Monomial orderings and Gröbner bases

Abstract

Let there be given a set of monomials in n variables and some order relations between them. The following fundamental problem of monomial ordering is considered. Is it possible to decide whether these ordering relations are consistent and if so to extend them to an admissible ordering for all monomials? The answer is given in terms of the algorithm MACOT which constructs a matrix of so called cotes which establishes the desired ordering relations. The main area of application of this algorithm, i.e. the construction of Gröbner bases for different orderings and of universal Gröbner bases is treated in the last section.