lattice in ${ℝ}^n$

Definition.

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

Example: The following is an example of a lattice $ℒ\subset {ℝ}^2$, generated by $w_1=(1,2),w_2=(4,1)$.

$$ℒ=\{\alpha w_1+\beta w_2\mid \alpha ,\beta \in \mathbb{Z}\}$$