Two Types Of Special Ordered Semigroup

Algebraic theory of partially ordered semigroups is still one of the most active branches in Algebra. We present a thorough study on the properties and characteristics of two classes of partially ordered semigroups. Furthermore, we also clarify some relations among the filters, ideals and certain ordered semi-groups. The main results can be showed as follows:1. We study the fundamental properties of the principal ideal and how to characterize the Green's relation. In the meantime, we introduce four types of Archimedean ordered semigroups and discuss their properties based on the properties of the principal ideal and the Green's relation.2. The relations and characteristics of some ideals in l(r)-Archimedean or-dered semigroups are obtained. Some Archimedean ordered semigroups can be described by simple ordered semigroups and intra-regular ordered semigroup. Furthermore, under certain conditions, we find four types of Archimedean or-dered semigroups are equivalent.3. We study another class of ordered semigroups-π-intra-regular ordered semigroup andπ-regular ordered semigroup. We discuss the properties of the two types of semigroups. We obtain characteristics of these semigroups based on the properties of their ideals and filters.