  In this paper, the concept of toric difference varieties is defined and four
equivalent descriptions for toric difference varieties are presented in terms
of difference rational parametrization, difference coordinate rings, toric
difference ideals, and group actions by difference tori. Connections between
toric difference varieties and affine N[x]-semimodules are established by
proving the correspondence between the irreducible invariant difference
subvarieties and the faces of the N[x]-submodules and the orbit-face
correspondence. Finally, an algorithm is given to decide whether a binomial
difference ideal represented by a Z[x]-lattice defines a toric difference
variety.
