What is the weakest idempotent Maltsev condition that implies that abelian tolerances generate abelian congruences?
Keith A. Kearnes, Emil W. Kiss
TL;DR
The paper resolves a flaw in The Shape of Congruence Lattices by identifying that the claimed implication from a Taylor term to abelian-tolerance-generated abelian congruences fails without the stronger hypothesis of a weak difference term. The main result establishes the precise equivalence ${\mathscr W} \Leftrightarrow (\mathscr T \land \mathscr A)$, meaning that the conjunction of a Taylor term and the abelian-tolerance-to-congruence property is exactly captured by the existence of a weak difference term. Consequently, abelian-tolerance consequences are Maltsev-definable relative to ${\mathscr T}$ via ${\mathscr W}$, and the prior false claim about ${\mathscr T}$ alone is corrected. The work also delineates the corrected consequences for related lattice-theoretic results and clarifies how ${\mathscr W}$ relates to ${\mathscr T}$ in both general and locally finite contexts.
Abstract
We answer the question in the title. In the process, we correct an error in our AMS Memoir The Shape of Congruence Lattices.