Characterization of (m, n)-regular Ordered Semihypergroups

Abstract

We, first, characterize (m, n)-hyperideals of ordered semihypergroups in terms of its (m, 0)-hyperideals and (0, n)-hyperideals as well as minimality of (m, n)-hyperideals of ordered semihypergroups in terms of its minimal (m, 0)-hyperideals and minimal (0, n)-hyperideals. Further characterization (m, n)-
regular ordered semihypergroups in terms of its (m, 0)-hyperideals, (0, n)-hyperideals, (m, n)-hyperideals and (m, n)-quasi-hyperideals is studied. After introducing relations Bnm and Qnm on ordered semihypergroups,
we prove that Bnm ⊆ Qnm and provide conditions for the equality to hold in this inclusion. Finally we show that, in any ordered semihypergroups, Qnm = Hnm
and, in any (m, n)-hypersimple ordered semihypergroup,S, Bnm = Qnm = Hnm
= S × S.