rigina tems, investigated by Davvaz [6], also see [7,15,18,21,24,25,35]. In [44], the notions of rough prime ideals and rough fuzzy prime ideals in a semigroup were introduced. In [38], by considering the notion of an MV-algebra, Rasouli and Davvaz considered a relationship between rough sets andMV-algebra theory. They introduced the notion of rough ideal with respect to an ideal of an MV-algebra, which is an extended notion of ideal in an MV-algebra. In [39], rough approximations of Cayley graphs are studied, and rough edge Cayley graphs are introduced. Also, see [19,46–49]. 0020-0255/$ - see front matter � 2012 Elsevier Inc. All rights reserved. ⇑ Corresponding author. E-mail addresses:
[email protected] (A.A. Estaji),
[email protected] (M.R. Hooshmandasl),
[email protected] (B. Davvaz). Information Sciences 200 (2012) 108–122 Contents lists available at SciVerse ScienceDirect Information Sciences http://dx.doi.org/10.1016/j.ins.2012.02.060 example, see [10,12,13,22,26,28,36,37,40,41]. Rough set theory is an extension of set theory in which a subset of a universe is described by a pair of ordinary sets called lower and upper approximations. A key concept in Pawlak’s rough set model is an equivalence relation. The equivalence classes are the building blocks for the construction of the lower and upper approxima- tions. It soon invoked a natural question concerning a possible connection between rough sets and algebraic systems. Kuroki in [16] introduced the notion of a rough ideal in a semigroup. Kuroki and Wang [17] gave some properties of the lower and upper approximations with respect to the normal subgroups. Mordeson [29] used the covers of the universal set to define approximation operators on the power set of a given set. In [5,6], Davvaz dealt with a relationship between rough sets and ring theory and considered a ring as a universal set and introduced the notion of rough ideals and rough subrings with respect to an ideal of a ring. In [14], Kazanci and Davvaz introduced the notions of rough prime (primary) ideals and rough fuzzy prime (primary) ideals in a ring and gave some properties of such ideals, also see [8,9]. Rough modules have been Lattice Approximation space Rough set Ideal Filter Lower and upper approximations 1. Introduction The concept of a rough set was o plete information in information sys lly proposed by Pawlak [32] as a formal tool for modeling and processing incom- also see [33,34]. Since then, the subject has been investigated in many studies. For Rough set theory applied to lattice theory A.A. Estaji a, M.R. Hooshmandasl b, B. Davvaz c,⇑ aDepartment of Mathematics, Sabzevar Tarbiat Moallem University, Sabzevar, Iran bDepartment of Computer Science, Yazd University, Yazd, Iran cDepartment of Mathematics, Yazd University, Yazd, Iran a r t i c l e i n f o Article history: Received 18 May 2010 Received in revised form 31 January 2011 Accepted 26 February 2012 Available online 20 March 2012 Keywords: a b s t r a c t In this paper, we intend to study a connection between rough sets and lattice theory. We introduce the concepts of upper and lower rough ideals (filters) in a lattice. Then, we offer some of their properties with regard to prime ideals (filters), the set of all fixed points, compact elements, and homomorphisms. � 2012 Elsevier Inc. All rights reserved. journal homepage: www.elsevier .com/locate / ins We know from elementary arithmetic that for any two natural numbers a and b there is a largest number dwhich divides both a and b, namely the greatest common divisor gcd(a, b) of a and b. Also, there is a smallest numbermwhich is multiple of both a and b, namely the least common multiple lcm(a, b). This is pictured in Fig. 1. Turning to another situation, given two statements a and b. There is a ‘‘weakest’’ statement implying both a and b, namely the statement ‘‘a and b’’, which we write as a ^ b. Similarly, there is a ‘‘strongest’’ statement which is implied by a and b, namely ‘‘a or b’’, written as a _ b. This pic- tured in Fig. 2. A third situation arises when we study sets A and B. Again, there is a largest set contained in A and B, the intersection A \ B, and a smallest one containing both A and B, the union A [ B. We obtain similar diagram in Fig. 3. It is typ- ical for modern mathematics that seemingly different areas lead to very similar situations. Then, the idea is to construct the common features in these examples, to study these features, and to apply the resulting theory to many different areas. In this paper, we discuss a general mathematical concept called a lattice, which includes all those examples and many others as special cases. The paper has addressed a connection between two research topics, namely, rough sets and lattice theory, both of which have applications across a wide variety of fields. The paper is in line with a series of algebraic studies reported in the literature. So far, a good number of researchers have dealt with a great host of aspects and issues in this field. In particular, the first papers on connections between rough sets and algebraic structures appeared in Information Sciences, for example [16,17], also see [4,5,21,44]. Lattice theory play an important role in many disciplines of computer science and engineering. For example, they have