EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 13, No. 4, 2020, 730-738 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Another Look at Topological BCH-algebras Jemil D. Mancao1,∗, Sergio R. Canoy, Jr.1 1 Department of Mathematics and Statistics, College of Science and Mathematics, Center of Graph Theory, Algebra and Analysis, Premier Research Institute of Science and Mathematics, Mindanao State University-Iligan Institute of Technology, 9200 Iligan City, Philippines Abstract. A BCH-algebra (H, ∗, 0) furnished with a topology τ on H (also called a BCH-topology on H) is called a topological BCH-algebra (or TBCH-algebra) if the function ∗ : H × H → H, defined by ∗((x, y)) = x ∗ y for any x, y ∈ H, is continuous, where the Cartesian product topology on H × H is furnished by τ . In this paper, we give other structural properties of topological BCH-algebras. 2020 Mathematics Subject Classifications: 06F35, 03G25 Key Words and Phrases: BCH-algebra, topology, TBCH-algebra, separation axioms 1. Introduction In 1983, Hu and Li [5, 6] introduced the notion of a BCH-algebra which is a generalization of BCK and BCI-algebras. In the same paper, the concept of associative BCH-algebra was also introduced. Dar, K. H., and Akram, M. [2] defined the concepts of BCH-ideal, BCH-subalgebra, ∗-commutative, left and right mappings on a BCH-algebra and some properties structures were investigated. In [8] and [4], the concepts of topological BCK-algebra and topological BCI-algebra were defined and some properties of each newly defined concepts were investigated. In 2017, M. Jansi and V. Thiruveni [7] introduced the concept of topological BCH-algebra (or TBCH-algebra) and investigated some of its algebraic and topological properties. The aim of this paper is to give other structural properties of topological BCH-algebras. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v13i4.3842 Email addresses: jemil.mancao@g.msuiit.edu.ph (J. Mancao), sergio.canoy@g.msuiit.edu.ph (S. Canoy) https://www.ejpam.com 730 c© 2020 EJPAM All rights reserved. J. Mancao, S. Canoy / Eur. J. Pure Appl. Math, 13 (4) (2020), 730-738 731 2. Preliminaries and Known Results Definition 1. [3] Let (X, τ) be a topological space and let x ∈ X. Any set U ∈ τ containing x is called a neighborhood (sometimes written as nbhd or τ -nbhd) of x. Definition 2. [3] Let (X, τ) be a topological space. Then (i) (X, τ) is a T0-space if for any x, y ∈ X with x 6= y, there exists an open set U containing one but not the other; (ii) (X, τ) is a T1-space if for any x, y ∈ X with x 6= y, there exist nbhds U and V of x and y, respectively, such that x /∈ V and y /∈ U ; (iii) (X, τ) is a T2-space (or Hausdorff space) if for any x, y ∈ X with x 6= y, there exist disjoint nbhds U and V of x and y, respectively. Remark 1. [3] T2 ⇒ T1 ⇒ T0 but not conversely. Theorem 1. [3] Let (X, τ) be a topological space. X is a T1-space if and only if for each