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 <A;@B| is a left-conflictor, which is why it points to the left. The “A” part of this is the effect of this conflictor. The right-conflictor is symmetric with the left-conflictor. Together, left and right conflictors are referred to as \emph{ordinary conflictors}. There are a couple more fields in an ordinary conflictor, but I won’t discuss them quite yet. To the conflictor experts they’ll be easy to guess.

The third sort of conflictor is the “middle-conflict”, which is written as |(A);@B;(C)|. Its identity is B, and conflicts with patches on both the right and the left. Its effect is B^.

Two non-commuting patches plus inverse

A        A^       B           %   a  ia   b

<–>AA^ <A;@A^| |@A;A^> B % ia1 a1 b <–>AB <A;@A^| <(A);A^@B| |@A;A^B> % ia1 b1 a2 <–>A^B B <B^A;@A^| |@A;A^B> % b2 ia2 a2 <–>A^A B <B^;@A| |@A^;B> % b2 a3 ia3 <–>BA A <A^;@B| |@A^;B> % a b3 ia3

Permutation of three noncommuting patches

A         B            C

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

Permutation of two commuting patches with one non-commuter

A          B            C      % a  b  c

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