This is a [cross-post from MSE.](https://math.stackexchange.com/questions/4376145/affine-semigroup-generating-a-lattice)


Everything is assumed to be finite-dimensional. Let $S$ be a finitely generated affine semigroup (i.e. a subsemigroup of a lattice $N$ of a Euclidean space). 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.



Affine semigroup generating a lattice
(Write $\mathbf{N}=\{0,1,2,\dots\}$.)


What about the submonoid of $\mathbf{Z}^2$ generated by $\{(2,0),(3,0),(0,1)\}$? It equals $\mathbf{N}^2\smallsetminus (\{1\}\times\mathbf{N}$). So here the complement is infinite.


