Danielle Bowerman, a doctoral candidate in mathematics, will defend their dissertation titled “Generalizations of Finiteness Conditions and Extension Monads in Algebras with Infinitely Many or Infinitary Operations.” Their advisor, Dr. Matt Insall, is an associate professor in mathematics and statistics department. The dissertation abstract is provided below.

In this work, we extend the results of finiteness conditions and extension monads found in \cite{Insall} from finitely many finitary operations to infinitely many finitary operations, as well as touching on infinitary operations. We also examine varieties of algebras, including the notion of strong varieties introduced in \cite{Insall}, and common constructions of extension monads in varieties of algebras. We see that for finite collections of algebras of the same signature, the extension monad operation on a variety of algebras commutes with the direct product operation, and all retractions from an enlargement or extension monad are trivial. We also see that for any two varieties of algebras which are categorically equivalent, one is strong if and only if the other is strong.