On right units of special inverse monoids

We study the class of monoids that arise as the submonoid of right units of finitely presented special inverse monoids. Classical results of Makanin show that the monoids of right units in finitely presented special monoids decompose as a free product of the group of units and a finite rank free monoid. Gray and Ruskuc (2024) gave the first example of a finitely presented special inverse monoid whose submonoid of right units does not admit such a decomposition, which left open the question of determining the structure of such monoids. 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 special inverse monoid can admit such a free product decomposition is when their group of units is finitely presented. This implies that the typical behaviour of the right units is not to admit such a free product decomposition. In showing this, we establish some general results about finite generation and presentability of subgroups of special inverse monoids. 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 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 special inverse monoids and prefix monoids of finitely presented groups are independent, in the sense that neither of them is contained in the other. 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 special inverse monoid 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 special inverse monoids 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.