EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 14, No. 2, 2021, 590-600 ISSN 1307-5543 – ejpam.com Published by New York Business Global A Topology on a Hyper BCI-algebra Generated by a Hyper-order Michelle T. Panganduyon1, Sergio R. Canoy, Jr.2, Bijan Davvaz3 1 College of Arts and Sciences, Surigao State College of Technology, 8400 Surigao City, Surigao del Norte, Philippines 2 Department of Mathematics and Statistics, College of Science and Mathematics, Center of Graph Theory, Algebra and Analysis-PRISM, Mindanao State University-Iligan Institute of Technology, 9200 Iligan City, Philippines 3 Department of Mathematics, Yadz University, Yadz, Iran Abstract. In this paper, we introduce an operator on a hyper BCI-algebra via application of a left hyper-order. The family consisting of the images of subsets under the operator turns out to be a base for some topology on the hyper BCI-algebra. We investigate some important properties of the induced topology on certain hyper BCI-algebras. In particular, we show that the generated topology on a non-trivial hyper subalgebra of an ordered hyper BCI-algebra coincides with the relative topology on this hyper subalgebra. 2020 Mathematics Subject Classifications: 20M14, 05C25 Key Words and Phrases: Hyper BCI-algebra, topology, hyper-order, hyperatom 1. Introduction The notion of BCK-algebras was proposed by Y. Imai and K. Iséki in 1966. In the same year, K. Iséki [3] introduced the notion of a BCI-algebra which is a generalization of BCK-algebra. R. A. Alo and E. Y. Deeba [1] attempted to study the topological aspects of the BCK-structures. They studied and investigated various topologies on BCK-algebras analogous to that which had already been studied on lattices. In [4], Y. B. Jun et al. initiated the study of topological BCI-algebras (briefly, TBCI-algebras). In their study a BCI-algebra (H, ∗, 0) is furnished with a topology in such a way that the associated operation ∗ : H ×H → H of the BCI-algebra is continuous, where the Cartesian product H ×H is furnished with the product topology. During the 8th Congress of Scandinavian Mathematicians, F. Marty [6] introduced the theory of hyperstructure (sometimes called multialgebras). Following its introduction, various algebraic hyperstructures have been defined and many important results have DOI: https://doi.org/10.29020/nybg.ejpam.v14i2.3970 Email addresses: mpanganduyon@ssct.edu.ph (M. Panganduyon), sergio.canoy@g.msuiit.edu.ph (S. Canoy, Jr.), bdavvaz@yahoo.com (B. Davvaz) http://www.ejpam.com 590 c© 2021 EJPAM All rights reserved. M. Panganduyon, S. Canoy, Jr., B. Davvaz / Eur. J. Pure Appl. Math, 14 (2) (2021), 590-600 591 appeared. Some recent studies on hyperstructures are on soft hypervector spaces and hyper-deductive systems done by Muhiuddin et al. in [8], [9], and [10]. As one may find, these hyperstructures have many applications in both pure and applied sciences. In [5], Y.B. Jun et