We study a construction of an HNN extension for inverse semigroups with zero. We prove a normal form for the elements of the universal group of an inverse semigroup that is categorical at zero, and use it to establish structural results for the universal group of an HNN extension. Our main application of the HNN construction is to show that graph inverse semigroups – including the polycyclic monoids – admit HNN decompositions in a natural way, and that this leads to concise presentations for them.
- inverse semigroups
- HNN extensions