# Theory/Permutivity

This is a page for discussing and recording various permutivity scenarios. If you aren’t familiar with the new conflictor work (and most people aren’t), you probably won’t benefit much from reading this page. You may also wish to get the source code and latex versions of some of this work at

`darcs get http://darcs.net/darcs-patch-theory`

In particular, this will let you see how conflictors really ought to look.

My latest idea for conflictor notation consists of three conflictor forms, a “left-conflict” a “right-conflict” and a “middle-conflict”. An ordinary conflicting commute of two primitive patches leads to a left and right conflict:

`AB <-> <A;@B| |@A;B>`

The “@B” portion of the conflictor indicates the identity of the patch—A is now on the right, and B on the left. The conflictor <–>AA^ B % ia1 a1 b <–>AB |@A;A^B> % ia1 b1 a2 <–>A^B B % b2 ia2 a2 > <–>A^A B % b2 a3 ia3 <–>BA A |@A^;B> % a b3 ia3

# Permutation of three noncommuting patches

`A B C`

<–>AB C <–>AC <–>BC <–>BA |@B;C> <–>CA A

# Permutation of two commuting patches with one non-commuter

`A B C % a b c`

<–>AB B’ A’ C % b’ a’ c <–>AC B’ % b’ c1 a2 <–>BC |@A’;C> % c2 b2 a2 <–>BA |@B;C> % c2 a3 b3 <–>CA A % a c3 b3

# Three parallel conflicting patches

Here we have `A \/ B \/ C`

. Note that the symmetry between `B`

and `C`

leads to a sort of “top-bottom” symmetry in the permutations. And even the symmetry between `A^`

and `C`

is apparent in the middle scenario.

`A^ B <B^;@C|`

<–>AB AC <–>BC <–>BA CA A^ C <C^;@B|

# Permutation of four noncommuting patches

This is just a sketch, feel free to add in more. I’m using the notation {a[BC]} to mean “the patch A, which has effect BC”. Feel free to adjust the notation.

```
a b c d = a[A] b[B] c[C] d[D]
b1 a1 c d = b[A] a[B] c[C] d[D]
b1 c1 a2 d = b[A] c[] a[BC] d[D]
c2 b2 a2 d = c[AB] b[B^] a[BC] d[D]
c2 a3 b3 d = c[AB] a[] b[C] d[D]
a c3 b3 d = a[A] c[B] b[C] d[D]
(so far it's just the three-way commute, and we can copy stuff over from above)
a b d4 c4 = a[A] b[B] d[C] c[D]
a d5 b4 c4
a d5 c5 b5
a c3 d3 b5
a c3 b3 d
(again, it's trivial stuff)
d6 a4 b4 c4 = d[ABC] a[] b[] c[D]
d6 b6 a5 c4 = d[ABC] b[?] a[?] c[D]
d6 b6 c6 a6 = d[ABC] b[C^] c[B^] a[BCD] ?
d6 c7 b7 a6 = d[ABC] c[C^] b[B^] a[BCD] ??
b1 a1 d4 c4 = b[A] a[B] d[C] c[D]
b1 a1 c d = b[A] a[B] c[C] c[D]
b1 d7 a7 c4 = b[A] d[B]? a[C]? c[D] ???
(I haven't covered all the permutations, but that's enough for now)
```

Arjan’s table:

`|~(G) ~(G)H ~(G)hI ~(G)HiJ ~(G)hIjK ~(G)HiJkL ~(G)hIjKlM`

———–+——————————————————- (G)~ | G H - I - J - F(G)~ | F G^ H^ - I^ - J^ ~ Ef(G)~ | - F^ G H - I -

- DeF(G)~ | E - F G^ H^ - I^
- CdEf(G)~ | - E^ - F^ G H -
BcDeF(G)~ | D - E - F G^ H^

AbCdEf(G)~ | - D^ - E^ - F^ G