Pregroups and Graphs of Groups

The main goal of this work is to construct a graph of groups for a pregroup P from the order tree for P; to construct a pregroup from the resulting graph of groups and to generalize this idea to construct a pregroup from any arbitrary graph of groups; to show that the universal group of the pregroup U(P) is isomorphic to the fundamental group of the graph of groups constructed above; also to define a natural length function on the fundamental group of a graph of groups.By this way we have proved that any group has Length function Has an up-down pregroup.