On a Graph Induced by a Hyper BCI-algebra EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 12, No. 1, 2019, 146-158 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global On a Graph Induced by a Hyper BCI-algebra Michelle T. Panganduyon1, Sergio R. Canoy, Jr.1,∗ 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. This paper introduces the notion of the zero divisor graph of a hyper BCI-algebra and investigates some of its properties. 2010 Mathematics Subject Classifications: 20M14, 05C25 Key Words and Phrases: Zero divisor graph, hyper BCI-algebra, hyperatom, complete, star 1. Introduction Graph Theory and Abstract Algebra have been profoundly studied by mathematicians because of the interesting topics laid upon these branches of mathematics. Indeed, some authors studied graph theory to build connections with certain algebraic structures such as commutative semigroups, commutative rings, and non-commutative rings. Beck, in his work in [1], associated to any commutative ring R its zero divisor graph G(R) whose vertices are the zero divisors of R (including an element 0 of R) and where adjacency between two distinct elements of R is defined as follows: two vertices x, y are adjacent if and only if xy = 0. In 2002, DeMeyer et al. [2] also pioneered the notion of zero-divisor graph of commutative semigroup S with 0. They associated an undirected graph Γ(S) to any commutative semigroup S with 0 whose vertices are the nonzero zero divisors of S, such that two vertices x, y are adjacent if and only if xy = 0. More recently, Y. B. Jun and K. J. Lee [5] introduced the concept of associated graph of BCI-algebra and verified some properties of the graph. Motivated by these works, in this paper, we shall introduce the notion of the zero divisor graph of a hyper BCI-algebra and investigate some of its properties. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v12i1.3362 Email addresses: michelle.panganduyon@g.msuiit.edu.ph (M. Panganduyon), sergio.canoy@g.msuiit.edu.ph (S. Canoy) http://www.ejpam.com 146 c© 2019 EJPAM All rights reserved. M. Panganduyon, S. Canoy / Eur. J. Pure Appl. Math, 12 (1) (2019), 146-158 147 2. Preliminaries The concepts on Graph Theory are taken from [4]: A graph G is an ordered pair (V (G), E(G)), where V (G) is a finite nonempty set called the vertex set of G and E(G) is a set of unordered pairs {u, v} (or simply uv) of distinct elements from V (G) called the edge set of G. The elements of V (G) are called vertices and the cardinality |V (G)| of V (G) is the order of G. The elements of E(G) are called edges and the cardinality |E(G)| of E(G) is the size of G. A graph K = (V (K), E(K)) is a subgraph of a graph G = (V (G), E(G)) if V (K) ⊆ V (G) and E(K) ⊆ E(G). Two vertices u, v of a graph G are adjacent, or neighbors, if uv is an edge of G. The set of neighbors of a vertex v of G is denoted by NG(v) and the degree of v in G, denoted deg v, is equal to |NG(v)|. The degree of G, denoted by ∆(G), is equal to the largest degree of a vertex of G. A vertex w of G is called an isolated vertex if degG(w) = |NG(w)| = 0. The