The Ursini commutator as normalized Smith-Pedicchio commutator

Sandra Mantovani

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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