Quasiparallelism in finite planar spaces (Q1394816)

The author deals with finite planar spaces \(S=(P,L,P^*)\), where \((P,L)\) is a linear space and \(P^*\) is a non-empty family of proper subspaces of \((P,L)\), called planes, such that the following conditions hold: (i) through any three non-collinear points there is a unique plane and it is the smallest subspace containing them, (ii) every plane has at least three non-collinear points, (iii) there are at least two planes. For every plane \(\alpha\in P^*\) is denoted by \(L_\alpha\) the family of lines on the plane \(\alpha\). The subspaces \(X\) and \(X'\) of \(S\) are called quasiparallel if \(|X\cap l|= |X'\cap l|\) for all lines \(l\not\subset X\cup X'\). The author is interested in finite planar spaces \(S=(P,L,P^*)\) with the following properties: (A) \(\forall\alpha\), \(s\in P^*\), \(P\neq\alpha\cup \beta\), (B) \(\forall \alpha\in P^*\), \(\forall r,s\in L_\alpha\), \(\alpha\neq r\cup s\). In particular, he studies quasiparallelism relation in finite planar spaces fulfilling the conditions (A) and (B). Furthermore, 3-dimensional projective spaces minus a point or minus a line are characterized with help of quasiparallelism relation.