A relational description of higher commutators in Mal'cev varieties

TL;DR

This paper provides a relational framework for higher commutators in Mal'cev varieties and constructs a maximal clone preserving key algebraic properties, enhancing understanding of algebraic structures with Mal'cev terms.

Contribution

It introduces a relational description of higher commutators and constructs a largest clone with preserved properties in Mal'cev algebras.

Findings

Relational description of higher commutators in Mal'cev varieties

Existence of a largest clone containing the Mal'cev operation

Preservation of congruence lattice and higher commutators

Abstract

We give a relational description of higher commutator operators, which were introduced by Bulatov, in varieties with a Mal'cev term. Furthermore, we use this result to prove that for every algebra with a Mal'cev term there exists a largest clone on the same underlying set containing the Mal'cev operation and having the same congruence lattice and the same higher commutator operators as the original algebra. A local variant of this theorem is given.