A topological approach to leading monomial ideals

TL;DR

This paper introduces a topological framework on monomial orderings to establish finiteness and existence results for leading monomial ideals and Groebner bases in algebraic structures.

Contribution

It develops a natural topology on monomial orderings and uses it to prove finiteness of leading ideals and the existence of universal Groebner bases, independent of combinatorial methods.

Findings

Finiteness of minimal leading monomial ideals

Finiteness of leading ideals under degree orderings

Existence of universal Groebner bases

Abstract

We introduce a very natural topology on the set of total orderings of monomials of any algebra having a countable basis over a field. This topological space and some notable subspaces are compact. This topological framework allows us to deduce some finiteness results about leading monomial ideals of any fixed ideal, namely: (1) the number of minimal leading monomial ideals with respect to total orderings is finite; (2) the number of leading monomial ideals with respect to degree orderings is finite; (3) the number of leading monomial ideals with respect to admissible orderings is finite under some multiplicativity assumptions on the considered algebra. Finally we are able to infer the existence of universal Groebner bases from the topological properties of degree and admissible orderings in a class of algebras that includes at least the algebras of solvable type. These existence results turn out to be independent from the finiteness results mentioned above, in contrast to the typical situation that occurs with "classical" more combinatorial proofs.