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On Certain Linear Congruence Class Geometries

DOI: 10.1016/s0304-0208(08)73327-9
  • Mathematics


Summary A subset of the points of a finite affine desarguesian geometry, equipped with the induced subspace structure and the induced parallelism, is called a representable geometry. The question is answered as to which repre-sentable geometries are congruence class geometries in the sense of Wille, i.e. allow “enough” dilatations to generate the induced subspace structure and the parallelism. Each geometry induced on a subset of the points of an affine desarguesian geometry satisfies the Exchange Axiom for subspaces, and hence is linear, i.e. every line is determined uniquely by any two of its points. Thus, using results of Wille, Pasini and Herzer on finite linear congruence class geometries, one easily obtains an answer to the above question (Theorem 3.2.5). Moreover, the problem is solved as to which finite congruence class geometries have the same points and subspaces (but possibly a different parallelism) as a representable geometry (Theorem 3.2.6). Corollary 3.2.7 is a consequence of this theorem, stating that a finite congruence class geometry satisfies the Exchange Axiom, if all its subplanes do.

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