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Jun 21, 2008, 3:18:19 AM6/21/08

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From Osher Doctorow

The Jacobson Radical, which I recently discussed in this thread as

related to Probable Causation/Influence (PI), as well as my earlier

discussions, turns out to be expressible in terms of ternions in the

paper by Hans Havlicek (Technische U. Wien) and Metod Saniga

(Astronomical Institute Slovak Republic), "Vectors, cyclic submodules

and projective spaces linked with ternions," arXiv: 0806.3153 v1 [math-

ph] 19 Jun 2008, 10 pages. Saniga has 36 papers in arXiv.

Ternions are upper triangular 2 x 2 matrices over a commutative field

F, with usual matrix + and matrix multiplication and multiplicative

and additive identities respectively I and 0 matrices.

Readers will notice the number 1, both in the forms -1 and +1, and the

matrices I and 0, very frequently in Havlicek and Saniga's paper, as

well as often q^n, q^2n, q^(n+1), q^(2n + 2) for q the number of

elements of a Galois field (when F is a Galois field). The numbers 5

and 6 (6 isn't in the PI Fundamental Set, though) also occur in

respectively one important case (5 for the number 5 + /F/ of orbits).

Projective spaces over rings and projective lines in particular are

related to Quantum physics and in particular Quantum information

theory, and applications of projective geometries over the ternions

are expected to be fruitful.

They get formulas yielding a combinatorial approach to lines and

points of PG(n, q) for n > = 2 in terms of non-unimodular free

submodules of R^(n+1) where R is the ring of ternions, and also in

terms of vectors, and achieve some important classification results.

Some especially useful matrices among ternions turn out to have the

first column or the second row composed of all zeros, and the Jacobson

Radical of R turns out to equal a set of matrices with 3 out of 4

elements 0 (0 elements on the diagonal and one other).

Osher Doctorow

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