al. introduced and studied the concept of a hyper BCK-algebra. In [7], Muhiuddin et al. studied fuzzy soft hyper BCK-ideals in hyper BCK-algebras. In [13], Xin applied hyperstructures to BCI-algebras giving rise to the the concept of a hyper BCI-algebra. A study on a graph induced by a hyper BCI-algebra is done in [11]. Previous studies on the topological aspects of certain algebraic hyperstructures mo- tivated us to study the topological structure of a hyper BCI-algebra when it carries a topology other the one considered in earlier studies. In this study, we purposely use the hyper-order associated with the hyperstructure to topologize it. Specifically, we topologize a given hyper BCI-algebra by considering a family of subsets which will form a base for some topology on the hyper BCI-algebra. These subsets are generated via left applica- tion of the hyper-order associated with the hyper BCI-algebra. Topological properties of the resulting space are investigated in various aspects. In particular, we show that the topology generated on a non-trivial hyper subalgebra of an ordered hyper BCI-algebra coincides with the relative (subspace) topology. 2. Preliminaries A hyperoperation on a nonempty set H is a map from H×H into the nonempty subsets of H, P ∗(H) = P (H) \ {∅}. Let ~ be a hyperoperation on H and (x, y) ∈ H ×H. Then its image under ~, denoted by x~y, is called the hyperproduct of x and y. If A and B are nonempty subsets of H, then A ∗B is given by A~B = ⋃ a∈A,b∈B a~ b. We shall use x~ y instead of x~{y}, {x}~y, or {x}~{y}. When A ⊆ 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 hyper BCI-algebra (H,~, 0) (see [5]) is a nonempty set H endowed with a hyperop- eration “ ~ ” and a constant 0 such that: for all x, y, z ∈ H, (B1) ((x~ z) ~ (y ~ z))� x~ y, (B2) (x~ y) ~ z = (x~ z) ~ y, (B3) x� x, (B4) x� y and y � x imply x = y, (B5) 0 ~ (0 ~ x)� x, x 6= 0, 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. A hyper BCI-algebra (H,~, 0) is said to be ordered if for x, y, z ∈ H, x� y and y � z implies x� z. All throughout, we denote a hyper BCI-algebra (H,~, 0) by H, unless otherwise spec- ified. M. Panganduyon, S. Canoy, Jr., B. Davvaz / Eur. J. Pure Appl. Math, 14 (2) (2021), 590-600 592 Let H be a hyper BCI-algebra and A ⊆ H. In [11], the set LH(A) is given by LH(A) = {x ∈ H | x � a,∀ a ∈ A} = {x ∈ H | 0 ∈ x ~ a,∀ a ∈ A}. If A = {a}, we write LH({a}) = LH(a). An element a of H is called a hyperatom if for each x ∈ H, x � a implies x = 0 or x = a. Denote by A(H) the set of all hyperatoms of H, and by A∗(H) the set of all nonzero hyperatoms of H; that is, A∗(H) = A(H) \ {0}. H is said to be hyperatomic if each element of H is a hyperatom, that is, A(H) = H. It is shown in [11] that H is hyperatomic if and only if LH(x) = {x} or LH(x) = {0, x} for each x ∈ H. 3. Results The following result gives some properties of the operator LH . Proposition 1. [11] Let A and B be subsets of H. Then the