fom - 06 - involutions on the connectivity algebra

[fom]

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.