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Approximation operators on complete completely distributive lattices
Information Sciences 247 (2013) 123–130
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Approximation operators on complete completely distributive lattices Keyun Qin a,⇑, Zheng Pei b, Jilin Yang c, Yang Xu a a
College of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan 610031, China School of Mathematics & Computer Engineering, Xihua University, Chengdu, Sichuan 610039, China c College of Fundamental Education, Sichuan Normal University, Chengdu, Sichuan 610068, China b
a r t i c l e
i n f o
Article history: Received 6 January 2010 Received in revised form 17 April 2013 Accepted 11 June 2013 Available online 20 June 2013 Keywords: Rough set CCD lattice Neighborhood Lower approximation operator Upper approximation operator
a b s t r a c t Rough set, a tool for data mining, deals with the vagueness and granularity in information systems. In 2006, Chen et al. initiated the study of rough approximations on a complete completely distributive lattice (CCD lattice for short) and brought generalizations of rough sets into a unified framework. In this paper, we discuss the approximation operators on a CCD lattice. Based on the concept of neighborhood, three kinds of upper approximation operators and a kind of lower approximation operator are constructed. Basic properties of lower and upper approximation operators are examined. Furthermore, the relationships among these operators are analyzed. Ó 2013 Published by Elsevier Inc.
1. Introduction The theory of rough sets was proposed by Pawlak [13,14]. It is an extension of set theory and provides a systematic method for dealing with vague concepts caused by indiscernibility in situations with insufficient and incomplete information. The concepts of upper and lower approximations in rough set theory allow us to discover the knowledge hidden in information systems and express it in the form of decision rules. In Pawlak’s rough set model, equivalence relation is a key and primitive notion. The equivalence classes are building blocks for constructing the lower and upper approximations. This equivalence relation, however, seems to be a very stringent condition that may limit the application domain of the rough set model. To solve this problem, generalizations of rough sets were considered by some scholars. One generalization approach is to consider a similarity or tolerance relation [3,11,16,19,20,28,29] rather than an equivalence relation. Another generalization approach is to extend the partition of the universe to a cover [1,2,15,18,30–34]. Fuzzy rough set models are proposed by using fuzzy relations instead of crisp binary relations on the universe [5–7,10,12,17,21,22,25]. They provide a useful tool for dealing with data sets with both vagueness and fuzziness. However, these generalizations have not been interconnected with each other [4]. In order to incorporate the generalized rough set models into a unified framework, in paper [4], the complete completely distributive lattice (CCD lattice for short) is selected as the mathematical foundation for defining upper and lower approximations. These definitions, which result from the concept of cover introduced on a CCD lattice, have improved the approximations of the existing generalizations of rough sets.
⇑ Corresponding author. Tel.: +86 28 87602468; fax: +86 28 87600764. E-mail addresses: [email protected] (K. Qin), [email protected] (Z. Pei), [email protected] (J. Yang), [email protected] (Y. Xu). 0020-0255/$ - see front matter Ó 2013 Published by Elsevier Inc. http://dx.doi.org/10.1016/j.ins.2013.06.032
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Neighborhood system is a tool for investigating rough sets [8,9,26,27]. In a CCD lattice, join-irreducible elements are ‘point-like’ elements and they have the identity of basic units [23,24]. The intersection of all elements in a cover which is larger than or equal to a join-irreducible element is non-zero. These elements, called neighborhoods, can be employed to approximate other elements. In this paper, we discuss the approximation operators on a CCD lattice. Based on the concept of neighborhood, three kinds of upper approximation operators and a kind of lower approximation operator are constructed with their basic properties being discussed. These definitions improve the approximations on a CCD lattice. Furthermore, the relationship among these operators are investigated. In the following section, we recall some terms and theorems to be used in the paper. In Section 3, the definition of neighborhood with respect to a cover is proposed and the lower approximation operator is studied. In Section 4, the upper approximation operators are constructed. The relationships among these operators are also investigated. Section 5 concludes the paper. 2. Preliminaries This section presents a review of some fundamental notions of rough approximations on a complete completely distributive lattice. We refer to reference [4] for details. A lattice is a partially ordered set in which any two elements a and b have a least upper bound a _ b and a greatest lower W V bound a ^ b. A lattice L is complete if any subset A # L has a least upper bound A and a greatest lower bound A in L. An element e in a lattice L is called join-irreducible if b, c 2 L and e = b _ c imply that e = b or e = c. A non-zero join-irreducible element is said to be a molecule. A lattice L is completely distributive if the following conditions hold:
^_ ð aij Þ ¼
_
i2I j2Ji
f2
Q
i2I j2Ji
f2
Q
_^ ð aij Þ ¼
J i2I i
^
J i2I i
ð
^ aif ðiÞ Þ; i2I
_ ð aif ðiÞ Þ; i2I
Q where I and Ji are nonempty index sets, aij 2 L, and f 2 i2I J i means f is a mapping f:I ? [i2IJi such that f(i) 2 Ji for every i 2 I. For example, ([0, 1], _, ^, 0, 1) is a complete completely distributive lattice, or CCD lattice. If X is a nonempty set, L = P(X), then (L, [, \, ;, X) is a CCD lattice. Let F(X) be the collection of fuzzy sets in X. Then (F(X), [, \, ;, X) is a CCD lattice. In a CCD lattice, join-irreducible elements have the identity of units. Each element of a CCD is a join of join-irreducible elements [23,24]. Let (L, _, ^, 0, 1) be a CCD lattice. 0 and 1 are the least and greatest element of L respectively. We denote by M(L) the set of all molecules of L. W Definition 2.1 [4]. For every C # L {0}, C is called a cover of L if C satisfies (1) a2Ca = 1; (2) for every p 2 M(L), there exists a 2 C such that p 6 a, and for any other b 2 C, if p 6 b, then b 6 a does not hold. a is called the minimal element of p with respect to C, the collection of minimal elements of p in C is denoted as Cp. Definition 2.2 [4]. Let C be a cover of L. C is said to be reduced if for every a 2 C, there exists p 2 M(L) such that p 6 a and for each b 2 C, b – a, p ^ b = 0 holds. Definition 2.3 [4]. Suppose C # L {0} satisfies a ^ b = 0.
W
a2Ca
= 1. C is called a partition of L if for every a, b 2 C, a – b implies
Theorem 2.1 [4]. If C is a partition of L, then C is a reduced cover of L. Definition 2.4 [4]. Let C be a cover of L. Define s; s : L ! L, for every x 2 L,
sðxÞ ¼ _p6x _a2C p a; sðxÞ ¼ _fa 2 C; a 6 xg: s and s are called an upper approximation and a lower approximation of L with respect to C respectively. sðxÞ and s(x) are called upper approximation and lower approximation of x. This definition presents a more precise upper approximation than the existing generalizations of rough sets by similarity relation and cover approaches. Furthermore, it is a generalization of fuzzy rough lower and upper approximations when Tsimilarity relation is considered [4]. The following two theorems summarize basic properties of lower and upper approximations induced by a cover. Theorem 2.2 [4]. Let C be a cover of L. For every x,y 2 L, (1) x 6 sðxÞ, s(x) 6 x.
