Tbilisi Mathematical Journal

Structural properties for $(m,n)$-quasi-hyperideals in ordered semihypergroups

Ahsan Mahboob, Noor Mohammad Khan, and Bijan Davvaz

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Abstract

In this paper, we first introduce the notion of an $(m,n)$-quasi-hyperideal in an ordered semihypergroup and, then, study some properties of $(m,n)$-quasi-hyperideals for any positive integers $m$ and $n$. Thereafter, we characterize the minimality of an $(m,n)$-quasi-hyperideal in terms of $(m,0)$-hyperideals and $(0,n)$-hyperideals respectively. The relation $\mathcal{Q}_m^n$ on an ordered semihypergroup is, then, introduced for any positive integers $m$ and $n$ and proved that the relation $\mathcal{Q}_m^n$ is contained in the relation $\mathcal{Q}=\mathcal{Q}_1^1$. We also show that, in an $(m,n)$-regular ordered semihypergroup, the relation $\mathcal{Q}_m^n$ coincides with the relation $\mathcal{Q}$. Finally, the notion of an $(m,n)$-quasi-hypersimple ordered semihypergroup is introduced and some properties of $(m,n)$-quasi-hypersimple ordered semihypergroups are studied. We further show that, on any $(m,n)$-quasi-hypersimple ordered semihypergroup, the relations $\mathcal{Q}_m^n$ and $\mathcal{Q}$ are equal and are universal relations.

Article information

Source
Tbilisi Math. J., Volume 11, Issue 4 (2018), 145-163.

Dates
Received: 6 June 2018
Accepted: 8 October 2018
First available in Project Euclid: 4 January 2019

Permanent link to this document
https://projecteuclid.org/euclid.tbilisi/1546570891

Digital Object Identifier
doi:10.32513/tbilisi/1546570891

Mathematical Reviews number (MathSciNet)
MR3954213

Subjects
Primary: 20N20: Hypergroups

Keywords
ordered semihypergroup $(m,0)$-hyperideal $(0,n)$-hyperideal $(m,n)$-quasi-hyperideal

Citation

Mahboob, Ahsan; Khan, Noor Mohammad; Davvaz, Bijan. Structural properties for $(m,n)$-quasi-hyperideals in ordered semihypergroups. Tbilisi Math. J. 11 (2018), no. 4, 145--163. doi:10.32513/tbilisi/1546570891. https://projecteuclid.org/euclid.tbilisi/1546570891


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References

  • T. Changphas and B. Davvaz, Bi-hyperdeals and quasi-hyperideals in ordered semihypergroups, Ital. J. Pure Appl. Math. 35 (2015) 493-508.
  • T. Changphas and B. Davvaz, Properties of hyperideals in ordered semihypergroups, Ital. J. Pure Appl. Math. 33 (2014) 425-43.
  • B. Davvaz, Semihypergroup Theory, Elsevier, 2016.
  • B. Davvaz, Some results on congruences in semihypergroups. Bull. Malays. Math. Soc. 23(2) (2000) 53-58.
  • B. Davvaz and V. Leoreanu-Fotea, Binary relations on ternary semihypergroups, Comm. Algebra 38 (2010) 3621-3636.
  • B. Davvaz, P. Corsini and T. Changphas, Relationship between ordered semihypergroups and ordered semigroups by using pseudoorder, European J. Combinatorics 44 (2015) 208-217.
  • K. Hila, B. Davvaz and K. Naka, On quasi-hyperideals in semihypergroups, Comm. Algebra. 39 (2011) 4183-4194.
  • D. Heidari and B. Davvaz, On ordered hyperstructures, Politehn. Univ. Bucharest Sci. Bull, Ser. A, Appl. Math. Phys. 73(2) (2011) 85-96.
  • S. Lajos, Generalized ideals in semigroups, Acta Sci. Math. 22 (1961) 217-222.
  • S. Lajos, Notes on $(m,n)$-ideals I, Proc. Japan Acad. 39 (1963) 419-421.
  • S. Lajos, Notes on $(m,n)$-ideals II, Proc. Japan Acad. 40 (1964) 631-632.
  • S. Lajos, Notes on $(m,n)$-ideals III, Proc. Japan Acad. 41 (1965) 383-385.
  • F. Marty, Sur une generalization de la notion de group, Stockholm: 8th Congres Math. Scandinaves, 1934, 45-49.
  • B. Pibaljommee and B. Davvaz, On fuzzy bi-hyperideals in ordered semihypergroups, Journal Intelligent and Fuzzy Systems 28 (2015) 2141-2148.
  • S. Omidi and B. Davvaz, A short note on the relation $\mathcal{N}$ in ordered semihypergroups, Gazi University Journal of Science 29(3) (2016) 659-662.
  • B. Pibaljommee, K. Wannatong and B. Davvaz, An investigation on fuzzy hyperideals of ordered semihypergroups, Quasigroups and Related Systems 23 (2015) 297-308.
  • J. Tang, B. Davvaz and Y.F. Luo, Hyperfilters and fuzzy hyperfilters of ordered semihypergroups, J. Intell. Fuzzy Systems 29(1) (2015) 75-84.
  • J. Tang, B. Davvaz, and X.Y. Xie, An investigation on hyper $S$-posets over ordered semihypergroups, Open Mathematics, 15 (2017) 37-56.
  • O. Steinfeld, On the ideals quotients and prime ideals, Acta Math. Acad. Sci. Hung. 4 (1953) 289-298.
  • O. Steinfeld, Über die Quasiideale von Halbgruppen, Publ. Math. Debrecen 4 (1956) 262-275 (Mr 18, 790).