following hold: (i) LH(∅) = H (ii) LH({0}) = {0} (iii) If A ⊆ B, then LH(B) ⊆ LH(A). (iv) LH(A) = ⋂ a∈A LH({a}) (v) If x ∈ H, then x ∈ LH({x}). Furthermore, LH({x}) = {0} if and only if x = 0. Theorem 1. [2] Let (X, τ) be a topological space and (Y, τY ) be a subspace. If {Uα |α ∈ A } is a basis (subbasis) for τ , {Y ∩ Uα |α ∈ A } is a basis (subbasis) for τY . Lemma 1. Let {Aα : α ∈ I} be a collection of subsets of a hyper BCI-algebra H. Then ⋂ α∈I LH(Aα) = LH (⋃ α∈I Aα ) . Proof. If ⋂ α∈I LH(Aα) = ∅, then by Proposition 1(iii), LH (⋃ α∈I Aα ) ⊆ ⋂ α∈I LH(Aα) = ∅. Thus, LH (⋃ α∈I Aα ) = ∅. If ⋂ α∈I LH(Aα) 6= ∅, 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 ) . This proves the assertion. M. Panganduyon, S. Canoy, Jr., B. Davvaz / Eur. J. Pure Appl. Math, 14 (2) (2021), 590-600 593 Theorem 2. Let H be a hyper BCI-algebra. Then the family BL(H) = {LH(A) : ∅ 6= A ⊆ H} is a basis for some topology on H. Proof. Clearly, H = ⋃ a∈H LH(a). Let A and B be nonempty subsets of H. Then by Lemma 1, LH(A) ∩ LH(B) = LH(A ∪B) ∈ BL(H). Therefore, BL(H) is a basis for some topology on H. Denote by τL(H) the topology generated by BL(H). Example 1. Consider H := [0,∞) with the hyperoperation “ ~ ”, defined in [13]: x~ y :=  [0, x], if x ≤ y, (0, y], if x > y 6= 0, {x}, if y = 0 for all x, y ∈ H. Then (H,~, 0) is a hyper BCI-algebra. Now, let k ∈ H. Then LH(k) = [0, k]. Let ∅ 6= A ⊆ H and let p = inf A. Since LH(A) = ⋂ a∈A LH(a) = LH(p), it follows that LH(A) = [0, p] = LH(p). Let ∅ 6= G ∈ τL(H). Then G = ⋃ p∈K LH(p), where K ⊆ H. Suppose first that |G| < ∞ and let q = supG. Then G = LH(q). Suppose q > 0. Then G = LH(q) = [0, q], a contradiction. Thus, q = 0, that is, G = LH(0) = {0}. Next, suppose that G is an infinite set. If K is infinite, then G = ⋃ p∈K [0, p] = H. Suppose K is finite. Since G is infinite, 0 < m = maxK. Hence, G = [0,m]. Consequently, τL(H) = {∅, H} ∪ {[0, p] : p ∈ H}. Example 2. Consider H = {0, a, b} with the hyperoperation “ ~ ” defined as follows: ~ 0 a b 0 {0, a} {0, a} {b} a {a} {0, a} {b} b {b} {b} {0, a} Then H is a hyper BCI-algebra. By Theorem 2, BL(H) = {LH(A) : ∅ 6= A ⊆ H} = {{0}, {0, a}, {b},∅}. Thus, τL(H) = {{0}, {0, a}, {b}, {0, b},∅, H}. Observe that in Example 1, (H, τL(H)) is connected, however, in Example 2, H = {0, a} ∪ {b}. Hence, (H, τL(H)) is disconnected. Lemma 2. Let H be an ordered hyper BCI-algebra and let x ∈ H. If z ∈ LH(x), then LH(z) ⊆ LH(x). Proof. Suppose that z ∈ LH(x) and let w ∈ LH(z). Then w � z. Since z � x and H is ordered, w � x; that is, w ∈ LH(x). Therefore, LH(z) ⊆ LH(x). An ordered hyper BCI-algebra H is said to be LH -0 hereditary if 0 ∈ LH(z) for all z ∈ LH(x) whenever x ∈ H with 0 ∈ LH(x). M. Panganduyon, S. Canoy, Jr., B. Davvaz / Eur. J. Pure Appl. Math, 14 (2) (2021), 590-600 594 Example 3. The hyper BCI-algebra H in Example 2 is LH -0 hereditary. Theorem 3. Let H be an LH-0 hereditary hyper BCI-algebra. Then (H, τL(H)) is connected if