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.

Minimal presentations of shifted numerical monoids

“…A plot depicting the Betti elements of M n for S = 6, 9, 20 and n ≤ 250. Each point (n, b) indicates b ∈ Betti(M n ).As a consequence of Corollary 3.5, we obtain Corollary 5.7, which offers an improved bound over[6, Theorem 2.2].…”

Beyond Coins, Stamps, and Chicken McNuggets: An Invitation to Numerical Semigroups

“…The publication of [3] led to a long series of papers devoted to the study of delta sets and related properties in numerical semigroups, which approach delta sets from both theoretical and computational standpoints. In our bibliography, we offer a subset of this list of papers that include undergraduate co-authors ( [1,4,5,7,8,9,16]).…”

“…Given how central the functions that compute Z(n) and L(n) are, these functions have undergone numerous improvements since the early days of the numericalsgps package, and now run surprisingly fast even for reasonably large input. McN); [ 3,7,8,9,10 ] gap> LengthsOfFactorizationsElementWRTNumericalSemigroup(150, McN); [ 10,11,13,14,15,16,17,18,19,20,21,22,23,24,25 ] The numericalsgps package can also compute delta sets, both of numerical semigroups and of their elements. The original implementation of the latter function used Theorem 5 to compute the delta set of every element up to N, and only more recently was a more direct algorithm developed [14].…”

Factoring in the Chicken McNugget Monoid

“…Intuitively, the delta set records the "gaps" in (or "missing") factorization lengths. There is a wealth of recent work concerning the computation of the delta set of a numerical monoid [5,7,10,13,15,16,18,22]. For numerical monoids with three generators, the computation of the delta set is tightly related to Euclid's extended greatest common divisor algorithm [23,24].…”