June 24, 2024

"The Structure of Finite Algebras" Course

Graduate course on the structure of finite algebras
taught by M. Valeriote (McMaster)

The first part of this course will deal with the structure theory developed by Hobby and McKenzie for finite algebraic structures. The main source for this part of the course will be the book "The Structure of Finite Algebras" by David Hobby and Ralph McKenzie and published in the Contemporary Mathematics Series of the American Mathematical Society. The name that is commonly attached to this structure theory is "Tame Congruence Theory". We will see that tame congruence theory provides a means to analyze the local structure of finite algebras and also to obtain useful information about the varieties they generate.

The second part of the course will be concerned with applications of tame congruence theory to varieties of algebras. We will examine the structure of varieties whose first order theories are recursive (decidable varieties) and in particular will discuss the results of McKenzie-Valeriote and Idziak-Jeong.

Tame congruence theory has turned out to be a useful tool in trying to understand the residual character of varieties. We will look at results of Hobby, Kearnes, McKenzie and others on this subject and hopefully will have time at the end to discuss McKenzie's solution to Tarski's finite basis problem.