and only if 0 ∈ LH(x) for all x ∈ H. Proof. Suppose that (H, τL(H)) is connected and suppose that there exists x ∈ H \{0} such that 0 /∈ LH(x). Set D1 = {z ∈ H : 0 /∈ LH(z)} and D2 = H \ D1. Since 0 /∈ LH(x) and 0 ∈ LH(0), D1 6= ∅ and D2 6= ∅. Let z ∈ D1 and let w ∈ LH(z). Then LH(w) ⊆ LH(z) by Lemma 2. Since 0 /∈ LH(z), 0 /∈ LH(w); that is, w ∈ D1. Thus, z ∈ LH(z) ⊆ D1 and so, D1 is τL(H)-open. Next, let y ∈ D2 and let v ∈ LH(y). Since 0 ∈ LH(y) and H is LH -0 hereditary, it follows that 0 ∈ LH(x); that is, v ∈ D2. Hence, y ∈ LH(y) ⊆ D2 and so, D2 is τL(H)-open. Since D1 ∩ D2 = ∅ and D1 ∪ D2 = H, the space is disconnected, contrary to our assumption. For the converse, let G be a non-empty open subset of H. Then there exists A ⊆ H such that LH(A) ⊆ G. Since 0 ∈ LH(x) for all x ∈ H, 0 ∈ LH(A). Thus, 0 ∈ G. It follows that (H, τL(H)) is connected. The next result follows from Theorem 2 and the definition of discrete topology. Proposition 2. Let H be a hyper BCI-algebra. Then τL(H) is the discrete topology D on H if and only if for each x ∈ H, there exists Ax ⊆ H such that LH(Ax) = {x}. Corollary 1. Let H be a hyper BCI-algebra. If LH(x) = {x} for each x ∈ H, then τL(H) is the discrete topology D on H. In particular, BL(H) = {{a} : a ∈ H}. Proof. Suppose that for each x ∈ H, LH(x) = {x}. Then by Proposition 2, τL(H) is the discrete topology D on H. Furthermore, for any A ⊆ H with |A| ≥ 2, LH(A) = ∅. Therefore, BL(H) = {{a} : a ∈ H}. Example 4. Consider H = {0, a, b} with the hyperoperation “ ~ ” defined as follows: ~ 0 a b 0 {0} {b} {a} a {a} {0} {b} b {b} {a} {0} Then H is a hyper BCI-algebra. By Theorem 2, BL(H) = {LH(A) : ∅ 6= A ⊆ H} = {{0}, {a}, {b},∅}. Thus, τL(H) = {{0}, {a}, {b}, {0, a}, {0, b}, {a, b},∅, H} = D . Theorem 4. If H is a finite hyper BCI-algebra, then the family SL(H) = {LH(a) : a ∈ H} is a subbase of τL(H). Proof. That SL(H) ⊆ τL(H) is evident. Since LH(A) = ⋂ a∈A LH({a}) for each nonempty A ⊆ H, it follows that every element of BL(H) is a finite intersection of members of SL(H). Hence, SL(H) is a subbase of τL(H). M. Panganduyon, S. Canoy, Jr., B. Davvaz / Eur. J. Pure Appl. Math, 14 (2) (2021), 590-600 595 Proposition 3. Let H be a hyper BCI-algebra with |H| ≥ 2. Then BL(H) ={{a} : a ∈ A∗(H), 0 /∈ LH(a)} ∪ {{0, a} : a ∈ A∗(H), 0 ∈ LH(a)} ∪ {LH(A) : A ∩A∗(H) = ∅}. Proof. For each a ∈ A∗(H), either LH(a) = {a} or LH(a) = {0, a}. Let A be a nonempty subset of H such that A∩A∗(H) 6= ∅, say q ∈ A∩A∗(H). Since LH(A) ⊆ LH(q) (by Proposition 1(iii)) and LH(q) ∈ {{q}, {0, q}}, it follows that LH(A) ∈ {{0}, {q}, {0, q}}. Corollary 2. Let H be a hyper BCI-algebra such that 0 ∈ LH(x) for each x ∈ H \ {0} with |H| ≥ 2. Then BL(H) = {{0, a} : a ∈ A∗(H)} ∪ {LH(A) : A ∩A∗(H) = ∅}. Corollary 3. Let H be a hyper BCI-algebra such that 0 ∈ LH(x) for each x ∈ H \ {0} with |H| ≥ 2. If A∗(H) = {a}, then BL(H) = {{0, a}} ∪ {LH(A) : a /∈ A}. Theorem 5. Let H be a hyper BCI-algebra with |H| ≥ 