Arithmetic properties encoded in undermonoids
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233_2025_Article_10578.pdf
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Author(s)
Gotti, Felix
Li, Bangzheng
Date Issued
September 19, 2025
Journal
Semigroup Forum
Publisher
Springer US
Citation
Gotti, F., Li, B. Arithmetic properties encoded in undermonoids. Semigroup Forum (2025).
Version
Final published version
Abstract
Let M be a cancellative and commutative monoid. A submonoid N of M is called an undermonoid if the Grothendieck groups of M and N coincide. For a given property p , we are interested in providing an answer to the following main question: does it suffice to check that all undermonoids of M satisfy p to conclude that all submonoids of M satisfy p ? In this paper, we give a positive answer to this question for the property of being atomic, and then we prove that if M is hereditarily atomic (i.e., every submonoid of M is atomic), then M must satisfy the ACCP, proving a recent conjecture posed by Vulakh and the first author. We also give positive answers to our main question for the following well-studied factorization properties: the bounded factorization property, half-factoriality, and length-factoriality. Finally, we determine all the monoids whose submonoids/undermonoids are half-factorial/length-factorial.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1007/s00233-025-10578-3
