EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6940 ISSN 1307-5543 – ejpam.com Published by New York Business Global On Edge Q-Algebras Ananya Anantayasethi1,∗, Kittisak Saengsura1, Yeni Susanti2, Napaporn Sarasit3,∗ 1 Department of Mathematics, Faculty of Science, Mahasarakham University, Mahasarakham, 44150, Thailand 2 Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Gadjah Mada, Yogyakarta 55281, Indonesia 3 Division of Mathematics, Faculty of Engineering, Rajamangala University of Technology Isan, Khon Kaen 40000, Thailand Abstract. The concept of an edge in Q-algebra is introduced in this work. We explore some properties of edge Q-algebras. The characterization of subsets of an edge Q-algebra to be subalge- bras is provided. We show that for an edge Q-algebra X of order n, there are 2n−1 subalgebras of X. Moreover, the set of all subalgebras forms a semigroup with a right identity. Beside this, the concept of ideal is discussed. We obtain that the set of all ideals forms a left zero semigroup and a simple semigroup. Finally, we describe all possible structures of edge Q-algebras and enumerate all members of a class of all edge Q-algebras. We prove that there are exactly 2n 2−3n+2 edge Q- algebras of order n. Finally, we show a connection between Q-algebras and d-algebras. We obtain that every edge d-algebra is a Q-algebra. Precisely, every edge d-algebra is an edge Q-algebra. 2020 Mathematics Subject Classifications: 03G25, 03G27, 06F35, 20K01 Key Words and Phrases: Q-algebra, subalgebra, ideal, edge, edge Q-algebra, semigroup 1. Introduction and Preliminaries Back in the period of the late 20th century, K. Iseki and Y. Imai introduced two classes of logical algebras which are called BCK-algebra and BCI-algebra in 1966 [1, 2]. The class of BCK-algebras is a proper subclass of the class of BCI-algebras. Later, in 1978 K. Iseki and S. Tanaka discussed the theory of BCK-algebras in [3]. Since then, many new kinds of algebras which are related to BCK/BCI- algebras are introduced. In 1983, Q. P. Hu and X. Li introduced the notion of BCH-algebra which is a generalization of BCK/BCI-algebras [4, 5]. In 1984, a class of BCC-algebras was presented by Y. Komori [6], and then in 1992 W.A. Dudek discussed some properties of this algebra in [7]. ∗Corresponding author. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6940 Email addresses: ananya.a@msu.ac.th (A. Anantayasethi), kittisak.s@msu.ac.th (K. Saengsura), yeni-math@ugm.ac.id (Y. Susanti), napaporn.sr@rmuti.ac.th (N. Sarasit) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) A. Anantayasethi et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6940 2 of 14 A d-algebra, another generalization of BCK/BCI-algebras, was appeared in 1999 by J. Neggers and H.S. Kim [8]. The authors explored relations between d-algebras and BCK- algebras. The concept of edge in d-algebras is also provided. They showed that every d-transitive edge d-algebra is a BCK-algebra. Moreover, they obtained a correspondence between oriented digraphs and edge d-algebras such that every edge d-algebra produces an oriented digraph. Some years later, Neggers and Kim also introduced the notion of Q-algebras and B-algebras. In 2001, an algebra which related to BCK/BCI-algebras was emerged, so called Q-algebra, by J. Neggers, S. Ahn and H. S. Kim [9]. A Q-algebra is an algebraic system (X; ∗, 0) consists of a non-empty set X, a constant 0 ∈ X and a binary operation ∗ defined on X that yields the following three conditions (Q1), (Q2) and (Q3) as the following: For any x, y, z ∈ X, (Q1) x ∗ x = 0, (Q2) x ∗ 0 = x, (Q3) (x ∗ y) ∗ z = (x ∗ z) ∗ y. We omit the symbol ∗ for a convenient reason. Let us mention here, later on we will denote the symbol X as a Q-algebra (X; ∗, 0) unless otherwise specified. In [9], the authors presented some connections of BCK/BCI/BCH-algebras and Q-algebras. They showed that a Q-algebra X satisfying the condition (A): for all x, y ∈ X, xy = 0 and yx = 0 implies x = y, is a BCH-algebra. A Q-algebra X satisfying the conditions (A) and (B): (xy)(xz) = zy for all x, y, z ∈ X, is a BCI-algebra. A Q-algebra X is a BCK-algebra if the conditions (A), (B) and (C): for all x, y, z ∈ X, (xy)x = 0, are hold. The concepts of subalgebra, G-part and ideal were also offered in [9]. A non-empty subset S of X is a subalgebra if ab ∈ S for any a and b in S. It is easy to see that a subset {0} is a subalgebra of a Q-algebra X since 00 = 0 by (Q1). A non-empty subset I of X is an ideal of X if the following conditions (I1) and (I2) are hold: (I1) 0 ∈ I; (I2) for x, y ∈ X, if xy ∈ I, y ∈ I, then x ∈ I. The subsets {0} and X are obviously ideals of X. An ideal I of X is called a zero ideal if I = {0}, otherwise I is a non-zero ideal of X . The G-part of X , denoted by G(X), is defined by G(X) = {a ∈ X | 0a = a}. It is easy to see that 0 ∈ G(X) since 00 = 0. The authors in [9] obtained the charac- terization of G(X) which is an ideal of X when |X| ≤ 3. They also provided that every subalgebra S of X, G(X) ∩ S = G(S) where G(S) = {x ∈ S | 0x = x}. Example 1. Let X = {0, a, b, c, d, f}. Define a binary operations ∗ on X as the following table: A. Anantayasethi et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6940 3 of 14 ∗ 0 a b c d f 0 0 a c b c b a a 0 b c b c b b c 0 a 0 a c c d a 0 a 0 d d c 0 a 0 a f f d a 0 a 0 It is a routine to check that X is a Q-algebra. A subset S = {0, a, b, c, d} is a subalgebra of X but a subset T = {0, f} is not a subalgebra since 0 ∗ f = b /∈ T . Moreover, a subset T is not an ideal since c ∗ f = 0 ∈ T and f ∈ T but c /∈ T . Let consider a subset I = {0, a}. It is not difficult to check that I is an ideal and I = G(X). In 2004, S. S. Ahn, H. Kim and H. D. Lee discussed the homomorphisms of Q-algebra in [10]. They introduced the self-maps of Q-algebras which are called a right map and a left map. They obtained that a right map is an endomorphism whenever X is a positive implicative Q-algebra, i.e. (xy)(xz) = xyz for all x, y, z ∈ X. In 2011, another self-map of a Q-algebra X, called a right fixed map, is discussed and examined their properties in [11] by S. M. Lee. The author showed that the set of all right fixed maps of X is a Q-algebra under a binary operation which is induced from a binary operation on X. In 2010, S. S. Ahn and S. E. Kang proposed the concept of atom in Q-algebras. An element w of X is an atom if xw = 0 implies x = w for all x ∈ X. They showed that any subalgebra of X is an ideal of X whenever every non-zero element of X is an atom [12]. In 2024 the authors in [13] examined some properties of atoms in Q-algebras. They showed some relations between atoms and the set G-part which is related to the concept of ideal. They proved that every element of G(X) is an atom whenever G(X) is an ideal. The notion of strong atoms was also offered in [13], which was inspired from the concept of strong atom in