Self-dual axioms for many-dimensional projective geometry

Esser, Martinus

doi:10.1090/S0002-9947-1973-0320871-5

Self-dual axioms for many-dimensional projective geometry
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Trans. Amer. Math. Soc. 177 (1973), 221-236 Request permission

Abstract:

Proposed and compared are four equivalent sets R, S, T, D of self-dual axioms for projective geometries, using points, hyperplanes and incidence as primitive elements and relation. The set R is inductive on the number of dimensions. The sets S, T, D all include the axiom “on every n points there is a plane", the dual of this axiom, one axiom on the existence of a certain configuration, and one or several axioms on the impossibility of certain configurations. These configurations consist of $(n + 1)$ points and $(n + 1)$ planes for sets S, T, but of $(n + 2)$ points and $(n + 2)$ planes for set D. Partial results are obtained by a preliminary study of self-dual axioms for simplicial spaces (spaces which may have fewer than 3 points per line).

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