@UNPUBLISHED{publication1, title = "Variations of the Shifting Lemma and Goursat categories", author = "{GRAN, Marino} and {RODELO, Diana} and {NGUEFEU, Idriss Tchoffo}", year = "2018-09-26", abstract = "We prove that Mal\’tsev and Goursat categories may be characterised through stronger variations of the Shifting Lemma, that is classically expressed in terms of three congruences R, S and T, and characterises congruence modular varieties. We first show that a regular category is a Mal\’tsev category if and only if the Shifting Lemma holds for reflexive relations on the same object in C. Moreover, we prove that a regular category 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 C. In particular this provides a new characterisation of 2-permutable and 3-permutable varieties and quasi-varieties of universal algebras. ", url = "http://www.mat.uc.pt/preprints/eng\_2018.html", note = "" }