EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 12, No. 3, 2019, 821-833 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global On Varieties of Pseudo Hyper GR-ideals of Pseudo Hyper GR-algebras Ramises G. Manzano, Jr.1, Gaudencio C. Petalcorin, Jr.2,∗ 1 Mathematics Department, College of Science, University of the Philippines Cebu, 6000 Cebu City, Philippines 2 Department of Mathematics and Statistics, College of Science and Mathematics, Mindanao State University-Iligan Institute of Technology, 9200 Iligan City, Philippines Abstract. This study is based on the structure of hyper GR-algebras, an algebra that is par- tially related on some class of hyper BCI-algebras. This allows us to create a new structure and investigate how this two algebras are related to each other. A pseudo hyper GR-algebra involves two hyper operations and a set of axioms that come in pairs or a combination of both making it interesting like some algebras established. This paper focuses on some properties of pseudo hyper GR-algebras and its ideals. Moreover, pseudo hyper GR-ideals were defined and classified to determine their relationship to each other. 2010 Mathematics Subject Classifications: 14L17, 20N20, 03G25 Key Words and Phrases: pseudo hyper GR-algebras, pseudo hyper GR-ideals 1. Introduction Algebraic hyperstructures were introduced by a French mathematician, Marty [7], in 1934. They represent a natural extension of classical hyperstructures in which the composition of two elements of a given set is a set, instead of an element. Afterwards, this new idea was expanded rapidly and showed itself as a new view of sets. The introduction of hyperstructure theory led to the study of several problems of noncommutative algebra. Algebraic hyperstructure theory has multiple applications to other fields such as: geometry, graphs and hypergraphs, binary relations, lattices, groups, relation algebras, artificial intelligence, probabilities, and so on. In 1966, Y. Imai and K. Iséki [4] initiated the notion of BCK-algebra as a generalization of the concept of set-theoretic difference and propositional calculi. Furthermore, Y.B. Jun et al. [6] applied hyperstructure theory to BCK-algebras and introduced the notion of hyper BCK-algebras as a generalization of BCK-algebra. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v12i3.3431 Email addresses: rgmanzano@up.edu.ph (R. Manzano, Jr.), gaudencio.petalcorin@g.msuiit.edu.ph (G. Petalcorin, Jr.) http://www.ejpam.com 821 c© 2019 EJPAM All rights reserved. R. Manzano, Jr., G. Petalcorin, Jr. / Eur. J. Pure Appl. Math, 12 (3) (2019), 821-833 822 In order to extend BCK-algebra to a noncommutative form, G. Georgescu and A. Iorgulescu [3] introduced the notion of pseudo BCK-algebras and studied their properties. On the other hand, R. A. Borzooei, A. Rezazadeh and R. Ameri [1] introduced the concept of hyper pseudo BCK-algebra which is a generalization of pseudo BCK-algebra. R.A. Indangan and G.C. Petalcorin [5] defined a new class of algebraic hyperstructure called hyper GR-algebra. In this algebra, they presented a helpful understanding on how this hyper algebra differs from the rest. In this paper we define a pseudo hyper GR-algebra analogous to that of a hyper GR- algebra and its pseudo hyper GR-ideals and their relationships. 