Talk: Peter Schuster (Verona)

Title:

Taming the Big by the Small

(Based on joint work with Daniel Misselbeck-Wessel)

Abstract:

The Jacobson radical of a complete lattice is the meet of all maximal elements, which are typical ideal objects requiring transfinite methods. With the axiom of choice, the Jacobson radical equals the join of all small elements whenever the lattice is algebraic or compactly generated, in which case smallness can be tested by quantifying over compact or finite elements only. The latter observation is the keystone a proof-theoretic interpretation, in the vein of dynamical algebra, of fairly universal maximality principles such as the Teichmüller-Tukey lemma. For proof practice the Jacobson radical is given an inductive generation and a description by finite binary trees; the prime application in algebra is about bases for arbitrary vector spaces.