The class L ( M ) {\displaystyle {\mathcal {L}}(M)} of all flats, partially ordered by set inclusion, forms a matroid lattice . Conversely, every matroid lattice L {\displaystyle L} forms a matroid over its set E {\displaystyle E} of atoms under the following closure operator: for a set S {\displaystyle S} of atoms with join ⋁ S {\displaystyle \bigvee S} ,