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# Long Sets of Lengths with Maximal Elasticity

Published:2018-01-18
Printed: Dec 2018
• Alfred Geroldinger,
Institute for Mathematics and Scientific Computing, University of Graz, Graz, Austria
• Qinghai Zhong,
Institute for Mathematics and Scientific Computing, University of Graz, Graz, Austria
 Format: LaTeX MathJax PDF

## 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.
 Keywords: transfer Krull monoid, weakly Krull monoid, set of length, elasticity
 MSC Classifications: 13A05 - Divisibility; factorizations [See also 13F15] 13F05 - Dedekind, Prufer, Krull and Mori rings and their generalizations 16H10 - Orders in separable algebras 16U30 - Divisibility, noncommutative UFDs 20M13 - Arithmetic theory of monoids

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