Duality theory for distributive lattices

Let L be a bounded distributive lattice, and let X denote the set of prime filters of L. For each a ∈ L, let φ+(a) = {x∈ X : a ∈ x}.

In fact, each spectral space is homeomorphic to the prime spectrum of some bounded distributive lattice.

[4] Finally, let ≤ be set-theoretic inclusion on the set of prime filters of L and let τ = τ+∨ τ−.

Then the above three representations of bounded distributive lattices can be extended to dual equivalence[6] between Dist and the categories Spec, PStone, and Pries of spectral spaces with spectral maps, of pairwise Stone spaces with bi-continuous maps, and of Priestley spaces with Priestley morphisms, respectively: Thus, there are three equivalent ways of representing bounded distributive lattices.

Each one has its own motivation and advantages, but ultimately they all serve the same purpose of providing better understanding of bounded distributive lattices.