Algebra universalis

, 80:1 | Cite as

On supernilpotent algebras

  • Alexander WiresEmail author


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.


Higher commutator Supernilpotent Polynomial equivalence 

Mathematics Subject Classification

03C05 08B05 08A30 



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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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.School of Economic MathematicsSouthwestern University of Finance and EconomicsChengduChina

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