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(2) If x 6 y, then sðxÞ 6 sðyÞ, s(x) 6 s(y). (3) sðx _ yÞ ¼ sðxÞ _ sðyÞ, s(x ^ y) 6 s(x) ^ s(y). (4) sðx ^ yÞ 6 sðxÞ ^ sðyÞ. (5) s(x _ y) P s(x) _ s(y). (6) sðxÞ 6 sðsðxÞÞ, s(x) = s(s(x)).
Theorem 2.3 [4]. Let C be a cover of L. Then the following assertions are equivalent. (1) C is a partition of L. (2) C is reduced and for each x 2 L, sðxÞ ¼ sðsðxÞÞ (3) C is reduced and for each x 2 L, s(x ^ y) = s(x) ^ s(y). 3. The lower approximation operators This section discusses lower approximation operators. We suppose that L is a CCD lattice throughout the paper. Let C be a cover of L. For each p 2 M(L),
NðpÞ ¼ ^fa 2 C; p 6 ag is called the neighborhood of p with respect to C. Lemma 3.1 [4]. For every x, y 2 L, x 6 y if and only if for each p 2 M(L), p 6 x ) p 6 y. Lemma 3.2. Let C be a cover of L. (1) (2) (3) (4)
For each p 2 M(L), p 6 N(p). For each p 2 M(L), NðpÞ ¼ ^a2C p a. ’’For every p,q 2 M(L), p 6 N(q) ) N(p) 6 N(q). p 2 M(L) if and only if: for every x, y 2 L, p 6 x _ y ) p 6 x, or p 6 y.
Proof (1) and (2) are trivial. (3) Let p, q 2 M(L) and p 6 N(q). Then q 6 a ) p 6 a for each a 2 C. It follows that {a 2 C; q 6 a} # {a 2 C; p 6 a} and hence
NðqÞ ¼ ^fa 2 C; q 6 ag P ^fa 2 C; p 6 ag ¼ NðpÞ: (4) Suppose that p 2 M(L). For every x,y 2 L, if p 6 x _ y, then p = p ^ (x _ y) = (p ^ x) _ (p ^ y). Therefore p = p ^ x, or p = p ^ y, and consequently p 6 x, or p 6 y as required. Conversely, suppose that for every x, y 2 L, p 6 x _ y ) p 6 x, or p 6 y. If p = u _ v, then p 6 u _ v and thus p 6 u, or p 6 v. By u 6 p and v 6 p, we have p = u, or p = v and consequently p 2 M(L). h The neighborhoods with respect to a cover can be employed to approximate other elements in L. Definition 3.1. Let C be a cover of L. Define lower approximation operator s1: L ? L, for every x 2 L,
s1 ðxÞ ¼ _fp 2 MðLÞ; NðpÞ 6 xg: Theorem 3.3. Let C be a cover of L and x, y 2 L. (1) (2) (3) (4) (5) (6)
s1(1) = 1, s1(0) = 0. s1(x) 6 x. x 6 y implies s1(x) 6 s1(y). s1(x) = _ {N(p); N(p) 6 x}. s1(x ^ y) = s1(x) ^ s1(y). For every a 2 C, s1(a) = a.
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Proof. (1), (2) and (3) are trivial. (4) By (1) of Lemma 3.2, it follows that s1(x) 6 _ {N(p); N(p) 6 x}. On the other hand, suppose that p 2 M(L) and N(p) 6 x. For each q 2 M(L), q 6 N(p) implies N(q) 6 N(p) 6 x. It follows that q 6 _ {r 2 M(L); N(r) 6 x} = s1(x), and hence N(p) 6 s1(x) by Lemma 3.1. Consequently, _{N(p); N(p) 6 x} 6 s1(x). (5) s1(x ^ y) 6 s1(x) ^ s1(y) follows from (3). On the other hand,
s1 ðxÞ ^ s1 ðyÞ ¼ ð_fNðpÞ; NðpÞ 6 xgÞ ^ ð_fNðqÞ; NðqÞ 6 ygÞ ¼ _p;q2MðLÞ fNðpÞ ^ NðqÞ; NðpÞ 6 x; NðqÞ 6 yg: Suppose that N(p) 6 x and N(q) 6 y. For each r 2 M(L), if r 6 N(p) ^ N(q), then r 6 N(p), r 6 N(q) and thus N(r) 6 N(p), N(r) 6 N(q) by (3) of Lemma 3.2. It follows that N(r) 6 N(p) ^ N(q) 6 x ^ y. Consequently,
r 6 NðrÞ 6 _fNðuÞ; NðuÞ 6 x ^ yg ¼ s1 ðx ^ yÞ: Therefore N(p) ^ N(q) 6 s1(x ^ y) by Lemma 3.1. So, we have
s1 ðxÞ ^ s1 ðyÞ ¼ _p;q2MðLÞ fNðpÞ ^ NðqÞ; NðpÞ 6 x; NðqÞ 6 yg 6 s1 ðx ^ yÞ: (6) Let a 2 C, p 2 M(L) and p 6 a. Then N(p) 6 a and hence
p 6 _fr 2 MðLÞ; NðrÞ 6 ag ¼ s1 ðaÞ: Consequently, a 6 s1(a) and thus a = s1(a) by (2).
h
Remark. Generally speaking, there are two forms of lower approximation operators:
s1 ðxÞ ¼ _fp 2 MðLÞ; NðpÞ 6 xg; s2 ðxÞ ¼ _fNðpÞ; NðpÞ 6 xg; for every x 2 L. By Theorem 3.3(4), s1 and s2 are equivalent. Thereby in what follows we focus on operator s1. Theorem 3.4. Let C be a cover of L. For every x 2 L, s(x) 6 s1(x). Proof. Suppose that a 2 C and a 6 x. For each p 2 M(L), p 6 a implies N(p) 6 a 6 x. It follows that
p 6 NðpÞ 6 _fNðqÞ; NðqÞ 6 xg ¼ s1 ðxÞ: Consequently, a 6 s1(x) by Lemma 3.1 and hence
sðxÞ ¼ _fb 2 C; b 6 xg 6 s1 ðxÞ: The following example implies that the strict inequality in this theorem is possible. Example 3.1. Let U = {1, 2, 3}, L = P(U) the power set lattice of U, A = {1, 2}, B = {2, 3}. Then {A, B} is a cover of L and N({1}) = {1, 2}, N({2}) = {2}, N({3}) = {2, 3}. By direct computation we have
sðf2gÞ ¼ _fa 2 C; a 6 f2gg ¼ 0; s1 ðf2gÞ ¼ _fNðpÞ; NðpÞ 6 f2gg ¼ f2g: Theorem 3.5. Let C be a reduced cover of L. The following assertions are equivalent: (1) For each p 2 M(L), jCpj = 1. (2) For each x 2 L, s(x) = s1(x).
