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.