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On arithmetical numerical monoids with some generators omitted

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Abstract

Numerical monoids (cofinite, additive submonoids of the non-negative integers) arise frequently in additive combinatorics, and have recently been studied in the context of factorization theory. Arithmetical numerical monoids, which are minimally generated by arithmetic sequences, are particularly well-behaved, admitting closed forms for many invariants that are difficult to compute in the general case. In this paper, we answer the question “when does omitting generators from an arithmetical numerical monoid S preserve its (well-understood) set of length sets and/or Frobenius number?” in two extremal cases: (1) we characterize which individual generators can be omitted from S without changing the set of length sets or Frobenius number; and (2) we prove that under certain conditions, nearly every generator of S can be omitted without changing its set of length sets or Frobenius number.

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References

  1. Amos, J., Chapman, S., Hine, N., Paixão, J.: Sets of lengths do not characterize numerical monoids. Integers 7, A50 (2007)

    MathSciNet  MATH  Google Scholar 

  2. Barron, T., O’Neill, C., Pelayo, R.: On dymamic algorithms for factorization invariants in numerical monoids. Math. Comput. 86, 2429–2447 (2017)

    Article  MATH  Google Scholar 

  3. Chapman, S., Corrales, M., Miller, A., Miller, C., Patel, D.: The catenary degrees of elements in numerical monoids generated by arithmetic sequences, to appear. Commun. Algebra. 45, 5443–5452 (2017)

    Article  MATH  Google Scholar 

  4. Chapman, S., Kaplan, N., Lemburg, T., Niles, A., Zlogar, C.: Shifts of generators and delta sets of numerical monoids. Intern. J. Algebra Comput. 24(5), 655–669 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Conaway, R., Gotti, F., Horton, J., O’Neill, C., Pelayo, R., Williams, M., Wissman, B.: Minimal presentations of shifted numerical monoids. Int. J. Algebra Comput. arXiv:1701.08555 [math.AC] (to appear)

  6. Delgado, M., García-Sánchez, P., Morais, J.: NumericalSgps, a Package for Numerical Semigroups, version 1.1.0, (GAP package). https://gap-packages.github.io/numericalsgps/ (2017)

  7. Geroldinger, A., Halter-Koch, F.: Nonunique Factorization, Algebraic, Combinatorial and Analytic Theory, Pure and Applied Mathematics, vol. 278. Chapman & Hall/CRC, London (2006)

    MATH  Google Scholar 

  8. Narkiewicz, W.: Finite abelian groups and factorization problems. Colloq. Math. 42, 319–330 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  9. Omidali, M.: The catenary and tame degree of numerical monoids generated by generalized arithmetic sequences. Forum Math. 24(3), 627–640 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. O’Neill, C., Pelayo, R.: Apéry sets of shifted numerical monoids. Adv. Appl. Math. 97, 27–35 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ramírez Alfonsín, J.: The Diophantine Frobenius Problem, Oxford Lecture Series in Mathematics and its Applications, vol. 30. Oxford University Press, Oxford (2005)

    Book  MATH  Google Scholar 

  12. Rosales, J., García-Sánchez, P.: Numerical Semigroups, Developments in Mathematics, vol. 20. Springer, New York (2009)

    Book  Google Scholar 

  13. Schmid, W.: Characterization of class groups of Krull monoids via their systems of sets of lengths: a status report. In: Number Theory and Applications: Proceedings of the International Conferences on Number Theory and Cryptography, Hindustan Book Agency, pp. 189–212 (2009)

  14. Stanley, R.: Combinatorics and Commutative Algebra, Progress in Mathematics, vol. 41, 2nd edn. Birkhäuser Boston Inc., Boston, MA (1996)

    Google Scholar 

  15. The Sage Developers, SageMath, the Sage Mathematics Software System (version 7.2), http://www.sagemath.org (2016)

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Acknowledgements

The authors would like to thank Vadim Ponomarenko for many helpful conversations.

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Authors and Affiliations

  1. Mathematics Department, University of California Berkeley, Berkeley, CA, 94720, USA

    Sung Hyup Lee & Brandon Van Over

  2. Mathematics Department, University of California Davis, Davis, CA, 95616, USA

    Christopher O’Neill

Authors
  1. Sung Hyup Lee
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  2. Christopher O’Neill
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  3. Brandon Van Over
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Corresponding author

Correspondence to Christopher O’Neill.

Additional information

Communicated by Fernando Torres.

The third author would like to dedicate this paper to his late father, Thomas Van Over, who taught him the perseverence necessary to complete this paper.

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Lee, S.H., O’Neill, C. & Van Over, B. On arithmetical numerical monoids with some generators omitted. Semigroup Forum 98, 315–326 (2019). https://doi.org/10.1007/s00233-018-9952-3

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  • DOI: https://doi.org/10.1007/s00233-018-9952-3

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