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International Journal of Mathematics and Mathematical Sciences
Volume 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 Hindawi Publishing Corporation. 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 P, Q are disjoint ordered sets, we construct (all) the ordered sets V which have an ideal P which is isomorphic to P, and the complement of P in V is isomorphic to Q. Conversely, we prove that every extension of an ordered set P by an ordered set Q can be so constructed. Illustrative examples of the main theorem in case of finite ordered sets are given.