LAUSR

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.