Networks with asymmetric inputs: lattice of synchrony subspaces

We consider coupled cell networks with asymmetric inputs and study their lattice of synchrony subspaces. For the particular case of one-input regular coupled cell networks we describe the join-irreducible synchrony subspaces for their lattice of synchrony subspaces, first in terms of the eigenvectors and generalized eigenvectors that generate them, and then by giving a characterization of the possible patterns of the associated balanced colourings. The set of the join-irreducible synchrony subspaces is join-dense for the lattice, that is, the lattice can be obtained by sums of those join-irreducible elements (Aguiar et al 2011 J. Nonlinear Sci. 21 271-323), and we draw conclusions about the possible patterns of balanced colourings associated to the synchrony subspaces in the lattice. We also consider the disjoint union of two regular coupled cell networks with the same cell-type and the same edge-type. We show how to obtain the lattice of synchrony subspaces for the network union from the lattice of synchrony subspaces for the component networks. The lattice of synchrony subspaces for a homogeneous coupled cell network is given by the intersection of the lattice of synchrony subspaces for its identical-edge subnetworks for each edge-type (Aguiar and Dias 2014 J. Nonlinear Sci. 24 949-96). This, together with the results in this paper on the lattice of synchrony subspaces for one-input regular networks and on the lattice of synchrony subspaces for the disjoint union of networks, define a procedure to obtain the lattice of synchrony subspaces for homogeneous coupled cell networks with asymmetric inputs.