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

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 sub-groups 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(Pn) and E=E(Pn) arise from the partition monoid Pn. Specifically, we show that the maximal subgroup of PG(P(Pn)) corresponding to a projection of rank r≤n−2
is (isomorphic to) the symmetric group Sr. In IG(E(Pn)), the corresponding subgroup is the direct product Z×Sr. The appearance of the infinite cyclic group Z is explained by a connection to a certain twisted partition monoid PΦn, which has the same biordered set as Pn.