RATIO MATHEMATICA 29 (2015) 3-14 ISSN:1592-7415 Rough sets applied in sublattices and ideals of lattices R. Ameri1, H. Hedayati2,Z. Bandpey3 1School of Mathematics, Statistics and Computer Science, College of Sciences, University of Tehran, P.O.Box 14155-6455, Teheran, Iran rameri@ut.ac.ir 2Department of Mathematics, Faculty of Basic Science, University of Mazandaran, Babolsar, Iran zeinab bandpey@yahoo.com 3Department of Mathematics, Faculty of Basic Science, Babol University of Technology, Babol, Iran h.hedayati@nit.ac.ir Abstract The purpose of this paper is the study of rough hyperlattice. In this regards we introduce rough sublattice and rough ideals of lattices. We will proceed by obtaining lower and upper approximations in these lattices. Keywords: rough set, lower approximation, upper approxima- tion, rough sublattice, rough ideal doi: 10.23755/rm.v29i1.18 1 Introduction Never in the history of mathematics has a mathematical theory been the object of such vociferous vituperation as lattice theory (for more details see[3, 13]). Lattices are partially ordered sets in which least upper bounds and greatest lower bounds of any two elements exist. A lattice is a set on which two operations are defined, called join and meet and denoted by ∨ 3 R. Ameri, H. Hedayati and Z. Bandpey and ∧, which satisfy the idempotent, commutative and associative laws, as well as the absorption laws: a∨ (b∧a) = a, a∧ (b∨a) = a. Lattices are better behaved than partially ordered sets lacking upper or lower bounds. The concept of rough set was originally proposed by Pawlak [21, 22] as a formal tool for modeling and processing incomplete information in informa- tion systems. Since then the subject has been investigated in many papers (see [20, 23, 24]). The theory of rough set is an extension of set theory, in which a subset of a universe is described by a pair of ordinary sets called the lower and upper approximations. A key notion in Pawlak rough set model is an equivalence relation. The equivalence classes are the building blocks for the construction of the lower and upper approximations. The lower approx- imation of a given set is the union of all the equivalence classes which are subsets of the set, and the upper approximation is the union of all the equiva- lence classes which have a non-empty intersection with the set. Some authors, for example, Bonikowaski [5], Iwinski [15], and Pomykala and Pomykala [24] studied algebraic properties of rough sets. The lattice theoretical approach has been suggested by Iwinski [15]. In this paper we concentrates on the relationship between rough sets and lattice theory. We introduce the notion of rough sublattices (resp. ideals) of lattices, and investigate some properties of lower and upper approximations in lattices. 