set of all isolated vertices of G will be denoted by I(G). A graph G is called an empty graph, denoted by K |V (G)|, if E(G) = ∅, that is, I(G) = V (G). A walk of a graph G is an alternating sequence of vertices and edges, beginning and ending with vertices, v0, e1, v1, . . . , vn−1, en, vn, in which each edge is incident with the two vertices immediately preceding and following it. This walk joins v0 and vn; and is sometimes called a v0-vn walk. It is closed if v0 = vn and is open otherwise. It is a path if all the vertices (and thus necessarily all the edges) are distinct. If the walk is closed, then it is a cycle provided its n vertices are distinct and n ≥ 3. We denote by Cn the graph consisting of a cycle with n vertices and by Pn a path with n vertices. A graph is connected if every pair of vertices are joined by a path. A maximal connected subgraph of G is called a component of G. The complete graph Kp has every pair of its p vertices adjacent. A bipartite graph G is a graph whose vertex set V can be partitioned into two subsets V1 and V2 such that every edge of G joins V1 with V2. If G contains every edge joining V1 and V2, then G is a complete bipartite. If V1 and V2 have m and n vertices, respectively, then we write G = Km,n. A star is a complete bipartite K1,n. The Kronecker product G ⊗ K of two graphs G and K is the graph with vertex set V (G ⊗ K) = V (G) × V (K) and edge set E(G ⊗ K) satisfying the following conditions: (x, u)(y, v) ∈ E(G⊗K) if and only if xy ∈ E(G) and uv ∈ E(K). Let G and K be graphs and let f : V (G) → V (K) be a function. Then f is a graph homomorphism if f(x)f(y) ∈ E(K) whenever xy ∈ E(G). Two graphs G and K are isomorphic (written as G ∼= K) if there exists a one-to-one correspondence between the vertex sets which preserves adjacency. A hyperoperation on a nonempty set H is a map from H×H into 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. M. Panganduyon, S. Canoy / Eur. J. Pure Appl. Math, 12 (1) (2019), 146-158 148 A hyper BCI-algebra (H,~, 0) is a nonempty set H endowed with a hyperoperation “~ ” and a constant 0 satisfying the following axioms: 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 (see [7]). A hyper BCI-algebra (H,~, 0) is said to be ordered if for each x, y, z ∈ H, x� y and y � z imply x� z. Example 1. [7] Let H = {0, 1, 2}. Define the hyperoperation “ ~ ” by the Cayley table shown below. ~ 0 1 2 0 {0, 1} {0, 1} {0, 1} 1 {1} {0, 1} {0, 1} 2 {2} {1, 2} {0, 1, 2} Then by routine calculations, (H,~, 0) is a hyper BCI-algebra. Further, H is ordered. Let (H1,~1, 01) and (H2,~2, 02) be two hyper BCI-algebras. Consider a mapping f : H1 → H2. Then f is said to be a homomorphism if f(x ~1 y) = f(x) ~2 f(y), for all x, y ∈ H1. If f is a homomorphism and f(01) = 02, then we call f a hyper homomorphism. If f is a homomorphism, one-to-one, and onto, we say that f is an isomorphism and (H1,~1, 01) and (H2,~2, 02) are isomorphic, denoted by H1 ∼= H2 (see [6]). Let f : H1 → H2 be a hyper homomorphism of hyper BCI-algebras. If f is one to one (resp. onto) we say f is a hyper monomorphism (resp. hyper