We introduce an intrinsic description of the Ursini commutator [Urs81, GU84] 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 [R H;R K]SP of the equivalence relations generated by H and K, extending the result valid for ideal determined varieties given in [Urs81, GU84].
The Ursini commutator as normalized Smith-Pedicchio commutator / S. Mantovani. - In: THEORY AND APPLICATIONS OF CATEGORIES. - ISSN 1201-561X. - 27:8(2012 Sep), pp. 174-188.
The Ursini commutator as normalized Smith-Pedicchio commutator
S. MantovaniPrimo
2012
Abstract
We introduce an intrinsic description of the Ursini commutator [Urs81, GU84] 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 [R H;R K]SP of the equivalence relations generated by H and K, extending the result valid for ideal determined varieties given in [Urs81, GU84].
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