2. Preliminaries Let H be a nonempty set endowed with a hyperoperation “ ∗ ”, that is, “ ∗ ” is a function from H × H to P ∗(H) = P (H) \ {∅}. For two nonempty subsets A and B of H, A ∗ B = ⋃ a∈A,b∈B a ∗ b. We shall use x ∗ y instead of x ∗ {y}, {x} ∗ y or {x} ∗ {y}. When A is a nonempty subset of H and x ∈ H, we agree to write A ∗x instead of A ∗ {x}. Similarly, we write x ∗A for {x} ∗A. In effect, A ∗x = ⋃ a∈A a ∗ x and x ∗A = ⋃ a∈A x ∗ a. A set H endowed with a family Γ of hyperoperations is called a hyperstructure. If Γ is singleton, that is, Γ = {f}, then the hyperstructure is called a hypergroupoid. Definition 2.1. [2] Let x, y ∈ H and A,B ⊆ H. Then (i) x� y if and only if 0 ∈ x~ y; and (ii) A� B if and only if for any a ∈ A, there exists b ∈ B such that a� b. We call � a hyperorder on H. Remark 2.2. [2] For all A,B ⊆ H, A� B implies 0 ∈ A~B. Definition 2.3. [5] Let H be a nonempty set with a hyperoperation“~” on H. Then (H;~, 0) is called a hyper GR-algebra if it contains a constant 0 ∈ H and for all x, y, z ∈ H, the following conditions are satisfied: [HGR1] (x~ z)~ (y ~ z)� x~ y; [HGR2] (x~ y)~ z = (x~ z)~ y; [HGR3] x� x; [HGR4] 0~ (0~ x)� x, for all x 6= 0; and [HGR5] (x~ y)~ z � y ~ z. Example 2.4. [5] Let H = {0, 1, 2}. Define the operation “~” by the Cayley table shown below. R. Manzano, Jr., G. Petalcorin, Jr. / Eur. J. Pure Appl. Math, 12 (3) (2019), 821-833 823 ~ 0 1 2 0 {0} {0} {0} 1 {0, 1, 2} {0, 1} {0, 1} 2 {0, 2} {0, 1, 2} {0, 2} By routine calculations, (H;~, 0) is a hyper GR-algebra. Definition 2.5. [5] A hyper GR-algebra H is faithful if for all A,B ⊆ H, 0 ∈ A ~ B implies A� B. Definition 2.6. [5] Let H be a hyper GR-algebra and S be a subset of H containing 0. If S is a hyper GR-algebra with respect to the hyperoperation ~ on H, then we say that S is a hyper subGR-algebra of H. Theorem 2.7. [5] (Hyper SubGR-algebra Criterion) Let H be a hyper GR-algebra and S be a nonempty subset of H. Then S is a hyper subGR-algebra of H if and only if x~ y ⊆ S, for all x, y ∈ S. Definition 2.8. [5] Let I be a subset of a hyper GR-algebra H such that 0 ∈ I. Then (i) I is a hyper GR-ideal of H if for all x, y ∈ H, x~ y ⊆ I and y ∈ I imply that x ∈ I; (ii) if H is faithful such that x~ x� I for all x ∈ H, then I is GR-reflexive in H; (iii) I is hyper left (resp. hyper right) stable in H if x~ a� I (resp. a~ x� I) for all x ∈ H and for all a ∈ I; (iv) I is hyper stable in H if I is both hyper left and hyper right stable in H; (v) I is hyper left (resp. hyper right) stable GR-ideal of H if (a) I is hyper left (resp. hyper right) stable in H; and (b) I is a hyper GR-ideal of H. (vi) I is a hyper stable GR-ideal of H if I is both hyper left and hyper right stable GR-ideal of H. Theorem 2.9. [5] If {Ii|i ∈ Λ} is a nonempty collection of hyper GR-ideals of a hyper GR-algebra H, then so is ⋂ i∈Λ Ii. Definition 2.10. [5] Let H be a hyper GR-algebra, X a nonempty proper subset of H, and I a subset of H such that 0 ∈ I. Then (i) I is a hyper GR-ideal of H related to X if for all x, y ∈ X, x~y ⊆ I and y ∈ I imply that x ∈ I; (ii) I is hyper left (resp. hyper right) stable in H related to