The classical synthetic notion of projective plane consists of a set of points, a set of lines, and a relation of incidence between the two, such that any two distinct points lie on a unique line and any two distinct lines intersect in a unique point (plus some nondegeneracy assumptions). There are similar notions of projective 3-space, $n$-space, and so on — but 1-dimensional projective space seems harder to capture synthetically, since there is no "room", dimensionally, for subspaces in between the points and the entire space.

Has anyone attempted to define a synthetic notion of "projective line"? Ideally such a definition would have properties like the following:

• The space $P^1(k)$ is naturally a projective line for any division ring $k$, and from any projective line $L$ satisfying enough axioms we can construct a skew field $c(L)$ such that $c(P^1(k)) \cong k$ and $P^1(c(L))\cong L$ (unnaturally). The corresponding facts for Desarguesian projective planes are classical.

• Any line in a projective plane is a projective line, and any projective line satisfying enough axioms can be embedded as a line in some projective plane. This would be analogous to how any plane in a projective 3-space is a Desarguesian projective plane, while any Desarguesian projective plane can be embedded in a projective 3-space.

I have an idea for how one might do this, by axiomatizing the "quadrangular hexad" relation on a line in a projective plane; but before I try very hard, I'm looking for references where something like it has been tried before.