Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-27T09:14:53.221Z Has data issue: false hasContentIssue false
  • English
  • Français

Elements in a Numerical Semigroup with Factorizations of the Same Length

Published online by Cambridge University Press:  20 November 2018

S. T. Chapman ,
P. A. García-Sánchez ,
D. Llena  and
J. Marshall
S. T. Chapman
Affiliation:
Sam Houston State University, Department of Mathematics and Statistics, Huntsville, TX, U.S.A.e-mail: scott.chapman@shsu.edu
P. A. García-Sánchez
Affiliation:
Departamento de Álgebra, Universidad de Granada, Granada, Españae-mail: pedro@ugr.es
D. Llena
Affiliation:
Departamento de Geometría, Topología y Química Orgánica, Universidad de Almería, Almería, Españae-mail: dllena@ual.es
J. Marshall
Affiliation:
Sandia National Laboratories, P.O. Box 5800, Albuquerque, NM, U.S.A.e-mail: jormars@sandia.gov
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Questions concerning the lengths of factorizations into irreducible elements in numerical monoids have gained much attention in the recent literature. In this note, we show that a numerical monoid has an element with two different irreducible factorizations of the same length if and only if its embedding dimension is greater than two. We find formulas in embedding dimension three for the smallest element with two different irreducible factorizations of the same length and the largest element whose different irreducible factorizations all have distinct lengths. We show that these formulas do not naturally extend to higher embedding dimensions.

Keywords

20M1420D6011B75numerical monoidnumerical semigroupnon-unique factorization
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 54 , Issue 1 , 01 March 2011 , pp. 39 - 43
Copyright
Copyright © Canadian Mathematical Society 2011

References

[1] Amos, J., Chapman, S. T., Hine, N., and Paixão, J., Sets of lengths do not characterize numerical monoids. Integers 7(2007), A50.Google Scholar
[2] Bowles, C., Chapman, S. T., Kaplan, N., and Reiser, D., On Delta sets of numerical monoids. J. Algebra Appl. 5(2006), no. 5, 695–718. doi:10.1142/S0219498806001958Google Scholar
[3] Chapman, S. T., García-Sánchez, P. A., and Llena, D., The catenary and tame degree of numerical monoids. Forum Math. 21(2009), no. 1, 117–129. doi:10.1515/FORUM.2009.006Google Scholar
[4] Chapman, S. T., Holden, M., and Moore, T., Full elasticity in atomic monoids and integral domains. Rocky Mountain J. Math. 36(2006), no. 5, 1437–1455. doi:10.1216/rmjm/1181069375Google Scholar
[5] Chapman, S. T., Hoyer, R., and Kaplan, N., Delta sets of numerical monoids are eventually periodic. Aequationes Math. 77(2009), no. 3, 273–279. doi:10.1007/s00010-008-2948-4Google Scholar
[6] Chapman, S. T., Kaplan, N., Lemburg, T., Niles, A., and Zlogar, C., Shifts of generators and delta sets of numerical monoids. To appear, J. Comm. Algebra.Google Scholar
[7] Delgado, M., García-Sánchez, P. A., Morais, J., “numericalsgps”: a GAP package on numerical semigroups. http://www.gap-system.org/Packages/numericalsgps.html.Google Scholar
[8] Geroldinger, A. and Halter-Koch, F., Non-unique factorizations: Algebraic, combinatorial and analytic theory. Pure and Applied Mathematics, 278, Chapman & Hall/CRC, Boca Raton, FL, 2006.Google Scholar