On Some Properties for Cofiniteness of Submonoids and Ideals of an Affine Semigroup

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 $S\setminus I$.