The Realization Problem for Delta Sets of Numerical Semigroups

Abstract

The delta set of a numerical semigroup $S$, denoted $\Delta (S)$, is a factorization invariant that measures the complexity of the sets of lengths of elements in $S$. We study the following problem: Which finite sets occur as the delta set of a numerical semigroup $S$? It is known that $\min \Delta (S)=\gcd \Delta (S)$ is a necessary condition. For any two-element set $\{d,td\}$ we produce a semigroup $S$ with this delta set. We then show that for $t\ge 2$, the set $\{d,td\}$ occurs as the delta set of some numerical semigroup of embedding dimension three if and only if $t=2$.

13 pages

References in corpus (2)

• On dynamic algorithms for factorization invariants in numerical monoids
• Delta sets for numerical semigroups with embedding dimension three

Cited by in corpus (8)

• On strongly primary monoids, with a focus on Puiseux monoids
• Systems of sets of lengths: Transfer Krull monoids versus weakly Krull monoids
• Sets of Arithmetical Invariants in Transfer Krull Monoids
• Factorization invariants in numerical monoids
• Realizable sets of catenary degrees of numerical monoids
• Delta sets for numerical semigroups with embedding dimension three
• Minimal relations and catenary degrees in Krull monoids
• The set of distances in seminormal weakly Krull monoids