Laws of Triple Form: 8 of 12

     Chapter 7 : Pivot

        7. Pivot definition and laws

We can derive S₃ from the “pivot” operator. Define the pivot x#y thus:

x#y = z  
if and only if
x=y=z=x
or
x ≠ y ≠ z ≠ x.

So x#y is ‘the same or the third”. Pivot is unique to three-valued systems. Here is its table:

x#y  y   {}  <>  []
x          
{}       {}  []  <>
<>       []  <>  {}
[]       <> {} []

x#y y    0   6   1
x          
0        0   1   6
6        1   6   0
1        6   0   1

{x}    =   {} # x   =    0 # x
<x>   =   [] # x     =    1# x
[x]     =  <> # x    =    6 # x

Pivot has these laws:

Recall:                 x#x  = x

Commutativity:    x#y = y#x

Cancellation:       x#(x#y)  =  y

4-associativity:     (x#y)#(z#w)  =  (x#z)#(y#w)

Self-distribution:  x#(y#z)  =  (x#y)#(x#z)

Therefore “Permutation distribution”:

p(x#y)  =  p(x)#p(y)   for any permutation p

          This is because any permutation is a composition of pivots, and pivot distributes over itself.

Theorem:  Average Mod Three

x#y    =       (x+y) / 2      mod 3

x#y    =       - (x+y)         mod 3 

for any matching of the three forms to Z mod 3.