BCK-algebra (see [14]). They proved that X does not contain a strong atom whenever X contains a non-zero ideal G(X). The concept of fuzzy set on Q-algebra can be found in [15] and [16]. The authors provided some properties of fuzzy Q-ideals, fuzzy prime ideals and fuzzy relations of Q-algebras. Recently, in 2025 the authors in [17] discussed the concept of ideal in Q-algebra. They provided a characterization of ideals which is related to the G-part. They showed that every G-part that is an ideal, is an abelian group. In this work, we introduce the notion of edge Q-algebras. We describe all possible edge Q-algebras of order n. We examine some properties of edge Q-algebras. The connection between d-algebras and edge Q-algebras is provided. The concepts of subalgebras and ideals are discussed in edge Q-algebras. We give necessary and sufficient conditions for subsets of an edge Q-algebra to be subalgebras. We also provide some properties in edge Q-algebras which are related to the concept of ideals. We obtain that the set of all ideals forms a right zero semigroup and a simple semigroup. Finally, we describe all possible structures of edge Q-algebras and enumerate all members of a class of all edge Q-algebras. A. Anantayasethi et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6940 4 of 14 2. Edge Q-algebras Let A and B be non-empty subsets of a Q-algebra X. We define a product AB in a usual way as follow: AB = {ab | a ∈ A, b ∈ B}. If B = {x}, we denote AB and BA by Ax and xA, respectively. Proposition 1. Let A,B and C be non-empty subsets of a Q-algebra X. Then the following properties are valid: (i) (AB)C = (AC)B. (ii) A{0} = A0 = A. (iii) If A ⊆ B, then AC ⊆ BC and CA ⊆ CB. (iv) If 0 ∈ B, then A ⊆ AB. (v) If A ∩B ̸= ∅, then 0 ∈ AB. Proof. (i) Follows directly from the condition (Q3). (ii) By the condition (Q2), we get A{0} = {a0 | a ∈ A} = {a | a ∈ A} = A. (iii) It is obvious. (iv) By (ii), (iii) and since 0 ∈ B, then A = A{0} ⊆ AB. (v) Let x ∈ A ∩B. Then by the condition (Q1), 0 = xx ∈ AB. Concerning to the concept of subalgebra in algebras, in general a subalgebra is a non-empty subset which is closed. In Q-algebras, we get a sharp condition as seen in the following proposition: Proposition 2. Let A and B be non-empty subsets of a Q-algebra X. Then (i) A is a subalgebra of X if and only if AA = A. (ii) If A ⊆ B and B is a subalgebra, then AB ⊆ B. Proof. (i) Assume that A is a subalgebra of X , then AA ⊆ A. Since 0 ∈ A, then by Proposition 1(iv), A ⊆ AA. Thus, AA = A. The converse direction is clear. (ii) Assume that B is a subalgebra of X and ∅ ̸= A ⊆ B. Then by (i) and Proposition 1(iii), we get AB ⊆ BB ⊆ B. Next, we will introduce the notion of an edge Q-algebra. In d-algebra, J. Neggers and H. S. Kim introduced the concept of edge d-algebras in 1999 [8]. A d-algebra consists of a non-empty set X with a constant 0 ∈ X together with a binary operation on X satisfying the following axioms: for all x, y, z ∈ X, (d1) xx = 0, (d2) x0 = 0, (d3) xy = 0 and yx = 0 imply x = y. A. Anantayasethi et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6940 5 of 14 Both d-algebras and Q-algebras are generalizations of BCI/BCK/BCH-algebras. But both of them are independent, i.e. a Q-algebra need not be a d-algebra and vice versa. The following example shows this fact: Example 2. Let X = {0, a, b, c}, Y = {0, x, y, z} and T = {0, γ, β, ν}. Define binary operations ∗ on X, • on Y and ◦ on T as the