Tripolar Fuzzy (m, n)-Quasi-Ideals of Ordered Semigroups

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DOI: https://doi.org/10.28924/2291-8639-24-2026-156
Published: May 20, 2026
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Anothai Phukhaengsi, Pannawit Khamrot, Thiti Gaketem, Tripolar Fuzzy (m, n)-Quasi-Ideals of Ordered Semigroups, Int. J. Anal. Appl., 24 (2026), 156.

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Anothai Phukhaengsi, Pannawit Khamrot, Thiti Gaketem

Abstract

In 2009, M. A. Ansari et al. introduced the concept of \((m,n)\)-quasi-ideals in semigroup theory. Nearly a decade later, in 2018, Rao proposed the framework of tripolar fuzzy sets, which generalizes the classical fuzzy, bipolar fuzzy, and intuitionistic fuzzy set theories. Building upon these developments, this paper introduces and studies the notion of tripolar fuzzy \((m,n)\)-quasi-ideals in semigroups. We explore their fundamental characteristics, analyze their connections with conventional \((m,n)\)-quasi-ideals, and establish relationships between these two structures. In addition, we investigate the notions of minimality, primeness, and semiprimeness within this newly defined framework.

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References

  1. M.A. Ansari, M.R. Khan, J.P. Kaushik, A Note on $(m, n)$-Quasi-Ideals in Semigroups, Int. J. Math. Anal. 3 (2009), 1853–1858.
  2. K.T. Atanassov, Intuitionistic Fuzzy Sets, Fuzzy Sets Syst. 20 (1986), 87–96. https://doi.org/10.1016/s0165-0114(86)80034-3.
  3. T. Changphas, On $(m, n)$-Ideals of an Ordered Semigroup, Int. J. Pure Appl. Math. 100 (2015), 1–5. https://doi.org/10.12732/ijpam.v100i1.1.
  4. V. Chinnadurai, K. Arulmozhi, Characterization of Bipolar Fuzzy Ideals in Ordered Gamma Semigroups, J. Int. Math. Virtual Inst. 8 (2018), 141–156.
  5. T. Gaketem, P. Khamrot, On Some Semigroups Characterized in Terms of Bipolar Fuzzy Weakly Interior Ideals, IAENG Int. J. Comput. Sci. 48 (2021), 250–256.
  6. C.S. Kim, J.G. Kang, J.M. Kang, Ideal Theory of Semigroups Based on the Bipolar Valued Fuzzy Set Theory, Ann. Fuzzy Math. Inform. 2 (2012), 193–206.
  7. P. Khamrot, M. Siripitukdet, On Properties of Generalized Bipolar Fuzzy Semigroups, Songklanakarin J. Sci. Technol. 41 (2019), 405–413.
  8. N. Kuroki, On Fuzzy Ideals and Fuzzy Bi-Ideals in Semigroups, Fuzzy Sets Syst. 5 (1981), 203–215. https://doi.org/10.1016/0165-0114(81)90018-x.
  9. P. Khamrot, A. Iampun, T. Gaketem, Tripolar Fuzzy Ideals Which Coincide in Semigroups, IAENG Int. J. Appl. Math. 55 (2025), 361–368.
  10. P. Khamrot, S. Saekuai, T. Phongthai, T. Gaketem, Ordered Almost $(m, n)$-Ideal and Fuzzy Ordered Almost $(m, n)$-Ideals in Ordered Semigroups, IAENG Int. J. Appl. Math. 55 (2025), 2342–2348.
  11. T. Promai, A. Iampun, T. Gaketem, Tripolar Fuzzy Ideals in Semigroups, IAENG Int. J. Appl. Math. 54 (2024), 2775–2782.
  12. P. Khamrot, A. Iampun, T. Gaketem, Tripolar Fuzzy Bi-Ideals Which Coincide in Semigroups, IAENG Int. J. Appl. Math. 55 (2025), 1972–1979.
  13. T. Gaketem, P. Khamrot, Generalized Interval Valued Fuzzy Ideals in Semigroups, J. Math. Comput. Sci. 34 (2024), 116–127. https://doi.org/10.22436/jmcs.034.02.02.
  14. P. Khramrot, A. Iampan, T. Gaketem, Fuzzy $(m,n)$-Ideals and (n)-Interior Ideals in Ordered Semigroups, Eur. J. Pure Appl. Math. 18 (2025), 5596. https://doi.org/10.29020/nybg.ejpam.v18i1.5596.
  15. P. Khamrot, A. Iampan, T. Gaketem, Bipolar Fuzzy $(m,n)$-Ideals and $n$-Interior Ideals of Semigroups, IAENG Int. J. Comput. Sci. 52 (2025), 598–605.
  16. P. Khramrot, P. Chaisuwan, P. Keawton, C. Wangsamphao, T. Gaketem, Almost $(m,n)$-Quasi-Ideals and Fuzzy Almost $(m,n)$-Quasi-Ideals in Ordered Semigroups, Eur. J. Pure Appl. Math. 18 (2025), 5837. https://doi.org/10.29020/nybg.ejpam.v18i2.5837.
  17. A. Mahboob, M. Al-Tahan, G. Muhiuddin, Characterizations of Ordered Semigroups in Terms of Fuzzy $(m,n)$-Substructures, Soft Comput. 28 (2024), 10827–10834. https://doi.org/10.1007/s00500-024-09880-z.
  18. G. Muhiuddin, A. Mahboob, N.M. Khan, D. Al-Kadi, New Types of Fuzzy $(m,n)$-Quasi-Ideals in Ordered Semigroups, J. Intell. Fuzzy Syst. 41 (2021), 6561–6574.
  19. M.M.K. Rao, Tripolar Fuzzy Interior Ideals of $Gamma$-Semigroup, Ann. Fuzzy Math. Inform. 15 (2018), 199–206. https://doi.org/10.30948/afmi.2018.15.2.199.
  20. M.M.K. Rao, Tripolar Fuzzy Ideals of a $Gamma$-Semiring, Asia Pac. J. Math. 15 (2018), 192–207.
  21. N. Wattanasiripong, J. Mekwian, H. Sanpan, S. Lekkoksung, On Tripolar Fuzzy Pure Ideals in Ordered Semigroups, Int. J. Anal. Appl. 20 (2022), 49. https://doi.org/10.28924/2291-8639-20-2022-49.
  22. N. Wattanasiripong, N. Lekkoksung, S. Lekkoksung, On Tripolar Fuzzy Interior Ideals in Ordered Semigroups, Int. J. Innov. Comput. Inf. Control 18 (2022), 1291–1304.
  23. N. Wattanasiripong, N. Lekkoksung, S. Lekkoksung, On Tripolar Fuzzy Interior Ideals in Ordered Semigroups, Int. J. Innov. Comput. Inf. Control 18 (2022), 1291–1304.
  24. L.A. Zadeh, Fuzzy Sets, Inf. Control. 8 (1965), 338–353. https://doi.org/10.1016/s0019-9958(65)90241-x.
  25. W.R. Zhang, Bipolar Fuzzy Sets and Relations: A Computational Framework for Cognitive Modeling and Multiagent Decision Analysis, in: Proceedings of the First International Joint Conference of The North American Fuzzy Information Processing Society Biannual Conference, IEEE, pp. 305–309. https://doi.org/10.1109/ijcf.1994.375115.