On Supernilpotent Algebras
Alexander Wires
TL;DR
The paper develops a unified framework for higher commutators in Mal'cev varieties and characterizes $n$-supernilpotent algebras via polynomial representations to $n$-type loops, generalizing the affine structure observed in abelian cases. It demonstrates that, under a finiteness condition on the 2-generated free algebra, neutrality of higher commutators is equivalent to congruence meet-semidistributivity, and that the ability to interpret a Mal'cev term in every supernilpotent algebra is equivalent to the existence of a weak difference term. It further analyzes how absorbing polynomials control the higher commutator and shows how, in varieties with weak $n$-difference terms, restricted versions of key commutator properties hold, enabling a broader Mal'cev-theoretic characterization. These results connect structural properties of congruence lattices with term conditions, offering Mal'cev-style criteria for when complex higher-arity commutator behavior reduces to more familiar algebraic frameworks with potential applications to tame congruence theory and finite-model theory.
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 3-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.