Hardness results for the subpower membership problem

The main result of this paper shows that if [Formula: see text] is a consistent strong linear Maltsev condition which does not imply the existence of a cube term, then for any finite algebra [Formula: see text] there exists a new finite algebra [Formula: see text] which satisfies the Maltsev condition [Formula: see text], and whose subpower membership problem is at least as hard as the subpower membership problem for [Formula: see text]. 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 [Formula: see text]-permutable ([Formula: see text]) whose subpower membership problem is EXPTIME-complete.

Deciding absorption

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.

Growth rates of algebras, II: Wiegold dichotomy

We investigate the function dA(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 Ak is finitely generated, then dA(n) ∈ O( log (n)) if A is perfect and dA(n) ∈ O(n) if A is imperfect. When A is finite, then one may replace "Big O" with "Big Theta" in these estimates.

COMPUTATIONAL COMPLEXITY OF VARIOUS MAL'CEV CONDITIONS

This paper examines the computational complexity of determining whether or not an algebra satisfies a certain Mal'Cev condition. First, we define a class of Mal'Cev conditions whose satisfaction can be determined in polynomial time (special cube term satisfying the DCP) when the algebra in question is idempotent and provide an algorithm through which this determination may be made. The aforementioned class notably includes near unanimity terms and edge terms of fixed arity. Second, we define a different class of Mal'Cev conditions whose satisfaction, in general, requires exponential time to determine (Mal'Cev conditions satisfiable by CPB0 operations).