On groups of units of special and one-relator inverse monoids

TL;DR

This paper explores the structure of groups of units in one-relator and special inverse monoids, providing new methods for their presentation and revealing cases where classical results do not apply.

Contribution

It introduces novel approaches to present the groups of units in special inverse monoids and establishes conditions for these groups to be one-relator, extending classical results.

Findings

Conditions under which the group of units is a one-relator group

Existence of a one-relator special inverse monoid with a non-one-relator group of units

Existence of a finitely presented special inverse monoid with a non-finitely presented group of units

Abstract

We investigate the groups of units of one-relator and special inverse monoids. These are inverse monoids which are defined by presentations where all the defining relations are of the form $r=1$. We develop new approaches for finding presentations for the group of units of a special inverse monoid, and apply these methods to give conditions under which the group admits a presentation with the same number of defining relations as the monoid. In particular our results give sufficient conditions for the group of units of a one-relator inverse monoid to be a one-relator group. When these conditions are satisfied these results give inverse semigroup theoretic analogues of classical results of Adjan for one-relator monoids, and Makanin for special monoids. In contrast, we show that in general these classical results do not hold for one-relator and special inverse monoids. In particular, we show that there exists a one-relator special inverse monoid whose group of units is not a one-relator group (with respect to any generating set), and we show that there exists a finitely presented special inverse monoid whose group of units is not finitely presented.