Maltsev Conditions, Term Conditions, and The Shape of Congruence Lattices
by
Keith Kearnes
University of Colorado

During the 1980's, David Hobby and Ralph McKenzie developed a localization theory called tame congruence theory for algebras in locally finite varieties. The point of view of the theory is that information about the local algebraic structure of algebras in a locally finite variety is encoded in the shapes of the congruence lattices of the members of the variety. Hobby and McKenzie used intricate combinatorial arguments concerning operations on finite sets to successfully link certain varietal properties to properties of local algebras in locally finite varieties. Through these links, they were able to derive new and deep theorems - expressible without reference to tame congruence theory - about the interrelationships between Maltsev conditions, term conditions, and congruence lattice shapes.

The localization theory of Hobby and McKenzie does not extend intact to varieties that are not locally finite. Despite this fact, Emil Kiss and I have discovered that nearly every interrelationship between Maltsev conditions, term conditions, and congruence lattice shapes that was proved by Hobby and McKenzie for locally finite varieties using tame congruence theory actually holds for any variety. The purpose of my talk is to outline our results and methods.

Date received: March 5, 2003