nLab Birkhoff's theorem

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).

Some named results pertaining to Garrett Birkhoff (mainly algebraist) are

• the Birkhoff–von Neumann theorem on Birkhoff-von-Neumann algorithm for decomposing a bistochastic matrix into a convex combination of permutation matrices
• Birkhoff Duality of distributive lattices
• Birkhoff's HSP theorem or Birkhoff's variety theorem in universal algebra
• Birkhoff's subdirect representation theorem in universal algebra

Named results pertaining to George David Birkhoff (mainly analyst) include

• Birkhoff decomposition (or factorization)

References

On Birkhoff's subdirect representation theorem:

• Garrett Birkhoff, Subdirect unions in universal algebra, Bull. Amer. Math. Soc. 50 (1944), 764–768 pdf

• Wolfram Mathworld

William Lawvere considers this Birkhoff’s theorem through the eyes of Isbell duality and cohesive toposes in

• William Lawvere, Birkhoff’s Theorem from a geometric perspective: A simple example, to appear in Categories and General Algebraic Structures with Applications (2016), journal page.