---
tag: N56DHWH283
kind: definition
name: Ordered geometry
msc: [51G05]
defines: T7G8X5AD5G
uses: [0MP9BCF6DP, CE8GX6ADZS, KJT98G16Y4]
sources:
  - url: https://en.wikipedia.org/wiki/Ordered_geometry
---

## Definition

Let $\Pi$ be a set of points with a family of subsets called [[lines|0MP9BCF6DP]].
Let $[ABC]$ be a ternary relation on $\Pi$, read "$B$ lies [[between|CE8GX6ADZS]] $A$ and $C$".
Then $\Pi$ is an *ordered plane* if:

1. Two distinct points lie on exactly one line, each line has at least two points, and there are three points not on one line.
2. If $[ABC]$, then $A$, $B$, $C$ are three distinct points of one line, and $[CBA]$.
3. For distinct points $A$ and $C$, there is a point $B$ with $[ACB]$.
4. Of three distinct points of a line, at most one lies between the other two.
5. [[Pasch's axiom|KJT98G16Y4]] holds.

*Ordered geometry* is the study of the consequences of these axioms, and of their analogues in higher dimensions.

## Remarks

Example: the plane $F^2$ over an ordered field $F$, with the usual lines and with $[ABC]$ meaning $B = A + t(C - A)$ for some $t \in F$ with $0 < t < 1$, is an ordered plane.
Non-example: take $\R^2$ with the usual lines, and let $[ABC]$ mean that $A$, $B$, $C$ are three distinct points of one line.
This fails clause 4, since each of three collinear points then lies between the other two.
One proves from the axioms that, of three distinct collinear points, exactly one lies between the other two.
The Sylvester–Gallai theorem holds in every ordered plane.