RATIO MATHEMATICA 26 (2014), 03–20 ISSN:1592-7415 Rough Set Theory Applied To Hyper BCK-Algebra R. Ameria, R. Moradianb and R. A. Borzooeic aSchool of Mathematics, Statistics and Computer Science, College of Sciences, University of Tehran, P.O. Box 14155-6455, Teheran, Iran ameri@ut.ac.ir bDepartment of Mathematics, Payam Noor University, Tehran, Iran rmoradian58@yahoo.com cDepartment of Mathematics, Shahid Beheshti University, Tehran, Iran borzooei@sbu.ac.ir Abstract The aim of this paper is to introduce the notions of lower and up- per approximation of a subset of a hyper BCK-algebra with respect to a hyper BCK-ideal. We give the notion of rough hyper subalgebra and rough hyper BCK-ideal, too, and we investigate their properties. Key words: rough set, rough (weak, strong) hyper BCK-ideal, rough hyper subalgebra, regular congruence relation. MSC 2010: 20N20, 20N25. 1 Introduction In 1966, Y. Imai and K. Iseki [2] introduced a new notion, called a BCK- algebra. The hyper structure theory (called also multi algebras ) was intro- duced in 1934 by F. Marty [6] at the 8th Congress of Scandinavian Math- ematicians. In [3], Y. B. Jun, M. M. Zahedi, X. L. Xin, R. A. Borzooei applied the hyper structures to BCK-algebras and they introduced the no- tion of hyper BCK-algebra (resp. hyper K-algebra) which is a generalization of BCK-algebra (resp. hyper BCK-algebra). They also introduced the no- tion of hyper BCK-ideal, weak hyper BCK-ideal, hyper K-ideal and weak 3 Ameri, Moradian, Borzooei hyper K-ideal and gave relations among them. In 1982, Pawlak introduced the concept of rough set (see [7]). Recently Jun [5] applied rough set theory to BCK-algebras. In this paper, we apply the rough set theory to hyper BCK-algebras. 2 Preliminaries Let U be a universal set. For an equivalence relation Θ on U, the set of elements of U that are related to x ∈ U, is called the equivalence class of x and is denoted by [x]Θ. Moreover, let U/Θ denote the family of all equivalence classes induced on U by Θ. For any X ⊆ U, we write Xc to denote the complement of X in U, that is the set U\X. A pair (U, Θ) where U 6= φ and Θ is an equivalence relation on U is called an approximation space. The interpretation in rough set theory is that our knowledge of the objects in U extends only up to membership in the class of Θ and our knowledge about a subset X of U is limited to the class of Θ and their unions. This leads to the following definition. Definition 2.1. [7] For an approximation space (U, Θ), by a rough approxi- mation in (U, Θ) we mean a mapping Apr : P(U) −→ P(U) ×P(U) defined for every X ∈ P(U) by Apr(X) = (Apr(X),Apr(X)), where Apr(X) = {x ∈ U|[x]Θ ⊆ X}, Apr(X) = {x ∈ U|[x]Θ ∩X 6= φ}. Apr(X) is called a lower rough approximation of X in (U, Θ), whereas Apr(X) is called an upper rough approximation of X in (U, Θ). Definition 2.2. [7] Given an approximation space (U, Θ), a pair (A,B) ∈ P(U) ×P(U) is called a rough set in (U, Θ) if and only if (A,B) = Apr(X) for some X ∈ P(U). Definition 2.3. ([7]) Let (U, Θ) be an approximation space and X be a non-empty subset of U. (i) If Apr(X) = Apr(X), then X is called definable. (ii) If Apr(X) = φ, then X is called empty interior. 