Anticommutativity and the triangular lemma

TL;DR

This paper links the triangular lemma in universal algebra to a categorical anticommutativity condition, showing their equivalence and implications for the structure of algebraic varieties.

Contribution

It establishes that anticommutativity and the triangular lemma are equivalent Mal'tsev conditions, extending understanding of algebraic structures and their congruence properties.

Findings

Anticommutativity is a Mal'tsev condition.

The local version of anticommutativity is equivalent to the triangular lemma.

Locally anticommutative varieties have directly decomposable congruence classes.

Abstract

For a variety $\mathcal{V}$, it has been recently shown that binary products commute with arbitrary coequalizers locally, i.e., in every fibre of the fibration of points $\pi: \mathrm{Pt} (\mathbb{C}) \rightarrow \mathbb{C}$, if and only if Gumm's shifting lemma holds on pullbacks in $\mathcal{V}$. In this paper, we establish a similar result connecting the so-called triangular lemma in universal algebra with a certain categorical $\textit{anticommutativity}$ condition. In particular, we show that this anticommutativity and its local version are Mal'tsev conditions, the local version being equivalent to the triangular lemma on pullbacks. As a corollary, every locally anticommutative variety $\mathcal{V}$ has directly decomposable congruence classes in the sense of Duda, and the converse holds if $\mathcal{V}$ is idempotent.