Dual cone (${\sigma }^{\vee }$)

For a cone $\sigma \subseteq N_{\mathbb{R}}$, its dual cone is

For a rational cone $\sigma$, the lattice points ${\sigma }^{\vee }\cap M$ form a finitely generated additive monoid.

The Hilbert basis is the unique minimal generating set of the additive monoid of lattice points in a pointed rational cone.