@MISC{TRANSGEO.MIZ,
  AUTHOR = {Oryszczyszyn, Henryk and Pra\.zmowski, Krzysztof},
  TITLE = {Transformations in Affine Spaces},
  NOTE = {Supported by RPBP.III-24.C2.},
  DAY = {31},
  MONTH = {May},
  YEAR = {1990},
  ADDRESS1 = {Warsaw University\\Bia{\l}ystok},
  ADDRESS2 = {Warsaw University\\Bia{\l}ystok},
  SUMMARY = {Two  classes of bijections of its point universe
are correlated with every affine structure. The first class consists of the 
transformations, called formal isometries, which map every segment onto 
congruent segment, the second class consists of the automorphisms of such 
a structure. Each of these two classes of bijections forms a group for a 
given affine structure, if it satisfies a very weak axiom system (models of 
these axioms are called congruence spaces); formal isometries form a normal 
subgroup in the group of automorphism. In particular ordered affine spaces
and affine spaces are congruence spaces; therefore formal
isometries of these structures can be considered. 
They are called positive dilatations and
dilatations, resp. For convenience the class of negative 
dilatations, transformations which map every ``vector" onto parallel ``vector",
but with opposite sense, is singled out. 
The class of translations is distinguished as well.
Basic facts concerning all these types of transformations are established, 
like rigidity, decomposition principle, introductory group-theoretical
properties. At the end collineations of affine spaces and
their properties are investigated; for affine planes it is proved 
that the class of 
collineations coincides with the class of bijections preserving lines.}}

