Consider an n+1 dimensional vector space V over the reals.
Consider some n dimensional subspace S of V and some vector x that is not in S.
The set P = {x+y|y∊S} is not a subspace for 0 ∉ P.
Most subspaces of V intersect P however and P forms a stage for conventional projective geometry of n dimensions.
Just as Euclidean geometry needs an extra line at infinity to suit the axioms of projective geometry, so must ideal elements be added to P.
A subspace that intersects P is uniquely determined by that intersection.
There is such an ideal element for each subspace of V that does not intersect P.
Each non-ideal member of the projective space corresponds to a subspace of V which intersects P in just that member.
Intersections in P correspond to intersections of the vector subspaces.
The whole projective geometry is thus mapped, one to one, onto the subspaces of V.
To make the projective space a lattice we include two unconventional elements of the projective space:
The empty set matching the subspace {0}, and all of P matching V.