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arXiv · 2311.16662

On the arithmetic of ultraproducts of commutative cancellative monoids

Daniel Windisch
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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 $1$ 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.

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Daniel Windisch. 2023-11-28. On the arithmetic of ultraproducts of commutative cancellative monoids. https://arxiv.org/abs/2311.16662

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