# On the Fine Structure of the Projective Line Over GF2 x GF2 x GF2

1 Astronomical Institute of the Slovak Academy of Sciences 2 FEMTO-ST - Franche-Comté Électronique Mécanique, Thermique et Optique - Sciences et Technologies

Abstract : The paper gives a succinct appraisal of the properties of the projective line defined over the direct product ring $R {\triangle} \equiv$ GF2$\otimes$GF2$\otimes$GF2. The ring is remarkable in that except for unity, all the remaining seven elements are zero-divisors, the non-trivial ones forming two distinct sets of three; elementary -slim- and composite -fat-. Due to this fact, the line in question is endowed with a very intricate structure. It contains twenty-seven points, every point has eighteen neighbour points, the neighbourhoods of two distant points share twelve points and those of three pairwise distant points have six points in common - namely those with coordinates having both the entries `fat- zero-divisors. Algebraically, the points of the line can be partitioned into three groups: a the group comprising three distinguished points of the ordinary projective line of order two the -nucleus-, b the group composed of twelve points whose coordinates feature both the unity and a zero-divisor the -inner shell- and c the group of twelve points whose coordinates have both the entries zero-divisors the -outer shell-. The points of the last two groups can further be split into two subgroups of six points each; while in the former case there is a perfect symmetry between the two subsets, in the latter case the subgroups have a different footing, reflecting the existence of the two kinds of a zero-divisor. The structure of the two shells, the way how they are interconnected and their link with the nucleus are all fully revealed and illustrated in terms of the neighbour-distant relation. Possible applications of this finite ring geometry are also mentioned.

Keywords : Finite Product Rings Projective Ring Lines Neighbour-Distant Relation

Author: Metod Saniga - Michel Planat -

Source: https://hal.archives-ouvertes.fr/