Abstract

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 ℐ-trivial ordered semigroups which are defined via the Green's relation ℐ, and with the nil and Δ-ordered semigroups. We prove that every nil ordered semigroup is ℐ-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.