A generalization of the Birkhoff–Whitney theorem for hereditary systems

The notion of hereditary system is a natural generalization of the notion of matroid. We prove the following generalization of the Birkhoff–Whitney theorem for hereditary systems: the lattice of closed sets of any hereditary system does not contain infinite chains and, vice versa, any nonempty lattice without infinite chains is isomorphic to the lattice of closed sets of some hereditary system. In particular, any finite nonempty lattice is isomorphic to the lattice of closed sets of some finite hereditary system.