# On Supernilpotent Algebras

## TL;DR

This paper characterizes supernilpotent Mal'cev algebras, explores their properties, and links algebraic conditions like neutrality of higher commutators to structural properties of varieties, extending previous results.

## Contribution

It generalizes the structure of supernilpotent Mal'cev algebras and connects higher commutator properties with variety characteristics.

## Key findings

- Neutrality of higher commutators is equivalent to congruence meet-semidistributivity.
- Varieties interpreting a Mal'cev term in supernilpotent algebras have a weak difference term.
- Properties of higher commutators are established in specific variety classes.

## 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.