2. Then BL(H) = {{0}} ∪ {{a} : a ∈ H \{0}, 0 /∈ LH(a)}∪{{0, a} : a ∈ H \{0}, 0 ∈ LH(a)} if and only if H is hyperatomic. Proof. Suppose H is hyperatomic. Then for any nonempty subset A of H such that A 6= {0}, A ∩ A∗(H) 6= ∅. Thus, {LH(A) : A 6= ∅ and A ∩ A∗(H) = ∅} = {{0}}. The result then follows from Proposition 3. For the converse, suppose that BL(H) is the given family of subsets of H. Let a ∈ H \ {0}. Then either LH(a) = {a} or LH(a) = {0, a}. Hence, if x ∈ H and x � a, then either x = a or x = 0. Thus, a ∈ A(H). Therefore, H is hyperatomic. Example 5. Refer to Example 2. It is easy to verify that H is hyperatomic. Corollary 4. Let H be a hyper BCI-algebra such that 0 ∈ LH(x) for each x ∈ H with |H| ≥ 2. Then BL(H) = {{0}} ∪ {{0, a} : a ∈ H \ {0}} if and only if H is hyperatomic. Theorem 6. Let H be a hyperatomic hyper BCI-algebra. Then A ∈ τL(H) if and only if A = ∅ or 0 ∈ A or 0 /∈ LH(a) for all a ∈ A. Proof. Let A ∈ τL(H) \ {∅} and let a ∈ A. Since BL(H) is a basis for τL(H), there exists Ba ⊆ H such that a ∈ LH(Ba) ⊆ A. Since H is hyperatomic, LH(b) = {b} or {0, b} for each b ∈ Ba. If a = 0, then 0 ∈ A. Suppose that a 6= 0 and let b ∈ Ba. Then b 6= 0 and a ∈ LH(b). Hence, a = b; that is, Ba = {a}. Thus, LH(Ba) = LH(a) = {a}. Therefore, either 0 ∈ A or 0 /∈ A and LH(a) = {a} for each a ∈ A. For the converse, suppose first that 0 /∈ LH(a) for each a ∈ A. Then A = ⋃ a∈A LH(a) ∈ τL(H). Next, suppose that 0 ∈ A. Since LH(x) = {x} or {0, x} for all x ∈ H, it follows that LH(a) ⊆ A for all a ∈ A. Thus, A =  ⋃ a∈A 0∈LH(a) LH(a) ⋃  ⋃ a∈A 0/∈LH(a) LH(a)  ∈ τL(H). M. Panganduyon, S. Canoy, Jr., B. Davvaz / Eur. J. Pure Appl. Math, 14 (2) (2021), 590-600 596 This proves the assertion. Recall that for a nonempty set X and a fixed p ∈ X, the topology τp given by τp = {∅} ∪ {A ⊆ X : p ∈ A} is called the particular point p topology on X (see [12]). The next result gives a characterization of τL(H) involving a particular point topology. Theorem 7. Let H be a hyper BCI-algebra such that LH({x, y}) = {0} for every pair of distinct points x and y of H. Then τL(H) is the particular point 0 topology τ0 on H if and only if H is hyperatomic. Proof. Suppose that H is hyperatomic. Then by Corollary 4, BL(H) = {{0}}∪{{0, a} : a ∈ H \ {0}}. Since BL(H) is a basis for τL(H), A ∈ τL(H) ⇔ A = ∅ or A = {0} or A = ⋃ a∈A {0, a} ⇔ A = ∅ or 0 ∈ A ⇔ A ∈ τ0. Thus, τL(H) = τ0. For the converse, suppose that τL(H) = τ0 and let x ∈ H \ {0}. Then {0, x} ∈ τL(H). Since BL(H) is a basis for τL(H), there exists a subset A of H such that x ∈ LH(A) ⊆ {0, x}. Hence, LH(A) = {x} or LH(A) = {0, x}. Now, since 0 ∈ LH(a) for each a ∈ A, LH(A) = {0, x}. If A = ∅, then by Proposition 1(i), LH(A) = H = {0, x}. Hence, H is hyperatomic. If A 6= ∅, then |A| = 1 (otherwise, LH(A) = {0} which is a contradiction). Therefore, since y ∈ LH(y) for each y ∈ H, A = {x}, that is, LH(A) = LH(x) = {0, x}. This shows that H is hyperatomic. Remark 1. The