Laws of Triple Form: 9 of 12

     Chapter 8 : Orderings

          8A. 6 orders on the three forms

Juxtaposition is the minimum operator in the order   []  <  <>  <  {}

There are six minimum operators, one for each ordering of the three forms:

xy                         =    minimum on 1<6<0

{[ [{x}]  [{y}] ]}   =    minimum on 0<1<6

[{ {[x]}  {[y]} }]   =    minimum on 6<0<1

These are the rotated juxtapositions.

[[x][y]]                   =     minimum on 0<6<1

{{x}{y}}                        =     minimum on 6<1<0

<<x><y>>             =     minimum on 1<0<6

These are the reflected juxtapositions.

Call each of these operators  &_ab, where a is the attractor and b is the identity.

xy                                  =     minimum on 1<6<0      =     x  &₁₀  y

{[ [{x}]  [(y)] ]}    =     minimum on 0<1<6      =     x  &₀₆  y

[{ {[x]}  {[y]} }]   =     minimum on 6<0<1      =     x  &₆₁  y

[[x][y]]                  =     minimum on 0<6<1      =     x  &₀₁  y

{{x}{y}}              =     minimum on 6<1<0      =     x  &₆₀  y

<<x><y>>            =     minimum on 1<0<6      =     x  &₁₆  y

They have these laws:

Attractor:             x &_ab a   =   a

          Identity:               x &_ab b   =   x

          Recall:                  x &_ab x   =   x

Commutativity:   x &_ab y   =  y &_ab x

Associativity:       x &_ab (y &_ab z)     =       (x &_ab y) &_ab z

These identities define each operator’s truth table.

x &_ab y           a   a#b   b  _            
a                    a     a     a

a#b                a   a#b   a#b

b                    a   a#b   b

x &_FT y       F       I         T               
F                 F       F       F

I                  F       I         I

T                 F       I         T

x &_TF y       F       I         T    

F                 F       I         T

I                  I         I         T

T                 T       T       T

x &_TI y       F       I         T    

F                 F       F       T

I                  F       I         T

T                 T       T       T

x &_IT y       F       I         T    

F                 F       I         F

I                  I         I         I

T                 F       I         T

x &_IF y        F       I         T    

F                 F       I         T

I                  I         I         I

T                 T       I         T

x &_FI y        F       I         T    

F                 F       F       F

I                  F       I         T

T                 F       T       T

Generalized De Morgan Law:

If p(x) is any permutation of the three forms, then for all x and y;

p( x  &_ab  y  )   =  p(x)  &_p(a),p(b)  p(y) 

Proof is by cases.