small point but if we define the matrices to have the boolean \\(\Phi(x, y)\\) at position row \\(y\\) and column \\(x\\) (ie the transpose of the above) then our matrix multiplication rule matches up nicely with composition, ie we get \\(\text{M}(\Psi\Phi) = \text{M}(\Psi)\text{M}(\Phi)\\).