  We establish that, under certain closure assumptions on a pseudovariety of
semigroups, the corresponding relatively free profinite semigroups freely
generated by a non-singleton finite set act faithfully on their minimum ideals.
As applications, we enlarge the scope of several previous join irreducibility
results for pseudovarieties of semigroups, which turn out to be even join
irreducible in the lattice of pseudovarieties of ordered semigroups, so that,
in particular, they are not generated by proper subpseudovarieties of ordered
semigroups. We also prove the stronger form of join irreducibility for the
Krohn-Rhodes complexity pseudovarieties, thereby solving a problem proposed by
Rhodes and Steinberg.
