On right units of special inverse monoids

TL;DR

This paper investigates the structure of right units in finitely presented special inverse monoids, providing characterizations, conditions for finite generation, and extending classification results with implications for monoid theory.

Contribution

It offers a comprehensive analysis of submonoids of right units in SIMs, including new characterizations and a generalization of existing classification results.

Findings

Only finitely presented group of units allows free product decomposition.

Characterization of finitely generated subgroups via connectedness of cosets.

Every finitely generated submonoid of a finitely RC-presented monoid embeds into a SIM.

Abstract

We study the class of monoids that arise as the submonoid of right units of finitely presented special inverse monoids (SIMs). Gray and Ru\v{s}kuc (2024) gave the first example of a finitely presented SIM whose submonoid of right units does not admit a decomposition into a free product of the group of units and a finite rank free monoid. In the first part of this paper we prove a general result which shows that the only instances where the right units of a finitely presented SIM can admit such a free product decomposition is when their group of units is finitely presented. In showing this, we establish some general results about finite generation and presentability of subgroups of SIMs. In particular, we give an exact characterisation of when an arbitrary subgroup is finitely generated in terms of connectedness properties of unions of its cosets in its $\mathscr{R}$-class, and also a characterisation of when an arbitrary subgroup is finitely presented. We also give a sufficient condition for finite generation and presentability of an arbitrary subgroup given in terms of a geometric finiteness property called boundary width. As a consequence, we show that the classes of monoids of right units of finitely presented SIMs and prefix monoids of finitely presented groups are independent. In the second part of the paper, we show that every finitely generated submonoid of a finitely RC-presented monoid is isomorphic to a submonoid $N$ of a finitely presented SIM $M$ such that $N$ is a submonoid of the right units of $M$, and $N$ contains the group of units of $M$. This result generalises and extends the classification of groups of units of finitely presented SIMs recently obtained by Gray and Kambites (2025). From this, we derive a number of surprising properties of RC-presentations for right cancellative monoids contrasting the classical theory of monoid presentations.