Longest and Shortest Factorizations in Embedding Dimension Three

TL;DR

This paper characterizes when the identities for longest and shortest factorization lengths hold universally in numerical monoids with embedding dimension three, advancing understanding of their factorization properties.

Contribution

It provides a complete characterization of when these factorization length identities are valid for all elements in three-dimensional numerical monoids.

Findings

Identifies conditions for universal validity of factorization length identities

Characterizes properties of numerical monoids of embedding dimension three

Enhances understanding of factorization structure in numerical monoids

Abstract

For a numerical monoid $\langle n_1, \dots, n_k \rangle$ minimally generated by $n_1, \dots, n_k \in \mathbb{N}$ with $n_1 < \cdots < n_k$, the longest and shortest factorization lengths of an element $x$, denoted as $L(x)$ and $\ell(x)$, respectively, follow the identities $L(x+n_1) = L(x) + 1$ and $\ell(x+n_k) = \ell(x) + 1$ for sufficiently large elements $x$. We characterize when these identities hold for all elements of numerical monoids of embedding dimension three.