  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.
