Semi-biplanes and antiregular generalized quadrangles
A semi-biplane is a connected symmetric finite block design such that every pair of points is incident with 0 or 2 blocks and, dually, every pair of blocks is incident with 0 or 2 points. A generalized quadrangle is an incidence structure with a set \(P\) of points and a set \(\mathcal L\) of lines such that any two distinct points have at most one joining line, for every anti-flag \((p,K)\) there exists exactly one flag \((q,L)\) such that \((p,L)\) and \((q,K)\) are flags, and every point (line) is incident with at least three lines (points). The set of points collinear to \(p\) is denoted by \(p^\perp\). A set of pairwise non-collinear points is called a triad. For any triad \(\{p,q,r\}\) the intersection \(p^\perp\cap q^\perp\cap r^\perp\) is called the centre of \(p\), \(q\) and \(r\). A generalized quadrangle is called anti-regular, if the center of every triad is either empty or contains precisely two points. In this pleasant paper, which is intended to be the first one of a series of papers about incidence geometries as described above, the authors construct semi-biplanes out of anti-regular generalized quadrangles \(Q\). This is done in the finite case as well as in the topological case. In the latter case the semi-biplanes so obtained are always topological and divisible, i.e. the geometric operations of join and intersection are continuous and both parallel relations are transitive.
scientific article; zbMATH DE number 1129443