condition LH({x, y}) = {0} for each pair (x, y) ∈ H ×H, where x 6= y, cannot be omitted. The hyper BCI-algebra in Example 2 is hyperatomic but does not satisfy this condition. Hence, τL(H) 6= τ0. Theorem 8. Let H be a hyperatomic hyper BCI-algebra and let A,F ⊆ H. Then with respect to τL(H), (i) int(A) =  A if A = ∅ or 0 ∈ A or 0 /∈ LH(a) ∀ a ∈ A, A \ {a ∈ A : 0 ∈ LH(a)} otherwise; and (ii) F =  F if 0 ∈ F and 0 /∈ LH(x) ∀x ∈ H \ F or 0 /∈ F, F ∪ {x ∈ H \ F : 0 ∈ LH(x)} otherwise. M. Panganduyon, S. Canoy, Jr., B. Davvaz / Eur. J. Pure Appl. Math, 14 (2) (2021), 590-600 597 Proof. (i) If A = ∅ or 0 ∈ A or 0 /∈ LH(a) for all a ∈ A, then A ∈ τL(H) by Theorem 6. Thus, intA = A. Now, suppose A /∈ τL(H). Then A 6= ∅, 0 /∈ A, and there exists a ∈ A such that 0 ∈ LH(a) by Theorem 6. Let BA = A \ {x ∈ A : 0 ∈ LH(x)}. Clearly, BA ( A. Let z ∈ BA. Then 0 /∈ LH(z). By Theorem 6, BA ∈ τL(H). Next, let G ∈ τL(H) such that G ⊆ A and let v ∈ G. Since 0 /∈ A, 0 /∈ G. Hence, by Theorem 6, 0 /∈ LH(v), that is, v ∈ BA. Therefore, intA = BA. (ii) Suppose first that 0 /∈ F . Then 0 ∈ H \ F = F c; hence F c ∈ τL(H) by Theorem 6. If 0 ∈ F and 0 /∈ LH(x) for all x ∈ F c, then by Theorem 6, F c ∈ τL(H). Thus, in both cases, F is a τL(H)-closed set. Therefore, F = F . Next, suppose that 0 ∈ F and there exists x ∈ H \ F such that 0 ∈ LH(x). Let Q = F ∪ {z ∈ H \ F : 0 ∈ LH(z)} and let q ∈ Qc. Then 0 /∈ Qc and 0 /∈ LH(q). By Theorem 6, Qc ∈ τL(H), that is, Q is τL(H)-closed. Now, let w ∈ H \ F such that 0 /∈ LH(w). Then LH(w) = {w} is a neighborhood of w with LH(w)∩F = ∅. Thus, w /∈ F . Therefore, the smallest closed set containing F is Q, that is, F = Q. Theorem 9. Let H be a hyper BCI-algebra and let D ( H. (i) If 0 ∈ LH(x) for all x ∈ H, then D is dense in H if and only if 0 ∈ D. (ii) If H is hyperatomic, then D is dense if and only if 0 ∈ D and 0 ∈ LH(x) for all x ∈ H \D. Proof. (i) If D is dense in H, then LH(0) ∩D 6= ∅. Hence, 0 ∈ D. Next, suppose that 0 ∈ D and A ⊆ H with LH(A) 6= ∅. Since 0 ∈ LH(a) for all a ∈ A, 0 ∈ LH(A). Thus, LH(A) ∩D 6= ∅. Therefore, D is dense in H. (ii) Suppose D is dense in H. Then 0 ∈ D. Since D 6= H, D is not τL(H)-closed (otherwise, D = D 6= H, a contradiction.) Thus, by Theorem 8 and the assumption that D is dense, D = D ∪ {x ∈ H \ D : 0 ∈ LH(x)} = H. Therefore, 0 ∈ LH(x) for all x ∈ H \ D. For the converse, suppose that the given conditions hold. By Theorem 8, D = H. Thus, D is dense in H. Lemma 3. Let K be a hyper subalgebra of a hyper BCI-algebra H. Then (i) A∗(H) ∩K ⊆ A∗(K); and (ii) LK(D) = LH(D) ∩K for every D ⊆ K. (iii) LH(A) ∩K ⊆ LH(A ∩K) for any A ⊆ H. M. Panganduyon, S. Canoy, Jr., B. Davvaz / Eur. J. Pure Appl. Math, 14 (2) (2021), 590-600 598 Proof. (i) Let a ∈ A∗(H) ∩K. Then a ∈ K and for all x ∈ H, x � a implies that x = a or x = 0. In particular, for all y ∈ K, y � a implies y = 0 or y = a. Thus, a ∈ A∗(K). (ii) Let D ⊆ K. Then z ∈ LK(D) if and only if z ∈ K and z � d for