Proof (1) ) (2). For each p 2 M(L), by jCpj = 1 we have NðpÞ ¼ ^a2C p a 2 C and hence
s1 ðxÞ ¼ _fNðpÞ; NðpÞ 6 xg 6 _fb 2 C; b 6 xg ¼ sðxÞ: Consequently, s(x) = s1(x) by Theorem 3.4. (2) ) (1). Assume that there exists p 2 M(L) such that jCpj > 1. Thus there exist a, b 2 Cp such that a – b. By
s1 ðNðpÞÞ ¼ _fNðrÞ; NðrÞ 6 NðpÞg ¼ NðpÞ;
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it follows that
sðNðpÞÞ ¼ _fd 2 C; d 6 NðpÞg ¼ NðpÞ – 0; and hence there exists d 2 C such that d 6 N(p). Consequently, d 6 a, d 6 b. Therefore d – a. For every q 2 M(L), q 6 d implies that q ^ a P q ^ d = q – 0. This is a contradiction with the fact that C is reduced. h
Corollary 3.1. Let C be a cover of L. The following assertions are equivalent: (1) C is reduced and for each x 2 L, s(x) = s1(x). (2) C is a partition of L.
Proof. (1) ) (2). For every x, y 2 L, we have
sðx ^ yÞ ¼ s1 ðx ^ yÞ ¼ s1 ðxÞ ^ s1 ðyÞ ¼ sðxÞ ^ sðyÞ: By Theorem 2.3, C is a partition of L. (2) ) (1). Suppose that C is a partition of L. It follows that jCpj = 1 for each p 2 M(L). Hence, (1) holds by Theorem 3.5. h 4. The upper approximation operators In this section, we discuss the upper approximation operators by using neighborhoods. Definition 4.1. Let C be a cover of L. Define upper approximation operators s1 ; s2 ; s3 : L ! L, for every x 2 L,
s1 ðxÞ ¼ _fNðpÞ; NðpÞ ^ x – 0g; s2 ðxÞ ¼ _fp 2 MðLÞ; NðpÞ ^ x – 0g; s3 ðxÞ ¼ _fNðpÞ; p 6 xg: Let U be a nonempty set and L = P(U). It follows that, for every X # U,
s1 ðXÞ ¼ [fNðpÞ; NðpÞ \ X – ;g; s2 ðXÞ ¼ fp 2 U; NðpÞ \ X – ;g; s3 ðXÞ ¼ [fNðpÞ; p 2 Xg: Hence, Definition 4.1 is a generalization of covering approximation operators in [15,31]. Theorem 4.1. Let C be a cover of L. (1) (2) (3) (4)
s1 ð1Þ ¼ 1, s1 ð0Þ ¼ 0. For every x, y 2 L, x 6 y implies s1 ðxÞ 6 s1 ðyÞ. For every x 2 L, x 6 s1 ðxÞ. For any xi 2 L, i 2 I, s1 ð_i2I xi Þ ¼ _i2I s1 ðxi Þ. Where I is an index set.
Proof (1) and (2) follow from Definition 4.1 immediately. (3) For each p 6 x, by p 6 N(p), it follows that N(p) ^ x P p – 0, and hence
p 6 NðpÞ 6 _fNðqÞ; NðqÞ ^ x – 0g ¼ s1 ðxÞ: Consequently, x 6 s1 ðxÞ by Lemma 3.1. (4) s1 ð_i2I xi Þ P _i2I s1 ðxi Þ follows from (2). On the other hand, for every p 2 M(L), N(p) ^ (_i2Ixi) – 0 implies _i2I(N(p) ^ xi) – 0, thus there exists i 2 I such that N(p) ^ xi – 0. Hence NðpÞ 6 s1 ðxi Þ 6 _i2I s1 ðxi Þ. Consequently,
s1 ð_i2I xi Þ ¼ _fNðqÞ; NðqÞ ^ ð_i2I xi Þ – 0g 6 _i2I s1 ðxi Þ:
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Similarly, we have: Theorem 4.2. Let C be a cover of L. (1) (2) (3) (4)
s2 ð1Þ ¼ 1, s2 ð0Þ ¼ 0. For every x 2 L, x 6 s2 ðxÞ. For every x,y 2 L, x 6 y implies s2 ðxÞ 6 s2 ðyÞ. For any xi 2 L, i 2 I, s2 ð_i2I xi Þ ¼ _i2I s2 ðxi Þ. Where I is an index set.
Theorem 4.3. Let C be a cover of L and x,y 2 L. (1) s3 ð1Þ ¼ 1, s3 ð0Þ ¼ 0. (2) x 6 s3 ðxÞ. (3) x 6 y implies s3 ðxÞ 6 s3 ðyÞ. (4) s3 ðx _ yÞ ¼ s3 ðxÞ _ s3 ðyÞ. (5) For every a 2 C, s3 ðaÞ ¼ a. The proof of the following theorem is straightforward. Theorem 4.4. Let C be a cover of L. (1) For every x 2 L, s3 ðxÞ 6 s1 ðxÞ. (2) For every x 2 L, s2 ðxÞ 6 s1 ðxÞ. (3) For every x 2 L, s3 ðxÞ 6 sðxÞ. The following example shows that the strict inequality in (1), (2) and (3) of Theorem 4.4 is possible. Example 4.1. Let U = {x, y, z} and L = P(U). We know L is a CCD lattice. Let a = {x, y}, b = {y, z}. Then C = {a, b} is a cover of L. Clearly, N({x}) = {x, y}, N({y}) = {y}, N({z}) = {y, z}. (1) Let X = {y}. Then
s1 ðXÞ ¼ _fNðpÞ; NðpÞ ^ X – 0g ¼ NðfxgÞ _ NðfygÞ _ NðfzgÞ ¼ fx; y; zg; s3 ðXÞ ¼ _fNðpÞ; p 6 Xg ¼ NðfygÞ ¼ fyg; sðXÞ ¼ _d2C fyg d ¼ fx; yg _ fy; zg ¼ fx; y; zg: It follows that s1 ðXÞ – s3 ðXÞ and sðXÞ – s3 ðXÞ. (2) Let X = {x, z}. Then
s1 ðXÞ ¼ _fNðpÞ; NðpÞ ^ X – 0g ¼ NðfxgÞ _ NðfzgÞ ¼ fx; y; zg; s2 ðXÞ ¼ _fp; NðpÞ ^ X – 0g ¼ fx; zg; and s1 ðXÞ – s2 ðXÞ. (3) Let X = {y}, Y = {x}. Then
s2 ðXÞ ¼ _fp; NðpÞ ^ X – 0g ¼ fx; y; zg; s3 ðXÞ ¼ _fNðpÞ; p 6 Xg ¼ NðfygÞ ¼ fyg; s2 ðYÞ ¼ _fp; NðpÞ ^ Y – 0g ¼ fxg; s3 ðYÞ ¼ _fNðpÞ; p 6 Yg ¼ NðfxgÞ ¼ fx; yg: Therefore s2 is3 and s3 is2 .
