On the Rank of Disjunctive Cuts

Let L be a family of lattice-free polyhedra in R containing the splits. Given a polyhedron P in R, we characterize when a valid inequality for P ∩(Z×R) can be obtained with a finite number of disjunctive cuts corresponding to the polyhedra in L. We also characterize the lattice-free polyhedra M such that all the disjunctive cuts corresponding to M can be obtained with a finite number of disjunctive cuts corresponding to the polyhedra in L, for every polyhedron P . Our results imply interesting consequences, related to split rank and to integral lattice-free polyhedra, that extend recent research findings.