Am 02.07.20 um 00:29 schrieb James Cook: > Definition: A /patch address/ is a patch sequence address where the > sequence (nj) has length one. > > Sorry, there are a lot of definitions, but maybe you missed that one?
Oops. Yes, sorry. I was jumping around, trying to get a grip on the idea. >> Indeed, given a minimal patch sequence address in its general form (a, >> b, (Qi), (nj), X, Y), is this now a new *singular* patch in the extended >> theory, regardless of how long the sequence Qi is? I don't see how how >> this can work. Isn't that universe much larger than what was intended? > > I intended only to include patch addresses, i.e. where (nj) has length > one. On the other hand, (Qi) is not restricted in length. Okay, thanks. >> I think calculating this for a small but non-trivial example would be >> highly instructive. Say we have primitive patches A;B;C, where neither >> A;B nor B;C commute. What does the extended universe look like here? > > I think that primitive patch theory has four contexts. Call them a, b, > c, d: a A b B c C d. > > The new patch theory should have 2^3=8 contexts. Here are the > remaining four. (I'm using patches and their names interchangeably.) > > (1) After just applying B: (a, c, [A, B], {B}, {A}). (This is the > simple case of switching two patches.) > > (2) After just applying C: (a, d, [A, B, C], {C}, {A, B}). (We can't > get C without A and B, so we need everything here.) > > (3) After applying A and C (missing B): (b, d, [B, C], {C}, {B}). > (This is another example of the simple case of switching two patches.) > > (4) After applying B and C: (a, d, [A, B, C], {B, C}, {A}). (This > situation is similar to (2), but with everything reversed.) > > The new patch theory should have 12 patches (a cube has 12 edges). > Three are primitive, so there are 9 left to add. > > (a) B in the sequence B;A;C. (In other words, commute A and B and look > at B.) This patch's starting context is a, its ending context is (1) > above, and as a canonical patch address, it is represented as: (a, c, > [A, B], [B], {}, {A}). The first three elements of the tuple tell us > this patch lives somewhere on the square of edges between a and c, and > the last three elements locate the patch within the square: the patch > is named B; nothing comes before the patch; and A comes after the > patch. > > (b) C in the sequence C;A;B (or C;B;A) is (a, d, [A, B, C], [C], {}, {A, B}). > > (c) A in the sequence B;A;C is (a, c, [A, B], {B}, {}) > > (d) A in the sequence C;A;B is (a, d, [A, B, C], {C}, {B}). > > (e) B in the sequence C;B;A is (a, d, [A, B, C], [B], {C}, {A}). > > (f) C in the sequence A;C;B is (b, d, [B, C], {}, {B}). > > (g) C in the sequence B;C;A is (a, d, [A, B, C], {B}, {A}). > > (h) B in the sequence A,C;B is (b, d, [B, C], [B], {C}, {}). > > (i) A in the sequence B;C;A (or C;B;A) is (a, d, [A, B, C], [A], {B, C}, {}). Thanks for writing this down. I think you should add this example to your text as it is a good illustration. I cannot comment yet on the rest because I still haven't read those later sections. Cheers Ben _______________________________________________ darcs-users mailing list darcs-users@osuosl.org https://lists.osuosl.org/mailman/listinfo/darcs-users