Ratio Mathematica Vol. 34, 2018, pp. 15-33 ISSN: 1592-7415 eISSN: 2282-8214 On Rough Sets and Hyperlattices Ali Akbar Estaji∗, Fereshteh Bayati† Received: 08-10-2017. Accepted: 28-01-2018. Published: 30-06-2018 doi:10.23755/rm.v34i0.350 c©Ali Akbar Estaji et al. Abstract In this paper, we introduce the concepts of upper and lower rough hyper fuzzy ideals (filters) in a hyperlattice and their basic properties are discussed. Let θ be a hyper congruence relation on L. We show that if µ is a fuzzy subset of L, then θ(< µ >) = θ(< θ(µ) >) and θ(µ∗) = θ((θ(µ))∗), where < µ > is the least hyper fuzzy ideal of L containing µ and µ∗(x) = sup{α ∈ [0,1] : x ∈ I(µα)} for all x ∈ L. Next, we prove that if µ is a hyper fuzzy ideal of L, then µ is an upper rough fuzzy ideal. Also, if θ is a ∧−complete on L and µ is a hyper fuzzy prime ideal of L such that θ(µ) is a proper fuzzy subset of L, then µ is an upper rough fuzzy prime ideal. Furthermore, let θ be a ∨-complete congruence relation on L. If µ is a hyper fuzzy ideal, then µ is a lower rough fuzzy ideal and if µ is a hyper fuzzy prime ideal such that θ(µ) is a proper fuzzy subset of L, then µ is a lower rough fuzzy prime ideal. Keywords: rough set, upper and lower approximations, hyperlattice, hyper fuzzy prime ideal, hyper fuzzy prime filter. 2010 AMS subject classifications: 03G10, 03E72, 46H10, 06D50, 08A72. ∗Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran. aaestaji@hsu.ac.ir †Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.fereshte.bayati@yahoo.com 15 A. A. Estaji and F. Bayati 1 Introduction In this paper, we are using three basic notions: hyperlattice, rough set and fuzzy subset. Hyperstructure theory was born in 1934 when Marty [14] defined hypergroups as a generalization of groups. Extending lattices (also called hyper- lattices) have been recently studied by a number of authors, in particular, Koguep, Nkuimi and Lele [12], Feng and Zou [8], Guo and Xin [9], Rahnamai-Barghi [19], etc. Rough set theory was introduced by Pawlak in 1982 [17]. Many authors have studied the general properties of generalized rough sets [1, 4, 5]. The concept of fuzzy subsets was first introduced by Zadeh [22] in 1965 and then the fuzzy subsets have been used in the reconsideration of classical math- ematics. The relationships between fuzzy subsets and algebraic hyperstructures had been already considered by many researchers (for example [2, 21]). Also, there have been many papers studying the connections and differences of fuzzy subset theory and rough set theory [3, 15, 18]. In recent years, many efforts have been made to compare and combine the three theories [6, 7]. This paper is structured as follows. After the introduction, in Section 2, we recall some basic notions and results on hyperlattices, rough sets and fuzzy sub- sets. In Section 3, the notions of hyper congruence relation on a hyperlattice are introduced. Next, some important properties of θ-upper approximations of a fuzzy subset will be studied. Also by an example, we show that Theorem 2.15 in [12] is incorrect (see Example 3.20) and a corrected version is considered, Proposi- tion 3.21. Finally, in Section 4, θ-lower approximations of a fuzzy subset on a hyperlattice will be studied. 