Geometric lattice

The following definitions apply to posets in general, not just lattices, except where otherwise stated.
When a graded poset has a bottom element, one may assume, without loss of generality, that its rank is zero. In this case, the atoms are the elements with rank one.

The geometric lattices are cryptomorphic to (finite, simple) matroids, and the matroid lattices are cryptomorphic to simple matroids without the assumption of finiteness.