International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 61286, 8 pages

Ideals and Green's relations in ordered semigroups

Niovi Kehayopulu

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.


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.