Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Cohen, Frederick R. / Droste, Manfred / Darmon, Henri / Duzaar, Frank / Echterhoff, Siegfried / Gordina, Maria / Neeb, Karl-Hermann / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna


IMPACT FACTOR 2018: 0.867

CiteScore 2018: 0.71

SCImago Journal Rank (SJR) 2018: 0.898
Source Normalized Impact per Paper (SNIP) 2018: 0.964

Mathematical Citation Quotient (MCQ) 2018: 0.71

Online
ISSN
1435-5337
See all formats and pricing
More options …
Volume 30, Issue 5

Issues

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
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Wolfgang Alexander Schmid
  • Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8, CNRS, UMR 7539, F-93526, Saint-Denis, France
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2018-02-07 | DOI: https://doi.org/10.1515/forum-2017-0180

Abstract

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

References

  • [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.

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
Scott T. Chapman, Felix Gotti, and Marly Gotti
Communications in Algebra, 2019, Page 1

Comments (0)

Please log in or register to comment.
Log in