THREE LECTURES ON THE RS PROBLEM

Abstract. In these lectures we return to the RS Problem discussed in E. Kiss’s article (this volume). We discuss the current status of the prob-lem and describe some results which arose from the study of the problem, including the solution to Tarski’s finite basis problem. A subtitle for these lectures might be “Some recent results in general algebra, mostly due to R. McKenzie.” 1. First lecture: The residual character of an equational class These three lectures are about a cluster of problems concerning equational classes of algebras, that is, classes of models (in a language having no re-lation symbols) which can be axiomatized by a set of universally quanti-fied equations. (Such classes are known to general algebraists as varieties). These lectures are particularly concerned with the smallest equational class containing a given algebra A; this class is denoted by HSP(A), because it coincides with the closure of {A} first under arbitrary Cartesian powers, then under substructures, finally under homomorphic images. HSP(A) i