Hostname: page-component-5b777bbd6c-v47t2 Total loading time: 0 Render date: 2025-06-20T06:18:50.352Z Has data issue: false hasContentIssue false
  • English
  • Français

NUMERICAL SEMIGROUPS FROM RATIONAL MATRICES II: MATRICIAL DIMENSION DOES NOT EXCEED MULTIPLICITY

Part of: Algebraic combinatorics Semigroups

Published online by Cambridge University Press:  12 December 2024

ARSH CHHABRA Open the ORCID record for ARSH CHHABRA [Opens in a new window] ,
STEPHAN RAMON GARCIA Open the ORCID record for STEPHAN RAMON GARCIA [Opens in a new window]  and
CHRISTOPHER O’NEILL Open the ORCID record for CHRISTOPHER O’NEILL [Opens in a new window]
ARSH CHHABRA
Affiliation:
Department of Mathematics and Statistics, Pomona College, 610 N. College Ave., Claremont, CA 91711, USA e-mail: acaa2021@mymail.pomona.edu
STEPHAN RAMON GARCIA
Affiliation:
Department of Mathematics and Statistics, Pomona College, 610 N. College Ave., Claremont, CA 91711, USA e-mail: stephan.garcia@pomona.edu
CHRISTOPHER O’NEILL*
Affiliation:
Mathematics Department, San Diego State University, San Diego, CA 92182, USA
*
e-mail: cdoneill@sdsu.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We continue our study of exponent semigroups of rational matrices. Our main result is that the matricial dimension of a numerical semigroup is at most its multiplicity (the least generator), greatly improving upon the previous upper bound (the conductor). For many numerical semigroups, including all symmetric numerical semigroups, our upper bound is tight. Our construction uses combinatorially structured matrices and is parametrised by Kunz coordinates, which are central to enumerative problems in the study of numerical semigroups.

Keywords

numerical semigrouprational matrixconductor

MSC classification

Primary: 20M14: Commutative semigroups
Secondary: 05E40: Combinatorial aspects of commutative algebra
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , First View , pp. 1 - 8
Check for updates
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

The second author is partially supported by NSF grant DMS-2054002.

References

Alhajjar, E., Russell, T. and Steward, M., ‘Numerical semigroups and Kunz polytopes’, Semigroup Forum 99 (2019), 153–168.CrossRefGoogle Scholar
Apéry, R., ‘Sur les branches superlinéaires des courbes algébriques’, C. R. Acad. Sci. Paris 222 (1946), 1198–1200.Google Scholar
Assi, A., D’Anna, M. and García-Sánchez, P. A., Numerical Semigroups and Applications, 2nd edn, RSME Springer Series, 3 (Springer, Cham, 2020).Google Scholar
Barucci, V., Dobbs, D. E. and Fontana, M., Maximality Properties in Numerical Semigroups and Applications to One-dimensional Analytically Irreducible Local Domains, Memoirs of the American Mathematical Society, 125(598) (American Mathematical Society, Providence, RI, 1997).CrossRefGoogle Scholar
Bogart, T. and Fakhari, S. A. S., ‘Unboundedness of irreducible decompositions of numerical semigroups’, Comm. Algebra, to appear. Published online (15 October 2024).CrossRefGoogle Scholar
Chhabra, A., Garcia, S. R., Zhang, F. and Zhang, H., ‘Numerical semigroups from rational matrices I: power-integral matrices and nilpotent representations’, Comm. Algebra, to appear. Published online (30 September 2024).CrossRefGoogle Scholar
Elmacioglu, C., Hilmer, K., O’Neill, C., Okandan, M. and Park-Kaufmann, H., ‘On the cardinality of minimal presentations of numerical semigroups’, Algebr. Comb. 7(3) (2024), 753–771.Google Scholar
Kaplan, N., ‘Counting numerical semigroups’, Amer. Math. Monthly 124(9) (2017), 862–875.CrossRefGoogle Scholar
Kaplan, N. and O’Neill, C., ‘Numerical semigroups, polyhedra, and posets I: the group cone’, Combin. Theory 1 (2021), Article no. 19, 23 pages.CrossRefGoogle Scholar
Kunz, E., ‘The value-semigroup of a one-dimensional Gorenstein ring’, Proc. Amer. Math. Soc. 25 (1970), 748–751.CrossRefGoogle Scholar
Kunz, E., Über die Klassifikation numerischer Halbgruppen, Regensburger mathematische Schriften, 11 (Fakultät Mathematik der Universität Regensburg, 1987).Google Scholar
Rosales, J. C., ‘Numerical semigroups with Apéry sets of unique expression’, J. Algebra 226(1) (2000), 479–487.CrossRefGoogle Scholar
Rosales, J. C. and Branco, M. B., ‘Decomposition of a numerical semigroup as an intersection of irreducible numerical semigroups’, Bull. Belg. Math. Soc. Simon Stevin 9(3) (2002), 373–381.Google Scholar
Rosales, J. C. and Branco, M. B., ‘Irreducible numerical semigroups’, Pacific J. Math. 209(1) (2003), 131–143.CrossRefGoogle Scholar
Rosales, J. C. and García-Sánchez, P. A., Numerical Semigroups, Developments in Mathematics, 20 (Springer, New York, 2009).CrossRefGoogle Scholar
Rosales, J. C., García-Sánchez, P. A., García-García, J. I. and Branco, M. B., ‘Systems of inequalities and numerical semigroups’, J. Lond. Math. Soc. (2) 65(3) (2002), 611–623.CrossRefGoogle Scholar
Wilf, H. S., ‘A circle-of-lights algorithm for the “money-changing problem”’, Amer. Math. Monthly 85(7) (1978), 562–565.Google Scholar