Everything is assumed to be finite-dimensional. Let $S$ be a finitely generated affine semigroup (i.e. a subsemigroup of a lattice $N$). Assume that $S$ generates $N$ as a group. Is it true that it contains all but a finite number of points in $\mathbb{R}_{\ge 0} S\cap N?$
I believe the result should be known, however, I am not a specialist in convex geometry so book reference will be much appreciated.