Jónsson's lemma

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.

Going deeper

For a student

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 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 $M \in \mathbf{HS}(N)$.