Structure of Some Pregroups and Length Functions

The concept of Pregroups was introduced by Stallings in 1971. Subsequently the concept of Pregroups was developed by many other researchers. Stallings originally defined a set with a binary operation satisfying five axioms, namely, P1, P2, P3, P4, and P5. It has been proved later that P3 is a consequence of the other axioms. Stallings has also linked this construction of a Pregroup to Free Product of Groups.

This construction is developed to include a new axiom called P6, which enabled to define a length function on the universal group of Pregroups. Applications of Pregroups with length functions led to direct proof of many other problems in combinatorial group theory.