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