One configurational characterization of Ostrom nets. (English) Zbl 0737.51002

An Ostrom net \(N\) of the title is defined via a lift vector space \(V\) over a skew field \(F\). The point set of \(N\) is \(V\times V\). The lines of \(N\) are \(\ell_ c=\{(c,y)\mid y\in V\}\), for \(c\) in \(V\), and \(\ell_{\alpha,v}=\{(x,\alpha x+v)\mid x\in V\}\), for \((\alpha,v)\) in \(F\times V\). A net with point set \(P\) and line set \(L\) satisfies the quadrangular condition (not quadrilateral as the author would have it) if any four points \(A_ 1\), \(A_ 2\), \(A_ 3\), \(A_ 4\) with no three on the same line are such that if each of five of the six pairs are collinear then so also is the sixth. By an argument of some considerable detail, it is shown that a net satisfying the quadrangular is an Ostrom set and vice versa.


51A25 Algebraization in linear incidence geometry
51A20 Configuration theorems in linear incidence geometry