  This paper presents a framework based on matrices of monoids for the study of
coupled cell networks. We formally prove within the proposed framework, that
the set of results about invariant synchrony patterns for unweighted networks
also holds for the weighted case. Moreover, the approach described allows us to
reason about any multiedge and multiedge-type network as if it was single edge
and single-edge-type. Several examples illustrate the concepts described.
Additionally, an improvement of the coarsest invariant refinement algorithm to
find balanced partitions is presented that exhibits a worst-case complexity of
$ \mathbf{O}(\vert\mathcal{C}\vert^3) $, where $\mathcal{C}$ denotes the set of
cells.
