> It would be very interesting to see if there are any results giving a means of producing a dcpo from a preorder! My hunch is that it would only really make sense for posets, and even then there might need to be some extra conditions before such a procedure gives meaningful results.

I know of one result.

A related way to the Cauchy sequence construction of the reals from the rational numbers is the method of *Dedekind cuts*. In lattice theory, this is generalized to the [*Dedekind-MacNeil Completion*](https://en.wikipedia.org/wiki/Dedekind%E2%80%93MacNeille_completion) of a preorder. The Dedekind-MacNeil completion of a preorder \\(\preceq\\) is the smallest complete lattice that embeds \\(\preceq\\). Your intuition is very close to the construction. The members of the lattice are the lower sets of the upper sets of \\(\preceq\\). See [Davey and Priestley (2002), Â§7.38 The Dedekindâ€“MacNeille completion](https://books.google.com/books?id=vVVTxeuiyvQC&pg=PA166#v=onepage&q&f=false) for more info.

> It would probably make more sense to consider the _lower set_ \\(\operatorname{\downarrow}X\\), rather than the upper set of upper bounds of \\(X\\), too. Seems like an [open set](https://en.wikipedia.org/wiki/Open_set) / [closed set](https://en.wikipedia.org/wiki/Closed_set) kind of distinction, but it might matter here.

Your intuition is right on the money once again! Every preorder \\(\preceq\\) gives rise to an [Alexandrov topology](https://en.wikipedia.org/wiki/Alexandrov_topology). The open sets of that topology are upper sets and the closed sets are lower sets. Every Alexandrov topology in turn gives rise to a preorder via its [specialization order](https://en.wikipedia.org/wiki/Specialization_(pre)order). The specialization order of the Alexandrov topology for a preorder \\(\preceq\\) is isomorphic to \\(\preceq\\) itself. In fact there is a full blown categorical duality where \\(\textsf{PreOrder} \cong \textsf{AlexandrovTop}^{op}\\).

I know of one result.

A related way to the Cauchy sequence construction of the reals from the rational numbers is the method of *Dedekind cuts*. In lattice theory, this is generalized to the [*Dedekind-MacNeil Completion*](https://en.wikipedia.org/wiki/Dedekind%E2%80%93MacNeille_completion) of a preorder. The Dedekind-MacNeil completion of a preorder \\(\preceq\\) is the smallest complete lattice that embeds \\(\preceq\\). Your intuition is very close to the construction. The members of the lattice are the lower sets of the upper sets of \\(\preceq\\). See [Davey and Priestley (2002), Â§7.38 The Dedekindâ€“MacNeille completion](https://books.google.com/books?id=vVVTxeuiyvQC&pg=PA166#v=onepage&q&f=false) for more info.

> It would probably make more sense to consider the _lower set_ \\(\operatorname{\downarrow}X\\), rather than the upper set of upper bounds of \\(X\\), too. Seems like an [open set](https://en.wikipedia.org/wiki/Open_set) / [closed set](https://en.wikipedia.org/wiki/Closed_set) kind of distinction, but it might matter here.

Your intuition is right on the money once again! Every preorder \\(\preceq\\) gives rise to an [Alexandrov topology](https://en.wikipedia.org/wiki/Alexandrov_topology). The open sets of that topology are upper sets and the closed sets are lower sets. Every Alexandrov topology in turn gives rise to a preorder via its [specialization order](https://en.wikipedia.org/wiki/Specialization_(pre)order). The specialization order of the Alexandrov topology for a preorder \\(\preceq\\) is isomorphic to \\(\preceq\\) itself. In fact there is a full blown categorical duality where \\(\textsf{PreOrder} \cong \textsf{AlexandrovTop}^{op}\\).