4 Rough Set Theory Applied To Hyper BCK-Algebra (iii) If Apr(X) = U, then X is called empty exterior. Let H be a non-empty set endowed with a hyper operation “◦”, that is ◦ is a function from H ×H to P∗(H) = P(H) −{φ}. For two subsets A and B of H, denote by A◦B the set ⋃ a∈A,b∈B a◦ b. We shall use x◦ y instead of x◦{y}, {x}◦y, or {x}◦{y}. Definition 2.4. ([3]) By a hyper BCK-algebra we mean a non- empty set H endowed with a hyper operation “◦”and a constant 0 satisfying the following axioms: (HK1) (x◦z) ◦ (y ◦z) � x◦y, (HK2) (x◦y) ◦z = (x◦z) ◦y, (HK3) x◦H �{x}, (HK4) x � y and y � x imply x = y, for all x,y,z ∈ H, where x � y is defined by 0 ∈ x◦y and for every A,B ⊆ H, A � B is defined by ∀a ∈ A,∃b ∈ B such that a � b. In such case, we call “�”the hyper order in H. Theorem 2.5. ([3]) In any hyper BCK-algebra H, the following hold: (a1) 0 ◦ 0 = {0}, (a2) 0 � x, (a3) x � x, (a4) A � A, (a5) A � 0 implies A = {0}, (a6) A ⊆ B implies A � B, (a7) 0 ◦x = {0}, (a8) x◦y � x, (a9) x◦ 0 = {x}, (a10) y � z implies x◦z � x◦y, (a11) x◦y = {0} implies (x◦z) ◦ (y ◦z) = {0} and x◦z � y ◦z, (a12) A◦{0} = {0} implies A = {0}, for all x,y,z ∈ H and for all non-empty subsets A and B of H. 5 Ameri, Moradian, Borzooei Definition 2.6. ([3]) Let H be a hyper BCK-algebra and let S be a subset of H containing 0. If S be a hyper BCK-algebra with respect to the hyper operation “◦”on H, we say that S is a hyper subalgebra of H. Theorem 2.7. ([3]) Let S be a non-empty subset of hyper BCK-algebra H. Then S is a hyper subalgebra of H if and only if x◦y ⊆ S, for all x,y ∈ S. Definition 2.8. ([3]) Let I be a non-empty subset of hyper BCK-algebra H and 0 ∈ I. (i) I is said to be a hyper BCK-ideal of H if x◦ y � I and y ∈ I imply x ∈ I for all x,y ∈ H. (ii) I is said to be a weak hyper BCK-ideal of H if x ◦ y ⊆ I and y ∈ I imply x ∈ I for all x,y ∈ H. (iii) I is called a strong hyper BCK-ideal of H if (x◦y) ∩ I 6= φ and y ∈ I imply x ∈ I for all x,y ∈ H. Theorem 2.9. ([3]) If H be a hyper BCK-algebra, then (i) every hyper BCK-ideal of H is a weak hyper BCK-ideal of H. (ii) every strong hyper BCK-ideal of H is a hyper BCK-ideal of H. Definition 2.10. ([4]) Let H be a hyper BCK-algebra. A hyper BCK- ideal I of H is called reflexive if x◦x ⊆ I for all x ∈ H. Definition 2.11. ([1]) Let Θ be an equivalence relation on hyper BCK- algebra H and A,B ⊆ H. Then, (i) AΘB means that, there exist a ∈ A and b ∈ B such that aΘb, (ii) AΘ̄B means that, for all a ∈ A there exists b ∈ B such that aΘb and for all b ∈ B there exists a ∈ A such that aΘb, (iii) Θ is called a congruence relation on H, if xΘy and x′Θy′ imply x ◦ x′Θ̄y ◦y′ for all x,y,x′,y′ ∈ H. (iv) Θ is called a regular relation on H, if x◦yΘ{0} and y ◦xΘ{0} imply xΘy for all x,y ∈ H. 6 Rough Set Theory Applied To Hyper BCK-Algebra Example 2.12. Let H1 = {0, 1, 2}, H2 = {0,a,b} and hyper operations “◦1”and “◦2”on H1 and H2 are defined respectively, as follow: ◦1 0 1 2 0 {0} {0} {0} 1 {1} {0} {1} 2 {2} {2} {0, 2} ◦2 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {a,b} {0,b} Then (H1,◦1) and (H2,◦2) are hyper BCK-algebras. Define the equivalence relation Θ1 and Θ2 on H1 and H2, respectively, as Θ1 = {(0, 0), (1, 1), (2, 2), (0, 2), (2, 0)}, and Θ2 = {(0, 0), (a,a), (b,b), (0,a), (a, 0)}. It is easily checked that Θ1 is a congruence relation on H1. But Θ2 is not a congruence relation on H2, since bΘ2b and 0Θ2a but b◦ 0Θ̄2b◦a is not true. Example 2.13. Let (H1,◦1) be a hyper BCK-algebra as Example 2.12. Let H2 = {0,a,b,c} and define the hyper operation “◦2”on H2 as follow: ◦2 0 a b c 0 {0} {0} {0} {0} a {a} {0,a} {0} {a} b {b} {b} {0,a} {b} c {c} {c} {c} {0,c} Then (H2,◦2) is a hyper BCK-algebra. Define the congruence relation Θ1 and Θ2 on H1 and H2, respectively, by Θ1 = {(0, 0), (1, 1), (2, 2), (0, 1), (1, 0)}, and Θ2 = {(0, 0), (a,a), (b,b), (c,c), (0,b), (b, 0)}. It is easily checked that Θ1 is a regular congruence relation on H1, but Θ2 is not a regular relation on H2, since a◦bΘ2{0} and b◦aΘ2{0} but (a,b) 6∈ Θ2. Theorem 2.14. ([1]) Let Θ be a regular congruence relation on hyper BCK-algebra H. Then [0]Θ is a hyper BCK-ideal of H. 7 Ameri, Moradian, Borzooei Theorem 2.15. ([1]) Let Θ be a regular congruence relation on H,I = [0]Θ and H I = {Ix : x ∈ H}, where Ix = [x]Θ for all x ∈ H. Then HI with hyper operation “◦”and hyper order “<”which is defined as follow, is a hyper BCK- algebra which is called quotient hyper BCK-algebra, Ix ◦ Iy = {Iz : z ∈ x◦y}, and Ix < Iy ⇐⇒ I ∈ Ix ◦ Iy. Theorem 2.16. ([1]) Let I be a reflexive hyper BCK-ideal of H and rela- tion Θ on H be defined as follow: xΘy ⇐⇒ x◦y ⊆ I and y ◦x ⊆ I for all x,y ∈ H. Then Θ is a regular congruence relation on H and I = [0]Θ. 3 Rough hyper BCK-ideals Throughout this section H is a hyper BCK-algebra. In this section first we define lower and upper approximation of the subset A of H with respect to hyper BCK-ideal of H and prove some properties. Then we give the definition of (weak, strong) rough hyper BCK-ideals and investigate the relation between them and (weak, strong) hyper BCK-ideals of H. Definition 3.1. Let Θ be a regular congruence relation on hyper BCK- algebra H, I = [0]Θ, Ix = [x]Θ and A be a non-empty subset of H. Then the sets Apr I (A) = {x ∈ H|Ix ⊆ A}, AprI (A) = {x ∈ H|Ix ∩A 6= φ}. are called lower and upper approximation of the set A with respect to the hyper BCK-ideal I, respectively. Proposition 3.2. For every approximation space (H, Θ) and every subsets A,B ⊆ H, we have: (1) Apr I (A) ⊆ A ⊆ AprI (A), (2) Apr I (φ) = φ = AprI (φ), 8 Rough Set Theory Applied To Hyper BCK-Algebra (3) Apr I (H) = H = AprI (H), (4) if A ⊆ B, then Apr I (A) ⊆ Apr I (B) and AprI (A) ⊆ AprI (B), (5) Apr I (Apr I (A)) = Apr I (A), (6) AprI (AprI (A)) = AprI (A), (7) AprI (AprI (A)) = AprI (A), (8) Apr I (AprI (A)) = AprI (A), (9) Apr I (A) = (AprI (A c))c, (10) AprI (A) = (AprI (A c))c, (11) AprI (A∩B) ⊆ AprI (A) ∩AprI (B), (12) Apr I (A∩B) = Apr I (A) ∩Apr I (B), (13) AprI (A∪B) = AprI (A) ∪AprI (B), (14) Apr I (A∪B) ⊇ Apr I (A) ∪Apr I (B), (15) Apr I (Ix) = H = AprI (Ix) for all x ∈ H. Proof. (1), (2) and (3) are straightforward. (4) For any x ∈ Apr I (A) we have Ix ⊆ A ⊆ B and so x ∈ AprI (B). Now, suppose that x ∈ AprI (A). Then Ix ∩A 6= φ and so Ix ∩B 6= φ. Hence x ∈ AprI (B). (5) Since Apr I (A) ⊆ A, by (4) we have Apr I (Apr I (A)) ⊆ Apr I (A). Now, let x ∈ Apr I (A). Then Ix ⊆ A. Since for any y ∈ Ix, we have Ix = Iy, then Iy ⊆ A and so y ∈ AprI (A). Therefore, Ix ⊆ AprI (A) and then we obtain x ∈ Apr I (Apr I (A)). (6) By (1) and (4), AprI (A) ⊆ AprI (AprI (A)). On the other hand, we assume that x ∈ AprI (AprI (A)). Then we have Ix ∩AprI (A) 6= φ and so there exist a ∈ Ix and a ∈ AprI (A). Hence Ia = Ix and Ia ∩A 6= φ which imply Ix ∩A 6= φ. Therefore, x ∈ AprI (A). 9 Ameri, Moradian, Borzooei (7) By (1), we have Apr I (A) ⊆ AprI (AprI (A)). Now, let x ∈ AprI (AprI (A)). Then Ix ∩ AprI (A) 6= φ and so there exist a ∈ Ix and a ∈ AprI (A). Hence Ia = Ix and Ia ⊆ A which imply Ix ⊆ A. Therefore, x ∈ Apr I (A). (8) By (1), we have Apr I (AprI (A)) ⊆ AprI (A). Now, we assume that x ∈ AprI (A). Then Ix ∩ A 6= φ. For every y ∈ Ix, we have Iy = Ix and so Iy ∩ A 6= φ. Hence y ∈ AprI (A) which implies Ix ⊆ AprI (A). Therefore, x ∈ Apr I (AprI (A)). (9) For any subset A of H we have: (AprI (A c))c = {x ∈ H : x 6∈ AprI (A c)} = {x ∈ H : Ix ∩Ac = φ} = {x ∈ H : Ix ⊆ A} = {x ∈ H : x ∈ Apr I (A)} = Apr I (A). (10) For any subset A of H we have: (Apr I (Ac))c = {x ∈ H : x 6∈ Apr I (Ac)} = {x ∈ H : Ix 6⊂ Ac} = {x ∈ H : Ix ∩A 6= φ} = {x ∈ H : x ∈ AprI (A)} = AprI (A). (11) Since A∩B ⊆ A and A∩B ⊆ B, then by (4), AprI (A∩B) ⊆ AprI (A) and AprI (A∩B) ⊆ AprI (B). Hence AprI (A∩B) ⊆ AprI (A)∩AprI (B). 10 Rough Set Theory Applied To Hyper BCK-Algebra (12) For any subset A and B of H we have: x ∈ Apr I (A∩B) ⇐⇒ Ix ⊆ A∩B ⇐⇒ Ix ⊆ A and Ix ⊆ B ⇐⇒ x ∈ Apr I (A) and x ∈ Apr I (B) ⇐⇒ x ∈ Apr I (A) ∩Apr I (B). (13) For any subset A and B of H we have x ∈ AprI (A∪B) ⇐⇒ Ix ∩ (A∪B) 6= φ ⇐⇒ (Ix ∩A) ∪ (Ix ∩B) 6= φ ⇐⇒ Ix ∩A 6= φ or Ix ∩B 6= φ ⇐⇒ x ∈ AprI (A) or x ∈ AprI (B) ⇐⇒ x ∈ AprI (A) ∪AprI (B). (14) Since A ⊆ A∪B and B ⊆ A∪B, then by (4), Apr I (A) ⊆ Apr I (A∪B) and Apr I (B) ⊆ Apr I (A∪B), which imply that Apr I (A) ∪Apr I (B) ⊆ Apr I (A∪B). (15) The proof is straightforward. Corollary 3.3. Let (H, Θ) be