  A partitioned process theory, as defined by Coecke, Fritz, and Spekkens, is a
symmetric monoidal category together with an all-object-including symmetric
monoidal subcategory. We think of the morphisms of this category as processes,
and the morphisms of the subcategory as those processes that are freely
executable. Via a construction we refer to as parallel-combinable processes
with discarding, we obtain from this data a partially ordered monoid on the set
of processes, with f > g if one can use the free processes to construct g from
f. The structure of this partial order can then be probed using additive
monotones: order-preserving monoid homomorphisms with values in the real
numbers under addition. We first characterise these additive monotones in terms
of the corresponding partitioned process theory.
  Given enough monotones, we might hope to be able to reconstruct the order on
the monoid. If so, we say that we have a complete family of monotones. In
general, however, when we require our monotones to be additive monotones, such
families do not exist or are hard to compute. We show the existence of complete
families of additive monotones for various partitioned process theories based
on the category of finite sets, in order to shed light on the way such families
can be constructed.
