How to Define Global Lie Group Actions on Functions
  A particularly easy, even if for long overlooked way is presented for defining globally arbitrary Lie group actions on smooth functions on Euclidean domains. This way is based on the appropriate use of the usual parametric representation of functions. As a further benefit of this way, one can define large classes of genuine Lie semigroup actions. Here "genuine" means that, unlike in the literature, such Lie semigroups need no longer be sub-semigroups of Lie groups, and instead, can contain arbitrary noninvertible smooth functions on Euclidean domains. 
Elemer E Rosinger