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