Journal of Commutative Algebra

The realization problem for delta sets of numerical semigroups

Stefan Colton and Nathan Kaplan

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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

Permanent link to this document
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

Subjects
Primary: 11B75: Other combinatorial number theory 20M13: Arithmetic theory of monoids 20M14: Commutative semigroups

Keywords
Numerical semigroup delta set factorization theory non-unique factorization

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


Export citation

References

  • D.F. Anderson, S. Chapman, N. Kaplan and D. Torkornoo, An algorithm to compute $\omega$-primality in a numerical monoid, Semigroup Forum 82 (2011), 96–108.
  • A. Assi and P.A. García-Sánchez, Constructing the set of complete intersection numerical semigroups with a given Frobenius number, Appl. Alg. Eng. Comm. Comp. 24 (2013), 133–148.
  • P. Baginski, S. Chapman, R. Rodriguez, G. Schaeffer and Y. She, On the delta set and catenary degree of Krull monoids with infinite cyclic divisor class group, J. Pure Appl. Alg. 214 (2010), 1334–1339.
  • P. Baginski, S. Chapman and G. Schaeffer, On the delta set of a singular arithmetical congruence monoid, J. Th. Nombr. Bordeaux 20 (2008), 45–59.
  • T. Barron, C. O'Neill and R. Pelayo, On dynamic algorithms for factorization invariants in numerical monoids, Math. Comp. 86 (2017), 2429–2447.
  • C. Bibby, S. Chapman, C. Leverson, A. Malyshev and D. Steinberg, Determining delta sets of numerical monoids, preprint.
  • C. Bowles, S. Chapman, N. Kaplan and D. Reiser, On delta sets of numerical monoids, J. Alg. Appl. 5 (2006), 1–24.
  • L. Bryant, J. Hamblin and L. Jones, Maximal denumerant of a numerical semigroup with embedding dimension less than four, J. Commutative Algebra 4 (2012), 489–503.
  • S. Chapman, J. Daigle, R. Hoyer and N. Kaplan, Delta sets of numerical monoids using non-minimal sets of generators, Comm. Alg. 38 (2010), 2622–2634.
  • S. Chapman, P.A. García-Sánchez and D. Llena, The catenary and tame degree of numerical monoids, Forum Math. 21 (2009), 117–129.
  • S. Chapman, P. A. García-Sánchez, D. Llena, A. Malyshev and D. Steinberg, On the delta set and the betti elements of a BF-monoid, Arab. J. Math. 1 (2012), 53–61.
  • S. Chapman, F. Gotti and R. Pelayo, On delta sets and their realizable subsets in Krull monoids with cyclic class groups, Colloq. Math. 137 (2014), 137–146.
  • S. Chapman, R. Hoyer and N. Kaplan, Delta sets of numerical monoids are eventually periodic, Aequat. Math. 77 (2009), 273–279.
  • M. D'Anna, V. Micale and A. Sammartino, Classes of complete intersection numerical semigroups, Semigroup Forum 88 (2014), 453–467.
  • M. Delgado, P.A. García-Sánchez and J. Morais, “numericalsgps": A gap package on numerical semigroups, http://www.gap-system.org/Packages/numericalsgps.html.
  • J.I. García-García, M.A. Moreno-Frías and A. Vigneron-Tenorio, A computation of the $\omega$-primality and asymptotic $\omega$-primality with applications to numerical semigroups, Israel J. Math. 206 (2015), 395–411.
  • ––––, Computation of delta sets of numerical monoids, Monatsh. Math. 178 (2015), 457–472.
  • P.A. García-Sánchez and M.J. Leamer, Huneke-Wiegand conjecture for complete intersection numerical semigroup rings, J. Algebra 391 (2013), 114–124.
  • P.A. García-Sánchez, D. Llena and A. Moscariello, Delta sets for numerical semigroups with embedding dimension three, Forum Math., to appear, https://doi.org/10.1515/forum-2015-0065.
  • P.A. García-Sánchez and I. Ojeda, Uniquely presented finitely generated commutative monoids, Pacific J. Math. 248 (2010), 91–105.
  • P.A. García-Sánchez and J. Rosales, Numerical semigroups, Develop. Math. 20, Springer, New York, 2009.
  • ––––, Finitely generated commutative monoids, Nova Science Publishers, Inc., Commack, NY, 1999.
  • A. Geroldinger, A structure theorem for sets of lengths, Colloq. Math. 78 (1998), 225–259.
  • A. Geroldinger and F. Halter-Koch, Non-unique factorizations: Algebraic, combinatorial and analytic theory, Pure Appl. Math. 278, Chapman & Hall/CRC, Boca Raton, 2006.
  • M. Omidali, The catenary and tame degree of numerical monoids generated by generalized arithmetic sequences, Forum Math. 24 (2012), 627–640.
  • A. Philipp, A characterization of arithmetical invariants by the monoid of relations, Semigroup Forum 81 (2010), 424–434.
  • ––––, A characterization of arithmetical invariants by the monoid of relations II: The monotone catenary degree and applications to semigroup rings, Semigroup Forum 90 (2015), 220–250.
  • W. Schmid, A realization theorem for sets of lengths, J. Num. Th. 129 (2009), 990–999.