following tables: ∗ 0 a b c 0 0 0 0 0 a a 0 c b b c 0 0 a c c a c 0 • 0 x y z 0 0 x z y x x 0 y z y y z 0 x z z y x 0 ◦ 0 γ β ν 0 0 0 0 0 γ γ 0 γ 0 β β 0 0 β ν ν ν ν 0 Then (X; ∗, 0) is a d-algebra and (Y ; •, 0) is a Q-algebra. Since 0 ∗ b = c ̸= 0, then the condition (Q2) is not satisfied in X. Then (X; ∗, 0) is not a Q-algebra. Moreover, (Y ; •, 0) is not a d-algebra since 0 • y = z ̸= 0, i.e. the condition (d2) is not valid. The algebra (T ; ◦, 0) is a d-algebra and a Q-algebra. A d-algebra X is said to be edge if xX = {0, x} for all x ∈ X. From Example 2, a d-algebra (T ; ◦, 0) is an edge d-algebra. In [8] the authors obtained some properties of edge d-algebras as the following: Proposition 3. [8] Let X be an edge d-algebra. Then x0 = x for all x ∈ X. Now we describe the concept of edge in Q-algebra motivated by [8]. Definition 1. A Q-algebra X is said to be edge if aX = {0, a} for all a ∈ X. Example 3. Let X = {0, x, y, z} and Y = {0, a, b, c, d, f}. Define binary operations ∗ on X and • on Y as the following tables: ∗ 0 x y z 0 0 0 0 0 x x 0 0 0 y y y 0 y z z 0 0 0 • 0 a b c d f 0 0 a f b b b a a 0 b f f f b b c 0 a a a c c b a 0 0 0 d d b a 0 0 0 f f b a 0 0 0 Then (X; ∗, 0) and (Y ; •, 0) are Q-algebras. Since 0X = {0}, xX = {0, x}, yX = {0, y} and zX = {0, z}, then X is an edge Q-algebra. Since aY = {0, a, b, f} ̸= {0, a}, then Y is not edge. A. Anantayasethi et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6940 6 of 14 Next proposition provides a connection between d-algebras and Q-algebras. We show a sufficient condition for a d-algebra to be a Q-algebra. Proposition 4. Every edge d-algebra is a Q-algebra, more specific it is an edge Q-algebra. Proof. Let X be an edge d-algebra. We want to show that X is a Q-algebra. The condition (Q1) and (Q2) follow from the condition (d1) and Proposition 3, respectively. Let x, y, z ∈ X. We calculate (xy)z and (xz)y. Since X is an edge d-algebra, then xy ∈ xX = {0, x}, i.e. xy = 0 or xy = x. If xy = 0, then by (d2) (xy)z = 0z = 0. Since xz ∈ xX = {0, x}, there follows (xz)y ∈ {0y, xy} = {0}. Thus, (xy)z = (xz)y. Now we assume xy = x. Then (xy)z = xz ∈ xX = {0, x}. If xz = 0, then by (d2) (xy)z = xz = 0 = 0y = (xz)y. If xz = x, then (xy)z = xz = x = xy = (xz)y. Therefore, (xy)z = (xz)y for all x, y, z ∈ X. Hence, the condition (Q3) is fulfilled. Thus, X is a Q-algebra. Since X is edge, then X is an edge Q-algebra. The converse of Proposition 4 is not true, i.e. there is an edge Q-algebra which is not an edge d-algebra as shown in the following example. Example 4. Let X = {0, α, η, µ} and a binary operation ∗ be defined on X as the following table. ∗ 0 α η µ 0 0 0 0 0 α α 0 α 0 η η 0 0 η µ µ 0 µ 0 Then (X; ∗, 0) is an edge Q-algebra. Since α ∗ µ = 0 and µ ∗ α = 0 but α ̸= µ, then the condition (d3) is not satisfied. Thus, (X; ∗, 0) is not a d-algebra. Next, we examine some properties of edge Q-algebras. Proposition 5. Let X be an edge Q-algebra and let A and B be non-empty subsets of X. Then the following properties are valid. (i) 0X = {0}. (ii) If 0 ∈ A, then 0 ∈ AB. Proof. (i) It is clear. (ii) Let 0 ∈ A and b ∈ B, then by (i) there follows that 0 = 0b ∈ 0B ⊆ AB. For any a ∈ X, a subset aX normally is not necessarily a subalgebra of X. For example, a set aY = {0, a, b, f} in Example 3 is not a subalgebra of (Y ; •, 0) since b • a = c ̸∈ aY . But for edge Q-algebras we get the following positive result. Proposition 6. If X is an edge