2 Preliminaries of hyperlattices, rough sets and fuzzy subsets In the remainder of the paper we use some notation and results from the theory of hyperlattices, rough sets and fuzzy subsets. We present a few basic definitions here. Let L be a set partially ordered by the binary relation ≤. The poset (L,≤) is a meet-semilattice if for all elements x and y of L, the greatest lower bound or the meet of the set {x,y}, denoted by x∧y, exists. For x and y in a meet-semilattice L, x ≤ y ⇔ x = x∧y. Replacing greatest lower bound with least upper bound results in the dual concept of a join-semilattice. The least upper bound of {x,y} is called the join of x and y and is denoted by x ∨ y. A poset L is a lattice if and only if it is both a meet- and a join-semilattice. 16 On Rough Sets and Hyperlattices In this paper, we use the following notion of a hyperlattice. Let L be a non-empty meet-semilattice and ∨ : L×L → P(L)∗ be a hyper- operation, where P(L) is the power set of L and P(L)∗ = P(L) \ {∅}. Then (L,∧,∨) is a hyperlattice [19], if for all a,b,c ∈ L: 1. a ∈ a∨a and a = a∧a. 2. a∨ b = b∨a and a∧ b = b∧a. 3. (a∨ b) ∨ c = a∨ (b∨ c) and (a∧ b) ∧ c = a∧ (b∧ c). 4. a ∈ [a∧ (a∨ b)] ∩ [a∨ (a∧ b)]. 5. a ∈ a∨ b ⇔ a∧ b = b. Where for all non-empty subsets A and B of L, A∧B = {a∧b|a ∈ A,b ∈ B}, A∨B = ⋃ {a∨ b|a ∈ A,b ∈ B}. Throughout this paper, L is a hyperlattice with the least element 0 and the greatest element 1. For X ⊆ L and x ∈ L we write: 1. ↓ X = {y ∈ L : y ≤ x for some x ∈ X}. 2. ↑ X = {y ∈ L : y ≥ x for some x ∈ X}. 3. ↓ x =↓ {x}. 4. ↑ x =↑ {x}. A pair (L,θ), where θ is an equivalence relation on L, is called an approxima- tion space [17] and for a ∈ L, the equivalence class (or coset) of a modulo θ is the set [a]θ = {x ∈ L|(a,x) ∈ θ} and also for A ⊆ L, we put [A]θ = ⋃ a∈A[a]θ. For an approximation space (L,θ), by an upper rough approximation in (L,θ) we mean a mapping Apr : P(L) → P(L) which is defined for every X ∈ P(L) by Apr(X) = {a ∈ L : [a]θ ∩X 6= ∅}. Also, by a lower rough approximation in (L,θ) we mean a mapping Apr : P(L) → P(L) defined for every X ∈P(L) by Apr(X) = {a ∈ L : [a]θ ⊆ X}. Then Apr(X) = (Apr(X),Apr(X)) is called a rough subset in (L,θ) if Apr(X) 6= Apr(X). The following proposition is well known and easily seen. 17 A. A. Estaji and F. Bayati Proposition 2.1. Let (L,θ) be an approximation space. For every subsets X,Y ⊆ L, we have 1. Apr(X) ⊆ X ⊆ Apr(X). 2. If X ⊆ Y , then Apr(X) ⊆ Apr(Y ) and Apr(X) ⊆ Apr(Y ). 3. Apr(X∪Y ) = Apr(X)∪Apr(Y ) and Apr(X∩Y ) ⊆ Apr(X)∩Apr(Y ). 4. Apr(X∩Y ) = Apr(X)∩Apr(Y ) and Apr(X∪Y ) ⊇ Apr(X)∪Apr(Y ). 5. Apr ( Apr(X) ) = Apr(X) and Apr ( Apr(X) ) = Apr(X). Proof. See [13]. Proposition 2.2. [12] Let (L,∨,∧) be a hyperlattice.Then, for each pair (a,b) ∈ L×L there exist a1,b1 ∈ a∨ b, such that a ≤ a1 and b ≤ b1. Definition 2.3. [19] A nonempty subset J of L is called an ideal of L if for all x,y ∈ L 1. x,y ∈ J implies x∨y ⊆ J. 2. If x ∈ J, then ↓ x ⊆ J. Definition 2.4. [19] A nonempty subset F of L is called a filter of L if for all x,y ∈ L 1. x,y ∈ F implies x∧y ∈ F . 2. If x ∈ F and x ≤ y, then y ∈ F . Given a hyperlattice L and a set X ⊆ L, let I(X) denote the least ideal con- taining X, called the ideal generated by X. A fuzzy subset of X is any function from X into [0, 1]. Let F(L) be the set of all fuzzy subsets of L. For µ,λ ∈ F(X), we say µ ⊆ λ if and only if µ(x) ≤ λ(x) for all x ∈ X. Definition 2.5. [12] Let µ be a fuzzy subset of L. Then 1. µ is a hyper fuzzy ideal of L if, for all x,y ∈ L, (a) ∧ a∈x∨y µ(a) ≥ µ(x) ∧µ(y). (b) x ≤ y ⇒ µ(x) ≥ µ(y). 