Cube term blockers without finiteness

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 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 + ∑ i = 1 k ( n i - 1 ) . This upper bound on the dimension of a minimal-dimension cube term for 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.