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Variations of the Shifting Lemma and Goursat categories

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Abstract

We prove that Mal’tsev and Goursat categories may be characterized through variations of the Shifting Lemma, that is classically expressed in terms of three congruences R, S and T, and characterizes congruence modular varieties. We first show that a regular category \({\mathbb {C}}\) is a Mal’tsev category if and only if the Shifting Lemma holds for reflexive relations on the same object in \({\mathbb {C}}\). Moreover, we prove that a regular category \({\mathbb {C}}\) is a Goursat category if and only if the Shifting Lemma holds for a reflexive relation S and reflexive and positive relations R and T in \({\mathbb {C}}\). In particular this provides a new characterization of 2-permutable and 3-permutable varieties and quasi-varieties of universal algebras.

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Authors and Affiliations

  1. Institut de Recherche en Mathématique et Physique, Université Catholique de Louvain, Chemin du Cyclotron 2, 1348, Louvain-la-Neuve, Belgium

    Marino Gran & Idriss Tchoffo Nguefeu

  2. Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade do Algarve, Campus de Gambelas, 8005-139, Faro, Portugal

    Diana Rodelo

  3. CMUC, Department of Mathematics, University of Coimbra, 3001-501, Coimbra, Portugal

    Diana Rodelo

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  1. Marino Gran
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  2. Diana Rodelo
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  3. Idriss Tchoffo Nguefeu
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Correspondence to Marino Gran.

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D. Rodelo acknowledges partial financial assistance by Centro de Matemática da Universidade de Coimbra—UID/MAT/00324/2013, funded by the Portuguese Government through FCT/MCTES and co-funded by the European Regional Development Fund through the Partnership Agreement PT2020. I. Tchoffo Nguefeu acknowledges financial assistance by Fonds de la Recherche Scientifique-FNRS Crédit Bref Séjour à l’étranger 2018/V 3/5/033-IB/JN-11440.

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Gran, M., Rodelo, D. & Nguefeu, I.T. Variations of the Shifting Lemma and Goursat categories. Algebra Univers. 80, 2 (2019). https://doi.org/10.1007/s00012-018-0575-z

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  • DOI: https://doi.org/10.1007/s00012-018-0575-z

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