International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 61286, 8 pages
doi:10.1155/IJMMS/2006/61286
Ideals and Green's relations in ordered semigroups
Department of Mathematics, University of Athens, Panepistimiopolis 15784, Greece
Received 25 November 2005; Revised 3 March 2006; Accepted 12 March 2006
Copyright © 2006 Niovi Kehayopulu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.