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A finite factored set is "just" a set  with a specific choice of decomposition as a product of sets . I'm not sure what definition of phase space you're using, but for a sufficiently general definition of dynamical system (e.g. I don't think that the phase space necessarily has coordinates in this way. The position / momentum phase space example is a special case, where your phase space happens to look like a product of copies of the real numbers, which is then getting back to the "factored" part of a finite factored set. I'm not convinced that there's a deep connection here, though am not very familiar with either concept so could easily be missing something here!

Thanks, that makes sense! Could you say a little about why the weak union axiom holds? I've been struggling to prove that from your definitions. I was hoping that  would hold, but I don't think that  satisfies the second condition in the definition of conditional history for .

I'm confused by the definition of conditional history, because it doesn't seem to be a generalisation of history. I would expect , but both of the conditions in the definition of  are vacuously true if . This is independent of what  is, so . Am I missing something?