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

A. Mahboob, N. M. Khan, B. Davvaz

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 Ǫmn on an ordered semihypergroup is, then, introduced for any positive integers m and n and proved that the relation Ǫmn is contained in the relation Ǫ=Ǫ11. We also show that, in an (m,n)-regular ordered semihypergroup, the relation Ǫmn coincides with the relation Ǫ. 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 Ǫmn and Ǫ are equal and are universal relations.

Tbilisi Mathematical Journal, Vol. 11(4) (2018), 145-163