  A synchrony subspace of R^n is defined by setting certain components of the
vectors equal according to an equivalence relation. Synchrony subspaces
invariant under a given set of square matrices form a lattice. Applications of
these invariant synchrony subspaces include equitable and almost equitable
partitions of the vertices of a graph used in many areas of graph theory,
balanced and exo-balanced partitions of coupled cell networks, and coset
partitions of Cayley graphs. We study the basic properties of invariant
synchrony subspaces and provide many examples of the applications. We also
present what we call the split and cir algorithm for finding the lattice of
invariant synchrony subspaces. Our theory and algorithm is further generalized
for non-square matrices. This leads to the notion of tactical decompositions
studied for its application in design theory.
