# Monoid generalizations of the Richard Thompson groups

## TL;DR

This paper introduces monoid and inverse monoid generalizations of Richard Thompson groups, proves their simplicity and characterizes their Green relations, and analyzes their computational complexity and algebraic properties.

## Contribution

It defines and studies the properties of monoids M_{k,1} and Inv_{k,1} as generalizations of Thompson groups, including simplicity, Green relations, and complexity results.

## Key findings

- M_{k,1} and Inv_{k,1} are congruence-simple for all k.
- They are J-0-simple with k-1 non-zero D-classes.
- Their word problem is in P over finite sets and coNP-complete over certain infinite sets.

## Abstract

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.