A well known result about the natural numbers $\mathbb{N}$ says that for any finite subset $A \subset \mathbb{N}$ there exists $R \ge 0$ such that if $n$ is in the subgroup of $\mathbb{Z}$ generated by $A$ and if $n \ge R$ then $n$ is in the semigroup generated by $A$. 


Are there any references to a higher dimensional version of this result? 


The version I want goes like this. 


* Take a finite subset $U$ of $\mathbb{N}^n$. Let $C\_U$ be the smallest closed cone in $\mathbb{R}^n$ containing $U$, i.e. all non-negative real linear combinations of $U$. Let $G\_U$ be the subgroup of $\mathbb{N}^n$ generated by $U$, i.e. all integer linear combinations. Let $S\_U$ be the subsemigroup generated by $U$, i.e. all non-negative integer linear combinations. Then there exists $R>0$ such that for every $v \in G\_U$, if the ball around $v$ of radius $R$ is contained in $C\_U$ then $v \in S\_U$.



Reference for subsemigroups of $\mathbb{N}^n$
This combines comments of myself and Lee Mosher.


Exercise 7.15 of Miller and Sturmfels combinatorial commutative algebra book proves the following. Let $N=C\_U\cap G\_U$. Then $N$ is a finitely generated semigroup and there exists according to exercise 7.15 an element $a$ of $S\_U$ with $a+N\subseteq S\_U$. Now take $R>|a|$. If $B\_R(v)\subseteq C\_U$ then $v−a\in C\_U\cap G\_U=N$ so $v\in a+N\subseteq S\_U$.


