Some characterizations of the commutator

We start with a characterization of the modular commutator that was given by E. Kiss and the first author in [DK] and explore some of its consequences. Implicit in this characterization is that both the shifting lemma and the cube lemma from [Gu] yield descriptions of the commutator in terms of implications of identities. The shifting lemma translates into the well known term condition and the cube lemma yields a similar condition involving two terms. We give some applications, improving a result of [Gu] and propose to define the commutator in non-modular varieties using the construction from [DK]. Varieties satisfying [α,β]=0 and those satisfying [α,α]=α are then characterized. This work was done when the second author visited Lakehead University in February and March of 1986. E. Kiss has, through a different approach, independently obtained a result very similar to Theorem 3.2 in a preprint of November 1986. The second author expresses his thanks to A. Day and the National Research Council of Canada for making his stay at Lakehead University both possible and pleasant.