Lemma 4.5. Let C be a cover of L and CN = {N(p); p 2 M(L)}. (1) CN is a cover of L and (CN)p = {N(p)} for every p 2 M(L). (2) CN is a partition of L if and only if for every p, q 2 M(L), p 6 N(q) ) q 6 N(p). Proof (1) _p2M(L)N(p) P _ p2M(L)p = 1 holds. For each p 2 M(L), we have p 6 N(p). If p 6 N(q), then N(p) 6 N(q). Therefore N(p) is the unique minimal element of p with respect to CN. Hence CN is a cover of L and (CN)p = {N(p)}.
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(2) Suppose CN is a partition of L. For every p, q 2 M(L), p 6 N(q) implies N(p) 6 N(q). It follows that N(p) ^ N(q) = N(p) – 0, and hence N(p) = N(q). Consequently, q 6 N(q) = N(p). Conversely, suppose that N(p) ^ N(q) – 0. Then there exists r 2 M(L) such that r 6 N(p) ^ N(q). It follows that r 6 N(p), r 6 N(q) and hence p 6 N(r), q 6 N(r). Consequently, N(p) = N(r) = N(q). Thus CN is a partition of L. h Theorem 4.6. Let C be a cover of L. s1 ¼ s3 if and only if CN is a partition of L. Proof ()): Suppose that s1 ðxÞ ¼ s3 ðxÞ for every x 2 L. If p, q 2 M(L), p 6 N(q), then N(q) ^ p = p – 0 and hence
s1 ðpÞ ¼ _fNðrÞ; NðrÞ ^ p – 0g P NðqÞ: It follows that
q 6 NðqÞ 6 s1 ðpÞ ¼ s3 ðpÞ ¼ _fNðrÞ; r 6 pg ¼ NðpÞ: Consequently, by Lemma 4.5, CN is a partition of L. (Ü): Suppose that CN is a partition of L. For every x 2 L, N(p) ^ x – 0 implies there exists r 2 M(L) such that r 6 N(p) ^ x. Thus r 6 N(p) and r 6 x. Consequently,
NðpÞ ¼ NðrÞ 6 _fNðqÞ; q 6 xg ¼ s3 ðxÞ: So, we have s1 ðxÞ 6 s3 ðxÞ.
h
Theorem 4.7. Let C be a cover of L. s2 ¼ s3 if and only if CN is a partition of L. Proof. ()): Suppose that s2 ðxÞ ¼ s3 ðxÞ for every x 2 L. If p,q 2 M(L), N(p) ^ N(q) – 0, then there exists r 2 M(L) such that r 6 N(p) ^ N(q). By N(p) ^ r = r – 0, we have
s2 ðrÞ ¼ _ft 2 MðLÞ; NðtÞ ^ r – 0g P p; s3 ðrÞ ¼ _fNðtÞ; t 6 rg ¼ NðrÞ: Consequently, p 6 s2 ðrÞ ¼ s3 ðrÞ ¼ NðrÞ, and hence N(p) 6 N(r) 6 N(p). Analogically one can show that N(q) = N(r). So, we have N(p) = N(q). Therefore CN is a partition of L. (Ü): Suppose that CN is a partition of L and x 2 L. Let p 2 M(L) and p 6 x. For every r 2 M(L), r 6 N(p) implies N(r) 6 N(p), and hence N(r) = N(p). It follows that N(r) ^ x = N(p) ^ x P p – 0. We have r 6 _fq 2 MðLÞ; NðqÞ ^ x – 0g ¼ s2 ðxÞ, and consequently NðpÞ 6 s2 ðxÞ by Lemma 3.1. Thus s3 ðxÞ 6 s2 ðxÞ. On the other hand, let p 2 M(L) and N(p) ^ x – 0. Then there exists r 2 M(L) such that r 6 N(p) ^ x. It follows that r 6 x, r 6 N(p). Since CN is a partition of L, then N(r) = N(p). Consequently,
p 6 NðpÞ ¼ NðrÞ 6 _fNðqÞ; q 6 xg ¼ s3 ðxÞ; and thus
s2 ðxÞ ¼ _fq; NðqÞ ^ x – 0g 6 s3 ðxÞ: In the last of this section, we discuss the relationship between C and CN. Example 4.2. Let U = {x, y, z} and L = P(U). Let a = {x, y}, b = {y, z}, c = {x, z}. Then C = {a, b, c} is a cover of L. Clearly, N({x}) = {x}, N({y}) = {y}, N({z}) = {z}. That is to say, CN is a partition of L and C is not a partition.
Theorem 4.8. Let C be a cover of L. The following assertions are equivalent: (1) C is a partition of L. (2) C is a reduced cover and CN is a partition of L. Proof (1) ) (2). Let p 2 M(L). Since C is a partition, then there exists unique a 2 C such that p 6 a, and hence N(p) = a 2 C. Consequently, CN # C. It follows that N(p) – N(q) implies N(p) ^ N(q) = 0, and therefore CN is a partition of L.
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(2) ) (1). For every a 2 C, there exists p 2 M(L) such that p 6 a and for each b 2 C, b – a, p ^ b = 0 holds. It follows that p i b and hence N(p) = a. That is to say, C # CN. Consequently, if a, b 2 C, a – b, then a ^ b = 0, and C is a partition as required. h Corollary 4.1. Let C be a partition of L. Then s1 ¼ s2 ¼ s3 ¼ s. 5. Conclusions In this paper, we discuss the approximation operators on a CCD lattice. Based on the concept of neighborhood, three kinds of upper approximation operators and one kind of lower approximation operator are constructed with their basic properties being discussed. These operators are extensions of approximation operators in a covering rough set model. Furthermore, the relationship among these operators are investigated. However, it is still an open question regarding the duality of lower and upper approximation operators on a CCD lattice. It is worth noticing that the duality of lower and upper approximation operators cannot be discussed in a general CCD lattice, inasmuch as there is no complement operation in it. We can study this problem in a special CCD lattice which is equipped with a complement operation. Another further study is the topological structure of approximation operators. Acknowledgements This work has been supported by the National Natural Science Foundation of China (Grant Nos. 61175044, 61175055) and the Sichuan Key Technology Research and Development Program (Grant No. 2011FZ0051). References [1] [2] [3] [4] [5] [6] [7] [8] [9]
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Information Sciences journal homepage: www.elsevier.com/locate/ins
Approximation operators on complete completely distributive lattices Keyun Qin a,⇑, Zheng Pei b, Jilin Yang c, Yang Xu a a
College of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan 610031, China School of Mathematics & Computer Engineering, Xihua University, Chengdu, Sichuan 610039, China c College of Fundamental Education, Sichuan Normal University, Chengdu, Sichuan 610068, China b
a r t i c l e
i n f o
Article history: Received 6 January 2010 Received in revised form 17 April 2013 Accepted 11 June 2013 Available online 20 June 2013 Keywords: Rough set CCD lattice Neighborhood Lower approximation operator Upper approximation operator
a b s t r a c t Rough set, a tool for data mining, deals with the vagueness and granularity in information systems. In 2006, Chen et al. initiated the study of rough approximations on a complete completely distributive lattice (CCD lattice for short) and brought generalizations of rough sets into a unified framework. In this paper, we discuss the approximation operators on a CCD lattice. Based on the concept of neighborhood, three kinds of upper approximation operators and a kind of lower approximation operator are constructed. Basic properties of lower and upper approximation operators are examined. Furthermore, the relationships among these operators are analyzed. Ó 2013 Published by Elsevier Inc.
