LolWolf, thank you very much for 'defending' the algebra much better than I could possibly have defended it myself :-)

Just to add: I don't understand why we should compare the algebra and the matrix encoding suggested above. The latter is just one possible implementation (a model) of the former. It's a good, useful model. But the point of having an algebra is that we can abstract of details of particular implementation and instead focus on the laws that all such implementations satisfy. Sometimes these laws can be interesting by themselves (at least they are interesting to me).