Cube term blockers without finiteness

Abstract

We show that an idempotent variety has a d-dimensional cube term if and only if its free algebra on two generators has no d-ary compatible cross. We employ Hall’s Marriage Theorem to show that an idempotent variety $${\mathcal{V}}$$ of finite signature whose fundamental operations have arities n 1, . . . , n k, has a d-dimensional cube term for some d if and only if it has one of dimension $${1 + \sum_{i=1}^{k} (n_{i} - 1)}$$ . This upper bound on the dimension of a minimal-dimension cube term for $${\mathcal{V}}$$ is shown to be sharp. We show that a pure cyclic term variety has a cube term if and only if it contains no 2- element semilattice. We prove that the Maltsev condition “existence of a cube term” is join prime in the lattice of idempotent Maltsev conditions.