2 Preliminaries Suppose that U is a non-empty set. A partition or classification of U is a family P of non-empty subsets of U such that each element of U is contained in exactly one element of P . Recall that an equivalence relation on a set U is a reflexive, symmetric, and transitive binary relation on U. Each partition P induces an equivalence relation θ on U by setting: xθy ⇔ x and y are in the same class of P . Conversely, each equivalence relation θ on U induces a partition P of U whose classes have the form [x]θ = {y ∈ U | xθy}. Given a non-empty universe U, by P(U) we will denote the power set on U. If θ is an equivalence relation on U then for every x ∈ U, [x]θ denotes the equivalence class of θ determined by x. For any X ⊆ U, we write Xc to denote the complementation of X in U, that is the set U \X. Definition 2.1. [8] A pair(U,θ); where U 6= ∅ and θ is an equivalence 4 Rough sets applied in sublattices and ideals of lattices relation on U, is called an approximation space. Definition 2.2. [8] For an approximation space (U,θ), by a rough approximation in (U,θ) we mean a mapping A : P(U) → P(U) × P(U) defined by for every X ∈ P(U), A(X) = (A(X),A(X)) where A(X) = {x ∈ X | [x]θ ⊆ X}, A(x) = {x ∈ X | [x]θ ∩ X 6= ∅}. A(X) is called a lower rough approximation of X in (U,θ), where as A(X) is called upper rough approximation of X in (U,θ). Definition 2.3. [8] Given an approximation space (U,θ) a pair (A,B) ∈ P(U) × P(U) is called a rough set in (U,θ) iff (A,B) = A(X) for some X ∈ P(U). For the sake of illustration, let (U,θ) is an approximation space, where: U = {x1,x2,x3,x4,x5,x6,x7,x8}, and an equivalence relation θ with the following equivalence classes: E1 = {x1,x4,x8}, E2 = {x2,x5,x7}, E3 = {x3}, E4 = {x6}, Let X = {x3,x5}, then A(X) = {x3} and A(X) = {x2,x3,x5,x7} and so ({x3},{x2,x3,x5,x7}) = A(X) is a rough set. The reader will find in [18,21-25] a deep study of rough set theory. Definition 2.4. [7] A subset Xof U is called definable if A(X) = A(X). If X ⊆ U given by a predicate P and x ∈ U, then: 1. x ∈ A(X) means that x certainly has property P , 2. x ∈ A(X) means that x possibly has property P , 3. x ∈ U \A(X) means that x definitely does not have property P . When A(A) v A(B), we say that A(A) is a rough subset of A(B). Thus in the case of rough sets A(A) and A(B), A(A) v A(B) if and only if A(A) ⊆ A(B) and A(A) ⊆ A(B). This property of rough inclusion has all the properties of set inclusion. The rough complement of A(A) denoted by Ac(A) is defined by: Ac(A) = (U \ A(A),U \ A(A)). Also, we can define A(A) \A(B) as follows: A(A) \A(B) = A(A) uAc(B) = (A(A) \A(B),A(A) \A(B)). Let L be a lattice and S ⊆ L, If S is a