## Tbilisi Mathematical Journal

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

#### 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
Accepted: 8 October 2018
First available in Project Euclid: 4 January 2019

https://projecteuclid.org/euclid.tbilisi/1546570891

Digital Object Identifier
doi:10.32513/tbilisi/1546570891

Mathematical Reviews number (MathSciNet)
MR3954213

Subjects
Primary: 20N20: Hypergroups

#### 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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