  Based on the monoid classifier, we give an alternative axiomatization of
Freyd's paracategories, which can be interpreted in any bicategory of partial
maps. Assuming furthermore a free-monoid monad T in our ambient category, and
coequalisers satisfying some exactness conditions, we give an abstract envelope
construction, putting paramonoids (and paracategories) in the more general
context of partial algebras. We introduce for the latter the crucial notion of
saturation, which characterises those partial algebras which are isomorphic to
the ones obtained from their enveloping algebras. We also set up a
factorisation system for partial algebras, via epimorphisms and (monic) Kleene
morphisms and relate the latter to saturation.