lattice, then S is called a sublattice of L. A sublattice I is called an ideal of L, if a ∈ L and x ∈ I imply a∧x ∈ L 5 R. Ameri, H. Hedayati and Z. Bandpey (see[2]). Let ρ be an equivalence relation on L and x, y, z ∈ L. (1) ρ is called a congruence relation if xρy implies (x ∨ z)ρ(y ∨ z) and (x∧z)ρ(y ∧z). (2) ρ is called a complete congruence relation if [x]ρ∨ [y]ρ = [x∨y]ρ, and [x]ρ ∧ [y]ρ = [x∧y]ρ. If ρ is a congruence relation on L, then it is easy to verify that [x]ρ∨[y]ρ ⊆ [x∨y]ρ, [x]ρ ∧ [y]ρ ⊆ [x∧y]ρ. 3 Rough ideals of lattices Throughout this paper L denotes a lattice. Let ρ be an equivalence rela- tion on L and X be a non-empty subset of L. When U = L and θ is the above equivalence relation, then we use the pair (L,ρ) instead of the approximation space (U,θ). Also, in this case we use the symbols Aρ(X) and Aρ(X) instead of A(X) and A(X). Proposition 3.1. For every approximation space (L,ρ), where ρ is an equivalence relation, and every subsets A, B ⊆ L, we have: (1) Aρ(A) ⊆ A ⊆ Aρ(A); (2) Aρ(∅) = ∅ = Aρ(∅); (3) Aρ(L) = L = Aρ(L); (4) If A ⊆ B, then Aρ(A) ⊆ Aρ(B), and Aρ(A) ⊆ Aρ(B); (5) Aρ(Aρ(A)) = Aρ(A); (6) Aρ(Aρ(A)) = Aρ(A); (7) Aρ(Aρ(A)) = Aρ(A); (8) Aρ(Aρ(A)) = Aρ(A); (9) Aρ(A) = (Aρ(A c))c; (10)Aρ(A) = (Aρ(A c))c; (11)Aρ(A∩B) = Aρ(A) ∩Aρ(B); (12)Aρ(A∩B) ⊆ Aρ(A) ∩Aρ(B) ; (13)Aρ(A∪B) ⊇ Aρ(A) ∪Aρ(B); (14)Aρ(A∪B) = Aρ(A) ∪Aρ(B); (15)Aρ([x]ρ) = Aρ([x]ρ) for all x ∈ L; Proof. (15) Aρ([x]ρ) = {y ∈ L | [y]ρ ⊆ [x]ρ} = [x]ρ, and Aρ([x]ρ) = {y ∈ L | [y]ρ ∩ [x]ρ 6= ∅} = [x]ρ. Hence Aρ([x]ρ) = Aρ([x]ρ). 6 Rough sets applied in sublattices and ideals of lattices The other parts of the proof is similar to the [17, Theorem 2.1] and [7, Proposition 4.1]. 2 The following example shows that the converse of (12) and (13) in Propo- sition 3.1 are not true. Example 3.2. Let L = {1, 2, ..., 8}, Then (L,∧,∨) is a lattice, where ∀a,b ∈ L, a ∧ b = min{a,b}, a ∨ b = max{a,b}. Let ρ be an equivalence relation on L with the following equivalence classes: [1]ρ = {1, 4, 8}, [2]ρ = {2, 5, 7}, [3]ρ = {3}, [6]ρ = {6}, and A = {3, 5, 7}, B = {2, 6}. Then: Aρ(A) = {3}, Aρ(B) = {6}, Aρ(A∪B) = {2, 3, 5, 6, 7}, Aρ(A) = {2, 3, 5, 7}, Aρ(B) = {2, 5, 6, 7}, Aρ(A∩B) = ∅, and so Aρ(A) ∩Aρ(B) * Aρ(A∩B), Aρ(A∪B) * Aρ(A) ∪Aρ(B). Corollary 3.3. For every approximation space (L,ρ), (i) For every A ⊆ L, Aρ(A) and Aρ(A) are definable sets, (ii) For every x ∈ L, [x]ρ is definable set. Proof. It is immediately by Proposition 3.1 (parts (5), (6), (7), (8) and (15)). 