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

Abstract

Let [Formula: see text] 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 [Formula: see text], let [Formula: see text] denote the set of all [Formula: see text] with the property that there are atoms [Formula: see text] such that [Formula: see text] (thus, [Formula: see text] is the union of all sets of lengths containing [Formula: see text]). The Structure Theorem for Unions states that, for all sufficiently large [Formula: see text], the sets [Formula: see text] 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.