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| Main Authors: | , , , , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2508.11319 |
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| _version_ | 1866917531872133120 |
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| author | Dani, Jiya Deng, Anna Gotti, Marly Li, Bryan Paladiya, Arav Vulakh, Joseph Zeng, Jason |
| author_facet | Dani, Jiya Deng, Anna Gotti, Marly Li, Bryan Paladiya, Arav Vulakh, Joseph Zeng, Jason |
| contents | Let $M$ be a cancellative and commutative monoid. A non-invertible element of $M$ is called an atom (or irreducible element) if it cannot be factored into two non-invertible elements, while an atom $a$ of $M$ is called strong if $a^n$ has a unique factorization in $M$ for every $n \in \mathbb{N}$. The monoid $M$ is atomic if every non-invertible element factors into finitely many atoms (repetitions allowed). For an algebraic number $α$, we let $M_α$ denote the additive monoid of the subsemiring $\mathbb{N}_0[α]$ of $\mathbb{C}$. The atomic structure of $M_α$ reflects intricate interactions between algebraic number theory and additive semigroup theory. For $m, n \in \mathbb{N}_0 \cup \{ \infty \}$ (with $m \le n$), the pair $(m,n)$ is called realizable if there exists an algebraic number $α\in \mathbb{C}$ such that $M_α$ has $m$ strong atoms and $n$ atoms. Our primary goal is to identify classes of realizable pairs with the long-term goal of providing a complete description of the full set of realizable pairs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_11319 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the set of atoms and strong atoms in additive monoids of cyclic semidomains Dani, Jiya Deng, Anna Gotti, Marly Li, Bryan Paladiya, Arav Vulakh, Joseph Zeng, Jason Commutative Algebra Let $M$ be a cancellative and commutative monoid. A non-invertible element of $M$ is called an atom (or irreducible element) if it cannot be factored into two non-invertible elements, while an atom $a$ of $M$ is called strong if $a^n$ has a unique factorization in $M$ for every $n \in \mathbb{N}$. The monoid $M$ is atomic if every non-invertible element factors into finitely many atoms (repetitions allowed). For an algebraic number $α$, we let $M_α$ denote the additive monoid of the subsemiring $\mathbb{N}_0[α]$ of $\mathbb{C}$. The atomic structure of $M_α$ reflects intricate interactions between algebraic number theory and additive semigroup theory. For $m, n \in \mathbb{N}_0 \cup \{ \infty \}$ (with $m \le n$), the pair $(m,n)$ is called realizable if there exists an algebraic number $α\in \mathbb{C}$ such that $M_α$ has $m$ strong atoms and $n$ atoms. Our primary goal is to identify classes of realizable pairs with the long-term goal of providing a complete description of the full set of realizable pairs. |
| title | On the set of atoms and strong atoms in additive monoids of cyclic semidomains |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2508.11319 |