The space of preorders on a commutative monoid
Abstract
For a finitely generated commutative monoid , we present a constructive description of all (total) preorders on that are compatible with the monoid structure. Equipped with a natural topology, these preorders form an irreducible spectral space, which we show can be covered by a countable union of admissible sets: subsets of of the form where is semialgebraic and is a countable union of hyperplanes, both defined over the rational numbers. As a consequence of this description, we show that the universal theory of commutative monoids with a total order is decidable. Our proofs use a divide-and-conquer technique that requires establishing all of our results in the greater generality of sets on which acts with finitely many orbits. As a by-product, we find a new description of all monomoial orders on free modules over a polynomial ring.
Keywords
Cite
@article{arxiv.2607.07473,
title = {The space of preorders on a commutative monoid},
author = {Jan Draisma and George Metcalfe and Simon Santschi},
journal= {arXiv preprint arXiv:2607.07473},
year = {2026}
}