EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 12, No. 4, 2019, 1524-1532 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Topologies on a Hyper Sum and Hyper Product of Two Hyper BCK-algebras Rachel M. Patangan1, Sergio R. Canoy, Jr.2,∗ 1 Department of Applied Mathematics, College of Arts and Science, Agusan del Sur State College of Agriculture and Technology, Bunawan, Agusan del Sur, Philippines 2 Department of Mathematics and Statistics, College of Science and Mathematics, Center for Graph Theory, Algebra, and Analysis-PRISM, Mindanao State University - Iligan Institute of Technology, 9200, Iligan City, Philippines Abstract. Given a hyper BCK-algebra (H, ∗, 0), each of the families BL(H) = {LH(A) : ∅ 6= A ⊆ H} and BR(H) = {RH(A) : ∅ 6= A ⊆ H} forms a base for some topology on H, where LH(A) = {x ∈ H : x � a, ∀a ∈ A} and RH(A) = {x ∈ H : a � x, ∀a ∈ A} for any subset A of H. In this paper, we determine the bases of the topologies induced by the hyper sum H1 ⊕H2 and hyper product H1 ×H2, where (H1, ∗1, 01) and (H2, ∗2, 02) are two hyper BCK-algebras. 2010 Mathematics Subject Classifications: 06F35, 03G25 Key Words and Phrases: Hyper BCK-algebra, hyper sum, hyper product 1. Introduction Although algebra and topology seem to differ generally in their nature, they appear together in some areas of mathematics such as functional analysis, dynamical systems, and representation theory. Previous studies (see [2]) would show the blend of algebraic and of topological structures. Indeed, there are various ways of introducing a a topological structure in a given algebraic structure. For example, in the definition of a topological group, the requirement imposed is that the topology on a given group is the one that makes the multiplication and inversion maps continuous. However, given an algebraic structure (or hyperstructure), it may be possible to find some family of subsets of the underlying set that will serve as base for some topology on the set. This approach can then give rise to a structure that is both algebraic and topological. The present study considers an algebraic structure which is a decendant of BCK- algebra, an algebraic structure that was introduced and investigated by Y. Imai and K. Iséki [5] in 1966. This variant of BCK-algebra utilizes the hyperstructure theory introduced ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v12i4.3559 Email addresses: rhapsodistchelar@gmail.com (R. Patangan), sergio.canoy@g.msuiit.edu.ph (S. Canoy, Jr.) http://www.ejpam.com 1524 c© 2019 EJPAM All rights reserved. R. Patangan, S. Canoy, Jr. / Eur. J. Pure Appl. Math, 12 (4) (2019), 1524-1532 1525 by F. Marty [7] at the 8th Congress of Scandinavian Mathematicians in 1934. Specifically, Y.B. Jun et al. [6] applied the hyperstructure theory to BCK-algebras and introduced the notion of a hyper BCK-algebra. Recently, Patangan and Canoy [8, 9] showed that the families BL(H) = {LH(A) : ∅ 6= A ⊆ H} and BR(H) = {RH(A) : ∅ 6= A ⊆ H}, where LH(A) = {x ∈ H : x � a, ∀a ∈ A} and RH(A) = {x ∈ H : a � x, ∀a ∈ A} for any subset A of H, are bases for some topologies on a hyper BCK-algebra (H, ∗, 0). Thus, given a hyper BCK-algebra, two different topological structures are generated and investigated. A hyper BCK-algebra is a nonempty set H endowed with a hyperoperation “ ∗ ” and a constant 0 satisfying the following axioms: for all x, y, z ∈ H, (H1) (x ∗ z) ∗ (y ∗ z)� x ∗ y, (H2) (x ∗ y) ∗ z = (x ∗ z) ∗ y, (H3) x ∗H � x, (H4) x� y and y � x imply x = y, where for every A,B ⊆ H, A � B if and only if for each a ∈ A, there exists b ∈ B such that 0 ∈ a ∗ b. In particular, for every x, y ∈ H, x � y if and only if 0 ∈ x ∗ y. In such case, we call “� ” the hyper order in H. Throughout this study, (H1, ∗1, 01) (or simply H1) and (H2, ∗2, 02) (or simply H2) are hyper BCK-algebras. Let H be a hyper BCK-algebra and A ⊆ H. The sets LH(A) and RH(A) are given as follows: LH(A) := {x ∈ H | x� a ∀a ∈ A} = {x ∈ H | 0 ∈ x ∗ a ∀a ∈ A} and RH(A) := {x ∈ H | a� x ∀a ∈ A} = {x ∈ H | 0 ∈ a ∗ x ∀a ∈ A}. If A = {a}, we write LH({a}) = LH(a) and RH({a}) = RH(a). Let (H1, ∗1, 0) and (H2, ∗2, 0) be hyper BCK-algebras such that H1 ∩ H2 = {0} and H = H1 ∪ H2. Then (H, ∗, 0) is a hyper BCK-algebra denoted by H1 ⊕ H2, called the hyper sum, where the hyperoperation “ ∗ ” on H is defined for all x, y ∈ H by, x ∗ y =  x ∗1 y if x, y ∈ H1 x ∗2 y if x, y ∈ H2 {x} otherwise. Let (H1, ∗1, 01) and (H2, ∗2, 02) be hyper BCK-algebras and H = H1 ×H2. Define a hyperoperation “∗” on H as follows: for all (a1, b1), (a2, b2) ∈ H, (a1, b1)∗ (a2, b2) = (a1 ∗1 a2, b1 ∗2 b2). For A ⊆ H1 and B ⊆ H2, by (A,B) we mean (A,B) = {(a, b) : a ∈ A, b ∈ B}, 0 = (01, 02) and (a1, b1) � (a2, b2) ⇐⇒ a1 � a2 and b1 � b2. Then (H, ∗, 0) is a hyper BCK-algebra, and it is called the hyper product of H1 and H2. 2. Known Results Proposition 2.1. [1] Let A and B be subsets of a hyper BCK-algebra H. Then the following hold: R. Patangan, S. Canoy, Jr. / Eur. J. Pure Appl. Math, 12 (4) (2019), 1524-1532 1526 (i) LH(∅) = H. (ii) LH(A) = ⋂ a∈A LH(a). (iii) For any A ⊆ H, 0 ∈ LH(A). If 0 ∈ A, then LH(A) = {0}. Proposition 2.2. [8] Let H be a hyper BCK-algebra and A ⊆ H. Then the following hold: (i) RH(A) = ⋂ a∈A RH(a). (ii) For any ∅ 6= A ⊆ H such that A 6= {0}, 0 /∈ RH(A). (iii) RH(x) 6= ∅ ∀x ∈ H. In particular, x ∈ RH(x). Furthermore, RH(x) = H if and only if x = 0 ∀x ∈ H. Theorem 2.3. [9] The family BL(H) = {LH(A) : ∅ 6= A ⊆ H} where H is a hyper BCK- algebra, is a basis for some topology on H. Theorem 2.4. [8] The family BR(H) = {RH(A) : ∅ 6= A ⊆ H} where H is a hyper BCK- algebra, is a basis for some topology on H. 3. Bases of τL(H1 ⊕H2) and τR(H1 ⊕H2) Theorem 3.1. Let H be a hyper sum of hyper BCK-algebras H1 and H2 with |H1| ≥ 2 and |H2| ≥ 2. Then BL(H) = BL(H1 ⊕H2) = BL(H1) ∪ BL(H2). Proof: Since BL(H1) ⊆ BL(H) and BL(H2) ⊆ BL(H), it follows that BL(H1) ∪ BL(H2) ⊆ BL(H). Next, let V ∈ BL(H). Then there exists a nonempty set B ⊆ H such