Laws of Triple Form: 5 of 12

     Chapter 4 : Inner Order

  

          Majority

          Min

          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  

t

<  

i       

<  

f

          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, ... , x_n )

x  <  y   if and only if   ( x_i < y_i ) 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.