On the Mِbius Algebra of Geometric Lattices

The Mِbius algebra of a poset was introduced by Solomon and also studied by Greene in the special case of lattices. Let L denote a geometric lattice. Our key idea is to consider all the characteristic polynomials of upper intervals in L as components of one object, which is multiplicative. Now, by simple algebraic means we obtain new identities involving the characteristic polynomial and the Tutte polynomial of a geometric lattice.