Geometry of free cyclic submodules over ternions

Abstract

Given the algebra T of ternions (upper triangular 2×2 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.