A -polyhedron can be defined as the image of a polyhedron by an
invertible, integral function. Consider, for instance, the lattice
and the polyhedron
. Then
is
a -polyhedron (see figure ). can also be expressed as:

which is the image of polyhedron by the function . is obtained by taking the preimage of by the function defining the lattice: .

As Polylib operates on *domains* made out of
unions of polyhedra, it is natural to define a similar
object for -polyhedra. A -domain is a finite union of intersections
between a domain and a full dimensionnal integral affine lattice.
The set of -domains is closed under union, intersection,
difference, and image by integral invertible function.

The image and pre-image by rational functions is also implemented in Polylib, but the result of the image (or pre-image) is always intersected with the canonical lattice (). See [NRi00] for further detail on this subject.

From the implementation point of view, a -polyhedron is represented internally as the image of a polyhedron by an affine, invertible mapping. Hence, storing a -polyhedron amounts to storing a domain and a matrix.