---
tag: DASF939439
kind: concept
name: Subdirectly irreducible algebra
msc: [08B26]
sources:
  - url: https://en.wikipedia.org/wiki/Subdirectly_irreducible_algebra
same_as:
  wikidata: Q7631005
  wikipedia: Subdirectly irreducible algebra
---

## Summary

A subdirectly irreducible algebra is a nontrivial [[algebra|QJGPT2W0FT]] that cannot be split as a [[subdirect product|88WN3AD2VQ]] in a nontrivial way.
In every subdirect representation of it, one of the projections is already an isomorphism.
Equivalently, among its [[congruences|DCSXXT4SER]] other than equality there is a smallest one, called the monolith.
By [[Birkhoff's subdirect representation theorem|CZF5696TPY]], these algebras are the building blocks of all algebras, and a variety is determined by its subdirectly irreducible members.

## Explanations

### L1 human

Any nontrivial gluing of a subdirectly irreducible algebra, however small, must glue a certain fixed pair of elements.
So the algebra cannot be recovered from several smaller pictures of it, since each picture loses that pair.
Every [[simple algebra|9CJCXNYNK8]] is subdirectly irreducible.
The group $\Z/4\Z$ is subdirectly irreducible but not simple: every nontrivial gluing identifies $0$ and $2$.
The group $\Z/6\Z$ is not, because it is the product of $\Z/2\Z$ and $\Z/3\Z$.