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Sets of lengths in atomic unit-cancellative finitely presented monoids

Volume 151 / 2018

Alfred Geroldinger, Emil Daniel Schwab Colloquium Mathematicum 151 (2018), 171-187 MSC: Primary 20M13, 20M05; Secondary 13A05. DOI: 10.4064/cm7242-6-2017 Published online: 20 November 2017

Abstract

For an element $a$ of a monoid $H$, its set of lengths $\mathsf L (a) \subset \mathbb N$ is the set of all positive integers $k$ for which there is a factorization $a=u_1 \cdot \ldots \cdot u_k$ into $k$ atoms. We study the system $\mathcal L (H) = \{\mathsf L (a) \mid a \in H \}$ with a focus on the unions $\mathcal U_k (H) \subset \mathbb N$ of all sets of lengths containing a given $k \in \mathbb N$. The Structure Theorem for Unions—stating that for all sufficiently large $k$, the sets $\mathcal U_k (H)$ are almost arithmetical progressions with the same difference and global bound—has attracted much attention for commutative monoids and domains. We show that it holds true for the not necessarily commutative monoids in the title satisfying suitable algebraic finiteness conditions. Furthermore, we give an explicit description of the system of sets of lengths of the monoids $B_{n} = \langle a,b \mid ba=b^{n} \rangle $ for $n \in \mathbb N_{\ge 2}$. Based on this description, we show that the monoids $B_n$ are not transfer Krull.

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