caa a22 4500 469026774 CHVBK 20180323132732.0 cr unu---uuuuu 170328e19920301xx s 000 0 eng 10.1007/BF01190756 doi (NATIONALLICENCE)springer-10.1007/BF01190756 Some characterizations of the commutator [Elektronische Daten] [A. Day, H. Gumm] 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. Birkhäuser Verlag, 1992 Day A. Department of Mathematics, SUNY at New Paltz, 12561, New Paltz, New York, USA aut Gumm H. Department of Mathematics, SUNY at New Paltz, 12561, New Paltz, New York, USA aut algebra universalis Birkhäuser-Verlag 29/1(1992-03-01), 61-78 0002-5240 29:1<61 1992 29 12 https://doi.org/10.1007/BF01190756 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/BF01190756 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Day A. Department of Mathematics, SUNY at New Paltz, 12561, New Paltz, New York, USA aut NATIONALLICENCE 700 1- Gumm H. Department of Mathematics, SUNY at New Paltz, 12561, New Paltz, New York, USA aut NATIONALLICENCE 773 0- algebra universalis Birkhäuser-Verlag 29/1(1992-03-01), 61-78 0002-5240 29:1<61 1992 29 12 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer