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Monatshefte für Mathematik

, Volume 178, Issue 3, pp 457–472 | Cite as

Computation of Delta sets of numerical monoids

  • J. I. García-García
  • M. A. Moreno-Frías
  • A. Vigneron-Tenorio
Article

Abstract

Let \(\{a_1,\ldots ,a_p\}\) be the minimal generating set of a numerical monoid S. For any \(s\in S\), its Delta set is defined by \(\Delta (s)=\{l_{i}-l_{i-1}\mid i=2,\ldots ,k\}\) where \(\{l_1<\dots <l_k\}\) is the set \(\{\sum _{i=1}^px_i\mid s=\sum _{i=1}^px_ia_i \text { and } x_i\in \mathbb {N}\text { for all }i\}.\) The Delta set of a numerical monoid S, denoted by \(\Delta (S)\), is the union of all the sets \(\Delta (s)\) with \(s\in S.\) As proved in Chapman et al. (Aequationes Math. 77(3):273–279, 2009), there exists a bound N such that \(\Delta (S)\) is the union of the sets \(\Delta (s)\) with \(s\in S\) and \(s<N\). In this work, we obtain a sharpened bound and we present an algorithm for the computation of \(\Delta (S)\) that requires only the factorizations of \(a_1\) elements.

Keywords

Delta set Non-unique factorization Numerical monoid Numerical semigroup 

Mathematics Subject Classification

Primary 20M14 Secondary 20M05 

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Copyright information

© Springer-Verlag Wien 2015

Authors and Affiliations

  1. 1.Departamento de MatemáticasUniversidad de CádizCadizSpain
  2. 2.Departamento de MatemáticasUniversidad de CádizCadizSpain

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