Birkhoff's HSP theorem

Remarks

One direction is a direct check: identities are preserved by homomorphic images, by subalgebras and by direct products. For the converse, let $\Sigma$ be the set of identities that hold in every member of $K$, and let $A$ satisfy $\Sigma$. The free algebra of $HSP(K)$ on a set of the size of $A$ is a subalgebra of a product of members of $K$. Since $A$ satisfies $\Sigma$, the algebra $A$ is a homomorphic image of it. So $A \in HSP(K) = K$. Example: groups form a variety, and the theorem confirms that subgroups, quotient groups and products of groups are groups. Closure under products cannot be dropped. The class of groups with at most two elements is closed under $H$ and $S$, but it does not contain $\Z / 2\Z \times \Z / 2\Z$.

Going deeper

For a student

The theorem lets one decide whether a class is a variety without writing identities. To show that a class is not equationally definable, one finds a product, subalgebra or image that leaves the class. For example, torsion-free abelian groups are not a variety, because their quotient $\Z / 2\Z$ has torsion. The proof introduces free algebras of $HSP(K)$, which are the main tool of the theory. A finer form, Tarski's formula $V(K) = HSP(K)$, describes the variety generated by any class.