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.