Growth Rates of Algebras, II: Wiegold Dichotomy

Abstract

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.