[jgilias]

But that's basically what the conclusion is, isn't it? The author set out with 'obviously the monoid must be commutative', fuzzed it and came to the second point you mention. This all may be very obvious to some, but I found it an interesting read that reminds that one should never fail to question one's assumptions and understanding.

[dan-robertson]

I think that confusion stems from thinking about an operation that ought to be defined on sets, mapreduce, as one defined on lists. Your monoid obviously needs to be commutative for the former and not the latter. And the implementations are pretty similar in serial but quite different in performance in parallel.