  Algebraic relations are established that determine the invariance of the
transformed number after several transformations. The restrictions that
determine the group structure of these relationships are analyzed, as is the
case of the Klein group. Parametric Kr functions associated with the existence
of cycles are presented, as well as the role of the number of their links in
the grouping of numbers in higher order equivalence classes. For this we have
developed a methodology based on binary equivalence relations and the complete
parameterization of the Kaprekar routine using Ki functions of parametric
transformation.