Q-algebra, then aX is a subalgebra of X for all a ∈ X. Proof. Let a ∈ X. Then aX = {0, a}. It is easy to verify that 00, a0 and aa are elements in aX. Now we calculate 0a. Since 0a ∈ 0X, by Proposition 5(i) we get 0a = 0 ∈ aX. Therefore, aX is closed and then aX is a subalgebra of X. A. Anantayasethi et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6940 7 of 14 Next, we will examine some conditions that lead the product of subsets of a Q-algebra X to be a subalgebra of X. Proposition 7. Let X be an edge Q-algebra and let ∅ ̸= A,B ⊆ X. If 0 ∈ A or A∩B ̸= ∅ then AB is a subalgebra of X. Proof. Assume that 0 ∈ A. Then by Proposition 5(ii), 0 ∈ AB. Let x, y ∈ AB. Since X is an edge Q-algebra, then xy ∈ xX = {0, x}. There follows that xy = 0 or xy = x. Thus, xy ∈ AB. Therefore, AB is a subalgebra. Assume now that A ∩ B ̸= ∅. By Proposition 1(v), 0 ∈ AB. Using the same argument, we get AB is closed. Hence, AB is a subalgebra of X. The converse of Proposition 7 is not true as shown in the following example. Example 5. Let X = {0, a, b, c} and let a binary operation ∗ be defined on X as follow: ∗ 0 a b c 0 0 0 0 0 a a 0 a 0 b b 0 0 0 c c c c 0 It is a routine to check that (X; ∗, 0) is a Q-algebra. Since 0X = {0}, aX = {0, a}, bX = {0, b} and cX = {0, c}, then X is an edge. Let A = {b} and B = {c}. Then AB = {b ∗ c} = {0} is a subalgebra. But neither 0 ∈ A nor A ∩ B ̸= ∅. Thus, the converse of Proposition 7 is not true. Corollary 1. Let X be an edge Q-algebra and let A,B be non-empty subsets of X. Then the following properties are true. (i) If A is a subalgebra of X, then AB is a subalgebra of X. (ii) If A is an ideal of X, then AB is a subalgebra of X. Proof. (i) Follows directly from Proposition 7 since 0 is a member of A. (ii) Assume that A is an ideal of X. By (I1), 0 is a member of I there follows by Proposition 7 that AB is a subalgebra of X. Corollary 2. Let X be an edge Q-algebra. Then the following properties are true. (i) For any a, b ∈ X, aX ∪ bX is a subalgebra of X. (ii) Let ∅ ̸= Λ ⊆ X. Then ∪ a∈Λ aX is a subalgebra of X. A. Anantayasethi et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6940 8 of 14 Proof. (i) Let a, b ∈ X. Since X is an edge Q-algebra, then aX∪bX = {0, a}∪{0, b} = {0, a, b}. Set A = {0, a, b} and B = {0}, then by Proposition 1(ii), AB = A = {0, a, b} = aX ∪ bX. Since 0 ∈ A, then by Proposition 7, AB = aX ∪ bX is a subalgebra of X. (ii) Let a ∈ Λ ⊆ X. Since X is an edge Q-algebra, then aX = {0, a}. There follows ∪ a∈Λ aX = Λ ∪ {0}. Set A = Λ ∪ {0} and B = {0}. Then AB = Λ ∪ {0} = ∪ a∈Λ aX. Since 0 ∈ A, by Proposition 7 AB = ∪ a∈Λ aX is a subalgebra of X. For a Q-algebra X, we denote the set of all subalgebras of X by Sub(X). Next proposition provides the characterization of subalgebras of an edge Q-algebra and enumerate all of subalgebras of any edge Q-algebra X. Proposition 8. Let X be an edge Q-algebra and let |X| = n for some positive integer n. Then the following conditions are hold: (i) For ∅ ̸= S ⊆ X, S is a subalgebra of X if and only if 0 ∈ S. (ii) |Sub(X)| = 2n−1. Proof. (i) (⇒) It is clear. (⇐) Let S be a non-empty subset of X and assume 0 ∈ S. By Proposition 1(ii), S{0} = S. Set A = S,B = {0} and then by Proposition 7 there follows AB = S{0} = S is a subalgebra of X. (ii) From (i) we conclude that any subset of X containing a constant 0 is a subalgebra of X. There follows that |Sub(X)| = n−1∑ i=0 ( n− 1 i ) = 2n−1. Proposition 9. Let X be an edge