@[Matthew](https://forum.azimuthproject.org/discussion/comment/18252/#Comment_18252): oh man, that's _awesome_! I was thinking of mentioning Dedekind cuts, but I wasn't sure which one would make the most sense in context, so I stuck to Cauchy sequences. I'll definitely have to read a bit more into the Dedekind-MacNeille completion.
(Incidentally, the Wikipedia page suggests it's only defined for posets rather than all preorders, which lines up with my other hunch. Is that accurate?)
**EDIT:** Also, digging a little further, it looks like the concept we wanted after all was just that of a [complete lattice](https://en.wikipedia.org/wiki/Complete_lattice). Lattices assign _all_ sets suprema, but DCPOs only assign _directed_ sets suprema. DCPOs are neat, and quite useful for [denotational semantics](https://en.wikipedia.org/wiki/Denotational_semantics) of programming languages, but aren't _quite_ what we wanted here.