Factorizations of the same length in abelian monoids

Abstract

Let ${{\mathcal {S}}}\subseteq {{\mathbb {Z}}}^m \oplus T$ be a finitely generated and reduced monoid. In this paper we develop a general strategy to study the set of elements in ${\mathcal {S}}$ having at least two factorizations of the same length, namely the ideal ${\mathcal {L}}_{{\mathcal {S}}}$. To this end, we work with a certain (lattice) ideal associated to the monoid ${\mathcal {S}}$. Our study can be seen as a new approach generalizing [9], which only studies the case of numerical semigroups. When ${{\mathcal {S}}}$ is a numerical semigroup we give three main results: (1) we compute explicitly a set of generators of the ideal ${\mathcal {L}}_{\mathcal S}$ when ${\mathcal {S}}$ is minimally generated by an almost arithmetic sequence; (2) we provide an infinite family of numerical semigroups such that ${\mathcal {L}}_{{\mathcal {S}}}$ is a principal ideal; (3) we classify the computational problem of determining the largest integer not in ${\mathcal {L}}_{{\mathcal {S}}}$ as an $\mathcal {NP}$-hard problem.