Re: How to extend a patch theory to fully commute

On Thu, 2 Jul 2020 at 07:53, Ben Franksen <[email protected]> wrote:
> Am 02.07.20 um 00:37 schrieb James Cook:
> >> The construction works by inductively taking any pair of incommutable
> >> patches (in sequence, i.e. with a common middle state) and then
> >> "formally" commuting it. The new node is represented as that pair of
> >> patches. Is that, essentially, the idea?
> >
> > Sorry, I should add one note here: I'm not sure it's accurate to say
> > the construction "inductively" makes pairs of incommutable patches
> > commute. That sounds like you're saying everything's built out of
> > transpositions. Maybe there's some equivalent formulation that works
> > that way, but e.g. if you want to commute A;B;C to C;B;A there's
> > nothing in my construction that says it's done one transposition at a
> > time. Instead, you identify each patch in the sequence C;B;A with a
> > "patch address", then simplify that patch address as much as you can.
>
> I see. But if you can prove permutivity (and from skimming through to
> the end of your story it loks like you can) then it should be possible
> to arrive at the same result(s) by doing it one transposition at a time.
>
> That is, we start with A;B;C, then first commute B;C to C';B', then A;C'
> to C'';A'. The result C'';A';B' should be the same as if we do it in one
> stroke and simplify as in the definition.

Yes, once we've defined the extended patch universe, you can achieve
any permutation through transpositions.

I was concerned about the construction of the patch universe itself.
If you start with the primitive patch theory, and then extend the
theory one failed commutation at a time until you have a theory where
all commutations are possible, then it probably ends up being the same
theory but I'm not sure.

James