Quasi-uniform completions of partially ordered spaces

Abstract

In this paper we define partially ordered quasi-uniform spaces (X, U, ≤) (PO-quasi-uniform spaces) as those spaces with a biconvex quasi-uni- formity U on the poset (X, ≤) and give a construction of a (transitive) biconvex compatible quasi-uniformity on a partially ordered topological space when its topology satisfies certain natural conditions. We also show that under certain conditions on the topology τU∗ of a PO-quasi-uniform space (X, U, ≤), the bi- completion (X, e Ue) of (X, U) is also a PO-quasi-uniform space (X, e Ue, ) with a partial order on Xe that extends ≤ in a natural way.