A
discrete (
additive)
subgroup of
Rd. A lattice is always
isomorphic to
Zd', for some
d' <
d. This fact is non-trivial to prove, and depends heavily on the discreteness of the lattice.
The form of the isomorphism is also decidedly nontrivial (and nonunique); some very hard problems arise from lattices, e.g. in computational number theory.