  The concept of diagrammatic combinatorial Hopf algebras in the form
introduced for describing the Heisenberg-Weyl algebra
in~\cite{blasiak2010combinatorial} is extended to the case of so-called rule
diagrams that present graph rewriting rules and their composites. The resulting
rule diagram algebra may then be suitably restricted in four different ways to
what we call the rule algebras, which are non-commutative, unital associative
algebras that implement the algebra of compositions of graph rewriting rules.
Notably, our framework reveals that there exist two more types of graph
rewriting systems than previously known in the literature, and we present an
analysis of the structure of the rule algebras as well as a form of
Poincar\'{e}-Birkhoff-Witt theorem for the rule diagram algebra. Our work lays
the foundation for a fundamentally new way of analyzing graph transformation
systems, and embeds this very important concept from theoretical computer
science firmly into the realm of mathematical combinatorics and statistical
physics.