18 On Rough Sets and Hyperlattices 2. µ is a hyper fuzzy filter of L if, for all x,y ∈ L, (a) µ(x∧y) ≥ µ(x) ∧µ(y). (b) x ≤ y ⇒ µ(x) ≤ µ(y). Definition 2.6. [12] Let µ be a proper hyper fuzzy ideal of L. 1. µ is called a hyper fuzzy prime ideal, if µ(x ∧ y) ≤ µ(x) ∨ µ(y), for all x,y ∈ L. 2. µ is called a hyper fuzzy prime filter, if ∧ a∈x∨y µ(a) ≤ µ(x) ∨µ(y), for all x,y ∈ L. Definition 2.7. [6] Let θ be an equivalence relation on L and µ a fuzzy subset of L. Then we define the fuzzy subsets θ(µ) and θ(µ) as follows: θ(µ)(x) = ∨ a∈[x]θ µ(a) and θ(µ)(x) = ∧ a∈[x]θ µ(a). The fuzzy subsets θ(µ) and θ(µ) are, respectively, called the θ-upper and θ-lower approximation of the fuzzy subset µ. Then θ(µ) = (θ(µ),θ(µ)) is called a rough fuzzy subset with respect to µ if θ(µ) 6= θ(µ). Proposition 2.8. [6] Let θ be an equivalence relation on L and µ,λ ∈ F(L). Then 1. θ(µ) ≤ µ ≤ θ(µ). 2. If µ ⊆ λ, then θ(µ) ≤ θ(λ) and θ(µ) ≤ θ(λ). 3. θ θ(µ) = θ(µ) and θ θ(µ) = θ(µ). 4. θ(µ)(x) = θ(µ)(a) and θ(µ)(x) = θ(µ)(a), for all x ∈ L and a ∈ [x]θ. 5. θθ(µ) = θ(µ) and θθ(µ) = θ(µ). The proofs of the following propositions are straightforward. Proposition 2.9. Let θ be an equivalence relation on set A. Then the following statements hold: 1. For each X ∈P(A), Apr(X) = ⋂{ Y ∈P(A) : X ⊆ Apr(Y ) } = Min { Y ∈P(A) : X ⊆ Apr(Y ) } . 19 A. A. Estaji and F. Bayati 2. For each X ∈P(A), Apr(X) = ⋃{ Y ∈P(L) : Apr(Y ) ⊆ X } = Max { Y ∈P(L) : Apr(Y ) ⊆ X } . Proposition 2.10. Let θ be an equivalence relation on set A. Then the following statements hold: 1. For each µ ∈F(A), θ(µ) = ∧ {λ ∈F(A) : θ(λ) ≥ µ} = Min{λ ∈F(A) : θ(λ) ≥ µ} . 2. For each µ ∈F(A), θ(µ) = ∨{ λ ∈F(L) : θ(λ) ≤ µ } = Max { λ ∈F(L) : θ(λ) ≤ µ } . 3 Upper approximations of a fuzzy subset In this section we give some important properties of θ with many examples, starting with the following definition. Definition 3.1. [16, 20] Let θ be an equivalence relation on a hyperlattice L. Then θ is called a hyper congruence relation if (a,b) ∈ θ implies that (a∨x)×(b∨x) ⊆ θ and (a∧x,b∧x) ∈ θ for all x ∈ L. It is clear that if L is a hyperlattice L and θ = L × L, then θ is a hyper congruence relation. Also, in Example 3.9, we’ll provide a non-trivial example. Lemma 3.2. Let θ be a hyper congruence relation on L. Then, for every a,b,c,d ∈ L, 1. If (a,b) ∈ θ and (c,d) ∈ θ, then (a∧c,b∧d) ∈ θ and (a∨c)×(b∨d) ⊆ θ. 2. [a]θ ∨ [b]θ ⊆ [a∨ b]θ. 3. [a]θ ∧ [b]θ ⊆ [a∧ b]θ. Proof. Evident. Proposition 3.3. Let θ be an equivalence relation on L and X ⊆ L. If µ ∈F(L) is a hyper fuzzy ideal of L, then 1. µ(↓ X) ⊆↑ µ(X). 20 On Rough Sets and Hyperlattices 2. µ(↑ X) ⊆↓ µ(X). 3. µ(Apr(↓ X)) ⊆↑ µ(Apr(X)). 4. µ(Apr(↑ X)) ⊆↓ µ(Apr(X)). 5. θ(µ)(Apr(↓ X)) ⊆↑ θ(µ)(X). Proof. (1) Let a ∈↓ X. Then there exists x ∈ X such that a ≤ x. Since µ is a hyper fuzzy ideal, we conclude that µ(x) ≤ µ(a) which implies that µ(a) ∈↑ µ(X) and the proof is now complete. (2) The proof is similar to the proof of (1). (3) For each X ⊆ L, since ↓ Apr(X) = Apr(↓ X) =↓ Apr(↓ X), we can then conclude from (1) that µ(Apr(↓ X)) ⊆↑ µ(Apr(X)). (4) For each X ⊆ L, we have ↑ Apr(X) = Apr(↑ X) =↑ Apr(↑ X). By (2), µ(Apr(↑ X)) ⊆↓ µ(Apr(X)). (5) Since θ(µ)(Apr(↓ X)) = θ(µ)(↓ X), we can then conclude from (1) that θ(µ)(Apr(↓ X)) ⊆↑ θ(µ)(X). Proposition 3.4. Let θ be a hyper congruence relation on L and x,y ∈ L. If µ ∈F(L) is a hyper fuzzy ideal of L, then∨ a∈[x]θ,b∈[y]θ ∨ µ(a∨ b) ≤ ∨ θ(µ)(x∨y). Proof. By Lemma 3.2,∨ a∈[x]θ,b∈[y]θ ∨ µ(a∨ b) ≤ ∨ z∈[x∨y]θ µ(z) = ∨ z∈ ⋃ a∈x∨y[a]θ µ(z) = ∨ a∈x∨y ∨ z∈[a]θ µ(z) = ∨ a∈x∨y θ(µ)(a) = ∨ θ(µ)(x∨y). Lemma 3.5. Let θ be a hyper congruence relation on L and x,y ∈ L. If µ ∈F(L) is a hyper fuzzy filter of L, then θ(µ)(x∧y) = ∨ a∈[x]θ,b∈[y]θ µ(a∧ b). Proof. By Lemma 3.2, ∨ a∈[x]θ,b∈[y]θ µ(a ∧ b) ≤ ∨ z∈[x∧y]θ µ(z) = θ(µ)(x ∧ y). Now, assume that z ∈ [x∧y]θ. By Lemma 3.2, (x∨ z) ×{x} ⊆ (x∨ z) × (x∨ (x∧y)) ⊆ θ and (y∨z)×{y}⊆ (y∨z)×(y∨(x∧y)) ⊆ θ. Also, by Proposition 2.2, there exist zx ∈ x ∨ z and zy ∈ y ∨ z such that z ≤ zx and z ≤ zy. Since z ≤ zx∧zy and µ is a hyper fuzzy filter of L, we conclude that µ(z) ≤ µ(zx∧zy). Therefore, θ(µ)(x∧y) = ∨ z∈[x∧y]θ µ(z) ≤ ∨ a∈[x]θ,b∈[y]θ µ(a∧b) and the proof is now complete. 