‘the Behaviour of the Idempotents of an Endomorphism

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 continue the study of the free idempotent generated semigroup IG(E) over the biordered set E of a wreath product G ≀ Tn. The latter monoid 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 ε ∈ EndFn(G) is the number of (free) generators in its image. For rather straightforward reasons it is known that if rank ε = n − 1 (respectively, n), then the maximal subgroup of ε in IG(E) is free (respectively, trivial). We show that if 1 ≤ rank ε ≤ n−2, then the maximal subgroup of ε in IG(E) is isomorphic to that in EndFn(G) and hence to G ≀ Sn. We have previously shown this result in the case rank r = 1; however, for higher rank, a more sophisticated approach is needed.