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The characterizations of upper approximation operators based on special coverings

  • Pei Wang and Qingguo Li EMAIL logo
Published/Copyright: March 4, 2017
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Open Mathematics
From the journal Open Mathematics Volume 15 Issue 1

Abstract

In this paper, we discuss the approximation operators apr¯NS and 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.

Keywords: Upper approximation operator; Closure operator; Base; Topology
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 apr¯S . Based on their works, we will investigate the properties of NS(U) and S. We also give characterizations for apr¯NS , 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 apr¯NS being a closure operator, while Section 5 considers the properties of S and apr¯S and obtains the general and topological characterizations of special covering S for covering-based upper approximation operator 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 P(U) denotes the family of all subsets of U. If C is a family of subsets of U, none of sets in C is empty, and ∪C=U , then C is called a covering of U.

Definition 2.1

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

Definition 2.2

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

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

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

Definition 2.3

(Covering approximation space [23]). If U is an universe and C is a covering of U, then we call U together with covering C a covering approximation space, denoted by (U,C) .

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 x ∈ U 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 x ∈ U and n(x) ∈ NS(x), there exists an n1(x) ∈ NS(x) satisfying thatfor any y ∈ n1(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 U1 ∩ U2 ∈ τ; (c)If A⊆τ , then ∪A∈τ.τ 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 B⊆τ is called a base for (U, τ) if for every non-empty open subset O of U and each x ∈ O, there exists a set B∈B such that x ∈ B ⊆ O. Equivalently, a family B⊆τ if every non-empty open subset O of U can be represented as the union of a subfamily of B .

  4. For any x ∈ U, a family B⊆τ is called alocal base at x for (U τ) if x ∈ B for each B∈B , and for every open subset O of U with x ∈ O, there exists a set B∈B such that B ⊆ O.

  5. If P is a partition of U, the topology τ = { O ⊆ U O} is the union of some members of P∪{∅} 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, y ∈ U with x ≠ y, there exist open sets O, O' such that x ∈ O, y ∈ O′ and O ∩ O′ = ∅, 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 X ⊆ U. It is easy to check that τ′ = {O∩ X : 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, Y ⊆ U,

(H1)H(X∪Y)=H(X)∪H(Y) ;

(H2)X⊆H(X) ;

(H3)H(∅)=∅ ;

(H4)H(H(X))=H(X) .

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, Y ⊆ U,

(I1)I(X∩Y)=I(X)∩I(Y) ;

(I2)I(X)⊆X ;

(I3)I(U)=U ;

(I4)I(I(X))=I(X) .

Definition 2.9

(Dual operator). Assume that H, I: P(U) → P(U) are two operators on U. If for any X ⊆ U, 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. apr¯NS(X∪Y)=apr¯NS(X)∪apr¯NS(Y)forallX,Y⊆U ;

  2. apr_NS(X∪Y)=apr_NS(X)∪apr_NS(Y)forallX,Y⊆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. apr¯NS(apr¯NS(X))⊆apr¯NS(X)forallX⊆U

  2. apr_NS(X)⊆apr_NS(apr_NS(Y))forallX⊆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, y ∈ U, and n(x) ∈ NS(x), y ∈ n(x) ⇒ n(x) ⊆ n(y).

Proposition 3.2

Let NS(U) be a neighborhood system of U. If NS(U) is Euclidean, then apr¯NS(apr_NS(X))⊆apr_NS(X) and apr¯NS(X)⊆apr_NS(apr¯NS(X)) for any X ⊆ U.

Proof

For any x ∈ U, if x∈apr¯NS(apr_NS(X)) and n(x)∈ NS(U), We have n(x)∩apr_NS(X)≠∅ . Pick y∈n(x)∩apr_NS(X) , 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∈apr_NS(X) , therefore apr¯NS(apr_NS(X))⊆apr_NS(X) .

We can get apr¯NS(X)⊆apr_NS(apr¯NS(X)) by the duality between apr¯NS and 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 apr¯NS(apr_NS(X))⊆apr_NS(X) for any X ⊆ U. Since a ∈ n(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, y ∈ U, if x ∈ n(y), then n(x)⊆ n(y).

Proof

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

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

Proposition 3.5

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

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

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

Proof

(1) ⇒ (2) For any x,y ∈ U and y ∈ n(x). Since {n(x) : x ∈ U} forms a partition of U and n : U → P(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, y ∈ U, if n(x) ≠ n(y), then n(x)>∩ n(y) = ∅. Otherwise, we take a z ∈ n(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 x ∈ U.

Proof

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

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

4 Characterization of NS(U) for apr¯NS being a closure operator

In Section 3, we discuss the properties of NS(U). One natural question thus arise: When is apr¯NS a closure operator?

