Monoid generalizations of the Richard Thompson groups

AbstractThe groups Gk,1 of Richard Thompson and Graham Higman can be generalized in a natural way to monoids, that we call Mk,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,1 have connections with circuit complexity (studied in other papers). Here we prove that Mk,1 and Invk,1 are congruence-simple for all k. Their Green relations J and D are characterized: Mk,1 and Invk,1 are J-0-simple, and they have k−1 non-zero D-classes. They are submonoids of the multiplicative part of the Cuntz algebra Ok. 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