Monoid generalizations of the Richard Thompson groups

The groups Gk,1Gk,1 of Richard Thompson and Graham Higman can be generalized in a natural way to monoids, that we call Mk,1Mk,1, and to inverse monoids, called Invk,1; this is done by simply generalizing bijections to partial functions or partial injective functions. The monoids Mk,1Mk,1 have connections with circuit complexity (studied in other papers). Here we prove that Mk,1Mk,1 and Invk,1 are congruence-simple for all kk. Their Green relations JJ and DD are characterized: Mk,1Mk,1 and Invk,1 are JJ-0-simple, and they have k−1k−1 non-zero DD-classes. They are submonoids of the multiplicative part of the Cuntz algebra OkOk. They are finitely generated, and their word problem over any finite generating set is in P. Their word problem is coNP-complete over certain infinite generating sets.