Date: Dec 7, 2012 4:32 AM
Author: fom
Subject: fom - 06 - involutions on the connectivity algebra


Given truth tables, it becomes possible
to represent the three collineations
of the 21-point projective plane
as involutions on the 16 element
domain of the connectivity algebra.


The representation of truth functionality
being used here has three columns.


It is a simple matter to recognize that
negation corresponds to simply exchanging
the symbols in the rightmost row


So, this truth table transforms


   FIX  LET  |  IF
   ----------|-----
             |
    T    T   |  T
    T    F   |  T
    F    F   |  T
    F    T   |  F


into


   FIX  LET  |  NIF
   ----------|-----
             |
    T    T   |  F
    T    F   |  F
    F    F   |  F
    F    T   |  T



On the other hand, one rarely realizes
that conjugation is a symbol
exchange on all three columns

So, the truth table



   FIX  LET  |  IF
   ----------|-----
             |
    T    T   |  T
    T    F   |  T
    F    F   |  T
    F    T   |  F


becomes


   FIX  LET  | NIMP
   ----------|-----
             |
    F    F   |  F
    F    T   |  F
    T    T   |  F
    T    F   |  T



Finally, if the two leftmost columns
have their symbols exchanged, one
obtains contraposition


   FIX  LET  |  IF
   ----------|-----
             |
    T    T   |  T
    T    F   |  T
    F    F   |  T
    F    T   |  F


transforms into


   FIX  LET  |  IMP
   ----------|-----
             |
    F    F   |  T
    F    T   |  T
    T    T   |  T
    T    F   |  F



These three operations form a commutative diagram.
That is,



Negation(Contraposition) = Conjugation

and

Contraposition(Negation) = Conjugation


The truth tables make it a simple matter to see
why this is the case.  One also has



Conjugation(Contraposition) = Negation

and

Contraposition(Conjugation) = Negation


as well as



Conjugation(Negation) = Contraposition

and

Negation(Conjugation) = Contraposition



However, only the conjugation is a fundamental
invariant on the connectivity algebra.  The
axioms are preserved under conjugation


For example,


OR (NOR,AND) = LEQ


transforms under conjugation into


AND (NAND,OR) = XOR


which is still an axiom.