In this paper, we will learn about the Groebner basis of a normal binomial $\sigma$-ideal and the condition when it is finite. A normal binomial ideal in a differnce ring corresponds to a parital character in $\mathbb{Z}[x]^{n}$. Thus this question can be converted into another one aboout a $\mathbb{Z}[x]$-submodule. We will give an equivalent condition when the Groebner basis of the saturation of an ideal generated by a binomial in $k\{y\}$, and then promote this conclusion to the situation about many generators or $k\{y_1,y_2,\ldots,y_n\}$. We research the aforementioned condition furtherly, and then give such many necessary or sufficient conditions. Finnally we discuss the Groebner basis about a quadratic, and give a completely determining condition. \newline Keywords: Differece algebra, Binomial ideal, Groebner basis, $\mathbb{Z}[x]$-lattice