Sublattice (Euclidean geometry)

A sublattice is a subgroup of a Euclidean lattice. It is a Euclidean lattice in its real linear span, which may have smaller dimension than the original ambient Euclidean space. The equal-parity integer points in ${\mathbb{Z}}^d$ form a full-rank sublattice of index $2^{d-1}$: reduction of all coordinate differences modulo two gives the quotient.