an approximation space. Then (i) for every A ⊆ H, Apr I (A) and AprI (A) are definable sets, (ii) for every x ∈ H,Ix is definable set. Proof. (i) By proposition 3.2 (5) and (7), we have Apr I (Apr I (A)) = Apr I (A) = AprI (AprI (A)). Hence AprI (A) is a definable set. On the other hand by proposition 3.2 (6) and (8), we have AprI (AprI (A)) = AprI (A) = Apr I (AprI (A)). Therefore AprI (A) is a definable set. (ii) By proposition 3.2 (15) the proof is clear. 11 Ameri, Moradian, Borzooei Theorem 3.4. Let Θ be a regular congruence relation on H, I = [0]Θ be a hyper BCK-ideal of H and A,B are non-empty subsets of H. Then (i) AprI (A) ◦AprI (B) = AprI (A◦B), (ii) Apr I (A) ◦Apr I (B) ⊆ Apr I (A◦B). Proof. (i) Let z ∈ AprI (A) ◦AprI (B). Then there exist a ∈ AprI (A) and b ∈ AprI (B) such that z ∈ a◦b. Hence Ia ∩A 6= φ and Ib ∩B 6= φ and so there exist c ∈ Ia ∩A and d ∈ Ib ∩B such that aΘc and bΘd. Since Θ is a congruence relation on H, then we have a◦bΘ̄c◦d and because z ∈ a ◦ b, then there exist y ∈ c ◦ d such that zΘy. Hence y ∈ Iz. On the other hand, y ∈ c◦d ⊆ A◦B which implies Iz ∩ (A◦B) 6= φ and so z ∈ AprI (A◦B). Therefore AprI (A)◦AprI (B) ⊆ AprI (A◦B). Now, suppose that x ∈ AprI (A ◦ B). Then Ix ∩ (A ◦ B) 6= φ. Let z ∈ Ix ∩ (A◦B), then there exist a ∈ A and b ∈ B such that z ∈ a◦ b and Ix = Iz. Thus we have Iz ∈ Ia ◦ Ib and so Ix ∈ Ia ◦ Ib. Hence x ∈ a ◦ b ⊆ A ◦ B ⊆ AprI (A) ◦ AprI (B). Therefore, AprI (A ◦ B) ⊆ AprI (A) ◦AprI (B). 2 (ii) Let z ∈ Apr I (A) ◦ Apr I (B). Then there exist a ∈ Apr I (A) and b ∈ Apr I (B) such that z ∈ a◦ b, Ia ⊆ A and Ib ⊆ B. For every y ∈ Iz, we have Iz = Iy ∈ Ia ◦Ib and so y ∈ a◦b ⊆ A◦B. Then y ∈ A◦B and so Iz ⊆ A◦B. Therefore z ∈ AprI (A◦B). Example 3.5. Let H = {0, 1, 2} and define the hyper operation “◦”on H as follow: ◦ 0 1 2 0 {0} {0} {0} 1 {1} {0} {1} 2 {2} {2} {0, 2} Then (H,◦) is a hyper BCK-algebra. Define the equivalence relation Θ by Θ = {(0, 0), (1, 1), (2, 2), (0, 1), (1, 0)}. Then Θ is a regular congruence relation on H and so we have: I = [0]Θ = {0, 1},I1 = [1]Θ = {0, 1},I2 = [2]Θ = {2}. 12 Rough Set Theory Applied To Hyper BCK-Algebra Now, if we let A = {1, 2} and B = {0, 2}, then we have A◦B = {0, 1, 2} and so Apr I (A) = {x ∈ H|Ix ⊆ A} = {2}, AprI (A) = {x ∈ H|Ix ∩A 6= φ} = {0, 1, 2}, Apr I (B) = {x ∈ H|Ix ⊆ B} = {2}, AprI (B) = {x ∈ H|Ix ∩B 6= φ} = {0, 1, 2}, Apr I (A◦B) = {x ∈ H|Ix ⊆ A◦B} = {0, 1, 2}, AprI (A◦B) = {x ∈ H|Ix ∩ (A◦B) 6= φ} = {0, 1, 2}, AprI (A) ◦AprI (B) = {0, 1, 2}, Apr I (A) ◦Apr I (B) = {0, 2}. Therefore, we see that Apr I (A) ◦ Apr I (B) 6= Apr I (A ◦ B) but AprI (A) ◦ AprI (B) = AprI (A◦B). Definition 3.6. Let Θ be a regular congruence relation on H, I = [0]Θ be a hyper BCK-ideal of H and A be a non-empty subset of H. If Apr I (A) and AprI (A) are hyper subalgebra of H, then A is called a rough hyper subalgebra of H. Theorem 3.7. If I be a hyper