---
tag: VT72G3BA99
kind: statement
role: theorem
name: Jónsson's lemma
msc: [08B10, 08B26]
uses: [9GZ7F6ETC6, DASF939439, S7MJVSP79N, AX9Q86GB8M, GJAYCVYZYF]
about: [9GZ7F6ETC6, DASF939439]
sources:
  - url: https://en.wikipedia.org/wiki/Variety_(universal_algebra)
---

## Statement

Let $\mathcal{K}$ be a class of algebras of one signature.
Assume that the variety $\mathcal{V} = \mathbf{HSP}(\mathcal{K})$ is a [[congruence-distributive variety|9GZ7F6ETC6]].
Then every [[subdirectly irreducible algebra|DASF939439]] in $\mathcal{V}$ belongs to $\mathbf{HSP}_U(\mathcal{K})$.
Here $\mathbf{H}$, $\mathbf{S}$ and $\mathbf{P}_U$ are the [[class operators|S7MJVSP79N]] of homomorphic images, subalgebras and [[ultraproducts|AX9Q86GB8M]].
In particular, if $\mathcal{K}$ is a finite set of finite algebras, then every subdirectly irreducible algebra in $\mathcal{V}$ belongs to $\mathbf{HS}(\mathcal{K})$.

## Remarks

A subdirectly irreducible $B \in \mathcal{V}$ has the form $C / \psi$, where $C$ is a subalgebra of a product $\prod_{i \in I} A_i$ with $A_i \in \mathcal{K}$.
For each subset $J \subseteq I$, the kernel of the projection of $C$ onto $\prod_{i \in J} A_i$ is a congruence $\eta_J$ of $C$.
Distributivity of the congruence lattice of $C$ and the irreducibility of $\psi$ show that the sets $J$ with $\eta_J \le \psi$ contain an ultrafilter $U$ on $I$.
Then $\psi$ contains the congruence induced by $U$, so $B$ is a homomorphic image of a subalgebra of the ultraproduct $\prod_{i \in I} A_i / U$.
When $\mathcal{K}$ is a finite set of finite algebras, every ultraproduct of members of $\mathcal{K}$ is isomorphic to a member of $\mathcal{K}$, which gives the second claim.

Example: the variety of distributive lattices is generated by the two-element lattice $\mathbf{2}$.
Lattices are congruence distributive, so every subdirectly irreducible distributive lattice lies in $\mathbf{HS}(\mathbf{2})$.
The only such lattice with more than one element is $\mathbf{2}$ itself.

## Explanations

### L1 human

Jónsson's lemma controls the building blocks of a variety whose congruences are distributive.
The irreducible algebras of the variety cannot be new: they all come from the generating algebras by ultraproducts, subalgebras and quotients.
For a variety generated by one finite algebra, they are quotients of subalgebras of that algebra, so up to isomorphism there are only finitely many of them.

### L2 human

By Birkhoff's subdirect representation theorem, a variety is determined by its subdirectly irreducible members, so Jónsson's lemma describes the whole variety.
It gives a description of the [[lattice of subvarieties|GJAYCVYZYF]] of a congruence-distributive variety: each subvariety is generated by its subdirectly irreducible members.
For example, for a finite lattice $L$, the subvarieties of $\mathbf{HSP}(L)$ correspond to the down-sets of the finitely many subdirectly irreducible lattices in $\mathbf{HS}(L)$, up to isomorphism.
Here $M$ lies below $N$ when $M \in \mathbf{HS}(N)$.