Structural properties of subadditive families with applications to factorization theory

Abstract

Let $H$ be a multiplicatively written monoid. Given $k\in {\mathbf{N}}^{+}$, we denote by ${\mathcal{U}}_k$ the set of all $\ell \in {\mathbf{N}}^{+}$ such that $a_1\cdots a_k=b_1\cdots b_{\ell }$ for some atoms $a_1,\dots ,a_k,b_1,\dots ,b_{\ell }\in H$. The sets ${\mathcal{U}}_k$ 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 $k$, namely, $H$ satisfies the Structure Theorem for Unions. The present paper improves the current state of the art on this problem. More precisely, we show that, under mild assumptions on $H$, not only does the Structure Theorem for Unions hold, but there also exists $\mu \in {\mathbf{N}}^{+}$ such that, for every $M\in \mathbf{N}$, the sequences $$(({\mathcal{U}}_k-\inf {\mathcal{U}}_k)\cap [[0,M]]{)}_{k\ge 1}\text{and}((\sup {\mathcal{U}}_k-{\mathcal{U}}_k)\cap [[0,M]]{)}_{k\ge 1}$$ are $\mu$-periodic from some point on. The result applies, e.g., 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", by inquiring into the properties of what we call a subadditive family, i.e., a collection $\mathcal{L}$ of subsets of $\mathbf{N}$ such that, for all $L_1,L_2\in \mathcal{L}$, there is $L\in \mathcal{L}$ with $L_1+L_2\subseteq L$.

22 pp., no figures. Fixed a few typos and updated statements and definitions after realizing that what is proved in the paper is slighly stronger than what claimed in the previous version. To appear in Israel Journal of Mathematics

References in corpus (5)

• Power monoids: A bridge between Factorization Theory and Arithmetic Combinatorics
• Systems of sets of lengths: Transfer Krull monoids versus weakly Krull monoids
• Arithmetic of commutative semigroups with a focus on semigroups of ideals and modules
• Long sets of lengths with maximal elasticity
• Products of $k$ atoms in Krull monoids

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• Abstract Factorization Theorems with Applications to Idempotent Factorizations
• Lattices over Bass rings and graph agglomerations
• On the finiteness of certain factorization invariants
• Definite orders with locally free cancellation
• Factorization in monoids and rings
• Factorization in Finitely-Presented Monoids
• On product-one sequences over dihedral groups
• Factorization Theory in Commutative Monoids