In this paper we introduce
the concept of o-completely regular filters on a completely regular ordered
space in order to characterize compact ordered spaces and show that for
any completely regular ordered space X, Nachbin compactification of X is
precisely the strict extension of X with all maximal o-completely regular
filters as the filter trace. With this concept, we also define k-compact ordered
spaces for every infinite cardinal k, which give rise to a chain of extensive
subcategories of the category of completely regular ordered spaces, analogous to the
chain given by the category of k-compact spaces, for the different cardinals
k.
