Classes of algebraic structures that are defined by equational laws are called *varieties* or *equational classes*. A variety is *finitely generated* if it is defined by the laws that hold in some fixed finite algebra. We show that every subvariety of a finitely generated congruence permutable variety is finitely generated, thereby giving a partial solution to Problem 10.1 from "Sixty-four problems in universal algebra" by McKenzie et al. Our result applies in particular to all varieties of groups, loops, quasigroups, and their expansions (e.g., modules, rings, ...). In the proof, we represent equational theories by certain sets of finitary functions called *clonoids*.
This is joint work with Peter Mayr (U Colorado Boulder) and supported by the Austrian Science Fund FWF: P29931.