The intended audience for this paper is mathematicians and those theoretical computer scientists working on the denotational semantics of parallel or concurrent programs. To model recursive or infinite behavior of programs, complete partial orders (cpos) have traditionally been used. However, in concurrent programming, complete metric spaces have been applied very successfully.

The authors investigate the connection between these two completion techniques. Specifically, they present the conditions under which an isometry exists that embeds the metric completion in the ideal completion (“ideal” in the algebraic sense).

Given a partially ordered set D with a minimal element, there exists a weight function. The authors prove that this function makes the ideal completion IDL(D) of D a metric subspace of the complete metric space of downward closed subsets of D. A weight function is a length such that the set of all elements of length at most n, that are below some q, has a greatest element. Given this weight function, we can define a metric based on it.

Note that IDL(D) itself is only complete under certain conditions. If it is, the metric completion and the ideal completion are isometric.

Apart from the notions of cpo and metric, the paper is self-contained; it supplies all necessary notions and definitions.