Flats

A flat (closed set) is a subset $F \subseteq E$ such that adding any element outside $F$ increases the rank. Ordered by inclusion, the flats form a geometric lattice, the lattice of flats. For a vector configuration, flats correspond to the subspaces spanned by subsets of the vectors.

The smallest flat is the set of loops and the largest is $E$. Flats of rank 1 are the points (parallel classes), flats of rank 2 are lines (see lines), and flats of rank $r-1$ are the hyperplanes.