# Positive cone of a subgroup of $\mathbb{Z}^n$

This question sounds like it should be very well known, but for some reason I failed to find a decent answer anywhere. Let $$G\subset\mathbb{Z}^n$$ be a subgroup, and $$G_+=G\cap\mathbb{Z}_{\ge0}^n$$ be a cone of elements in $$G$$ whose coordinates are all nonnegative. The semigroup $$G_+$$ is finitely generated; it can be proved in a couple of ways. My question is, is there an effective upper bound on the number of generators in terms of $$n$$, or, even better, the rank of $$G$$?

No, there is no upper bound on the number of generators in terms of $$n$$.
Let $$k$$ be arbitrary positive integer. Consider the subgroup $$G=\{(x,y)\in\mathbb Z^2\mid x+y\equiv 0\pmod k\}.$$ Any set of generators of $$G_+$$ contains all the elements $$(x,y)$$ such that $$x,y\ge 0$$ and $$x+y=k$$. Therefore the number of generators of $$G_+$$ is $$k+1$$.
• OK, that's discouraging but convincing enough. I am definitely accepting it, and will post another question if I will figure out a way to put a restrictions on $G$'s that I had in mind. Thanks! – Vladimir Dotsenko Jul 14 '11 at 20:52