A difference between finite geometries and (e.g.) Euclidean space is that "lines" in finite geometries are *unordered* subsets of the universe, while "lines" in Euclidean space are *ordered* subsets of the universe, at least insofar as they have an implicit betweenness relation attached. For example, in Euclidean space, we can meaningfully say that point $a$ is between points $b$ and $c$ on line $L$, but this statement makes no sense in a finite projective plane.


Have finite geometries with lines ordered in this way been studied? Are any interesting results gained from this additional structure? And if so, are there any good resources for learning about this?



Is there literature on finite geometries with ordered lines?
Yes, this has been studied and is indeed known as *ordered geometry* or the study of *betweenness spaces*:


<https://en.m.wikipedia.org/wiki/Ordered_geometry>