21 A. A. Estaji and F. Bayati Definition 3.6. Let θ be an equivalence relation on L and µ ∈ F(L). Then, µ is called an upper rough fuzzy (prime) filter if θ(µ) is a hyper fuzzy (prime) filter of L. Proposition 3.7. Let θ be a hyper congruence relation on L. If µ ∈ F(L) is a hyper fuzzy filter, then µ is an upper rough fuzzy filter. Proof. Let x,y ∈ L and x ≤ y. If z ∈ [x]θ, then (z,x) ∈ θ and (z∨y)×(x∨y) ⊆ θ. Since y ∈ x ∨ y, we conclude that (z ∨ y) ×{y} ⊆ θ. Also, by Proposition 2.2, there exists z1 ∈ z ∨y such that z ≤ z1, and so µ(z) ≤ µ(z1) ≤ ∨ µ(z ∨y). Therefore, θ(µ)(x) = ∨ z∈[x]θ µ(z) ≤ ∨ z∈[x]θ ∨ µ(z ∨y) ≤ ∨ t∈[y]θ µ(t) = θ(µ)(y). Now, If x,y ∈ L, then θ(µ)(x∧y) = ∨ a∈[x]θ,b∈[y]θ µ(a∧ b) by Lemma 3.5 ≥ ∨ a∈[x]θ,b∈[y]θ µ(a) ∧µ(b) µ is a hyper fuzzy filter = ∨ a∈[x]θ µ(a) ∧ ∨ b∈[y]θ µ(b) = θ(µ)(x) ∧θ(µ)(y). Hence θ(µ) is a hyper fuzzy filter. Example 3.8. Let L = {0,a,b,c,d, 1} and define ∧ and ∨ by the following Cay- ley tables: ∧ 0 a b c d 1 0 0 0 0 0 0 0 a 0 a a a a a b 0 a b a b b c 0 a a c c c d 0 a b c d d 1 0 a b c d 1 ∨ 0 a b c d 1 0 {0} {a} {b} {c} {d} {1} a {a} {a} {b} {c} {d} {1} b {b} {b} {b} {d} {d} {1} c {c} {c} {d} {c} {d} {1} d {d} {d} {d} {d} {d} {1} 1 {1} {1} {1} {1} {1} {d, 1} It is easy to see that the operations ∧ and ∨ on L are well-defined and L is a hyperlattice. Let θ be an equivalence relation on the lattice L with the following equivalence classes: [0]θ = {0,a,b}; [d]θ = {d}; [c]θ = {c}; [1]θ = {1}. It is clear that θ is not a hyper congruence relation on the lattice L. If µ = ( 0 a b c d 1 0.1 0.2 0.7 0.2 0.7 0.9 ) , 22 On Rough Sets and Hyperlattices then µ is a hyper fuzzy filter and θ(µ) = ( 0 a b c d 1 0.7 0.7 0.7 0.2 0.7 0.9 ) . Since 0 ≤ c and θ(µ)(0) = 0.7 6≤ 0.2 = θ(µ)(c), we conclude that θ(µ) is not a hyper fuzzy filter. Example 3.9. Let the hyperlattice L be as in example 3.8. Let θ be an equivalence relation on the lattice L with the following equivalence classes: [0]θ = {0} and [1]θ = {a,b,c,d, 1}. It is clear that θ is a hyper congruence relation on the lattice L. Lemma 3.10. Let θ be a hyper congruence relation on L and x,y ∈ L.Then 1. If a ∈ [x]θ and b ∈ [y]θ, then [α]θ = [β]θ for every α ∈ [x ∨ y]θ and β ∈ [a∨ b]θ. 2. If µ is a fuzzy subset of L, then ∨ a∈[x]θ,b∈[y]θ ∧ z∈a∨b µ(z) ≤ ∧ z∈x∨y θ(µ)(z). Proof. (1) Let α ∈ [x ∨ y]θ and β ∈ [a ∨ b]θ. Then, by Lemma 3.2, (α,β) ∈ (a∨ b) × (x∨y) ⊆ θ, it follows that [α]θ = [β]θ. (2) By statement (1), we have µ(z) ≤ ∨ d∈[z]θ µ(d) = ∨ d∈[z′]θ µ(d) for ev- ery z ∈ a ∨ b and z′ ∈ x ∨ y. Hence µ(z) ≤ ∧ z ′∈x∨y ∨ d∈[z′]θ µ(d) for every z ∈ a ∨ b. Therefore ∧ z∈a∨b µ(z) ≤ ∧ z ′∈x∨y ∨ d∈[z′]θ µ(d), which follows that∨ a∈[x]θ,b∈[y]θ ∧ z∈a∨b µ(z) ≤ ∧ z∈x∨y θ(µ)(z). Lemma 3.11. Let θ be a hyper congruence relation on L and x,y ∈ L. If µ is a hyper fuzzy ideal of L and x ≤ y, then θ(µ)(x) = ∨ a∈[x]θ,b∈[y]θ µ(a∧ b). Proof. It is clear that {a∧ b : a ∈ [x]θ,b ∈ [y]θ}⊆ [x]θ. Therefore,∨ a∈[x]θ,b∈[y]θ µ(a∧ b) ≤ ∨ a∈[x]θ µ(a). Now, we suppose that a ∈ [x]θ. Then a ∧ y ∈ [x]θ and since µ is a hyper fuzzy ideal of L, we conclude that µ(a) ≤ µ(a∧y). Therefore θ(µ)(x) = ∨ a∈[x]θ µ(a) ≤ ∨ a∈[x]θ µ(a∧y) ≤ ∨ a∈[x]θ,b∈[y]θ µ(a∧ b). Hence θ(µ)(x) = ∨ a∈[x]θ,b∈[y]θ µ(a∧ b). 