1. Introduction The theory of rough sets was proposed by Pawlak [13,14]. It is an extension of set theory and provides a systematic method for dealing with vague concepts caused by indiscernibility in situations with insufficient and incomplete information. The concepts of upper and lower approximations in rough set theory allow us to discover the knowledge hidden in information systems and express it in the form of decision rules. In Pawlak’s rough set model, equivalence relation is a key and primitive notion. The equivalence classes are building blocks for constructing the lower and upper approximations. This equivalence relation, however, seems to be a very stringent condition that may limit the application domain of the rough set model. To solve this problem, generalizations of rough sets were considered by some scholars. One generalization approach is to consider a similarity or tolerance relation [3,11,16,19,20,28,29] rather than an equivalence relation. Another generalization approach is to extend the partition of the universe to a cover [1,2,15,18,30–34]. Fuzzy rough set models are proposed by using fuzzy relations instead of crisp binary relations on the universe [5–7,10,12,17,21,22,25]. They provide a useful tool for dealing with data sets with both vagueness and fuzziness. However, these generalizations have not been interconnected with each other [4]. In order to incorporate the generalized rough set models into a unified framework, in paper [4], the complete completely distributive lattice (CCD lattice for short) is selected as the mathematical foundation for defining upper and lower approximations. These definitions, which result from the concept of cover introduced on a CCD lattice, have improved the approximations of the existing generalizations of rough sets.
⇑ Corresponding author. Tel.: +86 28 87602468; fax: +86 28 87600764. E-mail addresses: [email protected] (K. Qin), [email protected] (Z. Pei), [email protected] (J. Yang), [email protected] (Y. Xu). 0020-0255/$ - see front matter Ó 2013 Published by Elsevier Inc. http://dx.doi.org/10.1016/j.ins.2013.06.032
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Neighborhood system is a tool for investigating rough sets [8,9,26,27]. In a CCD lattice, join-irreducible elements are ‘point-like’ elements and they have the identity of basic units [23,24]. The intersection of all elements in a cover which is larger than or equal to a join-irreducible element is non-zero. These elements, called neighborhoods, can be employed to approximate other elements. In this paper, we discuss the approximation operators on a CCD lattice. Based on the concept of neighborhood, three kinds of upper approximation operators and a kind of lower approximation operator are constructed with their basic properties being discussed. These definitions improve the approximations on a CCD lattice. Furthermore, the relationship among these operators are investigated. In the following section, we recall some terms and theorems to be used in the paper. In Section 3, the definition of neighborhood with respect to a cover is proposed and the lower approximation operator is studied. In Section 4, the upper approximation operators are constructed. The relationships among these operators are also investigated. Section 5 concludes the paper. 2. Preliminaries This section presents a review of some fundamental notions of rough approximations on a complete completely distributive lattice. We refer to reference [4] for details. A lattice is a partially ordered set in which any two elements a and b have a least upper bound a _ b and a greatest lower W V bound a ^ b. A lattice L is complete if any subset A # L has a least upper bound A and a greatest lower bound A in L. An element e in a lattice L is called join-irreducible if b, c 2 L and e = b _ c imply that e = b or e = c. A non-zero join-irreducible element is said to be a molecule. A lattice L is completely distributive if the following conditions hold:
^_ ð aij Þ ¼
_
i2I j2Ji
f2
Q
i2I j2Ji
f2
Q
_^ ð aij Þ ¼
J i2I i
^
J i2I i
ð
^ aif ðiÞ Þ; i2I
_ ð aif ðiÞ Þ; i2I
Q where I and Ji are nonempty index sets, aij 2 L, and f 2 i2I J i means f is a mapping f:I ? [i2IJi such that f(i) 2 Ji for every i 2 I. For example, ([0, 1], _, ^, 0, 1) is a complete completely distributive lattice, or CCD lattice. If X is a nonempty set, L = P(X), then (L, [, \, ;, X) is a CCD lattice. Let F(X) be the collection of fuzzy sets in X. Then (F(X), [, \, ;, X) is a CCD lattice. In a CCD lattice, join-irreducible elements have the identity of units. Each element of a CCD is a join of join-irreducible elements [23,24]. Let (L, _, ^, 0, 1) be a CCD lattice. 0 and 1 are the least and greatest element of L respectively. We denote by M(L) the set of all molecules of L. W Definition 2.1 [4]. For every C # L {0}, C is called a cover of L if C satisfies (1) a2Ca = 1; (2) for every p 2 M(L), there exists a 2 C such that p 6 a, and for any other b 2 C, if p 6 b, then b 6 a does not hold. a is called the minimal element of p with respect to C, the collection of minimal elements of p in C is denoted as Cp. Definition 2.2 [4]. Let C be a cover of L. C is said to be reduced if for every a 2 C, there exists p 2 M(L) such that p 6 a and for each b 2 C, b – a, p ^ b = 0 holds. Definition 2.3 [4]. Suppose C # L {0} satisfies a ^ b = 0.
W
a2Ca
= 1. C is called a partition of L if for every a, b 2 C, a – b implies
Theorem 2.1 [4]. If C is a partition of L, then C is a reduced cover of L. Definition 2.4 [4]. Let C be a cover of L. Define s; s : L ! L, for every x 2 L,
sðxÞ ¼ _p6x _a2C p a; sðxÞ ¼ _fa 2 C; a 6 xg: s and s are called an upper approximation and a lower approximation of L with respect to C respectively. sðxÞ and s(x) are called upper approximation and lower approximation of x. This definition presents a more precise upper approximation than the existing generalizations of rough sets by similarity relation and cover approaches. Furthermore, it is a generalization of fuzzy rough lower and upper approximations when Tsimilarity relation is considered [4]. The following two theorems summarize basic properties of lower and upper approximations induced by a cover. Theorem 2.2 [4]. Let C be a cover of L. For every x,y 2 L, (1) x 6 sðxÞ, s(x) 6 x.
