Hardness Results for the Subpower Membership Problem

The main result of this paper shows that if $\mathcal{M}$ is a consistent strong linear Maltsev condition which does not imply the existence of a cube term, then for any finite algebra $\mathbb{A}$ there exists a new finite algebra $\mathbb{A}_\mathcal{M}$ which satisfies the Maltsev condition $\mathcal{M}$, and whose subpower membership problem is at least as hard as the subpower membership problem for $\mathbb{A}$. We characterize consistent strong linear Maltsev conditions which do not imply the existence of a cube term, and show that there are finite algebras in varieties that are congruence distributive and congruence $k$-permutable ($k \geq 3$) whose subpower membership problem is EXPTIME-complete.