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

TL;DR

This paper derives a formula for the conductor of a semimodule over a two-generator numerical semigroup, linking it to the syzygy module and dual semimodule, revealing structural relationships.

Contribution

It introduces a new explicit formula for the conductor of semimodules over two-generator numerical semigroups, connecting it to syzygy modules and dual semimodules.

Findings

The difference between the conductors of the semimodule and the semigroup is in the semigroup.

A formula for the conductor of the semimodule in terms of the dual semimodule is established.

The paper characterizes the conductor using the syzygy module of the semimodule.

Abstract

We provide an expression for the conductor $c(\Delta)$ of a semimodule $\Delta$ of a numerical semigroup $\Gamma$ with two generators in terms of the syzygy module of $\Delta$ 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 $\Gamma$, as well as a formula for $c(\Delta)$ in terms of the dual semimodule of $\Delta$.