We introduce an intrinsic description of the Ursini commutator in any ideal determined category and we compare it with the Higgins and Huq commutators. After describing also the Smith-Pedicchio commutator by means of canonical arrows from a coproduct, we compare the two notions, showing that in any exact Mal'tsev normal category the Ursini commutator $[H,K]_{U}$ of two subobjects $H, K$ of $A$ is the normalization of the Smith-Pedicchio commutator of the equivalence relations generated by $H$ and $K$, extending the result valid for ideal determined varieties given by Ursini and Gumm.
Keywords: commutator, ideal determined category, exact Mal'tsev category
2010 MSC: 18C05, 08A30
Theory and Applications of Categories, Vol. 27, 2012, No. 8, pp 174-188.
Published 2012-09-20.
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