Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 4 (2008), 072, 7 pages      arXiv:0807.1790      https://doi.org/10.3842/SIGMA.2008.072

A Jacobson Radical Decomposition of the Fano-Snowflake Configuration

Metod Saniga a and Petr Pracna b
a) Astronomical Institute, Slovak Academy of Sciences, SK-05960 Tatranská Lomnica, Slovak Republic
b) J. Heyrovský Institute of Physical Chemistry, v.v.i., Academy of Sciences of the Czech Republic, Dolejskova 3, CZ-18223 Prague 8, Czech Republic

Received July 14, 2008, in final form October 17, 2008; Published online October 24, 2008

Abstract
The Fano-Snowflake, a specific configuration associated with the smallest ring of ternions Rà (arXiv:0803.4436 and arXiv:0806.3153), admits an interesting partitioning with respect to the Jacobson radical of Rà. The totality of 21 free cyclic submodules generated by non-unimodular vectors of the free left Rà-module Rà3 is shown to split into three disjoint sets of cardinalities 9, 9 and 3 according as the number of Jacobson radical entries in the generating vector is 2, 1 or 0, respectively. The corresponding ''ternion-induced'' factorization of the lines of the Fano plane sitting in the middle of the Fano-Snowflake is found to differ fundamentally from the natural one, i.e., from that with respect to the Jacobson radical of the Galois field of two elements.

Key words: non-unimodular geometry over rings; smallest ring of ternions; Fano plane.

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References

  1. Brehm U., Greferath M., Schmidt S.E., Projective geometry on modular lattices, in Handbook of Incidence Geometry, Editor F. Buekenhout, Elsevier, Amsterdam, 1995, 1115-1142.
  2. Saniga M., Havlicek H., Planat M., Pracna P., Twin "Fano-Snowflakes" over the smallest ring of ternions, SIGMA 4 (2008), 050, 7 pages, arXiv:0803.4436.
  3. Havlicek H., Saniga M., Vectors, cyclic submodules and projective spaces linked with ternions, arXiv:0806.3153.
  4. Veldkamp F.D., Projective planes over rings of stable rang 2, Geom. Dedicata 11 (1981), 285-308.
  5. Veldkamp F.D., Geometry over rings, in Handbook of Incidence Geometry, Editor F. Buekenhout, Elsevier, Amsterdam, 1995, 1033-1084.
  6. Herzer A., Chain geometries, in Handbook of Incidence Geometry, Editor F. Buekenhout, Elsevier, Amsterdam, 1995, 781-842.
  7. Blunck A., Herzer A., Kettengeometrien - Eine Einführung, Shaker Verlag, Aachen, 2005.
  8. Brown E., The many names of (7,3,1), Math. Mag. 75 (2002), 83-94.

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