#
On the ternary commutator, I:
Exact Mal'tsev categories

##
Cyrille Sandry Simeu and Tim Van der Linden

In this first article on the Bulatov commutator, we introduce a ternary commutator of equivalence relations in the context of an exact Mal'tsev category with binary coproducts. We prove that, for Mal'tsev varieties, our notion is a particular case (where n=3) of the n-fold commutator introduced (originally in the context of Mal'tsev algebras) by A. Bulatov. We study its basic stability properties as well as the relationship with the (binary) Smith-Pedicchio commutator.
In a forthcoming second article, we restrict the context to algebraically coherent semi-abelian categories, where we prove that the commutator introduced here corresponds to the ternary Higgins commutator of M. Hartl and the second author.

Keywords:
Higher commutator; exact Mal'tsev category

2020 MSC:
18E13

*Theory and Applications of Categories,*
Vol. 36, 2021,
No. 15, pp 379-422.

Published 2021-06-23.

http://www.tac.mta.ca/tac/volumes/36/15/36-15.pdf

TAC Home