(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.