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.



Synthetic projective lines
Building on previous work by Paul Libois, and related to work by Libois'
student Jean van Buggenhaut from 1969, Francis Buekenhout considered and
solved this question in "Foundations of one Dimensional Projective Geometry based on Perspectivities" Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 43 (1975) 21-29. Note that his approach also adapts to lines in Moufang planes. See also F. Buekenhout and A. Cohen Diagram geometry Springer 2013, Section 6.2 Projective lines.


