Maximal subgroups of free projection- and idempotent-generated semigroups with applications to partition monoids

TL;DR

This paper characterizes the maximal subgroups of free projection- and idempotent-generated semigroups, revealing their structure in relation to partition monoids and providing explicit group computations.

Contribution

It provides explicit presentations and computations of maximal subgroups in free projection- and idempotent-generated semigroups derived from partition monoids.

Findings

Maximal subgroup of PG(P_n) is isomorphic to S_r for rank r

Maximal subgroup of IG(E(P_n)) is Z × S_r

Connection established between twisted partition monoid and biordered set

Abstract

This paper investigates the maximal subgroups of a free projection-generated regular $*$-semigroup $PG(P)$ over a projection algebra $P$, and their relationship to the maximal subgroups of the free idempotent-generated semigroup $IG(E)$ over the corresponding biordered set $E = E(P)$. In the first part of the paper we obtain a number of general presentations by generators and defining relations, in each case reflecting salient combinatorial/topological properties of the groups. In the second part we apply these to explicitly compute the groups when $P = P(P_n)$ and $E = E(P_n)$ arise from the partition monoid $P_n$. Specifically, we show that the maximal subgroup of $PG(P(P_n))$ corresponding to a projection of rank $r\leq n-2$ is (isomorphic to) the symmetric group $S_r$. In $IG(E(P_n))$, the corresponding subgroup is the direct product $Z \times S_r$. The appearance of the infinite cyclic group $Z$ is explained by a connection to a certain twisted partition monoid $P_n^\Phi$, which has the same biordered set as $P_n$.