Arithmetic of commutative semigroups with a focus on semigroups of ideals and modules

Let $H$ be a commutative semigroup with unit element such that every non-unit can be written as a finite product of irreducible elements (atoms). For every $k\in \mathbb{N}$, let ${\mathcal{U}}_k(H)$ denote the set of all $\ell \in \mathbb N$withthepropertythatthereareatoms$u_1, \ldots, u_k, v_1, \ldots, v_{\ell}$suchthat$u_1 \cdot \ldots \cdot u_k = v_1 \cdot \ldots \cdot v_{\ell}$(thus,$\mathscr U_k (H)$ is the union of all sets of lengths containing $k$). The Structure Theorem for Unions states that, for all sufficiently large $k$, the sets ${\mathcal{U}}_k(H)$ are almost arithmetical progressions with the same difference and global bound. We present a new approach to this result in the framework of arithmetic combinatorics, by deriving, for suitably defined families of subsets of the non-negative integers, a characterization of when the Structure Theorem holds. This abstract approach allows us to verify, for the first time, the Structure Theorem for a variety of possibly non-cancellative semigroups, including semigroups of (not necessarily invertible) ideals and semigroups of modules. Furthermore, we provide the very first example of a semigroup (actually, a locally tame Krull monoid) that does not satisfy the Structure Theorem.