\(\{\) 1,2\(\}\)-semiaffine planar spaces (Q1118810)

Let (S,L) be a finite linear space, that is a finite set S whose elements we call points, L a family of parts in S, whose elements we call lines, such that any line has at least two points, two distinct points are contained in just a line and \(| L| \geq 2\). A subspace in (S,L) is a subset \(S'\) in S such that for any \(X,Y\in S'\), \(X\neq Y\), the line joining them belongs to \(S'\). Suppose a family P of subspaces in (S,L) exists such that \(| P| \geq 2\), every \(n\in P\) contains three non- collinear points and through three non-collinear points there is only one element of P. The triple (S,L,P) is called planar space, the elements of P are called planes. Let (X,t) be a pair consisting of a point \(X\in S\) and a line \(t\in L\) with \(X\not\in t\); let \(\pi\) (X,t) be the number of lines on X not meeting t and let \(H:=\{\pi (X,t)|\) \(X\in S\), \(t\in L\), \(X\not\in t\}\). (S,L) is also called H-semiaffine plane. Let \(n+1\) be the maximum number of lines on a point; then the integer n is called the order of (S,L,P). In this paper the following result is proved. Let (S,L,P) be a finite planar space such that every plane of P is a \(\{\) 1,2\(\}\)-semiaffine plane of order \(\geq 5\) and \(n+1\) is the number of planes through every line of L. Then (S,L,P) is one of the following examples: (a) PG(3,n)\(| \pi\); (b) PG(3,n)\(| \{\pi \cup X\}\), with \(X\not\in \pi\); (c) PG(3,n)\(| (\{\pi \cup t\}\), with \(t\subset \pi\); (d) \(PG(3,n)| \{\pi \cup \pi '\}\), where X, t, \(\pi\) are a point, a line, a plane of PG(3,n), respectively, and \(\pi '\) is a plane of PG(3,n) different from \(\pi\).