The relationship between two commutators
  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. 
Keith A. Kearnes (University of Arkansas), {\AA}gnes Szendrei (Bolyai   Institute)