> My one persnickety objection is that people don't call \\(\textbf{Bool}\\) 'Boolean algebra'. The phrase 'Boolean algebra' means a couple of things:

That's fair... I remember studying something like a combination of 1) and 2) in a Discrete math class a few years ago. But the description of \\(\mathbf{Bool}\\) as a rig seems straight forward. That would be the minimum definition for the matrix multiplication formula to make sense (or at least be well-defined).