NS}\left(X\right)\right)$ and n(x)∈ NS(U), We have $n\left(x\right)\cap {\underset{_}{apr}}_{NS}\left(X\right)\ne \mathrm{\varnothing }$. Pick $y\in n\left(x\right)\cap {\underset{_}{apr}}_{NS}\left(X\right)$, then there exists an n(y) ∈ NS(y) such that n(y) ⊆ X. Since NS(U) is Euclidean, we have n(x) ⊆ n(y). It follows that n(x) ⊆ X. From the Definition 2.4, We have $x\in {\underset{_}{apr}}_{NS}\left(X\right)$, therefore ${\overline{apr}}_{NS}\left({\underset{_}{apr}}_{NS}\left(X\right)\right)\subseteq {\underset{_}{apr}}_{NS}\left(X\right)$.

We can get ${\overline{apr}}_{NS}\left(X\right)\subseteq {\underset{_}{apr}}_{NS}\left({\overline{apr}}_{NS}\left(X\right)\right)$ by the duality between ${\overline{apr}}_{NS}$ and ${\underset{_}{apr}}_{NS}$. □

The inverse of Proposition 2.10 does not hold. An example is given as follows:

#### Example 3.3

Let U = {a, b, c}, NS(a) = {{a}}, NS(b) = {{b}, {a, b}} and NS(c) = {{c}}. Then:

$apr_NS(∅)=∅=apr¯NS(apr_NS(∅));apr_NS({a})={a}=apr¯NS(apr_NS({a}));apr_NS({b})={b}=apr¯NS(apr_NS({b}));apr_NS({c})={c}=apr¯NS(apr_NS({c}));apr_NS({a,b})={a,b}=apr¯NS(apr_NS({a,b}));apr_NS({a,c})={a,c}=apr¯NS(apr_NS({a,c}));apr_NS({b,c})={b,c}=apr¯NS(apr_NS({b,c}));apr_NS(U)=U=apr¯NS(apr_NS(U)).$

Hence ${\overline{apr}}_{NS}\left({\underset{_}{apr}}_{NS}\left(X\right)\right)\subseteq {\underset{_}{apr}}_{NS}\left(X\right)$ for any XU. Since an(b) = {a, b} ∈ NS(b) and n(b) ⊈ n(a). We obtain that NS(U) is not Euclidean.

#### Proposition 3.4

Let NS(U) be a neighborhood system of U. Then the following are equivalent:

1. NS(U) is transitive;

2. For any x, yU, if xn(y), then n(x)⊆ n(y).

#### Proof

(1) ⇒ (2) For any x, yU and xn(y), if n(x) ⊈ n(y), there exists pn(x) and pn(y). Since xn(y) and NS(U) is transitive, we have pn(y). It is contradictory to pn(y).

(2) ⇒ (1) for any x, y, zU, n(y) ∈ NS(y) and n(z) ∈ NS(z), xn(y) and yn(z). It is easy to prove NS(U) is transitive. □

#### Proposition 3.5

Let NS(U) be a neighborhood system of U and n : UP(U) is a reflexive mapping. Then the following are equivalent:

1. {n(x) : xU} forms a partition of U;

2. NS(U) is reflexive, transitive and Euclidean.

#### Proof

(1) ⇒ (2) For any x,yU and yn(x). Since {n(x) : xU} forms a partition of U and n : UP(U) is a reflexive mapping, then n(x) = n(y). By the Definition 3.1 and Proposition 2.11, it is easy to prove NS(U) is reflexive, transitive and Euclidean.

(2) ⇒ (1) for any x, yU, if n(x) ≠ n(y), then n(x)>∩ n(y) = ∅. Otherwise, we take a zn(x)∩ n(y), since NS(U) is reflexive, transitive and Euclidean, then n(z) = n(x) = n(y). It is contradictory to n(x) ≠ n(y). □

#### Proposition 3.6

Let NS(U) be a reflexive neighborhood system of U. NS(U) is weak-unary if and only if there is a topology on U such that NS(x) is a local base for any xU.

