# Delta sets of numerical monoids are eventually periodic

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## 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## Preview

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