Presentations for singular wreath products

Abstract

For a monoid M and a subsemigroup S of the full transformation semigroup Tn, the wreath product M≀S is defined to be the semidirect product Mn⋊S, with the coordinatewise action of S on Mn. The full wreath product M≀Tn is isomorphic to the endomorphism monoid of the free M-act on n generators. Here we are particularly interested in the case that S=Singn is the singular part of Tn, consisting of all non-invertible transformations. Our main results are presentations for M≀Singn in terms of certain natural generating sets, and we prove these via general results on semidirect products and wreath products. We re-prove a classical result of Bulman-Fleming that M≀Singn is idempotent-generated if and only if the set M/L of L-classes of M forms a chain under the usual ordering of L-classes, and we give a presentation for M≀Singn in terms of idempotent generators for such a monoid M. Among other results, we also give estimates for the minimal size of a generating set for M≀Singn, as well as exact values in some cases (including the case that M is finite and M/L is a chain, in which case we also calculate the minimal size of an idempotent generating set). As an application of our results, we obtain a presentation (with idempotent generators) for the idempotent-generated subsemigroup of the endomorphism monoid of a uniform partition of a finite set.