On the finiteness of certain factorization invariants

Let $H$ be a monoid, $\mathscr F(X)$ be the free monoid on a set $X$, and $\pi_H$ be the unique extension of the identity map on $H$ to a monoid homomorphism $\mathscr F(H) \to H$. Given $A \subseteq H$, an $A$-word $\mathfrak z$ (i.e., an element of $\mathscr F(A)$) is minimal if $\pi_H(\mathfrak z) \ne \pi_H(\mathfrak z')$ for every permutation $\mathfrak z'$ of a proper subword of $\mathfrak z$. 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 yields 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 (f.g.) modulo units and $A$ is the set $\mathscr A(H)$ of its atoms. We also check that commutativity is somewhat essential here, by proving the existence of an atomic, cancellative, f.g. monoid with trivial group of units whose minimal $\mathscr A(H)$-elasticity is infinite.