Q-algebra. Then Sub(X) is a semigroup with a right identity {0}. Proof. Let A,B,C ∈ Sub(X). By Corollary 1, AB is a subalgebra and then Sub(X) is closed. Since 0 ∈ B, by Proposition 1(iv) we get A ⊆ AB. Since X is edge and 0 ∈ A, then AB ⊆ AX = A ∪ {0} = A. Therefore, A ⊆ AB ⊆ A. Thus, AB = A. There follows (AB)C = AC = A = AB = A(BC). This gives an associative law. Hence, Sub(X) is a semigroup. Moreover, by Proposition 1(ii) A{0} = A this shows that {0} is a right identity. From Proposition 6, we get an information that a set aX is a subalgebra for any element a in an edge Q-algebra X. But this kind of subset need not be an ideal. For example, a subset aX of X in Example 5 is not an ideal since b ∗ a = 0 ∈ aX, a ∈ aX but b ̸∈ aX. Next, we will show a condition for a subset of an edge Q-algebra X in the form aX, a ∈ X to be an ideal of X. A. Anantayasethi et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6940 9 of 14 Proposition 10. Let X be an edge Q-algebra and let a ∈ X. Then aX is an ideal if and only if wa = w for all w ∈ X\{a}. Proof. Let a ∈ X, and assume that aX is an ideal of X. Suppose that there is an element w ∈ X,w ̸= a such that wa ̸= w. Since wa ∈ wX = {0, w} and wa ̸= w, then wa = 0. There follows wa ∈ aX. Since wa ∈ aX, a ∈ aX and aX is an ideal, then w ∈ aX. This gives a contradiction. Hence, for all w ∈ X\{a}, wa = w. For the converse direction, assume wa = w for all w ∈ X\{a}. We want to show that aX = {0, a} is an ideal. If a = 0, then it is obvious that aX is an ideal of X. Assume now a ̸= 0. Let xy ∈ aX and y ∈ aX. If y = 0, then x = x0 = xy ∈ aX. If y = a, then xa ∈ aX. There follows xa = 0 or xa = a. Suppose xa = a. Then x ̸= a otherwise a = xa = aa = 0, a contradiction. Hence, xa = 0. Then by assumption x ̸∈ X\{a}. Therefore, x = a ∈ aX. Altogether, aX is an ideal of X. In a Q-algebra X, a product I1I2 of ideals I1 and I2 of X is not necessarily an ideal as can be seen in the following example. Example 6. Let X = {0, a, b, c, d, f}. Define a binary operation ∗ on X as the following table: ∗ 0 a b c d f 0 0 a c d c c a a 0 d c d d b b c 0 a 0 0 c c b a 0 a a d d c 0 a 0 0 f f c 0 a 0 0 Then (X; ∗, 0) is a Q-algebra. The subsets I1 = {0, a} and I2 = {0, b, d, f} are ideals of (X; ∗, 0). Let consider a product I1I2 = {0, c, d}. Since a ∗ c = c ∈ I1I2 and c ∈ I1I2 but a /∈ I1I2, then I1I2 is not an ideal of (X; ∗, 0). Therefore, the set of all ideals of a Q-algebra is not necessarily closed. In an edge Q-algebra, we get a good outcome, i.e. the product of ideals is again an ideal as obtained in the following proposition. Proposition 11. Let X be an edge Q-algebra. If A and B are ideals of X, then AB is an ideal. Proof. Let A and B be ideals of X. Let a ∈ A. Since 0 ∈ B, by Proposition 1(iv), A ⊆ AB. Since X is an edge, aB ⊆ aX = {0, a}. Thus, we can conclude that AB ⊆ A ∪ {0} = A. Hence, AB = A so that AB is an ideal of X. As a direct consequence of Proposition 11 we have the subsequence corollary. Corollary 3. If A and B are ideals of an edge Q-algebra X, then AB = A. For a Q-algebra X, we denote I(X) as the set of all ideals of X. By Proposition 11 and Corollary 3, we obtain that the set of all ideals of an edge Q-algebra forms a semigroup. A. Anantayasethi et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6940 10 of 14 Proposition 12. Let X be an edge Q-algebra. Then I(X) is a semigroup. Proposition 13. Let X be an edge Q-algebra. Then I(X) is a subsemigroup