Structural properties of subadditive families with applications to factorization theory

Abstract

Let H be a multiplicatively written monoid. Given k ∈ N + , we denote by U k the set of all ℓ ∈ N + such that a 1 · · · a k = b 1 · · · b ℓ for some atoms (or irreducible elements) a 1 , . . . , a k , b 1 , . . . , b ℓ ∈ H. The sets 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 sequencesare 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 L 1 , L 2 ∈ L , there is L ∈ L with L 1 + L 2 ⊆ L.

Factorization invariants of Puiseux monoids generated by geometric sequences

“…Wishing to have a full picture of the sets of elasticities of cyclic rational semirings, we propose the following conjecture. [4,7,15,43]. In particular, the unions of sets of lengths and the local elasticities of Puiseux monoids have been considered in [34].…”

On the Arithmetic of Mori Monoids and Domains

“…We proceed in two steps. First we show that ρ k (H) = ∞ for all sufficiently large k ∈ N. In a second step we show that the sets of distances ∆(U k (H)) of all U k (H) are finite for all sufficiently large k ∈ N. These two results imply, by [23,Theorem 2.20], that H satisfies the Structure Theorem for Unions (Note that the fact ρ k (H) = ∞ for all sufficiently large k ∈ N implies that for any fixed ℓ ∈ N, the interval [ρ k−ℓ , ρ k ] is empty for all sufficient large k ∈ N).…”

Lattices over Bass Rings and Graph Agglomerations

“…The Strong Structure Theorem for Unions also gives information about the initial and final segments of the U k (A(G)). It was first proved by Tringali in [48] and later again using a more general framework in [47]. (4) This result is due to Geroldinger; a proof can be found in [28,Theorem 4.4.11].…”