Subgroups of $E$-unitary and $R_1$-injective special inverse monoids

TL;DR

This paper investigates the structure of maximal subgroups in special inverse monoids, showing that under certain conditions, these subgroups are closely related to the group of units, with new constructions illustrating possible subgroup configurations.

Contribution

The paper demonstrates that in $E$-unitary or $ _1$-injective special inverse monoids, maximal subgroups are governed by the group of units, and constructs examples where any finite group appears as a maximal subgroup.

Findings

Maximal subgroups in $E$-unitary and $ _1$-injective special inverse monoids are controlled by the group of units.

Every finite group can be realized as a maximal subgroup with trivial group of units in an $ _1$-injective special inverse monoid.

Open problem: whether any group and finite index subgroup can be realized as maximal subgroup and group of units.

Abstract

We continue the study of the structure of general subgroups (in particular maximal subgroups, also known as group $\mathcal{H}$-classes) of special inverse monoids. Recent research of the authors has established that these can be quite wild, but in this paper we show that if we restrict to special inverse monoids which are $E$-unitary (or have a weaker property we call $\mathcal{R}_1$-injectivity), the maximal subgroups are strongly governed by the group of units. In particular, every maximal subgroup has a finite index subgroup which embeds in the group of units. We give a construction to show that every finite group can arise as a maximal subgroup in an $\mathcal{R}_1$-injective special inverse monoid with trivial group of units. It remains open whether every combination of a group $G$ and finite index subgroup $H$ can arise as maximal subgroup and group of units.