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.