#### Proof

(1) ⇒ (2) For any xU, n1(x), n2(x) ∈ NS(x) and xn1(x) ∩ n2(x), since NS(U) is weak-unary, there exists n3(x) such that n3(x) ⊆ n1(x) ∩ n2(x). We have xn3(x) because NS(U) is a reflexive neighborhood system of U. So NS(x) is alocal base for any xU.

(2) ⇒ (1) By the definition of local base, it is easy to prove NS(U) is weak-unary. □

## 4 Characterization of NS(U) for ${\overline{apr}}_{NS}$ being a closure operator

In Section 3, we discuss the properties of NS(U). One natural question thus arise: When is ${\overline{apr}}_{NS}$ a closure operator?

#### Theorem 4.1

(General characterization of of NS(U) for ${\overline{apr}}_{NS}$ being a closure operator). Let NS(U) be a neigh- borhood system of U. ${\overline{apr}}_{NS}$ is a closure operator if and only if NS(U) is reflexive, weak-unary and weak- transitive.

#### Proof

(⊆) Assume that ${\overline{apr}}_{NS}$ is a closure operator. By the Definition 2.7, we get $X\subseteq {\overline{apr}}_{NS}\left(X\right)$, ${\overline{apr}}_{NS}\left({\overline{apr}}_{NS}\left(X\right)\right)={\overline{apr}}_{NS}\left(X\right)$ and ${\overline{apr}}_{NS}\left(X\cup Y\right)={\overline{apr}}_{NS}\left(X\right)\cup {\overline{apr}}_{NS}\left(Y\right)$ for all X, YU. According to Proposition 6 [49] and Proposition 7 [49], we obtain NS(U) is weak-unary and weak-transitive.

(⇐) Assume that NS(U) is weak-unary and weak-transitive. We prove that ${\overline{apr}}_{NS}$ satisfies the conditions (Hi)(for i = 1, 2, 3, 4). By the Definition of ${\overline{apr}}_{NS}$, it is easy to check that ${\overline{apr}}_{NS}\left(\mathrm{\varnothing }\right)=\mathrm{\varnothing }$. Hence (H3) is satisfied. We prove (H1) and (H4). Since NS(U) is weak-unary and weak-transitive, by the Proposition 6 [49] and Proposition 7 [49], we obtain ${\overline{apr}}_{NS}\left({\overline{apr}}_{NS}\left(X\right)\right)\subseteq {\overline{apr}}_{NS}\left(X\right)$ and ${\overline{apr}}_{NS}\left(X\cup Y\right)={\overline{apr}}_{NS}\left(X\right)\cup {\overline{apr}}_{NS}\left(Y\right)$ for all X, YU. Since NS(U) is a reflexive neighborhood system of U, so ${\overline{apr}}_{NS}\left(X\right)\subseteq {\overline{apr}}_{NS}\left({\overline{apr}}_{NS}\left(X\right)\right)$ for any XU. Therefore ${\overline{apr}}_{NS}\left(X\right)={\overline{apr}}_{NS}\left({\overline{apr}}_{NS}\left(X\right)\right)$ for any XU. It is obvious that (H2) holds. Thus ${\overline{apr}}_{NS}$ is a closure operator. □

#### Theorem 4.2

(Topological characterization of of NS(U) for ${\overline{apr}}_{NS}$ being a closure operator). Let NS(U) be a reflexive neighborhood system of U. Then ${\overline{apr}}_{NS}$ is a closure operator if and only if $\mathcal{N}=\left\{n\left(x\right):n\left(x\right)\in NS\left(x\right),x\in U\right\}$ is a base for some topology τ on U.

#### Proof

(⇒) Assume that ${\overline{apr}}_{NS}$ is a closure operator. We prove $\mathcal{N}=\left\{n\left(x\right)\phantom{\rule{thickmathspace}{0ex}}:\phantom{\rule{thickmathspace}{0ex}}n\left(x\right)\in NS\left(x\right),\phantom{\rule{thickmathspace}{0ex}}x\in U\right\}$ is a base for some topology τ on U. Since NS(U) is a reflexive neighborhood system of U. It is easy to prove xn(x) for each n(x) ∈ NS(x) and xU. Thus $\mathcal{N}$ is a cover of U. For any x, y, zU, n(y) ∈ NS(y), n(z) ∈ NS(z) and xn(y) ∩ n(z). Since ${\overline{apr}}_{NS}$ is a closure operator, then NS(U) is weak-unary. Therefore, there exists a n0(x) ∈ NS(x) such that n0 (x) ⊆ n(y) ∩ (z). By the definition of base, we have $\mathcal{N}=\left\{n\left(x\right)\phantom{\rule{thickmathspace}{0ex}}:\phantom{\rule{thickmathspace}{0ex}}n\left(x\right)\in NS\left(x\right),\phantom{\rule{thickmathspace}{0ex}}x\in U\right\}$ is a base for some topology τ on U.

