Math Forum Discussions

fom

Posts: 1,098
Registered: 12/4/12

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

Plain Text

Given that the 20-element 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 four-atom Boolean block
has an independent interpretation
that is isomorphic to the free
Boolean lattice on two generators
associated with truth-functional
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 four-atom
Boolean block with the elements
of the 16-element DeMorgan lattice.

Self-conjugate 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)