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Lattices in the sense of discrete subgroups of Euclidean spaces, as used in number theory, discrete geometry, Lie groups, etc. (Not to be confused with lattice theory or lattices as used in physics! For lattices (ordered sets), use the tag: [lattice-theory])
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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 …