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.
Categories: math.RA