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Aequationes mathematicae

, Volume 77, Issue 3, pp 273–279 | Cite as

Delta sets of numerical monoids are eventually periodic

  • Scott T. ChapmanEmail author
  • Rolf Hoyer
  • Nathan Kaplan
Article

Summary.

Let M be a numerical monoid (i.e., an additive submonoid of \({\mathbb{N}}_0\)) with minimal generating set \({\langle}n_1, . . . , n_t\rangle\). For \(m \in M\), if \(m = \sum_{i=1}^{t} x_{i}n_{i}\), then \(\sum_{i=1}^{t} x_{i}\) is called a factorization length of m. We denote by \({\mathfrak{L}}(m) = \{m_1, . . . ,m_k\}\) (where \(m_i < m_{i+1} {\rm for\, each}\, 1 \leq i < k\)) the set of all possible factorization lengths of m. The Delta set of m is defined by \(\Delta(m) =\{m_{i+1}-m_i|1 \leq i < k \}\) and the Delta set of M by \(\Delta(M) = \cup_{0 \neq m \in M}\Delta(m)\). In this paper, we expand on the study of Δ(M) begun in [2] and [3] by showing that the delta sets of a numerical monoid are eventually periodic. More specifically, we show for all \(x \geq 2kn_{2}n^{2}_{k}\) in M that \(\Delta(x) = \Delta(x + n_{1}n_k)\).

Mathematics Subject Classification (2000).

20M14 20D60 11B75 

Keywords.

Numerical monoid non-unique factorization delta set 

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Copyright information

© Birkhäuser Verlag, Basel 2009

Authors and Affiliations

  • Scott T. Chapman
    • 1
    Email author
  • Rolf Hoyer
    • 2
    • 3
  • Nathan Kaplan
    • 4
    • 5
  1. 1.Department of Mathematics and StatisticsSam Houston State UniversityHuntsvilleUSA
  2. 2.Department of MathematicsGrinnell CollegeGrinnell, IowaUSA
  3. 3.Department of MathematicsUniversity of ChicagoChicagoUSA
  4. 4.Department of MathematicsPrinceton UniversityPrincetonUSA
  5. 5.Department of MathematicsHarvard UniversityCambridgeUSA

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