# lattice in $\mathbb{R}^{n}$

###### Definition.

A lattice in ${\mathbb{R}}^{n}$ is an $n$-dimensional additive free group over $\mathbb{Z}$ which generates ${\mathbb{R}}^{n}$ over $\mathbb{R}$.

Example: The following is an example of a lattice $\mathcal{L}\subset{\mathbb{R}}^{2}$, generated by $w_{1}=(1,2),w_{2}=(4,1)$.

 $\mathcal{L}=\{\alpha w_{1}+\beta w_{2}\mid\alpha,\beta\in\mathbb{Z}\}$
Title lattice in $\mathbb{R}^{n}$ LatticeInmathbbRn 2013-03-22 13:52:14 2013-03-22 13:52:14 alozano (2414) alozano (2414) 7 alozano (2414) Definition msc 11H06 lattice grid MinkowskisTheorem PicksTheorem ProductOfPosets lattice in $\mathbb{R}^{n}$