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Topological characterizations of covering for special covering-based upper approximation operators
Information Sciences 204 (2012) 70–81
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Topological characterizations of covering for special covering-based upper approximation operators q Xun Ge a,b, Xiaole Bai c, Ziqiu Yun a,⇑ a
School of Mathematical Sciences, Soochow University, Suzhou 215006, PR China Zhangjiagang College, Jiangsu University of Science and Technology, Zhangjiagang 215600, PR China c Alliance Data Systems Corp., Columbus, OH 43219, USA b
a r t i c l e
i n f o
Article history: Received 22 June 2011 Received in revised form 19 January 2012 Accepted 1 April 2012 Available online 11 April 2012 Keywords: Covering-based rough set Unary covering Closure operator Base of a topology
a b s t r a c t The relationships among properties of covering approximation operators and their corresponding coverings have attracted intensive research in recent years. In particular, those among topological properties have drawn special attention because of their important applications in rough set theory. In this paper, we give topological characterizations of covering C for covering-based upper approximation operators FH, SH, TH and RH to be closure operators. We also give intuitive characterizations of covering C, and describe coveringbased approximation space (U, C) as certain types of information exchange systems when SH or RH is a closure operator. By applying our new characterizations, we give inequalities about the relationship between the number of members in C and the number of elements in U, and discuss relationships among conditions for different covering based upper approximation operators to be closure operators. To the best of our knowledge, it is the first time that such characterizations, descriptions, inequalities and discussions are systematically considered in the literature of rough set theory. Furthermore, in this paper we also give several characterizations of unary coverings, an important type of coverings in studying relationships among basic concepts in covering-based rough sets, by the relationships among different types of covering-base approximation operators. 2012 Elsevier Inc. All rights reserved.
1. Introduction Rough set theory has been acknowledged as a useful and powerful tool in data analysis particularly for dealing with granularity and vagueness [2,10,11,14,16,20–25,29,35–37,39,41–45,47,48,51,54,57]. The classical rough set theory is based on partitions of a universe [7,15,17,19,21,26,27,52,53,64], which imposes restrictions and limitations on many applications [56,67]. Zakowski generalized the classical rough set theory using coverings of a universe instead of partitions [56]. Such generalization leads to various covering approximation operators that are of both theoretical and practical importance [4,5,8,18,31,34,38,40,49,50,60,61,65]. In recent years, as data mining gets increasingly popular, the relationships between properties of covering-based approximation operators and their corresponding coverings have attracted intensive research [1,30,31,55,63,65–67,69,70]. It is worth noting that topological approaches have provided a valuable perspective and played an important role in rough set theory study [3,9,12,13,28,32,33,55,58,63]. Hence, topological properties have obtained a lot of attention [3,12,13,46,52,55,59,62,63]. In [66,68], Zhu and Wang discussed the relationship between properties of four types of covering-based upper approximation operators and their corresponding coverings. These four operators are named, q
This paper is supported by NSFC (Nos. 61070245, 10971185 and 11061004).
⇑ Corresponding author. Tel.: +86 512 65235165.
E-mail addresses: [email protected] (X. Ge), [email protected] (X. Bai), [email protected] (Z. Yun). 0020-0255/$ - see front matter 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.ins.2012.04.005
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respectively, the first, the second, the third, and the fourth type of covering-based upper approximation operators, and are denoted by FH, SH, TH and RH respectively (definitions of these operators are provided in the next section). Specifically, Zhu and Wang gave characterizations of covering C for FH, SH, and TH to be closure operators by the following results. Proposition 1.1 ([66,68]). FH is a closure operator if and only if C satisfies one of the following conditions: (1) "K1, K2 2 C, K1 \ K2 is a union of elements of C; (2) C is unary.
Proposition 1.2 ([68,70]). SH is a closure operator if and only if C satisfies the following condition: "x 2 U and K 2 C, either K # Friends(x) or K \ Friends(x) = /. Proposition 1.3 ([68,70]). TH is a closure operator if and only if C satisfies the following condition: "x 2 U and K 2 C, K \ CFriends(x) is a union of elements of C. These results are important and interesting, but not complete. There are several important yet open problems following this direction. First, to the best of our knowledge, so far there is no result on characterizations of covering C for RH to be a closure operator (only a necessary condition of covering C for RH to be a closure operator was presented in [66]). Hence, it is natural to ask: Question 1.4. Can we find characterizations of covering C for RH to be a closure operator? Notice that the relationship between definitions of RH and FH is similar to that between definitions of SH and TH; the latter two definitions (of FH and TH) just use CFriends instead of Friends as in the former ones (of RH and SH). It is important to know: Question 1.5. If the answer for Question 1.4 is ‘‘yes’’, then what is the relationship among characterizations of covering C for the aforementioned four types of operators to be closure ones? Second, the known characterizations of covering C that were presented in [66,68,70] for operators FH, SH and TH to be closure operators have nothing to do with topology, while being a closure operator is a notable topological property. It is then reasonable to consider: Question 1.6. Do topological characterizations of covering C exist for FH, SH, TH and RH to be closure operators? In this paper, we not only give answers to all these questions, but also give intuitive characterizations of covering C. By applying these characterizations, we describe covering-based approximation space (U, C) as certain types of information exchange systems when SH or RH is a closure operator, give two inequalities on the relationship between cardinalities of C and U when SH or RH is a closure operator, and discuss relationships among conditions for different covering based upper approximation operators to be closure ones. To the best of our knowledge, it is the first time that such characterizations, descriptions, inequalities and discussions are systematically considered in the literature of rough set theory. Giving characterizations of unary coverings is useful, since unary coverings are important coverings in discussing relationships among basic concepts in covering-based rough sets [65,66,68,70]. In particular, a characterization of unary coverings was presented in [68] using the equality of two types of covering-based upper approximation operators: a covering C is a unary covering if and only if TH = FH. In this paper, we first give a topological characterization of unary coverings without using any covering-based upper approximation operators. As applications of this characterization, we give sufficient and necessary conditions from the perspective of topology for FH or RH to be a closure operator. Then we give several characterizations of unary coverings by the relationships among different types of covering-based approximation operators. The rest of the paper is organized as follows. After giving fundamental concepts in Section 2, we present our main results in Sections 3–6. In these sections, we provide topological characterizations for the aforementioned four types of coveringbased upper approximation operators to be closure operators, intuitive characterizations of covering C, descriptions of covering-based approximation space (U, C) using some special types of information exchange systems, and the inequalities about the relationship between the cardinalities of C and U. In Section 7, we discuss relationships among conditions for different covering based upper approximation operators to be closure operators. In Section 8, we give characterizations of unary covering by relationships among different types of covering-based approximation operators. This paper concludes in Section 9 with remarks and questions we shall consider in our future study. 2. Basic definitions In this section, we present some basic concepts that are used in this paper. In the following discussion, unless it is mentioned specially, the universe of discourse U is considered finite. P(U) denotes the family of all subsets of U. C is a family of S subsets of U. If none of subsets in C is empty, and C = U, then C is called a covering of U. We call ordered pair (U, C) a covering-based approximation space.
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Let (U, C) be a covering-based approximation space and K 2 C, x 2 K. x is called a representative element of K if "S 2 C(x 2 S ) K # S). S Let C be a covering of U, x 2 U. Denote Md(x) = {K 2 C:x 2 K ^ ("S 2 C ^ x 2 S ^ S # K ) K = S)}, Friends(x) = {K 2 C:x 2 K}, S T CFriends(x) = Md(x), and N(x) = {K 2 C:x 2 K}. Let C be a covering of U. C is called unary if "x 2 U, jMd(x)j = 1. Remark. The concept of a representative element of K first appeared in [1]. Md(x), Friends(x), CFriends(x) and N(x), which are called the minimal description of x, the indiscernible neighborhood of x, closed friends of x, and the neighborhood of x, respectively, first appeared in [1,68,63]. The concept of unary covering was first proposed in [68] to characterize the condition for FH and TH to be equal. Let (U, C) be a covering-based approximation space. Our discussion in this paper involves the following types of coveringbased approximation operators. Definition 2.1 (Covering-based lower approximation operators CL and C2). Let C be a covering of U. The lower approximation operators CL and C2 are defined as follows, respectively: S "X 2 P(U), CL(X) = {K 2 C:K # X} and C2(X) = {x 2 U:N(x) # X}.
Definition 2.2 (Covering-based upper approximation operators FH, SH, TH and RH). Let C be a covering of U. Operators FH, SH, TH, RH:P(U) ? P(U) are defined as follows: "X 2 P(U), S S S FH(X) = CL(X) [ ( { Md(x):x 2 (X CL(X))}) = CL(X) [ ( {CFriends(x):x 2 (X CL(X))}), S S SH(X) = {K 2 C:K \ X – ;} = {Friends(x):x 2 X}, S S S TH(X) = { Md(x):x 2 X} = {CFriends(x):x 2 X}, and S S RH(X) = CL(X) [ ( {K 2 C:K \ (X CL(X)) – ;}) = CL(X) [ ( {Friends(x):x 2 (X CL(X))}).
Remark. Covering-based lower approximation operators CL and C2 were first defined in [56,31], respectively. The coveringbased upper approximation operators FH, SH and TH were called the first, the second, and the third type of covering-based upper approximation operators respectively in [68]. RH was called the fourth type of covering-based upper approximation operator in [66]. The following topological concepts and facts are elementary and can be found in [6]. We list them below for the purpose of this paper being self-contained. A topological space is a pair (U, s) consisting of a set U and a family s of subsets of U satisfying the following conditions: S (1) ; 2 s and U 2 s; (2) If U1 2 s and U2 2 s, then U1 \ U2 2 s; and (3) If A # s, then A 2 s. s is called a topology on U and the members of s are called open sets of (U, s). A family B # s is called a base for (U, s) if for every non-empty open subset O of U and each x 2 O, there exists a set B 2 B such that x 2 B # O. Equivalently, a family B # s is called a base for (U, s) if every non-empty open subset O of U can be represented as union of a subfamily of B. "x 2 U, a family B # s is called a local base at x for (U, s) if x 2 B for each B 2 B, and for every open subset O of U with x 2 O, there exists a set B 2 B such that B # O. For any set U consisting of no less than two points, there are more than one topologies on it. Let s and s0 be two topologies on U. If s # s0 , then we say that s is coarser than s0 or s0 is finer than s. Among all the topologies on U, there is a finest one which is called the discrete topology. A topology s is the discrete topology on U if and only if "x 2 U, {x} 2 s. A topology space (U, s) is called a discrete space if s is the discrete topology on U. If P is a partition of U, then it is easy to verify that s = {O # U:O is union of members of P} [ {/} is a topology of U, and was called pseudo-discrete topology in [13] (or called closed-open topology in [12]). Assume (U, s) is a topology space and X # U. It is easy to check that s0 = {O \ X:O 2 s} is a topology on X. s0 is called the induced topology, and topology space (X, s0 ) is called a subspace of (U, s). Definition 2.3 (Closure operator). An operator c:P(U) ? P(U) is called a closure operator on U if it satisfies the following axioms: "X, Y # U, Axiom Axiom Axiom Axiom
1. 2. 3. 4.
c(X [ Y) = c(X) [ c(Y); X # c(X); c(;) = ;; c(c(X)) = c(X).
