Friday, July 13, 2018

Laws of Triple Form: 5 of 12

     Chapter 4 : Inner Order
          Inner Order

          4A. Majority

Let M(x,y,z) denote the form [[xy][yz][zx]].
In Kleenean terms:
M(x,y,z)      =       (x & y) V (y & z) V (z & x)
M(x,y,z)      =       (x V y) & (y V z) & (z V x)

Exercise for the student:
Derive these theorems:

Majority:            [[xy][yz][zx]]        =       [[x][y]] [[y][z]] [[z][x]]
Symmetry: M(x,y,z) =  M(z,y,x)  =   M(x,z,y)  =   other permutations
Coalition:            M(x, x, y)             =      x
Transmission:       [ M(x,y,z) ]          =       M([x],[y],[z])
Distribution:        x M(a,b,c)             =       M(xa, b, xc)
Redistribution:  [[x][M(a,b,c)]]        =       M([[x][a]], b, [[x][c]])
Collection:                     M(x,y,z))    =       [[x][y]] [ [xy] [z] ]
General Distribution:  M(x,y,M(a,b,c))   =    M(M(x,y,a),b,M(x,y,c))
General Associativity:  M(x,a,M(y,a,z))   =   M(M(x,a,y), a, z)
Mediocrity:          M(0,6,1)               =       6

Therefore these operators:
M(x,[[]],y)            =       [[x][y]]  
M(x,[],y)     =       xy  
M(x,6,y)     =       [[x6][y6][xy]] 
have these properties: recall; commutativity; associativity; attractors ( [[]], [] and 6 respectively);  and mutual distribution.

          4B. Min

Define the operator min this way:

x min y  =  M(x,6,y)

x min y   F   I   T
F         F   I   I
I         I   I   I
T         I   I   T

It has these laws:

Recall:                                     x min x       =       x
Commutativity:                      x min y       =       y min x
Attractor:                                x min 6       =       6
Opposition:                                      x min (~ x)  =       6
Associativity:                 x min (y min z)     =       (x min y) min z
Transmission:                  not(x min y)      =       (not x) min (not y)
Distribution:                 x min (y & z)        =       (x min y) & (x min z)
                                      x min (y V z)        =       (x min y) V (x min z)
                                      x & (y min z)        =       (x & y) min (x & z)
                                      x V (y min z)        =       (x V y) min (x V z)

4C. Inner order

Now let’s define “inner order”:
x < y        iff         x min y  =  x  


          Exercises for the student:
          Prove these Theorems:

Theorem:   min is the minimum operator for < ;
(X min Y) < X ;      (X min Y) < Y ;
and    Z < (X min Y) ,    if  Z<X and Z<Y

Theorem: < is transitive and antisymmetric:
a < b    and    b < c       implies          a < c
a < b    and    b < a       implies         a = b

Theorem: < is preserved by disjunction and conjunction:
a < b     implies    a V c   <   b V c
  and           a & c   <   b & c

Theorem: < is preserved by negation:
a < b     implies   ~a  <  ~b .

Theorem: < is preserved by any Kleenean function:
a < b     implies    F(a)  <  F(b)

Theorem: For any Kleenean f;
f(x min y)  <  f(x) min f(y)

Extend < to ordered form vectors:
x   =    ( x1, x2, x3, ... , xn )
x  <  y   if and only if   ( xi < yi ) for all i

Theorem:  < has “limited chains”, with limit N.
That is, if x n is an ordered chain of finite form vectors;
And x1 < x2 < x3 < ...
and if N is the dimension of these vectors,
then for all n > N,  x n  =   x N .

Proof: Any given component of the x’s can move at most one step left before ending up at I, or at most one step right before ending up at T or F; then that component stops moving. For N components, this implies at most N steps in an ordered chain before it stops moving.

Given any Kleenean function f(x), define
a right seed for f is any vector a such that    a < f(a).
a fixedpoint for f is any vector a such that    a  = f(a).
A fixedpoint is also a right seed.

Right seeds generate fixedpoints, thus:
If a is a right seed for f, then a < f(a). Since f is Kleenean, it preserves inner order; so f (a) <  f2(a); and f 2(a) < f 3(a); and so on:
a  <  f(a) < f 2(a) <  f 3(a) < f 4(a)  < ...
This ascending sequence must reach its upper bound within n steps, if n is the number of components of f. Therefore f n(a) is a fixedpoint for f:
f(f n(a))  =   f n(a)
Right seeds grow rightwards towards fixedpoints.
If b is a fixedpoint right of a, then it’s also right of fn(b) for all n, including the fixedpoint. Therefore fn(a) is the leftmost fixedpoint right of a.

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