2 If A and B are non-empty subsets of L, let A∧B and A∨B denotes the following sets: A∧B = {a∧ b | a ∈ A,b ∈ B}, A∨B = {a∨ b | a ∈ A,b ∈ B}. Proposition 3.4. Let ρ be a complete congruence relation on L, and A, B non-empty subsets of L, then Aρ(A) ∧Aρ(B) = Aρ(A∧B). Proof. Suppose z be any element of Aρ(A) ∧ Aρ(B), then z = a ∧ b for some a ∈ Aρ(A), b ∈ Aρ(B), hence [a]ρ ∩ A 6= ∅ and [b]ρ ∩ B 6= ∅ and so there exist x ∈ [a]ρ ∩ A and y ∈ [b]ρ ∩ B. Therefore x ∧ y ∈ A ∧ B 7 R. Ameri, H. Hedayati and Z. Bandpey and x ∧ y ∈ [a]ρ ∧ [b]ρ = [a ∧ b]ρ hence [a ∧ b]ρ ∩ (A ∧ B) 6= ∅ and so Aρ(A) ∧Aρ(B) ⊆ Aρ(A∧B). Conversely, let x ∈ Aρ(A∧B) then [x]ρ ∩ (A∧B) 6= ∅ hence there exists y ∈ [x]ρ and y ∈ A ∧ B and so y = a ∧ b for some a ∈ A and b ∈ B. Now we have x ∈ [y]ρ = [a ∧ b]ρ = [a]ρ ∧ [b]ρ. Then there exist x′ ∈ [a]ρ and y′ ∈ [b]ρ such that x = x′ ∧ y′. Since a ∈ [x′]ρ ∩A and b ∈ [y′]ρ ∩B, hence x′ ∈ Aρ(A) and y′ ∈ Aρ(B), which yields that x = x′ ∧ y′ ∈ Aρ(A) ∧ Aρ(B) and so Aρ(A∧B) ⊆ Aρ(A) ∧Aρ(B). 2 Proposition 3.5. Let ρ be a complete congruence relation on L, and A, B non-empty subsets of L, then Aρ(A) ∨Aρ(B) = Aρ(A∨B). Proof. The proof is similar to the proof of Proposition 3.4, by consider- ing the suitable modification by using the definition of A∨B. 2 Proposition 3.6. Let ρ be a complete congruence relation on L, and A, B non-empty subsets of L, then Aρ(A) ∧Aρ(B) ⊆ Aρ(A∧B). Proof. Suppose x be any element of Aρ(A) ∧ Aρ(B) then x = a ∧ b for some a ∈ Aρ(A) and b ∈ Aρ(B). Hence [a]ρ ⊆ A and [b]ρ ⊆ B. Since [a∧b]ρ = [a]ρ∧[b]ρ ⊆ A∧B, we get a∧b ∈ Aρ(A∧B) and so x ∈ Aρ(A∧B). 2 The following example shows that the converse of Proposition 3.6 is not true. Example 3.7. Let L = {0, 1, 2, ..., 11}, Then (L,∧,∨) is a lattice, where ∀a,b ∈ L, a ∧ b = min{a,b}, a ∨ b = max{a,b}. Let ρ be a complete congruence relation on L with the following equivalence classes: [0]ρ = {0, 1, 2}, [3]ρ = {3, 4, 5}, [6]ρ = {6, 7, 8}, [9]ρ = {9, 10, 11}, and A = {1, 3, 4, 5}, B = {0, 1, 2, 6, 8}. Then: Aρ(A) = {3, 4, 5}, Aρ(B) = {0, 1, 2}, A∧B = {0, 1, 2, 3, 4, 5} Aρ(A∧B) = {0, 1, 2, 3, 4, 5}, Aρ(A) ∧Aρ(B) = {0, 1, 2} and so Aρ(A∧B) * Aρ(A) ∧Aρ(B). 8 Rough sets applied in sublattices and ideals of lattices Proposition 3.8. Let ρ be a complete congruence relation on L, and A, B non-empty subsets of L, then Aρ(A) ∨Aρ(B) ⊆ Aρ(A∨B). Proof. The proof is similar to the proof of Proposition 3.6, by consider- ing the suitable modification by using the definition of A∨B. 2 The following example shows that Aρ(A∨B) ⊆ Aρ(A) ∨Aρ(B) does not hold in general. Example 3.9. Let L = {0, 1, 2, ..., 8}, Then (L,∧,∨) is a lattice, where ∀a,b ∈ L, a ∧ b = min{a,b}, a ∨ b = max{a,b}. Let ρ be a complete congruence relation on L with the following equivalence classes: [0]ρ = {0, 1, 2}, [3]ρ = {3, 4}, [5]ρ = {5, 6, 7, 8}, and A = {3, 4, 5, 7}, B = {0, 1, 2, 3, 6, 8}. Then: Aρ(A) = {3, 4}, Aρ(B) = {0, 1, 2}, A∨B = {3, 4, 5, 6, 7, 8}, Aρ(A∨B) = {3, 4, 5, 6, 7, 8}, Aρ(A) ∨Aρ(B) = {3, 4}, and so Aρ(A∨B) * Aρ(A) ∨Aρ(B) Lemma 3.10. Let ρ1 and ρ2 be two complete congruence relations on L such that ρ1 ⊆ ρ2 and let A be a non-empty subset of L, then: (i) Aρ2 (A) ⊆ Aρ1 (A), (ii) Aρ1 (A) ⊆ Aρ2 (A). Proof. It is straightforward. 