EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 4, 2024, 3268-3276 ISSN 1307-5543 – ejpam.com Published by New York Business Global Relations between G-part and Atoms in Q-algebras Ananya Anantayasethi1,∗, Tanabat Kunawat1, Panuwat Moonnipa1 1 Department of Mathematics, Science Faculty , Mahasarakham University, Mahasarakham, 44150, Thailand Abstract. In this work the concepts of G-part G(X), atoms and strong atoms in Q-algebras are discussed. We provide some connections among G(X), set of all atoms and set of all strong atoms of X which related to the concept of ideals. We prove that a Q-algebra X does not contain a strong atom whenever it contains a non-zero ideal G(X). In addition, we provide some conditions that make a set of atoms an abelian group. 2020 Mathematics Subject Classifications: 03G25, 03G27, 06F35, 20K01 Key Words and Phrases: Q-algebra, ideals, G-part, G(X), A(X), atoms, strong atom, abelian group 1. Introduction and Preliminaries In 1996, two Japanese mathematicians Y. Imai and K. Iseki [6] introduced a class of logical algebra which is called a BCK-algebra. In the same year the notion of BCI- algebra was introduced by K. Iseki [7], which is a super class of BCK-algebra. For more informations of BCK-algebra and BCI-algebra see also [ [16], [8]]. It is natural to study a generalization of these algebras. Later on there is a rich literature involved with BCK- algebra and BCI-algebra. A BCH-algebra was emerged in 1983 by Q. P. Hu and X. Li which is a generalization of BCK, BCI-algebras. Later, J. Neggers et al. introduced many algebras which related to BCK, BCI-algebras such as d-algebra, B-algebra and Q- algebra. They examined some relations and some properties of theses algebras. In 2001, J. Neggers et al. [9] introduced a new generalization of BCI-algebra and BCK-algebra. This new algebra was known as Q-algebra which is also a generalization of BCH-algebra. In [9] the authors generalized some properties and theorems discussed in BCI-algebra. The concept of quadratic Q-algebra is also offered in [9]. A Q-algebra consists of a non- empty set X and a constant 0 ∈ X together with a binary operation ∗ on X that yields the following: for all x, y, z ∈ X (Q1) x ∗ x = 0, ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5403 Email addresses: ananya.a@msu.ac.th (A. Anantayasethi), 64010213029@msu.ac.th (T. Kunawat), 64010213047@msu.ac.th (P. Moolnipa) https://www.ejpam.com 3268 Copyright: © 2024 The Author(s). (CC BY-NC 4.0) A. Anantayasethi, T. Kunawat, P. Moolnipa / Eur. J. Pure Appl. Math, 17 (4) (2024), 3268-3276 3269 (Q2) x ∗ 0 = x, (Q3) (x ∗ y) ∗ z = (x ∗ z) ∗ y. For convenience, we write xy instead of x ∗ y. Since then, there are many authors working on Q-algebras [ see [2], [1], [3], [14], [12], [10], [11], [13]]. In [9], the concepts of ideal and G-part were established. A non-empty subset I of a Q-algebra X is an ideal if the following conditions were fulfilled: (I1) 0 ∈ I, (I2) xy ∈ I and y ∈ I imply x ∈ I. A set {0} and X are always ideals of X. A subset G(X) := {x ∈ X | 0x = x} of a Q- algebra X is called a G-part of X. In 2004, S. S. Ahn et al. [14] introduced the notion of implicative Q-algebra which is a Q-algebra with (xy)(yz) = (xy)z for all x, y, z ∈ X. Many mathematicians from Korea and Egypt studied on mappings of Q-algebras, namely: R-maps, L-maps, right fixed maps and fuzzy set [ see [11], [14], [13]]. In 2010, S. S. Ahn and K. So [4] considered homomorphims and congruence in Q-algebras. The authors in [4] provided some decompositions of ideals in Q-algebras. Recently, the concept of ideal is again in a spotlight. Various kinds of ideals were