On Semi Quadrangles

Abstract

We introduce semi quadrangles, which are finite partial linear spaces with a constant number of points on each line, having no ordinary triangles and containing, as minimal circuits, ordinary quadrangles and pentagons, with the additional property that every two non-collinear points are collinear with at least one other point of the geometry. A semi quadrangle is called thick if every point is incident with at least three lines and if every line is incident with at least three points. Thick semi quadrangles generalize (thick) partial quadrangles (see [4]). We will emphasize the special situation of the semi quadrangles which are subgeometries of finite generalized quadrangles. Some particular geometries arise in a natural way in the theory of symmetries of finite generalized quadrangles and in the theory of translation generalized quadrangles, as certain subgeometries of generalized quadrangles with concurrent axes of symmetry; these subgeometries have interesting automorphism groups, see [17] and also [19]. Semi quadrangles axiomatize these geometries. We will present several examples of semi quadrangles, most of them arising from generalized quadrangles or partial quadrangles. We will prove an inequality for semi quadrangles which generalizes the inequality of Cameron [4] for partial quadrangles, and the inequality of Higman [7,8] for generalized quadrangles. The proof also gives information about the equality. Some other inequalities and divisibility conditions are computed. Also, we will characterize the linear representations of the semi quadrangles, and we will have a look at the point graphs of semi quadrangles.