English

The space of preorders on a commutative monoid

Commutative Algebra 2026-07-08 v1 Logic

Abstract

For a finitely generated commutative monoid Π\Pi, we present a constructive description of all (total) preorders on Π\Pi 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 RN\mathbb{R}^N of the form AHA \setminus H where AA is semialgebraic and HH 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 Π\Pi 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}
}