(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,۰)-regularity, and (۰,n)-regularity of an ordered semihypergroup in terms of its (m,n)-hyperideals, (m,۰)-hyperideals and (۰,n)-hyperideals, respectively. The relations {_m\mathcal{I}}, \mathcal{I}_n, \mathcal{H}_m^n, and \mathcal{B}_m^n on an ordered semihypergroup are, then, introduced. We prove that \mathcal{B}_m^n \subseteq \mathcal{H}_m^n on an ordered semihypergroup and provide a condition under which equality holds in the above inclusion. We also show that the (m,۰)-regularity [(۰,n)-regularity] of an element induce the (m,۰)-regularity [(۰,n)-regularity] of the whole \mathcal{H}_m^n-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 \mathcal{B}_m^n-class and \mathcal{H}_m^n-class containing that element, respectively.

Keywords:

Ordered semihypergroups , (m , ۰)-hyperideals , (۰ , n)-hyperideals