applications in distributed computing, concurrency theory, programming language semantics and data mining. They are also A.A. Estaji et al. / Information Sciences 200 (2012) 108–122 109 useful in other disciplines of mathematics such as combinatorics, number theory and group theory, see [1,3,20,23,27,30,31,42,43,45]. This paper is organized as follows: In Section 2, we review some basic notions and properties of lattice and rough set. In Section 3, the concept of upper rough ideal on a lattice is introduced and its basic properties are discussed. In Section 4, the concept of lower rough ideal on a lattice is introduced, and we show that ðApr;AprÞ, where Apr;Apr : PðLÞ ! PðLÞ is a Galois connection and also if h is a _-complete full equivalence relation on L and Apr;Apr : IdðLÞ [ f;g ! IdðLÞ [ f;g, then Apr;Apr � � is a Galois connection. 2. Preliminaries of lattice and rough set In this section, we make some general remarks on the concepts of a lattice and rough set theory. The following definitions and preliminaries are required in the sequel of our work and, hence, presented in brief. A poset L is a lattice if and only if for every a and b in L both sup{a, b} and inf{a, b} exist in L. Throughout this paper, L is a lattice with the least element 0 and the greatest element 1. For X # L and x 2 L we write: 1. ;X = {y 2 L:y 6 x for some x 2 X}, 2. "X = {y 2 L:yP x for some x 2 X}, 3. ;x = ;{x}, 4. "x = "{x}. We also say that: 5. X is a lower set if and only if X = ;X, 6. X is an upper set if and only if X = "X. A subset D of L is directed if it is non-empty and every finite subset of D has an upper bound in D (aside from non-emp- tiness, it is sufficient to assume that every pair of elements in L has an upper bound in L). Dually, we call a non-empty subset F of L filtered if every finite subset of F has a lower bound in F. We also say F is a filter if and only if it is a filtered upper set. Fig. 1. a, b are two natural numbers, gcdða; bÞ is the greatest common divisor of a, b, and lcmða; bÞ is the least common multiple of a, b. 110 A.A. Estaji et al. / Information Sciences 200 (2012) 108–122 Re a supr L is for ev A l frame A n denot Propo Ik = T k In par lattice Proof Th dually Re relatio Fig. 2. a, b are two statements, a ^ b is the weakest statement and a _ b is the strongest statement. call that c 2 L is compact if c 6Wi2I xi implies that c 6Wi2Fxi, for a suitable finite subset F of I. L is algebraic if each x 2 L is emum of compact elements. n(L) stands for the set of compact elements of L. If 1 is compact it is said that L is compact. a frame if for every S # L, W S exists in L and the following distributive law holds: a ^ _ S � � ¼ _ s2S ða ^ sÞ; ery a 2 L. attice L is distributive if for every a,b,c 2 L, a ^ (b _ c) = (a ^ b) _ (a ^ c). It is well known that an algebraic lattice is a as long as it is distributive. on-empty subset I of a lattice L is an ideal if and only if it is a directed lower set. Consider the set of all ideals of L, ed as Id(L). sition 2.1. Id(L) is a poset under set inclusion and, as a poset, it is a lattice. Also, if {Ik}k2K is a set of ideals of L, then V k2K 2KIk and x 2 W k2KIk if and only if there exist indices k1, . . . , kn 2K and xki 2 Iki such that x 6 xk1 _ � � � _ xkn : ticular, Id(L) is an algebraic lattice and for every a,b 2 L, ;a ^ ;b = ;(a ^ b) and ;a _ ;b = ;(a _ b). Also, if L is a distributive , then Id(L) is a frame. . See [11]. h e function g:L?M between lattices L and M is a lattice homomorphism if, for each a, b 2 L, g(a ^ b) = g(a) ^ g(b), and for supremum. A lattice isomorphism which is also a bijection is said to be a lattice isomorphism. call that an equivalence relation h on L is a reflexive, symmetric, and transitive binary relation on L. If h is an equivalence n on L, then the equivalence class of a 2 L is the set {b 2 L:(a, b) 2 h}. We denote it as [a]h. Fig. 3. A, B are sets. define 1. AprðXÞ#X#AprðXÞ, 1. if X is a lower set, then AprðXÞ is a lower set, (2) A.A. Estaji et al. / Information Sciences 200 (2012) 108–122 111 i¼1 i¼1 By hypothesis, there exists d 2 D such that for every 1 6 i 6 n, di 6 d. Hence, by Lemma 3.2, _n di; _n ai 2 h: (3) Suppose that a1; . . . ; an 2 AprðDÞ. Then, there exist d1, . . . , dn 2 D such that for every 1 6 i6 n, (di, ai) 2 h andby Lemma3.2, ! This is similar to the statement (1). 2. if X is an upper set, then AprðXÞ is an upper set, 3. if D is a directed set, then AprðDÞ is a directed set, 4. if F is a filtered set, then AprðFÞ is a filtered set, 5. if F is a filter set, then AprðFÞ is a filter set. Proof (1) It is clear that AprðXÞ# # AprðXÞ. Now, assume that a 2# AprðXÞ. Then there exists b 2 AprðXÞ such that a 6 b. So there exists x 2 X such that (x, b) 2 h and by Definition 3.1, (a ^ x, a) = (a ^ x,a ^ b) 2 h. Since X is a lower set and a ^ x 6 x 2 X, we infer that a ^ x 2 [a]h \ X. Therefore, a 2 AprðXÞ and the proof is completed. Definition 3.1. Let h be an equivalence relation on L. Then h is called a full congruence relation if (a, b) 2 h implies that (a _ x,b _ x) 2 h and (a ^ x,b ^ x) 2 h for all x 2 L. Throughout this paper, h is a full congruence relation on L. The following lemma is exactly obtained from Definition 3.1. Lemma 3.2. For every a, b, c, d 2 L, 1. if (a, b) 2 h and (c, d) 2 h, then (a ^ c, b ^ d) 2 h and (a _ c, b _ d) 2 h, 2. {x _ y:x 2 [a]h and y 2 [b]h} # [a _ b]h, 3. {x ^ y:x 2 [a]h and y 2 [b]h} # [a ^ b]h. Proof. (1) Suppose that (a, b) 2 h and (c, d) 2 h. Then (a _ c, b _ c) 2 h and (b _ c,b _ d) 2 h. Since h is an equivalence relation on L, there holds (a _ c, b _ d) 2 h. Also, in a similar way, (a ^ c, b ^ d) 2 h. In order to prove (2) and (3), assume that x 2 [a]h and y 2 [b]h. By statement (1), (x ^ y, a ^ b) 2 h and (x _ y, a _ b) 2 h and the proof is completed. h Proposition 3.3. If X, D and F are subsets of L, then 2. if X # Y, then Apr(X) # Apr(Y) and AprðXÞ#AprðYÞ, 3. AprðX [ YÞ ¼ AprðXÞ [ AprðYÞ and AprðX \ YÞ#AprðXÞ \ AprðYÞ, 4. Apr(X \ Y) = Apr(X) \ Apr(Y) and Apr(X [ Y) � Apr(X) [ Apr(Y), 5. Apr(Apr(X)) = Apr(X) and AprðAprðXÞÞ ¼ AprðXÞ. Proof. It is straightforward. h 3. Upper rough ideals Our main objective in this section is upper rough approximations. We begin by introducing a full congruence relation that will be used throughout this paper. Also, by a lower rough approximation in (L, h) we mean a mapping Apr : PðLÞ ! PðLÞ defined for every X 2 PðLÞ by AprðXÞ ¼ fa 2 L : ½a�h#Xg: One may interprets Apr, Apr : PðLÞ ! PðLÞ as two unary set-theoretic operators called approximation operators. The system ðPðLÞ; c;Apr;Apr;\;[Þ is called a rough set algebra. The following proposition is well known and easily seen. Proposition 2.2. Let (L, h) be an approximation space. For every subsets X, Y # L, we have d for every X 2 PðLÞ by AprðXÞ ¼ fa 2 L : ½a�h \ X – ;g: The pair (L, h), where h is an equivalence relation on L, is called an approximation space [32]. For an approximation space (L, h), by an upper rough approximation in (L, h), we mean a mapping Apr : PðLÞ ! PðLÞ i¼1 i¼1 i¼1W� � y 6 x. By Definition 3.1, (x, z _ x) = (y _ x, z _ x) 2 h. Then z 6 z _ x 2 AprðXÞ and we conclude that z 2# AprðXÞ. Propo upper 112 A.A. Estaji et al. / Information Sciences 200 (2012) 108–122 sition 3.11. Let h be a ^-complete full equivalence relation on L. If P is a prime ideal of L such that AprðPÞ– L, then P is an rough prime ideal of L. Recall that a proper ideal I of L is called a prime ideal of L if for every a, b 2 R, a ^ b 2 I implies that a 2 I or b 2 I. Definition 3.10. A non-empty subset X of L is called an upper rough prime ideal if AprðXÞ is a prime ideal of L. If I is an ideal of L, then AprðIÞ is an ideal of L. Definition 3.9. A non-empty subset X of L is called an upper rough ideal if AprðXÞ is an ideal of L. Proposition 3.8. Let h be a _-complete full equivalence relation on L. Every prime filter of L is an upper rough prime filter. Proof. Suppose that F is a prime filter of L. Then, by Proposition 3.3, AprðFÞ is a filter set of L. Assume that a, b 2 L and a _ b 2 AprðFÞ. Then [a _ b]h \ F– ;. Since h is a _-complete, there exist x 2 [a]h and y 2 [b]h such that x _ y 2 F. We have x 2 F or y 2 F since F is a prime filter of L. Hence, x 2 [a]h \ F or y 2 [b]h \ F, i.e., a 2 AprðFÞ or b 2 AprðFÞ, and the proof is completed. h (2)) (1) By Lemma 3.2, it is clear. The equivalence of (1w) and (2w) is proved dually. h Definition 3.6. Let h be a full equivalence relation on L. Then h is called a _-complete if [a _ b]h = {x _ y:x 2 [a]h and y 2 [b]h} for all a,b 2 L. Likewise, h is called a ^-complete if [a ^ b]h = {x ^ y:x 2 [a]h and y 2 [b]h} for all a, b 2 L. Recall that a filter F of L is called a prime filter of L if for every a, b 2 L, a _ b 2 F implies that a 2 F or b 2 F. Definition 3.7. A non-empty subset X of L is called an upper rough prime filter, if AprðXÞ is a prime filter of L. Likewise, if F is a filter of L and a,b 2 L, then the following are equivalent: (1w) [a ^ b]h \ F– ;, (2w) {x ^ y:x 2 [a]h and y 2 [b]h} \ F– ;. Proof (1)) (2) Let x 2 L be any element of [a _ b]h \ I. Hence, (x, a _ b) 2 h. And by Definition 3.1, (x ^ a, a) = (x ^ a, (a _ b) ^ a) 2 h. Since x ^ a 6 x 2 I and I is an ideal of L, we infer that x ^ a 2 [a]h \ I. Also, in a similar way, x ^ b 2 [b]h \ I. Therefore, (x ^ a) _ (x ^ b) 2 {x _ y:x 2 [a]h and y 2 [b]h} \ I and the proof is completed. Lemma 3.5. If I is an ideal of L and a, b 2 L, then the following are equivalent: 1. [a _ b]h \ I– ;, 2. {x _ y:x 2 [a]h and y 2 [b]h} \ I– ;. The rest is similar. h Proof. Since ;X is a lower set, we can then conclude from Proposition 3.3 that Aprð# XÞ is a lower set, that is, Aprð# XÞ ¼# Aprð# XÞ. It is clear that AprðXÞ#Aprð# XÞ. Thus, since Aprð# XÞ is a lower set, we conclude that # AprðXÞ#Aprð# XÞ. Now, suppose that z 2 Aprð# XÞ. Then, there exists y 2 ;X such that (y, z) 2 h. It follows that there exists x 2 X such that Therefore, d _ ni¼1ai 2 AprðDÞ is an upper bound for faigni¼1. (4) The proof is similar to the proof of statement (3). (5) It is straightforward. h Corollary 3.4. For each X # L, we have # AprðXÞ ¼ Aprð# XÞ ¼# Aprð# XÞ and " AprðXÞ ¼ Aprð" XÞ ¼" Aprð" XÞ. d;d _ _n ai ! ! ¼ d _ _n di ! ;d _ _n ai ! ! 