# Sets of lengths in atomic unit-cancellative finitely presented monoids

## TL;DR

This paper extends the Structure Theorem for Unions to certain non-commutative monoids, providing explicit descriptions of their sets of lengths and demonstrating their differences from commutative cases.

## Contribution

It proves the Structure Theorem for Unions for non-commutative monoids under algebraic finiteness conditions and describes the sets of lengths for specific monoids $B_n$, showing they are not transfer Krull.

## Key findings

- The Structure Theorem for Unions holds for certain non-commutative monoids.
- Explicit descriptions of sets of lengths for monoids $B_n$ are provided.
- Monoids $B_n$ are shown to be not transfer Krull, distinguishing their length systems.

## 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 $\mathcal U_k (H) \subset \mathbb N$ which are the unions 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 found 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 monoids $B_{n} = \langle a,b \mid ba=b^{n} \rangle$ for $n \in \N_{\ge 2}$. Based on this description, we show that the monoids $B_n$ are not transfer Krull, which implies that their systems $\mathcal L (B_n)$ are distinct from systems of sets of lengths of commutative Krull monoids and others.