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(2) If x 6 y, then sðxÞ 6 sðyÞ, s(x) 6 s(y). (3) sðx _ yÞ ¼ sðxÞ _ sðyÞ, s(x ^ y) 6 s(x) ^ s(y). (4) sðx ^ yÞ 6 sðxÞ ^ sðyÞ. (5) s(x _ y) P s(x) _ s(y). (6) sðxÞ 6 sðsðxÞÞ, s(x) = s(s(x)).
Theorem 2.3 [4]. Let C be a cover of L. Then the following assertions are equivalent. (1) C is a partition of L. (2) C is reduced and for each x 2 L, sðxÞ ¼ sðsðxÞÞ (3) C is reduced and for each x 2 L, s(x ^ y) = s(x) ^ s(y). 3. The lower approximation operators This section discusses lower approximation operators. We suppose that L is a CCD lattice throughout the paper. Let C be a cover of L. For each p 2 M(L),
NðpÞ ¼ ^fa 2 C; p 6 ag is called the neighborhood of p with respect to C. Lemma 3.1 [4]. For every x, y 2 L, x 6 y if and only if for each p 2 M(L), p 6 x ) p 6 y. Lemma 3.2. Let C be a cover of L. (1) (2) (3) (4)
For each p 2 M(L), p 6 N(p). For each p 2 M(L), NðpÞ ¼ ^a2C p a. ’’For every p,q 2 M(L), p 6 N(q) ) N(p) 6 N(q). p 2 M(L) if and only if: for every x, y 2 L, p 6 x _ y ) p 6 x, or p 6 y.
Proof (1) and (2) are trivial. (3) Let p, q 2 M(L) and p 6 N(q). Then q 6 a ) p 6 a for each a 2 C. It follows that {a 2 C; q 6 a} # {a 2 C; p 6 a} and hence
NðqÞ ¼ ^fa 2 C; q 6 ag P ^fa 2 C; p 6 ag ¼ NðpÞ: (4) Suppose that p 2 M(L). For every x,y 2 L, if p 6 x _ y, then p = p ^ (x _ y) = (p ^ x) _ (p ^ y). Therefore p = p ^ x, or p = p ^ y, and consequently p 6 x, or p 6 y as required. Conversely, suppose that for every x, y 2 L, p 6 x _ y ) p 6 x, or p 6 y. If p = u _ v, then p 6 u _ v and thus p 6 u, or p 6 v. By u 6 p and v 6 p, we have p = u, or p = v and consequently p 2 M(L). h The neighborhoods with respect to a cover can be employed to approximate other elements in L. Definition 3.1. Let C be a cover of L. Define lower approximation operator s1: L ? L, for every x 2 L,
s1 ðxÞ ¼ _fp 2 MðLÞ; NðpÞ 6 xg: Theorem 3.3. Let C be a cover of L and x, y 2 L. (1) (2) (3) (4) (5) (6)
s1(1) = 1, s1(0) = 0. s1(x) 6 x. x 6 y implies s1(x) 6 s1(y). s1(x) = _ {N(p); N(p) 6 x}. s1(x ^ y) = s1(x) ^ s1(y). For every a 2 C, s1(a) = a.
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Proof. (1), (2) and (3) are trivial. (4) By (1) of Lemma 3.2, it follows that s1(x) 6 _ {N(p); N(p) 6 x}. On the other hand, suppose that p 2 M(L) and N(p) 6 x. For each q 2 M(L), q 6 N(p) implies N(q) 6 N(p) 6 x. It follows that q 6 _ {r 2 M(L); N(r) 6 x} = s1(x), and hence N(p) 6 s1(x) by Lemma 3.1. Consequently, _{N(p); N(p) 6 x} 6 s1(x). (5) s1(x ^ y) 6 s1(x) ^ s1(y) follows from (3). On the other hand,
s1 ðxÞ ^ s1 ðyÞ ¼ ð_fNðpÞ; NðpÞ 6 xgÞ ^ ð_fNðqÞ; NðqÞ 6 ygÞ ¼ _p;q2MðLÞ fNðpÞ ^ NðqÞ; NðpÞ 6 x; NðqÞ 6 yg: Suppose that N(p) 6 x and N(q) 6 y. For each r 2 M(L), if r 6 N(p) ^ N(q), then r 6 N(p), r 6 N(q) and thus N(r) 6 N(p), N(r) 6 N(q) by (3) of Lemma 3.2. It follows that N(r) 6 N(p) ^ N(q) 6 x ^ y. Consequently,
r 6 NðrÞ 6 _fNðuÞ; NðuÞ 6 x ^ yg ¼ s1 ðx ^ yÞ: Therefore N(p) ^ N(q) 6 s1(x ^ y) by Lemma 3.1. So, we have
s1 ðxÞ ^ s1 ðyÞ ¼ _p;q2MðLÞ fNðpÞ ^ NðqÞ; NðpÞ 6 x; NðqÞ 6 yg 6 s1 ðx ^ yÞ: (6) Let a 2 C, p 2 M(L) and p 6 a. Then N(p) 6 a and hence
p 6 _fr 2 MðLÞ; NðrÞ 6 ag ¼ s1 ðaÞ: Consequently, a 6 s1(a) and thus a = s1(a) by (2).
h
Remark. Generally speaking, there are two forms of lower approximation operators:
s1 ðxÞ ¼ _fp 2 MðLÞ; NðpÞ 6 xg; s2 ðxÞ ¼ _fNðpÞ; NðpÞ 6 xg; for every x 2 L. By Theorem 3.3(4), s1 and s2 are equivalent. Thereby in what follows we focus on operator s1. Theorem 3.4. Let C be a cover of L. For every x 2 L, s(x) 6 s1(x). Proof. Suppose that a 2 C and a 6 x. For each p 2 M(L), p 6 a implies N(p) 6 a 6 x. It follows that
p 6 NðpÞ 6 _fNðqÞ; NðqÞ 6 xg ¼ s1 ðxÞ: Consequently, a 6 s1(x) by Lemma 3.1 and hence
sðxÞ ¼ _fb 2 C; b 6 xg 6 s1 ðxÞ: The following example implies that the strict inequality in this theorem is possible. Example 3.1. Let U = {1, 2, 3}, L = P(U) the power set lattice of U, A = {1, 2}, B = {2, 3}. Then {A, B} is a cover of L and N({1}) = {1, 2}, N({2}) = {2}, N({3}) = {2, 3}. By direct computation we have
sðf2gÞ ¼ _fa 2 C; a 6 f2gg ¼ 0; s1 ðf2gÞ ¼ _fNðpÞ; NðpÞ 6 f2gg ¼ f2g: Theorem 3.5. Let C be a reduced cover of L. The following assertions are equivalent: (1) For each p 2 M(L), jCpj = 1. (2) For each x 2 L, s(x) = s1(x).
