Computation of Delta sets of numerical monoids

Let $$\{a_1,\ldots ,a_p\}$${a1,…,ap} be the minimal generating set of a numerical monoid S. For any $$s\in S$$s∈S, its Delta set is defined by $$\Delta (s)=\{l_{i}-l_{i-1}\mid i=2,\ldots ,k\}$$Δ(s)={li-li-1∣i=2,…,k} where $$\{l_1<\dots <l_k\}$${l1<⋯<lk} is the set $$\{\sum _{i=1}^px_i\mid s=\sum _{i=1}^px_ia_i \text { and } x_i\in \mathbb {N}\text { for all }i\}.$${∑i=1pxi∣s=∑i=1pxiaiandxi∈Nfor alli}. The Delta set of a numerical monoid S, denoted by $$\Delta (S)$$Δ(S), is the union of all the sets $$\Delta (s)$$Δ(s) with $$s\in S.$$s∈S. As proved in Chapman et al. (Aequationes Math. 77(3):273–279, 2009), there exists a bound N such that $$\Delta (S)$$Δ(S) is the union of the sets $$\Delta (s)$$Δ(s) with $$s\in S$$s∈S and $$s<N$$s<N. In this work, we obtain a sharpened bound and we present an algorithm for the computation of $$\Delta (S)$$Δ(S) that requires only the factorizations of $$a_1$$a1 elements.