A monoid version of the Brin-Higman-Thompson groups

TL;DR

This paper extends the Brin-Higman-Thompson groups to monoids by replacing bijections with partial functions, revealing new algebraic properties and computational complexities, and exploring higher-dimensional joinless codes.

Contribution

It introduces the monoid versions of these groups, proves their simplicity and finite generation, and analyzes their word problem complexity and higher-dimensional codes.

Findings

The monoids are congruence-simple and finitely generated.

The word problem for n M_{k,1} is coNP-complete for n ≥ 2.

New results on higher-dimensional joinless codes.

Abstract

We generalize the Brin-Higman-Thompson groups $n G_{k,1}$ to monoids $n M_{k,1}$, for $n \ge 1$ and $k \ge 2$, by replacing bijections by partial functions. The monoid $n M_{k,1}$ has $n G_{k,1}$ as its group of units, and is congruence-simple. Moreover, $n M_{k,1}$ is finitely generated, and for $n \ge 2$ its word problem is {\sf coNP}-complete. We also present new results about higher-dimensional joinless codes.