Fuzzy $(m,n)$-Hyperideals and Regularity in Ordered Semihypergroups

Year 2026, Volume: 14 Issue: 1 , 108 - 119 , 30.04.2026

Zainab Usmani , Ahsan Mahboob , Mohammad Aasim Khan

https://izlik.org/JA35YT24SB

Abstract

This paper introduces a novel framework for studying the structure of ordered semihypergroups through the lens of fuzzy set theory, focusing on the concept of fuzzy $(m,n)$-hyperideals. We define and characterize fuzzy left $n$-hyperideals and fuzzy right $m$-hyperideals, including their minimal and maximal forms. A key contribution is the introduction of $(m,n)$-intra-regular ordered semihypergroups, along with the exploration of their properties using fuzzy hyperideals. Additionally, we examine the interplay between fuzzy right $m$-hyperideals, fuzzy left $n$-hyperideals, and fuzzy $(m,n)$-quasi-hyperideals within the setting of $(m,n)$-regular ordered semihypergroups. This study offers new insights into the structural behavior of ordered semihypergroups and lays the groundwork for further investigations in fuzzy hyperstructure theory.

Keywords

Ordered semihypergroup , fuzzy left n-hyperideal , fuzzy right m-hyperideal , fuzzy (m,n) quasi hyperideal

Ethical Statement

It is declared that during the preparation process of this study, scientific and ethical principles were followed and all the studies benefited from are stated in the bibliography.

Thanks

The first and third authors would like to acknowledge Integral University, Lucknow, India, for providing the manuscript number IU/R&D/2025-MCN0003502

References

• [1] Al-Tahan, M. and Davvaz, B., On (m;n)-hyperideals in ordered semihyperrings: Applications to ordered semirings, Journal of Algebra and its Applications, 21(5) (2022). https://doi.org/10.1142/S0219498822501018
• [2] Al-Tahan, M., Davvaz, B., Mahboob, A. and Khan, N., On a generalization of fuzzy filters of ordered semigroups, New Mathematics and Natural Computation, 19(02) (2023), 489-502. https://doi.org/10.1142/S1793005723500187
• [3] Basar, A. and Abbasi, M. Y., On generalized G-hyperideals in ordered G-semihypergroups, Fundamental Journal of Mathematics and Applications, (2) (1) (2019), 18-23.
• [4] Changphas, T. and Davvaz, B., Bi-hyperideals and quasi-hyperideals in ordered semihypergroups, Ital. J. Pure Appl Math, 35 (2015), 493-508.
• [5] Changphas, T., On 0-minimal (m;n)-ideals in an ordered semigroup,Int J Pure and Appl Math. 89(1) (2013), 71-78.
• [6] Davvaz, B., Fuzzy hyperideals in semihypergroups, Italian J. Pure and Appl Math. 8 (2000),67-74.
• [7] Heidari, D. and Davvaz, B., On ordered hyperstructures, Politehn. Univ. Bucharest Sci. Bull, Ser. A, Appl. Math. Phys. 73(2) (2011),85-96.
• [8] Kehayopulu, N. and Tsingelis, M., Fuzzy sets in ordered groupoids, Semigroup Forum 65 (2002),128-132.
• [9] Lajos, S., Notes on (m;n)-ideals I, Proc. Japan Acad. 39 (1963),419-421.
• [10] Lajos, S., Notes on (m;n)-ideals II, Proc. Japan Acad. 40 (1964),631-632.
• [11] Mahboob, A., Khan, N.M. and Davvaz, B., Structural properties for (m,n)-quasi-hyperideals in ordered semihypergroups, Tbilisi Math. J. 11(4) (2018), 145-163.
• [12] Mahboob, A., Al-Tahan, M. and Muhiuddin, G., Characterizations of ordered semigroups in terms of fuzzy (m;n)-substructures, Soft Computing, 28 (2024), 10827–10834. https://doi.org/10.1007/s00500-024-09880-z
• [13] Mahboob, A., Davvaz, B. and Khan, N.M., Fuzzy (m;n)-ideals in semigroups, Computational and Applied Mathematics, 38, 189 (2019), 1-18.
• [14] Marty, F., Sur une generalization de la notion de group, Stockholm, 8th Congres Math. Scandinaves, Stockholm (H.Ohlssons boktryckeri) (1934), 45-49.
• [15] Muhiuddin, G., Mahboob, A., Khan, N. M. and Al-Kadi, D., New types of fuzzy (m;n)-ideals in ordered semigroups, Journal of Intelligent & Fuzzy Systems, 41(6) (2021), 6561-6574.
• [16] Pibaljommee, B. and Davvaz, B., Characterizations of (fuzzy) Bi-hyperideals in Ordered Semihypergroups, Journal of Intelligent & Fuzzy Systems, 28 (2015), 2141–2148.
• [17] Rosenfeld, A., Fuzzy groups, J Math Anal Appl. 35 (1971), 512-517.
• [18] Usmani, Z., Mahboob, A., Al-Tahan, M. and Khan, M. A., Fuzzy (m;n)-Quasi-Hyperideals in Ordered Semihypergroups: A Novel Approach, New Mathematics and Natural Computation, Vol. 22, No. 2 (2026), 659–674. https://doi.org/10.1142/S1793005726500328
• [19] Usmani, Z., Muhiuddin, G., Mahboob, A. and Khan, M.A., Interval Valued m-Polar Fuzzy Ideals in Ordered Semigroups: A Structural Analysis, Journal of Fuzzy Extension and Applications, (2025). https://doi.org/10.22105/jfea.2025.470331.1567
• [20] Zadeh, L. A., Fuzzy sets, Inf Control 8 (1965), 338-353.