of Sub(X). Proof. Let I ∈ I(X). By (I1), 0 ∈ I, so that by Proposition 8(i), I is a subalgebra. Thus, I(X) ⊆ Sub(X). Since I(X) is a semigroup and I(X) ⊆ Sub(X), then I(X) is a subsemigroup of Sub(X). We will recall some concepts of semigroup theory. A semigroup S is a left zero semigroup if za = z for all z, a ∈ S. A non-empty subset I of S is a left semigroup ideal (right semigroup ideal) if SI ⊆ I (IS ⊆ I, respectively). If I is both a left semigroup ideal and a right semigroup ideal, then I is a semigroup ideal. A semigroup ideal (left semigroup ideal, right semigroup ideal) I such that I ̸= S is called a proper semigroup ideal (left semigroup ideal, right semigroup ideal). A semigroup S is a left simple semigroup if S has no proper left semigroup ideals. A right simple semigroup and a simple semigroup are defined in an analogous way. For more intensive details in semigroup theory we refer to [18]. Proposition 14. Let X be an edge Q-algebra. Then I(X) is a left zero semigroup. Proof. By Proposition 12, I(X) is a semigroup. Let A,B ∈ I(X). Then by Corollary 3, AW = A. Therefore, I(X) is a left zero semigroup. Moreover, I(X) is a simple semigroup. Proposition 15. Let X be an edge Q-algebra. Then I(X) is a left simple semigroup. Proof. Let P be a left semigroup ideal of a semigroup I(X). Then I(X)P ⊆ P . Let A ∈ I(X) and B ∈ P . Then AB = A by Corollary 3. There follows that I(X) ⊆ I(X)P ⊆ P . Hence, I(X) = P . Therefore, I(X) does not contain a proper semigroup left ideal. Thus, I(X) is a left simple semigroup. Corollary 4. Let X be an edge Q-algebra. Then I(X) is a simple semigroup. Proof. It follows from Proposition 15. 3. Enumeration of Edge Q-algebras In this section, we describe all possible structures of edge Q-algebras of order n, for any positive integer n. To do this we need to construct a Q-algebra as follows: Construction(∗): Let Xn = {x1, x2, x3, . . . , xn} be a set of order n. We define a binary operation on Xn as follow: For xi, xj ∈ Xn, xixj =  x1 if i = j, x1 if i = 1, xi if j = 1, a ∈ {x1, xi} if otherwise. A. Anantayasethi et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6940 11 of 14 From the Construction (∗), in the case i ̸= j and i, j ̸= 1, the product xixj ∈ {x1, xi}, i.e. the product xixj is either x1 or xi. We denote here x ∨ y by ” either x or y”. Then we obtain the following Cayley table: x1 x2 x3 … xi−1 xi xi+1 … xn x1 x1 x1 x1 … x1 x1 x1 … x1 x2 x2 x1 x1 ∨ x2 … x1 ∨ x2 x1 ∨ x2 x1 ∨ x2 … x1 ∨ x2 x3 x3 x1 ∨ x3 x1 … x1 ∨ x3 x1 ∨ x3 x1 ∨ x3 … x1 ∨ x3 ... ... ... ... ... ... ... ... ... ... xi xi x1 ∨ xi x1 ∨ xi … x1 ∨ xi x1 x1 ∨ xi … x1 ∨ xi ... ... ... ... ... ... ... ... ... ... xn xn x1 ∨ xn x1 ∨ xn … x1 ∨ xn x1 ∨ xn x1 ∨ xn … x1 For the set Xn, the Construction (∗) allows us to get 2n 2−3n+2 algebraic struc- tures. Let EQ(Xn) be the set of all algebras which obtained from the Construction (∗). Example 7. Let X3 = {x1, x2, x3}. The following algebras A,B,C and D are obtained from the Construction (∗): x1 x2 x3 x1 x1 x1 x1 x2 x2 x1 x1 x3 x3 x1 x1 x1 x2 x3 x1 x1 x1 x1 x2 x2 x1 x1 x3 x3 x3 x1 A B x1 x2 x3 x1 x1 x1 x1 x2 x2 x1 x2 x3 x3 x1 x1 x1 x2 x3 x1 x1 x1 x1 x2 x2 x1 x2 x3 x3 x3 x1 C D It is not difficult to check that all above tables A,B,C and D are Q-algebras with x1 acts as a constant 0. Let consider the table A. Since x1X3 = {x1}, x2X3 = {x1, x2} and x3X3 = {x1, x3}, X3 is an edge Q-algebra. Similarly, we get that tables B,C and D are edge