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fom
Posts:
1,969
Registered:
12/4/12


Re: fom  10  CORRECTED  a fundamental demorgan algebra
Posted:
Dec 11, 2012 9:33 PM


Given that the 20element ortholattice has been constructed from the lines of our affine geometry, the 16 functions of our original connectivity algebra form the extensional basis of the construction.
However, the namespace of the ortholattice has been obtained in such a way that
NTRU has been replaced with NOT
The next construction embeds a DeMorgan algebra into the lattice so that the fouratom Boolean block has an independent interpretation that is isomorphic to the free Boolean lattice on two generators associated with truthfunctional logic.
Without negation, it is difficult to convey the four forms
Ax, Ax, Ex, Ex
The namespace is formulated so that
ALL corresponds with Ax NO corresponds with Ax SOME corresponds with Ex OTHER corresponds with Ex
We fix the relationships of these names, relative to the use of negation, by taking ALL and SOME as fixed and NO and OTHER as conjugate.
This choice reflects the fixing of an object type from an arbitrary domain on the basis of negative properties presumed to partition the arbitrary domain rather than positive properties that might be impredicative.
This DeMorgan transformation is made precise by the subdirectly irreducible DeMorgan algebra on four elements whose involution is given by
ALL > ALL NO > OTHER OTHER > NO SOME > SOME
It helps to visualize this as a lattice, with the exchanging elements positioned as if reflecting through a line.
OTHER
/ \ / \ / \ / \
SOME ALL
\ / \ / \ / \ /
NO
The product of this algebra with itself has sixteen elements.
We now correllate those line names used for the fouratom Boolean block with the elements of the 16element DeMorgan lattice.
Selfconjugate pairs:
FIX > (SOME,ALL)
FLIP > (ALL,SOME)
LET > (ALL,ALL)
DENY > (SOME,SOME)
Conjugate pairs:
NOR > (OTHER,SOME) NAND > (NO,SOME)
AND > (OTHER,ALL) OR > (NO,ALL)
NIF > (ALL,OTHER) IMP > (ALL,NO)
NIMP > (SOME,OTHER) IF > (SOME,NO)
LEQ > (OTHER, NO) XOR > (NO, OTHER)
TRU > (NO,NO) NOT > (OTHER,OTHER)


Date

Subject

Author

12/8/12


fom

12/11/12


fom


