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Long sets of lengths with maximal elasticity

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