  In this paper, we introduce the concept of P-difference varieties and study
the properties of toric P-difference varieties. Toric P-difference varieties
are analogues of toric varieties in difference algebra geometry. The category
of affine toric P-difference varieties with toric morphisms is shown to be
antiequivalent to the category of affine P[x]-semimodules with P[x]-semimodule
morphisms. Moreover, there is a one-to-one correspondence between the
irreducible invariant P-difference subvarieties of an affine toric P-difference
variety and the faces of the corresponding affine P[x]-semimodule. We also
define abstract toric P-difference varieties associated with fans by gluing
affine toric P-difference varieties. The irreducible invariant P-difference
subvarieties-faces correspondence is generalized to abstract toric P-difference
varieties. By virtue of this correspondence, a divisor theory for abstract
toric P-difference varieties is developed.
