Ideals and Green's relations in ordered semigroups

Exactly as in semigroups, Green’s relations play an important role in the theory of ordered semigroups—especially for decompositions of such semigroups. In this paper we deal with the -trivial ordered semigroups which are defined via the Green’s relation , and with the nil and Δ-ordered semigroups. We prove that every nil ordered semigroup is -trivial which means that there is no ordered semigroup which is 0-simple and nil at the same time. We show that in nil ordered semigroups which are chains with respect to the divisibility ordering, every complete congruence is a Rees congruence, and that this type of ordered semigroups are -ordered semigroups, that is, ordered semigroups for which the complete congruences form a chain. Moreover, the homomorphic images of -ordered semigroups are -ordered semigroups as well. Finally, we prove that the ideals of a nil ordered semigroup S form a chain under inclusion if and only if S is a chain with respect to the divisibility ordering.