(Preliminary version)

Abstract for the Introductory Course on Universal Algebra

Matt Valeriote

The following topics will be dealt with during the 4 lectures of this course:

Basic operations and invariants of algebras: The notion of a general algebra will be introduced. The basic (and important) operations of taking subalgebras, homomorphic images and direct products of algebras will be discussed. Two invariants of an algebra will be defined: the congruence lattice, and the subuniverse lattice.

Varieties: Classes of algebras defined by a set of equations will be studied. Birkhoff's Theorem, which deals with the connection between equations and the algebraic operations introduced in the first lecture, will be discussed. Additionally the notion of a free algebra will be introduced.

The nuts and bolts of varieties: Another theorem of Birkhoff's dealing with a certain kind of irreducible algebra will be discussed. Varieties will be classified according to the equations which are satisfied by the congruence lattices of the algebras they contain. A more general scheme for classification using Mal'cev conditions will be introduced.

Finite algebras: Combinatorial properties of finite algebras will be dealt with in this lecture. The notion of a minimal algebra will be introduced in order to give some indication of how the local structure of a finite algebra can be described. Palfy's theorem on minimal algebras will be discussed.