Duality theory: past, present and future

This talk will give a survey of Priestley duality for varieties of distributive lattices with additional operations and discuss its relationship to the use of canonical extensions. These extensions were pioneered by Jónsson and Tarski in 1951 for Boolean algebras with operators and the theory has recently been greatly extended by Gehrke and Jónsson.

For many years, lattice theorists, through duality theory, and logicians and computer scientists, through correspondence theory, have investigated semantic models for non-classical logics. The models are provided respectively by

It is the fundamental notion of a canonical extension that unites these approaches. The Priestley dualities may be viewed as yielding `topologized Kripke semantics´ for the associated logics.

Knowledge of duality theory and algebraic logic will not be expected, but familiarity with elementary facts and definitions about ordered sets and lattices will be assumed.

H.A. Priestley

University of Oxford