Birkhoff's variety theorem (HSP theorem) says that a class of algebras of the same signature is a variety of algebras iff it is closed under homomorphic images (H), subalgebras (S) and small products (P), as proved in

- Garrett Birkhoff,
*On the structure of abstract algebras*, Proc. Cambridge Philos. Soc. 31 (4): 433–454 (1935) doi pdf

The Birkhoff's subdirect representation theorem is proved in

- Garrett Birkhoff,
*Subdirect unions in universal algebra*, Bull. Amer. Math. Soc. 50 (1944), 764-768.

On monoids (*not* on *groupoids* in the modern sense of that term):

- Garrett Birkhoff,
*Hausdorff Groupoids*, Annals of Mathematics, Second Series**35**2 (1934) 351-360 [jstor:1968437, doi:10.2307/1968437]

On quantum logic:

- Garrett Birkhoff, John von Neumann:
*The logic of quantum mechanics*, Annals of Mathematics**37**823-843 (1936) [doi:10.2307/1968621, pdf]

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