Arithmetic properties encoded in undermonoids
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arXiv
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| Format: | Preprint |
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2024
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| author | Gotti, Felix Li, Bangzheng |
| author_facet | Gotti, Felix Li, Bangzheng |
| contents | 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 $\mathfrak{p}$, we are interested in providing an answer to the following main question: does it suffice to check that all undermonoids of $M$ satisfy $\mathfrak{p}$ to conclude that all submonoids of $M$ satisfy $\mathfrak{p}$? 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 (or length-factorial). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_11199 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Arithmetic properties encoded in undermonoids Gotti, Felix Li, Bangzheng Commutative Algebra Primary: 13F15, 13A05, Secondary: 20M13, 13F05 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 $\mathfrak{p}$, we are interested in providing an answer to the following main question: does it suffice to check that all undermonoids of $M$ satisfy $\mathfrak{p}$ to conclude that all submonoids of $M$ satisfy $\mathfrak{p}$? 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 (or length-factorial). |
| title | Arithmetic properties encoded in undermonoids |
| topic | Commutative Algebra Primary: 13F15, 13A05, Secondary: 20M13, 13F05 |
| url | https://arxiv.org/abs/2412.11199 |