Full-rank Euclidean lattice

A full-rank Euclidean lattice in $\mathbb R^n$ is a set

\Lambda=B\mathbb Z^n=\{Bm:m\in\mathbb Z^n\},

where the columns of $B$ form a basis of $\mathbb R^n$ . Thus $B$ is invertible. The lattice is an additive subgroup and is discrete: $|Bm|\ge |m|/\|B^{-1}\|$ , so its nonzero points have a positive lower bound on their distance from zero.

Basis choices

Different bases can describe the same lattice. Replacing $B$ by $BU$ , where $U$ is an integer matrix with integer inverse, preserves $\Lambda$ . The standard example is $\mathbb Z^n$ . Here “lattice” refers to a discrete additive set in Euclidean space; an order-theoretic lattice is a different structure.