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 Q(m)(n) on an ordered semihypergroup is, then, introduced for any positive integers m and n and proved that the relation Q(m)(n) is contained in the relation Q = Q(1)(1). We also show that, in an (m, n)-regular ordered semihypergroup, the relation Q(m)(n) coincides with the relation 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 Q(m)(n) and Q are equal and are universal relations.