Canonical extension and canonicity via DCPO presentations

The canonical extension of a lattice is in an essential way a two-sided completion. Domain theory, on the contrary, is primarily concerned with one-sided completeness. In this paper, we show two things. Firstly, that the canonical extension of a lattice can be given an asymmetric description in two stages: a free co-directed meet completion, followed by a completion by \emph{selected} directed joins. Secondly, we show that the general techniques for dcpo presentations of dcpo algebras used in the second stage of the construction immediately give us the well-known canonicity result for bounded lattices with operators.