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

Let S and C be affine semigroups in Nd such that S⊆C. We provide a characterization for the set C∖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∖I is a finite set. If so, we provide some procedures to compute the set S∖I.