in this thesis, we show that if we have a z2 action on a latticeordered dimension group, then it will arise as an inductive limit of z2 actions on simplicial groups. this work was motivated by the range of the invariant problem in elliott and su’s classification of af type z2 actions. in order to show this, we modify the proof of effros-handelmanshen theorem to include z2 actions at every stage of the arguments.