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 21point 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.