The relationship between two commutators

Abstract

We clarify the relationship between the linear commutator and the ordinary commutator by showing that in any variety satisfying a nontrivial idempotent Mal'cev condition the linear commutator is definable in terms of the centralizer relation. We derive from this that abelian algebras are quasi-affine in such varieties. We refine this by showing that if A is an abelian algebra and V(A) satifies an idempotent Mal'cev condition which fails to hold in the variety of semilattices, then A is affine.