Projective Rectangles: Harmonic Conjugation

Abstract

A projective rectangle is like a projective plane that has different lengths in two directions. We develop harmonic conjugation in projective rectangles. We construct projective rectangles in some harmonic matroids (matroids where harmonic conjugation is defined on every collinear point triple), such as Desarguesian projective planes of finite characteristic, by harmonic conjugation from extended lift matroids based on finite fields. Similar results follow for countable fields with characteristic $0$. We also show that projective rectangles are almost harmonic matroids.