Since the calculation of the Grobner basis is based on the termination of division algorithm, so far, only a monomial-ordering can be used to calculate the Gr?bner basis. In fact, in the study of the division algorithm by Reeves and Strumfels, it was shown that any reduction process on a set of polynomials terminates if and only if the leading-term of the elements of the set are selected by a monomialordering. Consequently, if the leading-terms of a set are not selected based on a monomial-ordering, a reduction process can be found that the division algorithm does not terminate. Note that using Reeves and Sturmfels result we cannot decide about the existance of a Gr?bner basis.
