Page "Equivalence of categories" Paragraph 23

* In lattice theory, there are a number of dualities, based on representation theorems that connect certain classes of lattices to classes of topological spaces.

Probably the most well-known theorem of this kind is Stone's representation theorem for Boolean algebras, which is a special instance within the general scheme of Stone duality.

Each Boolean algebra is mapped to a specific topology on the set of ultrafilters of.

Conversely, for any topology the clopen ( i. e. closed and open ) subsets yield a Boolean algebra.

One obtains a duality between the category of Boolean algebras ( with their homomorphisms ) and Stone spaces ( with continuous mappings ).

Another case of Stone duality is Birkhoff's representation theorem stating a duality between finite partial orders and finite distributive lattices.