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 ∆xiai and xi ∈ N for all i}. The Delta set of a numerical monoid S, denoted by ∆(S), is the union of all the sets ∆(s) with s ∈ S. As proved in [5], there exists a bound N such that ∆(S) is the union of the sets ∆(s) with s ∈ S and s < N . In this work, we obtain a sharpened bound and we present an algorithm for the computation of ∆(S) that requires only the factorizations of a1 elements.

Which sets are sets of lengths in all numerical monoids?

“…The maximum of the set of distances is unknown (in terms of the atoms) and this question seems to have the same complexity as questions about the Frobenius number. For partial results and computational approaches we refer to [9,12,13,14].…”

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

“…The importance of Theorem 5 cannot be overstated, as it turns the problem of computing ∆ (S) into a finite time exercise. The bound N given in Theorem 5 has been drastically improved in [13] ( Table 1 in that paper shows exactly how drastic this improvement is). An alternate view of the computation of ∆ (S) using the Betti numbers of S can be found in [5], which is also an REU product.…”

“…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].…”