A tale of three idempotents. (Q1764619)

\textit{J. M. Howie, E. F. Robertson}, and \textit{B. M. Schein} [in Proc. R. Soc. Edinb., Sect. A 109, No.~3/4, 319-328 (1988; Zbl 0654.20080)], characterized those selfmaps on a finite set which are products of three idempotent selfmaps on that same set. \textit{I. Fleischer} [in J. Algebra 238, No.~2, 459-461 (2001; Zbl 0996.20048)], characterized those selfmaps of a set (not necessarily finite) which are products of three idempotent selfmaps of that set and he went on to attempt to show that this characterization agreed with that given by the previous three authors in the case where the set is finite. However, he concluded that ``the argument was awkward and not completely convincing''. He remedies this situation in the paper under review.