  We investigate the relationship between mapping cones and matrix ordered
*-vector spaces (i.e., abstract operator systems). We show that to every
mapping cone there is an associated operator system on the space of n-by-n
complex matrices, and furthermore we show that the associated operator system
is unique and has a certain homogeneity property. Conversely, we show that the
cone of completely positive maps on any operator system with that homogeneity
property is a mapping cone. We also consider several related problems, such as
characterizing cones that are closed under composition on the right by
completely positive maps, and cones that are also semigroups, in terms of
operator systems.