23 A. A. Estaji and F. Bayati Definition 3.12. Let θ be an equivalence relation on L and µ ∈F(L). Then, µ is called an upper rough fuzzy (prime) ideal if θ(µ) is a hyper fuzzy (prime) ideal of L. Proposition 3.13. Let θ be a hyper congruence relation on L. If µ is a hyper fuzzy ideal of L, then µ is an upper rough fuzzy ideal. Proof. Let x,y ∈ L. Then θ(µ)(x) ∧θ(µ)(y) = ∨ a∈[x]θ µ(a) ∧ ∨ b∈[y]θ µ(b) = ∨ a∈[x]θ,b∈[y]θ µ(a) ∧µ(b) ≤ ∨ a∈[x]θ,b∈[y]θ ∧ z∈a∨b µ(z) µ is a hyper fuzzy ideal of L ≤ ∧ z∈x∨y θ(µ)(z) by Lemma 3.10. Now, we suppose that x,y ∈ L and x ≤ y. Hence θ(µ)(x) = ∨ a∈[x]θ,b∈[y]θ µ(a∧ b) by Lemma 3.11 ≥ ∨ a∈[x]θ,b∈[y]θ µ(b) µ is a hyper fuzzy ideal of L = θ(µ)(y). Example 3.14. Let the hyperlattice L and the equivalence relation θ on L be as in example 3.8. If µ = ( 0 a b c d 1 0.3 0.3 0.2 0.3 0.2 0.1 ) , then µ is a hyper fuzzy ideal and θ(µ) = ( 0 a b c d 1 0.3 0.3 0.3 0.3 0.2 0.1 ) . Since ∧ x∈b∨c θ(µ)(x) = 0.2 6≥ 0.3 = θ(µ)(b) ∧θ(µ)(c), we conclude that θ(µ) is not a hyper fuzzy ideal. Definition 3.15. Let θ be a hyper congruence relation on L. Then θ is called ∨- complete if [a∨b]θ = [a]θ∨[b]θ for all a,b ∈ L. Likewise, θ is called ∧−complete if [a∧ b]θ = [a]θ ∧ [b]θ for all a,b ∈ L. A hyper congruence relation on L which is both ∨-complete and ∧−complete is called complete. Proposition 3.16. Let θ be a ∧−complete on L. If µ ∈ F(L) is a hyper fuzzy prime ideal of L such that θ(µ) is a proper fuzzy subset of L, then µ is an upper rough fuzzy prime ideal. 24 On Rough Sets and Hyperlattices Proof. If x,y ∈ L, then θ(µ)(x∧y) = ∨ a∈[x∧y]θ µ(a) = ∨ a∈[x]θ,b∈[y]θ µ(a∧ b) θ is ∧−complete ≤ ∨ a∈[x]θ,b∈[y]θ µ(a) ∨µ(b) µ is a hyper fuzzy prime ideal = ∨ a∈[x]θ µ(a) ∨ ∨ b∈[y]θ µ(b) = θ(µ)(x) ∨θ(µ)(y). Now, by Proposition 3.13, the proof is complete. Example 3.17. Let the lattice L be as in example 3.8. Let θ be a hyper congruence relation on the lattice L with the following equivalence classes: [0]θ = {0,a}; [b]θ = {b}; [c]θ = {c}; [1]θ = {1,d}. Since [b∧ c]θ = [a]θ = {0,a} 6= {a} = [b]θ ∧ [c]θ we conclude that θ is not ∧− complete. If µ = ( 0 a b c d 1 0.9 0.8 0.8 0.7 0.7 0.2 ) , then µ is a hyper fuzzy prime ideal and θ(µ) = ( 0 a b c d 1 0.9 0.9 0.8 0.7 0.7 0.7 ) . Also, the ideal θ(µ) is not hyper fuzzy prime, because θ(µ)(b∧ c) = 0.9 � 0.8 = θ(µ)(b) ∨θ(µ)(c). Definition 3.18. Let µ be a fuzzy subset of L. The least hyper fuzzy ideal of L containing µ is called a hyper fuzzy ideal of L induced by µ and is denoted by < µ >. By Remark 2.6 in [12], if µ is a fuzzy subset of L, then there exits < µ >. Definition 3.19. For every µ ∈F(L), we define µ∗(x) = sup{α ∈ [0, 1] : x ∈ I(µα)} for all x ∈ L. With the following example, we prove that Theorem 2.15 in [12] is incorrect. 25 A. A. Estaji and F. Bayati Example 3.20. Let the hyperlattice L be as in example 3.8. If µ = ( 0 a b c d 1 0.5 0.8 0.4 0.5 0.7 0.6 ) , then µ ∈F(L) and µ∗ = ( 0 a b c d 1 1 0.8 0.7 0.7 0.7 0.6 ) . If ν = ( 0 a b c d 1 0.9 0.8 0.7 0.7 0.7 0.6 ) , then ν is the hyper fuzzy ideal of L, µ ≤ ν and µ∗ 6≤ ν. Therefore, µ∗ is not the hyper fuzzy ideal induced by µ. Hence, Theorem 2.15 in [12] is incorrect. Also, if λn = 0.7 − 1n, for every n ∈ N, then b ∈ ⋂ n∈N I(µλn) =↓ d, but b 6∈ µλn for every n ≥ 10. Hence the last paragraph of the proof of Theorem 2.15 in [12] is incorrect. Now we give the correct version of Theorem 2.15 in [12]. Proposition 3.21. Let µ be a fuzzy subset of L. Then the fuzzy subset µ∗ of L is the hyper fuzzy ideal of L and 1. µ ≤ µ∗. 