all d ∈ D. Thus, z ∈ LK(D) if and only if z ∈ K ∩ LH(d) for each d ∈ D ⊆ K ⊆ H. Consequently, LK(D) = K ∩ LH(D). (iii) Let A ⊆ H. Since A ∩K ⊆ A, by Proposition 1(iii), LH(A) ⊆ LH(A ∩K). Thus, LH(A) ∩ K ⊆ LH(A ∩ K) ∩ K = LK(A ∩ K), by (ii). Hence, LH(A) ∩ K ⊆ LK(A ∩K). Lemma 4. Let K be a hyper subalgebra of an ordered hyper BCI-algebra H. Then for any ∅ 6= A ⊆ H, LH(A) ∩K = ⋃ x∈LH(A)∩K LK(x). Proof. Let ∅ 6= A ⊆ H and x ∈ LH(A) ∩ K. Then x � a for all a ∈ A and x ∈ K. Let y ∈ LH(x) ∩K. Then y � x and y ∈ K. Since H is ordered, y � a for all a ∈ A. Hence, y ∈ LH(A) ∩ K showing that LH(x) ∩ K ⊆ LH(A) ∩ K. Consequently,⋃ x∈LH(A)∩K (LH(x) ∩K) ⊆ LH(A) ∩K. Next, let z ∈ LH(A)∩K. By Proposition 1(v), z ∈ LH(z). It follows that z ∈ LH(z)∩K showing that LH(A) ∩ K ⊆ LH(z) ∩ K. Thus, LH(A) ∩ K ⊆ ⋃ x∈LH(A)∩K (LH(x) ∩ K). Therefore, by Lemma 3(ii), LH(A) ∩K = ⋃ x∈LH(A)∩K (LH(x) ∩K) = ⋃ x∈LH(A)∩K LK(x). This proves the assertion. Theorem 10. Let K be a hyper subalgebra of an ordered hyper BCI-algebra H with |K| ≥ 2. Then τL(K) coincides with the relative topology τK on K. Proof. By Theorem 1 and Theorem 2, bases for τK and τL(K) are given by the families BK = {LH(A) ∩K : ∅ 6= A ⊆ H} and BL(K) = {LK(A) : ∅ 6= A ⊆ K}, respectively. Let U = LH(A) ∩ K ∈ BK and let x ∈ U . Since x ∈ LH(x) and x ∈ K, x ∈ LH(x)∩K = LK(x) by Lemma 3. By Lemma 4, LK(x) ⊆ ⋃ y∈LH(A)∩K LK(y) = LH(A)∩K. Take U ′ = LH(x). It follows that τK ⊆ τL(K). To show the other inclusion, let U ∈ BL(K). Then there exists B ⊆ K such that U = LK(B). By Lemma 3, U = LK(B) = LH(B) ∩K ∈ BK . Hence, BL(K) ⊆ BK , that is, τL(K) ⊆ τK . Therefore, τL(K) = τK . REFERENCES 599 Conclusion: An operator on the power set of a hyper BCI-algebra into the family of its nonempty subsets had been defined via left application of the hyper-order associated with the hyper BCI-algebra. The collection of images of subsets under this operator turned out to be a basis for some topology on the given hyperstructure. The topological space generated in this way enabled us to look into the topological structure of hyper BCI-algebra in many ways. In particular, under some conditions on the hyper BCI-algebra, elementary concepts associated with the space such as open, closed, density, closure, interior, and relative space had been described or characterized. The topological space generated in this study may be studied further for other topo- logical aspects such as connectedness and compactness. Also, if it were possibe to define hyper-orders on the sum (or join) and product of two hyper BCI-algebras so as to obtain two hyper BCI-algebras, it would be interesting to know what the respective bases would be for the sum and product. Further, it may be worthwhile to investigate whether or not the right application of the hyper-order or the combination of the left and right applica- tions will also give rise to a topological space. 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