Toward a Theory of Monomial Preorders

TL;DR

This paper develops a theory of monomial preorders that include ties, providing a less degenerate leading ideal and new insights even in classical monomial order cases, aiding in solving problems where traditional orders fail.

Contribution

It introduces a novel theory of monomial preorders allowing ties, expanding the classical framework and offering new tools for algebraic problem-solving.

Findings

Monomial preorders generalize monomial orders by allowing ties.

Leading ideals under preorders are less degenerate, enabling new applications.

Some results are novel even in classical monomial order contexts.

Abstract

In this paper we develop a theory of monomial preorders, which differ from the classical notion of monomial orders in that they allow ties between monomials. Since for monomial preorders, the leading ideal is less degenerate than for monomial orders, our results can be used to study problems where monomial orders fail to give a solution. Some of our results are new even in the classical case of monomial orders and in the special case in which the leading ideal defines the tangent cone.