BCK-ideal and J be a hyper subalgebra of H, then (i) AprI (J) is a hyper subalgebra of H. (ii) If I ⊆ J, then Apr I (J) is a hyper subalgebra of H. Proof. (i) Since 0 ∈ J ⊆ AprI (J), then AprI (J) 6= φ. Now, we assume that x,y ∈ AprI (J). We must prove that x ◦ y ⊆ AprI (J). Since Ix ∩ J 6= φ and Iy ∩ J 6= φ, we can let t ∈ Ix ∩ J, s ∈ Iy ∩ J and z ∈ x◦ y. Hence Iz ∈ Ix ◦ Iy = It ◦ Is and so z ∈ t◦ s ⊆ J. Thus we have z ∈ J and z ∈ Iz and so Iz ∩J 6= φ. Therefore, z ∈ AprI (J) and so x◦y ⊆ AprI (J). (ii) Since I = I0 ⊆ J, we have 0 ∈ AprI (J) 6= φ. Now, suppose that a,b ∈ Apr I (J). Then Ia ⊆ J and Ib ⊆ J. For every z ∈ a◦b and every y ∈ Iz, we have Iz = Iy ∈ Ia ◦ Ib and so y ∈ a◦ b ⊆ J. Hence Iz ⊆ J, which implies that z ∈ Apr I (J). Therefore, a◦ b ⊆ Apr I (J). 13 Ameri, Moradian, Borzooei Theorem 3.8. Let Θ and Φ be two regular congruence relations on H and I = [0]Θ, J = [0]Φ be two hyper BCK-ideals of H such that I ⊆ J. Then for any nonempty subset A of H, we have: (i) Apr J (A) ⊆ Apr I (A), (ii) AprI (A) ⊆ AprJ (A). Proof. (i) First we show that if I ⊆ J, then Ix ⊆ Jx. Let y ∈ Ix. Then xΘy. Since Θ is a congruence relation on H and xΘx, then x◦xΘ̄x◦y. Since 0 ∈ x ◦ x, then there exist t ∈ x ◦ y such that 0Θt and so t ∈ [0]Θ = I ⊆ J = [0]Φ. Thus by hypothesis, t ∈ [0]Φ and so x◦yΦ{0}. By the similar way, we can show that y ◦xΦ{0}. Since Φ is a regular congruence relation, we get xΦy and so y ∈ [x]Φ = Jx. Therefore, Ix ⊆ Jx. Now, let x ∈ AprJ (A). Then Jx ⊆ A and so Ix ⊆ A which implies x ∈ Apr I (A). (ii) Assume that x ∈ AprI (A). Then Ix ∩A 6= φ. Since Ix ⊆ Jx, we have Jx ∩A 6= φ. Therefore, x ∈ AprJ (A). Corollary 3.9. Let Θ and Φ are two regular congruence relations on H, I = [0]Θ, J = [0]Φ be two hyper BCK-ideals of hyper BCK-algebra H and A be a non-empty subset of H. Then (i) Apr I (A) ∩Apr J (A) ⊆ Apr I∩J (A), (ii) AprI∩J (A) ⊆ AprI (A) ∩AprJ (A). Proof. By theorem 3.8, the proof is clear. Definition 3.10. Let Θ be a regular congruence relation on H, I = [0]Θ be a hyper BCK-ideal of H, A be a non-empty subset of H and AprI (A) = (Apr I (A),AprI (A)) be a rough set in the approximation space (H, Θ). If Apr I (A) and AprI (A) are hyper BCK-ideals (resp, weak, strong) of H, then A is called a rough hyper BCK-ideal (resp, weak, strong) of H. 14 Rough Set Theory Applied To Hyper BCK-Algebra Example 3.11. Let H = {0, 1, 2, 3} and hyper operation “◦”on H is de- fined as follow: ◦ 0 1 2 3 0 {0} {0} {0} {0} 1 {1} {0, 1} {0} {1} 2 {2} {2} {0, 1} {2} 3 {3} {3} {3} {0, 3} Then (H,◦, 0) is a hyper BCK-algebra. We define the regular congruence relation on H as follow: Θ = {(0, 0), (1, 1), (2, 2), (3, 3), (0, 1), (1, 0)}. So we have: I = I0 = I1 = {0, 1},I2 = {2},I3 = {3}. Now, let A = {0, 1, 3} be a subset of H, then Apr I (A) = {x ∈ H|Ix ⊆ A} = {0, 1, 3}, AprI (A) = {x ∈ H|Ix ∩A 6= φ} = {0, 1, 3}. Easily we give that Apr I (A) and