  The structure of cones of positive and k-positive maps acting on a
finite-dimensional Hilbert space is investigated. Special emphasis is given to
their duality relations to the sets of superpositive and k-superpositive maps.
We characterize k-positive and k-superpositive maps with regard to their
properties under taking compositions. A number of results obtained for maps are
also rephrased for the corresponding cones of block positive, k-block positive,
separable and k-separable operators, due to the Jamiolkowski-Choi isomorphism.
Generalizations to a situation where no such simple isomorphism is available
are also made, employing the idea of mapping cones. As a side result to our
discussion, we show that extreme entanglement witnesses, which are optimal,
should be of special interest in entanglement studies.
