  Let $S$ and $\mathcal{C}$ be affine semigroups in $\mathbb{N}^d$ such that
$S\subseteq \mathcal{C}$. We provide a characterization for the set
$\mathcal{C}\setminus S$ to be finite, together with a procedure and
computational tools to check whether such a set is finite and, if so, compute
its elements. As a consequence of this result, we provide a characterization
for an ideal $I$ of an affine semigroup $S$ so that $S\setminus I$ is a finite
set. If so, we provide some procedures to compute the set $S\setminus I$.