Proof (1) ) (2). For each p 2 M(L), by jCpj = 1 we have NðpÞ ¼ ^a2C p a 2 C and hence
s1 ðxÞ ¼ _fNðpÞ; NðpÞ 6 xg 6 _fb 2 C; b 6 xg ¼ sðxÞ: Consequently, s(x) = s1(x) by Theorem 3.4. (2) ) (1). Assume that there exists p 2 M(L) such that jCpj > 1. Thus there exist a, b 2 Cp such that a – b. By
s1 ðNðpÞÞ ¼ _fNðrÞ; NðrÞ 6 NðpÞg ¼ NðpÞ;
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it follows that
sðNðpÞÞ ¼ _fd 2 C; d 6 NðpÞg ¼ NðpÞ – 0; and hence there exists d 2 C such that d 6 N(p). Consequently, d 6 a, d 6 b. Therefore d – a. For every q 2 M(L), q 6 d implies that q ^ a P q ^ d = q – 0. This is a contradiction with the fact that C is reduced. h
Corollary 3.1. Let C be a cover of L. The following assertions are equivalent: (1) C is reduced and for each x 2 L, s(x) = s1(x). (2) C is a partition of L.
Proof. (1) ) (2). For every x, y 2 L, we have
sðx ^ yÞ ¼ s1 ðx ^ yÞ ¼ s1 ðxÞ ^ s1 ðyÞ ¼ sðxÞ ^ sðyÞ: By Theorem 2.3, C is a partition of L. (2) ) (1). Suppose that C is a partition of L. It follows that jCpj = 1 for each p 2 M(L). Hence, (1) holds by Theorem 3.5. h 4. The upper approximation operators In this section, we discuss the upper approximation operators by using neighborhoods. Definition 4.1. Let C be a cover of L. Define upper approximation operators s1 ; s2 ; s3 : L ! L, for every x 2 L,
s1 ðxÞ ¼ _fNðpÞ; NðpÞ ^ x – 0g; s2 ðxÞ ¼ _fp 2 MðLÞ; NðpÞ ^ x – 0g; s3 ðxÞ ¼ _fNðpÞ; p 6 xg: Let U be a nonempty set and L = P(U). It follows that, for every X # U,
s1 ðXÞ ¼ [fNðpÞ; NðpÞ \ X – ;g; s2 ðXÞ ¼ fp 2 U; NðpÞ \ X – ;g; s3 ðXÞ ¼ [fNðpÞ; p 2 Xg: Hence, Definition 4.1 is a generalization of covering approximation operators in [15,31]. Theorem 4.1. Let C be a cover of L. (1) (2) (3) (4)
s1 ð1Þ ¼ 1, s1 ð0Þ ¼ 0. For every x, y 2 L, x 6 y implies s1 ðxÞ 6 s1 ðyÞ. For every x 2 L, x 6 s1 ðxÞ. For any xi 2 L, i 2 I, s1 ð_i2I xi Þ ¼ _i2I s1 ðxi Þ. Where I is an index set.
Proof (1) and (2) follow from Definition 4.1 immediately. (3) For each p 6 x, by p 6 N(p), it follows that N(p) ^ x P p – 0, and hence
p 6 NðpÞ 6 _fNðqÞ; NðqÞ ^ x – 0g ¼ s1 ðxÞ: Consequently, x 6 s1 ðxÞ by Lemma 3.1. (4) s1 ð_i2I xi Þ P _i2I s1 ðxi Þ follows from (2). On the other hand, for every p 2 M(L), N(p) ^ (_i2Ixi) – 0 implies _i2I(N(p) ^ xi) – 0, thus there exists i 2 I such that N(p) ^ xi – 0. Hence NðpÞ 6 s1 ðxi Þ 6 _i2I s1 ðxi Þ. Consequently,
s1 ð_i2I xi Þ ¼ _fNðqÞ; NðqÞ ^ ð_i2I xi Þ – 0g 6 _i2I s1 ðxi Þ:
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Similarly, we have: Theorem 4.2. Let C be a cover of L. (1) (2) (3) (4)
s2 ð1Þ ¼ 1, s2 ð0Þ ¼ 0. For every x 2 L, x 6 s2 ðxÞ. For every x,y 2 L, x 6 y implies s2 ðxÞ 6 s2 ðyÞ. For any xi 2 L, i 2 I, s2 ð_i2I xi Þ ¼ _i2I s2 ðxi Þ. Where I is an index set.
Theorem 4.3. Let C be a cover of L and x,y 2 L. (1) s3 ð1Þ ¼ 1, s3 ð0Þ ¼ 0. (2) x 6 s3 ðxÞ. (3) x 6 y implies s3 ðxÞ 6 s3 ðyÞ. (4) s3 ðx _ yÞ ¼ s3 ðxÞ _ s3 ðyÞ. (5) For every a 2 C, s3 ðaÞ ¼ a. The proof of the following theorem is straightforward. Theorem 4.4. Let C be a cover of L. (1) For every x 2 L, s3 ðxÞ 6 s1 ðxÞ. (2) For every x 2 L, s2 ðxÞ 6 s1 ðxÞ. (3) For every x 2 L, s3 ðxÞ 6 sðxÞ. The following example shows that the strict inequality in (1), (2) and (3) of Theorem 4.4 is possible. Example 4.1. Let U = {x, y, z} and L = P(U). We know L is a CCD lattice. Let a = {x, y}, b = {y, z}. Then C = {a, b} is a cover of L. Clearly, N({x}) = {x, y}, N({y}) = {y}, N({z}) = {y, z}. (1) Let X = {y}. Then
s1 ðXÞ ¼ _fNðpÞ; NðpÞ ^ X – 0g ¼ NðfxgÞ _ NðfygÞ _ NðfzgÞ ¼ fx; y; zg; s3 ðXÞ ¼ _fNðpÞ; p 6 Xg ¼ NðfygÞ ¼ fyg; sðXÞ ¼ _d2C fyg d ¼ fx; yg _ fy; zg ¼ fx; y; zg: It follows that s1 ðXÞ – s3 ðXÞ and sðXÞ – s3 ðXÞ. (2) Let X = {x, z}. Then
s1 ðXÞ ¼ _fNðpÞ; NðpÞ ^ X – 0g ¼ NðfxgÞ _ NðfzgÞ ¼ fx; y; zg; s2 ðXÞ ¼ _fp; NðpÞ ^ X – 0g ¼ fx; zg; and s1 ðXÞ – s2 ðXÞ. (3) Let X = {y}, Y = {x}. Then
s2 ðXÞ ¼ _fp; NðpÞ ^ X – 0g ¼ fx; y; zg; s3 ðXÞ ¼ _fNðpÞ; p 6 Xg ¼ NðfygÞ ¼ fyg; s2 ðYÞ ¼ _fp; NðpÞ ^ Y – 0g ¼ fxg; s3 ðYÞ ¼ _fNðpÞ; p 6 Yg ¼ NðfxgÞ ¼ fx; yg: Therefore s2 is3 and s3 is2 .
