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

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)