Partition of complement of good ideals and Ap\'{e}ry sets

TL;DR

This paper develops a partition of the complements of good semigroup ideals in , enabling a comprehensive description of Apry sets for good semigroups, extending previous 2D results to higher dimensions.

Contribution

It generalizes the description of Apry sets from 2D to higher dimensions and introduces new results on good semigroups in .

Findings

Partition of complements of good semigroup ideals in 

Generalization of Apry set descriptions to any dimension 

New structural results on good semigroups in 

Abstract

Good semigroups form a class of submonoids of $\mathbb{N}^d$ containing the value semigroups of curve singularities. In this article, we describe a partition of the complements of good semigroup ideals, having as main application the description of the Ap\'{e}ry sets of good semigroups. This generalizes to any $d \geq 2$ the results of a recent paper of D'Anna, Guerrieri and Micale, which are proved in the case $d=2$ and only for the standard Ap\'{e}ry set with respect to the smallest nonzero element. Several new results describing good semigroups in $\mathbb{N}^d$ are also provided.