nLab Birkhoff's HSP theorem

Statement

Theorem

(Birkhoff’s HSP theorem)

Given a language $L$ generated by a set of (single-sorted) finitary operations, and a class $C$ of structures for $L$. Then $C$ is the class of models for a set of universally quantified equations between terms of $L$ (a Lawvere theory) if and only if

1. (H) The class is closed under homomorphic images (see below),

2. (S) The class is closed under subalgebras,

3. (P) The class is closed under taking products.

See also at Lawvere theory – Characterization of examples

Remark

Here “closed under homomorphic images” means that if $A$ and $B$ are structures in the class, and $\phi \colon A \to B$ is a homomorphism between them, then also its image $im(\phi) \hookrightarrow B$ is an element of the class.

Remark

The first-order analogue of HSP (theorem ) is the characterization (see e.g. Chang and Keisler’s original text (Chang-Keisler 66) on continuous model theory) of elementary classes of structures of structures: they’re precisely those closed under elementary substructures, elementary embeddings, ultraproducts, and ultraroots (if an ultrapower of something is in your class, that something was in your class.)

References

• Wikipedia, Birkhoff's theorem

• Chang, Keisler Continuous Model Theory, Princeton University Press, 1966. ISBN: 9780691079295