Growth Rates of Algebras, II: Wiegold Dichotomy

We investigate the function $d_{\mathbf{A}}(n)$, which gives the size of a least size generating set for ${\mathbf{A}}^n$, in the case where $\mathbf{A}$ has a cube term. We show that if $\mathbf{A}$ has a $k$-cube term and ${\mathbf{A}}^k$ is finitely generated, then $d_{\mathbf{A}}(n)\in O(\log (n))$ if $\mathbf{A}$ is perfect and $d_{\mathbf{A}}(n)\in O(n)$ if $\mathbf{A}$ is imperfect. When $\mathbf{A}$ is finite, then one may replace “Big Oh” with “Big Theta” in these estimates.