Translation plane
In mathematics, a translation plane is a particular kind of projective plane. Almost all non-Desarguesian planes are either translation planes or are related to this type of incidence structure.
In a projective plane, let P represent a point, and l represent a line. A central collineation with center P and axis l is a collineation fixing every point on l and every line through P. It is called an elation if P is on l, otherwise it is called a homology. The central collineations with centre P and axis l form a group.
A line l in a projective plane Π is a translation line if the group of elations with axis l acts transitively on the points of the affine plane obtained by removing l from the plane Π, Πl (the affine derivative of Π). A projective plane with a translation line is called a translation plane and the affine plane obtained by removing the translation line is called an affine translation plane.
While in general it is often easier to work with projective planes, in this context the affine planes are preferred and several authors simply use the term translation plane to mean affine translation plane.
A (projective) translation plane having at least three nonconcurrent translation lines is a Moufang plane. All the lines of a Moufang plane are translation lines. Every finite Moufang plane is desarguesian and every desarguesian plane is a Moufang plane, but there are infinite Moufang planes that are not desarguesian (such as the Cayley plane). Moufang planes are coordinatized by alternative division rings.
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