For large signatures $\Sigma$ we prove that Birkhoff's Variety Theorem holds (i.e., equationally presentable collections of $\Sigma$-algebras are precisely those closed under limits, subalgebras, and quotient algebras) iff the universe of small sets is not measurable. Under that limitation Birkhoff's Variety Theorem holds in fact for $F$-algebras of an arbitrary endofunctor $F$ of the category Class of classes and functions.
For endofunctors $F$ of Set, the category of small sets, Jan Reiterman proved that if $F$ is a varietor (i.e., if free $F$-algebras exist) then Birkhoff's Variety Theorem holds for $F$-algebras. We prove the converse, whenever $F$ preserves preimages: if $F$is not a varietor, Birkhoff's Variety Theorem does not hold. However, we also present a non-varietor satisfying Birkhoff's Variety Theorem. Our most surprising example is two varietors whose coproduct does not satisfy Birkhoff's Variety Theorem.
Keywords: variety, Birkhoff's Theorem
2000 MSC: 18C10
Theory and Applications of Categories,
Vol. 14, 2005,
No. 18, pp 424-450.