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3. Topological characterization of coverings C for FH to be a closure operator In this section, we first give a topological characterization of unary coverings without involving any approximation operators. As an application of this result, we obtain a topological characterization of coverings C for FH to be a closure operator. Lemma 3.1. Let C be a unary covering of U, then for any x 2 U, there exists a Kx 2 C such that Md(x) = {Kx} and x is a representative element of Kx. Proof. This lemma can be proved by the definitions of Md and the representative element, and by an argument similar to the proof for Theorem 3 in [66]. h Lemma 3.2. A covering C is unary if and only if "K1, K2 2 C and "x 2 K1 \ K2, there is a K 2 C such that x 2 K # K1 \ K2. Proof. Suppose C is unary. "K1, K2 2 C and "x 2 K1 \ K2, let Md(x) = {Kx}. By Lemma 3.1, x is a representative element of Kx, and it follows that x 2 Kx # K1 \ K2, where Kx 2 C. On the other hand, if C is not unary, there is a x 2 U and there are K1, K2 2 Md(x) such that K1 – K2. By our assumption, there is a K 2 C such that x 2 K # K1 \ K2. It contradicts the fact that K1, K2 2 Md(x). h Lemma 3.3. (Proposition 1.2.1 in 6). A covering C of U is a base for some topology s on U if and only if "K1, K2 2 C and "x 2 K1 \ K2, there is a K 2 C such that x 2 K # K1 \ K2. The following result is only a combination of Lemmas 3.2 and 3.3: Theorem 3.4. (Topological characterization of unary coverings). A covering C is unary if and only if there exists a topology s on U, such that C is a base for (U, s). Combining with Proposition 1.1, we have a topological characterization for FH to be a closure operator as follows. Theorem 3.5. (Topological characterization for FH to be a closure operator). Covering-based upper approximation operator FH is a closure operator if and only if there exists a topology s on U, such that C is a base for (U, s).
Remark. The following two examples show that Lemmas 3.1, 3.2, Theorems 3.4, and 3.5 do not hold when U is an arbitrary universe: Example 3.6. Let U be the closed interval [1, 1] and C ¼ ffxg : x 2 ½1; 1 f0gg [ 1n ; 1n : n ¼ 1; 2; 3; . . . [ ff1; 0; 1gg. Then C is a covering of U and it is easy to check Md(x) = {{x}} for x 2 [1, 1] {0} and Md(0) = {{1, 0, 1}}. It follows that C is unary. However, none of Lemmas 3.1 and 3.2, and Theorem 3.4 holds for (U, C). Example 3.7. Let U = R, where R is the set of real numbers. Let C ¼ x 1n ; x þ 1n : x 2 U and n ¼ 1; 2; 3; . . . . Then C is a base of the usual topology on U. It is easy to check {x} R C and Md(x) = / for each x 2 R, and hence FH({x}) = / for each x 2 U. Thus, FH is not a closure operator, which shows Theorem 3.5 does not hold for (U, C).
4. Characterizations of covering C for SH to be a closure operator Theorem 4.1. (Topological characterization for SH to be a closure operator). SH is a closure operator if and only if {Friends(x):x 2 U} forms a partition of U. Proof. Using the definitions of SH and the closure operator, it is easy to check that if {Friends(x):x 2 U} forms a partition of U, then SH is a closure operator. On the other hand, assume that SH is a closure operator. We only need to prove that "x, y 2 U, if Friends(x) – Friends(y), then Friends(x) \ Friends(y) = /. Otherwise, we can take a z 2 Friends(x) \ Friends(y). By the assumpS tion that SH is a closure operator, we have that Friends(z) # {Friends(t):t 2 Friends(y)} = Friends(y) and Friends(y) # S {Friends(t):t 2 Friends(z)} = Friends(z), and hence Friends(z) = Friends(y). Similarly, Friends(z) = Friends(x). It follows that h Friends(x) = Friends(y), which contradicts to the fact that Friends(x) – Friends(y). Theorem 4.1 has several applications. By Theorem 4.1, we can not only obtain a topological characterization and an intuitive characterization of covering C for SH to be a closure operator, but also describe covering-based approximation space (U, C) as a special information system when SH is a closure operator.
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Since a covering C of U forms a base of some pseudo-discrete topology on U if and only if C is a partition of U, we get the following topological characterization of the covering for SH to be a closure operator. Theorem 4.2. SH is a closure operator if and only if there is a pseudo-discrete topology s on U such that {Friends(x):x 2 U} is a base of s. Remark. Finiteness of U is not used in the proof of Theorem 4.1. Therefore, Theorems 4.1 and 4.2 hold when U is an arbitrary set. For an intuitive characterization, we need the following definition. Definition 4.3. (Triangle Chain with three points). Let C be a covering of U. For any x, y, z 2 U, if either there exist K1, K2, K3 2 C, such that x, y 2 K1, y, z 2 K2, and z, x 2 K3, then we say that there is a triangle chain with three points x, y, z. In the following Fig. 1, U1 gives an intuitive illustration of the Triangle Chain with three points. Theorem 4.4. (Triangle Chain Condition). SH is a closure operator if and only if covering C divides U into disjoint parts U1, U2, . . ., Un on U, such that for each Ui with 1 6 i 6 n and "x, y, z 2 Ui, there is a Triangle Chain with three points x, y, z. The description in Theorem 4.4 is illustrated in Fig. 1 via a simple example. In Fig. 1, U = {a, b, c, d, e, f, g} and K1 = {a, b}, K2 = {b, c}, K3 = {a, c}, K4 = {d, e, f}, K5 = {e, f, g}, and K6 = {d, g}. Covering C = {Ki:i = 1, 2, 3, 4, 5, 6} divides U into two disjoint parts U1 and U2. It is obvious that points of U1 or U2 satisfy the Triangle Chain Condition. Theorem 4.4 can be also stated in a brief way as follows. Theorem 4.5. SH is a closure operator if and only if "x, y 2 U, either Friends(x) \ Friends(y) = /, or x 2 Friends(y). Proof. The ‘‘only if’’ part holds directly following Theorem 4.1. Now we consider the ‘‘if’’ part. Assume that "x, y 2 U, either Friends(x) \ Friends(y) = /, or x 2 Friends(y). To prove that SH is a closure operator, it only needs to show that {Friends(x):x 2 U} forms a partition of U by Theorem 4.1. It then becomes enough for us to prove that Friends(x) = Friends(y) if "x, y 2 U with Friends(x) \ Friends(y) – /. We prove that Friends(y) # Friends(x) first. By Friends(x) \ Friends(y) – / and our assumption, we have x 2 Friends(y). This means there is a K 2 C, such that x, y 2 K, and hence K # Friends(x) \ Friends(y). "z 2 Friends(y), we have y 2 Friends(z), thus K \ Friends(z) – /. It follows that Friends(x) \ Friends(z) – /. By our assumption, z 2 Friends(x). Since z is arbitrary, Friends(y) # Friends(x). With a similar argument, we can prove Friends(x) # Friends(y). It follows that Friends(x) = Friends(y). h Remark. Put a social network analysis scenario, assuming that U represents a set of people. "x, y 2 U, if x 2 Friends(y), then we say x is a friend of y or x, y are friends. Theorem 4.5 can be explained as: SH is a closure operator if and only if people in U have the relationship that if two people have a common friend, then they are friends themselves; in other words, only friends can have a common friend. Moreover, when SH is a closure operator, we can describe covering-based approximation space (U, C) as one type of information exchange system as follows. Let U be a set of people. For each pair x, y 2 U, if there is a people group, e.g., club or community, K 2 C such that x, y 2 K, then we say that x and y can exchange information directly; If there are people t1, t2, . . ., tn and people groups K0, K1, K2, . . ., Kn 2 C, such that x, t1 2 K0, t1, t2 2 K1, . . ., tn1, tn 2 Kn1, tn, y 2 Kn, then we say that x and y can exchange information. With this description, we have the following result. Theorem 4.6. SH is a closure operator if and only if information exchange system (U, C) described as above satisfies that: as long as two people can exchange information, they can also directly do that.
Fig. 1. Illustration for the triangle chain condition.
