Products of Three Idempotent Transformations

In 1988 Howie, Robertson and Schein characterised the transformations of a finite set X that can be written as a product of two or of three idempotent transformations of X; and in 1989 Saito did the same for products of four idempotents. In 1998 Thomas extended the characterisation of two idempotents to arbitrary sets, and here we characterise products of three idempotents in general. We also define a notion of complexity for transformations of any set and use it to provide a different solution to the three-idempotent problem.