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