Finitely generated equational classes

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; in fact, we prove the more general result that if a finitely generated variety has an edge term, then all its subvarieties are finitely generated as well. That is, finitely generated varieties with edge term are hereditarily finitely generated. This applies in particular to all varieties of groups, loops, quasigroups and their expansions (e.g., modules, rings, Lie algebras, …).