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

Abstract

This paper investigates the maximal subgroups of a free projection-generated regular $\ast$-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 $r\le n-2$ 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 $P_n^{\hat{I}}¦$, which has the same biordered set as $P_n$.

V2: 73 pages, 25 figures, referee comments incorporated, to appear in Proc LMS. V1: 72 pages, 25 figures