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Proof. Assume that SH is a closure operator and two people x, y can exchange information. Let there be people t1, t2, . . ., tn and people groups K0, K1, K2, . . ., Kn, such that x, t1 2 K0, t1, t2 2 K1, . . ., tn1, tn 2 Kn1, tn, y 2 Kn. By the Triangle Chain Condition, people x, t2 can exchange information directly. Then using the Triangle Chain Condition again, people x, t3 can exchange information directly. Continue in this way, at the nth step, people x, y can exchange information directly. On the other hand, if it holds that two people can exchange information directly as long as they can exchange information, then the Triangle Chain Condition is satisfied, and then SH is a closure operator. h At the end of this section, we give an inequality on the relationship between the number of elements in C that cover Friends(x) and the number of elements in Friends(x). For x 2 U, let Cx = {K 2 C:K \ Friends(x) – /} = {K1, K2, . . . , Kn}. Theorem 4.7. If SH is a closure operator, then "x 2 U, either there exists a K 2 C such that K covers Friends(x), or Pn i¼1 jK i j P 2jFriendsðxÞj holds. Here for set X, jXj denotes the number of elements of X. Proof. When SH is a closure operator, consider that there does not exist a K 2 C such that K covers Friends(x). To prove the conclusion, it is enough to prove that each y 2 Friends(x) is contained in more than one member of Cx. Take i with 1 6 i 6 n such that y 2 Ki. Since Ki does not cover Friends(x), there exists z 2 Friends(x) such that z R Ki. By the Triangle Chain Condition, there is a j with 1 6 j 6 n such that z, y 2 Kj. Since z 2 Kj and z R Ki, j – i, and it follows that y is contained in at least two members of Cx. h The following simple example illustrates a scenario when the equivalence in the above inequality holds. Example 4.8. Let U = {x, y, z}, C = {{x, y}, {z, x}, {y, z}}. Since " X # U with X – /, SH(X) = U, SH is a closure operator. One can Pn check it easily that 8w 2 U; i¼1 jK i j ¼ 2jFriendsðwÞj ¼ 6. 5. Topological characterization of coverings C for TH to be a closure operator Lemma 5.1. Covering-based upper approximation operator TH is a closure operator if and only if "x 2 U, x is a representative element of CFriends(x) for covering {CFriends(y):y 2 U}. S Proof. Assume that TH is a closure operator. "x 2 U, if x 2 CFriends(y) for y 2 U, then CFriends(x) # {CFriends(z):z 2 CFriends(y)} = TH(CFriends(y)) = TH(TH(y)) = TH(y) = CFriends(y). It follows that x is a representative element of CFriends(x) for covering {CFriends(y):y 2 U}. Assume that "x 2 U, x is a representative element of CFriends(x) for covering {CFriends(y):y 2 U}. "z 2 TH({x}) = CFriends(x), since z is a representative element of CFriends(z) for covering {CFriends(y):y 2 U}, TH({z}) = CFriends(z) # S CFriends(x) = TH({x}), and it follows that TH(TH({x})) = {CFriends(z):z 2 CFriends(x)} # TH({x}) # TH(TH({x})). Thus TH(TH({x})) = TH({x}). Now it is routine to prove that TH is a closure operator. h Lemma 5.2. If Covering-based upper approximation operator TH is a closure operator, then covering {CFriends(y):y 2 U} is a unary covering of U. Since the proof of Lemma 5.2 is routine, we omit it here. Theorem 5.3 (Topological characterization of coverings for TH to be a closure operator). Covering-based upper approximation operator TH is a closure operator if and only if there exists a topology s on U, such that {CFriends(x):x 2 U} is a base for (U, s) and "x 2 U, {CFriends(x)} is a local base at x for (U, s). Proof. Assume that TH is a closure operator. By Lemmas 5.1 and 5.2, {CFriends(x):x 2 U} is a unary covering of U and " x 2 U, CMd(x) = {CFriends(x)}. By Theorem 3.4, there exists a topology s on U, such that {CFriends(x):x 2 U} is a base for (U, s). Now, we only need to prove that "x 2 U, {CFriends(x)} is a local base at x for (U, s). Let "O 2 s and x 2 O. Since {CFriends(x):x 2 U} is a base for (U, s), {CFriends(x)} # s and there is a y 2 U such that x 2 CFriends(y) # O. By Lemma 5.1, CMd(x) = {CFriends(x)} and x is a representative element of CFriends(x). It follows that CFriends(x) # CFriends(y) # O. Assume that there exists a topology s on U, such that {CFriends(x):x 2 U} is a base for (U, s) and " x 2 U, {CFriends(x)} is a base at x in (U, s). Since {CFriends(x)} is a local base at x for (U, s), x is the representative element of CFriends(x) in covering {CFriends(y):y 2 U}. Thus, by Lemma 5.1, TH is a closure operator. h The following example shows that neither the condition that {CFriends(x):x 2 U} is a unary covering of U nor the condition that {CFriends(x):x 2 U} is a base for some topological space (U, s) can imply that TH is a closure operator. Example 5.4. Let U = {a, b, c, d, e}, C = {{a}, {a, b}, {b, c, d}, {d, e}, {e}}. Then C is a covering of U. Obviously, {CFriends(x):x 2 U} = {{a}, {e}, {a, b, c, d}, {b, c, d, e}, {b, c, d}} is a unary covering of U. By Theorem 3.4, {CFriends(x):x 2 U} is also a base for some
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topological space (U, s). For d 2 U, TH({d}) = {b, c, d, e}. However, TH({b}) = {a, b, c, d}. Therefore, TH(TH({d})) – TH({d}). So, TH is not a closure operator.
Remark. Unlike the proof of Lemma 3.1, finiteness of U is not used in proof of Lemma 5.1. Therefore, Lemmas 5.1, 5.2, and Theorem 5.3 hold even when U is an arbitrary set. However, when U is an arbitrary set, we should modify the proof of Theorem 5.3 by using Lemma 3.3 instead of Theorem 3.4.
6. Characterizations of covering C for RH to be a closure operator First, we give a necessary and sufficient condition for RH to be a closure operator. Theorem 6.1. Characterization of coverings when RH is a closure operatorRH is a closure operator if and only if C satisfies the following condition: "K1, K2 2 C, if K1 – K2 and K1 \ K2 – /, then "x 2 K1 \ K2 {x} 2 C. Proof. Assume that RH is a closure operator and $K1, K2 2 C, K1 – K2, K1 \ K2 – / and $x 2 K1 \ K2 such that {x} R C. Then S CL({x}) = /, and RH({x}) = {K 2 C:x 2 K} K1 [ K2. Since RH is a closure operator and {x} # K1, RH({x}) # RH({x}) [ RH(K1) = RH({x} [ K1) = RH(K1). By K1 2 C, we get RH(K1) = K1 and it follows that K1 [ K2 # K1. Hence, K2 # K1. Similarly, we can prove that K1 # K2. Thus, K1 = K2, which contradicts the assumption that K1 – K2. Assume that "K1, K2 2 C, if K1 – K2 and K1 \ K2 – /, then "x 2 K1 \ K2, {x} 2 C. By the definition of RH, it is obvious that RH satisfies the Axioms 2, 3 and 4 of closure operators. Thus, to prove that RH is a closure operator, it is enough to verify RH satisfies Axiom 1 of closure operators. By Lemma 2 of [66], we only need to prove that "X, Y # U with X # Y, we have RH(X) # RH(Y). Assume the contrary. If there exist X, Y # U with X # Y such that RH(X)6 # RH(Y), then there exists a t 2 U S such that t 2 RH(X) and t R RH(Y). Since t 2 RH(X), by the definition of RH, t 2 CL(X) or t 2 {K 2 C:K \ (X CL(X)) – /}. If t 2 CL(X), by the definition of CL, then there exists a K 2 C such that t 2 K # X. Since X # Y, t 2 K # Y and it follows that S t 2 CL(Y) # RH(Y). This contradicts the fact that t R RH(Y). So, t 2 {K 2 C:K \ (X CL(X)) – /}, and it means that there exists a Kt 2 C such that t 2 Kt and Kt \ (X CL(X)) – /. We choose any z 2 Kt \ (X CL(X)). Since X # Y, z 2 Y. If z 2 Y CL(Y), then S t 2 Kt # {K 2 C:K \ (Y CL(Y)) – /} # RH(Y). This contradicts the fact that t R RH(Y). So, z 2 CL(Y). Thus, there exists a 0 K 2 C such that z 2 K0 # Y. Since t R RH(Y), t R K0 . So Kt – K0 . Since z 2 Kt \ K0 , by the assumption of this theorem, we know that {z} 2 C, and it implies that Md(z) = {{z}}. By z 2 X and the definition of CL(X), we get z 2 CL(X). This contradicts the fact that z 2 X CL(X). h By Theorem 6.1, we obtain the following intuitive characterization of approximation spaces (U, C) when RH is a closure operator. Corollary 6.2. (Intuitive characterization of (U, C) for RH to be a closure operator). RH is a closure operator if and only if universe U consists of two disjoint parts U1 and U2, such that " x 2 U1, {x} 2 C and for any K, K0 2 C with K – K0 , either K \ U2 = K0 \ U2 = /, or K \ U2 – K0 \ U2, and {K \ U2:K 2 C} is a partition of U2. The following Fig. 2 gives an simple example to illustrate the above characterization. In Fig. 2, U = {a, b, c, d, e, f, g} and U1 = {a, b}, U2 = {c, d, e, f, g}, K1 = {a}, K2 = {b}, K3 = {a, c, d}, K4 = {b, e}, and K5 = {a, f, g}. U1 is covered by a family consisting of singletons, and {Ki \ U2:i = 3, 4, 5} is a partition on U2. Combining Lemma 3.3 with Theorem 6.1, we obtain the following result. Lemma 6.3. If operator RH is a closure operator, then there exists a topology s on U, such that C is a base for (U, s). Notice that a family B in U is a base for the discrete topology on U if and only if "x 2 U, {x} 2 B, and a partition on set X forms a base of pseudo-discrete topology on X. In covering approximation space (U, C), take U1 = {x 2 U:x 2 K0 \ K00 with K0 ,
Fig. 2. An example to illustrate the description in Corollary 6.3.
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K00 2 C and K0 – K00 } and U2 = U U1. Using Theorem 3.4, we obtain a topological characterization of covering approximation space (U, C) when RH is a closure operator. Theorem 6.4. (Topological characterization of coverings C for RH to be a closure operator). RH is a closure operator if and only if C is a base for some topology s on U and topological space (U, s) consists of two disjoint subspaces U1 and U2 satisfying the following conditions: (1) For K, K0 2 C, K – K0 , we have K \ U2 = K0 \ U2 = / or K \ U2 – K0 \ U2, and {K \ U2:K 2 C} is a partition of U2; and (2) (U1, s1) is a discrete space and (U2, s2) is a pseudo-discrete space. Here si(i = 1, 2) are the subspace topologies induced by s.
Remark. Since finiteness of U is not used in proofs of the above results in this section, they hold when U is an arbitrary universe. The following example shows that the condition (1) in Theorem 6.4 cannot be ignored. Example 6.5. Let U = {a, b, c, d, e, f}, C = {{a}, {b}, {c, d}, {a, c, d}, {b, c, d}, {b, e, f}}. Then C is a base of a topology s on U, where s = C [ {/, {a, b}, {a, b, c, d}, {a, b, e, f}, {b, c, d, e, f}, U}. Set U1 = {a, b} and U2 = {c, d, e, f}. The topologies induced by s on subspace U1 and U2 are s1 = {/, {a}, {b}, U1} and s2 = {/, {c, d}, {e, f}, U2}, respectively. It is easy to verify that (U1, s1) is a discrete space and (U2, s2) is a pseudo-discrete space. However, RH is not a closure operator. This is because if we take K1 = {a, c, d}, K2 = {b, c, d}, then K1, K2 2 C, c 2 K1 \ K2, and {c} R C. By Theorem 6.1, RH is not a closure operator. We can also describe covering-based approximation space (U, C) as an information exchange system as follows when RH is a closure operator. Let U be a set of people and K be an information exchange group. For any K 2 C, people in the same K can exchange a certain type of information. For a people x 2 U, if x alone is a group and {x} 2 C, then x has some information that is not exchanged with any other people. It is natural to say that x is a person who can keep some secrets. With this description, we have the following result. Theorem 6.6. RH is a closure operator if and only if in information exchange system (U, C) described above, only the person who can keep some secrets joins more than one information exchange groups. At the end of this section, as an application of our characterization, we give an inequality on the relationship between the cardinalities of C and U: Theorem 6.7. Denote U1 = {x 2 U:{x} 2 C} and U2 = U U1. If RH is a closure operator, then
P
K2C jKj
6 jU 1 jjU 2 j þ jU 1 j þ jU 2 j.