2 The following Corollary follows from Lemma 3.10. Corollary 3.11. Let ρ1 and ρ2 be two complete congruence relations on L and A a non-empty subset of L, then: (i) Aρ1 (A) ∩Aρ2 (A) ⊆ A(ρ1∩ρ2)(A), (ii) A(ρ1∩ρ2)(A) ⊆ Aρ1 (A) ∩Aρ2 (A). Proposition 3.12. Let ρ be a congruence relation on L, and J be an ideal of L, then Aρ(J) is an ideal of L. 9 R. Ameri, H. Hedayati and Z. Bandpey Proof. Suppose a,b ∈ Aρ(J) and r ∈ L, then [a]ρ∩J 6= ∅ and [b]ρ∩J 6= ∅. So there exist x ∈ [a]ρ ∩J and y ∈ [a]ρ ∩J. Since J is an ideal of L, we have x∨y ∈ J and x∨y ∈ [a]ρ∨[b]ρ ⊆ [a∨b]ρ. Hence [a∨b]ρ∩J 6= ∅ which implies a∨ b ∈ Aρ(J). Also, we have r ∧x ∈ J and r ∧x ∈ [r]ρ ∧ [a]ρ ⊆ [r ∧a]ρ. So [r∧a]ρ∩J 6= ∅ which implies r∧a ∈ Aρ(J). Therefore Aρ(J) is an ideal of L. 2 Similarly, if ρ is a congruence relation on L and J is a sublattice of L, then Aρ(J) is a sublattice of L. Proposition 3.13. Let ρ be a complete congruence relation on L, and J be an ideal of L, then Aρ(J) is an ideal of L. Proof. Suppose a, b ∈ Aρ(J) and r ∈ L, then [a]ρ ⊆ J and [b]ρ ⊆ J. So [a∨ b]ρ = [a]ρ ∨ [b]ρ ⊆ J, and [r ∧a]ρ = [a]ρ ∧ [b]ρ ⊆ J. Hence a∨ b ∈ Aρ(J) and r ∧a ∈ Aρ(J). 2 Similarly, if ρ is a complete congruence relation on L and J is a sublattice of L, then Aρ(J) is a sublattice of L. Definition 3.14. Let ρ be a congruence relation on L and Aρ(A) = (Aρ(A),Aρ(A)) a rough set in the approximation space (L,ρ). If Aρ(A) and Aρ(A) are ideals (resp. sublattice) of L, then we call Aρ(A) a rough ideal (resp. sublattice). Note that a rough sublattice also is called a rough lattice. Corollary 3.15. (i) Let ρ, be a congruence relation on L, and I an ideal of L then Aρ(I) is a rough ideals. (ii) Let ρ be a complete congruence relation on L and J a sublattice of L, then Aρ(J) is a rough lattice. Proof. It is obtained by 3.12 and 3.13. 2 Let L and L′ be two lattices, a map f : L → L′ is said to be homomor− phism or (lattice homomorphism) if for all a, b ∈ L, f(a∧ b) = f(a) ∧f(b), and f(a∨ b) = f(a) ∨f(b). Now, let L and L′ be two lattices and f : L → L′ a homomorphism. It is well known, θ = {(a,b) ∈ L × L | f(a) = f(b)} ⊆ L × L is a congruence relation on L. Because if aθb then f(a) = f(b) and for all z ∈ L, we have f(a∧ z) = f(a) ∧f(z) = f(b) ∧f(z) = f(b∧ z). Therefor (a∧ z) θ (b∧ z), and similarly (a∨z) θ (b∨z). 