discussed. Q-ideal, prime ideal, fuzzy ideal, intuitionistic fuzzy prime ideal, G-part ideal were studied in [2], [12], [10], [13]. In the year 2001, D. Sun [15] introduced the concept of atom and strong atom in BCK- algebra. He proved that a set of all strong atoms and together with zero element is an ideal of BCK-algebra X. In 2010, S. S. Ahn and S. E. Kang [3] introduced the concepts of atoms in Q-algebra. An atom of X is an element a ∈ X satisfying: for x ∈ X, xa = 0 implies x = a. A set of all atoms of X is denoted by A(X). Some properties of atoms are provided in [3]. The authors showed that if every non-zero element of X is an atom, then any subalgebra of X is an ideal. A subalgebra of a Q-algebra X is a non-empty subset I of X with ab ∈ I for all a, b ∈ I. Moreover, the authors in [3] proved that if every non-zero element of X is an atom, then any subalgebra of X is an ideal of X. Example 1. Let X = {0, a, b, c, d} and Y = {0, a, b}. The binary operations ∗ and • be defined on X and Y as the following tables: ∗ 0 a b c d 0 0 a c b b a a 0 b c c b b c 0 a a c c b a 0 0 d d b a 0 0 • 0 a b 0 0 a a a a 0 0 b b 0 0 It is a routine to check that (X; ∗, 0) and (Y ; •, 0) are Q-algebras. It is easy to see that G(X) = {0, a}, A(X) = {0, a, b} and G(Y ) = {0, a}, A(Y ) = {0}. Moreover, we get that G(X) is an ideal and a subalgebra of X. In this paper, we examine the properties of atoms and strong atoms. We also show some relations between a set G-part G(X), a set of all atoms A(X) and a set of all strong atoms of a Q-algebra X that involve with ideal property. Now we will review some properties and theorems that we will use later. In [9] and [3] gave us some calculations and showed a left cancellation law in a Q-algebra X. Lemma 1. [9] Let X be a Q-algebra and a, b, c ∈ X. If ab = ac, then 0b = 0c. A. Anantayasethi, T. Kunawat, P. Moolnipa / Eur. J. Pure Appl. Math, 17 (4) (2024), 3268-3276 3270 Corollary 1. [9] A left cancellation law holds in G(X), i. e. for all a, b, c ∈ G(X), ab = ac implies b = c. Lemma 2. [3] Every Q-algebra X satisfies the following property: 0(xy) = (0x)(0y) for all x, y ∈ X. In [10], some informations and calculations in G(X) are presented. Proposition 1. [10] Let X be a Q-algebra and x ∈ X. Then 0x ∈ G(X) if and only if (0x)x = 0. Proposition 2. [10] Let X be a Q-algebra. If a, b ∈ G(X), then ab = ba. Proposition 3. [10] Let X be a Q-algebra and a, b, c ∈ G(X). Then the following three properties hold: (i) If a ̸= b, then ab /∈ {0, a, b} for a ̸= 0 and b ̸= 0. (ii) If ab = c, then ac = b and bc = a. (iii) xa ̸= x for all 0 ̸= x ∈ X and a ̸= 0. 2. G-part and Atoms In this section, we investigate some properties of a set G-part G(X), atoms and strong atoms of a Q-algebra X. We also present some connections among them. First, we will mention some results of atoms in [3]. Theorem 1. [3] Let X be a Q-algebra. Then for all x, z, u of X, the following conditions are equivalent: (i) x is atom; (ii) x = z(zx); (iii) (zu)(zx) = xu. Theorem 2. [3] Let X be a Q-algebra and x ∈ X. If x is an atom of X, then the following properties are satisfied: (iv) 0(zx) = xz for all z ∈ X. (v) 0(0x) = x. The converse of Theorem 2 is not true. The following example is a counterex- ample. Example 2. Consider a Q-algebra X from Example 1. For all z ∈ X, we get that 0(zc) = cz but an element c is not an atom of X. Hence, the converse of Theorem 2(iv) is not ture. Beside that, the converse of (v) is also not ture since 0(0c) = 0b = c but c is not an atom. From Theorem 2 we get that every atom of Q-algebra X is a product of 0 and some element of X as the following: A. Anantayasethi, T. Kunawat, P. Moolnipa / Eur. J. Pure Appl. Math, 17 (4) (2024), 3268-3276 3271 Corollary 2. Let X be a Q-algebra. If a is an atom of X, then a = 0x for some x ∈ X. The following proposition shows some more properties of atoms in Q-algebras. Proposition 4. Let X be a Q-algebra and let a, b be atoms of X. Then the following properties hold: (i) a(xb) = b(xa) for all x ∈ X. (ii) (ax)(yb) = (bx)(ya) for all x, y ∈ X. Proof. Assume that a and b are atoms of X. (i): Let x ∈ X. Then by Theorem 1(ii), we get that a = x(xa) and b = x(xb). Then by (Q3) there follows that a(xb) = (x(xa))(xb) = (x(xb))(xa) = b(xa). (ii): Let x, y ∈ X. Then by (Q3) and (i) we get (ax)(yb) = (a(yb))x = (b(ya))x = (bx)(ya). The converse of Proposition 4 is not true as seen in the following example. Example 3. Consider a Q-algebra X from Example 1. Let us focus on elements c and b of X. We get that b(xc) = c(xb) for all x ∈ X but c is not an atom of X. Hence, the converse of Proposition 4(i) is not true. Proposition 5. Let X be a Q-algebra. Every element of X is an atom if and only if a(xb) = b(xa) for all a, b, x ∈ X. Proof. (⇒) Follows from Proposition 4(i). (⇐) Let z ∈ X. Then by assumption we get that z(xx) = x(xz) for all x ∈ X. There follows that z = z0 = z(xx) = x(xz) for all x ∈ X. Then by Theorem 1(ii), z is an atom of X. In 2001, D. Sun [15] introduced the concept of strong atoms in BCK-algebra. We will apply a concept of strong atom to Q-algebras in a similar way. Let a be an atom of a Q-algebra X. An element a is called a strong atom if a ̸= 0 and ax = a for all x ∈ X and x ̸= a. We denote a set SA(X) as follows: SA(X) = { a ∈ A(X) | a is a strong atom of X } ∪ {0}. There is a connection between strong atoms and G-part of X. The following properties show that X does not contain any strong storm whenever X contains G-part which is an ideal with the cardinality greater or equal to 2. First, we need the following proposition: Proposition 6. [5] Let X be a Q-algebra with |X| = n and G(X) ̸= X. If G(X) is an ideal of X, then |G(X)| ≤ n 2 . Proposition 7. Let X be a Q-algebra. If G(X) is an ideal and |G(X)| = 2, then SA(X) = {0}. Proof. Assume that G(X) is an ideal of X and |G(X)| = 2. We assume that G(X) = {0, a}. Suppose that a is a strong atom ofX. SinceG(X) is an ideal, then by Proposition 6, A. Anantayasethi, T. Kunawat, P. Moolnipa / Eur. J. Pure Appl. Math, 17 (4) (2024), 3268-3276 3272 we get |G(X)| ≤ |X| 2 . Therefore, |X| ≥ 4. Let b ∈ X such that b /∈ G(X). Since a is a strong atom, then ab = a. It follows that 0b = (aa)b = (ab)a = aa = 0. Since b /∈ G(X), a ∈ G(X) and G(X) is an ideal, then ba /∈ G(X). Moreover, by Proposition 3(iii), we get that ba ̸= b. Therefore, ba /∈ {0, a, b}. We may assume that ba = c for some c ∈ X\{0, a, b}. Similarly, we get ac = a and ca /∈ {0, a, c}. Then we get cb = (ba)b = (bb)a = 0a = a. It follows that a = ac = (cb)c = (cc)b = 0b = 0, a contradiction. Hence, a /∈ SA(X). Let x ∈ X \G(X). Since a ∈ G(X) and x ̸= 0, then xa ̸= x by by Proposition 3(iii). It follows that x /∈ SA(X). Altogether, SA(X) = {0}. Proposition 8. Let X be a Q-algebra. If |G(X)| ≥ 3, then SA(X) = {0}. Proof. Assume that |G(X)| ≥ 3. Then there are a, b ∈ G(X) such that a, b /∈ {0} and a ̸= b. Let x ∈ X and x ̸= 0. If x = a, then xb = ab /∈ {0, a, b} by Proposition 3(i). Therefore, xb ̸= x there follows that x /∈ SA(X). If x ̸= a, then by Proposition 3(i), xa ̸= x. Thus, x /∈ SA(X). Altogether, we get SA(X) = {0}. Proposition 7 and Proposition 8 give the following theorem: Theorem 3. Let X be a Q-algebra and G(X) ̸= {0}. If G(X) is an ideal, then SA(X) = {0}. It is clear that a set of all atoms of a