Theorem 4.1

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

Proof

(⊆) Assume that apr¯NS is a closure operator. By the Definition 2.7, we get X⊆apr¯NS(X) , apr¯NS(apr¯NS(X))=apr¯NS(X) and apr¯NS(X∪Y)=apr¯NS(X)∪apr¯NS(Y) for all X, Y ⊆ U. 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 apr¯NS satisfies the conditions (Hi)(for i = 1, 2, 3, 4). By the Definition of apr¯NS , it is easy to check that apr¯NS(∅)=∅ . 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 apr¯NS(apr¯NS(X))⊆apr¯NS(X) and apr¯NS(X∪Y)=apr¯NS(X)∪apr¯NS(Y) for all X, Y ⊆ U. Since NS(U) is a reflexive neighborhood system of U, so apr¯NS(X)⊆apr¯NS(apr¯NS(X)) for any X ⊆ U. Therefore apr¯NS(X)=apr¯NS(apr¯NS(X)) for any X ⊆ U. It is obvious that (H2) holds. Thus apr¯NS is a closure operator. □

Theorem 4.2

(Topological characterization of of NS(U) for apr¯NS being a closure operator). Let NS(U) be a reflexive neighborhood system of U. Then apr¯NS is a closure operator if and only if N={n(x):n(x)∈NS(x),x∈U} is a base for some topology τ on U.

Proof

(⇒) Assume that apr¯NS is a closure operator. We prove N={n(x):n(x)∈NS(x),x∈U} is a base for some topology τ on U. Since NS(U) is a reflexive neighborhood system of U. It is easy to prove x ∈ n(x) for each n(x) ∈ NS(x) and x ∈ U. Thus N is a cover of U. For any x, y, z ∈ U, n(y) ∈ NS(y), n(z) ∈ NS(z) and x ∈ n(y) ∩ n(z). Since 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 N={n(x):n(x)∈NS(x),x∈U} is a base for some topology τ on U.

(⇐) Assume that N={n(x):n(x)∈NS(x),x∈U} is a base for some topology τ on U. We prove that apr¯NS satisfies the conditions (Hi)(i = 1,2,3,4). By Definition of apr¯NS , it is easy to check that apr¯NS(∅)=∅ . Hence (H3) is satisfied. So to prove (H1), we prove (H1) holds. Since N={n(x):n(x)∈NS(x),x∈U} is a base for some topology τ on U, by Definition 2.5, for any x ∈ U 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 apr¯NS(X∪Y)=apr¯NS(X)∪apr¯NS(Y) for all X, Y ⊆ U. Since NS(U) is a reflexive neighborhood system of U, apr¯NS(X)⊆apr¯NS(apr¯NS(X)) for any X ⊆ U. So to prove (H1), we only need to prove apr¯NS(apr¯NS(A))⊆apr¯NS(A) for any A ⊆ U. Let x∈apr¯NS(apr¯NS(A)) , by Definition of the apr¯NS , n(x)∩apr¯NS(A)≠∅ for any n(x) ∈ NS(x). Pick p∈n(x)∩apr¯NS(A) . Then p ∈ n(x) and n(p) ∩ A ≠ ∅ for any n(p) ∈ NS(p). From p ∈ n(x), we obtain n(p) ⊆ n(x), therefore n(x) ∩ A ≠ ∅. By the arbitrariness of n, we have x∈apr¯NS(A) . Thus apr¯NS(A)=apr¯NS(apr¯NS(A)) for any A ⊆ U. Since NS(U) is a reflexive neighborhood system of U, (H2) is obvious satisfied. Therefore apr¯NS is a closure operator. □

Corollary 4.3

Let NS(U) be a reflexive neighborhood system of U. Then apr¯NS is a closure operator if and only if N={∩NS(x):x∈U} 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 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 apr¯S and give the Characterization of covering S for 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 S¯ , one can define its dual system S_ as follows:

S_={−X:X∈S¯}

Definition 5.2

(Subsystem based definition [49]). Suppose S=(S¯,S_) is a pair of subsystems of P(U),S¯ is a closure system and S_ is the dual system of S¯ . A pair of lower and upper approximation operators (apr¯S,apr_S) with respect to S is defined as:

a p r ¯ S ( X ) = ∩ { K ∈ S ¯ : X ⊆ K } ; a p r _ S ( X ) = ∪ { K ∈ S _ : K ⊆ X } f o r a n y X ⊆ U .

Remark 5.3

  1. (S¯,⊆) is a complete lattice.

