What is the weakest idempotent Maltsev condition that implies that abelian tolerances generate abelian congruences?

TL;DR

This paper identifies the minimal idempotent Maltsev condition necessary to ensure abelian tolerances lead to abelian congruences, correcting previous errors in related work.

Contribution

It determines the weakest such Maltsev condition and rectifies an earlier mistake in the authors' prior publication.

Findings

Established the minimal idempotent Maltsev condition for abelian tolerances

Corrected an error in the authors' previous research on congruence lattices

Clarified the relationship between tolerances and congruences in algebraic structures

Abstract

We answer the question in the title. In the process, we correct an error in our AMS Memoir The Shape of Congruence Lattices.