Abstract
We establish a characterization of supernilpotent Mal’cev algebras which generalizes the affine structure of abelian Mal’cev algebras and the recent characterization of 2-supernilpotent Mal’cev algebras. We then show that for varieties in which the two-generated free algebra is finite: (1) neutrality of the higher commutators is equivalent to congruence meet-semidistributivity, and (2) the class of varieties which interpret a Mal’cev term in every supernilpotent algebra is equivalent to the existence of a weak difference term. We then establish properties of the higher commutator in the aforementioned second class of varieties.
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Acknowledgements
I would like to thank Andrew Moorhead and Jakub Opršal for intriguing and enthusiastic discussions about the higher commutator during the Vanderbilt Workshop on Structure and Complexity in Universal Algebra held September 19–30, 2016 in Nashville, TN.
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Presented by R. Willard.
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The author was supported in part by National Natural Science Foundation of China Research Fund for International Young Scientists #11650110429.
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Wires, A. On supernilpotent algebras. Algebra Univers. 80, 1 (2019). https://doi.org/10.1007/s00012-018-0574-0
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DOI: https://doi.org/10.1007/s00012-018-0574-0