Some Characterizations of the Commutator A Day

We start with a characterization of the modular commutator that was given by E Kiss and the rst 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 de ne the com mutator in non modular varieties using the construction from DK Varieties satisfying and those satisfying are then characterized This work was done when the second author visited Lakehead University in February and March of E Kiss has through a di erent approach independently obtained a result very similar to theorem in a preprint of November 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 Preliminaries Given a homomorphism A B and a congruence relation on A we write or simply for the congruence relation on B generated by the Research of both authors supported by N S E R C Operating Grant A direct images of pairs in conB f x y j x y g This notation is justi ed by the observation that given a homomorphism B C then for all Con A If Con B the inverse image congruence on A is denoted by