(⇐) Assume that $\mathcal{N}=\left\{n\left(x\right)\phantom{\rule{thickmathspace}{0ex}}:\phantom{\rule{thickmathspace}{0ex}}n\left(x\right)\in NS\left(x\right),\phantom{\rule{thickmathspace}{0ex}}x\in U\right\}$ is a base for some topology τ on U. We prove that ${\overline{apr}}_{NS}$ satisfies the conditions (Hi)(i = 1,2,3,4). By Definition of ${\overline{apr}}_{NS}$, it is easy to check that ${\overline{apr}}_{NS}\left(\mathrm{\varnothing }\right)=\mathrm{\varnothing }$. Hence (H3) is satisfied. So to prove (H1), we prove (H1) holds. Since $\mathcal{N}=\left\{n\left(x\right):n\left(x\right)\in NS\left(x\right),x\in U\right\}$ is a base for some topology τ on U, by Definition 2.5, for any xU and n1(x), n2(x) ∈ NS(x), there exists an n3(x) ∈ NS(x) such that n3(x) ⊆ n1(x) ∩ n2(x); so NS(U) is weak-unary. By Proposition 2.10, we have ${\overline{apr}}_{NS}\left(X\cup Y\right)={\overline{apr}}_{NS}\left(X\right)\cup {\overline{apr}}_{NS}\left(Y\right)$ for all X, YU. Since NS(U) is a reflexive neighborhood system of U, ${\overline{apr}}_{NS}\left(X\right)\subseteq {\overline{apr}}_{NS}\left({\overline{apr}}_{NS}\left(X\right)\right)$ for any XU. So to prove (H1), we only need to prove ${\overline{apr}}_{NS}\left({\overline{apr}}_{NS}\left(A\right)\right)\subseteq {\overline{apr}}_{NS}\left(A\right)$ for any AU. Let $x\in {\overline{apr}}_{NS}\left({\overline{apr}}_{NS}\left(A\right)\right)$, by Definition of the ${\overline{apr}}_{NS}$, $n\left(x\right)\cap {\overline{apr}}_{NS}\left(A\right)\ne \mathrm{\varnothing }$ for any n(x) ∈ NS(x). Pick $p\in n\left(x\right)\cap {\overline{apr}}_{NS}\left(A\right)$. Then pn(x) and n(p) ∩ A ≠ ∅ for any n(p) ∈ NS(p). From pn(x), we obtain n(p) ⊆ n(x), therefore n(x) ∩ A ≠ ∅. By the arbitrariness of n, we have $x\in {\overline{apr}}_{NS}\left(A\right)$. Thus ${\overline{apr}}_{NS}\left(A\right)={\overline{apr}}_{NS}\left({\overline{apr}}_{NS}\left(A\right)\right)$ for any AU. Since NS(U) is a reflexive neighborhood system of U, (H2) is obvious satisfied. Therefore ${\overline{apr}}_{NS}$ is a closure operator. □

#### Corollary 4.3

Let NS(U) be a reflexive neighborhood system of U. Then ${\overline{apr}}_{NS}$ is a closure operator if and only if $\mathcal{N}=\left\{\cap NS\left(x\right)\phantom{\rule{thickmathspace}{0ex}}:\phantom{\rule{thickmathspace}{0ex}}x\in U\right\}$ is a base for some topology τ on U and NS(U) is transitive.

#### Proof

It is easy to prove by Proposition 3.4 and Theorem 4.2. □

## 5 Characterization of covering S for ${\overline{apr}}_{S}$ being a closure operator

Zhang, Li and Lin defined a special covering S and investigated twenty-three types of covering-based rough sets proposed in [49] which can be treated as the generalized rough sets based on neighborhood systems. In this section, we will discuss the properties of ${\overline{apr}}_{S}$ and give the Characterization of covering S for ${\overline{apr}}_{S}$ being a closure operator.

