(m, n)-hyperideals in ordered semihypergroups

In this paper, first we introduce the notions of an (m, n)- hyperideal and a generalized (m, n)-hyperideal in an ordered semihypergroup, and then, some properties of these hyperideals are studied. thereafter, we characterize (m, n)-regularity, (m, 0)-regularity, and (0, n)-regularity of an ordered semihypergroup in terms of its (m, n)-hyperideals, (m, 0)-hyperideals and (0, n)-hyperideals, respectively. the relations mi, in, hn m, and b n m on an ordered semihypergroup are, then, introduced. we prove that b n m ⊆ hn m on an ordered semihypergroup and provide a condition under which equality holds in the above inclusion. we also show that the (m, 0)-regularity [(0, n)- regularity] of an element induce the (m, 0)-regularity [(0, n)-regularity] of the whole hn m-class containing that element as well as the fact that (m, n)- regularity and (m, n)-right weakly regularity of an element induce the (m, n)- regularity and (m, n)-right weakly regularity of the whole b n m-class and hn mclass containing that element, respectively.