x ∈ X, {x} is a closed set in X. Definition 3. [5] A BCH-algebra is a nonempty set H endowed with a operation “ ∗ ” and constant 0 satisfying the following axioms: for all x, y, z ∈ H, (B1) x ∗ x = 0, (B2) x ∗ y = 0 and y ∗ x = 0 implies x = y. (B3) (x ∗ y) ∗ z = (x ∗ z) ∗ y, Remark 2. [5, 6] In any BCH-algebra (X, ∗, 0), the following hold: (i) x ∗ 0 = x; (ii) x ∗ 0 = 0 implies x = 0; (iii) 0 ∗ (x ∗ y) = (0 ∗ x) ∗ (0 ∗ y); (iv) (x ∗ (x ∗ y)) ∗ y = 0. Definition 4. [7] Let (X, ∗, 0) be a BCH-algebra and U, V be any nonempty subsets of X. We define a subset U ∗ V of X by U ∗ V = {x ∗ y : x ∈ U, y ∈ V }. Remark 3. Let (X, ∗, 0) be a BCH-algebra. Then ∗(A × B) = A ∗ B for any nonempty subsets A and B of X. Remark 4. Let (X, ∗, 0) be a BCH-algebra and A,B ⊆ X. If A∩B 6= ∅, then 0 ∈ A ∗B. Definition 5. [2] Let (X, ∗, 0) be a BCH-algebra. A nonempty subset S of X is a BCH- subalgebra if for each x, y ∈ S, x ∗ y ∈ S. J. Mancao, S. Canoy / Eur. J. Pure Appl. Math, 13 (4) (2020), 730-738 732 Definition 6. [7] Let (H, ∗, 0) be a BCH-algebra. A topology τ furnished on H is called a BCH-topology on H. In addition, (H, τ) is called a topological BCH-algebra (or TBCH- algebra) if τ is a BCH-topology on H and the function ∗ : H × H → H defined as ∗((x, y)) = x∗y is continuous, where the Cartesian product topology on H×H is furnished by τ . Example 1. Let X = {0, 1, 2, 3, 4} and define ∗ as follows: ∗ 0 1 2 3 4 0 0 0 0 0 4 1 1 0 0 1 4 2 2 2 0 0 4 3 3 3 3 0 4 4 4 4 4 4 0 Then, (X, ∗, 0) is a BCH-algebra [1]. Let τ = {X,∅, {4} , {0, 1, 2, 3}}. Then τ is a BCH-topology on X. Moreover, ∗−1(X) = X ×X ∗−1(∅) = ∅ ∗−1({4}) = ({0, 1, 2, 3} × {4}) ∪ ({4} × {0, 1, 2, 3}) ∗−1({0, 1, 2, 3}) = ({0, 1, 2, 3} × {0, 1, 2, 3}) ∪ ({4} × {4}). This implies that ∗ is continuous. Thus, (X, τ) is a TBCH-algebra. 3. Results Throughout this study, we denote a BCH-algebra (X, ∗, 0) by X, unless otherwise specified. Theorem 2. Let τ be a BCH-topology on X. Then, (X, τ) is a TBCH-algebra if and only if for each x, y ∈ X and each nbhd W of x ∗ y, there exist nbhds U and V of x and y, respectively, such that U ∗ V ⊆W . Proof. Let X be a TBCH-algebra. Let x, y ∈ X and a nbhd W of x ∗ y. Since ∗ is continuous, ∗−1(W ) is a nbhd of (x, y) in X × X. By definition of Cartesian product topology, there exist nbhds U and V of x and y, respectively, such that U × V ⊆ ∗−1(W ). By Remark 3, U ∗ V = ∗(U × V ). It follows that U ∗ V ⊆ ∗(∗−1(W )) ⊆W . Conversely, suppose that for each x, y ∈ X and each nbhd W of x ∗ y, there are nbhds U and V of x and y, respectively, such that U ∗V ⊆W . By definition of Cartesian product topology, U × V is a nbhd of (x, y) in X × X. By Remark 3, ∗(U × V ) = U ∗ V ⊆ W . Therefore, ∗ is continuous. Corollary 1. Let X be a TBCH-algebra and A ⊆ X. If z is an interior point of A, then there exist elements x, y ∈ X and nbhds Nx, Ny and Nz of x, y and z, respectively, such that z = x ∗ y and Nx ∗Ny ⊆ Nz = Nx∗y. J. Mancao, S. Canoy / Eur. J. Pure Appl. Math, 13 (4) (2020), 730-738 733 Proof. Suppose z is an interior point of A. Then there exists a nbhd Nz of z such that Nz ⊆ A. Since z ∈ X, z = x ∗ y for some x, y ∈ X (say, x = z and y = 0). By Theorem 2, there exist nbhds Nx and Ny of x and y, respectively, such that Nx ∗Ny ⊆ Nz = Nx∗y. The next theorem asserts that the topology associated in a TBCH-algebra having {0} as an open set is the discrete topology. Theorem 3. Let X be a TBCH-algebra. Then {0} is an open set in X if and only if X is a discrete space. Proof. Suppose that {0} is an open set in X and let x ∈ X. Then, x ∗ x = 0 ∈ {0} by (B1). Since {0} is an open set in X, there exist nbhds U and V of x such that U ∗V = {0} by Theorem 2. Let W = U ∩ V . Then, W is a nbhd of x and W ∗W ⊆ U ∗ V . Hence, W ∗W = {0}. Let y ∈ W . Then x ∗ y = 0 = y ∗ x. By (B2), y = x. Thus, W = {x}, showing that X is a discrete space. Conversely, suppose X is the discrete space. Then, {0} is an open set in X. Corollary 2. If {0} is an open set in a TBCH-algebra X, then every subset of X is both open and closed set in X. In particular, if |X| ≥ 2, then X is a disconnected space. Remark 5. If a BCH-topological space X is a discrete space, then X is a TBCH-algebra. We now show that a BCH-subalgebra of a TBCH-algebra is also a TBCH-algebra. Theorem 4. Let X be a TBCH-algebra and H a BCH-subalgebra of X. Then (H, τH) is a TBCH-algebra, where τH is the relative topology on H. Proof. Let x, y ∈ H and a nbhd WH of x∗y in the subspace H. Note that WH may be written as the intersection with H of some nbhd W of x ∗ y in X, that is, WH = H ∩W . Since X is a TBCH-algebra, there exist nbhds U and V of x and y, respectively, such that U ∗ V ⊆ W by Theorem 2. Observe that UH = H ∩ U and VH = H ∩ V are nbhds of x and y, respectively, in the subspace H. Furthermore. UH ∗ VH = (H ∩ U) ∗ (H ∩ V ) ⊆ U ∗ V ⊆W. Since H is a BCH -subalgebra, UH ∗ VH ⊆ H ∗H ⊆ H so that UH ∗ VH ⊆ H ∩W = WH . By Theorem 2, (H, τH) is a TBCH-algebra. J. Mancao, S. Canoy / Eur. J. Pure Appl. Math, 13 (4) (2020), 730-738 734 Theorem 5. Let (H1, ∗1, 0) and (H2, ∗2, 0) be BCH-algebras such that H1∩H2 = {0} and H = H1 ∪H2. Then (H, ∗, 0) is a BCH-algebra, denoted by H1 ⊕H2, where the operation “ ∗ ” 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. Proof. Let x ∈ H. Then x ∗ x = { x ∗1 x if x ∈ H1 x ∗2 x if x ∈ H2. Since (H1, ∗1, 0) and (H2, ∗2, 0) are BCH-algebras, x ∗ x = 0 by property (B1). Next, let x, y ∈ H and suppose that x ∗ y = 0 and y ∗ x = 0. Consider the following cases: Case 1: x, y ∈ H1 (or x, y ∈ H2). Then x ∗ y = x ∗1 y = 0 and y ∗ x = y ∗1 x = 0. Since (H1, ∗1, 0) is a BCH-algebra, property (B2) yields x = y. Similarly, x = y if x, y ∈ H2. Case 2: x ∈ H1 and y ∈ H2 (or y ∈ H1 and x ∈ H2). Then 0 = x ∗ y = x and 0 = y ∗ x = y. Hence, x = 0 = y. Finally, let x, y, z ∈ H. Consider the following cases: Case 1: x, y ∈ H1 (or x, y ∈ H2) Then, by the definition of ∗, (x ∗ y) ∗ z = { (x ∗1 y) ∗1 z if z ∈ H1 x ∗1 y if z ∈ H2. and (x ∗ z) ∗ y = { (x ∗1 z) ∗1 y if z ∈ H1 x ∗1 y if z ∈ H2. Since (H1, ∗1, 0) is a BCH-algebra, (x∗1 y)∗1 z = (x∗1 z)∗1 y if z ∈ H1. Hence, (x∗y)∗z = (x ∗ z) ∗ y. Similarly, (x ∗ y) ∗ z = (x ∗ z) ∗ y whenever x, y ∈ H2. Case 2: x ∈ H1 and y ∈ H2 (or y ∈ H1 and x ∈ H2) Then, by the definition of ∗, (x ∗ y) ∗ z = { x ∗1 z if z ∈ H1 x if z ∈ H2. and (x ∗ z) ∗ y = { x ∗1 z if z ∈ H1 x if z ∈ H2. J. Mancao, S. Canoy / Eur. J. Pure Appl. Math, 13 (4) (2020), 730-738 735 Therefore, (x ∗ y) ∗ z = (x ∗ z) ∗ y. Equality is also obtained if y ∈ H1 and x ∈ H2. Accordingly, (H, ∗, 0) is a BCH-algebra. Lemma 1. Let (H, ∗1) and (H2, ∗2) be BCH-algebras such that H1 ∩ H2 = {0} and let (H, ∗) be the sum of H1 and H2 defined in Theorem 5. Then each of the following holds: (i) If U and V are subsets of H1 (U and V are subsets of H2), then U ∗1 V = U ∗ V (resp. U ∗2 V = U ∗ V ). (ii) If A,B ⊆ H1, C ⊆ H2, and 0 ∈ B, then A ⊆ A∗B and A∗(B∪C) = A∗1B = A∗B. Proof. (i) Suppose U and V are subsets ofH1. Let x ∈ U and y ∈ V . Since x∗y = x∗1y, x∗y ∈ U ∗V if and only if x∗1y ∈ U ∗1V . Hence, U ∗1V = U ∗V . Similarly, U ∗2V = U ∗V if U and V are subsets of H2. (ii) Let x ∈ A. Then x = x ∗ 0 ∈ A ∗B since 0 ∈ B. Hence, A ⊆ A ∗B = A ∗1 B. To establish the equality, first note that A ∗1B = A ∗B ⊆ A ∗ (B ∪C). Let a ∈ A and x ∈ (B∪C). If x ∈ B, then a∗x = a∗1 x ∈ A∗1B. If x ∈ C, then a∗x = a ∈ A ⊆ A∗1B. Thus, A ∗ (B ∪ C) = A ∗1 B = A ∗B. Theorem 6. Let (H1, ∗1, 0) and (H2, ∗2, 0) be BCH-algebras such that H1∩H2 = {0} and let (H, ∗, 0) be the sum of H1 and H2 (defined in Theorem 5). Then each of the following holds: (i) (H1, ∗1, 0) and (H2, ∗2, 0) are BCH-subalgebras of H. (ii) (H, τH1) and (H, τH2) are TBCH-algebras, where τH1 = {∅, H1∪H2, H1} and τH2 = {∅, H1 ∪H2, H2}. (iii) If (H, τ) is a TBCH-algebra and A,B ∈ τ for some set A ⊆ H1 and B ⊆ H2 with 0 ∈ A ∩ B, then τ is the discrete topology on H. In particular, if H1, H2 ∈ τ , then τ is the discrete topology on H. (iv) If (H, τ) is a TBCH-algebra and τ ⊆ P (H1)∪{H1∪H2} (or τ ⊆ P (H2)∪{H1∪H2}), where P (H1) and P (H2) are the power sets of H1 and H2, respectively, then 0 ∈W for every W ∈ τ \ {∅}. Proof. (i) Let x, y ∈ H1. Then x ∗ y = x ∗1 y ∈ H1 by Theorem 5 and the fact that (H1, ∗1, 0) is a BCH-algebra. Therefore, (H1, ∗1, 0) = (H1, ∗, 0) is a BCH-subalgebra of H. Similarly, (H2, ∗2, 0) is a BCH-subalgebra of H. (ii) Clearly, τH1 are τH2 are topologies on H. First, consider the space (X, τH1). Let x, y ∈ H and let W be a τH1-nbhd of x ∗ y. Consider the following cases: Case 1: x, y ∈ H1 Then x ∗ y = x ∗1 y ∈ H1. Hence, W = H1 or W = H1 ∪H2. Then H1 is a τH1-nbhd of both x and y, and by Lemma 1(i), H1 ∗H1 = H1 ∗1 H1 = H1 ⊂ H1 ∪H2. Case 2: x, y ∈ H2 or [x ∈ H2 and y ∈ H1] J. Mancao, S. Canoy / Eur. J. Pure Appl. Math, 13 (4) (2020), 730-738 736 If x, y ∈ H2, then x ∗ y = x ∗2 y ∈ H2. Hence, W = H1 ∪H2. The set V = H1 ∪H2 is a τH1-nbhd of both x and y, and V ∗ V = H1 ∪ H2. If x ∈ H2 and y ∈ H1, then x ∗ y = x ∈ H2. Again, W = H1 ∪H2, V = H1 ∪H1 is a τH1-nbhd of both x and y, and V ∗ V = H1 ∪H2. Case 3: x ∈ H1 and y ∈ H2 Then x ∗ y = x ∈ H1. Hence, W = H1 or W = H1 ∪ H2. Let Vx = H1 and Vy = H1 ∪ H2. Then Vx and Vy are τH1-nbhds of x and y, respectively, and by Lemma 1(ii), Vx ∗ Vy = H1 ∗ (H1 ∪H2) = H1 ∗1 H1 = H1 ⊂ H1 ∪H2. Therefore, (H, τH1) is a TBCH algebra. Similarly, (H, τH2) is a TBCH algebra. (iii) Suppose A ⊆ H1, B ⊆ H2, 0 ∈ A ∩ B, and A,B ∈ τ . Since H1 ∩ H2 = {0}, it follows that A∩B = {0}. Since A,B ∈ τ , {0} ∈ τ . Thus, by Theorem 3, τ is the discrete topology on H. (iv) Suppose that (H, τ) is a TBCH-algebra and that τ ⊆ P (H1) ∪ {H1 ∪ H2}. Let W ∈ τ \ {∅}. Pick any x ∈ W and y ∈ H2. Since x ∗ y = x, W is a nbhd of x ∗ y. By continuity of ∗, there exist nbhds Vx and Vy of x and y, respectively, such that Vx∗Vy ⊆W . Now, since τ ⊆ P (H1) ∪ {H1 ∪H2}, the only nbhd of y is H1 ∪H2. Hence, Vy = H1 ∪H2 and by Lemma 1(ii), Vx ∗ Vy = Vx ∗ (H1 ∪H2) = Vx ∗1H1. Since x ∈ H1, x ∗1 x = x ∗ x = 0 ∈ Vx ∗1 H1. Therefore, 0 ∈W . Theorem 7. Let X be a TBCH-algebra. Then {0} is a closed set in X if and only if X is a T2-space. Proof. Suppose {0} is a closed set in X. Let x, y ∈ X with x 6= y. Then, x ∗ y 6= 0 or y ∗x 6= 0. Without loss of generality, assume that x ∗ y 6= 0. Note that x ∗ y ∈ X \ {0}. By Theorem 2, there exist nbhds U and V of x and y, respectively, such that U ∗V ⊆ X \{0}. Suppose U ∩ V 6= ∅. Let z ∈ U ∩ V . Then, z ∈ U and z ∈ V . Hence, by (B1) 0 = z ∗ z ∈ U ∗ V ⊆ X \ {0} a contradiction. Thus, U ∩ V = ∅ and so X is a T2-space. Conversely, assume that X is a T2-space. Let x ∈ X \ {0}. Then, there exist nbhds U and V of x and 0, respectively, such that U ∩ V = ∅. Since 0 /∈ U , x ∈ U ⊆ X \ {0}. This shows that X \ {0} is open in X. Therefore, {0} is a closed set in X. The next theorem asserts that T0, T1 and T2 topological spaces are equivalent in a TBCH-algebra. Theorem 8. Let X be a TBCH-algebra. Then the following statements are equivalent: (i) X is a T0-space (ii) X is a T1-space (iii) X is a T2-space. J. Mancao, S. Canoy / Eur. J. Pure Appl. Math, 13 (4) (2020), 730-738 737 Proof. (i)⇒(ii): Suppose X is a T0-space. Let x, y ∈ X with x 6= y. Then x ∗ y 6= 0 or y ∗ x 6= 0 by (B2). Without loss of generality, assume that x ∗ y 6= 0. Since X is a T0-space, there exists an open set U such that x∗y ∈ U but 0 /∈ U or 0 ∈ U but x∗y /∈ U . Consider the following cases: Case 1. x ∗ y ∈ U (but 0 /∈ U) By Theorem 2, there exist nbhdsGx andHy of x and y, respectively, such thatGx∗Hy ⊆ U . Since 0 /∈ U , 0 /∈ Gx ∗Hy. By Remark 4, Gx ∩Hy = ∅. Thus, y /∈ Gx and x /∈ Hy. Case 2. 