#### Definition 5.1

([49]). A family of subsets of universe U is called a closure system over U if it contains U and is closed under set intersection. Given a closure system $\overline{S}$, one can define its dual system $\underset{_}{S}$ as follows:

$S_={−X:X∈S¯}$

#### Definition 5.2

(Subsystem based definition [49]). Suppose $S=\left(\overline{S},\underset{_}{S}\right)$ is a pair of subsystems of $\mathcal{P}\left(U\right),\overline{S}$ is a closure system and $\underset{_}{S}$ is the dual system of $\overline{S}$. A pair of lower and upper approximation operators $\left({\overline{apr}}_{S},{\underset{_}{apr}}_{S}\right)$ with respect to S is defined as:

$apr¯S(X)=∩{K∈S¯:X⊆K};apr_S(X)=∪{K∈S_:K⊆X}foranyX⊆U.$

#### Remark 5.3

1. $\left(\overline{S},\phantom{\rule{thickmathspace}{0ex}}\subseteq \right)$ is a complete lattice.

2. $\overline{S}$ may not have element ∅. Fig.?? gives an intuitive illustration (2) of the Remark 5.3:

Fig. 1

3. If $\overline{S}$ has no less than two single sets, then $\mathrm{\varnothing }\in \overline{S}$. The converse may not hold:

Fig. 2

By Definition 5.2, it is easy to obtain properties of lower and upper approximation operators as follows:

#### Proposition 5.4

Let S = (S̅, S̲ be a pair of subsystems of $\mathcal{P}\left(U\right)$, for any X, YU, we have:

1. ${\underset{_}{apr}}_{S}\left(\mathrm{\varnothing }\right)=\mathrm{\varnothing }$;

2. ${\overline{apr}}_{S}\left(U\right)=U$;

3. $X\subseteq Y⇒{\overline{apr}}_{S}\left(X\right)\subseteq {\overline{apr}}_{S}\left(Y\right),{\underset{_}{apr}}_{S}\left(X\right)\subseteq {\underset{_}{apr}}_{S}\left(Y\right)$;

4. ${\overline{apr}}_{S}\left({\overline{apr}}_{S}\left(X\right)\right)={\overline{apr}}_{S}\left(X\right)$ for any XU;

5. ${\overline{apr}}_{S}\left(X\right)=-{\underset{_}{apr}}_{S}\left(-X\right),{\underset{_}{apr}}_{S}\left(X\right)=-{\overline{apr}}_{S}\left(-X\right)$.

However, the following properties may not hold:

(1) ${\overline{apr}}_{S}\left(\mathrm{\varnothing }\right)=\mathrm{\varnothing }$;

#### Example 5.5

Let $U=\left\{a,\phantom{\rule{thickmathspace}{0ex}}b,\phantom{\rule{thickmathspace}{0ex}}c\right\},\overline{S}=\left\{\left\{a\right\},\phantom{\rule{thickmathspace}{0ex}}\left\{a,\phantom{\rule{thickmathspace}{0ex}}b\right\},\phantom{\rule{thickmathspace}{0ex}}U\right\}$. It is easy to see $\overline{S}$ contains U and is closed under set intersection. But ${\overline{apr}}_{S}\left(\mathrm{\varnothing }\right)=\left\{a\right\}$.

(2) ${\underset{_}{apr}}_{S}\left(U\right)=U$;

#### Example 5.6

Let $U=\left\{a,\phantom{\rule{thickmathspace}{0ex}}b,\phantom{\rule{thickmathspace}{0ex}}c\right\},\overline{S}=\left\{\left\{a\right\},\phantom{\rule{thickmathspace}{0ex}}\left\{a,\phantom{\rule{thickmathspace}{0ex}}b\right\},\phantom{\rule{thickmathspace}{0ex}}U\right\},\underset{_}{S}=\left\{\left\{b,\phantom{\rule{thickmathspace}{0ex}}c\right\},\phantom{\rule{thickmathspace}{0ex}}\left\{c\right\},\phantom{\rule{thickmathspace}{0ex}}\mathrm{\varnothing }\right\}$. Then ${\underset{_}{apr}}_{S}\left(U\right)=\left\{b,c\right\}\ne U$.

