Reference for subsemigroups of $\mathbb{N}^n$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T09:45:05Z http://mathoverflow.net/feeds/question/103810 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/103810/reference-for-subsemigroups-of-mathbbnn Reference for subsemigroups of $\mathbb{N}^n$ Lee Mosher 2012-08-02T18:44:40Z 2012-08-03T00:47:17Z <p>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 $A$. </p> <p>Are there any references to a higher dimensional version of this result? </p> <p>The version I want goes like this. </p> <ul> <li>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 $v \in S_U$.</li> </ul> http://mathoverflow.net/questions/103810/reference-for-subsemigroups-of-mathbbnn/103827#103827 Answer by Benjamin Steinberg for Reference for subsemigroups of $\mathbb{N}^n$ Benjamin Steinberg 2012-08-03T00:47:17Z 2012-08-03T00:47:17Z <p>This combines comments of myself and Lee Mosher.</p> <p>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 $v\in a+N\subseteq S_U$.</p>