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).