ABSTRACT
For many algorithms that require computations involving multivariate polynomials, the order placed on the monomials (called a term-ordering) often affects their practical efficiency. For this reason, term-orderings have been extensively studied. For calculations with multivariate partial differential polynomials, the order placed on the derivatives (called a ranking) plays a similar important role. In this paper, we give a structural decomposition and characterization of rankings using a generalized concept of Dedekind cuts. We give a general procedure to construct a wide class of commonly used rankings. This study of rankings is pursued first in the setting of general abelian groups, and then specialized to the cases of m-dimensional modules over the real numbers and the integers.
