A Spectral-style Duality for Distributive Posets

In this paper, we present a topological duality for a category of partially ordered sets that satisfy a distributivity condition studied by David and Erné. We call these posets mo-distributive. Our duality extends a duality given by David and Erné because our category of spaces has the same objects as theirs but the class of morphisms that we consider strictly includes their morphisms. As a consequence of our duality, the duality of David and Erné easily follows. Using the dual spaces of the mo-distributive posets we prove the existence of a particular Δ₁-completion for mo-distributive posets that might be different from the canonical extension. This allows us to show that the canonical extension of a distributive meet-semilattice is a completely distributive algebraic lattice.