2 h: Propo Proof x 2 Ap and th Propo a2I b2J a2I;b2J (2)) (1) It is straightforward. h A.A. Estaji et al. / Information Sciences 200 (2012) 108–122 113 ¼ AprðI _ JÞ: ¼ _ a2I;b2J Aprð# aÞ _ Aprð# bÞ ¼ _ Aprð# ða _ bÞ (5) Since I _ J# I _ AprðJÞ, we infer that AprðI _ JÞ#AprðI _ AprðJÞÞ. Now, suppose that x 2 AprðI _ AprðJÞÞ. Then, there exists y 2 I _ AprðJÞ such that (x,y) 2 h, which implies that a 2 I and b 2 AprðJÞ such that y 6 a _ b. Hence, there exists c 2 J such that (c, b) 2 h and by Definition 3.1, we have ((a _ c) ^ y, (a _ b) ^ y) = ((a _ c) ^ y, y) 2 h. Since h is an equivalence relation on L, we infer that ((a _ c) ^ y, x) 2 h. Therefore, x 2 AprðI _ JÞ, because I _ J is an ideal, and (a _ c) ^ y 2 I _ J. h AprðIÞ _ AprðJÞ#AprðI _ JÞ. In general, equality fails in this relation. We refer the reader to Example 4.2. Proposition 3.14. In a lattice Id(L), the following statements are equivalent: 1. For each a, b 2 L, Aprð# ða _ bÞÞ ¼ Aprð# aÞ _ Aprð# bÞ; 2. For each I,J 2 Id(L), AprðI _ JÞ ¼ AprðIÞ _ AprðJÞ. Proof (1)) (2) Let I,J 2 Id(L). By Proposition 3.12, AprðIÞ _ AprðJÞ ¼ _ Aprð# aÞ _ _ Aprð# bÞ AprðI _ JÞ#AprðAprðIÞ _ AprðJÞÞ: Since I _ J#AprðIÞ _ AprðJÞ, then AprðAprðIÞ _ AprðJÞÞ#AprðI _ JÞ: 5. AprðI _ AprðJÞÞ ¼ AprðI _ JÞ. Proof (1) It is clear that AprðI ^ JÞ#AprðIÞ ^ AprðJÞ. Let x 2 L be any element of AprðIÞ ^ AprðJÞ. Then, there exist a 2 I and b 2 J such that (x,a) 2 h and (x, b) 2 h and by Lemma 3.2, we have (x,a ^ b) 2 h. Since a ^ b 2 I ^ J, we infer that x 2 AprðI ^ JÞ. (2) Since ;(a ^ b) = ;a ^ ; b, by statement (1), the proof is completed. (3) It is straightforward. (4) By statement (3) and Proposition 2.2, we have 1. AprðI ^ JÞ ¼ AprðIÞ ^ AprðJÞ, 2. For every a, b 2 L, Aprð# ða ^ bÞÞ ¼ Aprð# aÞ ^ Aprð# bÞ, 3. AprðIÞ _ AprðJÞ#AprðI _ JÞ, 4. AprðAprðIÞ _ AprðJÞÞ ¼ AprðI _ JÞ, sition 3.12. For every I 2 Id(L), AprðIÞ ¼ Wa2IAprð# aÞ. . For every a 2 I, by Proposition 2.2, Aprð# aÞ#AprðIÞ, which implies that Wa2IAprð# aÞ#AprðIÞ. Now, suppose that rðIÞ. Then, there exists a 2 I such that (a,x) 2 h. Hence, a 2 [x]h \ ;a, i.e., x 2 Aprð# aÞ. Therefore, AprðIÞ# W a2IAprð# aÞ, e proof is completed. h sition 3.13. Let I and J be ideals of L. In the lattice Id(L), the following properties hold. Proof. The proof is similar to that of Proposition 3.8. h 1. h 6. Aprh1 ðAÞ is a prime ideal of L1 if and only if Aprh2 ðuðAÞÞ is a prime ideal of L2. since h2 is a full congruence relation on L2 and u is an epimorphism, we conclude that (u(a _ c), 2. fX# L : Aprh2 ðXÞ ¼ Xg# fX# L : Aprh1 ðXÞ ¼ Xg. 114 A.A. Estaji et al. / Information Sciences 200 (2012) 108–122 Proof (1)) (2) Let X # L and Aprh2 ðXÞ ¼ X. Then, X#Aprh1 ðXÞ#Aprh2 ðXÞ ¼ X. It follows that Aprh1 ðXÞ ¼ X. (2)) (1) Let X # L. Since Aprh2 ðXÞ 2 fX# L : Aprh2 ðXÞ ¼ Xg, we have Aprh2 ðXÞ 2 fX# L : Aprh1 ðXÞ ¼ Xg. Thus, Aprh1 ðXÞ#Aprh1 ðAprh2 ðXÞÞ ¼ Aprh2 ðXÞ: � An ideal I of L is called fixed-points of upper rough approximation of Id(L), if AprðIÞ ¼ I. Let FðLÞ denote the family of all fixed- points of upper rough approximation of Id(L). u(b _ c)) = (u(a) _ u(c), u(b) _ u(c)) 2 h2 and (u(a ^ c), u(b ^ c)) = (u(a) ^ u(c), u(b) ^ u(c)) 2 h2. It follows that (a _ c, b _ c) 2 h1 and (a ^ c, b ^ c) 2 h1, and the proof is completed. (2) Let a,b,c 2 L2 and c 2 ½a ^ b�h2 . Then, there exist x, y, z 2 L1 such that u(x) = a, u(y) = b, and u(z) = c. Hence, uðzÞ 2 ½uðxÞ ^uðyÞ�h2 ¼ ½uðx ^ yÞ�h2 . It follows that z 2 ½x ^ y�h1 . Since h1 is ^-complete, we conclude that there exist x1 2 ½x�h1 and y1 2 ½y�h1 such that z = x1 ^ y1. From this, we infer c = u(z) = u(x1) ^ u(y1), uðx1Þ 2 ½a�h2 , uðy1Þ 2 ½b�h2 , and the proof is completed. (3) Let x, y, z 2 L1 and z 2 ½x ^ y�h1 . Then, uðzÞ 2 ½uðx ^ yÞ�h2 ¼ ½uðxÞ ^uðyÞ�h2 . It follows that there exist a 2 ½uðxÞ�h2 and b 2 ½uðyÞ�h2 such that u(z) = a ^ b. Since u is an epimorphism, we conclude that there exist x1, y1 2 L1 such that u(x1) = a and u(y1) = b. Hence, u(z) =u(x1 ^ y1). From this, we infer z = x1 ^ y1. This completes the proof of part (3). (4) It is straightforward. (5) Necessity: Let a; b 2 Aprh2 ðuðAÞÞ. Then, there exist x,y 2 L1 and x1, y1 2 A such that u(x) = a, u(y) = b, (u(x), u(x1)) 2 h2, and (u(y),u(y1)) 2 h2. From this, we infer (x, x1) 2 h1, and (y, y1) 2 h1. That is, x; y 2 Aprh1 ðAÞ. Since Aprh1 ðAÞ is an ideal of L1, we conclude that x _ y 2 Aprh1 ðAÞ, and from this we infer a _ b 2 Aprh2 ðuðAÞÞ. Now, suppose that a,b 2 L2 and a 6 b 2 Aprh2 ðuðAÞÞ. Then there exist x, y 2 L1 and y1 2 A such that u(x) = a, u(y) = b, and (u(y), u(y1)) 2 h2. By Defini- tion 3.1, (u(x), u(x ^ y1)) 2 h2, this is, (x, x ^ y1) 2 h1. Again by Definition 3.1, (x _ y1, y1) 2 h1 and from this we infer x _ y1 2 Aprh1 ðAÞ. Since Aprh1 ðAÞ is an ideal of L1, we conclude that x 2 Aprh1 ðAÞ. Hence, a ¼ uðxÞ 2 Aprh2 ðuðAÞÞ. Sufficiency: Take x; y 2 Aprh1 ðAÞ. Then, uðxÞ;uðyÞ 2 Aprh2 ðuðAÞÞ. Since Aprh2 ðuðAÞÞ is an ideal of L2, we conclude that uðx _ yÞ ¼ uðxÞ _uðyÞ 2 Aprh2 ðuðAÞÞ. That is, there exists a 2 A such that (u(x _ y), u(a)) 2 h2. Hence, (x _ y, a) 2 h1. From this, we infer x _ y 2 Aprh1 ðAÞ. Now, suppose that x, y 2 L1 and x 6 y 2 Aprh1 ðAÞ. Then, there exists a1 2 A such that (y, a1) 2 h1 and by Definition 3.1, (x _ a1, a1) = ((x ^ y) _ a1, (x ^ a1) _ a1) 2 h1. Hence, (u(x) _ u(a1), u(a1)) 2 h2, and from this we infer uðxÞ _uða1Þ 2 Aprh2 ðuðAÞÞ. Since Aprh2 ðuðAÞÞ is an ideal of L2, we conclude that uðxÞ 2 Aprh2 ðuðAÞÞ. Thus, there exists a2 2 A such that (u(x), u(a2)) 2 h2. That is, (x, a2) 2 h1; in fact, x 2 Aprh1 ðAÞ. (6) Necessity: Take a, b 2 L2 and a ^ b 2 Aprh2 ðuðAÞÞ. Then, there exist x,y 2 L1 and z 2 A such that u(x) = a, u(y) = b, and (u(x ^ y), u(z)) 2 h2. From this, we infer (x ^ y, z) 2 h1. That is