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 $S\setminus I$.
Categories: math.AC