It was proved in 2018 by Geroldinger and Schmid [3] that for every nonempty subset Σ of  there exist a numerical monoid T and an element x of T such that a natural number n is the number of atoms in a decomposition of x into a sum of atoms if and only if n belongs to Σ. We give a completely different proof for  The different approach that we offer can shed some additional light on this and related problems, some of which are open.

numerical monoid, atoms, irreducible elements, factorization process, addendization.