Barker proved that the lattice of faces, (K), of a finite dimensional proper cone K is always complemented. The proof, however, contained a gap. In this paper we offer a correct proof of the result. At the same time new characterizations of lattices of finite length which are section complemented or relatively complemented are found. It is also proved that the lattice of exposed faces of a proper cone K is complemented. Finally, we show that the class of proper cones with a relatively complemented face lattice is strictly contained in the class of proper cones with a dual atomistic face lattice. For a few natural related questions counterexamples are given. It turns out that one of our examples also answers in the negative an open problem posed recently by the second author.
Linear Algebra and Its Applications 79(C), pp.195-207