# Supernilpotent Taylor algebras are nilpotent

## TL;DR

This paper introduces the hypercommutator, a new higher commutator operation for Taylor varieties, proving that supernilpotent Taylor algebras are necessarily nilpotent and characterizing certain algebraic varieties.

## Contribution

It defines the hypercommutator using higher dimensional congruences and establishes its properties, linking supernilpotency to nilpotency in Taylor algebras.

## Key findings

- Hypercommutator is symmetric and satisfies an inequality for nested terms.
- Supernilpotent Taylor algebras are proven to be nilpotent.
- Characterization of congruence meet-semidistributive varieties via the higher commutator.

## Abstract

We develop the theory of the higher commutator for Taylor varieties. A new higher commutator operation called the hypercommutator is defined using a type of invariant relation called a higher dimensional congruence. The hypercommutator is shown to be symmetric and satisfy an inequality relating nested terms. For a Taylor algebra the term condition higher commutator and the hypercommutator are equal when evaluated at a constant tuple, and it follows that every supernilpotent Taylor algebra is nilpotent. We end with a characterization of congruence meet-semidistributive varieties in terms of the neutrality of the higher commutator.