Q-algebra X is not closed. But if we focus on a set of strong atoms, we get that the product of strong atoms is again a strong atom. It follows that SA(X) is a subalgebra of X. Proposition 9. Let X be a Q-algebra. Then SA(X) is a subalgebra of X. Proof. If |SA(X)| ≤ 2, then it is clear that SA(X) is a subalgebra. Assume now that |SA(X)| ≥ 3. Let a, b ∈ SA(X). If b = 0, then ab = a0 = a ∈ SA(X). If a ̸= 0 and b ̸= 0, then ab = a since a is a strong atom. Therefore, ab = a ∈ SA(X). If a = 0 and b ̸= 0, then ab = 0b. Since |SA(X)| ≥ 3, then there is a strong atom c such that c /∈ {0, b}. Then cb = c. By Proposition 4(i) we get that ab = 0b = 0(bc) = c(b0) = cb = c. Therefore, ab ∈ SA(X). Altogether, we get SA(X) is a subalgebra of X. Next we will examine some properties of a set of all atoms A(X) of any Q- algebra X. In general, a set A(X) need not to be closed and also need not to be an ideal of X as the following example. Example 4. Let X = {0, a, b, c, d, f} and let a binary operation ∗ be defined on X as the following: ∗ 0 a b c d f 0 0 a c b b b a a 0 b c c c 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 A. Anantayasethi, T. Kunawat, P. Moolnipa / Eur. J. Pure Appl. Math, 17 (4) (2024), 3268-3276 3273 It is a routine to check that (X; ∗, 0) is a Q-algebra. It is easy to see that A(X) = {0, a, b}. We get that A(X) is not a subalgebra since 0, b ∈ A(X) but 0b = c /∈ A(X). Moreover, A(X) is not an ideal of X. Indeed, db = a ∈ A(X) and b ∈ A(X) but d /∈ A(X). From Example 4, let we mention some errors in [3], namely [ Corollary 3.6 ]: ” Let X be a Q-algebra. If a is an atom of X, then for all x in X, ax is an atom. Hence, A(X) is a subalgebra of X. For every x of X, there is an atom a such that ax = 0, i.e. every Q-algebra is generated by atoms.” is invalid. The mistakes show in Example 4. Next, we investigate some relations between G-part G(X) and set of all atoms A(X). We know that G(X)∩A(X) ̸= ∅ since an element 0 is an atom and 00 = 0 ∈ G(X). The set G(X) need not to be a subset of A(X) and vice versa. From Example 1 we have that G(X) ⊆ A(X) and A(Y ) ⊆ G(Y ). Our aim is to find some conditions that yield previous inclusions. Next proprosition shows a sufficient condition of an element of G(X) to be an atom of X. Proposition 10. Let X be a Q-algebra. If G(X) is an ideal of X, then G(X) ⊆ A(X). Proof. Assume that G(X) is an ideal of X. Let a ∈ G(X). If a = 0, then a ∈ A(X). Now we assume that a ̸= 0. Suppose that there is an element w ∈ X,w ̸= a such that wa = 0. Since a ∈ G(X), then 0a = a there follows that w ̸= 0. Since wa = 0 ∈ G(X), a ∈ G(X) and G(X) is ideal, then w ∈ G(X). Now, there are 0, a and w belong to G(X) and wa = 0, then by Proposition 3(ii) we get that a0 = w. Thus, by (Q2) we get that w = a0 = a, a contradiction. Hence, wa = 0 implies w = a. This gives a is an atom of X. Altogether, we get G(X) ⊆ A(X). The converse of Proposition 10 is not true, i.e. if all members of G(X) are atoms of X, then G(X) need not to be an ideal of X. The following example is the counterexample of the converse. Example 5. [13] Let consider a Q-algebra X, defined as the following table: ∗ 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 0 3 0 4 4 4 4 4 4 0 It is not difficult to verify that G(X) = {0, 4} and A(X) = {0, 4}. Then G(X) ⊆ A(X). But G(X) is not an ideal of X. Indeed, 2(4) = 2 ∈ G(X) and 4 ∈ G(X) but 2 /∈ G(X). Let A and B be non-empty subset of a Q-algebra X. We define AB as following: AB = {ab | a ∈ A, b ∈ B}. Then we get some important informations of G(X) and A(X): Remark 1. Let X be a Q-algebra. Then we get: (i) G(X) ⊆ G(X)G(X), A(X) ⊆ A(X)A(X) and G(X) ⊆ G(X)A(X). A. Anantayasethi, T. Kunawat, P. Moolnipa / Eur. J. Pure Appl. Math, 17 (4) (2024), 3268-3276 3274 (ii) G(X) ⊆ A(X)G(X), A(X) ⊆ A(X)G(X) and G(X) ∪A(X) ⊆ A(X)G(X). (iii) G(X) ⊆ G(X)A(X) ∩A(X)G(X). (iv) A(X) ⊆ A(X)A(X) ∩A(X)G(X) = A(X) ( A(X) ∩G(X) ) . A set of all atoms A(X) need not to be closed, i.e. in general A(X) ̸= A(X)A(X). From Remark 1(iv), we have that A(X) ⊆ A(X) ( A(X) ∩ G(X) ) . It follows that every atom of X can be written in the form of products of atoms. But the product za of atom z and atom a need not to be atom as seen from Example 4. Next lemma shows the condition that gives equality of Remark 1(iv). Proposition 11. Let X be a Q-algebra. If A(X) is an ideal of X, then A(X) = A(X) ( A(X) ∩G(X) ) . Proof. Assume that A(X) is an ideal of X. Let z ∈ A(X) and a ∈ A(X) ∩ G(X). Then we get a = 0a = (zz)a = (za)z. It follows that (za)z = a ∈ G(X) ⊆ A(X). Since (za)z ∈ A(X), a ∈ A(X) and A(X) is an ideal, then za ∈ A(X). Therefore, A(X) ( A(X) ∩G(X) ) ⊆ A(X). The inclusion A(X) ⊆ A(X) ( A(X) ∩G(X) ) follows from Remark 1(iv). Hence, A(X) = A(X) ( A(X) ∩G(X) ) . As a consequence of Proposition 11, the product ab of atom a and atom b with 0b = b is again an atom of X. Proposition 12. Let X be a Q-algebra. If A(X) is an ideal of X, then A(X) ∩G(X) is an abelian group. Proof. Let x, y, z ∈ A(X) ∩G(X). Then by Lemma 2 we get that 0(xy) = (0x)(0y) = xy. Therefore, xy ∈ G(X). Since x ∈ A(X) and y ∈ A(X)∩G(X), then by Proposition 11 we get xy ∈ A(X). Thus, xy ∈ A(X) ∩ G(X). The commutative property follows from Proposition 2. Since the commutative property is hold, then we get (xy)z = (yx)z = (yz)x = x(yz). Hence, an associative law is hold. Moreover, 0 ∈ A(X) ∩ G(X), by (Q1) and x ∈ G(X) we get x0 = x = 0x. Therefore, 0 is an identity of A(X) ∩ G(X). An inverse property follows from (Q2). Altogether, we get that A(X) ∩ G(X) is an abelian group. Proposition 13. Let X be a Q-algebra. If A(X) ⊆ G(X) and A(X) is an ideal of X, then (i) A(X) is a subalgebra of X (ii) A(X) is an abelian group Proof. (i) Since A(X) ⊆ G(X), then A(X) ∩G(X) = A(X). Then by Proposition 11 we get that A(X) = A(X) ( A(X) ∩G(X) ) = A(X)A(X). Hence, A(X) is a subalgebra of X. (ii) Since A(X) ∩ G(X) = A(X), then by Proposition 12 we get that A(X) is an abelian group. REFERENCES 3275 Proposition 14. Let X be a Q-algebra. If G(X) and A(X) are ideals of X, then A(X)G(X) is an ideal of X. Proof. Assume that G(X) and A(X) are ideals of X. Then by Proposition 10 we get that G(X) ⊆ A(X). Since A(X) is an ideal, then A(X) = A(X) ( A(X) ∩ G(X) ) . There follows that A(X) = A(X)G(X). Hence, A(X)G(X) is an ideal of X. 3. Conclusion The concept of ideal plays an important role in studying Q-algebra structures. Many mathematicians examine various subsets of a Q-algebra which are ideals. In this work, we obtain information that all elements of G(X) are atoms whenever G(X) is an ideal. Moreover, we get that a Q-algebra X such that G(X) is an ideal and G(X) ̸= {0}, does not contain a strong atom. For future study one can investigate when a set of all atoms A(X) is an ideal of X and which conditions that make X contains both non-zero atoms and strong atoms. Also, for any Q-algebra X one can find the sufficient condition of A(X) to be an ideal of X. Acknowledgements This research project was financially supported by Mahasarakham University, Thailand. References [1] H. K. Abdullah and M. Tach. Intuitionistic fuzzy prime ideal on q-algebras. Inter- national Journal of Academic and Applied Research, 4(10):66–78, 2020. [2] H. K. Abdullah and M. Tach. Prime ideal in q-algebra. 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