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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?

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

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