ON DELTA SETS OF NUMERICAL MONOIDS

Let S be a numerical monoid (i.e. an additive submonoid of ℕ0) with minimal generating set 〈n1,…,nt〉. For m ∈ S, if , then is called a factorization length of m. We denote by (where mi < mi+1 for each 1 ≤ i < k) the set of all possible factorization lengths of m. The Delta set of m is defined by Δ(m) = {mi+1 - mi|1 ≤ i < k} and the Delta set of S by Δ(S) = ∪m∈SΔ(m). In this paper, we address some basic questions concerning the structure of the set Δ(S). In Sec. 2, we find upper and lower bounds on Δ(S) by finding such bounds on the Delta set of any monoid S where the associated reduced monoid Sred is finitely generated. We prove in Sec. 3 that if S = 〈n, n + k, n + 2k,…,n + bk〉, then Δ(S) = {k}. In Sec. 4 we offer some specific constructions which yield for any k and v in ℕ a numerical monoid S with Δ(S) = {k, 2k,…,vk}. Moreover, we show that Delta sets of numerical monoids may contain natural "gaps" by arguing that Δ(〈n, n + 1, n2 - n - 1〉) = {1,2,…,n - 2, 2n - 5}.