X if x ~ a � I (resp. a~ x� I) for all x ∈ X and for all a ∈ I; R. Manzano, Jr., G. Petalcorin, Jr. / Eur. J. Pure Appl. Math, 12 (3) (2019), 821-833 824 (iii) I is hyper stable in H related to X if I is both hyper left and hyper right stable in H related to X; (iv) I is hyper left (resp. hyper right) stable GR-ideal of H related to X if (a) I is hyper left (resp. hyper right) stable in H related to X; and (b) I is a hyper GR-ideal of H related to X. (v) I is a hyper stable GR-ideal of H related to X if I is both hyper left and hyper right stable GR-ideal of H related to X. 3. Pseudo Hyper GR-ideals In this section we will define a pseudo hyper GR-algebra and the different types of pseudo hyper GR-ideals. Also, relationship among the twelve types of these ideals are discussed. Definition 3.1. Let H be a nonempty set with “~” and “◦” be the two hyperoperations on H. Then (H;~, ◦, 0) is called a pseudo hyper GR-algebra, if it contains a constant 0 ∈ H and for all x, y, z ∈ H, the following conditions are satisfied: [PHGR1] (x ◦ z) ◦ (y ◦ z)� x ◦ y and (x~ z)~ (y ~ z)� x~ y; [PHGR2] (x ◦ y)~ z = (x~ z) ◦ y; [PHGR3] 0 ∈ x~ x and 0 ∈ x ◦ x; [PHGR4] 0 ◦ (0~ x)� x, for all x 6= 0; and [PHGR5] (x~ y)~ z � y ◦ z. where x� y if and only if 0 ∈ x ◦ y and 0 ∈ x~ y, and for every A,B ⊆ H, A� B means that for every a ∈ A, there exists b ∈ B such that a� b. Throughout this chapter, we denote a pseudo hyper GR-algebra (H,~, ◦, 0) simply by H, unless otherwise stated. Example 3.2. Let H = {0, 1, 2, 3} and consider the following Cayley tables below. ~ 0 1 2 3 0 {0, 1} {0, 1} {0, 1} {0, 1} 1 {0, 1} {0, 1} {0, 1} {0, 1} 2 {0, 2} {0, 1, 2} {0, 2} {0, 1, 2} 3 {0, 1, 2} {0, 3} {0, 1, 3} {0, 3} R. Manzano, Jr., G. Petalcorin, Jr. / Eur. J. Pure Appl. Math, 12 (3) (2019), 821-833 825 ◦ 0 1 2 3 0 {0, 1} {0, 1} {0, 1} {0, 1} 1 {1} {0, 1} {0, 1} {0, 1} 2 {0, 2} {0, 2} {0, 1, 2} {0, 1, 2} 3 {0, 3} {0, 1, 3} {0, 1, 3} {0, 1, 3} By routine calculations, we see that (H;~, ◦, 0) is a pseudo hyper GR-algebra. Remark 3.3. In a pseudo hyper GR-algebra H, the following are evident: (i) x� x; (ii) (x ◦ y)~ z � (x~ z) ◦ y; (iii) (A ◦B)~ C = (A~ C) ◦B; and (iv) A ⊆ B implies A� B. Example 3.4. Let H = N ∪ {0} be the set of all nonnegative integers and let the hyper- operations “~” and “◦” be defined on H as follows: x~ y = {0, x} and x ◦ y = {0, x, y}. Then H is a pseudo hyper GR-algebra. To verify this, we need to check that the five conditions are satisfied. Note that {0, x, z} ◦ {0, y, z} = {0, x, y, z} � {0, x, y}. This means that (x ◦ z) ◦ (y ◦ z)� x ◦ y . On the other hand, {0, x} ~ {0, y} = {0, x} � {0, x} means that (x ~ z) ~ (y ~ z) � x ~ y. Thus, [PHGR1] holds. Now, (x ◦ y) ~ z = {0, x, y} ~ z = {0, x, y}, also (x ~ z) ◦ y = {0, x}◦y = {0, x, y} and so (x◦y)~z = (x~z)◦y, that is, [PHGR2] is satisfied. [PHGR3] follows immediately from the defined operations ◦ and ~ on H, that is, x ◦x = {0, x} and x~ x = {0, x} for all x ∈ H. Let x 6= 0, then 0 ◦ (0~ x) = 0 ◦ {0} = {0} � x, and thus, [PHGR4] holds. Finally, {0, x}~z = {0, x} � {0, y, z} = y ◦z. Hence, (x~y)~z � y ◦z, that is, [PHGR5] holds. Therefore, H is a pseudo hyper GR-algebra. Remark 3.5. Note that if the two hyperoperations are equal, that