(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.