  We introduce the concept of a bounded below set in a lattice. This can be
used to give a generalization of Rota's broken circuit theorem to any finite
lattice. We then show how this result can be used to compute and
combinatorially explain the M\"obius function in various examples including
non-crossing set partitions, shuffle posets, and integer partitions in
dominance order. Next we present a generalization of Stanley's theorem that the
characteristic polynomial of a semimodular supersolvable lattice factors over
the integers. We also give some applications of this second main theorem,
including the Tamari lattices.
