A square matrix on the Booleans is indeed also equivalent to a directed graph, as explained. It is also equivalent to a Chu space [0], specifically a sort of unitary Chu transformation.
We can specialize slightly. Let our square matrices be upper triangular; that is, let the lower triangle not including the diagonal be all 0/false. Then the corresponding graph is a DAG. (Apologies if this was in the article; I saw hints of it, but never the direct statement!) The corresponding Chu space has poset structure. These three facts are all related by fourth fact that the matrix's rows are labels for a topological sorting of the DAG, and the fifth fact that DAGs are equivalent to posets; we obtain a sixth fact that posets are equivalent to upper triangular Boolean matrices.