Enumeration in torus arrangements

This paper presents two new results of somewhat different flavors. The first result is a formula for the numbers of cells of each dimension determined by an arrangement of closed subgroups of a torus group. This is accompanied by a brief description of previous work on cell counting in torus groups. In the course of things, two distinct formulas yielding the numbers of cells are encountered. The question of how to reconcile these two formulas motivates a general result that applies to lattices whose elements are the closed sets of a closure system. This result gives the characteristic polynomial for the (lattice) dual of such a lattice as a sum of characteristic polynomials of lattices of closed sets of deleted minors of the closure system.