From Osher Doctorow
David Kruml of Czech Republic (Czech Republic Research Grant No.
201/06/0664), arXiv: 0801.0504 v1 [math.CT] 3 Jan 2008, 12 pages,
"Quantum Triads: an algebraic approach," explores and proves some
arguably fascinating relationships between the numbers 0 and 1 and
Quantum Logic, which although focused on different ideas, agrees with
the emphasis of Probable Causation/Influence (PI) on 0 and 1.
The Czech Republic, by the way, as well as Poland, are the two nations
of the former USSR's sphere of influence which have not only shown
most independence from modern Russia, but have contributed most to
Quantum Logic together with Austria, Switzerland, USA, and the U.K.,
although Hungary under the USSR (but not now) was the nation that gave
us John von Neumann who arguably contributed to the algebraic school
of Quantum Logic very highly.
Kruml's article uses Quantales, Quantum Triads, Sup-Lattices,
Bimorphisms, Rings, etc., concepts that at least for the first 3 may
be unfamiliar to most Readers.
If S is a sup-lattice (partially ordered by the supremum operation or
function), an example is the definition of a product:
1) xy = 0 if y < = x
1 if y is not < = x
for all x, y that are elements of S.
This and its machinery is generalized to a quantum frame with two
types of actions:
2) xz = cz - x ^ z (RT, TL)
3) xy = x o y (LR)
for x, y in frame F and z in the closure of T, where T refers to two-
sided ideals, R to right ideals, L to left ideals. In the simple two-
point case, say points P and Q, L and R can be thought of respectively
as right and left arrows between P and Q, T as a circular arrow
leading frm Q back to Q, and RT, TL, LR as successive applications of
the arrows in the indicated orders.
With the above facts and slow, careful reading, Readers should be able
to read the rest of the paper if they have minimal ability to
discriminate (know the difference between two things or ideas) and
integrate (know the similarity between two things or ideas) and
patience. I don't recommend it for one-line or one-word addicts of
the troll/graffiti artist types.
Osher Doctorow
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