Semantics
Premonoidal structure (together with an interpretation of base types
and constructors) is exactly what is required to give a semantics
of process terms. For example:
- V
C
StateZ(C),
where StateZ(C)[X,Y] is
C[X 
Z,Y Z].
- V
VT
VU,
where T is any commutative monad satisfying the monic condition,
U is any monad satisfying the monic condition, and there
is an appropriate natural transform from T to U.
- Term(
) Term(,
) Term(,,
).
- Graph() Graph(,) Graph(,,).
Proposition
Graph(,,) is the
initial strict symmetric premonoidal category over
over
with Graph(,) as its centre.