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arXiv · 2403.09478

A syntactic characterization of weakly Mal'tsev varieties

Nadja Egner·Pierre-Alain Jacqmin·Nelson Martins-Ferreira
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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.

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BibTeXRIS

Nadja Egner, Pierre-Alain Jacqmin, Nelson Martins-Ferreira. 2024-03-14. A syntactic characterization of weakly Mal'tsev varieties. https://arxiv.org/abs/2403.09478

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