Dualities in Lattice Theory

In this note we prove several duality theorems in lattice theory. We also discuss the connection between spectral spaces and Priestley spaces, and interpret Priestley duality in terms of spectral spaces. The organization of this note is as follows. In the first section we collect appropriate definitions and basic results common to many of the various topics. The next four sections consider Birkhoff duality between finite distributive lattices and finite posets, Stone’s duality between Boolean algebras and Stone spaces, Priestley duality between bounded distributive lattices and Priestley spaces, and Heyting between Heyting algebras and Heyting spaces. Finally, in Section 6 we consider the relation between the Stone topology and the spectral topology on the set of prime filters of a distributive lattice. The spectral topology more closely resembles the topology on the spectrum of a commutative ring, while the Stone topology yields a compact Hausdorff space. We will then interpret Priestley duality in terms of spectral spaces. As we will see, Priestley duality generalizes both Birkhoff and Stone duality, and Heyting duality considers special cases of bounded distributive lattices and Priestley spaces.