Birkhoff's variety theorem with and without free algebras

Jiri Adamek and Vera Trnkova

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.

http://www.tac.mta.ca/tac/volumes/14/18/14-18.dvi
http://www.tac.mta.ca/tac/volumes/14/18/14-18.ps
http://www.tac.mta.ca/tac/volumes/14/18/14-18.pdf
ftp://ftp.tac.mta.ca/pub/tac/html/volumes/14/18/14-18.dvi
ftp://ftp.tac.mta.ca/pub/tac/html/volumes/14/18/14-18.ps

TAC Home