A formula for the conductor of a semimodule of a numerical semigroup with two generators

TL;DR

This paper addresses the problem of computing the conductor $c(\Delta)$ of a $\langle \alpha,\beta\rangle$-semimodule, extending the Frobenius problem to semimodule structure. It proves a closed formula $c(\Delta)=M-\alpha-\beta+1$ where $M$ is the largest minimal generator of the syzygy semimodule $\mathrm{Syz}(\Delta)$, and relates it to $c(\Gamma)$ via the lattice coordinates of $M$. It further connects this to the Apéry set by showing $M=\max \mathrm{Ap}(\Delta,\alpha+\beta)$ and provides an equivalent dual formulation $c(\Delta)=\alpha\beta-\min\{x_i\}-\alpha-\beta+1$ using the dual semimodule $\Delta^{*}$; the two minimal generator sets are in bijection via $x\mapsto \alpha\beta-x$. The results yield a practical, unified framework for computing conductors in the two-generator case and reveal structural links between syzygies, Apéry sets, and dual semimodules.

Abstract

We provide an expression for the conductor $c(Δ)$ of a semimodule $Δ$ of a numerical semigroup $Γ$ with two generators in terms of the syzygy module of $Δ$ and the generators of the semigroup. In particular, we deduce that the difference between the conductor of the semimodule and the conductor of the semigroup is an element of $Γ$, as well as a formula for $c(Δ)$ in terms of the dual semimodule of $Δ$.

Figures (1)

• Figure 2.1: Lattice path for the $\langle 5,7 \rangle$-lean set $I=[0,9,11,8]$ and the corresponding syzygy minimal generators $J=[14,16,18,15]$. The biggest generator $M$ with respect to $\leq_{\mathbb{N}}$ is depicted bigger.

Theorems & Definitions (17)

• Proposition 2.1
• proof
• Definition 2.2
• Definition 2.3
• Lemma 2.4
• proof
• Example 2.5
• Theorem 3.1
• proof
• Example 3.2