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? 



Basis of cone lattice
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. 