(3) ${\overline{apr}}_{S}\left(X\cup Y\right)\subseteq {\overline{apr}}_{S}\left(X\right)\cup {\overline{apr}}_{S}\left(Y\right)$ for any X, YU.

#### Example 5.7

Let $U=\left\{a,\phantom{\rule{thickmathspace}{0ex}}b,\phantom{\rule{thickmathspace}{0ex}}c\right\},\overline{S}=\left\{\mathrm{\varnothing },\phantom{\rule{thickmathspace}{0ex}}\left\{a\right\},\phantom{\rule{thickmathspace}{0ex}}\left\{b\right\},\phantom{\rule{thickmathspace}{0ex}}U\right\}$. Let X = {a}, Y = {b}, then ${\overline{apr}}_{S}\left(X\right)=\left\{a\right\},{\overline{apr}}_{S}\left(Y\right)=\left\{b\right\},{\overline{apr}}_{S}\left(X\cup Y\right)=U$. Hence ${\overline{apr}}_{S}\left(X\cup Y\right)⊈{\overline{apr}}_{S}\left(X\right)\cup {\overline{apr}}_{S}\left(Y\right)$.

#### Definition 5.8

([53]). Suppose R is an arbitrary relation on U. With respect to R, we can define the left neighborhoods of an element x in U as follows:

$lR(x)={y∈U:yRx}$

#### Definition 5.9

([53]). For an arbitrary relation R, by substituting equivalence class [x]R with right neighborhood lR(x), we define the operators $\overline{R}$ and $\underset{_}{R}$(Lin, 1992) from P(U) to itselfas follows:

$R¯(X)={x∈U:lR(x)∩X≠∅}.$

#### Lemma 5.10

([53]). If S is another binary relation on U and $\overline{R}\left(X\right)=\overline{S}\left(X\right)$ for any XU, then R = S.

#### Proposition 5.11

Let $S=\left(\overline{S},\underset{_}{S}\right)$ be a pair of subsystems of $\mathcal{P}\left(U\right)$ and U is finite. If ${\overline{apr}}_{S}\left(X\cup Y\right)={\overline{apr}}_{S}\left(X\right)\cup {\overline{apr}}_{S}\left(Y\right)$ for any X, YU, then there exists a unique reflexive and transitive relation R on U such that ${\overline{apr}}_{S}\left(X\right)=\overline{R}\left(X\right)$ for any XU.

#### Proof

Using $\overline{S}$ of U, for any xU, we choose all $S\in \overline{S}$ such that {x} ⊆ S which forms the family S′, via left neighborhood of an element xU, we construct the binary R on U as follows:

lR(x) = {yU : y ∈ ∩ S It is clear that R is a reflexive relation. Since ${\overline{apr}}_{S}\left(\left\{x\right\}\right)={l}_{R}\left(x\right)$ and ${\overline{apr}}_{S}\left(X\cup Y\right)={\overline{apr}}_{S}\left(X\right)\cup {\overline{apr}}_{S}\left(Y\right)$ for any X, YU, we have ${\overline{apr}}_{S}\left(X\right)=\cup x\in x{\overline{apr}}_{S}\left(\left\{x\right\}\right)=\bigcup _{X\in X}{l}_{R}\left(x\right)=\overline{R}\left(X\right)$. Since ${\overline{apr}}_{S}\left({\overline{apr}}_{S}\left(X\right)\right)={\overline{apr}}_{S}\left(X\right)$ for any XU, this means that $\overline{R}\left(\overline{R}\left(X\right)\right)=\overline{R}\left(X\right)$. This implies R is transitive. Thus R is a reflexive and transitive relation. The unique R comes from Lemma 5.10. □

#### Proposition 5.12

Let S = (S̅ S̲) be a pair of subsystems of $\mathcal{P}\left(U\right)$, then the following are equivalent:

1. $\overline{S}$ has element ∅;

2. ${\overline{apr}}_{S}\left(\mathrm{\varnothing }\right)=\mathrm{\varnothing }$;

3. ${\underset{_}{apr}}_{S}\left(U\right)=U$.

