A generalization of semiconductor supersolvable lattices

Stanley (Algebra Universalis 2, 1972, 197–217) introduced the notion of a supersolvable lattice, L, in part to combinatorially explain the factorization of its characteristic polynomial over the integers when L is also semimodular. He did this by showing that the roots of the polynomial count certain sets of atoms of the lattice. In the present work we define an object called an atom decision tree. The class of semimodular lattices with atom decision trees strictly contains the class of supersolvable lattices, but their characteristic polynomials still factor for combinatorial reasons. We then apply this notion to prove the factorization of polynomials associated with various hyperplane arrangements having non-supersolvable lattices.