10 Rough sets applied in sublattices and ideals of lattices Theorem 3.16. Let L and L′ be two lattices and f : L → L′ a homo- morphism. If A is a non-empty subset of L, then f(Aθ(A)) = f(A). Proof. Since A ⊆ Aθ(A) it follows that f(A) ⊆ f(Aθ(A)). Conversely, let y ∈ f(Aθ(A)). Then there exists an element x ∈ Aθ(A), such that f(x) = y, so we have [x]θ ∩A 6= ∅. Thus there exists an element a ∈ [x]θ∩A. Then a ∈ [x]θ, hence xθa, and so f(x) = f(a) ∈ f(A), therefore f(Aθ(A)) ⊆ f(A). 2 Let f : L → L′ be a homomorphism and A a subset of L, Since Aθ(A) ⊆ A it follows that f(Aθ(A)) ⊆ f(A). But the following example shows that, in general, f(Aθ(A)) 6= f(A). Example 3.17. Let (L,∧,∨) and (L′,∧,∨) be two lattices where L = {1, 2, 3, 4}; and L′ = {5, 6, 7}; and for all s, t in L or L′, s∧t= min{s,t} and s∨ t = max{s,t}. The map f : L → L′ given by f(4) = f(3)= 7, f(2) = 6, f(1) = 5, is a homomorphism. We have θ = {3, 4}. Suppose A = {1, 2}, then f(A) = {5, 6}, Aθ(A) = ∅ and f(Aθ(A)) = ∅, and so f(Aθ(A)) 6= f(A). The lower and upper approximations can be presented in an equivalent form as follows: Let L be a lattice, ρ a congruence relation on L, and A a non-empty subset of L. Then we define ∨ and ∧ on L/ρ = {[x]ρ | x ∈ L}, by [x]ρ∨[y]ρ = [x∨y]ρ, [x]ρ∧[y]ρ = [x∧y]ρ. This relation is well-defined, since if [x1]ρ = [x2]ρ and [y1]ρ = [y2]ρ, then x1ρx2 and y1ρy2. Since ρ is a congruence relation we have (x1 ∨y1)ρ(x2 ∨y1) and (x2 ∨ y1)ρ(x2 ∨ y2). Then (x1 ∨ y1)ρ(x2 ∨ y2), so [x1 ∨ y1]ρ = [x2 ∨ y2]ρ. Therefore [x1]ρ∨[y1]ρ = [x2]ρ∨[y2]ρ. It is easy to see that (L/ρ,∨,∧), is a lattice. Also if A 6= ∅, and A ⊆ L put A ρ (A) = {[x]ρ ∈ L/ρ | [x]ρ ⊆ A} and Aρ(A) = {[x]ρ ∈ L/ρ | [x]ρ ∩A 6= ∅}. Proposition 3.18. Let ρ be a congruence relation on L and J be an ideal of L, then Aρ(J) is an ideal of L/ρ. Proof. Assume that [a]ρ, [b]ρ ∈ Aρ(J) and [r]ρ ∈ L/ρ. Then [a]ρ∩J 6= ∅ and [b]ρ∩J 6= ∅, so there exist x ∈ [a]ρ∩J and y ∈ [b]ρ∩J. Since J is an ideal of L, we have x∨y ∈ J and r∧x ∈ J. Also, we have x∨y ∈ [a]ρ∨[b]ρ ⊆ [a∨b]ρ, and r∧x ∈ [r]ρ∧[a]ρ ⊆ [r∧a]ρ. Therefore [a∨b]ρ∩J 6= ∅ and [r∧a]ρ∩J 6= ∅, 11 R. Ameri, H. Hedayati and Z. Bandpey which imply [a]ρ ∨ [b]ρ ∈ Aρ(J) and [r]ρ ∧ [a]ρ ∈ Aρ(J). Therefore Aρ(J) is an ideal of L/ρ. 2 Proposition 3.19. Let ρ be a complete congruence relation on L and J be an ideal of L, then A ρ (J) is an ideal of L/ρ. Proof. Assume that [a]ρ, [b]ρ ∈ Aρ(J) and [r]ρ ∈ L/ρ. Then [a]ρ ⊆ J and [b]ρ ⊆ J. Since J is an ideal of L, we have a ∨ b ∈ J and r ∧ a ∈ J Therefore [a]ρ ∨ [b]ρ = [a ∨ b]ρ ⊆ J ∨ J = J, and [r]ρ ∧ [a]ρ = [r ∧ a]ρ ⊆ J, which imply [a]ρ ∨ [b]ρ ∈ Aρ(J) and [r]ρ ∧ [a]ρ ∈ Aρ(J). Therefore Aρ(J) is an ideal of L/ρ. 2 Proposition 3.20. (i) Let ρ be a congruence relation on L and J a sublattice of L, then Aρ(J) is a sublattice of L/ρ. (ii) Let ρ be a complete congruence relation on L and J a sublattice of L, then A ρ (J) is a sublattice of L/ρ. Proof. 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