Preordered uniform Hjelmslev planes (Q2266228)

A point P is said to be a neighbour of a point Q if P and Q are incident with two distinct lines m and \(\ell\) (in symbols \(P\sim Q)\). \(P\sim \ell\) means P is a neighbour of some point of \(\ell.\) Let \(\ell\) and m be two lines, and let \(U\) be a point such that \(UI\ell\), \(U\nsim m\) (for the symbol ''I'' and the other special notions see \textit{C. Baker, N. D. Lane} and \textit{J. W. Lorimer}: J. Geom. 19, 8-42 (1982; Zbl 0505.51017). Then \(\frac{U}{\bigwedge}(\ell \to m)=\{(X,X')| XI\ell,\quad X'Im;X,X',U\quad collinear\}\) is called a projective relation. If \(p\nsim \ell\), then the map \(\ell \to^{p}m\) (X\(\rightsquigarrow PX\wedge m)\) is a bijective projective relation called a perspectivity. The projective relation \(\frac{U}{\bigwedge}\) is said to preserve order if it has the following property: for any W,X,Y,ZI\(\ell\) such that W,X,Y\(\nsim U\) and WX\(| YZ\) (the pair W, X separates the pair Y, Z) with (W,W'),(X,X'),(Y,Y'),(Z,Z')\(\in \frac{U}{\bigwedge}\) then either W'X'\(| Y'Z'\) or two or more of W',X',Y',Z' are equal. A projective Hjelmslev plane (PH) is called a preordered (ordered) PH- plane if all perspectivities \(\ell \to^{U}m\) (all projective relations \(\frac{U}{\bigwedge})\) preserve order. The following theorem is proved: A preordered uniform PH-plane is ordered.