epimorphism). If f is a hyper homomorphism and a bijection, f is said to be a hyper isomorphism, denoted by H1 ∼=H H2 (see [3]). Throughout this study, we denote a hyper BCI-algebra (H,~, 0) by H, unless otherwise specified. The following results generated previously give some of the properties of a hyper BCI- algebra. Proposition 1. [7] In any hyper BCI-algebra H, the following hold: (i) x� 0 implies x=0, (ii) 0 ∈ x~ (x~ 0), M. Panganduyon, S. Canoy / Eur. J. Pure Appl. Math, 12 (1) (2019), 146-158 149 (iii) x� x~ 0, (iv) 0~ (x~ y)� y ~ x, (v) A� A, (vi) A ⊆ B implies A� B, (vii) A� {0} implies A = {0}, (viii) x~ 0� {y} implies x� y, (ix) y � z implies x~ z � x~ y, (x) x~ y = {0} implies (x~ z)~ (y ~ z) = {0} and x~ z � y ~ z, (xi) A~A = {0} implies A is a singleton, (xii) A~ {0} = {0} implies A = {0}. for all x, y, z ∈ H and for all non-empty subsets A and B of H. Theorem 1. [3] Let f : H1 → H2 be a hyper homomorphism. Then the following hold: (i) If x� y, where x, y ∈ H1, then f(x)� f(y). (ii) If A,B ⊆ H1 such that A� B, then f(A)� f(B). 3. Zero Divisor Graph of a Hyper BCI-algebra Let H be a hyper BCI-algebra and A ⊆ H. We will use the notation LH(A) to denote the set LH(A) := {x ∈ H|x� a,∀ a ∈ A} = {x ∈ H|0 ∈ x~ a,∀ a ∈ A}. If A = {a}, we write LH({a}) = LH(a). For any x ∈ H, the set of zero divisors of x is Zx = {y ∈ H|LH({x, y}) = {0}}. Let H be a finite hyper BCI-algebra. The zero divisor graph Γ(H) of H is the graph whose vertex set V (Γ(H)) = H and edge set E(Γ(H)) satisfying the following condition: for every distinct x, y ∈ H, xy ∈ E(Γ(H)) if and only if LH({x, y}) = {0} (equivalently, x ∈ Zy or y ∈ Zx). Although there are infinite hyper BCI-algebras, this paper only considers zero divisor graphs of finite hyper BCI-algebras. Example 2. Consider the hyper BCI-algebra H defined in Example 1. Then LH({0, 1}) = LH({0, 2}) = {0} and LH({1, 2}) = {0, 1}. The zero divisors of x ∈ H are Z0 = {y ∈ H|LH({0, y}) = {0}} = {1, 2} and Z1 = {0} = Z2. Thus, the zero divisor graph Γ(H) of H is given by the following figure: M. Panganduyon, S. Canoy / Eur. J. Pure Appl. Math, 12 (1) (2019), 146-158 150 1 2 0 The next result gives some properties of the operator LH . Proposition 2. 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. Proof. (i) Suppose LH(∅) 6= H. Then ∃h ∈ H such that h /∈ LH(∅); i.e., ∃ a ∈ ∅ such that a 6� h, a contradiction. Therefore, LH(∅) = H. (ii) By definition, LH({0}) = {x ∈ H|x� 0} = {0}, by Proposition 1. (iii) Let x ∈ LH(B). Then x � b, ∀ b ∈ B. Since A ⊆ B, x � a, ∀ a ∈ A. Thus, x ∈ LH(A). Hence, LH(B) ⊆ LH(A). (iv) Follows from the definition of LH(A): LH(A) = {x ∈ H|x� a,∀ a ∈ A} = {x ∈ H|x ∈ LH({a}),∀ a ∈ A} = ⋂ a∈A LH({a}). (v) Let x ∈ H. By (B3), x� x. Hence, x ∈ LH({x}). Furthermore, x ∈ LH({x}) = {0} implies x = 0 and if x = 0, then LH({x}) = LH({0}) = {0} by (ii). � The zero divisor graph of a hyper BCI-algebra is not always connected: M. Panganduyon, S. Canoy / Eur. J. Pure Appl. Math, 12 (1) (2019), 146-158 151 Example 3. Consider the hyper BCI-algebra H with ‘~’ defined by the following Cayley table: ~ 0 