I known axiomatizations of projective spaces of dimension > 2 and also of projective planes (either those obeying the axiom of Pappus, which come from fields, or those obeying the axiom of Desargues, which come from division rings, or the fully general ones, which come from planar ternary rings).

How about projective lines? One might try to axiomatize at least those coming from fields using the cross-ratio. If $k$ is a field the cross-ratio gives at least a partially-defined 4-ary function

$$ kP^1 \times kP^1 \times kP^1 \times kP^1 \to kP^1 $$

where $kP^1$ is the corresponding projective line. Is the cross-ratio well-defined even when several of the four arguments coincide? Does it give a regular map of projective varieties? Does it obey enough equational laws that we can use these as axioms for an interesting concept of projective line?

Some related ideas are discussed here:

• Mike Shulman, Synthetic projective lines, Math Overflow.
• Jacob Lurie, Action of PGL(2) on projective space, Math Overflow.

The latter turned up this nice result: as is well known, $\mathrm{PGL}(2,k)$ acts in a sharply 3-transitive way on $kP^1$, but more generally, any sharply 3-transitive group arises from a 'KT-field', which is a kind of generalized field. I don't know how KT-fields are related to planar ternary rings.

However, all this is---at least superficially---a bit different than trying to axiomatize the cross-ratio. Perhaps it's just another viewpoint on the same thing.

I known axiomatizations of projective spaces of dimension > 2 and also of projective planes (either those obeying the axiom of Pappus, which come from fields, or those obeying the axiom of Desargues, which come from division rings, or the fully general ones, which come from planar ternary rings).

How about projective lines? One might try to axiomatize at least those coming from fields using the cross-ratio. If $k$ is a field the cross-ratio gives at least a partially-defined 4-ary function

$$ kP^1 \times kP^1 \times kP^1 \times kP^1 \to kP^1 $$

where $kP^1$ is the corresponding projective line. Is the cross-ratio well-defined even when several of the four arguments coincide? Does it give a regular map of projective varieties? Does it obey enough equational laws that we can use these as axioms for an interesting concept of projective line?

Some related ideas are discussed here:

• Mike Shulman, Synthetic projective lines, Math Overflow.
• Jacob Lurie, Action of PGL(2) on projective space, Math Overflow.

The latter turned up this nice result: as is well known, $\mathrm{PGL}(2,k)$ acts in a sharply 3-transitive way on $kP^1$, but more generally, any sharply 3-transitive group arises from a 'KT-field', which is a kind of generalized field. I don't know how KT-fields are related to planar ternary rings.

However, all this is---at least superficially---a bit different than trying to axiomatize the cross-ratio. Perhaps it's just another viewpoint on the same thing.