  It is shown that the lattices of flats of boolean representable simplicial
complexes are always atomistic, but semimodular if and only if the complex is a
matroid. A canonical construction is introduced for arbitrary finite atomistic
lattices, providing a characterization of the lattices of flats of boolean
representable simplicial complexes and a decidability condition. We remark that
every finite lattice occurs as the lattice of flats of some simplicial complex.
