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

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.