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

Simeu, Cyrille Sandry;Van der Linden, Tim
(2021) Theory and Applications of Categories — Vol. 36, n° 15, p. 379-422 (2021)

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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.
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Simeu, C. S., & Van der Linden, T. (2021). On the ternary commutator, I: Exact Mal’tsev categories,. Theory and Applications of Categories, 36(15), 379-422. https://hdl.handle.net/2078.5/109599 (Original work published 2021)