Abstract
We generalize Rota′s theorem characterizing the Möbius function of a geometric lattice in terms of subsets of atoms containing no broken circuit and give applications to the weak Bruhat order of a finite Coxeter group and the Tamari lattices. We also give a direct proof of the fact that in the geometric case any total order of the atoms can be used. Simple involutions are used in both proofs. Finally, we show how involutions can be used in similar situations, specifically in a special case of Rota′s Crosscut Theorem as well as in related proofs of Walker on Hall′s Theorem and Reiner on characteristic and Poincaré polynomials.
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