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 ‐semigroup over a projection algebra , and their relationship to the maximal subgroups of the free idempotent‐generated semigroup over the corresponding biordered set . 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 and arise from the partition monoid . Specifically, we show that the maximal subgroup of corresponding to a projection of rank is (isomorphic to) the symmetric group . In , the corresponding subgroup is the direct product . The appearance of the infinite cyclic group is explained by a connection to a certain twisted partition monoid , which has the same biordered set as .