Ideal extensions of ordered sets

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 toQ. 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.