A generalization of Stallings' pregroup

Recently, pregroups have received much attention and this has produced some important results. One such result is by Rimlinger who showed in [6] that a group G is the fundamental group of a finite graph of finite groups if and only if G is the universal group of a finite pregroup. A further result was obtained by Hoare [2] who showed that any “simplicial” pregroup P determines a Lyndon length function for G(P). New questions are now raised by our generalized pregroups. For example, does Hoare's length function, mentioned above, carry over into this new setting? There is also the question of whether one can further weaken Stallings' axiom [P5], that is, weaken our axioms [K] and [Q5], and still retain the property that P embeds as a set in G(P). The authors conjecture that one such set of axioms will include axiom [K] and some type of length axiom like [Tn].