Products of idempotents in certain semigroups of transformations

Abstract

In an earlier paper ( 5 ) a description was given in set-theoretic terms of the semigroup generated by the idempotents of a full transformation semigroup , one of the results being that if X is finite then every element of that is not bijective is expressible as a product of idempotents. In view of this it was natural to ask whether by analogy every singular square matrix is expressibleas a product of idempotent matrices. This is indeed the case, as was shown by J. A. Erdos ( 2 ). Magill ( 6 ) has considered products of idempotents in thesemigroup of all continuous self-maps of a topological space X , but a comparable characterization of products of idempotents in this case appears to be extremelydifficult, and no solution is available yet.