#### Proof

(1) ⇒ (2) By the Definition 5.2, it is easy to prove.

(2) ⇔ (3). It can be obtained by the duality.

(2) ⇒ (1) If $\mathrm{\varnothing }\notin \overline{S}$. We can choose $\overline{S}\subseteq \overline{S},\mathrm{\varnothing }\subseteq S$ for any SS̅. Thus ∅ ⊆ ∩ S̅. Since $\overline{S}$ is closed for intersection, then ∩S̅ ∈ S̅ and ∩S̅≠∅. From the Definition 5.2, we have ${\overline{apr}}_{S}\left(\mathrm{\varnothing }\right)\ne \mathrm{\varnothing }$. It is a contradiction to (2). □

#### Lemma 5.13

For any X, YU, ${\overline{apr}}_{S}\left(X\cup Y\right)={\overline{apr}}_{S}\left(X\right)\cup {\overline{apr}}_{S}\left(Y\right)$ if and only if ABS̅ for any A, BS̅.

#### Proof

(⇒) Suppose there exists S1, S2S̅, it is easy to prove S1S2S̅.

(⇐) For any X, YU, we have:

$apr¯S(X∪Y)=∩{S∈S¯:(X∪Y)⊆S}∈S¯;apr¯S(X)=∩{S∈S¯:X⊆S}=S1∈S¯;apr¯S(Y)=∩{S∈S¯:Y⊆S}=S2∈S¯.$

Thus, we have XYS1S2S̅. Hence ${\overline{apr}}_{S}\left(X\cup Y\right)\subseteq {S}_{1}\cup {S}_{2}\subseteq {\overline{apr}}_{S}\left(X\right)\cup {\overline{apr}}_{S}\left(Y\right).{\overline{apr}}_{S}\left(X\right)\cup {\overline{apr}}_{S}\left(Y\right)\subseteq {\overline{apr}}_{S}\left(X\cup Y\right)$ is obvious. Therefore ${\overline{apr}}_{S}\left(X\cup Y\right)={\overline{apr}}_{S}\left(X\right)\cup {\overline{apr}}_{S}\left(Y\right)$. □

#### Theorem 5.14

(General characterization of coverings S for ${\overline{apr}}_{S}$ being a closure operator). Let $S=\left(\overline{S},\underset{_}{S}\right)$ be a pair of subsystems of $\mathcal{P}\left(U\right)$. ${\overline{apr}}_{S}$ is a closure operator if and only if $\mathrm{\varnothing }\in \overline{S}$ and ${S}_{1}\cup {S}_{2}\in \overline{S}$ for any ${S}_{1},{S}_{2}\in \overline{S}$.

#### Proof

(⇒) Assume ${\overline{apr}}_{S}$ is a closure operator. By the Proposition 5.4 and Lemma 5.13, we have $\mathrm{\varnothing }\in \overline{S}$ and S1S2S̅ for any S1, S2S̅.

(⇐) Assume ∅ ∈ S̅ and S1S2S̅ for any S1, S2S̅. We need to prove ${\overline{apr}}_{S}$ is a closure operator. By the Proposition 5.12, we have ${\overline{apr}}_{S}\left(\mathrm{\varnothing }\right)=\mathrm{\varnothing }$. Hence, (H3) is satisfied. By the Lemma 5.13, we have ${\overline{apr}}_{S}\left(X\cup Y\right)={\overline{apr}}_{S}\left(X\right)\cup {\overline{apr}}_{S}\left(Y\right)$ for any X, YU. Thus, (H1) is satisfied. By the Definition 5.2, it is easy to prove $X\subseteq {\overline{apr}}_{S}\left(X\right)$ for any XU.

