A syntactic characterization of weakly Mal'tsev varieties
Abstract
The notion of a weakly Mal'tsev category, as it was introduced in 2008 by the third author, is a generalization of the classical notion of a Mal'tsev category. It is well-known that a variety of universal algebras is a Mal'tsev category if and only if its theory admits a Mal'tsev term. In the main theorem of this paper, we prove a syntactic characterization of the varieties that are weakly Mal'tsev categories. We apply our result to the variety of distributive lattices which was known to be a weakly Mal'tsev category before. By a result of Z. Janelidze and the third author, a finitely complete category is weakly Mal'tsev if and only if any internal strong reflexive relation is an equivalence relation. In the last part of this paper, we give a syntactic characterization of those varieties in which any regular reflexive relation is an equivalence relation.
Keywords
Cite
@article{arxiv.2403.09478,
title = {A syntactic characterization of weakly Mal'tsev varieties},
author = {Nadja Egner and Pierre-Alain Jacqmin and Nelson Martins-Ferreira},
journal= {arXiv preprint arXiv:2403.09478},
year = {2024}
}
Comments
41 pages, accepted to the special volume of Theory and Applications of Categories in honour of Pieter Hofstra