Dijkstra's Predicate Transformers and Priestley Duality from the Pleistocene Era: The Duals of Order-Preserving Maps between Distributive Lattices
by
Jonathan David Farley
University of Oxford and Vanderbilt University

You can model predicate transformers in computer science by using order-preserving maps between distributive lattices. A powerful tool for understanding distributive lattices (for example, Boolean algebras) is Priestley duality. Priestley duality associates to any distributive lattice a compact totally order-disconnected partially ordered topological space. Under this duality, lattice homomorphisms correspond functorially to continuous order-preserving maps between these spaces. Jónsson and Tarski (followed by Cignoli et al.) found that the duals of semilattice homomorphisms are continuous order-preserving relations or multifunctions. But what do ordinary order-preserving maps between distributive lattices correspond to? We show that the category of bounded distributive lattices with order-preserving maps is dually equivalent to the category of Priestley spaces with Priestley multirelations.

(The speaker used Priestley duality for special order-preserving maps to solve a problem that had been open since 1964.)

Most of this is ancient work from the speaker's doctoral dissertation at Oxford. It has some connection with Kripke frames, and may have some significance in computer science; but we hope not.

http://www.math.vanderbilt.edu/~farley

Date received: May 15, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cagw-54.