2023/11/28 by Daniel Windisch, Windisch, Daniel
Computer Science · Mathematics · #03C20 #03C60 #13A15 #Advanced Algebra and Logic #Commutative Algebra (math.AC) #FOS: Mathematics #Logic (math.LO) #Rings, Modules, and Algebras #primary: 13F15 #secondary: 13L05 #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2311.16662
openalex publication_date 2023/11/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.