We show that the free weakly E-ample monoid on a set X is a full submonoid of the free inverse monoid FIM(X) on X. Consequently, it is ample, and so coincides with
both the free weakly ample and the free ample monoid FAM(X) on X. We introduce the notion of a semidirect product Y ∗ T of a monoid T acting doubly on a semilattice Y with identity. We argue that the free monoid X∗ acts doubly on the semilattice Y of
idempotents of FIM(X) and that FAM(X) is embedded in
Y * X*. Finally we show that every weakly E-ample monoid has a proper ample cover.

CEMAT - Center for Computational and Stochastic Mathematics