Aequationes mathematicae

, Volume 77, Issue 3, pp 273–279

Delta sets of numerical monoids are eventually periodic

• Scott T. Chapman
• 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

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