1998/08/01 by Keith A. Kearnes, Ágnes Szendrei · 59 citations
Mathematics · Computer Science · #Advanced Topics in Algebra #Advanced Algebra and Logic #Rings, Modules, and Algebras #Mathematics #Commutator #Centralizer and normalizer #Affine transformation #Idempotence #Pure mathematics #Variety (cybernetics) #Abelian group #Algebra over a field
paper · doi:10.1142/s0218196798000247
published in International Journal of Algebra and Computation 08(04), 497-531 (World Scientific)
openalex publication_date 1998/08/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/15
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.