Semigroup Forum

, Volume 94, Issue 1, pp 37–50 | Cite as

On the set of elasticities in numerical monoids

  • Thomas Barron
  • Christopher O’Neill
  • Roberto PelayoEmail author
Research Article


In an atomic, cancellative, commutative monoid S, the elasticity of an element provides a coarse measure of its non-unique factorizations by comparing the largest and smallest values in its set of factorization lengths (called its length set). In this paper, we show that the set of length sets \({\mathcal {L}}(S)\) for any arithmetical numerical monoid S can be completely recovered from its set of elasticities R(S); therefore, R(S) is as strong a factorization invariant as \({\mathcal {L}}(S)\) in this setting. For general numerical monoids, we describe the set of elasticities as a specific collection of monotone increasing sequences with a common limit point of \(\max R(S)\).


Factorization Numerical monoid Elasticity Length set Arithmetic sequence 



The authors would like to thank Scott Chapman, Alfred Geroldinger, and Sherilyn Tamagawa for various helpful conversations and insights. The third author is funded by National Science Foundation Grant DMS-1045147.


  1. 1.
    Amos, J., Chapman, S., Hine, N., Paixão, J.: Sets of lengths do not characterize numerical monoids, Integers 7, #A50 (2007)Google Scholar
  2. 2.
    Bowles, C., Chapman, S., Kaplan, N., Reiser, D.: On delta sets of numerical monoids. J. Algebr. Appl. 5, 1–24 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chapman, S., García-Sánchez, P., Llena, D., Marshall, J.: Elements in a numerical semigroup with factorizations of the same length. Can. Math. Bull. 54(1), 39–43 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chapman, S.T., Holden, M.T., Moore, T.A.: Full elasticity in atomic monoids and integral domains. Rocky Mountain J. Math. 36, 1437–1455 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Geroldinger, A.: A structure theorem for sets of lengths. Colloq. Math. 78, 225–259 (1998)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Geroldinger, A., Halter-Koch, F.: Nonunique Factorization, Algebraic, Combinatorial and Analytic Theory, Pure and Applied Mathematics, vol. 278. Chapman & Hall/CRC, Boca Raton (2006)zbMATHGoogle Scholar
  7. 7.
    O’Neill, C., Pelayo, R.: On the linearity of \(\omega \)-primality in numerical monoids. J. Pure Appl. Algebr. 218, 1620–1627 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Schmid, W.: A realization theorem for sets of lengths. J. Number Theory 129(5), 990–999 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Thomas Barron
    • 1
  • Christopher O’Neill
    • 2
  • Roberto Pelayo
    • 3
    Email author
  1. 1.Mathematics DepartmentUniversity of KentuckyLexingtonUSA
  2. 2.Mathematics DepartmentTexas A&M UniversityCollege StationUSA
  3. 3.Mathematics DepartmentUniversity of Hawai‘i at HiloHiloUSA

Personalised recommendations