is, ~ = ◦, then a pseudo hyper-GR algebra H becomes a hyper GR-algebra. Definition 3.6. Let H be a pseudo hyper GR-algebra and S be a subset of H containing 0. If S itself is a pseudo hyper GR-algebra with respect to the hyperoperations ~ and ◦ on H, then S is called a pseudo hyper subGR-algebra of H. Theorem 3.7. (Pseudo Hyper SubGR-algebra Criterion) Let S be a nonempty subset of a pseudo hyper GR-algebra H. Then S is a pseudo hyper subGR-algebra if and only if both x~ y ⊆ S and x ◦ y ⊆ S for all x, y ∈ S. R. Manzano, Jr., G. Petalcorin, Jr. / Eur. J. Pure Appl. Math, 12 (3) (2019), 821-833 826 Proof. Suppose that S is a pseudo hyper subGR-algebra of H. By Definition 3.6, S is closed under the hyperoperations ~ and ◦ so that x~ y ⊆ S and x ◦ y ⊆ S for all x, y ∈ S. Conversely, suppose that S has the property x~ y ⊆ S and x ◦ y ⊆ S for all x, y ∈ S. Since S ⊆ H, all the axioms [PHGR1] to [PHGR5] of Definition 3.1 are all satisfied. It remains to show that S contains the element 0. From the above hypothesis, S is nonempty and thus, must contain an element, say c. Then by Definition 3.1 [PHGR3], 0 ∈ c~ c and 0 ∈ c ◦ c. Note that c~ c ⊆ S and c ◦ c ⊆ S. Thus, 0 ∈ S. � Example 3.8. For any pseudo hyper GR-algebra H, the set S = {0} is a pseudo hyper subGR-algebra of H. For any nonempty subset I of a pseudo hyper GR-algebra H and any element y of H, we introduce the following notations and their meanings: I�~,y = {x ∈ H |x~ y � I}. I⊆~,y = {x ∈ H |x~ y ⊆ I}. I�◦,y = {x ∈ H |x ◦ y � I}. I⊆◦,y = {x ∈ H |x ◦ y ⊆ I}. Definition 3.9. Let I be a nonempty subset of a pseudo hyper GR-algebra H such that 0 ∈ I. Then I is said to be a pseudo hyper-GR ideal of H if for any y ∈ I, I⊆~,y ⊆ I and I⊆◦,y ⊆ I. Example 3.10. Consider the pseudo hyper GR-algebra H in Example 3.2. Let I = {0, 2}. Observe that I⊆~,0 = {x ∈ H |x~ 0 ⊆ I} = {2} ⊆ I I⊆~,2 = {x ∈ H |x~ 2 ⊆ I} = {2} ⊆ I I⊆◦,0 = {x ∈ H |x ◦ 0 ⊆ I} = {2} ⊆ I I⊆◦,2 = {x ∈ H |x ◦ 2 ⊆ I} = ∅ ⊆ I. Thus, I is indeed a pseudo hyper GR-ideal. From now on, we shall call the ideal in Definition 3.9 as pseudo hyper GR-ideal of type 1 for we will be considering some forms of pseudo hyper GR-ideals which will be defined analogously as in Definition 3.9. R. Manzano, Jr., G. Petalcorin, Jr. / Eur. J. Pure Appl. Math, 12 (3) (2019), 821-833 827 Definition 3.11. Let I be a nonempty subset of a pseudo hyper GR-algebra H such that 0 ∈ I. Then I is said to be a pseudo hyper-GR ideal of H of : type 2, if for any y ∈ I, I⊆~,y ⊆ I and I�◦,y ⊆ I. type 3, if for any y ∈ I, I�~,y ⊆ I and I⊆◦,y ⊆ I. type 4, if for any y ∈ I, I�~,y ⊆ I and I�◦,y ⊆ I. type 5, if for any y ∈ I, I⊆~,y ⊆ I or I⊆◦,y ⊆ I. type 6, if for any y ∈ I, I⊆~,y ⊆ I or I�◦,y ⊆ I. type 7, if for any y ∈ I, I�~,y ⊆ I or I⊆◦,y ⊆ I. type 8, if for any y ∈ I, I�~,y ⊆ I or I�◦,y ⊆ I. type 9, if for any y ∈ I, I⊆~,y ∩ I⊆◦,y ⊆ I. type 10, if for any y ∈ I, I⊆~,y ∩ I�◦,y ⊆ I. type 11, if for any y ∈ I, I�~,y ∩ I⊆◦,y ⊆ I. type 12, if for any y ∈ I, I�~,y ∩ I�◦,y ⊆ I. Example 3.12. Let H = {0, 1, 2} with the hyperoperations ~ and ◦ on H given by the Cayley table below ~ 0 1 2 0 {0} {0} {0} 1 {1} {0} {0} 2 {2} {0, 2} {0} ◦ 0 1 2 0 {0} {0} {0} 1 {1} {0} {0} 2 {0, 2} {2} {0, 2} By routine calculations, H is