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Arithmetic of commutative semigroups with a focus on semigroups of ideals and modules

Yushuang Fan

Mathematical College, China University of Geosciences, Haidian District, Beijing, P. R. China

E-mail Address: fys@cugb.edu.cn

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Alfred Geroldinger

Institute for Mathematics and Scientific Computing, University of Graz Heinrichstr. 36, 8010 Graz, Austria

E-mail Address: alfred.geroldinger@uni-graz.at

Corresponding author.

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Florian Kainrath

Institute for Mathematics and Scientific Computing, University of Graz Heinrichstr. 36, 8010 Graz, Austria

E-mail Address: florian.kainrath@uni-graz.at

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Salvatore Tringali

Institute for Mathematics and Scientific Computing, University of Graz Heinrichstr. 36, 8010 Graz, Austria

E-mail Address: salvatore.tringali@uni-graz.at

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https://doi.org/10.1142/S0219498817502346Cited by:18 (Source: Crossref)

Abstract

Let $H$ $H$ be a commutative semigroup with unit element such that every non-unit can be written as a finite product of irreducible elements (atoms). For every $k \in ℕ$ , let $𝒰_{k} (H)$ denote the set of all $ℓ \in ℕ$ with the property that there are atoms $u_{1}, \dots, u_{k}, v_{1}, \dots, v_{ℓ}$ such that $u_{1} \cdot \dots \cdot u_{k} = v_{1} \cdot \dots \cdot v_{ℓ}$ (thus, $𝒰_{k} (H)$ is the union of all sets of lengths containing $k$ ). The Structure Theorem for Unions states that, for all sufficiently large $k$ , the sets $𝒰_{k} (H)$ are almost arithmetical progressions with the same difference and global bound. We present a new approach to this result in the framework of arithmetic combinatorics, by deriving, for suitably defined families of subsets of the non-negative integers, a characterization of when the Structure Theorem holds. This abstract approach allows us to verify, for the first time, the Structure Theorem for a variety of possibly non-cancellative semigroups, including semigroups of (not necessarily invertible) ideals and semigroups of modules. Furthermore, we provide the very first example of a semigroup (actually, a locally tame Krull monoid) that does not satisfy the Structure Theorem.

Communicated by A. Facchini

Keywords:

AMSC: 11P70, 13A05, 13F05, 16D70, 20M12, 20M13

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