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

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 to 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. We also apply these results to establish a close connection between the question of finite presentability of the groups of units of special one-relator inverse monoids, and the open problem of whether all one-relator groups are coherent. In addition, we give a new approach via regular languages for computing invertible pieces of relators in special inverse monoids. We prove that this algorithm outperforms the classical Adjan overlap algorithm when applied to special inverse monoids, in the sense that it discovers more invertible pieces of relators.

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

“…It can be shown that Λ generates the subgroup consisting of all units of M (see [22,Proposition 4.2], for the stronger form here stated see [19,Theorem 1.3]). This subgroup is denoted U (M ), and is called the group of units of M .…”

“…In particular, as already proved by Adian [3,Theorem 8], U (M ) is a onerelator group if M is a special one-relation monoid. Although we do not directly study the group of units in this article, we highlight the fact that by contrast, in the inverse case M = Inv A | w = 1 , it need not be the case that U (M ) is a one-relator group; Gray & Ruškuc [19,Theorem 7.1] provide an example in which U (M ) is isomorphic to the free product of two copies of the fundamental group of a surface of genus 2, which is not a one-relator group by [25,Proposition 5.13]. This demonstrates the contrast between special inverse and special "ordinary" monoids.…”

“…In view of the O'Hare monoid, there was, for some time, no natural candidate for an algorithm to compute the minimal invertible pieces (or indeed the group of units) of a special inverse onerelation monoid. To amend this, Gray & Ruškuc [19] recently introduced the Benois algorithm. We now give a full description of this rather short algorithm here, as we shall frequently use it in this article.…”

“…4 The name of the Benois algorithm stems from the decidability of this latter problem. Gray & Ruškuc proved (see [19,Theorem 4.5]) that the Benois algorithm detects any factorisation that Adian's overlap algorithm does, and that furthermore it detects the factorisation (2.2) in the O'Hare monoid, thus strictly outperforming Adian's algorithm.…”

“…One of the difficulties in attempting to mimic Adian's theory of special one-relation monoids in the inverse case comes from the present inability to understand the rôle of the group of units. Gray & Ruškuc [19] have recently demonstrated much pathological behaviour for the group of units of special one-relation inverse monoids. Indeed, there is currently no known algorithm for computing the group of units, given a presentation Inv A | w = 1 .…”