0 ∈ U (but x ∗ y /∈ U). By (B1), x ∗ x = 0 ∈ U . By Theorem 2, there exist nbhds Nx and Mx of x such that Nx ∗Mx ⊆ U . Since x ∗ y /∈ U , x ∗ y /∈ Nx ∗Mx. It follows that y /∈ Mx. Similarly, since y ∗ y = 0 ∈ U , there exist nbhds Ny and My of y such that Ny ∗My ⊆ U . Since x ∗ y /∈ U , x ∗ y /∈ Ny ∗My. It follows that x /∈ Ny. Hence, there exist nbhds Mx and Ny of x and y, respectively, such that y /∈Mx and x /∈ Ny. Therefore, X is a T1-space. (ii)⇒(iii): Suppose X is a T1-space. By Theorem 1, {0} is a closed set in X. By Theorem 7, X is a T2-space. By Remark 1, T2 ⇒ T1 ⇒ T0. Therefore, (i), (ii), and (iii) are equivalent. The following corollary follows from Theorems 7 and 8. Corollary 3. Let X be a TBCH-algebra. Then the following statements are equivalent: (i) X is a T0-space (ii) X is a T1-space (iii) X is a T2-space (iv) {0} is a closed set in X. Theorem 9. Let X be a TBCH-algebra. Then X is a T2-space if and only if for any x ∈ X with x 6= 0, there exists a nbhd U of x such that 0 /∈ U . Proof. Clearly, if X is a T2-space, then for any x ∈ X with x 6= 0, there exists a nbhd U of x such that 0 /∈ U . For the converse, suppose that for any x ∈ X with x 6= 0, there exists a nbhd U of x such that 0 /∈ U . Let a, b ∈ X with a 6= b. Then a∗b 6= 0 or b∗a 6= 0 by (B2). Without loss of generality, assume that a ∗ b 6= 0. Then, by assumption, there exists a nbhd W of a ∗ b such that 0 /∈ W . By Theorem 2, there exist nbhds Wa and Wb of a and b, respectively, such that Wa ∗Wb ⊆W . Since 0 /∈W , 0 /∈Wa ∗Wb. By Remark 4, Wa ∩Wb = ∅. Thus, X is a T2-space. Conclusion: Given two BCH-algebras H1 and H2 such that H1∩H2 = {0}, an operation “ ∗ ” can be defined on H = H1 ∪H2 so that (H, ∗) is a BCH-algebra and H1 and H2 are BCH-subalgebras. Further, it is shown that T0, T1 and T2 axioms are equivalent in any topological BCH-algebra. REFERENCES 738 Acknowledgements The authors would like to thank the referees for reviewing the initial paper and for the invaluable comments and suggestions that eventually led to this much improved version of the work. This research is funded by the Philippine Department of Science and Technology - Accelerated Science and Technology Human Resource Development Program (DOST- ASTHRDP). References [1] M.A. Chaudhry and H. Fakhar-Ud-Din. On some classes of BCH-algebras. Interna- tional Journal of Mathematics and Mathematical Sciences, 25(3):205–211, 2001. [2] K.H. Dar and M. Akram. On endomorphisms of BCH-algebras. Annals of the Univer- sity of Craiova-Mathematics and Computer Science Series, 33:227–234, 2006. [3] J. Dugundji. Topology. Allyn and Bacon, Inc., Boston, 1966. [4] Y.B. Jun et al. On Topological BCI-Algebras. Information Sciences, 116(2-4):253–261, 1999. [5] Q.P. Hu and X. Li. On BCH-Algebras. Math. Seminar Notes, 11(2):313–320, 1983. [6] Q.P. Hu and X. Li. On Proper BCH-algebras. Mathematica Japonica, 30(4):659–661, 1985. [7] M. Jansi and V. Thiruveni. Topological structures on BCH-algebras. Mathematica Japonicae, 6:22594–22600, 2017. [8] D. S. Lee and D. N. Ryu. Notes on topological BCK-algebras. Sci. Math, 1:231–235, 1998.