2. µ∗ = ∧ {λ ∈FI(L)|µ ≤ λ and λ(0) = 1}. Proof. For λ ∈ Im(µ∗), let λn = λ − 1n, for n ∈ N, and let x ∈ µ ∗ λ. Then µ∗(x) ≥ λ, which implies that µ∗(x) > λn. Hence there exists β ∈ {α ∈ [0, 1]|x ∈ I(µα)} such that β > λn. Thus µβ ⊆ µλn and so x ∈ I(µβ) ⊆ I(µλn) for all n ∈ N. Therefore, x ∈ ⋂ n∈N I(µλn). Conversely, if x ∈ ⋂ n∈N I(µλn), λn ∈ {α ∈ [0, 1] : x ∈ I(µα)}, for n ∈ N. Therefore, λn = λ − 1n ≤ ∨ {α ∈ [0, 1] : x ∈ I(µα)} = µ∗(x). Hence µ∗(x) ≥ λ, so that x ∈ µ∗λ. Then we have µ∗λ = ⋂ n∈N I(µλn) which is an ideal of L. For x ∈ L, let β ∈ {α ∈ [0, 1] : x ∈ µα}. Then x ∈ µβ, and so x ∈ I(µβ). Thus β ∈{α ∈ [0, 1] : x ∈ I(µα)}, which implies that µ(x) = ∨ {α ∈ [0, 1] : x ∈ µα}≤ ∨ {α ∈ [0, 1] : x ∈ I(µα)} = µ∗(x). Therefore, µ ≤ µ∗ (see [11, 12]). Now, let ν be a hyper fuzzy ideal of L containing µ such that ν(0) = 1. Then for every α ∈ [0, 1], since να 6= ∅, we conclude that I(µα) ≤ I(να) = να. Hence µ∗(x) = ∨ {α ∈ [0, 1] : x ∈ I(µα)}≤ ∨ {α ∈ [0, 1] : x ∈ να} = ν(x) 26 On Rough Sets and Hyperlattices for every x ∈ L. Also, for every α ∈ [0, 1], 0 ∈ I(µα) and we infer that µ∗(0) =∨ {α ∈ [0, 1] : 0 ∈ I(µα)} = 1. Therefore, µ∗ ∈{λ ∈FI(L)|µ ≤ λ and λ(0) = 1}. Finally, we have µ∗ = ∧ {λ ∈FI(L)|µ ≤ λ and λ(0) = 1}. Proposition 3.22. Let θ be a hyper congruence relation on L and µ ∈ F(L). Then θ(< µ >) = θ(< θ(µ) >) and θ(µ∗) = θ((θ(µ))∗). Proof. Since µ ≤< µ >≤ µ∗, we conclude from Proposition 2.8 that θ(µ) ≤ θ(< µ >) ≤ θ(µ∗). It is clear that θ(µ∗)(0) = 1 and by Propositions 3.13 and 3.21, we have < θ(µ) >≤ θ(< µ >) and (θ(µ))∗ ≤ θ(µ∗). Again, by Proposition 2.8, θ(< θ(µ) >) ≤ θ(< µ >) and θ((θ(µ))∗) ≤ θ(µ∗). Since µ ≤ θ(µ), we conclude that < µ >≤< θ(µ) > and µ∗ ≤ (θ(µ))∗ and by Proposition 2.8, θ(< µ >) ≤ θ(< θ(µ) >) and θ(µ∗) ≤ θ((θ(µ))∗). Finally, we have θ(< µ >) = θ(< θ(µ) >) and θ(µ∗) = θ((θ(µ))∗). By the following example, we prove that the condition for an equivalence relation on L does not imply θ((θ(µ))∗) = θ(µ∗). Example 3.23. Let the hyperlattice L and the equivalence relation θ on L be as in example 3.8. If µ = ( 0 a b c d 1 0.5 0.8 0.4 0.7 0.5 0.6 ) , then θ(µ∗) = ( 0 a b c d 1 1 1 1 0.7 0.6 0.6 ) , 27 A. A. Estaji and F. Bayati and θ((θ(µ))∗) = ( 0 a b c d 1 1 1 1 0.7 0.7 0.6 ) . Hence θ(µ∗) 6= θ((θ(µ))∗). Therefore, in general θ(µ∗) = θ((θ(µ))∗) doesn’t hold. 4 Lower approximations of a fuzzy subset In this section we give some important properties of θ with many examples. Lemma 4.1. Let θ be a hyper congruence relation on L and x,y ∈ L. If µ is a hyper fuzzy filter of L and x ≤ y, then θ(µ)(x) = ∧ a∈[x]θ,b∈[y]θ µ(a∧ b). Proof. Since x ≤ y, we can then conclude from Lemma 3.2 that {µ(a∧ b) : a ∈ [x]θ,b ∈ [y]θ}⊆{µ(z) : z ∈ [x]θ}. Hence θ(µ)(x) ≤ ∧ a∈[x]θ,b∈[y]θ µ(a∧b). Also, since µ is a hyper fuzzy filter of L, we conclude that µ(a ∧ b) ≤ µ(a) for every a ∈ [x]θ and b ∈ [y]θ, which follows that θ(µ)(x) ≥ ∧ a∈[x]θ,b∈[y]θ µ(a∧b) and the proof is now complete. Definition 4.2. Let θ be an equivalence relation on L and µ ∈ F(L). Then, µ is called a lower rough fuzzy (prime) ideal if θ(µ) is a hyper fuzzy (prime) ideal of L. Proposition 4.3. Let θ be a ∨-complete congruence relation on L. If µ ∈ F(L) is a hyper fuzzy ideal, then µ is a lower rough fuzzy ideal. Proof. Let x,y ∈ L. Since ∧ t∈x∨y µ(t) ∈ { ∧ t∈a∨b µ(t) : a ∈ [x]θ,b ∈ [y]θ}, we conclude that θ(µ)(x) ∧θ(µ)(y) = ∧ a∈[x]θ µ(a) ∧ ∧ b∈[y]θ µ(b) = ∧ a∈[x]θ,b∈[y]θ µ(a) ∧µ(b) ≤ ∧ a∈[x]θ,b∈[y]θ ∧ t∈a∨b µ(t) µ is a hyper fuzzy ideal = ∧ t∈[x]θ∨[y]θ µ(t) = ∧ t∈[x∨y]θ µ(t) θ is ∨-complete = ∧ z∈x∨y ∧ t∈[z]θ µ(t) = ∧ z∈x∨y θ(µ)(z). Let x,y ∈ L and x ≤ y. Hence θ(µ)(x) = ∧ a∈[x]θ,b∈[y]θ µ(a∧ b) by Lemma 4.1 ≥ ∧ b∈[y]θ µ(b) µ is a hyper fuzzy ideal = θ(µ)(y). 