We call a restriction
semigroup almost perfect if it is proper and the least congruence that identifies all its projections is perfect. We show that any such
semigroup is isomorphic to a ‘
W -product’
W(T,Y), where
T is a monoid,
Y is a semilattice and there is a homomorphism from
T into the
inverse semigroup TIY of isomorphisms between ideals of
Y. Conversely, all such
W-products are almost perfect. Since we also show that every restriction
semigroup has an easily computed cover of this type, the combination yields a ‘McAlister-type’ theorem for all restriction
semigroups. It is one of the theses of this work that almost perfection and perfection, the analogue of this definition for restriction monoids, are the appropriate settings for such a theorem. That these theorems do
not reduce to a general theorem for
inverse semigroups illustrates a second thesis of this work: that restriction (and, by extension, Ehresmann)
semigroups have a rich theory that does not consist merely of generalizations of
inverse semigroup theory. It is then with some ambivalence that we show that all the main results of this work easily generalize to encompass
all proper restriction
semigroups.
The notation W(T,Y) recognizes that it is a far-reaching generalization of a long-known similarly titled construction. As a result, our work generalizes Szendrei's description of almost factorizable semigroups while at the same time including certain classes of free restriction semigroups in its realm.