The relationship between two commutators
Rings and Algebras
2008-02-03 v1
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.
Cite
@article{arxiv.math/9604246,
title = {The relationship between two commutators},
author = {Keith A. Kearnes and Ågnes Szendrei},
journal= {arXiv preprint arXiv:math/9604246},
year = {2008}
}