Free spectrum of a variety

Summary

The free spectrum of a variety $\mathcal{V}$ is a function on the natural numbers. It sends $n$ to the number of elements of the free algebra of $\mathcal{V}$ on $n$ generators. It is finite for all $n$ exactly when $\mathcal{V}$ is locally finite. Its growth rate measures how much freedom the identities of $\mathcal{V}$ leave, and it is used to separate and classify locally finite varieties.