28 On Rough Sets and Hyperlattices Example 4.4. Let the hyperlattice L and the hyper congruence relation θ on L be as in example 3.17. Let µ = ( 0 a b c d 1 1 0.8 0.6 0.4 0.4 0 ) . It is clear that µ is a hyper fuzzy ideal on L and θ(µ) = ( 0 a b c d 1 0.8 0.8 0.6 0.4 0 0 ) . It is easy to see that θ is not ∨ - complete, because [b∨ c]θ = {1.d} 6= {d} = [b]θ ∨ [c]θ. Also, since θ(µ)(b) ∧θ(µ)(c) = 0.4 6≤ 0 = ∧ d∈b∨c θ(µ)(d) we conclude that θ(µ) is not a hyper fuzzy ideal. Definition 4.5. Let θ be an equivalence relation on L and µ ∈ F(L). Then, µ is called a lower rough fuzzy (prime) filter if θ(µ) is a hyper fuzzy (prime) filter of L. Proposition 4.6. Let θ be a ∧−complete congruence relation on L. If µ ∈F(L) is a hyper fuzzy filter, then µ is a lower rough fuzzy filter. Proof. Let x,y ∈ L. θ(µ)(x∧y) = ∧ a∈[x∧y]θ µ(a) = ∧ a∈[x]θ,b∈[y]θ µ(a∧ b) θ is ∧−complete ≥ ∧ a∈[x]θ,b∈[y]θ µ(a) ∧µ(b) µ is a hyper fuzzy filter = ∧ a∈[x]θ µ(a) ∧ ∧ b∈[y]θ µ(b) = θ(µ)(x) ∧θ(µ)(y). Let x,y ∈ L and x ≤ y. Hence θ(µ)(x) = ∧ a∈[x]θ,b∈[y]θ µ(a∧ b) by Lemma 4.1 ≤ ∧ b∈[y]θ µ(b) µ is a hyper fuzzy filter = θ(µ)(y). 29 A. A. Estaji and F. Bayati Example 4.7. Let the hyperlattice L and the hyper congruence relation θ on L be as in example 3.17. If µ = ( 0 a b c d 1 0.1 0.6 0.6 0.7 0.7 0.9 ) , then µ is a hyper fuzzy filter and θ(µ) = ( 0 a b c d 1 0.1 0.1 0.6 0.7 0.7 0.7 ) . Since [b∧ c]θ = {0,a} 6= {a} = [b]θ ∧ [c]θ, we conclude that θ is not ∧− complete. Also θ(ν) is not a hyper fuzzy filter, because θ(ν)(b∧ c) = θ(ν)(a) = 0.1 6≥ 0.6 = θ(ν)(b) ∧θ(ν)(c). Proposition 4.8. Let θ be a ∨-complete on L. If µ ∈F(L) is a hyper fuzzy prime ideal such that θ(µ) is a proper fuzzy subset of L, then µ is a lower rough fuzzy prime ideal. Proof. Let x,y ∈ L. θ(µ)(x∧y) = ∧ z∈[x∧y]θ µ(z) ≤ ∧ a∈[x]θ,b∈[y]θ µ(a∧ b) by Lemma 3.2 ≤ ∧ a∈[x]θ,b∈[y]θ µ(a) ∨µ(b) µ is a hyper fuzzy prime ideal = ∧ a∈[x]θ µ(a) ∨ ∧ b∈[y]θ µ(b) = θ(µ)(x) ∨θ(µ)(y). By Proposition 4.3, the proof is now complete. Example 4.9. Let the lattice L and the hyper congruence relation θ on L be as in example 3.17. If µ = ( 0 a b c d 1 0.9 0.8 0.8 0.7 0.7 0.2 ) , then µ is a hyper fuzzy prime ideal and θ(µ) = ( 0 a b c d 1 0.8 0.8 0.8 0.7 0.2 0.2 ) . Since [b∨ c]θ = {1.d} 6= {d} = [b]θ ∨ [c]θ, we conclude that θ is not ∨ - complete. Also θ(µ) is not hyper fuzzy ideal, because θ(µ)(b) ∧θ(µ)(c) = 0.7 6≤ 0.2 = ∧ d∈(b∨c) θ(µ)(d). 30 On Rough Sets and Hyperlattices Proposition 4.10. Let θ be a complete congruence relation on L. If µ ∈ F(L) is a hyper fuzzy prime filter such that θ(µ) is a proper fuzzy subset of L, then µ is a lower rough fuzzy prime filter. Proof. Let x,y ∈ L. θ(µ)(x) ∨θ(µ)(y) = ∧ a∈[x]θ µ(a) ∨ ∧ b∈[y]θ µ(b) = ∧ a∈[x]θ,b∈[y]θ µ(a) ∨µ(b) ≥ ∧ a∈[x]θ,b∈[y]θ ∧ t∈a∨b µ(t) µ is a hyper fuzzy prime filter = ∧ z∈x∨y ∧ t∈[z]θ µ(t) θ is ∨-complete = ∧ z∈x∨y θ(µ)(z). By Proposition 4.6, the proof is now complete. Example 4.11. Let the hyperlattice L and the hyper congruence relation θ on L be as in example 3.17. It is clear that µ = ( 0 a b c d 1 0.1 0.3 0.7 0.3 0.7 0.9 ) is a hyper fuzzy prime filter and θ(µ) = ( 0 a b c d 1 0.1 0.1 0.7 0.3 0.7 0.7 ) . Since θ(µ)(b) ∧θ(µ)(c) = 0.3 6≤ 0.1 = θ(µ)(b∧ c), we conclude that θ(µ) is not a hyper fuzzy ideal. Also, as we have seen in Example 4.4, θ is not ∨ - complete. 