Duality in stable planes and related closure and kernel operations
Stable planes are topological incidence geometries with a ``local'' planarity condition (stability) replacing the requirement that any two lines meet (or the parallel axiom). Examples include all geometries induced on open nonempty sets of points in topological projective planes, in particular, the hyperbolic planes. As usual, the paper under review concentrates on the locally compact case. The involution assigning to each projective plane its dual is extended, in two different ways, to subclasses of the class of stable planes. If \(E\) is an open subset of the point set \(P\) of a topological projective plane \((P,{\mathcal L})\), one obtains a stable plane \(E^{\text{dual}}=({\mathcal E},P)\) as subplane of \(({\mathcal L},P)\). The largest subset of \(P\) whose dual coincides with \(E^{\text{dual}}\) is denoted \(E^*\). The paper under review shows \(E^{**}=E^*\). For an arbitrary stable plane \((E,{\mathcal E})\) (without embedding into a projective plane) one considers the set \({\mathcal K}\) of all projective lines (that is, lines that meet all other lines). The author defines \(E^\circ\) as the set of all points lying on some projective line, and shows \(E^{\circ\circ}=E^\circ\). It is known that \(E^\circ\) is an open subset of~\(E\), so \(E^\circ\) is the point space of a stable plane \(E^{\text{opp}}:=({\mathcal K},E^\circ)\) if \(\mathcal K\) is not empty. In the paper under review, it is shown that \(\Aut(E^{\text{dual}})=\Aut(E^*)\) and that \(\Aut(E^{\text{opp}})=\Aut(E^\circ)\) if \(\mathcal K\) is not empty. These results are used to determine the automorphism groups of complex cylinder planes.
scientific article; zbMATH DE number 1292214