# Basis of cone lattice

I only want to know whether a construction that I use appears in literature and maybe has a name already.

Let $$V$$ be a $$\mathbb Q$$ vector space of dimension $$d\in\mathbb N$$. A subset $$C\subset V$$ is called a proper cone with $$d$$ sides if there are linearly independent $$\alpha_1,\dots,\alpha_d\in \mathrm{Hom}(V,{\mathbb Q})$$ such that $$C$$ is the set of all $$v\in V$$ with $$\alpha_j(v)>0$$ for every $$j=1,\dots,d$$. Let $$\Sigma$$ be a lattice in $$V$$, i.e., a finitely generated additive subgroup which spans $$V$$. Then it is not hard to show that there exists a finite subset $$E\subset\Sigma$$ and $$a_1,\dots,a_d\in\Sigma$$ such that $$C\cap\Sigma=E+{\mathbb N}_0a_1+\dots+{\mathbb N}_0a_d$$ and for each $$v\in C\cap\Sigma$$ the representation $$v=e+n_1a_1+\dots+n_da_d$$ is unique. Here $${\mathbb N}_0={\mathbb N}\cup\{0\}$$. Then the sets $$E$$ and $$\{a_1,\dots,a_d\}$$ are uniquely determined. Do they have a name?

• Looks very much like Gordan's Lemma, though that doesn't mention uniqueness. – Martin Bright Dec 6 '18 at 11:11
• Yes indeed, and it doesn't give a name. Maybe somebody else has? – Zero Dec 6 '18 at 11:17

I think that $$\alpha_1, \dots \alpha_d$$ are called lattice ray generators and $$E$$ is the set of lattice points in the fundamental domain. It is a special case of Gordan's lemma: the case when the cone is simplicial.