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

Part of: Semigroups Algebraic combinatorics

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
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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

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 112 , Issue 1 , August 2025 , pp. 155 - 162
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

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

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