Locally finite variety

Remarks

An equivalent condition is that the free algebra $F_{\mathcal{V}}(n)$ of $\mathcal{V}$ on $n$ generators is finite for every natural number $n$. This holds because every algebra of $\mathcal{V}$ generated by $n$ elements is a homomorphic image of $F_{\mathcal{V}}(n)$. Example: the variety of Boolean algebras is locally finite. A Boolean algebra generated by $n$ elements is an image of the free Boolean algebra on $n$ generators, which has $2^{2^n}$ elements. Non-example: the variety of groups is not locally finite. The group $(\Z, +)$ is generated by the single element $1$ and is infinite.