#
Birkhoff's variety theorem with and without free algebras

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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

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