Graduate Student Seminar

A Brief Introduction to Universal Algebra

Charlotte Aten

Friday, October 13th, 2017
5:00 PM - 6:00 PM
Hylan 1106A

Universal algebra is the branch of mathematics which deals with constructions common to all traditional branches of algebra. We introduce the formal definition of an algebra and of a homormophism of algebras as well as the concepts of subuniverse, congruence, quotient, and product. This allows us to prove a generalization of the Isomorphism Theorems which holds for all types of algebraic structures. We describe the class operators for taking homomorphic images, subalgebras, and products and define terms and identities. The study of general algebras began in earnest with Birkhoff’s Theorem, which says that a class is closed under the three aforementioned operators if and only if it is defined by a set of identities. Such a class is called a variety of algebras. Groups, rings, modules, lattices, and Boolean algebras are all familiar examples of varieties. Current work is directed towards understanding the lattice of varieties and the manner in which the congruence lattice of an algebra relates to the identities it satisfies.

Event contact: george dot grell at rochester dot edu