Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the cardinality of minimal presentations of numerical semigroups

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

TL;DR

The paper addresses the problem of determining the possible minimal presentation cardinalities $\eta(S)$ for numerical semigroups $S$ in terms of multiplicity $m$ and embedding dimension $e$, using Kunz nilsemigroups and outer Betti elements from a combinatorial poset perspective. It develops a self-contained introduction to Kunz nilsemigroups, establishes a general lower bound $\eta(S) \ge \binom{e}{2} - r$ with $r = m - e$, constructs an interval of attainable $\eta$ values, and provides partial upper bounds for small embedding codimension. The authors analyze the case $e=4$ in depth, supplying explicit families that realize a wide range of $\eta$ values and proving several sharp results; they also present computational data up to modest $m$ and $e$ and propose several open questions. Overall, the work demonstrates how Kunz nilsemigroups and outer Betti elements can yield precise information about $\eta(S)$ without requiring full minimal presentations, advancing the understanding of how $m$ and $e$ constrain $\eta$ in numerical semigroups.

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.

On the cardinality of minimal presentations of numerical semigroups

TL;DR

The paper addresses the problem of determining the possible minimal presentation cardinalities for numerical semigroups in terms of multiplicity and embedding dimension , using Kunz nilsemigroups and outer Betti elements from a combinatorial poset perspective. It develops a self-contained introduction to Kunz nilsemigroups, establishes a general lower bound with , constructs an interval of attainable values, and provides partial upper bounds for small embedding codimension. The authors analyze the case in depth, supplying explicit families that realize a wide range of values and proving several sharp results; they also present computational data up to modest and and propose several open questions. Overall, the work demonstrates how Kunz nilsemigroups and outer Betti elements can yield precise information about without requiring full minimal presentations, advancing the understanding of how and constrain in numerical semigroups.

Abstract

In this paper, we consider the following question: "given the multiplicity and embedding dimension of a numerical semigroup , what can be said about the cardinality of a minimal presentation of ?" 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.
Paper Structure (7 sections, 10 theorems, 60 equations, 7 figures)

This paper contains 7 sections, 10 theorems, 60 equations, 7 figures.

Table of Contents

  1. Introduction
  2. An overview of nilsemigroups and outer Betti elements
  3. A lower bound on minimal presentation cardinality
  4. An interval of attainable minimal presentation cardinalities
  5. A partial upper bound on minimal presentation cardinality
  6. Minimal presentation cardinalities in embedding dimension 4
  7. Some open questions

Key Result

Theorem 2.4

If $\rho$ is a minimal presentation for the Kunz nilsemigroup $N$ of a numerical semigroup $S$, then $\rho' \cup \rho"$ is a minimal presentation for $S$, where: In particular, $\eta(S) = \eta(N) = b(N) + |\rho|$.

Figures (7)

  • Figure 1: The values of $\eta(S) \le 26$ attained for $\mathsf m(S) \le 17$, with those attained for each $\mathsf e(S) \le 8$ outlined.
  • Figure 2: The Kunz posets of $S_1$ (left) and $S_2$ (right) from Example \ref{['e:kunzposets']}, with outer Betti elements depicted as in Example \ref{['e:outerbettis']}.
  • Figure 3: The Kunz posets of the semigroups $S_3 = \langle10,22,23,24\rangle$ (left) from Example \ref{['e:kunzposets']} and $S_4 = \langle6,7,8,9\rangle$ (right) from Example \ref{['e:lemma56']}.
  • Figure 4: The Kunz poset structure of $S$ with $\eta(S) = \binom{e}{2} - s$ from Theorem \ref{['t:intervalfamily']} (left) and the case $s = 0$ from Example \ref{['e:intervalfamily']} (right).
  • Figure 5: The Kunz poset structure of the numerical semigroup from Example \ref{['e:extrabettifamily']} (left) and that of $S$ with $\eta(S) = \binom{e}{2} + 1$ from Theorem \ref{['t:extrabettifamily']} (right).
  • ...and 2 more figures

Theorems & Definitions (35)

  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Theorem 2.4
  • Example 2.5
  • Example 2.6
  • Example 2.7
  • Remark 2.8
  • Definition 3.2
  • Lemma 3.3
  • ...and 25 more