On the finiteness of certain factorization invariants

$\def\F{\mathscr{F}}\def\z{\mathfrak{z}}\def\zprime{\mathfrak{z}^\prime}$ 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.

Keywords

atoms, elasticity, [minimal] factorizations, irreducibles, monoids, preorders, sets of lengths

2010 Mathematics Subject Classification

Primary 08A50, 13A05, 20M05, 20M13. Secondary 20M14.

Full Text (PDF format)

L. C. was supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 101021791, and by the FWF (Austrian Science Fund) project No. P33499-N. The Marie Skłodowska-Curie grant also financed S. T.’s visit at University of Graz in summer-fall 2022, during which the paper was written. The authors are both members of the Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni (GNSAGA) of the Italian Mathematics Research Institute (INdAM).

Received 28 February 2023

Received revised 5 October 2023

Accepted 16 October 2023

Published 1 June 2024