Non-finitely Generated Maximal Subgroups of Context-free Monoids

TL;DR

This paper presents an example of a context-free monoid with non-finitely generated maximal subgroups, answering an open question and highlighting differences between context-free monoids and groups.

Contribution

It provides the first known example of a context-free monoid with all maximal subgroups non-finitely generated, addressing a question about the structure of units.

Findings

Existence of a context-free monoid with non-finitely generated maximal subgroups

Contrasts between properties of context-free monoids and groups

Answers to previously open questions in algebra

Abstract

A finitely generated group or monoid is said to be context-free if it has context-free word problem. In this note, we give an example of a context-free monoid, none of whose maximal subgroups are finitely generated. This answers a question of Brough, Cain & Pfeiffer on whether the group of units of a context-free monoid is always finitely generated, and highlights some of the contrasts between context-free monoids and context-free groups.