  2. S¯ may not have element ∅. Fig.?? gives an intuitive illustration (2) of the Remark 5.3:

    Fig. 1
    Fig. 1

  3. If S¯ has no less than two single sets, then ∅∈S¯ . The converse may not hold:

    Fig. 2
    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 P(U) , for any X, Y ∈ U, we have:

  1. apr_S(∅)=∅ ;

  2. apr¯S(U)=U ;

  3. X⊆Y⇒apr¯S(X)⊆apr¯S(Y),apr_S(X)⊆apr_S(Y) ;

  4. apr¯S(apr¯S(X))=apr¯S(X) for any X ⊆ U;

  5. apr¯S(X)=−apr_S(−X),apr_S(X)=−apr¯S(−X) .

However, the following properties may not hold:

(1) apr¯S(∅)=∅ ;

Example 5.5

Let U={a,b,c},S¯={{a},{a,b},U} . It is easy to see S¯ contains U and is closed under set intersection. But apr¯S(∅)={a} .

(2) apr_S(U)=U ;

Example 5.6

Let U={a,b,c},S¯={{a},{a,b},U},S_={{b,c},{c},∅} . Then apr_S(U)={b,c}≠U .

(3) apr¯S(X∪Y)⊆apr¯S(X)∪apr¯S(Y) for any X, Y ⊆ U.

Example 5.7

Let U={a,b,c},S¯={∅,{a},{b},U} . Let X = {a}, Y = {b}, then apr¯S(X)={a},apr¯S(Y)={b},apr¯S(X∪Y)=U . Hence apr¯S(X∪Y)⊈apr¯S(X)∪apr¯S(Y) .

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 R¯ and 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 R¯(X)=S¯(X) for any X ⊆ U, then R = S.

Proposition 5.11

Let S=(S¯,S_) be a pair of subsystems of P(U) and U is finite. If apr¯S(X∪Y)=apr¯S(X)∪apr¯S(Y) for any X, Y ⊆ U, then there exists a unique reflexive and transitive relation R on U such that apr¯S(X)=R¯(X) for any X ⊆ U.

Proof

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

lR(x) = {y ∈ U : y ∈ ∩ S It is clear that R is a reflexive relation. Since apr¯S({x})=lR(x) and apr¯S(X∪Y)=apr¯S(X)∪apr¯S(Y) for any X, Y ⊆ U, we have a p r ¯ S ( X ) = ∪ x ∈ x a p r ¯ S ( { x } ) = ⋃ X ∈ X l R ( x ) = R ¯ ( X ) . Since apr¯S(apr¯S(X))=apr¯S(X) for any X ⊆ U, this means that R¯(R¯(X))=R¯(X) . 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 P(U) , then the following are equivalent:

  1. S¯ has element ∅;

  2. apr¯S(∅)=∅ ;

  3. apr_S(U)=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 ∅∉S¯ . We can choose S¯⊆S¯,∅⊆S for any S ∈ S̅. Thus ∅ ⊆ ∩ S̅. Since S¯ is closed for intersection, then ∩S̅ ∈ S̅ and ∩S̅≠∅. From the Definition 5.2, we have apr¯S(∅)≠∅ . It is a contradiction to (2). □

Lemma 5.13

For any X, Y ⊆ U, apr¯S(X∪Y)=apr¯S(X)∪apr¯S(Y) if and only if A ∪ B ∈ S̅ for any A, B ∈ S̅.

Proof

(⇒) Suppose there exists S1, S2 ∈ S̅, it is easy to prove S1 ∪ S2 ∈ S̅.

(⇐) For any X, Y ⊆ U, 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 X ∪ Y ⊆ S1 ∪ S2 ∈ S̅. Hence apr¯S(X∪Y)⊆S1∪S2⊆apr¯S(X)∪apr¯S(Y).apr¯S(X)∪apr¯S(Y)⊆apr¯S(X∪Y) is obvious. Therefore apr¯S(X∪Y)=apr¯S(X)∪apr¯S(Y) . □

Theorem 5.14

(General characterization of coverings S for apr¯S being a closure operator). Let S=(S¯,S_) be a pair of subsystems of P(U) . apr¯S is a closure operator if and only if ∅∈S¯ and S1∪S2∈S¯ for any S1,S2∈S¯ .

Proof

(⇒) Assume apr¯S is a closure operator. By the Proposition 5.4 and Lemma 5.13, we have ∅∈S¯ and S1 ∪ S2 ∈ S̅ for any S1, S2 ∈ S̅.

(⇐) Assume ∅ ∈ S̅ and S1 ∪ S2 ∈ S̅ for any S1, S2 ∈ S̅. We need to prove apr¯S is a closure operator. By the Proposition 5.12, we have apr¯S(∅)=∅ . Hence, (H3) is satisfied. By the Lemma 5.13, we have apr¯S(X∪Y)=apr¯S(X)∪apr¯S(Y) for any X, Y ⊆ U. Thus, (H1) is satisfied. By the Definition 5.2, it is easy to prove X⊆apr¯S(X) for any X ⊆ U.

