If the complete linearly ordered ternar in this theorem is obtained by the above extension of the ternar of a given betweenness plane, then the new betweenness plane, constructed according to this theorem, can be seen as an algebraic extension of the given plane.
If considering the projective plane obtained by algebraic extension, one has here a collineation in the sense of the theory of projective geometry.
If for a betweenness plane its algebraic extension is (z, A)-transitive for an arbitrary centre z [member of] A, then it is called the translation plane with respect to the axis A.
Let us consider the betweenness plane, whose algebraic extension gives the ordered projective plane (with the excluded line [L.
If the betweenness plane is such that its algebraic extension is a translation plane with respect to every line going through the point Y, then