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Forum Mathematicum

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Volume 30, Issue 5


A realization theorem for sets of lengths in numerical monoids

Alfred Geroldinger
  • Corresponding author
  • Institute for Mathematics and Scientific Computing, University of Graz, NAWI Graz, Heinrichstraße 36, 8010 Graz, Austria
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/ Wolfgang Alexander Schmid
  • Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8, CNRS, UMR 7539, F-93526, Saint-Denis, France
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Published Online: 2018-02-07 | DOI: https://doi.org/10.1515/forum-2017-0180


We show that for every finite nonempty subset of 2 there are a numerical monoid H and a squarefree element aH whose set of lengths 𝖫(a) is equal to L.

Keywords: Numerical monoids; numerical semigroup algebras; sets of lengths; sets of distances

MSC 2010: 20M13; 20M14


  • [1]

    A. Assi and P. A. García-Sánchez, Numerical Semigroups and Applications, RSME Springer Ser. 1, Springer, Cham, 2016. Google Scholar

  • [2]

    T. Barron, C. O’Neill and R. Pelayo, On the set of elasticities in numerical monoids, Semigroup Forum 94 (2017), no. 1, 37–50. Web of ScienceCrossrefGoogle Scholar

  • [3]

    V. Barucci, Numerical semigroup algebras, Multiplicative Ideal Theory in Commutative Algebra, Springer, New York, (2006), 39–53. Google Scholar

  • [4]

    V. Barucci, D. E. Dobbs and M. Fontana, Maximality properties in numerical semigroups and applications to one-dimensional analytically irreducible local domains, Mem. Amer. Math. Soc. 125 (1997), no. 598, 1–78. Google Scholar

  • [5]

    C. Bowles, S. T. Chapman, N. Kaplan and D. Reiser, On delta sets of numerical monoids, J. Algebra Appl. 5 (2006), no. 5, 695–718. CrossrefGoogle Scholar

  • [6]

    W. Bruns and J. Gubeladze, Polytopes, Rings, and K-Theory, Springer Monogr. Math., Springer, Dordrecht, 2009. Google Scholar

  • [7]

    S. T. Chapman, P. A. García-Sánchez and D. Llena, The catenary and tame degree of numerical monoids, Forum Math. 21 (2009), no. 1, 117–129. Web of ScienceGoogle Scholar

  • [8]

    S. T. Chapman, R. Hoyer and N. Kaplan, Delta sets of numerical monoids are eventually periodic, Aequationes Math. 77 (2009), no. 3, 273–279. CrossrefWeb of ScienceGoogle Scholar

  • [9]

    S. Colton and N. Kaplan, The realization problem for delta sets of numerical semigroups, J. Commut. Algebra 9 (2017), no. 3, 313–339. CrossrefWeb of ScienceGoogle Scholar

  • [10]

    M. Delgado, P. A. García-Sánchez and J. Morais, “Numericalsgps”: A gap package on numerical semigroups, http://www.gap-system.org/Packages/numericalsgps.html.

  • [11]

    S. Frisch, A construction of integer-valued polynomials with prescribed sets of lengths of factorizations, Monatsh. Math. 171 (2013), no. 3–4, 341–350. CrossrefWeb of ScienceGoogle Scholar

  • [12]

    S. Frisch, S. Nakato and R. Rissner, Integer-valued polynomials on rings of algebraic integers of number fields with prescribed sets of lengths of factorizations, preprint (2017), https://arxiv.org/abs/1710.06783.

  • [13]

    J. I. García-García, M. A. Moreno-Frías and A. Vigneron-Tenorio, Computation of delta sets of numerical monoids, Monatsh. Math. 178 (2015), no. 3, 457–472. CrossrefWeb of ScienceGoogle Scholar

  • [14]

    P. A. García-Sánchez, An overview of the computational aspects of nonunique factorization invariants, Multiplicative Ideal Theory and Factorization Theory, Springer Proc. Math. Stat. 170, Springer, Cham (2016), 159–181. Google Scholar

  • [15]

    A. Geroldinger and F. Halter-Koch, Non-Unique Factorizations. Algebraic, Combinatorial and Analytic Theory, Pure Appl. Math. (Boca Raton) 278, Chapman & Hall/CRC, Boca Rato, 2006. Google Scholar

  • [16]

    A. Geroldinger, W. Hassler and G. Lettl, On the arithmetic of strongly primary monoids, Semigroup Forum 75 (2007), no. 3, 568–588. Web of ScienceGoogle Scholar

  • [17]

    A. Geroldinger and W. A. Schmid, A realization theorem for sets of distances, J. Algebra 481 (2017), 188–198. CrossrefWeb of ScienceGoogle Scholar

  • [18]

    A. Geroldinger, W. A. Schmid and Q. Zhong, Systems of sets of lengths: Transfer Krull monoids versus weakly Krull monoids, Rings, Polynomials, and Modules, Springer, Cham (2017), 191–235. Google Scholar

  • [19]

    F. Gotti, On the atomic structure of Puiseux monoids, J. Algebra Appl. 16 (2017), no. 7, Article ID 1750126. Web of ScienceGoogle Scholar

  • [20]

    F. Kainrath, Factorization in Krull monoids with infinite class group, Colloq. Math. 80 (1999), no. 1, 23–30. CrossrefGoogle Scholar

  • [21]

    C. O’Neill and R. Pelayo, Realizable sets of catenary degrees of numerical monoids, Bull. Aust. Math. Soc. (2017), 10.1017/S0004972717000995, https://arxiv.org/abs/1705.04276.

  • [22]

    M. Omidali, The catenary and tame degree of numerical monoids generated by generalized arithmetic sequences, Forum Math. 24 (2012), no. 3, 627–640. Web of ScienceGoogle Scholar

  • [23]

    W. A. Schmid, A realization theorem for sets of lengths, J. Number Theory 129 (2009), no. 5, 990–999. Web of ScienceCrossrefGoogle Scholar

About the article

Received: 2017-08-27

Revised: 2018-01-03

Published Online: 2018-02-07

Published in Print: 2018-09-01

Funding Source: Austrian Science Fund

Award identifier / Grant number: Project Number P 28864-N35

This work was supported by the Austrian Science Fund FWF, Project Number P 28864-N35.

Citation Information: Forum Mathematicum, Volume 30, Issue 5, Pages 1111–1118, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2017-0180.

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