Q-algebras. Moreover, |EQ(X3)| = 23 2−3(3)+2 = 22 = 4 and EQ(X3) = {A,B,C,D}. Next proposition reveals that any algebraic structure obtained from the Con- struction (∗) is an edge Q-algebra. Proposition 16. Let Xn = {x1, x2, . . . , xn}. For any A ∈ EQ(Xn), A is an edge Q- algebra. Proof. Let A be any algebra in EQ(Xn). Let xi, xj , xk ∈ Xn. Then we get xix1 = xi and xixi = x1. Therefore, the conditions (Q1) and (Q2) hold and x1 acts as a constant 0. Next, we calculate (xixj)xk and (xixk)xj. Observe that xixj ∈ {x1, xi} and xixk ∈ {x1, xi}. A. Anantayasethi et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6940 12 of 14 If xixj = x1, then (xixj)xk = x1xk = x1 and (xixk)xj ∈ {x1xj , xixj} = {x1}. It follows that (xixj)xk = x1 = (xixk)xj. If xixj = xi, then (xixj)xk = xixk. If xixk = x1, then (xixj)xk = xixk = x1 = x1xj = (xixk)xj. If xixk = xi, then (xixj)xk = xixk = xi = xixj = (xixk)xj. Altogether, (Q3) is fulfilled. Since xiXn = {x1, xi} for all xi ∈ Xn, then the edge property is hold. Altogether, A is an edge Q-algebra. From Example 7, if we replace x1 by 0, then we get the following edge Q-algebras: 0 x2 x3 0 0 0 0 x2 x2 0 0 x3 x3 0 0 0 x2 x3 0 0 0 0 x2 x2 0 0 x3 x3 x3 0 A B 0 x2 x3 0 0 0 0 x2 x2 0 x2 x3 x3 0 0 0 x2 x3 0 0 0 0 x2 x2 0 x2 x3 x3 x3 0 C D Let consider an edge Q algebra X of order n, for any positive integer n. For any element a ∈ X, aX = {0, a}. There follows that ax ∈ {0, a} for all x ∈ X. Hence, there is an algebraic structure Y ∈ EQ(Xn) which is coinciding to X. Proposition 17. Let n be a positive integer and let Y be an edge Q-algebra of order n. Then Y is isomorphic to X for some X ∈ EQ(Xn). Combining Proposition 16 and Proposition 17 we get: Theorem 1. EQ(Xn) is the set of all edge Q-algebras of order n and hence, there are precisely 2n 2−3n+2 different edge Q-algebras of order n. 4. Conclusion We have introduced the concept of edge Q-algebras and explored their prop- erties. We obtained some results related to the concepts of subalgebras and ideals. The product of subalgebras of an edge Q-algebra is again a subalgebra, offered in Corollary 1. Similarly, the product of ideals of edge Q-algebra is also an ideal. Moreover, the set of all subalgebras of an edge Q-algebra X, Sub(X), and the set of all ideals, I(X) form a semigroup as shown in Proposition 9 and Proposition 12, respectively. We enumerated all of subalgebras of edge Q-algebra X. Proposition 8 shows the total number of all sub- algebras of X and |Sub(X)| = 2|X|−1. The construction of edge Q-algebras is presented. We also showed the total number of all structures of edge Q-algebras, as in Theorem 1. There are 2n 2−3n+2 structures of edge Q-algebras of order n. For further study, one can A. Anantayasethi et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6940 13 of 14 consider edge Q-algebras based on the following concepts: - Hyper algebras; - Filters and another kinds of ideals; - Fuzzy subalgebras, fuzzy ideals; - Homomorphisms and isomorphisms; - Connections with related algebras. Acknowledgements This research project was financially supported by Mahasarakham University, Thailand. References [1] Y. Imai and K. Iseki. On axiom system of proposaitional calculi. xiv. Proceedings of the Japan Academy, 42:19–22, 1966. [2] K. Iseki. An algebra related with a propositional calculus. Proceedings of the Japan Academy, 42:26–29, 1966. [3] K. Iseki and S. Tanaka. 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