Existence of cube terms in finite algebras

Abstract: We study the problem of whether a given finite algebra with finitely many basic operations contains a cube term; we give both structural and algorithmic results. We show that if such an algebra has a cube term then it has a cube term of dimension at most $N$, where the number $N$ depends on the arities of basic operations of the algebra and the size of the basic set. For finite idempotent algebras we give a tight bound on $N$ that, in the special case of algebras with more than $\binom{|A|}2$ basic operations, improves an earlier result of K. Kearnes and A. Szendrei. On the algorithmic side, we show that deciding the existence of cube terms is in P for idempotent algebras and in EXPTIME in general. Since an algebra contains a $k$-ary near unanimity operation if and only if it contains a $k$-dimensional cube term and generates a congruence distributive variety, our algorithm also lets us decide whether a given finite algebra has a near unanimity operation.