Algebra universalis

, 80:2 | Cite as

Variations of the Shifting Lemma and Goursat categories

  • Marino GranEmail author
  • Diana Rodelo
  • Idriss Tchoffo Nguefeu


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.


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

Mathematics Subject Classification

08C05 08B05 08A30 08B10 18C05 18B99 18E10 



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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Marino Gran
    • 1
    Email author
  • Diana Rodelo
    • 2
    • 3
  • Idriss Tchoffo Nguefeu
    • 1
  1. 1.Institut de Recherche en Mathématique et PhysiqueUniversité Catholique de LouvainLouvain-la-NeuveBelgium
  2. 2.Departamento de Matemática, Faculdade de Ciências e TecnologiaUniversidade do AlgarveFaroPortugal
  3. 3.CMUC, Department of MathematicsUniversity of CoimbraCoimbraPortugal

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