Geodesic
Birkhoff's variety theorem with and without free algebras
Theory and applications of categories, Tome 14 (2005), pp. 424-450.

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.
Classification : 18C10
Keywords: variety, Birkhoff's 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/}
}
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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/