An Order-Theoretic Property of the Commutator

We describe a new order-theoretic property of the commutator for finite algebras. As a corollary we show that any right nilpotent congruence on a finite algebra is left nilpotent. The result is false for infinite algebras and the converse is false even for finite algebras. We show further that any solvable E-minimal algebra is left nilpotent, any finite algebra whose congruence lattice contains a 0,1-sublattice isomorphic to M3 is left nilpotent and any homomorphic image of a finite abelian algebra is left and right nilpotent.