P P P P Since Proof. Let C1 = {{x}:{x} 2 C}, C2 = C C1 = {K1, K2, . . . , Kn}. Then K2C jKj ¼ K2C 1 jKj þ K2C 2 jKj ¼ jU 1 j þ K2C 2 jKj. P P Pn n 6 jU2j. It follows that {K \ U2:K 2 C2} is a partition of U 2 ; K2C 2 jK \ U 2 j ¼ jU 2 j and K2C 2 jKj ¼ i¼1 jK i j ¼ P Pn h K2C jKj 6 jU 1 jjU 2 j þ jU 1 j þ jU 2 j. i¼1 ðjK i \ U 1 j þ jK i \ U 2 jÞ 6 njU 1 j þ jU 2 j 6 jU 2 jjU 1 j þ jU 2 j ¼ jU 2 jðjU 1 j þ 1Þ, thus The following simple example illustrates a scenario when the equivalence in the above inequality holds. Example 6.8. Let U = {x, y, z}, C = {{x}, {y, x}, {z, x}}. Then it is easy to check RH is a closure operator by using Theorem 6.1. Also, P it is trivial to verify that K2C jKj ¼ jU 1 jjU 2 j þ jU 1 j þ jU 2 j ¼ 5. Remark. Definitions of FH and RH can be obtained from those of TH and SH by replacing Friends by CFriends. However, from the results we obtained from Sections 3–6, the relationships among characterizations of covering C for them to be closure operators are not simple.
7. Relationships among conditions for different covering-based upper approximation operators to be closure operators Our characterizations of coverings C in the previous four sections provide valuable tools. As one of their applications, we use them to obtain the following results on relationships among conditions for different covering-based upper approximation operators to be closure operators. Lemma 7.1 ([68]). Covering C is unary if and only if FH = TH. Theorem 7.2. RH being a closure operator )FH being a closure operator ) TH being a closure operator; and no other implications among conditions for different covering-based upper approximation operators to be closure operators.
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Proof. If RH is a closure operator, then by Theorem 6.1, "x 2 U, either {x} 2 C, or Md(x) = {K 2 C:x 2 K}, and one can easily see RH = FH by the definitions of RH and FH. Hence, RH being a closure operator ) FH being a closure operator. When FH is a closure operator, by Proposition 1.1, covering C is unary. By Lemma 7.1, FH = TH, and it follows that TH is a closure operator. The following four examples can show that no other implications among conditions for different covering-based upper approximation operators to be closure operators. Example 7.3 (A covering-based approximation space (U, C) in which TH is a closure operator but FH is not). Let (U, C) be a Triangle Chain with three points a, b, c, i.e., let U = {a, b, c} and C = {{a, b}, {a, c}, {c, b}}. Then it is easy to check that for (U, C), TH is a closure operator but FH is not a closure operator. Example 7.4 (A covering-based approximation space (U, C) in which FH is a closure operator but RH is not). Let U = {a, b, c, d,} and C = {{a, b}, {a, b, c}, {a, b, d}}. Then s = {/, U} [ C is a topology on U and C is a base of s, and hence FH is a closure operator by Theorem 3.5. On the other hand, since {a, b, c} \ {a, b, d} = {a, b} is not a union of singletons, by Theorem 6.1, RH is not a closure operator. Example 7.5 (A covering-based approximation space (U, C) in which SH is a closure operator but TH is not). Take any coveringbased approximation space (U, C) for which TH is not a closure operator. Let C0 = C [ {U}. Then in (U, C0 ), "X # U with X – /, SH(X) = U, and hence SH is a closure operator. On the other hand, TH is the same in (U, C0 ) and in (U, C). It follows that in (U, C0 ), TH is not a closure operator. Example 7.6 (A covering-based approximation space (U, C) in which RH is a closure operator but SH is not). Let U be the space in Fig. 2, i.e., U = {a, b, c, d, e, f, g}, K1 = {a}, K2 = {b}, K3 = {a, c, d}, K4 = {b, e}, K5 = {a, f, g} and C = {Ki:1 6 i 6 5}. As shown in Section 6, RH is a closure operator in (U, C). On the other hand, a 2 Friends(c) \ Friends(g) and c R Friends(g), by the Triangle Chain Condition, SH is not a closure operator in (U, C). h Remark. (a) The results of Theorem 7.2 are surprising and interesting. For any X # U, by definitions of FH and RH, it is clear that FH(X) # RH(X). We then have that RH being a closure operator ) FH being a closure operator, but implication in the opposite direction does not hold. Similarly, for any X # U, we have FH(X) # TH(X). In this case, that TH being a closure operator ) FH being a closure operator does not hold, but implication in the opposite direction holds. On the other hand, FH and TH can be obtained from RH and SH, respectively, just using CFriends instead of Friends. However, although we have RH being a closure operator ) FH being a closure operator, there is no implication between conditions for TH to be a closure operator and those for SH to be a closure operator. (b) In the proof of Theorem 7.2, when we prove that RH being a closure operator ) FH being a closure operator, finiteness of U is not used, and hence this result also holds when U is an arbitrary universe. Note that we know that FH being a closure operator ) TH being a closure operator holds even when U is an arbitrary universe because finiteness of U is needed neither in the ‘‘if’’ part of the proof of Proposition 1.1, nor in the proof of Lemma 7.1. 8. Characterizations of unary coverings using relationships among different covering approximation operators. By Lemma 3.1, we can obtain the following Lemma. Lemma 8.1. C is a unary covering of U if and only if for any x 2 U, N(x) 2 C. Theorem 8.2. C is a unary covering of U if and only if "X # U, TH(RH(X)) = RH(X). Proof. ‘‘Only if’’ part: Let X # U. By Proposition 21 in [70], we know that RH(X) # TH(RH(X)). Thus, we only need to prove that TH(RH(X)) # RH(X). We choose any x 2 TH(RH(X)). By the definition of TH, there exists a y 2 RH(X) such that there exists a Ky 2 Md(y) and x 2 Ky. Since C is unary, Md(y) = {Ky}. Since y 2 RH(X), by the definition of RH, y 2 CL(X) or y 2 S {K 2 C:K \ (X CL(X)) – /}. If y 2 CL(X), then there exists a K 2 C such that y 2 K # X. Since Md(y) = {Ky}, Ky # K. So, S x 2 CL(X) # RH(X). If y 2 {K 2 C:K \ (X CL(X)) – /}, then there exists a K 2 C such that K \ (X CL(X)) – / and y 2 K. Since S S Md(y) = {Ky}, Ky # K. Thus, Ky # K # {K 2 C:K \ (X CL(X)) – /}. So, x 2 {K 2 C:K \ (X CL(X)) – /} # RH(X). By arbitrariness of x, TH(RH(X)) # RH(X). ‘‘If ’’ part: Assume the contrary. Suppose C is not unary, then there exists an x 2 U such that Md(x) has at least two elements, say K1, K2 2 Md(x). By the definition of TH and RH, RH(K1) = K1 and K2 # TH(RH(K1)). Since K1 – K2, TH(RH(K1)) – RH(K1). This is a contradiction. Thus, C is unary. h With a similar argument, we can prove the following Theorem 8.3.
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Theorem 8.3. C is a unary covering of U if and only if "X # U, TH(SH(X)) = SH(X). Theorem 8.4. C is a unary covering of U if and only if CL = C2. Proof. ‘‘Only if’’ part: Let X # U. We prove CL(X) # C2(X) first. Otherwise, we can pick an x 2 CL(X) C2(X). By the definition of CL, there exists a K 2 C such that x 2 K # X. It follows that N(x) # K # X, and hence x 2 C2(X), a contradiction. This contradiction means that CL(X) # C2(X). Now we prove C2(X) # CL(X). Otherwise, we can pick an x 2 C2(X) CL(X). Since x 2 C2(X), N(x) # X. By Lemma 8.1, N(x) 2 C and hence N(x) # CL(X). It follows that x 2 CL(X), also a contradiction. ‘‘If ’’ part: Assume that C is not unary. Then by Lemma 8.1, there is an x 2 X such that N(x) R C. Take X = N(x). It is obvious that x 2 C2(X) and x R CL(X), which contradicts the assumption that CL = C2. h Theorem 8.5. C is a unary covering of U if and only if TH(C2) = CL. Proof. ‘‘Only if’’ part: If C is unary. Let X # U. By Theorem 8.4, CL(X) = C2(X). It is easy to check that CL(X) is either a union of finite union of members of C or an empty set, and in both cases we have TH(CL(X)) = CL(X) and hence CL(X) = TH(CL(X)) = TH(C2(X)). ‘‘If ’’ part: If C is not unary. Then by Lemma 8.1, there is an x 2 X such that N(x) R C. Take X = N(x). It is easy to check that C2(X) = X and hence x 2 X # TH(C2(X)). On the other hand, since "K 2 C with x 2 K, we have X # K and X – K, and it follows that x R CL(X). This contradicts the assumption that TH(C2) = CL. h Remark. The following example shows that none of the results provided in this section holds when U is an arbitrary universe. Example 8.6. Let (U, C) be the covering-based approximation space which is defined in Example 3.6. It was proved that C is unary there. For 0 2 U, N(0) = {0} R C. Let X = {0}. Then CL(X) = / and C2 = X and hence TH(C2(X)) = TH(X) = {1, 0, 1}. It follows that none of Lemma 8.1, Theorems 8.4, and 8.5 holds for (U, C). Take X = (1, 1). Since X 2 C, RH(X) = X. It is easy to check that TH(RH(X)) = [1, 1]. Hence, Theorem 8.2 does hold for (U, C). Take r 2 (1, 1) {0} and let X = {r}. Then SH(X) = (1, 1) and TH(SH(X)) = [1, 1]. Thus, Theorem 8.3 does not hold for (U, C).
9. Conclusions and questions In this paper, we give topological characterizations of covering C for four types of covering-based upper approximation operators FH, SH, TH and RH to be closure operators. We also give intuitive characterizations of covering C for SH or RH to be a closure operator. Using these characterizations, we describe covering-based approximation space (U, C) as some special types of information exchange systems when SH or RH is a closure operator. As applications of our characterizations, we give two inequalities about the relationship between the number of elements in C and the number of elements in U when SH or RH is a closure operator. The results we obtain in this paper when SH or RH is a closure operator are much richer than those when FH or TH is a closure operator. Following our study, it is natural to raise the following questions: Question 9.1. Does intuitive characterizations of covering C exist for FH or TH to be a closure operator? Question 9.2. What kind of information exchange systems does covering-based approximation space (U, C) represent when FH or TH is a closure operator? Question 9.3. What is the relationship between cardinalities of C and U when FH or TH is a closure operator? Besides FH, SH, TH and RH, there are several more covering-based upper approximation operators listed in [34]. Then, it is reasonable to ask the following questions: Question 9.4. What are characterizations, either general, topological or intuitive, of covering C for them to be closure operators? What kind of information exchange systems does covering-based approximation space (U, C) represent when any of them is a closure operator? We will look for answers of the above questions in our future research.