1 2 0 {0, 1} {0, 1} {2} 1 {1} {0, 1} {2} 2 {2} {2} {0, 1} Then LH({0, 1}) = {0}; LH({0, 2}) = ∅ = LH({1, 2}). Thus, the zero divisor graph Γ(H) of H is given below: 1 2 0 Example 4. Consider H defined by the following Cayley table: ~ 0 1 2 3 0 {0} {0} {2} {2} 1 {1} {0} {2} {2} 2 {2} {2} {0} {0} 3 {3} {2} {1} {0, 1} Then LH({0, 1}) = {0}, LH({0, 2}) = LH({0, 3}) = LH({1, 2}) = LH({1, 3}) = ∅, and LH({2, 3}) = {2}. The zero divisor graph Γ(H) of H is given below 1 3 0 2 Proposition 3. Let H be a hyper BCI-algebra with |H| ≥ 2. Then (i) degΓ(H)(0) = |{x ∈ H \ {0} : 0 ∈ LH(x)}| = ∆(Γ(H)); (ii) Γ(H) = K |H| if and only if 0 /∈ LH(x) for all x ∈ H \ {0}; and (iii) if Γ(H) 6= K |H|, then I(Γ(H)) = {x ∈ H \ {0} : 0 /∈ LH(x)}. Proof. M. Panganduyon, S. Canoy / Eur. J. Pure Appl. Math, 12 (1) (2019), 146-158 152 (i) Note that for any x ∈ H \{0}, 0x ∈ E(Γ(H)) if and only if LH({0, x}) = {0}. Hence, by Proposition 2(ii), 0x ∈ E(Γ(H)) if and only if 0 ∈ LH(x). Thus, degΓ(H) 0 = |{x ∈ H \ {0} : 0x ∈ E(Γ(H))}| = |{x ∈ H \ {0} : 0 ∈ LH(x)}|. Let x ∈ H \ {0} and let y ∈ NΓ(H)(x). Then LH({x, y}) = {0}. By Proposition 2(ii) and 2(iv), it follows that LH(0, y) = {0}, that is, y ∈ NΓ(H)(0). Thus, degΓ(H)(x) = |NΓ(H)(x)| ≤ |NΓ(H)(0)| = degΓ(H)(0). Since x was arbitrarily chosen, it follows that ∆(Γ(H))| = degΓ(H)(0). (ii) Suppose that Γ(H) = K |H|. That is, I(Γ(H)) = V (Γ(H)). This implies that degΓ(H) 0 = 0. Hence, 0x /∈ E(Γ(H)) for all x ∈ H. Thus, 0 /∈ LH(x) for all x ∈ H \ {0}. For the converse, suppose that 0 /∈ LH(x) for all x ∈ H \ {0}. By (i), it follows that degΓ(H) 0 = ∆(Γ(H)) = 0 . Therefore, Γ(H) = K |H|. (iii) Suppose that Γ(H) 6= K |H|. Then degΓ(H) 0 = ∆(Γ(H)) 6= 0, i.e., 0 /∈ I(Γ(H)). Let x ∈ H \ {0}. If 0 /∈ LH(x), then LH(x, y) 6= {0} for all y ∈ H \ {x}. Thus, degΓ(H)(x) = 0, i.e., x ∈ I(Γ(H)). Conversely, if x ∈ I(Γ(H)), then 0x /∈ E(Γ(H)), i.e., LH(0, x) 6= {0}. By Proposition 2(ii) and 2(iv), 0 /∈ LH(x). Therefore, I(Γ(H)) = {x ∈ H \ {0} : 0 /∈ LH(x)}. � Next, we give equivalent statements for connectedness of the zero divisor graph. Proposition 4. Let H be a hyper BCI-algebra with |H| ≥ 2. Then the following are equivalent: (i) Γ(H) is connected. (ii) LH({x, 0}) = {0} for all x ∈ H \ {0}. (iii) 0 ∈ LH(x) for all x ∈ H \ {0}. (iv) ∆(Γ(H)) = |H| − 1 (v) I(Γ(H)) = ∅ Proof. (i)⇔(ii) Suppose LH({x, 0}) 6= {0} for some x ∈ H. Then 0 /∈ LH(x), by Proposition 2(ii) and 2(iv). Thus, 0 /∈ LH({x, y}) = LH({x}) ∩ LH({y}) for all y ∈ H. That is, for all y ∈ H, xy /∈ E(Γ(H)). This implies that Γ(H) is disconnected. For the converse, suppose that LH({x, 0}) = {0} for all x ∈ H \ {0}. Then degΓ(H)(0) = |H| − 1. Therefore, Γ(H) is connected. M. Panganduyon, S. Canoy / Eur. J. Pure Appl. Math, 12 (1) (2019), 146-158 153 (ii)⇔(iii) This follows from Proposition 2(ii) and 2(iv). (iii)⇔(iv) This follows from Proposition 3(i). (iv)⇔(v) By Proposition 3(i), degΓ(H)(0) = |H| − 1. This implies that 0x ∈ E(Γ(H)) for each x ∈ H \ {0}. Hence, I(Γ(H)) = ∅. Conversely, suppose that I(Γ(H)) = ∅ and let x ∈ H \ {0}. Since x /∈ I(Γ(H)), there exists y ∈ H \ {x} such that