  Different commutative semigroups may have a common saturation. We consider
distinguishing semigroups with a common saturation based on their ``sparsity''.
We propose to qualitatively describe sparsity of a semigroup by considering
which faces of the corresponding rational polyhedral cone have saturation
points. For a commutative semigroup we give a necessary and sufficient
condition for determining which faces have saturation points. We also show that
we can construct a commutative semigroup with arbitrary consistent patterns of
faces with saturation points.
