An application of a localized version of an axiom of Ian Chiswell
Anthony Gaglione, Seymour Saul Lipschutz, Dennis Spellman · Contemporary mathematics - American Mathematical Society · 2015
Generalizing results of Rimlinger, Hoare showed that certain pregroups admit Lyndon length functions on their universal groups from which an asssociated graph of groups follows. Chiswell introduced axiom (P6) which characterizes precisely those pregroups admitting such length functions on their universal groups. The prototypical example of a (P6)-pregroup is the subset G 1 ∪ G 2 G_{1}\cup G_{2} of the amalgamated free product G 1 ∗ A G 2 G_{1}\ast _{A}G_{2} where A A~ is proper in each of G 1 G_{1} and G 2 G_{2} . If, for example, x ∈ G 1 ∖ A x\in G_{1}\backslash A and y ∈ G 2 ∖ A y\in G_{2}\backslash A , then x y xy is not defined in G 1 ∪ G 2 G_{1}\cup G_{2} . Note that, in the above event, both x a