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Geometry of free cyclic submodules over ternions

Hans Havlicek, Andrzej Matras, Mark Pankov

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.

Geometry of free cyclic submodules over ternions

Abstract

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

Paper Structure

This paper contains 6 sections, 11 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Cyclic submodules
  3. Representation of points
  4. Structure of ${{\mathcal{G}}}$
  5. Adjacency
  6. Acknowledgements.

Key Result

Proposition 1

The $\Phi$-images of distinct cyclic submodules of $T^2$ are distinct subspaces of $F^6$.

Theorems & Definitions (21)

  • Proposition 1
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Proposition 5
  • proof
  • Remark 1
  • ...and 11 more