399 research outputs found
A Classification of the Projective Lines over Small Rings
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
Let be a Segre variety that is -fold direct product of projective
lines of size three. Given two geometric hyperplanes and of
, let us call the triple the
Veldkamp line of . We shall demonstrate, for the sequence , that the properties of geometric hyperplanes of are fully
encoded in the properties of Veldkamp {\it lines} of . Using this
property, a complete classification of all types of geometric hyperplanes of
is provided. Employing the fact that, for , the
(ordinary part of) Veldkamp space of is , we shall
further describe which types of geometric hyperplanes of lie on a
certain hyperbolic quadric that
contains the and is invariant under its stabilizer group; in the
case we shall also single out those of them that correspond, via the
Lagrangian Grassmannian of type , to the set of 2295 maximal subspaces
of the symplectic polar space .Comment: 16 pages, 8 figures and 7 table
A Classification of the Projective Lines over Small Rings II. Non-Commutative Case
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
Logical Dreams
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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