On linear spaces in which every plane is either projective or affine

Abstract

A linear space is a set S whose elements are called points, provided with certain distinguished subsets called lines such that any two points are contained in one and only one line and every line contains at least two points. The line passing through two points x and y will be denoted by xy. A subspace of a linear space S is a subset V of S such that any line having at least two points in V is contained in IF. The subspace generated by a subset X of S will be the intersection of all subspaces containing .V and will be denoted by (X>. A plane of a linear space S is a subspace generated by a triangle, that is by three non-coUinear points of S. An affine plane is a linear space A containing at least three non-collinear points and such that for every line L of A and any point p ~L, there is one and only one line L' such that p~U and LnL' =0. In an affine plane A, we shall say that a line L is parallel to a line L' (and write L//L') if LnL' =0 or L =L'. A projective plane is a linear space P containing at least four points, no three of which are collinear, and such that any two lines of P have a point in common. It is well-known that in a projective or an affine plane all lines have the same cardinality. The purpose of this paper is to prove the following theorem: