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

East, James, Gray, Robert D., Muhammed, P. A. Azeef and Ruskuc, Nik (2026) Maximal subgroups of free projection- and idempotent-generated semigroups with applications to partition monoids. Proceedings of the London Mathematical Society. ISSN 0024-6115 (In Press)

PDF (EGPR2025b_PLMS_final) - Accepted Version Restricted to Repository staff only until 31 December 2099. Request a copy

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 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.

Item Type:	Article
Uncontrolled Keywords:	regular ∗-semigroup,projection algebra,free projection-generated regular ∗-semigroup,biordered set,free idempotent-generated semigroup,singular square,maximal subgroup,presentation,partition monoid,twisted partition monoid
Faculty \ School:	Faculty of Science > School of Engineering, Mathematics and Physics
UEA Research Groups:	Faculty of Science > Research Groups > Algebra, Number Theory, Logic, and Representations (ANTLR)
Depositing User:	LivePure Connector
Date Deposited:	10 Aug 2026 09:05
Last Modified:	10 Aug 2026 09:05
URI:	https://ueaeprints.uea.ac.uk/id/eprint/104061
DOI:	issn:0024-6115

Actions (login required)

View Item