Ideal completion

Summary

The ideal completion of a partially ordered set is the set of its order ideals, the nonempty directed down-closed subsets, ordered by inclusion. Each element $x$ becomes the principal ideal of all elements below $x$. The ideal completion is a directed-complete partial order in which every element is a directed union of principal ideals. It is the free way to add suprema of directed sets. It is a basic construction of domain theory.