An Introduction to the Theory of Natural Dualities
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
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.