Facets of congruence distributivity in Goursat categories

Abstract: We give new characterisations of regular Mal'tsev categories with distributive lattice of equivalence relations through variations of the so-called Triangular Lemma and Trapezoid Lemma in universal algebra. We then give new characterisations of equivalence distributive Goursat categories (which extend $3$-permutable varieties) through variations of the Triangular and Trapezoid Lemmas involving reflexive and positive relations.