LH(x, y) = {0}. Hence, 0 ∈ LH(x) by Proposition 2(iv). By Proposition 3(i), it follows that degΓ(H)(0) = ∆(Γ(H)) = |H| − 1. � Remark 1. Let H be a hyper BCI-algebra with |H| ≥ 2. If Γ(H) is connected, then (i) diam(Γ(H)) = 2; (ii) degΓ(H) 0 = |H| − 1 = ∆(Γ(H)). Proposition 5. Let H be a hyper BCI-algebra such that LH{x, 0} 6= ∅ for all x ∈ H. Then LH{x, 0} = {0}. Proof. Suppose that LH{x, 0} 6= ∅ for all x ∈ H. Then there exists y ∈ LH{x, 0}. Note that y ∈ LH{0} = {0} means that y = 0. It follows from Proposition 2(ii) and 2(iv) that LH{x, 0} = {0}. � Remark 2. Let H be a hyper BCI-algebra such that LH{x, 0} 6= ∅ for all x ∈ H. Then 0x ∈ E(Γ(H)) ∀ x ∈ H \ {0}. Proposition 6. If |H| > 3, then Γ(H) is neither a cycle nor a path. Proof. Case 1. ∃ x ∈ H \ {0} such that 0 /∈ LH(x). Then Γ(H) is disconnected, and the result follows. Case 2. 0 ∈ LH(x) ∀ x ∈ H. Then 0x ∈ E(Γ(H)) ∀ x ∈ H \ {0}. Evidently, Γ(H) is neither a cycle nor a path. � Corollary 1. If a graph G is a cycle or a path of order n ≥ 4, then there is no hyper BCI-algebra H such that Γ(H) ∼= G. Proof. Immediate from Proposition 6. � Theorem 2. Let H be a hyper BCI-algebra with |H| ≥ 2. Then G = Γ(H) cannot have two nontrivial components; that is, G can only have at most one non-trivial component. Proof. If G is connected, then we are done. Suppose that G is disconnected. Suppose further that G has two distinct non-trivial components, say G1 and G2. Let G3 be a component of G with 0 ∈ V (G3) (G3 may be G1 or G2). If G3 is different from G1, then 0 /∈ LH(x) for all x ∈ V (G1). Similarly, if G3 is not G2, then 0 /∈ LH(y) for all y ∈ V (G2). Hence, by Proposition 4, G1 or G2 is the trivial graph, a contradiction. � M. Panganduyon, S. Canoy / Eur. J. Pure Appl. Math, 12 (1) (2019), 146-158 154 Proposition 7. Let H be an ordered hyper BCI-algebra. Then the following hold: (i) For any subset A of H, LH(LH(A)) ⊆ LH(A). (ii) For any a, b ∈ H, if a� b, then LH({a}) ⊆ LH({b}) and Zb ⊆ Za. Proof. (i) Let x ∈ LH(LH(A)). Then x � b for all b ∈ LH(A). Since b � a for all a ∈ A and H is ordered, it follows that x � a for all a ∈ A. Thus, x ∈ LH(A) and the result follows. (ii) Suppose x ∈ LH({a}). Then x � a. Since H is ordered and a � b, x � b. That is, x ∈ LH({b}). Hence, LH({a}) ⊆ LH({b}). Now, suppose x ∈ Zb. Then LH({b, x}) = {0}. Since LH({a, x}) ⊆ LH({b, x}), we have LH({a, x}) = {0}. This means that x ∈ Za. Thus, Zb ⊆ Za. � Proposition 8. Let H be a hyper BCI-algebra. Then (i) degΓ(H) x = |Zx| for all nonzero x ∈ H. (ii) y ∈ Zx if and only if x ∈ Zy for all x, y ∈ H. Proof. Let x, y ∈ H. (i) if x 6= 0, then |Zx| = |{y ∈ H \ {x} : LH({x, y}) = {0}}| = |{y ∈ H : xy ∈ E(Γ(H))}| = degΓ(H) x. (ii) y ∈ Zx means that LH({x, y}) = {0}, which further means that x ∈ Zy. � Lemma 1. Let f : H1 → H2 be a hyper monomorphism of hyper BCI-algebras. Then for any x, y ∈ H1, x� y if and only if f(x)� f(y). Proof. The sufficiency part is done by Theorem 1(i). Now, suppose f(x)� f(y). Then 02 ∈ f(x) ~2 f(y) = f(x ~1 y). Thus, 01 = f−1(02) ∈ f−1f(x ~1 y) = x ~1 y. Hence, x� y. � Proposition 9. Let f : H1 → H2 be a hyper