  Two fundamental constructions operating on residuated lattices and partially
ordered monoids (pomonoids) are so-called nuclear images and conuclear images.
Nuclear images allow us to construct many of the ordered algebras which arise
in non-classical logic (such as pomonoids, semilattice-ordered monoids, and
residuated lattices) from cancellative ones. Conuclear images then allow us to
construct some of these cancellative algebras from partially ordered or
lattice-ordered groups. Among other things, we show that finite (commutative)
integral residuated lattices are precisely the finite nuclear images of
commutative cancellative integral residuated lattices and that (commutative)
integrally closed pomonoids are precisely the nuclear images of subpomonoids of
partially ordered (Abelian) groups. The key construction is the free nuclear
preimage of a pomonoid. As a by-product of our study of free nuclear preimages,
we obtain a syntactic characterization of quasivarieties of pomonoids and
semilattice-ordered monoids closed under nuclear images.
