On the finiteness of certain factorization invariants

Abstract

$\def\F{\mathscr{F}}\def\z{\mathfrak{z}}\def\zprime{\mathfrak{z}^\prime}$ Let $H$ be a monoid and $\pi_H$ be the unique extension of the identity map on $H $ to a monoid homomorphism $\F(H) \to H$, where we denote by $\F(X)$ the free monoid on a set $X$. Given $A \subseteq H$, an $A$-word $\z$ (i.e., an element of F(A)) is minimal if $\pi_H (\z) \neq \pi_H (\zprime)$ for every permutation $\zprime$ of a proper subword of $\z$. The minimal $A$-elasticity of $H$ is then the supremum of all rational numbers $m/n$ with $m, n \in \mathbb{N}^+$ such that there exist minimal $A$-words $\mathfrak{a}$ and $\mathfrak{b}$ of length $m$ and $n$, resp., with $\pi_H (\mathfrak{a}) = \pi_H (\mathfrak{b})$. Among other things, we show that if $H$ is commutative and $A$ is finite, then the minimal $A$-elasticity of $H$ is finite. This provides a non-trivial generalization of the finiteness part of a classical theorem of Anderson et al. from the case where $H$ is cancellative, commutative, and finitely generated modulo units, and $A$ is the set $\mathscr{A} (H)$ of atoms of $H$. We also demonstrate that commutativity is somewhat essential here, by proving the existence of an atomic, cancellative, finitely generated monoid with trivial group of units whose minimal $\mathscr{A} (H)$-elasticity is infinite.