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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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
Keywords
- Mal’tsev categories
- Goursat categories
- Shifting Lemma
- Congruence modular varieties
- 3-permutable varieties