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Hometranslation plane
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translation plane
Let $\pi$ be a projective plane. Recall that a central collineation on $\pi$ is a collineation $\rho$ with a center $C$ and an axis $\ell$. It is wellknown that $C$ and $\ell$ are uniquely determined. We also call $\rho$ a $(C,\ell)$collineation.
Definition. Let $\pi$ be a projective plane. We say that $\pi$ is $(C,\ell)$transitive if there is a point $C$ and a line $\ell$, such that for any points $P,Q$ where

$P,Q$ and $C$ are collinear and pairwise distinct,

$P,Q\notin\ell$,
there is a $(C,\ell)$collineation $\rho$ such that $\rho(P)=Q$.
It can be shown that $\pi$ if $(C,\ell)$transitive iff it is $(C,\ell)$Desarguesian; that is, if two triangles are perspective from point $C$, then they are perspective from line $\ell$. From this, it is easy to see that $\pi$ is a Desarguesian plane iff it is $(C,\ell)$transitive for any point $C$ and any line $\ell$, of $\pi$.
Now, suppose that $C$ lies on $\ell$. Then one can show that $\pi$ is $(C,\ell)$transitive iff it can be coordinatized by a linear ternary ring $R$ such that $R$ is a group with respect to the derived operation $+$ (addition). When $\pi$ is so coordinatized, $\ell$ is the line at infinity, and $C$ is the point whose coordinate is $(\infty)$.
This group is not necessarily abelian. So what condition(s) must be imposed on $\pi$ so that $(R,+)$ is an abelian group? The answer lies in the next definition:
Definition. Let $\pi$ be a projective plane. $\pi$ is said to be $(m,\ell)$transitive if there are lines $m,\ell$ such that $\pi$ is $(C,\ell)$transitive for all $C\in m$.
Definition. A projective plane $\pi$ is a translation plane if there is a line $\ell$ such that $\pi$ is $(\ell,\ell)$transitive. We also say that $\pi$ is a translation plane with respect to $\ell$. The line $\ell$ is called a translation line of $\pi$.
It can be shown that $\pi$ is a translation plane with respect to $\ell$ iff it can be coordinatized by a VeblenWedderburn system (thus implying that $(R,+)$ is abelian).
When $\pi$ is a translation plane with respect to two distinct lines $\ell$ and $m$, then it is not hard to see that it is a translation plane with respect to every line passing through $\ell\cap m$.
When $\pi$ is a translation plane with respect to three nonconcurrent lines, then it is a translation plane with respect to every line. A projective plane which is a translation plane with respect to every line is called a Moufang plane. An example of a translation plane that is not Moufang is the Hall plane, coordinatized by the Hall quasifield. An example of a projective plane that is not a translation plane is the Hughes plane.
Remark. There are also duals to the notions above: a projective plane $\pi$ is
1. $(P,Q)$transitive if there are points $P,Q$ such that $\pi$ is $(P,m)$transitive for any line $m$ passing through $Q$.
2. a dual translation plane if there is a point $P$ such that $\pi$ is $(P,P)$transitive. We also say that $\pi$ is a dual translation plane with respect to $P$, and that $P$ is a translation point of $\pi$.
If $\pi$ is a projective plane, then the following are true:

$\pi$ is translation plane with respect to some line $\ell$ and a dual translation plane with respect to some $P\in\ell$ iff $\pi$ can be coordinatized by a semifield. In this coordinatization, $\ell$ is the line at infinity and $P$ is the point with coordinate $(\infty)$.

$\pi$ is translation plane with respect to some line $PQ$ and $(P,Q)$ and $(Q,P)$transitive iff $\pi$ can be coordinatized by a nearfield. In this coordinatization, $PQ$ is the line at infinity where $P$ and $Q$ have coordinates $(0)$ and $(\infty)$ (or vice versa).
Remark. By removing the line at infinity from a translation plane, we obtain an affine translation plane. By the definition of a translation plane, an affine translation plane can be characterized as an affine plane where the minor affine Desarguesian property holds.
References
 1 R. Casse, Projective Geometry, An Introduction, Oxford University Press (2006)
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