x ^ y 2 Aprh1 ðAÞ. Since Aprh1 ðAÞ is a prime ideal of L1, we conclude that x 2 Aprh1 ðAÞ or y 2 Aprh1 ðAÞ. It follows that a 2 Aprh2 ðuðAÞÞ or b 2 Aprh2 ðuðAÞÞ. Hence by statement (5), Aprh2 ðuðAÞÞ is a prime ideal of L2. Sufficiency: Take x, y 2 L1 and x ^ y 2 Aprh1 ðAÞ. Then, uðxÞ ^uðyÞ 2 Aprh2 ðuðAÞÞ. Since Aprh2 ðuðAÞÞ is a prime ideal of L1, we conclude that uðxÞ 2 Aprh2 ðuðAÞÞ or uðyÞ 2 Aprh2 ðuðAÞÞ. Thus, there exists z 2 A such that (u(x), u(z)) 2 h2 or (u(y), u(z)) 2 h2. That is (x, z) 2 h1 or (y, z) 2 h1. From this, we infer x 2 Aprh1 ðAÞ or y 2 Aprh1 ðAÞ. Hence, by statement (5), Aprh1 ðAÞ is a prime ideal of L1. h Proposition 3.16. Let h1 and h2 be two equivalence relations on L. Then the following statements are equivalent in the lattice Id(L): 1. For each X # L, Aprh1 ðXÞ#Aprh2 ðXÞ, Proof (1) This is clear that h1 is an equivalence relation on L1. Now, suppose that (a,b) 2 h1 and c 2 L1. Then, (u(a), u(b)) 2 h2 and 1 1 1 2 1 2. If h1 is ^-complete, then h2 is ^-complete, 3. If h2 is ^-complete and u is one to one, then h1 is ^-complete, 4. uðAprh1 ðAÞÞ ¼ Aprh2 ðuðAÞÞ, 5. Aprh1 ðAÞ is an ideal of L1 if and only if Aprh2 ðuðAÞÞ is an ideal of L2, = {(a, b) 2 L � L :(u(a), u(b)) 2 h } is a full congruence relation on L , Proposition 3.15. Let u be an epimorphism of a lattice L1 to a lattice L2, and let h2 be a full congruence relation on L2. Then, for every A # L1, compl where WIdðLÞ WIdðLÞ Proof I ^WFk2 Propo Proof. Suppose thatm is an arbitrary element in A. Then, there exist K1; . . . ;Kn 2 A such that # m 6 i¼1Ki. By Proposition 2.1, w Hence, # a ^ Aprð# mÞ 6 AprðI^ # mÞ 6 J for each a 2 I. So by Proposition 3.12, I ^ Aprð# mÞ 6 J, i.e., Aprð# mÞ 2 A. We conclude and by Proposition 3.12, we have Apr WAð Þ 6 WA. Now, by Proposition 2.2, we have Apr WAð Þ ¼ WA. Since for every K 2 A, Rec Propo _FðLÞ By Lem k k A.A. Estaji et al. / Information Sciences 200 (2012) 108–122 115 k2K k2K We put J ¼ WIdðLÞk2K Ik, and we conclude from Proposition 3.12 that ma 3.17, _FðLÞ I ¼ Apr _IdðLÞ I ! : Aprð# aÞ 6 k2K Ik: for all a 2 L. In particular, for every I 2 FðLÞ, I ¼ a2I Aprð# aÞ. Proof. Since by Lemma 3.17, FðLÞ is a complete lattice, it suffices to show that each element of FðLÞ is a supremum of com- pact elements of FðLÞ. First, we show that for all a 2 L, Aprð# aÞ is a compact element in FðLÞ. Suppose that a is an arbitrary element in L. Let {Ik}k2K be a subset of FðLÞ such that all also that the compact elements of Id(L) are the ideals of the form ;a for every a 2 L. sition 3.20. FðLÞ is an algebraic complete lattice. In fact, the compact elements are precisely the ones of the form Aprð# aÞ,WFðLÞ K 6 AprðKÞ 6 WA, we conclude that WK2AAprðKÞ ¼ WA. h that Aprð# mÞ 6 WA. This proves that_ m2 W A Aprð# mÞ 6 _ A i¼1 i i¼1 i e have I^ # m 6 I ^ _n K ! ¼ _n ðI ^ K Þ 6 J: K2A W Wn sition 3.19. Let I 2 Id(L) and J 2 FðLÞ. If L is a distributive lattice and A ¼ fK 2 IdðLÞ : K ^ I 6 Jg, then in the lattice Id(L), Apr _ A � � ¼ _ AprðKÞ ¼ _ A: complete. h . If I 2 FðLÞ and fIkgk2K# FðLÞ, then by Lemma 3.17 and Propositions 2.1 and 3.13, we have ðLÞ KIk ¼ AprðIÞ ^ Apr WIdðLÞ k2K Ik � � ¼ Apr I ^WIdðLÞk2K Ik� � ¼ Apr WIdðLÞk2K ðI ^ IkÞ� � ¼ WFðLÞk2KðI ^ IkÞ, and by Lemma 3.17 the proof is now bound of {Ik}k2K. Since k2K Ik 6 K , we can then conclude from Proposition 2.2 that Aprð k2K IkÞ 6 AprðKÞ ¼ K. Therefore _FðLÞ k2K Ik ¼ Apr _IdðLÞ k2K Ik ! 2 FðLÞ: � Proposition 3.18. If L is a distributive lattice, then FðLÞ is a frame under set inclusion. Proof. It is clear that FðLÞ is poset under set inclusion. Let I and J be any elements in FðLÞ. By Proposition 3.13, AprðI ^ JÞ ¼ AprðIÞ ^ AprðJÞ ¼ I ^ J 2 FðLÞ. Now, we assume that fIkgk2K# FðLÞ. Since for every k 2K, Ik 6 WIdðLÞ k2K Ik, we can then conclude from Proposition 2.2 that Ik 6 AprðIkÞ 6 Apr WIdðLÞ k2K Ik � � . Therefore, WFðLÞ k2KIk 6 Apr WIdðLÞ k2K Ik � � . Let K 2 FðLÞ be an upper WId(L) denotes supremum in Id(L). k2K Ik ¼ Apr k2K Ik ; ete lattice. In particular, for every fIkgk2K# FðLÞ, _FðLÞ _IdðLÞ ! Lemma 3.17. The set FðLÞ of all fixed-points of upper rough approximation of Id(L) is poset under set inclusion and, as poset, it is a b 2 Iki : a2I No such t 1. For 2. Ap 4. Ap 5. For 6. Ap (1) (2) w 116 A.A. Estaji et al. / Information Sciences 200 (2012) 108–122 Apr AprH ^ A 6 ^ A; hich implies that (3) Let A ¼ fXkgk2K#PðLÞ. We have� �� � By definition of AprH, the proof is clear. Suppose that X;Y 2 PðLÞ such that X = Y. By the statement (1), AprðAprHðXÞÞ 6 X ¼ Y and by definition AprH, we have AprHðXÞ 6 AprHðYÞ. Similarly, AprHðYÞ 6 AprHðXÞ. Proof 7. if X is a lower set, then AprHðXÞ is a lower set, 8. if X is an upper set, then AprHðXÞ is an upper set. rH is order preserving, each X 2 PðLÞ, AprHðXÞ 6 X, rH Apr AprH ¼ AprH and Apr AprH Apr ¼ Apr, 3. AprH preserves arbitrary infima in lattice PðLÞ, each X 2 PðLÞ, AprðAprHðXÞÞ 6 X 6 AprHðAprðXÞÞ, rH is a map, I ¼ _ 16i6n Aprð# aiÞ ¼ Apr _ 16i6n # ai ¼ Apr # _ 16i6n ai : Therefore, I 2 FðLÞ is a compact element in FðLÞ if and only if there exists a 2 L such that I ¼ Aprð# aÞ. h Let AprH be a relation on PðLÞ that satisfies the following condition: X 6 AprHðYÞ () AprðXÞ 6 Y : We shall show this relation as a unique function. Proposition 3.21. The following properties hold: w, suppose that I is an arbitrary compact