Computation of Delta sets of numerical monoids

Abstract

Let {a1, . . . , ap} be the minimal generating set of a numerical monoid S. For any s ∈ S, its Delta set is defined by Δ(S) = {li − li−1 | i = 2, . . . , k} where {l1 < · · · < lk } is the set {∑^p i=1 xi | s = {∑^p i=1 xi ai and xi ∈ N for all i }. The Delta set of a numerical monoid S, denoted by Δ(S), is the union of all the sets Δ(s) with s ∈ S. As proved in Chapman et al. (Aequationes Math. 77(3):273–279, 2009), there exists a bound N such that Δ(S) is the union of the sets Δ(s) with s ∈ S and s < N. In this work, we obtain a sharpened bound and we present an algorithm for the computation of Δ(S) that requires only the factorizations of a1 elements.