Lattice (group)

Remarks

A subgroup $\Lambda$ of $\R^n$ is a lattice if and only if it is discrete and spans $\R^n$ as a real vector space. Two bases of the same lattice differ by an integer matrix of determinant $\pm 1$, so the covolume does not depend on the basis. It equals the volume of the fundamental parallelepiped $\set{t_1 v_1 + \dots + t_n v_n : 0 \le t_i < 1}$.

Example: $\Z^n$ is a lattice with basis the standard vectors $e_1, \dots, e_n$ and covolume $1$.

Non-example: $\Z + \Z\sqrt{2} \subset \R$ is a subgroup generated by two vectors, but two vectors in $\R^1$ are never linearly independent. In fact this subgroup is dense in $\R$, so it is not discrete. Another non-example: $\Z \times \set{0}$ in $\R^2$ is discrete but does not span $\R^2$.