Abstract
Let M be a commutative, cancellative, atomic monoid and x a nonunit in M. We define ω(x)=n if n is the smallest positive integer with the property that whenever x∣a 1⋅⋅⋅a t , where each a i is an atom, there is a T⊆{1,2,…,t} with |T|≤n such that x∣∏k∈T a k . The ω-function measures how far x is from being prime in M. In this paper, we give an algorithm for computing ω(x) in any numerical monoid. Simple formulas for ω(x) are given for numerical monoids of the form 〈n,n+1,…,2n−1〉, where n≥3, and 〈n,n+1,…,2n−2〉, where n≥4. The paper then focuses on the special case of 2-generator numerical monoids. We give a formula for computing ω(x) in this case and also necessary and sufficient conditions for determining when x is an atom. Finally, we analyze the asymptotic behavior of ω(x) by computing \(\lim_{x\rightarrow \infty}\frac{\omega(x)}{x}\).
Similar content being viewed by others
References
Anderson, D.F., Chapman, S.T.: How far is an element from being prime? J. Algebra Appl. 9, 1–11 (2010)
Bowles, C., Chapman, S.T., Kaplan, N., Reiser, D.: On delta sets of numerical monoids. J. Algebra Appl. 5, 1–24 (2006)
Chapman, S.T., Holden, M., Moore, T.: On full elasticity in atomic monoids and integral domains. Rocky Mt. J. Math. 36, 1437–1455 (2006)
Chapman, S.T., Kaplan, N., Hoyer, R.: Delta sets of numerical monoids are eventually periodic. Aequ. Math. 77, 273–279 (2009)
Chapman, S.T., Kaplan, N., Lemburg, T., Niles, A., Zlogar, C.: Shifts of generators and delta sets of numerical monoids. J. Commun. Algebra (to appear)
García-Sánchez, P.A., Rosales, J.C.: Numerical Semigroups. Developments in Mathematics, vol. 20. Springer, Berlin (2009)
Geroldinger, A., Halter-Koch, F.: Non-unique Factorizations: Algebraic, Combinatorial and Analytic Theory. Pure and Applied Mathematics, vol. 278. Chapman & Hall/CRC, London/Boca Raton (2006)
Geroldinger, A., Hassler, W.: Local tameness of v-noetherian monoids. J. Pure Appl. Algebra 212, 1509–1524 (2008)
Omidali, M.: The catenary and tame degree of certain numerical monoids. Forum Math. (to appear)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by J. Almeida.
The third and fourth authors received support from the National Science Foundation under grant DMS-0648390.
The authors wish to thank Rolf Hoyer, Jay Daigle and Terri Moore for discussions related to this work.
Rights and permissions
About this article
Cite this article
Anderson, D.F., Chapman, S.T., Kaplan, N. et al. An algorithm to compute ω-primality in a numerical monoid. Semigroup Forum 82, 96–108 (2011). https://doi.org/10.1007/s00233-010-9259-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00233-010-9259-5