The groups G_{k,1} of Richard Thompson and Graham Higman can be generalized in a natural way to monoids, that we call M_{k,1}, and to inverse monoids, called Inv_{k,1}; this is done by simply generalizing bijections to partial functions or partial injective functions. The monoids M_{k,1} have connections with circuit complexity (studied in another paper). Here we prove that M_{k,1} and Inv_{k,1} are congruence-simple for all k. Their Green relations J and D are characterized: M_{k,1} and Inv_{k,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 O_k. 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. Changes in this version: Section 4 has been thoroughly revised, and errors have been corrected; however, the main results of Section 4 do not change. Sections 1, 2, and 3 are unchanged, except for the proof of Theorem 2.3, which was incomplete; a complete proof was published in the Appendix of reference [6], and is also given here.
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