The Ap\'ery Set of a Good Semigroup

TL;DR

This paper explores the Apéry set of good semigroups in , extending properties known from numerical semigroups to semigroups associated with curve singularities with two branches, despite their infinite nature.

Contribution

It introduces a level-based partition of the Apéry set for good semigroups in , generalizing properties from numerical semigroups to a broader class.

Findings

Partition of Apéry set into levels enables property generalization.

The structure aids understanding of semigroups from curve singularities with two branches.

Framework extends classical results to infinite semigroups.

Abstract

We study the Ap\'ery set of good subsemigoups of $\mathbb N^2$, a class of semigroups containing the value semigroups of curve singularities with two branches. Even if this set in infinite, we show that, for the Ap\'ery set of such semigroups, we can define a partition in "levels" that allows to generalize many properties of the Ap\'ery set of numerical semigroups, i.e. value semigroups of one-branch singularities.