A Classification of the Projective Lines over Small RingsReport as inadecuate




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1 Astronomical Institute of the Slovak Academy of Sciences 2 FEMTO-ST - Franche-Comté Électronique Mécanique, Thermique et Optique - Sciences et Technologies 3 IPNL - Institut de Physique Nucléaire de Lyon 4 JH-INST - J. Heyrovsky Institute of Physical Chemistry

Abstract : A compact classification of the projective lines defined over commutative rings with unity of all orders up to thirty-one is given. There are altogether sixty-five different types of them. For each type we introduce the total number of points on the line, the number of points represented by coordinates with at least one entry being a unit, the cardinality of the neighbourhood of a generic point of the line as well as those of the intersections between the neighbourhoods of two and three mutually distant points, the number of `Jacobson- points per a neighbourhood, the maximum number of pairwise distant points and, finally, a list of representative-base rings. The classification is presented in form of a table in order to see readily not only the fine traits of the hierarchy, but also the changes in the structure of the lines as one goes from one type to the other. We hope this study will serve as an impetus to a search for possible applications of these remarkable geometries in physics, chemistry, biology and other natural sciences as well.

Keywords : Projective Ring Lines Rings of Order of a Power of Two





Author: Metod Saniga - Michel Planat - Maurice Kibler - Petr Pracna -

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



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