On the arithmetic of ultraproducts of commutative cancellative monoids
Commutative Algebra
2023-11-29 v1 Logic
Abstract
We develop first steps in the study of factorizations of elements in ultraproducts of commutative cancellative monoids into irreducible elements. A complete characterization of the (multi-)sets of lengths in such objects is given. As applications, we show that several important properties from factorization theory cannot be expressed as first-order statements in the language of monoids, and we construct integral domains that realize every multiset of integers larger as a multiset of lengths. Finally, we give a new proof (based on our ultraproduct techniques) of a theorem by Geroldinger, Schmid and Zhong from additive combinatorics and we propose a general method for applying ultraproducts in the setting of non-unique factorizations.
Keywords
Cite
@article{arxiv.2311.16662,
title = {On the arithmetic of ultraproducts of commutative cancellative monoids},
author = {Daniel Windisch},
journal= {arXiv preprint arXiv:2311.16662},
year = {2023}
}