The Relationship Between Two Commutators

@article{Kearnes1998TheRB,
  title={The Relationship Between Two Commutators},
  author={K. Kearnes and {\'A}. Szendrei},
  journal={Int. J. Algebra Comput.},
  year={1998},
  volume={8},
  pages={497-532}
}
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 (A) satisfies an idempotent Mal'cev condition which fails to hold in the variety of semilattices, then A is affine. 

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