that V = LH(B). Let B1 = B ∩H1 and B2 = B ∩H2. If V = {0}, then by Proposition 2.1(iii), V = LH1(0) ∈ BL(H1)∪BL(H2). So, suppose that V 6= {0}. Suppose further that B1 6= ∅ and B2 6= ∅. Choose x, y ∈ B such that x ∈ B1 and y ∈ B2. Pick u ∈ V \ {0}. Then u � x and u � y. If u ∈ H1, then u ∗ y = {u}. If u ∈ H2, u ∗ x = {u}. In both cases, we get a contradiction since u 6= 0. Therefore, either B1 = ∅ or B2 = ∅, say B2 = ∅. Then B = B1 ⊆ H1. Hence, V = LH(B) = LH1(B) ∈ BL(H1) ∪ BL(H2). Therefore, BL(H) = BL(H1) ∪ BL(H2). Theorem 3.2. Let H be a hyper sum of hyper BCK-algebras H1 and H2 with |H1| ≥ 2 and |H2| ≥ 2. Then BR(H) \ {∅, H} = (BR(H1) ∪ BR(H2)) \ {∅, H1, H2}. Proof: Let P ∈ BR(H1) \ {∅, H1}. Since P 6= H1, by Proposition 2.2(iii), there exists a nonempty set A ⊆ H1 \ {0} such that P = RH1(A). But A ⊆ H1 \ {0} ⊆ H, thus, P = RH1(A) = RH(A). Since A 6= {0} and P 6= ∅, by Theorem 2.2(iii) and definition of a hyper sum, P 6= H and P 6= ∅ in H. Consequently, P = RH(A) ∈ BR(H) \ {∅, H}. Similarly, if Q ∈ BR(H2) \ {∅, H2} then Q 6= H and Q 6= ∅ in H. Hence, Q = RH(B) ∈ BR(H) \ {∅, H}. Accordingly, (BR(H1) ∪ BR(H2)) \ {∅, H1, H2} ⊆ BR(H) \ {∅, H}. R. Patangan, S. Canoy, Jr. / Eur. J. Pure Appl. Math, 12 (4) (2019), 1524-1532 1527 Next, let U ∈ BR(H) \ {∅, H}. Since U 6= H, by Proposition 2.2(iii), there exists a nonempty subset D ⊆ H \ {0} such that U = RH(D). Let D1 = D ∩ (H1 \ {0}) and D2 = D ∩ (H2 \ {0}). Suppose that D1 6= ∅ and D2 6= ∅. Choose any x ∈ D1 and any y ∈ D2. Since x, y ∈ D, it follows that x � u and y � u for all u ∈ U . Pick w ∈ U . By the definition of a hyper sum, if w ∈ H1, then y ∗ w = {y} and if w ∈ H2, then x ∗w = {x}. Since x and y are nonzero, y 6� w and x 6� w, a contradiction. Thus, either D1 = ∅ or D2 = ∅, that is, either D = D1 or D = D2. If D = D1 then U = RH1(D1). Since D1 6= {0} and U 6= ∅ in H, by Theorem 2.2(iii), U 6= H1 and U 6= ∅ in H1. Thus, U = RH1(D1) ∈ BR(H1) \ {∅, H1} ⊆ [BR(H1) ∪ BR(H2)] \ {∅, H1, H2}. In the same way, if D = D2 then U = RH2(D2) ∈ BR(H2) \ {∅, H2} ⊆ [BR(H1) ∪ BR(H2)] \ {∅, H1, H2}. Therefore, BR(H) \ {∅, H} = (BR(H1) ∪ BR(H2)) \ {∅, H1, H2}. 4. Bases of τL(H1 ×H2) and τR(H1 ×H2) For any ∅ 6= D ⊆ H1×H2, the H1-projection and H2-projection of D are, respectively, the sets DH1 = {x ∈ H1 : (x, y) ∈ D for some y ∈ H2} and DH2 = {y ∈ H1 : (z, y) ∈ D for some z ∈ DH1}. Now, for each x ∈ S = DH1 , let Tx = {y ∈ DH2 : (x, y) ∈ D}. Then D = ⋃ x∈S [{x} × Tx]. Lemma 4.1. Let {Aα : α ∈ I} be a collection of subsets of a hyper BCK-algebra H. Then ⋂ α∈I LH(Aα) = LH (⋃ α∈I Aα ) . Proof: Let {Aα : α ∈ I} be a collection of subsets of H. Then x ∈ ⋂ α∈I LH(Aα)⇔ x ∈ LH(Aα) for all α ∈ I ⇔ x� a for all a ∈ Aα and for all α ∈ I ⇔ x� a for all a ∈ ⋃ α∈I Aα ⇔ x ∈ LH (⋃ α∈I Aα ) . Therefore, the equality is true. Theorem 4.2. Let H be a hyper product of hyper BCK-algebras H1 and H2. Then the following properties hold: (i) LH(A×B) = LH1(A)× LH2(B) for A ⊆ H1 and B ⊆ H2. (ii) If {Aα : α ∈ I} and {Bα : α ∈ I} are collections of subsets of H1 and H2, respectively, then⋂ α∈I [LH1(Aα)× LH2(Bα)] = LH1 (⋃ α∈I Aα ) × LH2 (⋃ α∈I Bα ) . R. Patangan, S. Canoy, Jr. / Eur. J. Pure Appl. Math, 12 (4) (2019), 1524-1532 1528 (iii) If D = ⋃ x∈S ({x} × Tx), where S ⊆ H1 and Tx ⊆ H2 for each x ∈ S, then LH(D) = ⋂ x∈S (LH1(x)× LH2(Tx)) = LH1(S)× LH2( ⋃ x∈S Tx). Proof: (i) Let A and B be subsets of H1 and H2, respectively. Then LH(A×B) = {(x, y) ∈ H1 ×H2 : (x, y)� (a, b) for all (a, b) ∈ A×B} = {(x, y) ∈ H1 ×H2 : x� a and y � b ∀a ∈ A and b ∈ B} = {x ∈ H1 : x� a ∀a ∈ A} × {y ∈ H2 : y � b ∀b ∈ B} = LH1(A)× LH2(B). (ii) Let {Aα : α ∈ I} and {Bα : α ∈ I} be collections of subsets of H1 and H2, respectively, and let K = ⋂ α∈I [LH1(Aα)× LH2(Bα)]. Then (x, y) ∈ K ⇔ (x, y) ∈ LH1(Aα)× LH2(Bα) ∀α ∈ I ⇔ x ∈ LH1(Aα) and y ∈ LH2(Bα) ∀α ∈ I ⇔ x� a ∀a ∈ Aα and y � b ∀b ∈ Bα and ∀α ∈ I ⇔ x� a ∀a ∈ ⋃ α∈I Aα and y � b ∀b ∈ ⋃ α∈I Bα ⇔ x ∈ LH1 (⋃ α∈I Aα ) and y ∈ LH2 (⋃ α∈I Bα ) ⇔ (x, y) ∈ LH1 (⋃ α∈I Aα ) × LH2 (⋃ α∈I Bα ) . Therefore, the assertion is true. (iii) Let D = ⋃ x∈S ({x}×Tx), where S ⊆ H1 and Tx ⊆ H2 for each x ∈ S. Then by Lemma 4.1, (i), and (ii), LH(D) = LH [⋃ x∈S ({x} × Tx) ] = ⋂ x∈S [LH({x} × Tx)] = ⋂ x∈S [LH1(x)× LH2(Tx)] = LH1(S)× LH2 (⋃ x∈S Tx ) . R. Patangan, S. Canoy, Jr. / Eur. J. Pure Appl. Math, 12 (4) (2019), 1524-1532 1529 Theorem 4.3. Let H be a hyper product of hyper BCK-algebras H1 and H2. Then BL(H) = BL(H1)× BL(H2). Proof: Let U ∈ BL(H). Then there exists a nonempty set D ⊆ H = H1 ×H2 such that U = LH(D). Let D = ⋃ x∈S ({x} × Tx) where S ⊆ H1 and Tx ⊆ H2 for each x ∈ S. Then LH(D) = LH1(S)×LH2( ⋃ x∈S Tx) by Theorem 4.2(iii). Hence, U ∈ BL(H1)×BL(H2), showing that BL(H) ⊆ BL(H1) × BL(H2). Next, let V ∈ BL(H1) × BL(H2). Then there exist nonempty sets A ⊆ H1 and B ⊆ H2 such that V = LH1(A)×LH2(B) = LH(A×B) ∈ BL(H) by Theorem 4.2(i). Thus, BL(H1) × BL(H2) ⊆ BL(H). Therefore, BL(H) = BL(H1)× BL(H2). Lemma 4.4. Let {Aα : α ∈ I} be a collection of subsets of a hyper BCK-algebra H. Then ⋂ α∈I RH(Aα) = RH (⋃ α∈I Aα ) . Proof: Let {Aα : α ∈ I} be a collection of subsets of H. Then x ∈ ⋂ α∈I RH(Aα)⇔ x ∈ RH(Aα) for all α ∈ I ⇔ a� x for all a ∈ Aα and for all α ∈ I ⇔ a� x for all a ∈ ⋃ α∈I Aα ⇔ x ∈ RH (⋃ α∈I Aα ) . Therefore, the equality holds. Theorem 4.5. Let H be a hyper product of hyper BCK-algebras H1 and H2. Then the following properties hold: (i) RH(A×B) = RH1(A)×RH2(B) for A ⊆ H1 and B ⊆ H2. (ii) If {Aα : α ∈ I} and {Bα : α ∈ I} are collections of subsets of H1 and H2, respectively, then⋂ α∈I [RH1(Aα)×RH2(Bα)] = RH1 (⋃ α∈I Aα ) ×RH2 (⋃ α∈I Bα ) . (iii) If E = ⋃ x∈P ({x} × Tx), where P ⊆ H1 and Tx ⊆ H2 for each x ∈ P , then RH(E) = ⋂ x∈P (RH1(x)×RH2(Tx)) = RH1(P )×RH2( ⋃ x∈P Tx). Proof: R. Patangan, S. Canoy, Jr. / Eur. J. Pure Appl. Math, 12 (4) (2019), 1524-1532 1530 (i) Let A and B be subsets of H1 and H2, respectively. Then RH(A×B) = {(x, y) ∈ H1 ×H2 : (a, b)� (x, y) for all (a, b) ∈ A×B} = {(x, y) ∈ H1 ×H2 : a� x and b� y ∀a ∈ A and b ∈ B} = {x ∈ H1 : a� x ∀a ∈ A} × {y ∈ H2 : b� y ∀b ∈ B} = RH1(A)×RH2(B). (ii) Let {Aα : α ∈ I} and {Bα : α ∈ I} be collections of subsets of H1 and H2, respectively, and let Q = ⋂ α∈I [RH1(Aα)×RH2(Bα)]. Then (x, y) ∈ Q⇔ (x, y) ∈ RH1(Aα)×RH2(Bα) ∀α ∈ I ⇔ x ∈ RH1(Aα) and y ∈ RH2(Bα) ∀α ∈ I ⇔ a� x ∀a ∈ Aα and b� y ∀b ∈ Bα and ∀α ∈ I ⇔ a� x ∀a ∈ ⋃ α∈I Aα and b� y ∀b ∈ ⋃ α∈I Bα ⇔ x ∈ RH1 (⋃ α∈I Aα ) and y ∈ RH2 (⋃ α∈I Bα ) ⇔ (x, y) ∈ RH1 (⋃ α∈I Aα ) ×RH2 (⋃ α∈I Bα ) . Therefore, the equality holds. (iii) Let E = ⋃ x∈P ({x} × Tx), where P ⊆ H1 and Tx ⊆ H2 for each x ∈ P . Then by Lemma 4.4, (i), and (ii), RH(E) = RH [⋃ x∈P ({x} × Tx) ] = ⋂ x∈P [RH({x} × Tx)] = ⋂ x∈P [RH1(x)×RH2(Tx)] = RH1(P )×RH2 (⋃ x∈P Tx ) . Theorem 4.6. Let H be a hyper product of hyper BCK-algebras H1 and H2. Then BR(H) = BR(H1)× BR(H2). Proof: Let D ∈ BR(H). Then there exists a nonempty set E ⊆ H = H1 ×H2 such that D = RH(E). Let E = ⋃ x∈P ({x} × Tx) where P ⊆ H1 and Tx ⊆ H2 for each x ∈ P . REFERENCES 1531 Then LH(E) = RH1(P ) × RH2( ⋃ x∈P Tx) ∈ BR(H1) × BR(H2) by Theorem 4.5(iii). Hence, BR(H) ⊆ BR(H1)×BR(H2). Next, suppose that F ∈ BR(H1)×BR(H2). Then there exist nonempty sets O ⊆ H1 and U ⊆ H2 such that F = RH1(O) × RH2(U) = RH(O × U) by Theorem 4.5(i). Thus, F ∈ BR(H), showing that BR(H1)×BR(H2) ⊆ BR(H). Therefore, BR(H) = BR(H1)× BR(H2). Conclusion: This study shows that, indeed, a topological structure may be generated from a given (hyper) algebraic structure by considering some family of subsets of the underlying set of the structure that would qualify as a base for some topology on the set. The topology generated in this way need not coincide with the topology for which continuity is imposed on some hyperoperations associated with the algebraic structure. In this study, the authors, using the construction of a topological structure they introduced, are able to determine the bases of the topologies generated by the hyper sum and hyper product of two hyper BCK-algebras. Acknowledgements This research is funded by the Philippine Department of Science and Technology - Accelerated Science and Technology Human Resource Development Program (DOST- ASTHRDP) and MSU-Iligan Institute of Technology. References [1] J Albaracin and J Vilela. Zero Divisor Graph of Finite Hyper BCK-algebra involving hyperatoms. Far East Journal of Mathematical Sciences, 103(4): 743-755, 2018. [2] A Arhangel’skii and M Tkachenko. Topological Groups and Related Structures. World Scientific, 2008. [3] R Borzooie, A Hasankhani, M Zahedi, and Y Jun. On Hyper K-algebras. Mathemat- icae Japonicae, 52(1): 113-121, 2000. [4] H Harizavi. On Direct Sum of Branches in Hyper BCK-algebras. Iranian Journal of Mathematical Sciences and Informatics, 11(2): 43-55, 2016. [5] Y Imai and K Iséki. On Axiom systems of Propositional Calculi XIV. Proc. Japan Academy, 42: 19-22, 1966. [6] Y Jun, M Zahedi, X Xin, and R Borzooei. On Hyper BCK-algebras. 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