Solution to a problem by FitzGerald

TL;DR

This paper demonstrates that four algebraic conditions thought to imply commuting idempotents in the monoid of endomorphisms are not sufficient, providing a counterexample and analyzing the conditions' dependence on the monoid structure.

Contribution

The paper provides the first counterexample to FitzGerald's conditions, clarifies their dependence on the monoid of endomorphisms, and identifies categories where the original question has a positive answer.

Findings

Counterexample disproves sufficiency of four conditions

Conditions depend only on the monoid of endomorphisms

Categories with affirmative answers identified

Abstract

FitzGerald identified four conditions (RI), (UR), (RI*) and (UR*) that are necessarily satisfied by an algebra, if its monoid of endomorphisms has commuting idempotents. We show that these conditions are not sufficient, by giving an example of an algebra satisfying the four properties, such that its monoid of endomorphisms does not have commuting idempotents. This settles a problem presented by Fitzgerald at the Conference and Workshop on General Algebra and Its Applications in 2013 and more recently at the workshop NCS 2018. After giving the counterexample, we show that the properties (UR), (RI*) and (UR*) depend only on the monoid of endomorphisms of the algebra, and that the counterexample we gave is in some sense the easiest possible. Finally, we list some categories in which FitzGerald's question has an affirmative answer.