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Information Sciences journal homepage: www.elsevier.com/locate/ins
Topological characterizations of covering for special covering-based upper approximation operators q Xun Ge a,b, Xiaole Bai c, Ziqiu Yun a,⇑ a
School of Mathematical Sciences, Soochow University, Suzhou 215006, PR China Zhangjiagang College, Jiangsu University of Science and Technology, Zhangjiagang 215600, PR China c Alliance Data Systems Corp., Columbus, OH 43219, USA b
a r t i c l e
i n f o
Article history: Received 22 June 2011 Received in revised form 19 January 2012 Accepted 1 April 2012 Available online 11 April 2012 Keywords: Covering-based rough set Unary covering Closure operator Base of a topology
a b s t r a c t The relationships among properties of covering approximation operators and their corresponding coverings have attracted intensive research in recent years. In particular, those among topological properties have drawn special attention because of their important applications in rough set theory. In this paper, we give topological characterizations of covering C for covering-based upper approximation operators FH, SH, TH and RH to be closure operators. We also give intuitive characterizations of covering C, and describe coveringbased approximation space (U, C) as certain types of information exchange systems when SH or RH is a closure operator. By applying our new characterizations, we give inequalities about the relationship between the number of members in C and the number of elements in U, and discuss relationships among conditions for different covering based upper approximation operators to be closure operators. To the best of our knowledge, it is the first time that such characterizations, descriptions, inequalities and discussions are systematically considered in the literature of rough set theory. Furthermore, in this paper we also give several characterizations of unary coverings, an important type of coverings in studying relationships among basic concepts in covering-based rough sets, by the relationships among different types of covering-base approximation operators. 2012 Elsevier Inc. All rights reserved.
1. Introduction Rough set theory has been acknowledged as a useful and powerful tool in data analysis particularly for dealing with granularity and vagueness [2,10,11,14,16,20–25,29,35–37,39,41–45,47,48,51,54,57]. The classical rough set theory is based on partitions of a universe [7,15,17,19,21,26,27,52,53,64], which imposes restrictions and limitations on many applications [56,67]. Zakowski generalized the classical rough set theory using coverings of a universe instead of partitions [56]. Such generalization leads to various covering approximation operators that are of both theoretical and practical importance [4,5,8,18,31,34,38,40,49,50,60,61,65]. In recent years, as data mining gets increasingly popular, the relationships between properties of covering-based approximation operators and their corresponding coverings have attracted intensive research [1,30,31,55,63,65–67,69,70]. It is worth noting that topological approaches have provided a valuable perspective and played an important role in rough set theory study [3,9,12,13,28,32,33,55,58,63]. Hence, topological properties have obtained a lot of attention [3,12,13,46,52,55,59,62,63]. In [66,68], Zhu and Wang discussed the relationship between properties of four types of covering-based upper approximation operators and their corresponding coverings. These four operators are named, q
This paper is supported by NSFC (Nos. 61070245, 10971185 and 11061004).
⇑ Corresponding author. Tel.: +86 512 65235165.
E-mail addresses: [email protected] (X. Ge), [email protected] (X. Bai), [email protected] (Z. Yun). 0020-0255/$ - see front matter 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.ins.2012.04.005
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respectively, the first, the second, the third, and the fourth type of covering-based upper approximation operators, and are denoted by FH, SH, TH and RH respectively (definitions of these operators are provided in the next section). Specifically, Zhu and Wang gave characterizations of covering C for FH, SH, and TH to be closure operators by the following results. Proposition 1.1 ([66,68]). FH is a closure operator if and only if C satisfies one of the following conditions: (1) "K1, K2 2 C, K1 \ K2 is a union of elements of C; (2) C is unary.
Proposition 1.2 ([68,70]). SH is a closure operator if and only if C satisfies the following condition: "x 2 U and K 2 C, either K # Friends(x) or K \ Friends(x) = /. Proposition 1.3 ([68,70]). TH is a closure operator if and only if C satisfies the following condition: "x 2 U and K 2 C, K \ CFriends(x) is a union of elements of C. These results are important and interesting, but not complete. There are several important yet open problems following this direction. First, to the best of our knowledge, so far there is no result on characterizations of covering C for RH to be a closure operator (only a necessary condition of covering C for RH to be a closure operator was presented in [66]). Hence, it is natural to ask: Question 1.4. Can we find characterizations of covering C for RH to be a closure operator? Notice that the relationship between definitions of RH and FH is similar to that between definitions of SH and TH; the latter two definitions (of FH and TH) just use CFriends instead of Friends as in the former ones (of RH and SH). It is important to know: Question 1.5. If the answer for Question 1.4 is ‘‘yes’’, then what is the relationship among characterizations of covering C for the aforementioned four types of operators to be closure ones? Second, the known characterizations of covering C that were presented in [66,68,70] for operators FH, SH and TH to be closure operators have nothing to do with topology, while being a closure operator is a notable topological property. It is then reasonable to consider: Question 1.6. Do topological characterizations of covering C exist for FH, SH, TH and RH to be closure operators? In this paper, we not only give answers to all these questions, but also give intuitive characterizations of covering C. By applying these characterizations, we describe covering-based approximation space (U, C) as certain types of information exchange systems when SH or RH is a closure operator, give two inequalities on the relationship between cardinalities of C and U when SH or RH is a closure operator, and discuss relationships among conditions for different covering based upper approximation operators to be closure ones. To the best of our knowledge, it is the first time that such characterizations, descriptions, inequalities and discussions are systematically considered in the literature of rough set theory. Giving characterizations of unary coverings is useful, since unary coverings are important coverings in discussing relationships among basic concepts in covering-based rough sets [65,66,68,70]. In particular, a characterization of unary coverings was presented in [68] using the equality of two types of covering-based upper approximation operators: a covering C is a unary covering if and only if TH = FH. In this paper, we first give a topological characterization of unary coverings without using any covering-based upper approximation operators. As applications of this characterization, we give sufficient and necessary conditions from the perspective of topology for FH or RH to be a closure operator. Then we give several characterizations of unary coverings by the relationships among different types of covering-based approximation operators. The rest of the paper is organized as follows. After giving fundamental concepts in Section 2, we present our main results in Sections 3–6. In these sections, we provide topological characterizations for the aforementioned four types of coveringbased upper approximation operators to be closure operators, intuitive characterizations of covering C, descriptions of covering-based approximation space (U, C) using some special types of information exchange systems, and the inequalities about the relationship between the cardinalities of C and U. In Section 7, we discuss relationships among conditions for different covering based upper approximation operators to be closure operators. In Section 8, we give characterizations of unary covering by relationships among different types of covering-based approximation operators. This paper concludes in Section 9 with remarks and questions we shall consider in our future study. 2. Basic definitions In this section, we present some basic concepts that are used in this paper. In the following discussion, unless it is mentioned specially, the universe of discourse U is considered finite. P(U) denotes the family of all subsets of U. C is a family of S subsets of U. If none of subsets in C is empty, and C = U, then C is called a covering of U. We call ordered pair (U, C) a covering-based approximation space.
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Let (U, C) be a covering-based approximation space and K 2 C, x 2 K. x is called a representative element of K if "S 2 C(x 2 S ) K # S). S Let C be a covering of U, x 2 U. Denote Md(x) = {K 2 C:x 2 K ^ ("S 2 C ^ x 2 S ^ S # K ) K = S)}, Friends(x) = {K 2 C:x 2 K}, S T CFriends(x) = Md(x), and N(x) = {K 2 C:x 2 K}. Let C be a covering of U. C is called unary if "x 2 U, jMd(x)j = 1. Remark. The concept of a representative element of K first appeared in [1]. Md(x), Friends(x), CFriends(x) and N(x), which are called the minimal description of x, the indiscernible neighborhood of x, closed friends of x, and the neighborhood of x, respectively, first appeared in [1,68,63]. The concept of unary covering was first proposed in [68] to characterize the condition for FH and TH to be equal. Let (U, C) be a covering-based approximation space. Our discussion in this paper involves the following types of coveringbased approximation operators. Definition 2.1 (Covering-based lower approximation operators CL and C2). Let C be a covering of U. The lower approximation operators CL and C2 are defined as follows, respectively: S "X 2 P(U), CL(X) = {K 2 C:K # X} and C2(X) = {x 2 U:N(x) # X}.
Definition 2.2 (Covering-based upper approximation operators FH, SH, TH and RH). Let C be a covering of U. Operators FH, SH, TH, RH:P(U) ? P(U) are defined as follows: "X 2 P(U), S S S FH(X) = CL(X) [ ( { Md(x):x 2 (X CL(X))}) = CL(X) [ ( {CFriends(x):x 2 (X CL(X))}), S S SH(X) = {K 2 C:K \ X – ;} = {Friends(x):x 2 X}, S S S TH(X) = { Md(x):x 2 X} = {CFriends(x):x 2 X}, and S S RH(X) = CL(X) [ ( {K 2 C:K \ (X CL(X)) – ;}) = CL(X) [ ( {Friends(x):x 2 (X CL(X))}).
Remark. Covering-based lower approximation operators CL and C2 were first defined in [56,31], respectively. The coveringbased upper approximation operators FH, SH and TH were called the first, the second, and the third type of covering-based upper approximation operators respectively in [68]. RH was called the fourth type of covering-based upper approximation operator in [66]. The following topological concepts and facts are elementary and can be found in [6]. We list them below for the purpose of this paper being self-contained. A topological space is a pair (U, s) consisting of a set U and a family s of subsets of U satisfying the following conditions: S (1) ; 2 s and U 2 s; (2) If U1 2 s and U2 2 s, then U1 \ U2 2 s; and (3) If A # s, then A 2 s. s is called a topology on U and the members of s are called open sets of (U, s). A family B # s is called a base for (U, s) if for every non-empty open subset O of U and each x 2 O, there exists a set B 2 B such that x 2 B # O. Equivalently, a family B # s is called a base for (U, s) if every non-empty open subset O of U can be represented as union of a subfamily of B. "x 2 U, a family B # s is called a local base at x for (U, s) if x 2 B for each B 2 B, and for every open subset O of U with x 2 O, there exists a set B 2 B such that B # O. For any set U consisting of no less than two points, there are more than one topologies on it. Let s and s0 be two topologies on U. If s # s0 , then we say that s is coarser than s0 or s0 is finer than s. Among all the topologies on U, there is a finest one which is called the discrete topology. A topology s is the discrete topology on U if and only if "x 2 U, {x} 2 s. A topology space (U, s) is called a discrete space if s is the discrete topology on U. If P is a partition of U, then it is easy to verify that s = {O # U:O is union of members of P} [ {/} is a topology of U, and was called pseudo-discrete topology in [13] (or called closed-open topology in [12]). Assume (U, s) is a topology space and X # U. It is easy to check that s0 = {O \ X:O 2 s} is a topology on X. s0 is called the induced topology, and topology space (X, s0 ) is called a subspace of (U, s). Definition 2.3 (Closure operator). An operator c:P(U) ? P(U) is called a closure operator on U if it satisfies the following axioms: "X, Y # U, Axiom Axiom Axiom Axiom
1. 2. 3. 4.
c(X [ Y) = c(X) [ c(Y); X # c(X); c(;) = ;; c(c(X)) = c(X).
