For an nonnegative matrix , an isomorphism is obtained between the lattice of initial subsets (of ) for and the lattice of -invariant faces of the nonnegative orthant . Motivated by this isomorphism, we generalize some of the known combinatorial spectral results on a nonnegative matrix that are given in terms of its classes to results for a cone-preserving map on a polyhedral cone, formulated in terms of its invariant faces. In particular, we obtain the following extension of the famous Rothblum index theorem for a nonnegative matrix: If leaves invariant a polyhedral cone , then for each distinguished eigenvalue of for , there is a chain of distinct -invariant join-irreducible faces of , each containing in its relative interior a generalized eigenvector of corresponding to (referred to as semi-distinguished -invariant faces associated with ), where is the maximal order of distinguished generalized eigenvectors of corresponding to , but there is no such chain with more than members. We introduce the important new concepts of semi-distinguished -invariant faces, and of spectral pairs of faces associated with a cone-preserving map, and obtain several properties of a cone-preserving map that mostly involve these two concepts, when the underlying cone is polyhedral, perfect, or strictly convex and/or smooth, or is the cone of all real polynomials of degree not exceeding that are nonnegative on a closed interval. Plentiful illustrative examples are provided. Some open problems are posed at the end.
