  The functionals on an ordered semigroup S in the category Cu--a category to
which the Cuntz semigroup of a C*-algebra naturally belongs--are investigated.
After appending a new axiom to the category Cu, it is shown that the
"realification" S_R of S has the same functionals as S and, moreover, is
recovered functorially from the cone of functionals of S. Furthermore, if S has
a weak Riesz decomposition property, then S_R has refinement and interpolation
properties which imply that the cone of functionals on S is a complete
distributive lattice. These results apply to the Cuntz semigroup of a
C*-algebra. At the level of C*-algebras, the operation of realification is
matched by tensoring with a certain stably projectionless C*-algebra.
