Geometry of free cyclic submodules over ternions

TL;DR

This paper explores the geometric structure of free cyclic submodules over the algebra of ternions, representing them as planes in projective space and analyzing their symmetries and automorphisms.

Contribution

It introduces a novel geometric model of the projective line over ternions as planes in projective space and studies its automorphisms and adjacency relations.

Findings

Model of the projective line over ternions as planes in projective space

Characterization of adjacency and automorphic collineations

The plane model does not admit any duality despite the algebra's antiautomorphism

Abstract

Given the algebra $T$ of ternions (upper triangular $2\times 2$ matrices) over a commutative field $F$ we consider as set of points of a projective line over $T$ the set of all free cyclic submodules of $T^2$. This set of points can be represented as a set of planes in the projective space over $F^6$. We exhibit this model, its adjacency relation, and its automorphic collineations. Despite the fact that $T$ admits an $F$-linear antiautomorphism, the plane model of our projective line does not admit any duality.