Partition of complement of good ideals and Apéry sets

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 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.