Duals of subdirectly irreducible modal algebras

We give a characterization of the simple, and of the subdirectly irreducible boolean algebras with operators (including modal algebras), in terms of the dual descriptive frame. These characterizations involve a special binary quasi-reachability relation on the dual structure; we call a point u a quasi-root of the dual structure if every ultrafilter is quasireachable from u. We prove that a boolean algebra with operators is simple iff every point in the dual structure is a quasi-root; and that it is subdirectly irreducible iff the collection of quasi-roots has measure nonzero in the Stone topology on the dual structure.