A note on free idempotent generated semigroups over the full monoid of partial transformations

TL;DR

This paper extends previous results on the structure of maximal subgroups in free idempotent generated semigroups from full transformation monoids to the broader context of partial transformation monoids, showing similar isomorphism to symmetric groups.

Contribution

It proves that the maximal subgroups over the full monoid of partial transformations are isomorphic to symmetric groups, generalizing earlier results from full to partial transformations.

Findings

Maximal subgroups over PT_n are isomorphic to symmetric groups S_k.

The result holds for transformations of rank k with k <= n-2.

Extends known structure results from T_n to PT_n.

Abstract

Recently, Gray and Ruskuc (arXiv:1101.1833) proved that if e is a rank k idempotent transformation of the set {1,...,n} to itself and k<=n-2, then the maximal subgroup of the free idempotent generated semigroup over the full transformation monoid T_n containing e is isomorphic to the symmetric group S_k. We prove that the same holds when T_n is replaced by PT_n, the full monoid of partial transformations on {1,...,n}.