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

A relational description of higher commutators in Mal'cev varieties