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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?

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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

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    $\begingroup$ From the definition, it does not look like this can work for a finite geometry (the second axiom in particular creates problems). $\endgroup$
    – xxxxxxxxx
    Commented Mar 19, 2017 at 12:26
  • $\begingroup$ Good point. I suppose not all the axioms need to be adopted in all studies, though $\endgroup$
    – Bjørn Kjos-Hanssen
    Commented Mar 19, 2017 at 16:43
  • $\begingroup$ Interesting book in this regard: Marvin J. Greenberg's Euclidean and non-Euclidean Geometries: Development and History (4th ed). $\endgroup$
    – Bjørn Kjos-Hanssen
    Commented Mar 19, 2017 at 17:07

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