AprI (A) are hyper BCK-ideals. Therefore, A is a rough hyper BCK-ideal of H. Example 3.12. Let H = {0,a,b,c}. By the following table (H,◦) is a hyper BCK-algebra. ◦ 0 a b c 0 {0} {0} {0} {0} a {a} {0,a} {0} {a} b {b} {b} {0,a} {b} c {c} {c} {c} {0,c} Now, let relation Θ on H is defined as follow: Θ = {(0, 0), (a,a), (b,b), (c,c), (0,b), (b, 0), (0,a), (a, 0), (a,b), (b,a)}. Then, I0 = Ia = Ib = {0,a,b},Ic = {c}. Let J1 = {0,c}, J2 = {0,b} and J3 = {c}. Then, Apr I (J1) = {c},AprI (J1) = {0,a,b,c}, Apr I (J2) = {},AprI (J2) = {0,a,b}, Apr I (J3) = {c},AprI (J3) = {c}. 15 Ameri, Moradian, Borzooei Hence we can see that J1 is a hyper BCK-ideal of H but AprI (J1) is not a hyper BCK-ideal. Moreover J2 is not a hyper BCK-ideal but AprI (J2) is a hyper BCK-ideal of H. In follows, J3 is not a hyper BCK-ideal and neither Apr I (J3) nor AprI (J3) is a hyper BCK-ideal of H. Theorem 3.13. Let Θ be a regular congruence relation on H and I = [0]Θ be a hyper BCK-ideal of H. Then (i) If J be a weak hyper BCK-ideal of H containing I, then Apr I (J) is a weak hyper BCK-ideal of H, (ii) If J be a hyper BCK-ideal of H containing I, then Apr I (J) is a hyper BCK-ideal of H, (iii) If J be a strong hyper BCK-ideal of H containing I, then Apr I (J) is a strong hyper BCK-ideal of H. Proof. (i) Since I = I0 ⊆ J, then 0 ∈ AprI (J). Now, Let x,y ∈ H be such that x ◦ y ⊆ Apr I (J) and y ∈ Apr I (J). We must prove that Ix ⊆ J. Let a ∈ Ix and b ∈ Iy. Then aΘx and bΘy. Since Θ is a congruence relation on H, we have a◦bΘx◦y and so for every z ∈ a◦b, there exist t ∈ x◦ y such that zΘt. Since x◦ y ⊆ Apr I (J), we have t ∈ Apr I (J) and so It = Iz ⊆ J which implies z ∈ J. Thus a ◦ b ⊆ J. On the other hand, b ∈ Iy ⊆ J. Since J is a weak hyper BCK-ideal, we have a ∈ J and so Ix ⊆ J. Hence x ∈ AprI (J). Therefore, AprI (J) is a weak hyper BCK-ideal of H. (ii) Let x,y ∈ H be such that x ◦ y � Apr I (J) and y ∈ Apr I (J). We must prove that Ix ⊆ J. Let a ∈ Ix and b ∈ Iy. Then aΘx and bΘy. Since Θ is a congruence relation on H, we have a◦ bΘx◦ y and so for every z ∈ a ◦ b, there exist z′ ∈ x ◦ y such that zΘz′. Since z′ ∈ x ◦ y � Apr I (J), then there exists t ∈ Apr I (J) ⊆ J such that z′ � t and so from zΘz′, we have I0 ∈ Iz′ ◦It = Iz ◦It. Hence 0 ∈ z◦ t and then z � t. Thus we have proved that for every z ∈ a ◦ b, there exist t ∈ J such that z � t which means that a◦b � J. On the other hand we have b ∈ Iy ⊆ J. Since J is a hyper BCK-ideal of H, we 16 Rough Set Theory Applied To Hyper BCK-Algebra have a ∈ J. Thus Ix ⊆ J which implies that x ∈ AprI (J). Therefore, Apr I (J) is a hyper BCK-ideal of H. (iii) Suppose that x,y ∈ H be such that (x ◦ y) ∩ Apr I (J) 6= φ and y ∈ Apr I (J). Let a ∈ Ix and b ∈ Iy. Then aΘx and bΘy. Since Θ is a congruence relation on H, we have a◦bΘx◦y. Since (x◦y)∩Apr I (J) 6= φ, then there exist t ∈ H such that t ∈ x ◦ y and t ∈ Apr I (J). Now, t ∈ x◦yΘa◦ b implies that there exist z ∈ a◦ b such that zΘt and so It = Iz ⊆ J. Hence z ∈ J and so (a◦b)∩J 6= φ. On the other hand, we have b ∈ Iy ⊆ J. Since J is a strong hyper BCK-ideal of H, then we have a ∈ J which implies Ix ⊆ J that means x ∈ AprI (J). Therefore, Apr I (J) is a strong hyper BCK-ideal of H. Theorem 3.14. Suppose that I be a hyper BCK-ideal of H and Θ be a regular congruence relation on H which is defined as follow: xΘy ⇔ x◦y ⊆ I and y ◦x ⊆ I. (i) If J be a weak hyper BCK-ideal of H containing I, then AprI (J) is a weak hyper BCK-ideal of H, (ii) If J be a hyper BCK-ideal of H containing I, then AprI (J) is a hyper BCK-ideal of H, (iii) If J be a strong hyper BCK-ideal of H containing I, then AprI (J) is a strong hyper BCK-ideal of H. Proof. (i) Since I ⊆ J ⊆ AprI (J), then we have 0 ∈ AprI (J). Let x,y ∈ H be such that x◦y ⊆ AprI (J) and y ∈ AprI (J). Then Iy ∩J 6= φ and for every z ∈ x◦y, we have z ∈ AprI (J) which means Iz ∩J 6= φ. Thus there exist a,b ∈ H such that a ∈ Iy ∩ J and b ∈ Iz ∩ J which imply that aΘy, bΘz and a,b ∈ J. Thus y◦a ⊆ I ⊆ J and z◦b ⊆ I ⊆ J and so we get y,z ∈ J, since J is a weak hyper BCK-ideal. Thus we have proved that for any z ∈ x◦y, we have z ∈ J and so x◦y ⊆ J. Since J is a weak hyper BCK-ideal and y ∈ J, obviously we have x ∈ J. Since x ∈ Ix, then Ix ∩ J 6= φ. Therefore x ∈ AprI (J) and so AprI (J) is a weak hyper BCK-ideal of H. 17 Ameri, Moradian, Borzooei (ii) Let x,y ∈ H be such that x ◦ y � AprI (J) and y ∈ AprI (J). Then Iy ∩J 6= φ and for every z ∈ x◦ y, there exist t ∈ AprI (J) such that z � t and It ∩J 6= φ. Thus, there exist c,d ∈ H such that c ∈ It ∩J and d ∈ Iy ∩ J and so cΘt, dΘy and c,d ∈ J. Hence t ◦ c ⊆ I ⊆ J and y ◦ d ⊆ I ⊆ J. Since J is a hyper BCK-ideal and c,d ∈ J, we have y,t ∈ J. Thus, we have proved that for every z ∈ x ◦ y, there exist t ∈ J such that z � t which means that x◦ y � J and so from y ∈ J we get x ∈ J. Consequently, Ix ∩ J 6= φ and so x ∈ AprI (J). Therefore, AprI (J) is a hyper BCK-ideal. (iii) Let x,y ∈ H be such that (x ◦ y) ∩ AprI (J) 6= φ and y ∈ AprI (J). Then Iy ∩ J 6= φ and so there exist z ∈ H such that z ∈ x ◦ y and z ∈ AprI (J). Hence Iz ∩J 6= φ and so there exist c,d ∈ H such that c ∈ Iz ∩J and d ∈ Iy ∩J. Hence cΘz and dΘy where c,d ∈ J. Thus we have z◦c ⊆ I ⊆ J and y◦d ⊆ I ⊆ J. Since J is a strong hyper BCK- ideal and c,d ∈ J, we have z ∈ J and y ∈ J. Thus we have proved that (x◦y) ∩J 6= φ and y ∈ J. Since J is a strong hyper BCK-ideal, we have x ∈ J and so Ix ∩J 6= φ which means that AprI (J) is a strong hyper BCK-ideal of H. 4 Conclusion This paper is intend to built up connection between rough sets and hy- per BCK-algebras. We have presented a definition of the lower and upper approximation of a subset of a hyper BCK-algebra with respect to a hyper BCK-ideal. 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