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November 2008 Unions of sets of lengths
Michael Freeze, Alfred Geroldinger
Funct. Approx. Comment. Math. 39(1): 149-162 (November 2008). DOI: 10.7169/facm/1229696561

Abstract

Let $H$ be a Krull monoid such that every class contains a prime (this includes the multiplicative monoids of rings of integers of algebraic number fields). For $k \in \mathbb{N}$ let $\mathcal{V}_k (H)$ denote the set of all $m \in \mathbb{N}$ \ with the following property{\rm \,:} There exist atoms (irreducible elements) \ $u_1, \ldots, u_k, v_1, \ldots, v_m \in H$ with $u_1 \cdot \ldots \cdot u_k = v_1 \cdot \ldots \cdot v_m$. We show that the sets $\mathcal{V}_k (H)$ are intervals for all $k \in \mathbb{N}$. This solves Problem 37 in [4].

Citation

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Michael Freeze. Alfred Geroldinger. "Unions of sets of lengths." Funct. Approx. Comment. Math. 39 (1) 149 - 162, November 2008. https://doi.org/10.7169/facm/1229696561

Information

Published: November 2008
First available in Project Euclid: 19 December 2008

zbMATH: 1228.20046
MathSciNet: MR2490095
Digital Object Identifier: 10.7169/facm/1229696561

Subjects:
Primary: 11R27
Secondary: 13A05, 13F05, 20M14

Rights: Copyright © 2008 Adam Mickiewicz University

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Vol.39 • No. 1 • November 2008
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