On Inherently Nonfinitely Based Varieties. (Paperback)

An algebra is a nonempty set equipped with finitary operations. The equational theory of an algebra is the set consisting of all equations true in that algebra. If there is a finite list of equations true in an algebra from which all equations true in the algebra can be deduced, we say that the equational theory of the algebra is finitely axiomatizable or that the algebra is finitely based. In 1996, Ralph McKenzie proved that there cannot be a list of necessary and sufficient conditions, recognizable by some algorithm, that describe finitely based finite algebras. This result shows the underlying difficulty in describing which finite algebras are finitely based and which are not. In this dissertation, we investigate properties of some algebras known to be nonfinitely based. According to a famous result of Garrett Birkhoff, every variety of algebras is determined by its subdirectly irreducible members. If all of the subdirectly irreducible algebras belonging to a variety are finite, we say that the variety is residually finite. Starting with a finite algebra, there is no guarantee that the variety it generates is finitely based nor that its variety is residually finite. We show that every algebra in a particular class of algebras known to be nonfinitely based must generate a variety that is not residually finite. We also provide a partial answer to a problem of Ježek and Quackenbush which asks whether all finite commutative directoids are finitely based. A commutative directoid is an algebra with an underlying partial ordering and one commutative operation that in most cases acts as a least upper bound. We construct a variety of commutative directoids which is nonfinitely based in a contagious way. In addition, we show that this variety has no finite residual bound.