  Dynamical systems with a coupled cell network structure can display
synchronous solutions, spectral degeneracies and anomalous bifurcation
behavior. We explain these phenomena here for homogeneous networks, by showing
that every homogeneous network dynamical system admits a semigroup of hidden
symmetries. The synchronous solutions lie in the symmetry spaces of this
semigroup and the spectral degeneracies of the network are determined by its
indecomposable representations. Under a condition on the semigroup
representation, we prove that a one-parameter synchrony breaking steady state
bifurcation in a coupled cell network must generically occur along an
absolutely indecomposable subrepresentation. We conclude with a classification
of generic one-parameter bifurcations in monoid networks with two or three
cells.
