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Quasi-uniform completions of partially ordered spaces

Authors:	David Buhagiar Tanja Telenta

Institution:	(1) Department of Mathematics Faculty of Science, University of Malta, Msida, MSD.06, Malta

Abstract:	In this paper we define partially ordered quasi-uniform spaces (X, $\mathfrak{U}$ , ≤) (PO-quasi-uniform spaces) as those space with a biconvex quasi-uniformity $\mathfrak{U}$ 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 $\tau _{\mathfrak{U}*}$ of a PO-quasi-uniform space (X, $\mathfrak{U}$ , ≤), the bicompletion $(\tilde X,\tilde {\mathfrak{U}})$ of (X, $\mathfrak{U}$ ) is also a PO-quasi-uniform space ( $(\tilde X,\tilde {\mathfrak{U}})$ , ⪯) with a partial order ⪯ on $\tilde X$ that extends ≤ in a natural way.

Keywords:	partially ordered set quasi uniform space uniform completion uniform bicompletion
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