Long sets of lengths with maximal elasticity

Abstract

We introduce a new invariant describing the structure of sets of lengths in atomic monoids and domains. For an atomic monoid $H$, let ${\Delta }_{\rho }(H)$ be the set of all positive integers $d$ which occur as differences of arbitrarily long arithmetical progressions contained in sets of lengths having maximal elasticity $\rho (H)$. We study ${\Delta }_{\rho }(H)$ for transfer Krull monoids of finite type (including commutative Krull domains with finite class group) with methods from additive combinatorics, and also for a class of weakly Krull domains (including orders in algebraic number fields) for which we use ideal theoretic methods.

to appear in Canadian Journal of Mathematics

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Cited by in corpus (6)

• On strongly primary monoids, with a focus on Puiseux monoids
• Structural properties of subadditive families with applications to factorization theory
• Systems of sets of lengths: Transfer Krull monoids versus weakly Krull monoids
• On the arithmetic of Mori monoids and domains
• On a zero-sum problem arising from factorization theory
• On product-one sequences over dihedral groups