Free idempotent generated semigroups and endomorphism monoids of free G-acts

Abstract

The study of the free idempotent generated semigroup IG(E) over a biordered set E began with the seminal work of Nambooripad in the 1970s and has seen a recent revival with a number of new approaches, both geometric and combinatorial. Here we study IG(E) in the case E is the biordered set of a wreath product G≀Tn, where G is a group and Tn is the full transformation monoid on n elements. This wreath product is isomorphic to the endomorphism monoid of the free G-act EndFn(G) on n generators, and this provides us with a convenient approach.We say that the rank of an element of EndFn(G) is the minimal number of (free) generators in its image. Let ε=ε2∈EndFn(G). For rather straightforward reasons it is known that if rankε=n−1 (respectively, n), then the maximal subgroup of IG(E) containing ε is free (respectively, trivial). We show that if rankε=r where 1≤r≤n−2, then the maximal subgroup of IG(E) containing ε is isomorphic to that in EndFn(G) and hence to G≀Sr, where Sr is the symmetric group on r elements. We have previously shown this result in the case r=1; however, for higher rank, a more sophisticated approach is needed. Our current proof subsumes the case r=1 and thus provides another approach to showing that any group occurs as the maximal subgroup of some IG(E). On the other hand, varying r again and taking G to be trivial, we obtain an alternative proof of the recent result of Gray and Ruškuc for the biordered set of idempotents of Tn.