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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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

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

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

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Mathematical Reviews number (MathSciNet)

Primary: 20N20: Hypergroups

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


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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