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Good semigroups form a class of submonoids of $\mathbb{N}^d$ containing thevalue semigroups of curve singularities. In this article, we describe apartition of the complements of good semigroup ideals, having as mainapplication the description of the Ap\'{e}ry sets of good semigroups. Thisgeneralizes 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 thestandard Ap\'{e}ry set with respect to the smallest nonzero element. Severalnew results describing good semigroups in $\mathbb{N}^d$ are also provided.

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