Title. Independence of multi-term commutators and centralizers
Author. Ross Willard
Institution. Department of Mathematics, University of Waterloo, Canada
Abstract. In 1977, Bill Lampe co-discovered the “term condition” and used it to prove the existence of algebraic lattices which cannot be represented as the congruence lattice of an algebra with just one basic operation. Lampe’s key observation was that there exist algebraic lattices whose shape forces every representing algebra to satisfy the term condition.
In the 1980s, a relativized version of the term condition came to form the standard foundation for what we now call the “usual” commutator. In modern language, Lampe’s observation was that there exist algebraic lattices whose shape forces the commutator operation of any representing algebra to be constantly zero. Hence the commutator is not independent of the congruence lattice on which it sits.
There exist “unusual” commutators, all of which coincide with the usual commutator in congruence modular varieties but outside of that context can be strictly bigger than the usual commutator. Each of these commutators is defined by systems of implications involving two or more terms. In 1997 Lampe, Keith Kearnes and I asked whether there might exist an algebraic lattice whose shape forces these larger commutators to be constantly zero. Thus began a “disastrous” collaboration which is now, finally, reaching consummation. It turns out that not only are these larger commutators independent of the congruence lattice (modulo a few obvious necessary conditions such as subadditivity), they are independent of each other (again modulo some obvious necessary restrictions). Stronger still, the corresponding centralizer relations can be abstractly axiomatized, and modulo this axiomatization, they are essentially independent of the congruence lattice and each other.
In this lecture I will try to connect these results to greed, Blackstone growlers, and Bill’s personal hygiene.