Birkhoff's subdirect representation theorem

Remarks

Proof idea. For each pair $a \ne b$ in $A$, Zorn's lemma gives a congruence $\theta_{a,b}$ that is maximal among the congruences not containing $(a, b)$. Every congruence of $\mathbf{A}/\theta_{a,b}$ other than equality comes from a strictly larger congruence, which contains $(a, b)$. So $\mathbf{A}/\theta_{a,b}$ is subdirectly irreducible, with monolith generated by the classes of $a$ and $b$. The intersection of all $\theta_{a,b}$ is equality, so $a \mapsto (a/\theta_{a,b})$ is a subdirect embedding.

Example: the ring $\Z$ is a subdirect product of the fields $\Z/p\Z$, $p$ prime, which are simple and hence subdirectly irreducible. Another example: the only subdirectly irreducible distributive lattice is the two-element chain, so every distributive lattice with at least two elements is a subdirect product of copies of it.

A subdirect product may need infinitely many factors even for a countable algebra, as for $\Z$. For a finite algebra, finitely many factors suffice, and Zorn's lemma is not needed.

Going deeper

For a student

The theorem reduces many questions about a variety to its subdirectly irreducible members. A variety is generated by its subdirectly irreducible algebras, so two varieties with the same subdirectly irreducible members are equal. For example, it shows that every Boolean algebra is a subdirect product of copies of the two-element Boolean algebra. This leads to notions such as residually finite algebras, which are subdirect products of finite algebras.