A compactification for convergence ordered spaces
Canadian mathematical bulletin
Compactifications are constructed for convergence ordered spaces and topological ordered spaces with extension properties that resemble those of the Stone-Cech compactification. 0. Introduction. One of the authors  introduced a convergence space compactification with an extension property similar to that of the topological Stone-Cech compactification. We later showed in  that the compactification of  gives rise to a topological compactification with an interesting lifting property.
... fting property. This work is concerned with convergence ordered spaces, a natural generalization of the topological ordered spaces of Nachbin . By "convergence ordered space" we mean a partially ordered set with a convergence structure generated by filters which have bases of convex sets. A preliminary section gives a brief introduction to such spaces. In Section 2, a convergence ordered compactification is constructed for an arbitrary convergence ordered space by defining an appropriate partial order on a class of filters and using a "Wallman-type" construction similar to that of  . The extension properties of this compactification are examined in Section 3; in addition to generalizing the extension results of , conditions are found subject to which ours is the largest convergence ordered compactification. The last section applies the results of the preceding sections to obtain a topological ordered compactification with similar lifting properties. Choe and Park  have constructed a Wallman ordered compactification for the topological setting. It is shown, under certain assumptions, that our topological ordered compactification is larger than that by Choe and Park. 1. Preliminaries. Let (X, <) be a partially ordered set (or poset) equipped with a convergence structure. A convergence structure on X is a relation-> between the set F(X) of all filters on X and X which satisfies the following