6. Application to construction of varieties

For a class ${\cal K}$ of similar algebras, let S$({\cal K})$, P$({\cal K})$, and H$({\cal K})$ denote the classes constructed from ${\cal K}$ by taking respectively subalgebras, products, and homomorphic images of members of ${\cal K}$.

Theorem (G. Birkhoff) A class ${\cal V}$ of similar algebras is a variety if and only if ${\cal V}$ is closed under S, P, and H.

Corollary (Birkhoff-Tarski) For any class ${\cal K}$ of similar algebras, Var$({\cal K})$ (the smallest variety containing ${\cal K}$) is obtainable as Var$({\cal K}) =$   HSP$({\cal K})$, meaning H$($S$($P$({\cal K})))$.