Then we only need to prove (H4) holds. For any XU, ${\overline{apr}}_{S}\left(X\right)\subseteq {\overline{apr}}_{S}\left({\overline{apr}}_{S}\left(X\right)\right)$ is obvious, we prove ${\overline{apr}}_{S}\left({\overline{apr}}_{S}\left(X\right)\right)\subseteq {\overline{apr}}_{S}\left(X\right)$ for any XU. By the Definition 5.2, we have ${\overline{apr}}_{S}\left(X\right)=\cap \left\{S\in \overline{S}\phantom{\rule{thickmathspace}{0ex}}:\phantom{\rule{thickmathspace}{0ex}}X\subseteq S\right\}$. Denote ${S}_{1}=\cap \left\{S\in \overline{S}\phantom{\rule{thickmathspace}{0ex}}:\phantom{\rule{thickmathspace}{0ex}}X\subseteq S\right\}$. Since $\overline{S}$ is closed for intersection, so ${\overline{apr}}_{S}\left({\overline{apr}}_{S}\left(X\right)\right)={S}_{1}$, thus S1S̅. Therefore ${\overline{apr}}_{S}\left({\overline{apr}}_{S}\left(X\right)\right)={\overline{apr}}_{S}\left(X\right)$ for any XU. Hence ${\overline{apr}}_{S}$ is a closure operator. □

#### Lemma 5.15

Let S = (S̅, S̲ be a pair of subsystems of $\mathcal{P}\left(U\right)$. ∅ ∈ S̅ and ABS̅ for any A, BS̅ if and only if $\stackrel{^}{S}=\left\{U\mathrm{\setminus }S:S\in \overline{S}\right\}$ is a topology on U.

#### Proof

(⇒) Assume $\overline{S}$ satisfies the condition, we need to prove ŝ is a topology on U. Since ∅, US̅, so ∅, U ∈ ŝ.

For any X, Y ∈ ŝ, we prove XY ∈ ŝ. There exists S1, S1S̅ such that X = U\ S1 and Y = U\ S2. Thus XY = (U\ S1) ∩ (U\ S2) = U\ (S1S2). By the condition, we obtain (S1S2) ∈ S̅. From the definition of ŝ, we have U\ (S1S2) ∈ ŝ.

Let {Ai : iI} ⊆ ŝ, there exists {Sj : iI} ⊆ S̅ such that Ai = U\ Sj for each iI, thus ${\bigcup }_{i\in I}{A}_{i}={\bigcup }_{i\in I}\left(U\mathrm{\setminus }{S}_{i}\right)=U\mathrm{\setminus }{\bigcap }_{i\in I}{S}_{i}$. Since $\overline{S}$ is closed for intersection, so ${\bigcap }_{{}_{i\in I}}{S}_{i}\in \overline{S}$, therefore $U\mathrm{\setminus }{\bigcap }_{{}_{i\in I}}{S}_{i}\in \stackrel{^}{S}$. Thus ŝ = {U S : SS̅} is a topology on U.

(⇐) Similarly, we can prove the converse. □

From the Lemma 5.15, we have the following conclusion:

#### Theorem 5.16

(Topological characterization of coverings S for ${\overline{apr}}_{S}$ being a closure operator). Let S = (S̅, S̲ is a pair of subsystems of $\mathcal{P}\left(U\right).{\overline{\phantom{\rule{thinmathspace}{0ex}}apr}}_{S}$ is a closure operator if and only if ŝ = {U\ S : SS̅} is a topology on U.

## 6 Conclusions

In this paper, we not only obtained the properties of NS(U) and S, but also investigated two type approximation operators. We give general characterization of covering S for covering-based upper approximation operator ${\overline{apr}}_{S}$ being a closure operator. Besides this, We obtain topological characterizations of two types of upper approximation operators ${\overline{apr}}_{NS}$ and ${\overline{apr}}_{S}$ to be a closure operators. In our future work, we will investigate the intuitive characterizations of covering S for ${\overline{apr}}_{S}$ or ${\overline{apr}}_{NS}$ to be a closure operator and describe covering-based approximation space as some special types of information exchange systems when ${\overline{apr}}_{S}$ or ${\overline{apr}}_{NS}$ is a closure operator respectively.

## Acknowledgement

This work is supported by the Natural Science Foundation of China (No.11371130), the Natural Science Foundation of Guangxi (No. 2014GXNSFBA118015), Guangxi Universities Key Lab of Complex System optimization and Big Data Processing (No.2016CSOBDP0004) and Key research and development of Hunan province (No.2016JC2014).

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Accepted: 2017-01-17

Published Online: 2017-03-04

Citation Information: Open Mathematics, Volume 15, Issue 1, Pages 193–202, ISSN (Online) 2391-5455,

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