a pseudo hyper GR-algebra. Let I = {0, 1}. Note that I⊆~,y = {0, 1} ⊆ I and I�◦,y = {0, 1} ⊆ I. Thus, I is pseudo hyper GR-ideal of type 2. Note also that I�~,y = {0, 1} ⊆ I and I⊆◦,y = {0, 1} ⊆ I. Thus, I is pseudo hyper GR-ideal of type 3. Moreover, R. Manzano, Jr., G. Petalcorin, Jr. / Eur. J. Pure Appl. Math, 12 (3) (2019), 821-833 828 I�~,y = {0, 1} ⊆ I and I�◦,y = {0, 1} ⊆ I, I�~,y = {0, 1} ⊆ I or I�◦,y = {0, 1} ⊆ I and I�~,y ∩ I�◦,y = {0, 1} ⊆ I. Therefore, I is pseudo hyper GR-ideal of type 4, 8 and 12 respectively. Example 3.13. Consider the pseudo hyper GR-algebra H in Example 3.2. Let I = {0, 3}. Note that for any y ∈ I, I⊆~,y = {3} ⊆ I. This is enough to categorize I as a pseudo hyper GR-ideal of type 6. Also for any y ∈ I, I�◦,y = {0, 1, 2, 3}. Even if I�◦,y 6⊆ I, I⊆~,y ∩ I�◦,y = {3} ⊆ I. Thus, I must be a pseudo hyper GR-ideal of type 10. Hence, I is an example of pseudo hyper GR-ideal of type 6 and 10 but not type 2 since I⊆~,y ⊆ I but I�◦,y 6⊆ I. Example 3.14. Consider the pseudo hyper GR-algebra H in Example 3.2. Let I = {0, 1}. By routine calculations, I is a pseudo hyper GR-ideal of type 5. Example 3.15. Consider the pseudo hyper GR-algebra H in Example 3.2. Let I = {0, 1, 3}. By routine calculations, I is a pseudo hyper GR-ideal of type 6. Example 3.16. Consider the pseudo hyper GR-algebra H in Example 3.2. Let I = {0, 2}. By routine calculations, I is a pseudo hyper GR-ideal of type 7. Example 3.17. Consider the pseudo hyper GR-algebra H in Example 3.4. Let H ′ = {0, 1, 2, 3}. Then H ′ together with the hyperoperations ~ and ◦ given by the Cayley table below is a pseudo hyper subGR-algebra of H. ~ 0 1 2 3 0 {0} {0} {0} {0} 1 {0, 1} {0, 1} {0, 1} {0, 1} 2 {0, 2} {0, 2} {0, 2} {0, 2} 3 {0, 3} {0, 3} {0, 3} {0, 3} ◦ 0 1 2 3 0 {0} {0, 1} {0, 2} {0, 3} 1 {0, 1} {0, 1} {0, 1, 2} {0, 1, 3} 2 {0, 2} {0, 1, 2} {0, 2} {0, 2, 3} 3 {0, 3} {0, 1, 3} {0, 2, 3} {0, 3} Consider I = {0, 2, 3}. Observe that I⊆~,y = {0, 2, 3} = I⊆◦,y. This means that I⊆~,y ∩ I⊆◦,y = {0, 2, 3} ⊆ I. Thus, I is a pseudo hyper GR-ideal of type 9. Let I = {0, 1, 2}. O bservier that I⊆~,y = {0, 1, 2} and I�◦,y = {0, 1, 2, 3}. Thus, we have I⊆~,y ∩ I�◦,y = {0, 1, 2} ⊆ I. Therefore, I is a pseudo hyper GR-ideal of type 10. Let I = {0, 1, 3}. Observe that I�~,y = {0, 1, 2, 3} and I⊆◦,y = {0, 1, 3}. Thus, we have I�~,y ∩ I⊆◦,y = {0, 1, 3} ⊆ I. Therefore, I is a pseudo hyper GR-ideal of type 11. R. Manzano, Jr., G. Petalcorin, Jr. / Eur. J. Pure Appl. Math, 12 (3) (2019), 821-833 829 Theorem 3.18. Every pseudo hyper GR-ideal in H of type 2 is a pseudo hyper GR-ideal in H of type 1. Proof. Let I be a pseudo hyper GR-ideal of type 2. Now, we will show that I is a pseudo hyper GR-ideal of type 1. It is enough to show that for any y ∈ I, I⊆◦,y ⊆ I. Let y ∈ I and x ∈ I⊆◦,y. Then, x ◦ y ⊆ I and by Remark 3.3 (iv), x ◦ y � I. Hence x ∈ I�◦,y. Since I is a pseudo hyper GR-ideal of type 2, I�◦,y ⊆ I and so x ∈ I. Therefore, I⊆◦,y ⊆ I. � Theorem 3.19. Every pseudo hyper GR-ideal in H of type 4 is a pseudo hyper GR-ideal in H of types 1, 2 and 8. Proof. Let I be a pseudo hyper GR-ideal in H of type 4. We will show that I is a pseudo hyper GR-ideal of type 2. It is enough to show that for any y ∈ I, I⊆~,y ⊆ I. Let y ∈ I and x ∈ I⊆~,y. Then, x ~ y ⊆ I and by Remark 3.3 (iv), x ~ y � I. Hence, x ∈ I�~,y. Since I is a pseudo hyper GR-ideal of type 4, I�~,y ⊆ I and so x ∈ I. Thus, I⊆~,y ⊆ I. Hence, I is a pseudo hyper GR-ideal of type 2 and by Theorem 3.18, I is a pseudo hyper GR-ideal of type 1. Furthermore, we will show that I is a pseudo hyper GR-ideal of type 8. That is, to show that for any y ∈ I, I�~,y ⊆ I or I�◦,y ⊆ I. Let y ∈ I and x ∈ I�~,y. Since I is a pseudo hyper GR-ideal of type 4, I�~,y ⊆ I and so, x ∈ I. Therefore, I�~,y ⊆ I. Similarly, we can show for the other case that I�◦,y ⊆ I. � Theorem 3.20. Every pseudo hyper GR-ideal in H of type 8 is a pseudo hyper GR-ideal in H of types 5, 6, 7 and 12. Proof. Let I be a pseudo hyper GR-ideal of type 8. We will show that I is a pseudo hyper GR-ideal of type 5. We will consider two cases : when I⊆◦,y ⊆ I and when I⊆◦,y 6⊆ I. If I⊆◦,y ⊆ I, then we are done. Suppose that I⊆◦,y 6⊆ I. Let x ∈ I⊆~,y, where y ∈ I. Then, x~ y ⊆ I, thus by Remark 3.3 (iv), x~ y � I. Hence, x ∈ I�~,y. Since I is a pseudo hyper GR-ideal of type 8, I�◦,y ⊆ I or I�~,y ⊆ I. Suppose that I�◦,y ⊆ I. The hypothesis I⊆◦,y 6⊆ I implies that there exists z ∈ I⊆◦,y such that z 6∈ I. Moreover, z ◦ y ⊆ I and by Remark 3.3 (iv), z ◦ y � I. Hence, z ∈ I�◦,y and so z ∈ I. A contradiction. Thus, I�◦,y 6⊆ I. Thus, I�~,y ⊆ I and so x ∈ I. Therefore, I⊆~,y ⊆ I. Next, we will prove that I is a pseudo hyper GR-ideal of type 6. If I�◦,y ⊆ I, then we are done. Suppose I�◦,y 6⊆ I. Let x ∈ I⊆~,y, where y ∈ I. Then, x~ y ⊆ I, thus by Remark 3.3 (iv), x ~ y � I. Hence, x ∈ I�~,y. Since I is a pseudo hyper GR-ideal of type 8 and I�◦,y 6⊆ I, then I�~,y ⊆ I and so x ∈ I. Therefore, I⊆~,y ⊆ I. The proof for type 7 follows similarly as in the case of type 6. Furthermore, we will prove that I is a pseudo hyper GR-ideal of type 12. Let y ∈ I and x ∈ I�~,y ∩ I�◦,y. Then x ∈ I�~,y and I�◦,y. Since I is a pseudo hyper GR-ideal of type 8, we have I�~,y ⊆ I or I�◦,y ⊆ I and so x ∈ I. Hence, I�~,y ∩ I�◦,y ⊆ I. � R. Manzano, Jr., G. Petalcorin, Jr. / Eur. J. Pure Appl. Math, 12 (3) (2019), 821-833 830 Theorem 3.21. Every pseudo hyper GR-ideal in H of type 6 is a pseudo hyper GR-ideal in H of types 5 and 10. Proof. Let I be a pseudo hyper GR-ideal of type 6. Now, we will show that I is a pseudo hyper GR-ideal of type 5. If I⊆~,y ⊆ I, then we are done. Suppose I⊆~,y 6⊆ I. Let x ∈ I⊆◦,y for any y ∈ I. Then x ◦ y ⊆ I and so by Remark 3.3 (iv), x ◦ y � I. Hence, x ∈ I�◦,y. Since I is a pseudo hyper GR-ideal of type 6 and I⊆~,y 6⊆ I, I�◦,y ⊆ I and thus, x ∈ I. Hence, I⊆◦,y ⊆ I. Next, we will show that I is a pseudo hyper GR-ideal of type 10. Let y ∈ I and x ∈ I⊆~,y ∩ I�◦,y. Then, x ∈ I⊆~,y and x ∈ I�◦,y. Since I is a pseudo hyper GR-ideal of type 6, we have I⊆~,y ⊆ I or I�◦,y ⊆ I and so x ∈ I. Hence, I⊆~,y ∩ I�◦,y ⊆ I. � Theorem 3.22. Every pseudo hyper GR-ideal in H of type 7 is a pseudo hyper GR-ideal in H of types 5 and 11. Proof. Let I be a pseudo hyper GR-ideal of type 7. Now, we will show that I is a pseudo hyper GR-ideal of type 5. If I⊆◦,y ⊆ I, then we are done. Suppose I⊆◦,y 6⊆ I. Let x ∈ I⊆~,y for any y ∈ I. Then x ~ y ⊆ I and so by Remark 3.3 (iv), x ~ y � I. Hence, x ∈ I�~,y. Since I is a pseudo hyper GR-ideal of type 7 and I⊆◦,y 6⊆ I, I�~,y ⊆ I and thus, x ∈ I. Hence, I⊆~,y ⊆ I. Next, we will show that I is a pseudo hyper GR-ideal of type 11. Let y ∈ I and x ∈ I�~,y ∩ I⊆◦,y. Then, x ∈ I�~,y and x ∈ I⊆◦,y. Since I is a pseudo hyper GR-ideal of type 7, we have I�~,y ⊆ I or I⊆◦,y ⊆ I and so x ∈ I. Hence, I�~,y ∩ I⊆◦,y ⊆ I. � Theorem 3.23. Every pseudo hyper GR-ideal in H of type 5 is a pseudo hyper GR-ideal in H of type 9. Proof. Suppose that I be a pseudo hyper GR-ideal of type 5. Now, we will show that I is a pseudo hyper GR-ideal of type 9. Let y ∈ I and x ∈ I⊆~,y ∩ I ⊆ ◦,y. Then, x ∈ I⊆~,y and x ∈ I⊆◦,y. Since I is a pseudo hyper GR-ideal of type 5, I⊆~,y ⊆ I or I⊆◦,y ⊆ I and so x ∈ I. Hence, I⊆~,y ∩ I⊆◦,y ⊆ I. � Theorem 3.24. Every pseudo hyper GR-ideal in H of type 12 is a pseudo hyper GR-ideal in H of types 9, 10 and 11. Proof. Suppose that I be a pseudo hyper GR-ideal of type 12. Now, we will show that I is a pseudo hyper GR-ideal of type 9. Let y ∈ I and x ∈ I⊆~,y ∩ I⊆◦,y. Then, x ∈ I⊆~,y and x ∈ I⊆◦,y. Thus, x ~ y ⊆ I and x ◦ y ⊆ I and and by Remark 3.3 (iv), x ~ y � I and x ◦ y � I. This means that x ∈ I�~,y and x ∈ I�◦,y or equivalently x ∈ I�~,y ∩ I�◦,y . Since I is a pseudo hyper GR-ideal of type 12, I�~,y ∩ I�◦,y ⊆ I, and so x ∈ I. Therefore, I⊆~,y ∩ I⊆◦,y ⊆ I. Next, we will show that I is a pseudo hyper GR-ideal of type 10. Let y ∈ I and x ∈ I⊆~,y ∩ I�◦,y. Then, x ∈ I⊆~,y and x ∈ I�◦,y. Thus, x ~ y ⊆ I and x ◦ y � I and and R. Manzano, Jr., G. Petalcorin, Jr. / Eur. J. Pure Appl. Math, 12 (3) (2019), 821-833 831 by Remark 3.3 (iv), x ~ y � I. This means that x ∈ I�~,y and x ∈ I�◦,y or equivalently x ∈ I�~,y ∩ I�◦,y . Since I is a pseudo hyper GR-ideal of type 12, I�~,y ∩ I�◦,y ⊆ I, and so x ∈ I. Therefore, I⊆~,y ∩ I�◦,y ⊆ I. The proof for type 11 follows similarly as of type 10 with some modifications. � Theorem 3.25. Every pseudo hyper GR-ideal in H of type 10 is a pseudo hyper GR-ideal in H of type 9. Proof. Suppose that I be a pseudo hyper GR-ideal of type 10. Now, we will show that I is a pseudo hyper GR-ideal of type 9. Let y ∈ I and x ∈ I⊆~,y ∩ I⊆◦,y. Then, x ∈ I⊆~,y and x ∈ I⊆◦,y. Thus, x ◦ y ⊆ I and and by Remark 3.3 (iv), x ◦ y � I which means that x ∈ I�◦,y. Thus, x ∈ I⊆~,y ∩ I�◦,y. Since I is a pseudo hyper GR-ideal of type 10, we have I⊆~,y ∩ I�◦,y ⊆ I and so x ∈ I. Hence, I⊆~,y ∩ I⊆◦,y ⊆ I. � Theorem 3.26. Every pseudo hyper GR-ideal in H of type 11 is a pseudo hyper GR-ideal in H of type 9. Proof. Suppose that I be a pseudo hyper GR-ideal of type 11. Now, we will show that I is a pseudo hyper GR-ideal of type 9. Let y ∈ I and x ∈ I⊆~,y ∩ I⊆◦,y. Then x ∈ I⊆~,y and x ∈ I⊆◦,y. Thus, x~ y ⊆ I and and by Remark 3.3 (iv), x~ y � I which means that x ∈ I�~,y. Thus, x ∈ I�~,y ∩ I⊆◦,y. Since I is a pseudo hyper GR-ideal of type 11, we have I�~,y ∩ I⊆◦,y ⊆ I and so, x ∈ I. Hence, I⊆~,y ∩ I⊆◦,y ⊆ I. � Theorem 3.27. Let {Iω|ω ∈ Ω} be a family of pseudo hyper GR-ideals of type i, 1 ≤ i ≤ 12, in H. Then ⋂ ω∈Ω Iω is also a pseudo hyper GR-ideal of type i, 1 ≤ i ≤ 12 in H. Proof. Assume that I = ⋂ ω∈Ω Iω. Let Iω be a pseudo hyper GR-ideal of specific type, say type 1, for any ω ∈ Ω. We will prove that I is a pseudo hyper