All the cardinal numbers of ideals G-part of Q-algebras

Koppitz, Jörg; Anantayasethi, Ananya

doi:10.47974/JDMSC-2609

Open Access Research Article

All the cardinal numbers of ideals G-part of Q-algebras

Jörg Koppitzkoppitz@math.bas.bgInstitute of Mathematics and InformaticsBulgarian Academy of SciencesSofia, 14476, BulgariaView full profile → , ^*Ananya AnantayasethiCorresponding authorananya.a@msu.ac.thAlgebra and Application Research UnitMahasarakham UniversityMahasarakham, 44150, ThailandView full profile →

* Corresponding author · click or hover a name for details

pp. 363–373Vol. 29Issue 1January 2026DOI: 10.47974/JDMSC-2609 Crossmark XML

Received:: 07 May 2025
Published Online:: 15 Jan 2026
Article type:: Research Article
Language:: EN
Article no.:: JDMSC-2609
Pages:: 363–373

Abstract

A Q-algebra (ξ; *, 0) is a generalization of the concepts of BCK, BCI-algebras, introduced by Neggers et al. [1]. As for BCK, BCI-algebras, the concept ideal is defined for Q-algebras [1]. In this paper, we show that for a given integer n, and for all k ∈ {0, 1, …, log2n – 1}, there is a Q-algebra ξ of cardinality n such that the G-part G(ξ) = {υ ∈ ξ : 0 * υ = υ} is an ideal of ξ with |G(ξ)|= 2k. Moreover, we show that if G(ξ) is an ideal of a Q-algebra ξ and |ξ|= n, then |G(ξ)|= 2k, for some k ∈ {0, 1, 2, …, log2 n – 1}.

Keywords

Q-algebra Ideals G-part Cardinality Abelian group

Subject Classifications

Primary 03G2503G2706F3520K01

References

[1] J. Neggers, S. Ahn, and H.S. Kim, “On Q-algebras,” Int. J. Math. Sci., vol. 27, no. 12, pp. 749-757 (2001).
[2] Y. Imai and K. Iseki, “On axiom systems of propositional calculi. XIV,” Proceeding of Japan Academy, vol. 42, no. 1, pp. 19-22 (1966).
[3] Y. Arai, K.Iseki, and S. Tanaka, “Characterizations of BCI, BCK-algebra,” Proceeding of Japan Academy, vol. 42, pp. 105-107 (1966).
[4] K. Iseki and S. Tanaka, “An introduction to theory of BCK-algebra,” Mathematics Japonica, vol. 23, pp. 1-26 (1978).
[5] Y. B. Jun, S. M. Hong, E. H. Roh, and J. Meng, “ On the BCI-G part of BCI-algebras (III),” Comm. Korean. Math. Soc., vol. 9, no. 3, pp. 531-538 (1994).
[6] M. M. Takallo, R. A. Borzooei, G. R. Rezaei, and Y. B. Jun, “True-False ideals of BCK/BCI-algebras,” Journal of Discrete Mathematical Sciences and Cryptography, vol. 27, no. 1, pp. 45-62 (2024). DOI: 10.47974/JDMSC-1544.
[7] S. Ahn and K. So, “On medial Q-algebras,” Commun. Korean. Math. Soc., vol. 35, no. 3, pp. 365-372, (2010).
[8] S. Ahn and S. E. Kang, “The role of T(X) in the ideal theory of Q-algebras,” Honam Mathematical J., vol. 32, pp. 515-523 (2010).
[9] S. M. Mostafa, M. A. Naby, and O. R. Elgendy, “Fuzzy Q-ideals in Q-algebras,” World Applied Programming, vol. 2, no. 2, pp. 69-80 (2012).
[10] H. K. Abdullah and M. Tach, “Prime ideal in Q-algebra,” International Journal of Academic and Applied Research, vol. 10, pp.79-87 (2020).
[11] A. Anantayasethi, T. Kunawat, and P. Moonnipa, “Relation between G-part and atoms in Q-algebras,” European Journal of Pure and Applied Mathematics, vol. 17, no. 4, pp. 3268-3276 (2024).
[12] A. Anantayasethi and J. Koppitz, Characterization of ideals of Q-algebras related to its G-part, Journal of Discrete Mathematical Sciences and Cryptography, vol. 28, no. 1, pp. 131-141 (2025).
[13] S. Das, R. Das, C. Granados, and A. Mukherjee, “Pentapartitioned neutrosophic Q-ideals of Q-algebra,” Neutrosophic Sets and Systems, vol. 41, pp. 52-63 (2021).
[14] H. K. Abdullah and M. Tach, “Intuitionistic fuzzy prime ideal on Q-algebra,” International Journal of Academic and Applied Research, vol.4, no. 10, pp.66-78 (2020).

Views: 151Downloads: 120Citations: 0