Computation of Delta sets of numerical monoids

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.