element in FðLÞ. Since I ¼ WFðLÞa2I Aprð# aÞ, we infer that there exist a1, . . . ,an 2 I hat FðLÞ IdðLÞ ! L ! ! # a 6 IdðLÞa2I # a 6 Apr IdðLÞa2I # a , we can then conclude from Proposition 2.2 that Aprð# aÞ 6 Apr IdðLÞa2I # a ¼ I, it follows that WFðLÞ a2I Aprð# aÞ 6 I, i.e., I ¼ WFðLÞ a2I Aprð# aÞ. Hence, FðLÞ is an algebraic lattice. # a 6 a2I Aprð# aÞ. Hence, by Lemma 3.17, I ¼ AprðIÞ ¼ Apr a2I # a 6 a2I Aprð# aÞ. Since for every a 2 I,W W� � W� � 16i6s It follows that # a 6 Aprð# bÞ 6 Apr _IdðLÞ 16i6s Iki ! : Hence, by Proposition 2.2 and Lemma 3.17, we have Aprð# aÞ 6 Apr _IdðLÞ 16i6s Iki ! ¼ _FðLÞ 16i6s Iki : Thus, Aprð# aÞ is a compact element in FðLÞ. Now, suppose that I is an arbitrary element in FðLÞ. Since for every a 2 I, # a 6 Aprð# aÞ 6 WFðLÞa2I Aprð# aÞ, we conclude thatWIdðLÞ WFðLÞ WIdðLÞ� � WFðLÞ Hence, there exist b1, . . . , bn 2 J such that # a 6 _IdðLÞ 16i6n Aprð# biÞ 6 Apr _IdðLÞ 16i6n Aprð# biÞ ! ¼ Apr # _L 16i6n bi ! ! : If we put b ¼ WL16i6nbi 2 J, then there exist k1 . . . , ks 2K such that_IdðLÞ # a 6 Aprð# aÞ 6 b2J Aprð# bÞ: _IdðLÞ AprH ^ A � � 6 AprHðX Þ f definition we have Apr AprHðXkÞ 6 Xk; k2K k2K H H (4) Suppose that X;Y 2 PðLÞ such that X 6 Y. Then X ^ Y = Y. By statement (3), we have AprHðXÞ ^ AprHðYÞ ¼ (7) It is clear that Apr ðXÞ# # Apr ðXÞ. Now, assume that a 2# Apr ðXÞ. Then, there exists b 2 Apr ðXÞ such that a 6 b. By we obtain a 2 AprHðXÞ. Propo fx _ y : x 2 ½a�h and y 2 ½b�hg ¼ ½a _ b�h: Proof and [a]h # I. It follows that [a _ b]h = {x _ y:x 2 [a]h and y 2 [b]h} # I. If c 2 Aprð# ða _ bÞ, then there exists d 6 a _ b such that ideal, Proposition 3.24. Let P 2 Id(L) and for each a, b 2 P, If P is H H H we in ideal o A.A. Estaji et al. / Information Sciences 200 (2012) 108–122 117 fer that Aprð# aÞ# P or Aprð# bÞ# P which implies that a 2# a#AprHðPÞ or b 2# b#AprHðPÞ. Hence, AprHðPÞ is a prime f L. h Proof. By Proposition 3.23, Apr ðIÞ is an ideal of L. Now, suppose that a, b 2 L and a ^ b 2 Apr ðPÞ. Then by Lemma 3.22, # ða ^ bÞ#AprHðPÞ, and by Proposition 3.13, we obtain Aprð# aÞ ^ Aprð# bÞ ¼ Aprð# ða ^ bÞÞ# P. Since P is a prime ideal of L, a prime ideal of L, then Apr ðPÞ is a prime ideal of L. fx _ y : x 2 ½a�h and y 2 ½b�hg ¼ ½a _ b�h: we conclude that c 2 I. Hence, Aprð# ða _ bÞ# I and we obtain a _ b 2# ða _ bÞ#Apr ðIÞ. h (d,c) 2 h, and by Definition 3.1, (b _ a, c _ b _ a) = (d _ b _ a, c _ b _ a) 2 h. It follows that c _ b _ a 2 [a _ b]h # I. Since I is an H . If a 2 AprHðIÞ and b 6 a, then by Lemma 3.22, b 2# a#AprHðIÞ. Now, suppose that a; b 2 AprHðIÞ. It is clear that [a]h # I Then, AprHðIÞ is an ideal of L. sition 3.23. Let I 2 Id(L) and for each a, b 2 I, (8) The proof is similar to that of statement (7). h Lemma 3.22. If I 2 Id(L) and a 2 AprHðIÞ, then # a#AprHðIÞ. Proof. It is clear that ;a # I and by Proposition 3.21, ½a�h ¼ AprðfagÞ#AprðAprHðIÞÞ# I. Now, suppose that b 2 Aprð# aÞ. Then, there exists c 2 ;a such that (b, c) 2 h and by Definition 3.1, (b _ a, c _ a) = (b _ a, a) 2 h. Thus b _ a 2 [a]h # I. Since I is an ideal, we conclude that b 2 I. Now we have # a#AprHðIÞ. h definition we have ½b�h ¼ AprðfbgÞ#X. Take x 2 [a]h. Then (x, a) 2 h and by Definition 3.1, (x _ b,b) = (x _ b, a _ b) 2 h. Hence, x _ b 2 [b]h # X. Since X is a lower set and x 6 x _ b 2 X, we conclude that x 2 X. Thus, ½a�h ¼ AprðfagÞ#X and so AprHðX ^ YÞ ¼ AprHðYÞ. Hence, AprHðXÞ 6 AprHðYÞ. (5) Take x 2 AprHðXÞ. Then, x 2 ½x�h ¼ AprðfxgÞ#AprðAprHðXÞÞ#X. (6) It is straightforward. H H H H k2K Apr ðXkÞ 6 Apr k2K Xk : V V� � and so we have k2K for each k 2K. Thus, we conclude that Apr V AprHðXkÞ � � 6 V Xk; V� � or each k 2K. Thus, AprH Að Þ 6 k2KAprHðXkÞ. Also, it is clear that k2KAprHðXkÞ 6 AprHðXkÞ for each k 2K and by k V V V Apr AprH ^ A � �� � 6 Xk for each k 2K and by definition, 2. For each X 2 PðLÞ, 2. If h Proof (1) Apr A 6 k2K AprðXkÞ: Propo 2. Ap 8. Ap 9. Apr is an identity map. Proof 118 A.A. Estaji et al. / Information Sciences 200 (2012) 108–122 (1)) (2) Suppose that X 2 PðLÞ. Then, there exists Y 2 PðLÞ such that AprðYÞ ¼ X. We conclude that AprðXÞ ¼ AprðAprðYÞÞ ¼ AprðYÞ ¼ X. (2)) (3) It is straightforward. rH is a surjective map, H 3. Apr is an injective map, 4. For each a 2 L, [a]h = {a}, 5. Apr AprH ¼ 1PðLÞ, 6. AprH is an injective map, 7. AprH Apr ¼ 1PðLÞ, r is an identity map, 1. Apr is a surjective map, sition 3.27. For Apr : PðLÞ ! PðLÞ, the following statements are equivalent: (2) The proof is similar to that of statement (1). h k2K AprðXkÞ 6 Apr A : Now, suppose that Y 2 PðLÞ is an another upper bound for fAprðXkÞgk2K. Hence, AprðXkÞ 6 Y for each k 2K. We conclude that Xk 6 AprHðYÞ for each k 2K which implies that WA 6 AprHðYÞ, i.e.,_� � _ It is clear that AprðXkÞ 6 Apr Að Þ for each k 2K. Then_ _� � W k2K k k2K k is a _-complete full equivalence relation on L, then in the lattice Id(L), we have Apr _ B � � ¼ W AprðI Þ and AprH ^B� � ¼ V AprHðI Þ: Proposition 3.26. Let A ¼ fXkgk2K#PðLÞ and B ¼ fIkgk2K# IdðLÞ. Then, the following statements hold: 1. In the lattice PðLÞ, Apr WAð Þ ¼ Wk2KAprðXkÞ; Proof. It is straightforward. h AprHðXÞ ¼ _ fY 2 PðLÞ : AprðYÞ 6 Xg ¼MaxfY 2 PðLÞ : AprðYÞ 6 Xg: Likewise, if h is a _-complete full equivalence relation on L, then for AprH;Apr : IdðLÞ [ f;g ! IdðLÞ [ f;g, the following statements hold: 1. For each I 2 Id(L), AprðIÞ ¼ ^ fJ 2 IdðLÞ : AprHðJÞP Ig ¼MinfJ 2 IdðLÞ : AprHðJÞP Ig; 2. For each I 2 Id(L), AprHðIÞ ¼ _ fJ 2 IdðLÞ : AprðJÞ 6 Ig ¼ MaxfJ 2 IdðLÞ : AprðJÞ 6 Ig: Proposition 3.25. For AprH;Apr : PðLÞ ! PðLÞ, the following statements hold: 1. For each X 2 PðLÞ, AprðXÞ ¼ ^ fY 2 PðLÞ : AprHðYÞP Xg ¼ MinfY 2 PðLÞ : AprHðYÞP Xg; h A.A. Estaji et al. / Information Sciences 200 (2012) 108–122 119 In view of the previous section and Proposition 4.1, we can derive the following conclusions: 1. If X is a lower set of L, then Apr(X) is a lower set of L, 2. If X is an upper set of L, then Apr(X) is an upper set of L, 3. Let hbe a _-complete full equivalence relation on L. If I 2 Id(L) and Apr(I)– ;, then Apr(I) is an ideal of L, 4. Let hbe a _-complete full equivalence relation on L. If I is a prime ideal of L and Apr(I)– ;, then Apr(I) is a prime ideal of L. Example 4.2. Take L = {0, a, b, c, d, 1}. We define the binary relation 6on L in the following figure. It is easy to see that the definition of 6 on L is well-defined and L is a lattice. Let h be a full congruence relation on the lattice L with the following equivalence classes: [0]h = {0, a}; [b]h = {b}; [c]h = {c}; [1]h = {1, d}. Take X = {a, b, d}, I = {0, a, b}, J = {0, a, c}, K = {0, a, b, c, d}, D = {b, c, d}, F1 = {b, c, d} and F2 = {a, b, c, d, 1}. Then the following properties hold. 1. I and J are ideals in L. Also, AprðIÞ _ AprðJÞ ¼ f0; a; b; c; dg � f0; a; b; c; d;1g ¼ AprðI _ JÞ. This shows that statement (3) of Proposition 3.13 does not necessarily impart an equality. 2. D is a directed set, but Apr(D) = {b, c} is not a directed set. 3. h is not a _-complete full equivalence relation on L. 4. h is not a ^-complete full equivalence relation on L. 5. I is a prime ideal of L, but Apr(I) = {0, a, b, c} is not an ideal of L. 6. F1 is a filtered set, but Apr(F1) = {b, c} is not a filtered set. 7. F2 is a filter set, but Apr(F2) = {b, c, d, 1} is not a filter set. 8. ;Apr(X) = {0, a, b} � {0, a, b, c} = Apr(;X) and "Apr(X) = {b, d,1} � {b, c, d,1} = Apr("X). (1)) (2) It is straightforward. (2)) (1) Let (0)– J 2 Id(L) and I ^ J = (0). Then, AprðIÞ ^ AprðJÞ ¼ AprðI ^ JÞ ¼ Aprðf0gÞ ¼ ½0�h ¼ ð0Þ. Since AprðIÞ is an essential ideal, we obtain J#AprðJÞ ¼ ð0Þ. It follows that J = (0), a contradiction. h 4. Lower rough ideals Our main objective in this section is a lower rough approximation. Proposition 4.1. If X is a subset of L, then AprðXÞ ¼ AprHðXÞ. Proof. Let a 2 L. Then a 2 AprHðXÞ if and only if ½a� ¼ AprðfagÞ#X if and only if a 2 Apr(X). h (3)) (4) Let a 2 L and x, y 2 [a]h. Then, AprðfxgÞ ¼ ½x�h ¼ ½y�h ¼ AprðfygÞ. Since Apr is an injective map, we conclude that x = y. (4)) (5) It is clear that for each X # L, AprðXÞ ¼ X. We conclude that X#AprHðXÞ. By Proposition 3.21, we have AprHðXÞ ¼ X. Hence, Apr AprH ¼ 1PðLÞ. (5)) (6) It is straightforward. (6)) (1) By Proposition 3.21, AprH Apr AprH ¼ AprH, and if AprH is injective, we have Apr AprH ¼ 1PðLÞ. Hence, Apr is a sur- jective map. (3)) (7) By Proposition 3.21, Apr AprH Apr ¼ Apr, and if Apr is injective, we have AprH Apr ¼ 1PðLÞ. (7)) (8) It is straightforward. (8)) (9) Let X;Y 2 PðLÞ and AprðXÞ ¼ AprðYÞ. Then, there exist X1;Y2 2 PðLÞ such that AprHðX1Þ ¼ X and AprHðY1Þ ¼ Y . Now, we conclude from Proposition 3.21 that X ¼ AprHðX1Þ ¼ AprH Apr AprHðX1Þ ¼ AprH Apr AprHðY1Þ ¼ AprHðY1Þ ¼ Y . Hence, Apr is an identity map and we infer that AprH is an identity map too. (9)) (3) It is straightforward. h In a similar way, we can express the above proposition for AprH;Apr : IdðLÞ [ f;g ! IdðLÞ [ f;g. Recall that an ideal I of L is an essential ideal of L if for every (0)– J 2 Id(L) implies that I ^ J– (0). We conclude this section with a proposition on essential ideals of L. Proposition 3.28. If [0]h = {0} and I 2 Id(L), then the following statements are equivalent: 1. I is an essential ideal. 2. AprðIÞ is an essential ideal. Proof. 2. Ap 3.25, AprHðXÞ 6 AprðXÞ. Now, if Y = Aprw(X), then by Proposition 4.3, Let S a (g, d) 120 A.A. Estaji et al. / Information Sciences 200 (2012) 108–122 2. the relations g(s)P t and sP d(t) are equivalent for all pairs of elements (s, t) 2 S � T. In an adjunction (g, d), the function g is called the upper adjoint and d the lower adjoint.Hence, ðApr;AprÞ where Apr;Apr : PðLÞ ! PðLÞ is a Galois connection. Also, if h is a _-complete full equivalence relation on L and Apr;Apr : IdðLÞ [ f;g ! IdðLÞ [ f;g, then ðApr;AprÞ is a Galois connection.An ideal I of L is called the fixed-points of the lower rough approximation of Id(L), if Apr(I) = I. Let F(L) denote the family of all the fixed-points of upper rough approximation of Id(L). Lemma 4.5. The set F(L) of all the fixed-points of upper rough approximation of Id(L) forms a lattice. Proof. It is straightforward. h Proposition 4.6. Let u be a homomorphism from lattice L1 to lattice L2 and let h2 be a full congruence relation on L2 and h1 = {(a, b) 2 L1 � L1: (u(a), u(b)) 2 h2}. Then for every A # L1, 1. uðAprh1 ðAÞÞ#Aprh2 ðuðAÞÞ; 2. If u is one to one, then uðAprh1 ðAÞÞ ¼ Aprh2 ðuðAÞÞ. Also, for each x 2 L1, x 2 Aprh1 ðAÞ if and only if uðxÞ 2 Aprh2 ðuðAÞÞ; 1. both g and d are monotone; nd T be two posets. Recall that g:S? T is monotone if and only if x 6 y implies that g(x) 6 g(y). We shall say that a pair of functions, g:S? T and d:T? S, is a Galois