Minimal generators in varieties

Given a variety V, the free V-algebra F 1 on one generator represents the canonical underlying functor from V to the category Set of sets. Hence one might ask, whether F 1 is, in some sense, a `canonical' generator of V. To make this question precise the notion of `minimal varietal generator' is introduced. It is shown that in many (though not all) varieties F 1 is a generator of this kind, and often even the unique one. The question whether every variety has a (unique) minimal varietal generator, remains on open problem.