On the construction of finite projective planes from homology semibiplanes
From a projective plane \(\Pi\) with involutory homology \(\tau\) one constructs an incidence system \(\Pi/\tau\) having as points and blocks the \(<\tau>\)-orbits of length 2 on the points and lines of \(\Pi\), and with incidence inherited from \(\Pi\). Such incidence systems satisfy certain properties which, when taken as axioms, define the class of homology semibiplanes. We describe how one determines, in principle, whether a given homology semibiplane \(\Sigma\) is realizable as \(\Pi/\tau\) for some \(\Pi\) and \(\tau\) and, moreover, how many non-equivalent pairs \((\Pi,\tau)\) yield \(\Sigma\). In case \(\Pi '\) is Desarguesian of prime order we show that \(\Pi'\) is characterized by its homology semibiplane; i.e. \(\Pi/\tau \simeq \Pi'/\tau'\) implies \(\Pi \simeq \Pi'\). The details are too involved to be described here. For the definition of a semibiplane we follow here that employed by \textit{D. R. Hughes} and \textit{F. C. Piper} [``Design Theory'' (1985; Zbl 0561.05009), pp. 196-211].
scientific article; zbMATH DE number 4205265