Then we only need to prove (H4) holds. For any X ⊆ U, apr¯S(X)⊆apr¯S(apr¯S(X)) is obvious, we prove apr¯S(apr¯S(X))⊆apr¯S(X) for any X ⊆ U. By the Definition 5.2, we have apr¯S(X)=∩{S∈S¯:X⊆S} . Denote S1=∩{S∈S¯:X⊆S} . Since S¯ is closed for intersection, so apr¯S(apr¯S(X))=S1 , thus S1 ∈ S̅. Therefore apr¯S(apr¯S(X))=apr¯S(X) for any X ⊆ U. Hence apr¯S is a closure operator. □

Lemma 5.15

Let S = (S̅, S̲ be a pair of subsystems of P(U) . ∅ ∈ S̅ and A ∪ B ∈ S̅ for any A, B ∈ S̅ if and only if S^={U∖S:S∈S¯} is a topology on U.

Proof

(⇒) Assume S¯ satisfies the condition, we need to prove ŝ is a topology on U. Since ∅, U ∈ S̅, so ∅, U ∈ ŝ.

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

Let {Ai : i ∈ I} ⊆ ŝ, there exists {Sj : i ∈ I} ⊆ S̅ such that Ai = U\ Sj for each i ∈ I, thus ⋃i∈IAi=⋃i∈I(U∖Si)=U∖⋂i∈ISi . Since S¯ is closed for intersection, so ⋂i∈ISi∈S¯ , therefore U∖⋂i∈ISi∈S^ . Thus ŝ = {U S : S ∈ S̅} 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 apr¯S being a closure operator). Let S = (S̅, S̲ is a pair of subsystems of P(U).apr¯S is a closure operator if and only if ŝ = {U\ S : S ∈ S̅} 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 apr¯S being a closure operator. Besides this, We obtain topological characterizations of two types of upper approximation operators apr¯NS and apr¯S to be a closure operators. In our future work, we will investigate the intuitive characterizations of covering S for apr¯S or apr¯NS to be a closure operator and describe covering-based approximation space as some special types of information exchange systems when apr¯S or 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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Received: 2016-7-23
Accepted: 2017-1-17
Published Online: 2017-3-4

© 2017 Wang and Li

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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https://doi.org/10.1515/math-2017-0015
Keywords for this article
Upper approximation operator; Closure operator; Base; Topology
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  239. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
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  242. Linear dynamics of semigroups generated by differential operators
  243. Special Issue on Recent Developments in Differential Equations
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  245. Special Issue on Recent Developments in Differential Equations
  246. Determination of a diffusion coefficient in a quasilinear parabolic equation
  247. Special Issue on Recent Developments in Differential Equations
  248. Homogeneous two-point problem for PDE of the second order in time variable and infinite order in spatial variables
  249. Special Issue on Recent Developments in Differential Equations
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  251. Special Issue on Recent Developments in Differential Equations
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  253. Special Issue on Recent Developments in Differential Equations
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  255. Special Issue on Recent Developments in Differential Equations
  256. Existence of solutions for delay evolution equations with nonlocal conditions
  257. Special Issue on Recent Developments in Differential Equations
  258. Comments on behaviour of solutions of elliptic quasi-linear problems in a neighbourhood of boundary singularities
  259. Special Issue on Recent Developments in Differential Equations
  260. Coupled fixed point theorems in complete metric spaces endowed with a directed graph and application
  261. Special Issue on Recent Developments in Differential Equations
  262. Existence of entropy solutions for nonlinear elliptic degenerate anisotropic equations
  263. Special Issue on Recent Developments in Differential Equations
  264. Integro-differential systems with variable exponents of nonlinearity
  265. Special Issue on Recent Developments in Differential Equations
  266. Elliptic operators on refined Sobolev scales on vector bundles
  267. Special Issue on Recent Developments in Differential Equations
  268. Multiplicity solutions of a class fractional Schrödinger equations
  269. Special Issue on Recent Developments in Differential Equations
  270. Determining of right-hand side of higher order ultraparabolic equation
  271. Special Issue on Recent Developments in Differential Equations
  272. Asymptotic approximation for the solution to a semi-linear elliptic problem in a thin aneurysm-type domain
  273. Topical Issue on Metaheuristics - Methods and Applications
  274. Learnheuristics: hybridizing metaheuristics with machine learning for optimization with dynamic inputs
  275. Topical Issue on Metaheuristics - Methods and Applications
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  277. Topical Issue on Cyber-security Mathematics
  278. Monomial codes seen as invariant subspaces
  279. Topical Issue on Cyber-security Mathematics
  280. Expert knowledge and data analysis for detecting advanced persistent threats
  281. Topical Issue on Cyber-security Mathematics
  282. Feedback equivalence of convolutional codes over finite rings
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