GR-ideal of type 1. Since every Iω is a pseudo hyper GR-ideal for each ω, 0 ∈ Iω, for all ω ∈ Ω and thus, 0 ∈ ⋂ ω∈Ω Iω = I. Let y ∈ I, x ∈ I⊆~,y and z ∈ I⊆◦,y. Then x ~ y ⊆ I and z ◦ y ⊆ I. This means that for any u ∈ x ~ y, u ∈ I. Thus, u ∈ Iω for any ω ∈ Ω and so, x ~ y ⊆ Iω. Hence, x ∈ I⊆ω,~,y and y ∈ Iω, for any ω ∈ Ω. Since Iω is a pseudo hyper GR-ideal of type 1, I⊆ω,~,y ⊆ Iω so that x ∈ Iω for any ω ∈ Ω. Hence, x ∈ I and thus,I⊆~,y ⊆ I. In a similar manner, we can also prove that z ∈ I so that I⊆◦,y ⊆ I. The proof for the remaining cases (i = 2, 3, . . . , 12) follows the same argument with some modifications. � Theorem 3.28. Let D be a nonempty subset of H. Let [D]i denote the intersection of all pseudo hyper GR-ideals of type i, 1 ≤ i ≤ 4 containing D. Then {x ∈ H|(...((x~ d1)~ d2)~ ...)~ dn = {0}, di ∈ D} ⊆ [D]i. REFERENCES 832 Proof. We will prove only the case for i = 1, that is for the case of pseudo hyper GR-ideals of type 1. Let x ∈ H and suppose that the condition (...((x~ d1)~ d2)~ ...)~ dn = {0} is satisfied for some d1, d2, ..., dn ∈ D. Note that 0 ∈ [D]1, hence (...((x~ d1)~ d2)~ ...)~ dn = {0} ⊆ [D]1 Thus, for each d ∈ (...((x~ d1)~ d2)~ ...)~ dn−1, we have d~ dn ⊆ [D]1, or equivalently, d ∈ ([D]1)⊆~,dn . Since [D]1 is a pseudo hyper GR-ideal of type 1, ([D]1)⊆~,dn ⊆ [D]1, and so, d ∈ [D]1. Thus, (...((x~ d1)~ d2)~ ...)~ dn−1 ⊆ [D]1 Continuing this process, we obtain {x} ∈ [D]1 and so, x ∈ [D]1. Therefore, {x ∈ H|(...((x~ d1)~ d2)~ ...)~ dn = {0}, di ∈ D} ⊆ [D]1. � The ideal [D]i in Theorem 3.28 is called the pseudo hyper GR-ideal generated by D. Acknowledgements The first author would like to thank the Philippine Department of Science and Technology- Accelerated Science and Technology Human Resource Development Program (DOST- ASTHRDP) for the financial support for this publication and for the three-year PhD scholarship grant in Mindanao State University-Iligan Institute of Technology. Lastly, the authors would like to thank the referees who gave their brilliant suggestions in refining this paper before publication to the European Journal of Pure and Applied Mathematics. References [1] A. Borzooei, A. Rezazadeh, and R. Ameri. On Hyper Pseudo BCK-algebras. Iranian Journal of Mathematical Sciences and Informatics, 9(1):13–29, 2014. [2] K. Hila et. al. Algebraic Hyperstructure of Soft Sets Associated with Ternary Semi- hypergroups. Italian Journal of Pure and Applied Mathematics, 30:349–372, 2013. [3] G. Georgescu and Iorgulescu. Pseudo BCK-algebras: an extension of BCK-algebras. In Springer, editor, Combinatorics, Computability and Logic, pages 97–114, London, 2001. Proceedings of DMTSC 01. [4] Y. Imai and K. Iseki. On Axiom Systems of Propositional Calculi XIV. pages 19–22, Japan, 1966. Proceedings of Japan Acedemy. [5] R. Indangan and G. Petalcorin. Some Results on Hyper GR-ideals of a Hyper GR- algebra. Journal of Algebra and Applied Mathematics, 14:101–119, 2016. REFERENCES 833 [6] Y. Jun, M. Zahedi, X. Xin, and R. Borzooei. On Hyper BCK-algebras. Journal of Algebra and Applied Mathematics, 8:127–136, 2000. [7] F. Marty. Sur une generalization de la notion de groupe. In Proceedings of the 8th Congress Math. Scandinaves, pages 45–49. Stockholm, 1934.