## Journal of Commutative Algebra

### 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~3 if and only if $t=2$.

#### Article information

Source
J. Commut. Algebra, Volume 9, Number 3 (2017), 313-339.

Dates
First available in Project Euclid: 1 August 2017

https://projecteuclid.org/euclid.jca/1501574425

Digital Object Identifier
doi:10.1216/JCA-2017-9-3-313

Mathematical Reviews number (MathSciNet)
MR3685046

Zentralblatt MATH identifier
06790173

#### Citation

Colton, Stefan; Kaplan, Nathan. The realization problem for delta sets of numerical semigroups. J. Commut. Algebra 9 (2017), no. 3, 313--339. doi:10.1216/JCA-2017-9-3-313. https://projecteuclid.org/euclid.jca/1501574425

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