International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 53, Pages 2847-2861
Ideal extensions of ordered sets
Department of Mathematics, University of Athens, Panepistimioupolis, Athens 15784, Greece
Received 30 January 2003
Copyright © 2004 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.
The ideal extensions of semigroups—without order—have been first considered by Clifford (1950). In this paper, we give the main theorem of the ideal extensions for ordered sets. If , are disjoint ordered sets, we construct (all) the ordered sets which have an ideal which is isomorphic to , and the complement of in is isomorphic to . Conversely, we prove that every extension of an ordered set by an ordered set can be so constructed. Illustrative examples of the main theorem in case of finite ordered sets are given.