Lemma 4.5. Let C be a cover of L and CN = {N(p); p 2 M(L)}. (1) CN is a cover of L and (CN)p = {N(p)} for every p 2 M(L). (2) CN is a partition of L if and only if for every p, q 2 M(L), p 6 N(q) ) q 6 N(p). Proof (1) _p2M(L)N(p) P _ p2M(L)p = 1 holds. For each p 2 M(L), we have p 6 N(p). If p 6 N(q), then N(p) 6 N(q). Therefore N(p) is the unique minimal element of p with respect to CN. Hence CN is a cover of L and (CN)p = {N(p)}.
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(2) Suppose CN is a partition of L. For every p, q 2 M(L), p 6 N(q) implies N(p) 6 N(q). It follows that N(p) ^ N(q) = N(p) – 0, and hence N(p) = N(q). Consequently, q 6 N(q) = N(p). Conversely, suppose that N(p) ^ N(q) – 0. Then there exists r 2 M(L) such that r 6 N(p) ^ N(q). It follows that r 6 N(p), r 6 N(q) and hence p 6 N(r), q 6 N(r). Consequently, N(p) = N(r) = N(q). Thus CN is a partition of L. h Theorem 4.6. Let C be a cover of L. s1 ¼ s3 if and only if CN is a partition of L. Proof ()): Suppose that s1 ðxÞ ¼ s3 ðxÞ for every x 2 L. If p, q 2 M(L), p 6 N(q), then N(q) ^ p = p – 0 and hence
s1 ðpÞ ¼ _fNðrÞ; NðrÞ ^ p – 0g P NðqÞ: It follows that
q 6 NðqÞ 6 s1 ðpÞ ¼ s3 ðpÞ ¼ _fNðrÞ; r 6 pg ¼ NðpÞ: Consequently, by Lemma 4.5, CN is a partition of L. (Ü): Suppose that CN is a partition of L. For every x 2 L, N(p) ^ x – 0 implies there exists r 2 M(L) such that r 6 N(p) ^ x. Thus r 6 N(p) and r 6 x. Consequently,
NðpÞ ¼ NðrÞ 6 _fNðqÞ; q 6 xg ¼ s3 ðxÞ: So, we have s1 ðxÞ 6 s3 ðxÞ.
h
Theorem 4.7. Let C be a cover of L. s2 ¼ s3 if and only if CN is a partition of L. Proof. ()): Suppose that s2 ðxÞ ¼ s3 ðxÞ for every x 2 L. If p,q 2 M(L), N(p) ^ N(q) – 0, then there exists r 2 M(L) such that r 6 N(p) ^ N(q). By N(p) ^ r = r – 0, we have
s2 ðrÞ ¼ _ft 2 MðLÞ; NðtÞ ^ r – 0g P p; s3 ðrÞ ¼ _fNðtÞ; t 6 rg ¼ NðrÞ: Consequently, p 6 s2 ðrÞ ¼ s3 ðrÞ ¼ NðrÞ, and hence N(p) 6 N(r) 6 N(p). Analogically one can show that N(q) = N(r). So, we have N(p) = N(q). Therefore CN is a partition of L. (Ü): Suppose that CN is a partition of L and x 2 L. Let p 2 M(L) and p 6 x. For every r 2 M(L), r 6 N(p) implies N(r) 6 N(p), and hence N(r) = N(p). It follows that N(r) ^ x = N(p) ^ x P p – 0. We have r 6 _fq 2 MðLÞ; NðqÞ ^ x – 0g ¼ s2 ðxÞ, and consequently NðpÞ 6 s2 ðxÞ by Lemma 3.1. Thus s3 ðxÞ 6 s2 ðxÞ. On the other hand, let p 2 M(L) and N(p) ^ x – 0. Then there exists r 2 M(L) such that r 6 N(p) ^ x. It follows that r 6 x, r 6 N(p). Since CN is a partition of L, then N(r) = N(p). Consequently,
p 6 NðpÞ ¼ NðrÞ 6 _fNðqÞ; q 6 xg ¼ s3 ðxÞ; and thus
s2 ðxÞ ¼ _fq; NðqÞ ^ x – 0g 6 s3 ðxÞ: In the last of this section, we discuss the relationship between C and CN. Example 4.2. Let U = {x, y, z} and L = P(U). Let a = {x, y}, b = {y, z}, c = {x, z}. Then C = {a, b, c} is a cover of L. Clearly, N({x}) = {x}, N({y}) = {y}, N({z}) = {z}. That is to say, CN is a partition of L and C is not a partition.
Theorem 4.8. Let C be a cover of L. The following assertions are equivalent: (1) C is a partition of L. (2) C is a reduced cover and CN is a partition of L. Proof (1) ) (2). Let p 2 M(L). Since C is a partition, then there exists unique a 2 C such that p 6 a, and hence N(p) = a 2 C. Consequently, CN # C. It follows that N(p) – N(q) implies N(p) ^ N(q) = 0, and therefore CN is a partition of L.
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(2) ) (1). For every a 2 C, there exists p 2 M(L) such that p 6 a and for each b 2 C, b – a, p ^ b = 0 holds. It follows that p i b and hence N(p) = a. That is to say, C # CN. Consequently, if a, b 2 C, a – b, then a ^ b = 0, and C is a partition as required. h Corollary 4.1. Let C be a partition of L. Then s1 ¼ s2 ¼ s3 ¼ s. 5. Conclusions In this paper, we discuss the approximation operators on a CCD lattice. Based on the concept of neighborhood, three kinds of upper approximation operators and one kind of lower approximation operator are constructed with their basic properties being discussed. These operators are extensions of approximation operators in a covering rough set model. Furthermore, the relationship among these operators are investigated. However, it is still an open question regarding the duality of lower and upper approximation operators on a CCD lattice. It is worth noticing that the duality of lower and upper approximation operators cannot be discussed in a general CCD lattice, inasmuch as there is no complement operation in it. We can study this problem in a special CCD lattice which is equipped with a complement operation. Another further study is the topological structure of approximation operators. Acknowledgements This work has been supported by the National Natural Science Foundation of China (Grant Nos. 61175044, 61175055) and the Sichuan Key Technology Research and Development Program (Grant No. 2011FZ0051). References [1] [2] [3] [4] [5] [6] [7] [8] [9]
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