  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.
