On the cardinality of minimal presentations  of numerical semigroups

Elmacioglu, Ceyhun; Hilmer, Kieran; O’Neill, Christopher; Okandan, Melin; Park-Kaufmann, Hannah

doi:10.5802/alco.354

On the cardinality of minimal presentations of numerical semigroups

Elmacioglu, Ceyhun ¹; Hilmer, Kieran ²; O’Neill, Christopher ³; Okandan, Melin ⁴; Park-Kaufmann, Hannah ⁵

¹ Mathematics Department Lafayette College Easton, PA 18042
² Mathematics Department Purdue University West Lafayette, IN 47907
³ Mathematics Department San Diego State University San Diego, CA 92182
⁴ Mathematics Department Koç University Istanbul, Turkey
⁵ Mathematics Department and Conservatory Bard College Annandale-on-Hudson, NY 12504

Algebraic Combinatorics, Volume 7 (2024) no. 3, pp. 753-771.

Abstract

In this paper, we consider the following question: “given the multiplicity $m$ and embedding dimension $e$ of a numerical semigroup $S$ , what can be said about the cardinality $η$ of a minimal presentation of $S$ ?” We approach this question from a combinatorial (poset-theoretic) perspective, utilizing the recently-introduced notion of a Kunz nilsemigroup. In addition to making significant headway on this question beyond what was previously known, in the form of both explicit constructions and general bounds, we provide a self-contained introduction to Kunz nilsemigroups that avoids the polyhedral geometry necessary for much of their source material.

Received: 2022-12-10
Revised: 2023-07-06
Accepted: 2024-01-12
Published online: 2024-06-27

DOI: 10.5802/alco.354

Classification: 20M14, 13F65, 05E40
Keywords: numerical semigroup, minimal presentation

Author's affiliations:

Elmacioglu, Ceyhun ¹; Hilmer, Kieran ²; O’Neill, Christopher ³; Okandan, Melin ⁴; Park-Kaufmann, Hannah ⁵

¹ Mathematics Department Lafayette College Easton, PA 18042
² Mathematics Department Purdue University West Lafayette, IN 47907
³ Mathematics Department San Diego State University San Diego, CA 92182
⁴ Mathematics Department Koç University Istanbul, Turkey
⁵ Mathematics Department and Conservatory Bard College Annandale-on-Hudson, NY 12504

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Elmacioglu, Ceyhun and Hilmer, Kieran and O{\textquoteright}Neill, Christopher and Okandan, Melin and Park-Kaufmann, Hannah},
     title = {On the cardinality of minimal presentations  of numerical semigroups},
     journal = {Algebraic Combinatorics},
     pages = {753--771},
     publisher = {The Combinatorics Consortium},
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Elmacioglu, Ceyhun; Hilmer, Kieran; O’Neill, Christopher; Okandan, Melin; Park-Kaufmann, Hannah. On the cardinality of minimal presentations  of numerical semigroups. Algebraic Combinatorics, Volume 7 (2024) no. 3, pp. 753-771. doi : 10.5802/alco.354. https://alco.centre-mersenne.org/articles/10.5802/alco.354/

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