Uniform ideal completions (Q2702770)

Let \(P\) be a poset. So-called Cauchy ideals of \(P\) are defined in such a way that every directed ideal of \(P\) is a Cauchy ideal of \(P\) and every Cauchy ideal of \(P\) is a Frink ideal of \(P\). In join-semilattices having a least element, all three types of ideals coincide. It is well known that the completion of \(P\) by Frink ideals (directed ideals, resp.) is the smallest algebraic complete lattice (smallest algebraic poset, resp.) containing \(P\) as a set of compact elements and that both types of ideal completions may be regarded as certain topological completions. In the paper under review it is shown that the completion of \(P\) by Cauchy ideals may be interpreted as a uniform completion of \(P\).