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The Relationship Between Two Commutators

Published:1998-08 Issue:04 Volume:08 Page:497-531
ISSN:0218-1967
Container-title:International Journal of Algebra and Computation
language:en
Short-container-title:Int. J. Algebra Comput.

Author:

Kearnes Keith A.¹,Szendrei Ágnes²

Affiliation:

1. Department of Mathematics, University of Louisville, Louisville, KY 40292, USA

2. Bolyai Institute, Aradi Vértanúk Tere 1, H-6720 Szeged, Hungary

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 [Formula: see text](A) satisfies an idempotent Mal'cev condition which fails to hold in the variety of semilattices, then A is affine.

Publisher

World Scientific Pub Co Pte Lt

Subject

General Mathematics

Link

https://www.worldscientific.com/doi/pdf/10.1142/S0218196798000247

Reference16 articles.

1. Some characterizations of the commutator

2. Congruence lattices of semilattices

3. A concrete ideal multiplication for algebraic systems and its relation to congruence distributivity

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