Sets of lengths in atomic unit-cancellative finitely presented monoids

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.