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In a projective plane endowed with a hyperbolic projective polarity, one can define an orthogonality relation on the set of lines which are not selfpolar (i.e. not passing through their poles), two lines being orthogonal if they pass through each other’s pole. The set of non-self-polar lines, together with the line-concurrence (three lines having a point in common are called concurrent) and orthogonality relations form a hyperbolic projective-metric plane (called a hyperbolic-metric plane in [7]). The motion group of this class of structures was first axiomatized in [1, §11], and the structures themselves were first axiomatized in [7] (see also [6]) in a mixed language, containing on the one hand line-reflections, and on the other the incidence and orthogonality structure defined in terms of line-reflections. The aim of this paper is to show that one can eliminate both the linereflection and the line-concurrence notions from Lingenberg’s axiom system, to obtain an axiom system for hyperbolic projective-metric planes with lines as the only individual variables and line-orthogonality as the only primitive notion.

An Axiomatics for Hyperbolic Projective-Metric Planes in Terms of Lines and Orthogonality