  For an $n \times n$ nonnegative matrix $P$, an isomorphism is obtained
between the lattice of initial subsets (of ${1,...,n}$) for $P$ and the lattice
of $P$-invariant faces of the nonnegative orthant $\IR^{n}_{+}$. 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 $A$ leaves invariant
a polyhedral cone $K$, then for each distinguished eigenvalue $\lambda$ of $A$
for $K$, there is a chain of $m_\lambda$ distinct $A$-invariant
join-irreducible faces of $K$, each containing in its relative interior a
generalized eigenvector of $A$ corresponding to $\lambda$ (referred to as
semi-distinguished $A$-invariant faces associated with $\lambda$), where
$m_\lambda$ is the maximal order of distinguished generalized eigenvectors of
$A$ corresponding to $\lambda$, but there is no such chain with more than
$m_\lambda$ members. We introduce the important new concepts of
semi-distinguished $A$-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 $n$ that are nonnegative on a
closed interval. Plentiful illustrative examples are provided. Some open
problems are posed at the end.
