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A syntactic characterization of weakly Mal'tsev varieties

Nadja Egner, Pierre-Alain Jacqmin, Nelson Martins-Ferreira

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.

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.
Paper Structure (6 sections, 14 theorems, 110 equations)

This paper contains 6 sections, 14 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Coproduct of two algebras
  4. (Weakly) Mal'tsev categories
  5. Weakly Mal'tsev varieties
  6. Another Mal'tsev-like property

Key Result

proposition 1

Let $A$ and $B$ be two non-empty algebras in $\mathbb{V}$.

Theorems & Definitions (26)

  • proposition 1
  • definition 1: Mal'tsev category carboni.pedicchio.pirovano:1992
  • proposition 2
  • proposition 3
  • definition 2: Weakly Mal'tsev category martins-ferreira:2008
  • proposition 4
  • theorem 1: Mal'tsev's theorem mal'tsev:1954mal'tsev:1963
  • lemma 1
  • proof
  • lemma 2
  • ...and 16 more