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3. Topological characterization of coverings C for FH to be a closure operator In this section, we first give a topological characterization of unary coverings without involving any approximation operators. As an application of this result, we obtain a topological characterization of coverings C for FH to be a closure operator. Lemma 3.1. Let C be a unary covering of U, then for any x 2 U, there exists a Kx 2 C such that Md(x) = {Kx} and x is a representative element of Kx. Proof. This lemma can be proved by the definitions of Md and the representative element, and by an argument similar to the proof for Theorem 3 in [66]. h Lemma 3.2. A covering C is unary if and only if "K1, K2 2 C and "x 2 K1 \ K2, there is a K 2 C such that x 2 K # K1 \ K2. Proof. Suppose C is unary. "K1, K2 2 C and "x 2 K1 \ K2, let Md(x) = {Kx}. By Lemma 3.1, x is a representative element of Kx, and it follows that x 2 Kx # K1 \ K2, where Kx 2 C. On the other hand, if C is not unary, there is a x 2 U and there are K1, K2 2 Md(x) such that K1 – K2. By our assumption, there is a K 2 C such that x 2 K # K1 \ K2. It contradicts the fact that K1, K2 2 Md(x). h Lemma 3.3. (Proposition 1.2.1 in 6). A covering C of U is a base for some topology s on U if and only if "K1, K2 2 C and "x 2 K1 \ K2, there is a K 2 C such that x 2 K # K1 \ K2. The following result is only a combination of Lemmas 3.2 and 3.3: Theorem 3.4. (Topological characterization of unary coverings). A covering C is unary if and only if there exists a topology s on U, such that C is a base for (U, s). Combining with Proposition 1.1, we have a topological characterization for FH to be a closure operator as follows. Theorem 3.5. (Topological characterization for FH to be a closure operator). Covering-based upper approximation operator FH is a closure operator if and only if there exists a topology s on U, such that C is a base for (U, s).
Remark. The following two examples show that Lemmas 3.1, 3.2, Theorems 3.4, and 3.5 do not hold when U is an arbitrary universe: Example 3.6. Let U be the closed interval [1, 1] and C ¼ ffxg : x 2 ½1; 1 f0gg [ 1n ; 1n : n ¼ 1; 2; 3; . . . [ ff1; 0; 1gg. Then C is a covering of U and it is easy to check Md(x) = {{x}} for x 2 [1, 1] {0} and Md(0) = {{1, 0, 1}}. It follows that C is unary. However, none of Lemmas 3.1 and 3.2, and Theorem 3.4 holds for (U, C). Example 3.7. Let U = R, where R is the set of real numbers. Let C ¼ x 1n ; x þ 1n : x 2 U and n ¼ 1; 2; 3; . . . . Then C is a base of the usual topology on U. It is easy to check {x} R C and Md(x) = / for each x 2 R, and hence FH({x}) = / for each x 2 U. Thus, FH is not a closure operator, which shows Theorem 3.5 does not hold for (U, C).
4. Characterizations of covering C for SH to be a closure operator Theorem 4.1. (Topological characterization for SH to be a closure operator). SH is a closure operator if and only if {Friends(x):x 2 U} forms a partition of U. Proof. Using the definitions of SH and the closure operator, it is easy to check that if {Friends(x):x 2 U} forms a partition of U, then SH is a closure operator. On the other hand, assume that SH is a closure operator. We only need to prove that "x, y 2 U, if Friends(x) – Friends(y), then Friends(x) \ Friends(y) = /. Otherwise, we can take a z 2 Friends(x) \ Friends(y). By the assumpS tion that SH is a closure operator, we have that Friends(z) # {Friends(t):t 2 Friends(y)} = Friends(y) and Friends(y) # S {Friends(t):t 2 Friends(z)} = Friends(z), and hence Friends(z) = Friends(y). Similarly, Friends(z) = Friends(x). It follows that h Friends(x) = Friends(y), which contradicts to the fact that Friends(x) – Friends(y). Theorem 4.1 has several applications. By Theorem 4.1, we can not only obtain a topological characterization and an intuitive characterization of covering C for SH to be a closure operator, but also describe covering-based approximation space (U, C) as a special information system when SH is a closure operator.
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Since a covering C of U forms a base of some pseudo-discrete topology on U if and only if C is a partition of U, we get the following topological characterization of the covering for SH to be a closure operator. Theorem 4.2. SH is a closure operator if and only if there is a pseudo-discrete topology s on U such that {Friends(x):x 2 U} is a base of s. Remark. Finiteness of U is not used in the proof of Theorem 4.1. Therefore, Theorems 4.1 and 4.2 hold when U is an arbitrary set. For an intuitive characterization, we need the following definition. Definition 4.3. (Triangle Chain with three points). Let C be a covering of U. For any x, y, z 2 U, if either there exist K1, K2, K3 2 C, such that x, y 2 K1, y, z 2 K2, and z, x 2 K3, then we say that there is a triangle chain with three points x, y, z. In the following Fig. 1, U1 gives an intuitive illustration of the Triangle Chain with three points. Theorem 4.4. (Triangle Chain Condition). SH is a closure operator if and only if covering C divides U into disjoint parts U1, U2, . . ., Un on U, such that for each Ui with 1 6 i 6 n and "x, y, z 2 Ui, there is a Triangle Chain with three points x, y, z. The description in Theorem 4.4 is illustrated in Fig. 1 via a simple example. In Fig. 1, U = {a, b, c, d, e, f, g} and K1 = {a, b}, K2 = {b, c}, K3 = {a, c}, K4 = {d, e, f}, K5 = {e, f, g}, and K6 = {d, g}. Covering C = {Ki:i = 1, 2, 3, 4, 5, 6} divides U into two disjoint parts U1 and U2. It is obvious that points of U1 or U2 satisfy the Triangle Chain Condition. Theorem 4.4 can be also stated in a brief way as follows. Theorem 4.5. SH is a closure operator if and only if "x, y 2 U, either Friends(x) \ Friends(y) = /, or x 2 Friends(y). Proof. The ‘‘only if’’ part holds directly following Theorem 4.1. Now we consider the ‘‘if’’ part. Assume that "x, y 2 U, either Friends(x) \ Friends(y) = /, or x 2 Friends(y). To prove that SH is a closure operator, it only needs to show that {Friends(x):x 2 U} forms a partition of U by Theorem 4.1. It then becomes enough for us to prove that Friends(x) = Friends(y) if "x, y 2 U with Friends(x) \ Friends(y) – /. We prove that Friends(y) # Friends(x) first. By Friends(x) \ Friends(y) – / and our assumption, we have x 2 Friends(y). This means there is a K 2 C, such that x, y 2 K, and hence K # Friends(x) \ Friends(y). "z 2 Friends(y), we have y 2 Friends(z), thus K \ Friends(z) – /. It follows that Friends(x) \ Friends(z) – /. By our assumption, z 2 Friends(x). Since z is arbitrary, Friends(y) # Friends(x). With a similar argument, we can prove Friends(x) # Friends(y). It follows that Friends(x) = Friends(y). h Remark. Put a social network analysis scenario, assuming that U represents a set of people. "x, y 2 U, if x 2 Friends(y), then we say x is a friend of y or x, y are friends. Theorem 4.5 can be explained as: SH is a closure operator if and only if people in U have the relationship that if two people have a common friend, then they are friends themselves; in other words, only friends can have a common friend. Moreover, when SH is a closure operator, we can describe covering-based approximation space (U, C) as one type of information exchange system as follows. Let U be a set of people. For each pair x, y 2 U, if there is a people group, e.g., club or community, K 2 C such that x, y 2 K, then we say that x and y can exchange information directly; If there are people t1, t2, . . ., tn and people groups K0, K1, K2, . . ., Kn 2 C, such that x, t1 2 K0, t1, t2 2 K1, . . ., tn1, tn 2 Kn1, tn, y 2 Kn, then we say that x and y can exchange information. With this description, we have the following result. Theorem 4.6. SH is a closure operator if and only if information exchange system (U, C) described as above satisfies that: as long as two people can exchange information, they can also directly do that.
Fig. 1. Illustration for the triangle chain condition.
