  In this paper, binomial difference ideals are studied. Three canonical
representations for Laurent binomial difference ideals are given in terms of
the reduced Groebner basis of Z[x]-lattices, regular and coherent difference
ascending chains, and partial characters over Z[x]-lattices, respectively.
Criteria for a Laurent binomial difference ideal to be reflexive, prime,
well-mixed, and perfect are given in terms of their support lattices. The
reflexive, well-mixed, and perfect closures of a Laurent binomial difference
ideal are shown to be binomial. Most of the properties of Laurent binomial
difference ideals are extended to the case of difference binomial ideals.
Finally, algorithms are given to check whether a given Laurent binomial
difference ideal I is reflexive, prime, well-mixed, or perfect, and in the
negative case, to compute the reflexive, well-mixed, and perfect closures of I.
An algorithm is given to decompose a finitely generated perfect binomial
difference ideal as the intersection of reflexive prime binomial difference
ideals.
