konuralp j. math. Konuralp Journal of Mathematics 2147-625X Mehmet Zeki SARIKAYA Applied Mathematics (Other) Uygulamalı Matematik (Diğer) Fuzzy $(m,n)$-Hyperideals and Regularity in Ordered Semihypergroups https://orcid.org/0009-0009-6617-9668 Usmani Zainab Integral University https://orcid.org/0000-0003-3305-2029 Mahboob Ahsan SR University https://orcid.org/0000-0002-1739-5198 Khan Mohammad Aasim Integral University, Lucknow 04 30 2026 14 1 108 119 01 13 2026 04 01 2026 Copyright © 2013, Konuralp Journal of Mathematics 2013 Konuralp Journal of Mathematics

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.

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