Structural properties of subadditive families with applications to factorization theory

Abstract

Let H be a multiplicatively written monoid. Given k ∈ N⁺, we denote by ${{\mathscr U}_k}$ the set of all ℓ ∈ N⁺ such that a₁ ··· a_k = b₁ ··· b_ℓ for some atoms (or irreducible elements) a₁, … a_k, b₁, …, b_ℓ ∈ H. The sets ${{\mathscr U}_k}$ are one of the most fundamental invariants studied in the theory of non-unique factorization, and understanding their structure is a basic problem in the field: In particular, it is known that, in many cases of interest, these sets are almost arithmetic progressions with the same difference and bound for all large k, which is usually expressed by saying that H satisfies the Structure Theorem for Unions. The present paper improves the current state of the art on this problem.

More precisely, we will show that, under mild assumptions on H, not only does the Structure Theorem for Unions hold, but there also exists μ ∈ N⁺ such that, for every M ∈ N, the sequences

$${\left({\left({{{\mathscr U}_k} - \inf \;{{\mathscr U}_k}} \right) \cap \left[\kern-0.15em\left[{0,\;M} \right]\kern-0.15em\right]} \right)_{k \ge 1}}\;\;\;\;{\rm{and}}\;\;\;\;{\left({\left({\sup \;{{\mathscr U}_k} - {{\mathscr U}_k}} \right) \cap \left[\kern-0.15em\left[{0,\;M} \right]\kern-0.15em\right]} \right)_{k \ge 1}}$$

are μ-periodic from some point on. The result applies, for instance, to (the multiplicative monoid of) all commutative Krull domains (e.g., Dedekind domains) with finite class group; a variety of weakly Krull commutative domains (including all orders in number fields with finite elasticity); some maximal orders in central simple algebras over global fields; and all numerical monoids.

Large parts of the proofs are worked out in a “purely additive model” (where no explicit reference to monoids or atoms is ever made), by inquiring into the properties of what we call a subadditive family, i.e., a collection ℒ of subsets of N such that, for all L₁, L₂ ∈ ℒ, there is L ∈ ℒ with L₁ + L₂ ⊆ L.