On Characterizing Injective Lineations in Desarguesian Affine Spaces

Abstract

Point functions that preserve collinearity are called lineations. In geometry and linear algebra collineations have been studied extensivly. However, there are only a few results concerning the characterization of injective lineations on subsets of affine spaces under mild hypotheses. David S. Carter and Andrew Vogt proved a representation of all lineations defined on a whole affine or projective Plane. Using these results, A. Brezuleanu and D.-C. Rădulescu characterized full lineations of desarguesian affine and projective spaces of finite dimension in projective spaces. In this paper a characterization of lineations, defined on three lines in general position which are injective at the intersection point of any two of the lines, is given, using the theorem of Menelaos. Similar results in the case of projective planes have been proved by J. Aczél and W. Benz. (Cf. also the theorem of Schaeffer). It is shown, that except of a few special cases a much simpler representation of the mapping can be given, if the domain is enlarged by only one point. Finally the upper results are used to characterize lineations in desargesian affine spaces of arbitrary dimension ≥ 2. Special lineations are given, induced by an endomorphism of the underlying division ring and some constants. These lineations need not be defined on the whole affine space. A representation theorem is presented that leads to a characterization of all "basisinjective" lineations of a desarguesian affine space into itself, i.e. lineations that map a given basis onto an independent set of points and fulfil some special conditions of injectivity. The theorem also gives a representation for many lineations only defined on a subset of an desarguesian affine space and describes the maximum domain of induced injective lineations.