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The relationship between two commutators. (English) Zbl 0923.08001
We clarify the relationship between the linear commutator and the ordinary commutator by showing that in any variety satisfying a nontrivial idempotent Mal’tsev 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 \({\mathcal V}(A)\) satisfies an idempotent Mal’tsev condition which fails to hold in the variety of semilattices, then \(A\) is affine.

MSC:
08A05 Structure theory of algebraic structures
08B05 Equational logic, Mal’tsev conditions
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