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Topic: fom - 06 - involutions on the connectivity algebra
Replies: 0

 fom Posts: 1,037 Registered: 12/4/12
fom - 06 - involutions on the connectivity algebra
Posted: Dec 7, 2012 4:32 AM

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.