connection or an adjunction between S and T f. AprHðYÞ ¼ AprðYÞ ¼ AprðAprHðXÞÞP X: We conclude from Proposition 3.25 that AprðXÞ 6 AprHðXÞ. h Proof. If there exists Y # L such that X 6 AprHðYÞ ¼ AprðYÞ, then by definition, we conclude that Aprw(X) 6 Y. By Proposition Proposition 4.4. If X is a subset of L, then AprHðXÞ ¼ AprðXÞ. Proof. The proof is similar to that of Proposition 3.21. h rw is a map. Proposition 4.3. We define AprH : PðLÞ ! PðLÞ by the following equivalence (which defines it unambiguously): AprHðXÞ 6 Y () X 6 AprðYÞ: Then the following properties hold. 1. For each X 2 PðLÞ, Aprw(Apr(X)) 6 X 6 Apr(Aprw(X)); [10] M. Diker, Textural approach to generalized rough sets based on relations, Information Sciences 180 (2010) 1418–1433. A.A. Estaji et al. / Information Sciences 200 (2012) 108–122 121 [11] Gorge Grätzer, General Lattice Theory, Academic Press, New York, San Francisco, 1978. [12] Q. Hu, S. An, D. Yu, Soft fuzzy rough sets for robust feature evaluation and selection, Information Sciences 180 (2010) 4384–4400. [13] Q. Hu, D. Yu, M. Guo, Fuzzy preference based rough sets, Information Sciences 180 (2010) 2003–2022. 3. If u is bijective, then Aprh1 ðAÞ is an ideal of L1 if and only if Aprh2 ðuðAÞÞ is an ideal of L2; 4. If u is bijective, then Aprh1 ðAÞ is a prime ideal of L1 if and only if Aprh2 ðuðAÞÞ is a prime ideal of L2. Proof (1) Take x 2 Aprh1 ðAÞ and b 2 ½uðxÞ�h2 . Then, u�1ðbÞ# ½x�h1 . From this, we infer b 2 u(A). Hence, uðxÞ 2 Aprh2 ðuðAÞÞ and the proof is completed. (2) Take x 2 Aprh2 ðuðAÞÞ. Then, ½x�h2 #uðAÞ. From this, we infer that there exists a 2 A such that x = u(a). Now, suppose that b 2 ½a�h1 . Then, uðbÞ 2 ½x�h2 #uðAÞ. It follows that u(b) = u(a0) for some a0 2 A. Since u is one to one, we conclude that b = a0 2 A. Hence, ½a�h1 #A, that is, x 2 uðAprh1 ðAÞÞ. Now, by statement (1), we have uðAprh1 ðAÞÞ ¼ Aprh2 ðuðAÞÞ. Now, suppose that uðxÞ 2 Aprh2 ðuðAÞÞ. Then, there exists y 2 Aprh1 ðAÞ such that u(x) = u(y). Since u is one to one, we obtain that x ¼ y 2 Aprh1 ðAÞ. Likewise, if x 2 Aprh1 ðAÞ, then it is clear that uðxÞ 2 Aprh2 ðuðAÞÞ. (3) Necessity: Take a; b 2 Aprh2 ðuðAÞÞ. Then, there exist x,y 2 L1 such that u(x) = a, u(y) = b, and by statement (2), we have x; y 2 Aprh1 ðAÞ. Since Aprh1 ðAÞ is an ideal of L1, we conclude that x _ y 2 Aprh1 ðAÞ. From statement (2), we infer a _ b 2 Aprh2 ðuðAÞÞ. Now, suppose that a, b 2 L2 and a 6 b 2 Aprh2 ðuðAÞÞ. Then, there exists x, y 2 L1 such that u(x) = a, u(y) = b, and u(x _ y) = u(y). Hence, by statement (2), we have x 6 x _ y 2 Aprh1 ðAÞ. Since Aprh1 ðAÞ is an ideal of L1, we conclude that x 2 Aprh1 ðAÞ, that is, a 2 Aprh2 ðuðAÞÞ. Sufficiency: It is straightforward. (4) Necessity: Take a, b 2 L2 and a ^ b 2 Aprh2 ðuðAÞÞ. Then, there exist x, y 2 L1 and z 2 Aprh1 ðAÞ such that u(x) = a, u(y) = b, andu(x ^ y) = u(z). By our hypothesis, x ^ y = z. It follows that x 2 Aprh1 ðAÞ or y 2 Aprh1 ðAÞ. Hence, by statement (2), we have a 2 Aprh2 ðuðAÞÞ or b 2 Aprh2 ðuðAÞÞ and by statement (3), the proof is completed. Sufficiency: Take x, y 2 L1 and x ^ y 2 Aprh1 ðAÞ. Then, by statement (2), we have uðxÞ ^uðyÞ 2 Aprh2 ðuðAÞÞ. From our hypothesis, we infer that uðxÞ 2 Aprh2 ðuðAÞÞ or uðyÞ 2 Aprh2 ðuðAÞÞ. Since u is bijective, we can then conclude from statement (2) that x 2 Aprh1 ðAÞ or y 2 Aprh1 ðAÞ and by statement (3), the proof is completed. h 5. Conclusion The present paper is in line with a series of algebraic studies reported in the literature. So far, a good number of research- ers have dealt with a great host of aspects and issues in this field. For example, Biswas [2], Davvaz [4–6,8], Kuroki [16,17], Kazancı [14,15], Qi [35], Xiao [44] andmany others studied algebraic rough sets (such as semigroups, groups, rings, and mod- ules). In this paper, we studied a general mathematical concept, called a lattice, which includes all those examples and many others as special cases. Moreover, lattices and ordered sets play an important role in many areas of computer science. These range from lattices as models for logics, which are fundamental to understanding computation, to the ordered sets as models for computation, to the role both lattices and ordered sets play in combinatorics, a fundamental aspect of computation. In addition, many applications utilize lattices and ordered sets in fundamental ways. These include such areas as knowledge representation, text categorization and data mining, where order plays a fundamental organizing principle, to the use of lat- tices and ordered sets to analyze crypto-protocols in security, to inductive logic programming, where ordered sets form basic models. This work focuses on rough set theory of lattice-theoretical foundations and for the new model some properties are given. We present a general framework for the study of approximations in lattices. In construction approach, a pair of lower and upper approximation operators is defined. It is our hope that this work would serve as a foundation for further study of the lattice theory. Acknowledgments The authors would like to express their sincere thanks to the referees for their valuable comments and suggestions which help a lot to improve the presentation of this paper. References [1] N. Ajmal, A. 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