Mِbius function on cross-section lattices

Let M be an irreducible algebraic monoid with a reductive unit group G. Then there is an idempotent cross-section Λ of G×G-orbits that preserves the Zariski closure ordering. The purpose of this paper is to compute the Möbius function on the cross-section lattice Λ. This is accomplished by analyzing an associated boolean family of face lattices of polytopes and then solving a resulting system of linear equations.