399 research outputs found

    A Classification of the Projective Lines over Small Rings

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    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.Comment: 7 pages, 1 figure; Version 2: classification extended up to order 20, references updated; Version 3: classification extended up to order 31, two more references added; Version 4: references updated, minor correctio

    Veldkamp-Space Aspects of a Sequence of Nested Binary Segre Varieties

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    Let S(N)≡PG(1, 2)×PG(1, 2)×⋯×PG(1, 2)S_{(N)} \equiv PG(1,\,2) \times PG(1,\,2) \times \cdots \times PG(1,\,2) be a Segre variety that is NN-fold direct product of projective lines of size three. Given two geometric hyperplanes H′H' and H′′H'' of S(N)S_{(N)}, let us call the triple {H′,H′′,H′ΔH′′‾}\{H', H'', \overline{H' \Delta H''}\} the Veldkamp line of S(N)S_{(N)}. We shall demonstrate, for the sequence 2≤N≤42 \leq N \leq 4, that the properties of geometric hyperplanes of S(N)S_{(N)} are fully encoded in the properties of Veldkamp {\it lines} of S(N−1)S_{(N-1)}. Using this property, a complete classification of all types of geometric hyperplanes of S(4)S_{(4)} is provided. Employing the fact that, for 2≤N≤42 \leq N \leq 4, the (ordinary part of) Veldkamp space of S(N)S_{(N)} is PG(2N−1,2)PG(2^N-1,2), we shall further describe which types of geometric hyperplanes of S(N)S_{(N)} lie on a certain hyperbolic quadric Q0+(2N−1,2)⊂PG(2N−1,2)\mathcal{Q}_0^+(2^N-1,2) \subset PG(2^N-1,2) that contains the S(N)S_{(N)} and is invariant under its stabilizer group; in the N=4N=4 case we shall also single out those of them that correspond, via the Lagrangian Grassmannian of type LG(4,8)LG(4,8), to the set of 2295 maximal subspaces of the symplectic polar space W(7,2)\mathcal{W}(7,2).Comment: 16 pages, 8 figures and 7 table

    A Classification of the Projective Lines over Small Rings II. Non-Commutative Case

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    A list of different types of a projective line over non-commutative rings with unity of order up to thirty-one inclusive is given. Eight different types of such a line are found. With a single exception, the basic characteristics of the lines are identical to those of their commutative counterparts. The exceptional projective line is that defined over the non-commutative ring of order sixteen that features ten zero-divisors and it most pronouncedly differs from its commutative sibling in the number of shared points by the neighbourhoods of three pairwise distant points (three versus zero), that of "Jacobson" points (zero versus five) and in the maximum number of mutually distant points (five versus three).Comment: 2 pages, 1 tabl

    Wadge Degrees and Pointclasses

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    Logical Dreams

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    We discuss the past and future of set theory, axiom systems and independence results. We deal in particular with cardinal arithmetic
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