Zoran Škoda writes:
There are several things called “Birkhoff’s theorem” in various field of mathematics and mathematical physics, and belong even to at least 2 different classical Birkhoff’s. Even wikipedia has pages for more than one such theorem. To me the first which comes to mind is Birkhoff’s factorization theorem, now also popular in Kreimer-Connes-Marcolli work and in connection to loop groups (cf. book by Segal and Pressley).
For now we have:
For the theorem as in universal algebra:
Garrett Birkhoff, Subdirect unions in universal algebra, Bull. Amer. Math. Soc. 50 (1944), 764-768.
William Lawvere considers this Birkhoff’s theorem through the eyes of Isbell duality and cohesive toposes in