3. Jónsson's Lemma

``Jónsson's Lemma'' would be called a theorem by most people, but it was called a lemma in the original paper and the name has stuck.

For a class ${\cal K}$ of similar algebras, let U$({\cal K})$ denote the class of algebras isomorphic¹ to ultraproducts of algebras in ${\cal K}$.

Theorem. (Jónsson's Lemma) Let ${\cal K}$ be a class of similar algebras such that Var$({\cal K})$ is congruence-distributive. If $B \in$   Var$({\cal K})$ is subdirectly irreducible, then $B \in$   HSU$({\cal K}).$

This theorem doesn't sound much different from the theorem that Var$({\cal K}) =$   H   S   P$({\cal K})$, but it is really substantially different, in that U preserves many more properties than P.

Corollary. For a finite algebra $A$, if Var$(A)$ is congruence-distributive, then for each subdirectly irreducible algebra $B \in$   Var$(A)$ we have $B \in$   HS$(A)$.

Notice that this Corollary is a little stronger than the Theorem of §1, since it is not assumed to start with that $B$ is finite. The conclusion is the same.