The patch construction is dual to algebraic DCPO representation (Q633791)

The author uses the parallel between the preframe and the suplattice approach to locale theory in order to prove that the patch construction, as an action on topologies, is the same thing as the process of recovering a discrete poset from its algebraic dcpo (ideal completion). The main theorem reads as follows: (i) There is a bijection between discrete posets and ideal completion locales; (ii) There is a bijection between ordered compact Hausdorff locales and stably locally compact locales.