Voir la notice de l'article provenant de la source Theory and Applications of Categories website
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.
@article{TAC_2005_14_a17,
author = {Jiri Adamek and Vera Trnkova},
title = {Birkhoff's variety theorem with and without free algebras},
journal = {Theory and applications of categories},
pages = {424--450},
publisher = {mathdoc},
volume = {14},
year = {2005},
language = {en},
url = {http://geodesic.mathdoc.fr/item/TAC_2005_14_a17/}
}
Jiri Adamek; Vera Trnkova. Birkhoff's variety theorem with and without free algebras. Theory and applications of categories, Tome 14 (2005), pp. 424-450. http://geodesic.mathdoc.fr/item/TAC_2005_14_a17/