Narrow Results

We describe a manageable set of relations that generates the finitary relational clone of an algebra with a parallelogram term. This result applies to any algebra with a Maltsev term and to any algebra with a near unanimity term. One consequence of the main result is that on any finite set and for any finite k there are only finitely many clones of algebras with a k-ary parallelogram term which generate residually small varieties.

The main result of this paper shows that if ${ℳ}$ is a consistent strong linear Maltsev condition which does not imply the existence of a cube term, then for any finite algebra $𝔸$ there exists a new finite algebra ${𝔸}_{{ℳ}}$ which satisfies the Maltsev condition ${ℳ}$, and whose subpower membership problem is at least as hard as the subpower membership problem for $𝔸$. 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 ($k\ge 3$) whose subpower membership problem is EXPTIME-complete.

We investigate the function d_A(n), which gives the size of a least size generating set for Aⁿ, in the case where A has a cube term. We show that if A has a k-cube term and A^k is finitely generated, then d_A(n) ∈ O(log(n)) if A is perfect and d_A(n) ∈ O(n) if A is imperfect. When A is finite, then one may replace "Big O" with "Big Theta" in these estimates.

We characterize absorption in finite idempotent algebras by means of Jónsson absorption and cube term blockers. As an application we show that it is decidable whether a given subset is an absorbing subuniverse of an algebra given by the tables of its basic operations.