Monoids that map onto the Thompson-Higman groups

A slight modification of the definition of the Thompson-Higman groups G _k,1 and F _k,1 leads to inverse monoids that map onto G _k,1 (respectively F _k,1), and that have interesting properties: they are finitely generated, and residually finite. These inverse monoids are closely related to the suffix expansion of G _k,1 (respectively F _k,1).