Canonical extensions and canonicity via dcpo presentations

The canonical extension of a lattice is in an essential way a two-sided completion. Domain theory, in contrast, is primarily concerned with one-sided completeness. In this paper, we show two things. First, we show 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 selected directed joins. Second, we show that the general techniques for directed complete partial order (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.