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Proof. Assume that SH is a closure operator and two people x, y can exchange information. Let there be people t1, t2, . . ., tn and people groups K0, K1, K2, . . ., Kn, such that x, t1 2 K0, t1, t2 2 K1, . . ., tn1, tn 2 Kn1, tn, y 2 Kn. By the Triangle Chain Condition, people x, t2 can exchange information directly. Then using the Triangle Chain Condition again, people x, t3 can exchange information directly. Continue in this way, at the nth step, people x, y can exchange information directly. On the other hand, if it holds that two people can exchange information directly as long as they can exchange information, then the Triangle Chain Condition is satisfied, and then SH is a closure operator. h At the end of this section, we give an inequality on the relationship between the number of elements in C that cover Friends(x) and the number of elements in Friends(x). For x 2 U, let Cx = {K 2 C:K \ Friends(x) – /} = {K1, K2, . . . , Kn}. Theorem 4.7. If SH is a closure operator, then "x 2 U, either there exists a K 2 C such that K covers Friends(x), or Pn i¼1 jK i j P 2jFriendsðxÞj holds. Here for set X, jXj denotes the number of elements of X. Proof. When SH is a closure operator, consider that there does not exist a K 2 C such that K covers Friends(x). To prove the conclusion, it is enough to prove that each y 2 Friends(x) is contained in more than one member of Cx. Take i with 1 6 i 6 n such that y 2 Ki. Since Ki does not cover Friends(x), there exists z 2 Friends(x) such that z R Ki. By the Triangle Chain Condition, there is a j with 1 6 j 6 n such that z, y 2 Kj. Since z 2 Kj and z R Ki, j – i, and it follows that y is contained in at least two members of Cx. h The following simple example illustrates a scenario when the equivalence in the above inequality holds. Example 4.8. Let U = {x, y, z}, C = {{x, y}, {z, x}, {y, z}}. Since " X # U with X – /, SH(X) = U, SH is a closure operator. One can Pn check it easily that 8w 2 U; i¼1 jK i j ¼ 2jFriendsðwÞj ¼ 6. 5. Topological characterization of coverings C for TH to be a closure operator Lemma 5.1. Covering-based upper approximation operator TH is a closure operator if and only if "x 2 U, x is a representative element of CFriends(x) for covering {CFriends(y):y 2 U}. S Proof. Assume that TH is a closure operator. "x 2 U, if x 2 CFriends(y) for y 2 U, then CFriends(x) # {CFriends(z):z 2 CFriends(y)} = TH(CFriends(y)) = TH(TH(y)) = TH(y) = CFriends(y). It follows that x is a representative element of CFriends(x) for covering {CFriends(y):y 2 U}. Assume that "x 2 U, x is a representative element of CFriends(x) for covering {CFriends(y):y 2 U}. "z 2 TH({x}) = CFriends(x), since z is a representative element of CFriends(z) for covering {CFriends(y):y 2 U}, TH({z}) = CFriends(z) # S CFriends(x) = TH({x}), and it follows that TH(TH({x})) = {CFriends(z):z 2 CFriends(x)} # TH({x}) # TH(TH({x})). Thus TH(TH({x})) = TH({x}). Now it is routine to prove that TH is a closure operator. h Lemma 5.2. If Covering-based upper approximation operator TH is a closure operator, then covering {CFriends(y):y 2 U} is a unary covering of U. Since the proof of Lemma 5.2 is routine, we omit it here. Theorem 5.3 (Topological characterization of coverings for TH to be a closure operator). Covering-based upper approximation operator TH is a closure operator if and only if there exists a topology s on U, such that {CFriends(x):x 2 U} is a base for (U, s) and "x 2 U, {CFriends(x)} is a local base at x for (U, s). Proof. Assume that TH is a closure operator. By Lemmas 5.1 and 5.2, {CFriends(x):x 2 U} is a unary covering of U and " x 2 U, CMd(x) = {CFriends(x)}. By Theorem 3.4, there exists a topology s on U, such that {CFriends(x):x 2 U} is a base for (U, s). Now, we only need to prove that "x 2 U, {CFriends(x)} is a local base at x for (U, s). Let "O 2 s and x 2 O. Since {CFriends(x):x 2 U} is a base for (U, s), {CFriends(x)} # s and there is a y 2 U such that x 2 CFriends(y) # O. By Lemma 5.1, CMd(x) = {CFriends(x)} and x is a representative element of CFriends(x). It follows that CFriends(x) # CFriends(y) # O. Assume that there exists a topology s on U, such that {CFriends(x):x 2 U} is a base for (U, s) and " x 2 U, {CFriends(x)} is a base at x in (U, s). Since {CFriends(x)} is a local base at x for (U, s), x is the representative element of CFriends(x) in covering {CFriends(y):y 2 U}. Thus, by Lemma 5.1, TH is a closure operator. h The following example shows that neither the condition that {CFriends(x):x 2 U} is a unary covering of U nor the condition that {CFriends(x):x 2 U} is a base for some topological space (U, s) can imply that TH is a closure operator. Example 5.4. Let U = {a, b, c, d, e}, C = {{a}, {a, b}, {b, c, d}, {d, e}, {e}}. Then C is a covering of U. Obviously, {CFriends(x):x 2 U} = {{a}, {e}, {a, b, c, d}, {b, c, d, e}, {b, c, d}} is a unary covering of U. By Theorem 3.4, {CFriends(x):x 2 U} is also a base for some
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topological space (U, s). For d 2 U, TH({d}) = {b, c, d, e}. However, TH({b}) = {a, b, c, d}. Therefore, TH(TH({d})) – TH({d}). So, TH is not a closure operator.
Remark. Unlike the proof of Lemma 3.1, finiteness of U is not used in proof of Lemma 5.1. Therefore, Lemmas 5.1, 5.2, and Theorem 5.3 hold even when U is an arbitrary set. However, when U is an arbitrary set, we should modify the proof of Theorem 5.3 by using Lemma 3.3 instead of Theorem 3.4.
6. Characterizations of covering C for RH to be a closure operator First, we give a necessary and sufficient condition for RH to be a closure operator. Theorem 6.1. Characterization of coverings when RH is a closure operatorRH is a closure operator if and only if C satisfies the following condition: "K1, K2 2 C, if K1 – K2 and K1 \ K2 – /, then "x 2 K1 \ K2 {x} 2 C. Proof. Assume that RH is a closure operator and $K1, K2 2 C, K1 – K2, K1 \ K2 – / and $x 2 K1 \ K2 such that {x} R C. Then S CL({x}) = /, and RH({x}) = {K 2 C:x 2 K} K1 [ K2. Since RH is a closure operator and {x} # K1, RH({x}) # RH({x}) [ RH(K1) = RH({x} [ K1) = RH(K1). By K1 2 C, we get RH(K1) = K1 and it follows that K1 [ K2 # K1. Hence, K2 # K1. Similarly, we can prove that K1 # K2. Thus, K1 = K2, which contradicts the assumption that K1 – K2. Assume that "K1, K2 2 C, if K1 – K2 and K1 \ K2 – /, then "x 2 K1 \ K2, {x} 2 C. By the definition of RH, it is obvious that RH satisfies the Axioms 2, 3 and 4 of closure operators. Thus, to prove that RH is a closure operator, it is enough to verify RH satisfies Axiom 1 of closure operators. By Lemma 2 of [66], we only need to prove that "X, Y # U with X # Y, we have RH(X) # RH(Y). Assume the contrary. If there exist X, Y # U with X # Y such that RH(X)6 # RH(Y), then there exists a t 2 U S such that t 2 RH(X) and t R RH(Y). Since t 2 RH(X), by the definition of RH, t 2 CL(X) or t 2 {K 2 C:K \ (X CL(X)) – /}. If t 2 CL(X), by the definition of CL, then there exists a K 2 C such that t 2 K # X. Since X # Y, t 2 K # Y and it follows that S t 2 CL(Y) # RH(Y). This contradicts the fact that t R RH(Y). So, t 2 {K 2 C:K \ (X CL(X)) – /}, and it means that there exists a Kt 2 C such that t 2 Kt and Kt \ (X CL(X)) – /. We choose any z 2 Kt \ (X CL(X)). Since X # Y, z 2 Y. If z 2 Y CL(Y), then S t 2 Kt # {K 2 C:K \ (Y CL(Y)) – /} # RH(Y). This contradicts the fact that t R RH(Y). So, z 2 CL(Y). Thus, there exists a 0 K 2 C such that z 2 K0 # Y. Since t R RH(Y), t R K0 . So Kt – K0 . Since z 2 Kt \ K0 , by the assumption of this theorem, we know that {z} 2 C, and it implies that Md(z) = {{z}}. By z 2 X and the definition of CL(X), we get z 2 CL(X). This contradicts the fact that z 2 X CL(X). h By Theorem 6.1, we obtain the following intuitive characterization of approximation spaces (U, C) when RH is a closure operator. Corollary 6.2. (Intuitive characterization of (U, C) for RH to be a closure operator). RH is a closure operator if and only if universe U consists of two disjoint parts U1 and U2, such that " x 2 U1, {x} 2 C and for any K, K0 2 C with K – K0 , either K \ U2 = K0 \ U2 = /, or K \ U2 – K0 \ U2, and {K \ U2:K 2 C} is a partition of U2. The following Fig. 2 gives an simple example to illustrate the above characterization. In Fig. 2, U = {a, b, c, d, e, f, g} and U1 = {a, b}, U2 = {c, d, e, f, g}, K1 = {a}, K2 = {b}, K3 = {a, c, d}, K4 = {b, e}, and K5 = {a, f, g}. U1 is covered by a family consisting of singletons, and {Ki \ U2:i = 3, 4, 5} is a partition on U2. Combining Lemma 3.3 with Theorem 6.1, we obtain the following result. Lemma 6.3. If operator RH is a closure operator, then there exists a topology s on U, such that C is a base for (U, s). Notice that a family B in U is a base for the discrete topology on U if and only if "x 2 U, {x} 2 B, and a partition on set X forms a base of pseudo-discrete topology on X. In covering approximation space (U, C), take U1 = {x 2 U:x 2 K0 \ K00 with K0 ,
Fig. 2. An example to illustrate the description in Corollary 6.3.
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K00 2 C and K0 – K00 } and U2 = U U1. Using Theorem 3.4, we obtain a topological characterization of covering approximation space (U, C) when RH is a closure operator. Theorem 6.4. (Topological characterization of coverings C for RH to be a closure operator). RH is a closure operator if and only if C is a base for some topology s on U and topological space (U, s) consists of two disjoint subspaces U1 and U2 satisfying the following conditions: (1) For K, K0 2 C, K – K0 , we have K \ U2 = K0 \ U2 = / or K \ U2 – K0 \ U2, and {K \ U2:K 2 C} is a partition of U2; and (2) (U1, s1) is a discrete space and (U2, s2) is a pseudo-discrete space. Here si(i = 1, 2) are the subspace topologies induced by s.
Remark. Since finiteness of U is not used in proofs of the above results in this section, they hold when U is an arbitrary universe. The following example shows that the condition (1) in Theorem 6.4 cannot be ignored. Example 6.5. Let U = {a, b, c, d, e, f}, C = {{a}, {b}, {c, d}, {a, c, d}, {b, c, d}, {b, e, f}}. Then C is a base of a topology s on U, where s = C [ {/, {a, b}, {a, b, c, d}, {a, b, e, f}, {b, c, d, e, f}, U}. Set U1 = {a, b} and U2 = {c, d, e, f}. The topologies induced by s on subspace U1 and U2 are s1 = {/, {a}, {b}, U1} and s2 = {/, {c, d}, {e, f}, U2}, respectively. It is easy to verify that (U1, s1) is a discrete space and (U2, s2) is a pseudo-discrete space. However, RH is not a closure operator. This is because if we take K1 = {a, c, d}, K2 = {b, c, d}, then K1, K2 2 C, c 2 K1 \ K2, and {c} R C. By Theorem 6.1, RH is not a closure operator. We can also describe covering-based approximation space (U, C) as an information exchange system as follows when RH is a closure operator. Let U be a set of people and K be an information exchange group. For any K 2 C, people in the same K can exchange a certain type of information. For a people x 2 U, if x alone is a group and {x} 2 C, then x has some information that is not exchanged with any other people. It is natural to say that x is a person who can keep some secrets. With this description, we have the following result. Theorem 6.6. RH is a closure operator if and only if in information exchange system (U, C) described above, only the person who can keep some secrets joins more than one information exchange groups. At the end of this section, as an application of our characterization, we give an inequality on the relationship between the cardinalities of C and U: Theorem 6.7. Denote U1 = {x 2 U:{x} 2 C} and U2 = U U1. If RH is a closure operator, then
P
K2C jKj
6 jU 1 jjU 2 j þ jU 1 j þ jU 2 j.
