Facets of congruence distributivity in Goursat categories

TL;DR

This paper characterizes certain algebraic categories, specifically regular Mal'tsev and Goursat categories, using variations of classical lemmas to understand their lattice of equivalence relations.

Contribution

It introduces new characterizations of regular Mal'tsev and Goursat categories via modified Triangular and Trapezoid Lemmas involving different types of relations.

Findings

New characterizations of regular Mal'tsev categories with distributive equivalence relations.

Extended characterizations of Goursat categories using variations of classical lemmas.

Connections between algebraic properties and lattice structures of relations.

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.