(preliminary version)

Exponentiation, duality, and the arithmetic of ordered sets and lattices

Jonathan Farley

This series of lectures is intended to pave the way towards an extension of Priestley duality for NON-distributive lattices. We will present several applications of this yet-to-be-developed theory by studying the structure of function lattices:

Let P be a poset and let Q be a partially ordered topological space. Then P^Q denotes the set of (continuous) order-preserving maps from Q to P, ordered pointwise, where P has the discrete topology. (In 1942, Birkhoff conjectured that if P, Q, and R are finite, non-empty posets such that P^Q is isomorphic to R^Q, then P is isomorphic to R.)

If P^Q is a lattice, then we call it a "function lattice." Expanding on a survey of Davey and Duffus, we will show how duality theory gives one a tool, called a logarithm, for understanding the structure of function lattices. If time permits, we will also discuss McKenzie's recent progress towards solving Birkhoff's conjecture.