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Long Sets of Lengths With Maximal Elasticity

Published online by Cambridge University Press:  20 November 2018

Alfred Geroldinger  and
Qinghai Zhong
Alfred Geroldinger
Affiliation:
Institute for Mathematics and Scientiûc Computing, University of Graz, NAWI Graz, Heinrichstraße 36, 8010 Graz, Austria, e-mail: alfred.geroldinger@uni-graz.at , qinghai.zhong@uni-graz.at
Qinghai Zhong
Affiliation:
Institute for Mathematics and Scientiûc Computing, University of Graz, NAWI Graz, Heinrichstraße 36, 8010 Graz, Austria, e-mail: alfred.geroldinger@uni-graz.at , qinghai.zhong@uni-graz.at
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Abstract

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We introduce a newinvariant describing the structure of sets of lengths in atomicmonoids and domains. For an atomic monoid $H$, let ${{\Delta }_{\rho }}\left( H \right)$ be the set of all positive integers d that occur as differences of arbitrarily long arithmetical progressions contained in sets of lengths havingmaximal elasticity $\rho \left( H \right)$. We study ${{\Delta }_{\rho }}\left( H \right)$ 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

13A0513F0516H1016U3020M13transfer Krull monoidweakly Krull monoidset of lengthelasticity
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 70 , Issue 6 , 01 December 2018 , pp. 1284 - 1318
Copyright
Copyright © Canadian Mathematical Society 2018

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