# Some notation for FPAs and HNN extensions

In order to define a free product with amalgamation

All three of these options require you to define some groups and some morphisms. You can define a group nicely by giving a presentation but defining maps usually takes a few lines. So, I’d like a quicker way of describing the kinds of maps we need here.

Let’s take the third option. If we have generating sets so that

So how about this as a presentation for an FPA?

The bits in angle brackets are presentations of

Because this is all on one line and delimited adequately, you can chain things together and use brackets and all that, like this contrived example:

For HNN extensions, you just need to say what the original group is, what the extending letter is, and what the identified subgroups are. So how about something like:

Apart from the different kinds of brackets being hard to distinguish visually, what have I missed?