Projective Rectangles: Harmonic Conjugation

TL;DR

This paper extends harmonic conjugation to projective rectangles, constructing them in harmonic matroids derived from finite fields and showing their relation to harmonic matroids.

Contribution

It introduces harmonic conjugation in projective rectangles and constructs these structures within harmonic matroids from finite fields.

Findings

Constructed projective rectangles in harmonic matroids from finite fields.

Extended harmonic conjugation to projective rectangles.

Showed projective rectangles are nearly harmonic matroids.

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.