---
tag: CZF5696TPY
kind: statement
role: theorem
name: Birkhoff's subdirect representation theorem
msc: [08B26]
uses: [QJGPT2W0FT, 88WN3AD2VQ, DASF939439, DCSXXT4SER, N5KE7FH6CX, 04RXTQA5ZA]
about: [88WN3AD2VQ, DASF939439]
sources:
  - url: https://en.wikipedia.org/wiki/Subdirectly_irreducible_algebra
---

## Statement

Let $\mathbf{A}$ be an [[algebra|QJGPT2W0FT]] with at least two elements.
Then $\mathbf{A}$ is isomorphic to a [[subdirect product|88WN3AD2VQ]] of a family of [[subdirectly irreducible algebras|DASF939439]].
Moreover, each factor of the family can be chosen to be a [[quotient|N5KE7FH6CX]] of $\mathbf{A}$.

## Remarks

Proof idea.
For each pair $a \ne b$ in $A$, [[Zorn's lemma|04RXTQA5ZA]] gives a [[congruence|DCSXXT4SER]] $\theta_{a,b}$ 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.

## Explanations

### L1 human

Every algebra can be taken apart into pieces that cannot be taken apart further.
The pieces are quotients of the algebra, and the algebra sits inside their product, with nothing lost.
It is like recovering an integer from its remainders modulo many numbers.

### L2 human

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|X9072A41NK]], which are subdirect products of finite algebras.