and are collinear and pairwise distinct,
there is a -collineation such that .
It can be shown that if -transitive iff it is -Desarguesian; that is, if two triangles are perspective from point , then they are perspective from line . From this, it is easy to see that is a Desarguesian plane iff it is -transitive for any point and any line , of .
Now, suppose that lies on . Then one can show that is -transitive iff it can be coordinatized by a linear ternary ring such that is a group with respect to the derived operation (addition). When is so coordinatized, is the line at infinity, and is the point whose coordinate is .
This group is not necessarily abelian. So what condition(s) must be imposed on so that is an abelian group? The answer lies in the next definition:
Definition. Let be a projective plane. is said to be -transitive if there are lines such that is -transitive for all .
Definition. A projective plane is a translation plane if there is a line such that is -transitive. We also say that is a translation plane with respect to . The line is called a translation line of .
It can be shown that is a translation plane with respect to iff it can be coordinatized by a Veblen-Wedderburn system (thus implying that is abelian).
When is a translation plane with respect to two distinct lines and , then it is not hard to see that it is a translation plane with respect to every line passing through .
When is a translation plane with respect to three non-concurrent 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 is
-transitive if there are points such that is -transitive for any line passing through .
a dual translation plane if there is a point such that is -transitive. We also say that is a dual translation plane with respect to , and that is a translation point of .
If is a projective plane, then the following are true:
is translation plane with respect to some line and a dual translation plane with respect to some iff can be coordinatized by a semifield. In this coordinatization, is the line at infinity and is the point with coordinate .
is translation plane with respect to some line and - and -transitive iff can be coordinatized by a nearfield. In this coordinatization, is the line at infinity where and have coordinates and (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.
|Date of creation||2013-03-22 19:15:15|
|Last modified on||2013-03-22 19:15:15|
|Last modified by||CWoo (3771)|
|Defines||dual translation plane|
|Defines||affine translation plane|