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 ∈ N, let Uk(H) denote the set of all ` ∈ N with the property that there are atoms u1, . . . , uk, v1, . . . , v` such that u1 · . . . · uk = v1 · . . . · v` (thus, Uk(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 Uk(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.