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

Given truth tables, it becomes possibleto represent the three collineationsof the 21-point projective planeas involutions on the 16 elementdomain of the connectivity algebra.The representation of truth functionalitybeing used here has three columns.It is a simple matter to recognize thatnegation corresponds to simply exchangingthe symbols in the rightmost rowSo, this truth table transforms   FIX  LET  |  IF   ----------|-----             |    T    T   |  T    T    F   |  T    F    F   |  T    F    T   |  Finto   FIX  LET  |  NIF   ----------|-----             |    T    T   |  F    T    F   |  F    F    F   |  F    F    T   |  TOn the other hand, one rarely realizesthat conjugation is a symbolexchange on all three columnsSo, the truth table   FIX  LET  |  IF   ----------|-----             |    T    T   |  T    T    F   |  T    F    F   |  T    F    T   |  Fbecomes   FIX  LET  | NIMP   ----------|-----             |    F    F   |  F    F    T   |  F    T    T   |  F    T    F   |  TFinally, if the two leftmost columnshave their symbols exchanged, oneobtains contraposition   FIX  LET  |  IF   ----------|-----             |    T    T   |  T    T    F   |  T    F    F   |  T    F    T   |  Ftransforms into   FIX  LET  |  IMP   ----------|-----             |    F    F   |  T    F    T   |  T    T    T   |  T    T    F   |  FThese three operations form a commutative diagram.That is,Negation(Contraposition) = ConjugationandContraposition(Negation) = ConjugationThe truth tables make it a simple matter to seewhy this is the case.  One also hasConjugation(Contraposition) = NegationandContraposition(Conjugation) = Negationas well asConjugation(Negation) = ContrapositionandNegation(Conjugation) = ContrapositionHowever, only the conjugation is a fundamentalinvariant on the connectivity algebra.  Theaxioms are preserved under conjugationFor example,OR (NOR,AND) = LEQtransforms under conjugation intoAND (NAND,OR) = XORwhich is still an axiom.
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