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Published online by Cambridge University Press: 04 November 2009
Abstract
This is a survey of all known rank three geometries belonging to a string diagram of type (c*, c)-geometry. There are three types of objects: points, lines, and blocks subject to axioms imposed by the diagram. There are several other formulations described here which are more convenient for presenting certain of the examples. All examples fall into these six classes:
Simplicial type, which can easily be characterized.
Fischer spaces with no affine planes.
Orthogonal types, whose points and lines are exterior points and tangent lines of certain low-dimensional quadrics.
Hall type, determined by alternating multilinear forms over the field of two elements.
Affine type, whose points are vectors in some d-dimensional space over the integers mod 2. Here, blocks are not subspaces.
A few special examples determined by coherent pairs: the construction of odd type of Cameron and Fisher, and two examples of Blokhuis and Brouwer.
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