  Biological systems exhibit processes on a wide range of time and length
scales. This work demonstrates that models, wherein the interaction between
system constituents is captured by algebraic operations, inherently allow for
successive coarse-graining operations through quotients of the algebra.
Thereby, the class of model is retained and all possible coarse-graining
operations are encoded in the lattice of congruences of the model. We analyze a
class of algebraic models generated by the subsequent and simultaneous
catalytic functions of chemicals within a reaction network. Our ansatz yields
coarse-graining operations that cover the network with local functional patches
and delete the information about the environment, and complementary operations
that resolve only the large-scale functional structure of the network. Finally,
we present a geometric interpretation of the algebraic models through an
analogy with classical models on vector fields. We then use the geometric
framework to show how a coarse-graining of the algebraic model naturally leads
to a coarse-graining of the state-space. The framework developed here is aimed
at the study of the functional structure of cellular reaction networks spanning
a wide range of scales.