P P P P Since Proof. Let C1 = {{x}:{x} 2 C}, C2 = C C1 = {K1, K2, . . . , Kn}. Then K2C jKj ¼ K2C 1 jKj þ K2C 2 jKj ¼ jU 1 j þ K2C 2 jKj. P P Pn n 6 jU2j. It follows that {K \ U2:K 2 C2} is a partition of U 2 ; K2C 2 jK \ U 2 j ¼ jU 2 j and K2C 2 jKj ¼ i¼1 jK i j ¼ P Pn h K2C jKj 6 jU 1 jjU 2 j þ jU 1 j þ jU 2 j. i¼1 ðjK i \ U 1 j þ jK i \ U 2 jÞ 6 njU 1 j þ jU 2 j 6 jU 2 jjU 1 j þ jU 2 j ¼ jU 2 jðjU 1 j þ 1Þ, thus The following simple example illustrates a scenario when the equivalence in the above inequality holds. Example 6.8. Let U = {x, y, z}, C = {{x}, {y, x}, {z, x}}. Then it is easy to check RH is a closure operator by using Theorem 6.1. Also, P it is trivial to verify that K2C jKj ¼ jU 1 jjU 2 j þ jU 1 j þ jU 2 j ¼ 5. Remark. Definitions of FH and RH can be obtained from those of TH and SH by replacing Friends by CFriends. However, from the results we obtained from Sections 3–6, the relationships among characterizations of covering C for them to be closure operators are not simple.
7. Relationships among conditions for different covering-based upper approximation operators to be closure operators Our characterizations of coverings C in the previous four sections provide valuable tools. As one of their applications, we use them to obtain the following results on relationships among conditions for different covering-based upper approximation operators to be closure operators. Lemma 7.1 ([68]). Covering C is unary if and only if FH = TH. Theorem 7.2. RH being a closure operator )FH being a closure operator ) TH being a closure operator; and no other implications among conditions for different covering-based upper approximation operators to be closure operators.
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Proof. If RH is a closure operator, then by Theorem 6.1, "x 2 U, either {x} 2 C, or Md(x) = {K 2 C:x 2 K}, and one can easily see RH = FH by the definitions of RH and FH. Hence, RH being a closure operator ) FH being a closure operator. When FH is a closure operator, by Proposition 1.1, covering C is unary. By Lemma 7.1, FH = TH, and it follows that TH is a closure operator. The following four examples can show that no other implications among conditions for different covering-based upper approximation operators to be closure operators. Example 7.3 (A covering-based approximation space (U, C) in which TH is a closure operator but FH is not). Let (U, C) be a Triangle Chain with three points a, b, c, i.e., let U = {a, b, c} and C = {{a, b}, {a, c}, {c, b}}. Then it is easy to check that for (U, C), TH is a closure operator but FH is not a closure operator. Example 7.4 (A covering-based approximation space (U, C) in which FH is a closure operator but RH is not). Let U = {a, b, c, d,} and C = {{a, b}, {a, b, c}, {a, b, d}}. Then s = {/, U} [ C is a topology on U and C is a base of s, and hence FH is a closure operator by Theorem 3.5. On the other hand, since {a, b, c} \ {a, b, d} = {a, b} is not a union of singletons, by Theorem 6.1, RH is not a closure operator. Example 7.5 (A covering-based approximation space (U, C) in which SH is a closure operator but TH is not). Take any coveringbased approximation space (U, C) for which TH is not a closure operator. Let C0 = C [ {U}. Then in (U, C0 ), "X # U with X – /, SH(X) = U, and hence SH is a closure operator. On the other hand, TH is the same in (U, C0 ) and in (U, C). It follows that in (U, C0 ), TH is not a closure operator. Example 7.6 (A covering-based approximation space (U, C) in which RH is a closure operator but SH is not). Let U be the space in Fig. 2, i.e., U = {a, b, c, d, e, f, g}, K1 = {a}, K2 = {b}, K3 = {a, c, d}, K4 = {b, e}, K5 = {a, f, g} and C = {Ki:1 6 i 6 5}. As shown in Section 6, RH is a closure operator in (U, C). On the other hand, a 2 Friends(c) \ Friends(g) and c R Friends(g), by the Triangle Chain Condition, SH is not a closure operator in (U, C). h Remark. (a) The results of Theorem 7.2 are surprising and interesting. For any X # U, by definitions of FH and RH, it is clear that FH(X) # RH(X). We then have that RH being a closure operator ) FH being a closure operator, but implication in the opposite direction does not hold. Similarly, for any X # U, we have FH(X) # TH(X). In this case, that TH being a closure operator ) FH being a closure operator does not hold, but implication in the opposite direction holds. On the other hand, FH and TH can be obtained from RH and SH, respectively, just using CFriends instead of Friends. However, although we have RH being a closure operator ) FH being a closure operator, there is no implication between conditions for TH to be a closure operator and those for SH to be a closure operator. (b) In the proof of Theorem 7.2, when we prove that RH being a closure operator ) FH being a closure operator, finiteness of U is not used, and hence this result also holds when U is an arbitrary universe. Note that we know that FH being a closure operator ) TH being a closure operator holds even when U is an arbitrary universe because finiteness of U is needed neither in the ‘‘if’’ part of the proof of Proposition 1.1, nor in the proof of Lemma 7.1. 8. Characterizations of unary coverings using relationships among different covering approximation operators. By Lemma 3.1, we can obtain the following Lemma. Lemma 8.1. C is a unary covering of U if and only if for any x 2 U, N(x) 2 C. Theorem 8.2. C is a unary covering of U if and only if "X # U, TH(RH(X)) = RH(X). Proof. ‘‘Only if’’ part: Let X # U. By Proposition 21 in [70], we know that RH(X) # TH(RH(X)). Thus, we only need to prove that TH(RH(X)) # RH(X). We choose any x 2 TH(RH(X)). By the definition of TH, there exists a y 2 RH(X) such that there exists a Ky 2 Md(y) and x 2 Ky. Since C is unary, Md(y) = {Ky}. Since y 2 RH(X), by the definition of RH, y 2 CL(X) or y 2 S {K 2 C:K \ (X CL(X)) – /}. If y 2 CL(X), then there exists a K 2 C such that y 2 K # X. Since Md(y) = {Ky}, Ky # K. So, S x 2 CL(X) # RH(X). If y 2 {K 2 C:K \ (X CL(X)) – /}, then there exists a K 2 C such that K \ (X CL(X)) – / and y 2 K. Since S S Md(y) = {Ky}, Ky # K. Thus, Ky # K # {K 2 C:K \ (X CL(X)) – /}. So, x 2 {K 2 C:K \ (X CL(X)) – /} # RH(X). By arbitrariness of x, TH(RH(X)) # RH(X). ‘‘If ’’ part: Assume the contrary. Suppose C is not unary, then there exists an x 2 U such that Md(x) has at least two elements, say K1, K2 2 Md(x). By the definition of TH and RH, RH(K1) = K1 and K2 # TH(RH(K1)). Since K1 – K2, TH(RH(K1)) – RH(K1). This is a contradiction. Thus, C is unary. h With a similar argument, we can prove the following Theorem 8.3.
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Theorem 8.3. C is a unary covering of U if and only if "X # U, TH(SH(X)) = SH(X). Theorem 8.4. C is a unary covering of U if and only if CL = C2. Proof. ‘‘Only if’’ part: Let X # U. We prove CL(X) # C2(X) first. Otherwise, we can pick an x 2 CL(X) C2(X). By the definition of CL, there exists a K 2 C such that x 2 K # X. It follows that N(x) # K # X, and hence x 2 C2(X), a contradiction. This contradiction means that CL(X) # C2(X). Now we prove C2(X) # CL(X). Otherwise, we can pick an x 2 C2(X) CL(X). Since x 2 C2(X), N(x) # X. By Lemma 8.1, N(x) 2 C and hence N(x) # CL(X). It follows that x 2 CL(X), also a contradiction. ‘‘If ’’ part: Assume that C is not unary. Then by Lemma 8.1, there is an x 2 X such that N(x) R C. Take X = N(x). It is obvious that x 2 C2(X) and x R CL(X), which contradicts the assumption that CL = C2. h Theorem 8.5. C is a unary covering of U if and only if TH(C2) = CL. Proof. ‘‘Only if’’ part: If C is unary. Let X # U. By Theorem 8.4, CL(X) = C2(X). It is easy to check that CL(X) is either a union of finite union of members of C or an empty set, and in both cases we have TH(CL(X)) = CL(X) and hence CL(X) = TH(CL(X)) = TH(C2(X)). ‘‘If ’’ part: If C is not unary. Then by Lemma 8.1, there is an x 2 X such that N(x) R C. Take X = N(x). It is easy to check that C2(X) = X and hence x 2 X # TH(C2(X)). On the other hand, since "K 2 C with x 2 K, we have X # K and X – K, and it follows that x R CL(X). This contradicts the assumption that TH(C2) = CL. h Remark. The following example shows that none of the results provided in this section holds when U is an arbitrary universe. Example 8.6. Let (U, C) be the covering-based approximation space which is defined in Example 3.6. It was proved that C is unary there. For 0 2 U, N(0) = {0} R C. Let X = {0}. Then CL(X) = / and C2 = X and hence TH(C2(X)) = TH(X) = {1, 0, 1}. It follows that none of Lemma 8.1, Theorems 8.4, and 8.5 holds for (U, C). Take X = (1, 1). Since X 2 C, RH(X) = X. It is easy to check that TH(RH(X)) = [1, 1]. Hence, Theorem 8.2 does hold for (U, C). Take r 2 (1, 1) {0} and let X = {r}. Then SH(X) = (1, 1) and TH(SH(X)) = [1, 1]. Thus, Theorem 8.3 does not hold for (U, C).
9. Conclusions and questions In this paper, we give topological characterizations of covering C for four types of covering-based upper approximation operators FH, SH, TH and RH to be closure operators. We also give intuitive characterizations of covering C for SH or RH to be a closure operator. Using these characterizations, we describe covering-based approximation space (U, C) as some special types of information exchange systems when SH or RH is a closure operator. As applications of our characterizations, we give two inequalities about the relationship between the number of elements in C and the number of elements in U when SH or RH is a closure operator. The results we obtain in this paper when SH or RH is a closure operator are much richer than those when FH or TH is a closure operator. Following our study, it is natural to raise the following questions: Question 9.1. Does intuitive characterizations of covering C exist for FH or TH to be a closure operator? Question 9.2. What kind of information exchange systems does covering-based approximation space (U, C) represent when FH or TH is a closure operator? Question 9.3. What is the relationship between cardinalities of C and U when FH or TH is a closure operator? Besides FH, SH, TH and RH, there are several more covering-based upper approximation operators listed in [34]. Then, it is reasonable to ask the following questions: Question 9.4. What are characterizations, either general, topological or intuitive, of covering C for them to be closure operators? What kind of information exchange systems does covering-based approximation space (U, C) represent when any of them is a closure operator? We will look for answers of the above questions in our future research.
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