Geometry of free cyclic submodules over ternions

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.

This work was carried out within the framework of the Scientific and Technological Cooperation Poland-Austria 2010--2011

References in corpus (5)

• Projective Representations I. Projective lines over rings
• Vectors, Cyclic Submodules and Projective Spaces Linked with Ternions
• The Dual of a Chain Geometry
• Twin "Fano-Snowflakes" Over the Smallest Ring of Ternions
• A Jacobson Radical Decomposition of the Fano-Snowflake Configuration

Cited by in corpus (2)

• A point model for the free cyclic submodules over ternions
• Projective Ring Lines and Their Generalisations