Shifts of generators and delta sets of numerical monoids

Abstract

Let S be a numerical monoid with minimal generating set 〈n1, …, nt〉. For m ∈ S, if [Formula: see text], then [Formula: see text] is called a factorization length of m. We denote by ℒ(m) = {m1, …, mk} (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 expand on the study of Δ(S) begun in [C. Bowles, S. T. Chapman, N. Kaplan and D. Reiser, On delta sets of numerical monoids, J. Algebra Appl. 5 (2006) 1–24] in the following manner. Let r1, r2, …, rt be an increasing sequence of positive integers and Mn = 〈n, n + r1, n + r2, …, n + rt〉 a numerical monoid where n is some positive integer. We prove that there exists a positive integer N such that if n > N then |Δ(Mn)| = 1. If t = 2 and r1 and r2 are relatively prime, then we determine a value for N which is sharp.