A construction of semimodular lattices

In this paper we prove that if is a finite lattice, and r is an integral valued function on satisfying some very natural conditions, then there exists a finite geometric (that is, semimodular and atomistic) lattice I containing as a sublattice such that r is the height function of restricted to . Moreover, we show that if, for all intervals e, f] of , semimodular lattices I, of length at most r(f)-r(e) are given, then I can be chosen to contain I in its interval e, f] as a cover preserving {0}-sublattice. As applications, we obtain results of R. P. Dilworth and D. T. Finkbeiner.