  A theory is developed which uses "networks" (directed acyclic graphs with
some extra structure) as a formalism for expressions in multilinear algebra. It
is shown that this formalism is valid for arbitrary PROPs (short for 'PROducts
and Permutations category'), and conversely that the PROP axioms are implicit
in the concept of evaluating a network. Ordinary terms and operads constitute
the special case that the graph underlying the network is a rooted tree.
  Furthermore a rewriting theory for networks is developed. Included in this is
a subexpression concept for which is given both algebraic and effective
graph-theoretical characterisations, a construction of reduction maps from
rewriting systems, and an analysis of the obstructions to confluence that can
occur. Several Diamond Lemmas for this rewriting theory are given.
  In addition there is much supporting material on various related subjects. In
particular there is a "toolbox" for the construction of custom orders on the
free PROP, so that an order can be tailored to suit a specific rewriting
system. Other subjects treated are the abstract index notation in a general
PROP context and the use of feedbacks (sometimes called traces) in PROPs.