monomorphism of hyper BCI-algebras. Then LH2(f(A)) = f(LH1(A)) where A ⊆ H1. M. Panganduyon, S. Canoy / Eur. J. Pure Appl. Math, 12 (1) (2019), 146-158 155 Proof. Let f : H1 → H2 be a hyper monomorphism. Let A ⊆ H1. y ∈ f(LH1(A)) ⇐⇒ f−1(y) ∈ LH1(A) ⇐⇒ f−1(y)� a for all a ∈ A ⇐⇒ y � f(a) for all a ∈ A, by Lemma 1 ⇐⇒ y ∈ LH2(f(A)) Therefore, LH2(f(A)) = f(LH1(A)). � Theorem 3. Let H1 and H2 be hyper BCI-algebras. If H1 ∼=H H2, then Γ(H1) ∼= Γ(H2). Proof. Suppose H1 ∼=H H2, say f : H1 → H2 is a hyper isomorphism. Since V (Γ(H1)) = H1 and V (Γ(H2)) = H2, there exists a one-to-one correspondence be- tween the vertex sets. Note that for any distinct elements x, y ∈ H1, xy ∈ E(Γ(H1)) if and only if LH1({x, y}) = {0}. By Proposition 9, xy ∈ E(Γ(H1)) if and only if LH2({f(x), f(y)}) = {0}. Thus, xy ∈ E(Γ(H1)) if and only if f(x)f(y) ∈ E(Γ(H2)). Consequently, Γ(H1) ∼= Γ(H2). � 3.1. On zero divisor graphs involving hyperatoms Definition 1. An element a of a hyper BCI-algebra 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; i.e., A∗(H) = A(H) \ {0}. Obviously, 0 ∈ A(H). Definition 2. A hyper BCI-algebra H is said to be hyperatomic if each element of H is a hyperatom, that is, A(H) = H. Remark 3. A hyper BCI-algebra H is hyperatomic if and only if LH(x) = {x} or LH(x) = {0, x} for each x ∈ H. Remark 4. A hyperatomic hyper BCI-algebra is ordered. Proof. Suppose H is a hyperatomic hyper BCI-algebra. Let x, y, z ∈ H such that x � y and y � z. Then by Remark 3, LH(z) = {z} or LH(z) = {0, z} . Thus, y � z implies that y = 0 or y = z. If y = 0, then x � 0 since x � y. By Proposition 1, x = 0. Since 0� z and x = 0, we have x� z. If y = z, then the assumption x� y implies that x� z. Hence, H is ordered. � Example 5. Consider the hyper BCI-algebra defined by the Cayley table: ~ 0 1 2 0 {0} {0, 1} {0, 1} 1 {1} {0, 1} {1} 2 {2} {2} {0, 1, 2} M. Panganduyon, S. Canoy / Eur. J. Pure Appl. Math, 12 (1) (2019), 146-158 156 H is hyperatomic since all its elements are hyperatoms. Example 6. The hyper BCI-algebra H in Example 1 is not hyperatomic since 2 is not a hyperatom of H: ∃x = 1 ∈ H with 1 � 2 but x = 1 6= 0 and x = 1 6= 2. However, the hyper BCI-algebra H in Example 3 is hyperatomic. Proposition 10. Let H be a hyper BCI-algebra such that |H| ≥ 2. If x and y are distinct nonzero hyperatoms of H, then LH({x, y}) = {0} or ∅. Proof. The result depends on whether or not 0 ∈ LH(x) for all x ∈ H. If 0 /∈ LH(x), then LH(x) = {x}. Hence, LH({x, y}) = ∅. If 0 ∈ LH({x}), then LH({x}) = {0, x}. Since LH({y}) = {0} or {0, y} by Remark 3, we have LH({x, y}) = {0} or ∅. � We have the following characterization for a complete graph: Proposition 11. Let H be a hyper BCI-algebra such that |H| ≥ 2. Then Γ(H) is a complete graph if and only if Γ(H) is connected and H is hyperatomic. Proof. If Γ(H) is disconnected, then Γ(H) is not a complete graph, and we are done. Assume that Γ(H) is connected. By Proposition 4, 0 ∈ LH(x) for all x ∈ H. Since x ∈ LH{x}, we now have 0, x ∈ LH(x). If H is not hyperatomic, then there exists z ∈ H \ {0} such that y � z with y /∈ {0, z}. Since y ∈ LH{y}, y ∈ LH{y, z}. This means that LH{y, z} 6= {0}, implying that yz /∈ E(Γ(H)). Therefore, Γ(H) is not complete. Conversely, suppose Γ(H) is connected and H is hyperatomic. Then by Remark 3, LH{x} = {0, x} for all x ∈ H \ {0}. Thus, for any distinct nonzero elements x, y of H = V (Γ(H)), LH{x} ∩ LH{y} = {0}, that is, xy ∈ E(Γ(H)). Consequently, Γ(H) is a complete graph. � Example 7. The hyper BCI-algebra in Example 5 has a complete zero divisor graph: 1 2 0 Remark 5. Given an ordered hyper BCI-algebra H, it is not always true that there exists a ∈ A∗(H) = A(H) \ {0} such that a� x for all x ∈ H \ {0}. Example 8. Consider the hyper BCI-algebra H defined in Example 3. H is hyperatomic and hence, ordered and A∗(H) = {1, 2}. Notice that neither 1 � x nor 2 � x for all x ∈ H \ {0}. But for each x ∈ H \ {0}, there exists a ∈ A∗(H) such that a� x. Theorem 4. Let H be an ordered hyper BCI-algebra with |H| ≥ 2. Then the following hold: M. Panganduyon, S. Canoy / Eur. J. Pure Appl. Math, 12 (1) (2019), 146-158 157 (i) For each x ∈ H, there is ax ∈ A∗(H) such that ax � x. In particular, A∗(H) 6= ∅. (ii) There exists a ∈ H\{0} such that a� x for all x ∈ H\{0} if and only if |A∗(H)| = 1 (that is, A∗(H) = {a}). Proof. (i) Let x ∈ H \ {0}. If x is a hyperatom, then take ax = x. If x is not a hyperatom, there exists x1 ∈ H \{0, x} such that x1 � x. Again, if x1 is a hyperatom, then take ax = x1. Otherwise, there exists x2 ∈ H \{0, x, x1} such that x2 � x1 � x. Since H is finite, continuing in this fashion yields a terminal point xn ∈ H\{0, x, x1, . . . , xn−1} with xn � xn−1 � · · · � x2 � x1 � x such that only z = 0 (provided 0 ∈ LH{x}) or z = xn satisfies z � x. This implies that ax = xn ∈ A∗(H) and ax � x. (ii) Suppose that there exists a ∈ H \ {0} such that a� x for all x ∈ H \ {0}. Choose any b ∈ A∗(H). Then a � b. Since b ∈ A(H) and a 6= 0, it follows that a = b. Thus a ∈ A∗(H). Since b was arbitrarily chosen, we have A∗(H) = {a}. Conversely, suppose that |A∗(H)| = 1, say A∗(H) = {a}. Let x ∈ H \ {0}. Then by (i), a� x. � Example 9. Consider the ordered hyper BCI-algebra H defined in Example 1. Note that 1 is the only nonzero hyperatom of H and the zero divisor graph Γ(H) of H is a star. As a generalization of Example 9, we have the following theorem. Theorem 5. Let H be an ordered hyper BCI-algebra with |H| ≥ 2. Then Γ(H) is a star if and only if Γ(H) is connected and |A∗(H)| = 1. Proof. Suppose that Γ(H) is a star. Then Γ(H) is connected. If |H| = 2, then clearly, |A∗(H)| = 1. Suppose that |H| ≥ 3. By Proposition 3(i), 0 is the central vertex of Γ(H). Suppose further that |A∗(H)| ≥ 2, say a, b ∈ A∗(H) with a 6= b. Since 0a, 0b ∈ E(Γ(H)), 0 ∈ LH(a) ∩ LH(b). By Proposition 10, ab ∈ E(Γ(H)). This implies that Γ(H) is not a star, a contradiction. Therefore, |A∗(H)| = 1. Conversely, suppose that Γ(H) is connected and |A∗(H)| = 1, say A∗(H) = {a}. If |H| = 2, then Γ(H) = P2, a star. Suppose that |H| ≥ 3 and let y, z ∈ H\{0}. By Theorem 4(ii), a� y and a� z. That is, a ∈ LH({y, z}). This implies that yz /∈ E(Γ(H)). 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