2025/07/09 by James East, James E. East, East, James +7
Mathematics · Computer Science · #Advanced Operator Algebra Research #semigroups and automata theory #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2507.06600
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(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 PnΦ, which has the same biordered set as Pn.