5 Conclusion Rough set, fuzzy set and hyperlattice are different aspects of set theory. Com- bining the three theories, one gets the rough concept fuzzy hyperlattice of a given context. We introduced the concepts of upper and lower rough hyper fuzzy ideals (filters) in a hyperlattice and its basic properties have been discussed. Also, we discussed the relations between hyper fuzzy (prime) ideal and hyper fuzzy (prime) filter with their upper and lower approximations, respectively. In addition, by an example we show that Theorem 2.15 in [12] is incorrect (see Example 3.20) and a corrected version is considered, Proposition 3.21. 31 A. A. Estaji and F. Bayati Acknowledgement We express our gratitude to Professor M. Mehdi Ebrahimi. References [1] S. M. Anvariyeh, S. Mirvakili and B. Davvaz, Pawlaks approximations in Γ−semihypergroups, Computers & Mathematics with Applications, Volume 60, Issue 1, July 2010, 45-53. [2] M. Asghari-Larimi, Hyperstructures with fuzzy subgroup and Fuzzy Ideal, International Mathematical Forum, 5, 2010, no. 10, 467-476. [3] K. Chakrabarty, R. Biswas and S. Nanda, Fuzziness in rough sets, Fuzzy Sets Syst. 110 (2) (2000) 247-251. [4] B. Davvaz, Approximations in hyperrings, Journal of Multiple-Valued Logic and Soft Computing 15 (2009) 471-488. [5] C. Degang, Z. Wenxiu, D. Yeung and E. C. C. Tsang, Rough approximations on a complete completely distributive lattice with applications to generalized rough sets, Information Sciences 176 (2006) 1829-1848. [6] A. A. Estaji, S. Khodaii and S. Bahrami, On rough set and fuzzy sublat- tice, Information Sciences, Volume 181, Issue 18, 15 September 2011, 3981- 3994. [7] A. A. Estaji, M. R. Hooshmandasl and B. Davvaz, Rough set theory applied to lattice theory, Information Sciences, Volume 200, 1 October 2012, 108- 122. [8] Y. Feng and L. Zou, On Product of Fuzzy Subhyperlattice and Interval- Valued Fuzzy Subhyperlattice, Discrete Mathematics, Algorithms and Ap- plications Vol. 2, No. 2 (2010) 239-246. [9] X. Z. Guo and X.L. Xin, On hyperlattice, Pure and Applied Mathematics 20 (1) (2004) 40-43. [10] O. Kazancı, S. Yamak and B. Davvaz, The lower and upper approximations in a quotient hypermodule with respect to fuzzy sets, Information Sciences 178 (2008) 2349-2359. [11] B. B. N. Koguep, C. Nkuimi and C. Lele, On fuzzy prime ideal of lattice, Samsa Journal of Pure and Applied Mathematics, 3 (2008) 1-11. 32 On Rough Sets and Hyperlattices [12] B. B. N. Koguep, C. Nkuimi and C. Lele, On Fuzzy Ideals of Hyperlattice, International Journal of Algebra, Vol. 2, 2008, no. 15, 739-750. [13] M. Kondo, On the structure of generalized rough sets, Information Sciences 176 (2006) 589-600. [14] F. Marty, Sur une generalization de la notion de group, 8th Congress Math. Scandenaves, Stockholm, (1934), 45-49. [15] J. N. Mordeson, Rough set theory applied to (fuzzy) ideal theory, Fuzzy Sets Syst. 121 (2) (2001) 315-324. [16] A. D. Lokhande and Aryani Gangadhara, Congruences in hypersemilattices, International Mathematical Forum, Vol. 7, 2012, no. 55, 2735 - 2742. [17] Z. Pawlak, Rough sets, International Journal of Computer and Information Sciences 11 (1982) 341-356. [18] A. M. Radzikowska and E. E. Kerre, A comparative study of fuzzy rough sets, Fuzzy Sets Syst. 126 (6) (2002) 137-156. [19] A. Rahnamai-Barghi, The prime ideal theorem for distributive hyperlattices, Ital. Journal of Pure and Applied Math., vol. 10, 2001 , 75-78. [20] M. Sambasiva Rao, Multipliers of Hypersemilattices, International Journal of Mathematics and Soft Computing Vol.3, No.1 (2013), 29 - 35. [21] S. Yamak, O. Kazancı, and B. Davvaz, Soft hyperstructure, Computers & Mathematics with Applications, Volume 62, Issue 2, July 2011, Pages 797- 803. [22] L. A. Zadeh, Fuzzy sets, Inform. and Control, 8 (1965) 338-353. 33