(preliminary version)

An Introduction to the Theory of Natural Dualities

Brian Davey

In this series of lectures we shall build on the general algebra from in the Introductory Courses. Motivated by dualities for Boolean algebras (Stone), abelian groups (Pontryagin), semilattices (Hoffman-Stralka-Mislove) and distributive lattices (Priestley), we shall introduce the theory of natural dualities for quasi-varieties generated by finite algebras. A natural duality provides a dual equivalence between the quasi-variety and a category of naturally defined topological structures. In particular, a natural duality yields a representation of each algebra in the quasi-variety as an algebra of continuous structure-preserving functions.

The basic theory will be developed but without proofs, for which we shall refer to the recent monograph

We shall see how it is possible to obtain a natural duality for the quasi-variety generated by a finite algebra without doing any topology and without doing any category theory! The existence of a natural duality is reduced to finite combinatorics.

Throughout the lectures we shall study the variety of Kleene algebras. (Kleene algebras are the algebraic formulation of a very natural 3-valued extension of classical logic which allows truth values of "true", "false" and "don't know".)

The relationship between natural dualities and restricted Priestley dualities for quasi-varieties of distributive-lattice-based algebras will be explored as such quasi-varieties often occur in the algebraic formulation of non-classical logics. In particular, we shall compare the natural and restricted Priestley duals for the variety of Kleene algebras.

The major unsolved problems in the theory will be discussed en route.