The semigroup of continuous lineations of a stable plane
Let \(({\mathcal P}, {\mathcal L})\) be a stable plane. The author considers the semigroup \(\Gamma\) of all continuous lineations of \(({\mathcal P}, {\mathcal L})\). A lineation is a map \({\mathcal P} \to{\mathcal P}\) with the property that the image of every line is contained in some line. A lineation is called collapsed if the image of the point space is contained in some line. Let \(\Pi\) denote the collection of all lineations which are constant, and let \(\Lambda\) denote the collection of all lineations which are collapses but not constant. Finally, let \(E\) denote the collection of all injective lineations. If the point space \({\mathcal P}\) is connected, then \(\Gamma= \Pi\sqcup \Lambda \sqcup E\). Moreover, each of these three subsets of \(\Gamma\) can be characterized intrinsically in the semigroup \(\Gamma\) without referring to the stable plane \(({\mathcal P},{\mathcal L})\). If all lines of \(({\mathcal P}, {\mathcal L})\) are connected, then the point space \({\mathcal P}\) can be identified with \(\Pi\), and the line space \({\mathcal L}\) can be identified with \(\Lambda/ \sim\), where \(\sim\) is an equivalence relation which can be described algebraically in \(\Gamma\). The incidence relation can also be described in terms of the semigroup \(\Gamma\). All these constructions are compatible with the topology on \(({\mathcal P}, {\mathcal L})\). Thus, the stable plane can be recovered uniquely from the semigroup \(\Gamma\). Finally, a counterexample is given to show that the assumption that the lines are connected is necessary.
scientific article; zbMATH DE number 1045581