Classification of general absolute planes by quasi-ends

Abstract

General (i.e. including non-continuous and non-Archimedean) absolute planes have been classified in different ways, e.g. by using Lambert–Saccheri quadrangles (cf. Greenberg, J Geom 12/1:45-64, 1979; Hartshorne, Geometry; Euclid and beyond, Springer, Berlin, 2000; Karzel and Marchi, Le Matematiche LXI:27–36, 2006; Rostamzadeh and Taherian, Results Math 63:171–182, 2013) or coordinate systems (cf. Pejas, Math Ann 143:212–235, 1961 and, for planes over Euclidean fields, Greenberg, J Geom 12/1:45-64, 1979). Here we consider the notion of quasi-end, a pencil determined by two lines which neither intersect nor have a common perpendicular (an ideal point of Greenberg, J Geom 12/1:45-64, 1979). The cardinality ω of the quasi-ends which are incident with a line is the same for all lines hence it is an invariant ${\omega_\mathcal{A}}$ of the plane ${\mathcal{A}}$ and can be used to classify absolute planes. We consider the case ${\omega_\mathcal{A}=0}$ and, for ${\omega_\mathcal{A} \geq 2}$ (it cannot be 1) we prove that in the singular case ${\omega_\mathcal{A}}$ must be infinite. Finally we prove that for hyperbolic planes, ends and quasi-ends are the same, so ${\omega_\mathcal{A}=2}$ .