On the structure of lower bounded HNN extensions

This paper studies the structure and preservational properties of lower bounded HNN extensions of inverse semigroups, as introduced by Jajcayová. We show that if $S^* = [ S;\; U_1,U_2;\; \phi ]$ is a lower bounded HNN extension then the maximal subgroups of $S^*$ may be described using Bass-Serre theory, as the fundamental groups of certain graphs of groups defined from the $\mathcal{D}$ -classes of $S$ , $U_1$ and $U_2$ . We then obtain a number of results concerning when inverse semigroup properties are preserved under the HNN extension construction. The properties considered are completely semisimpleness, having finite $\mathcal{R}$ -classes, residual finiteness, being $E$ -unitary, and $0$ - $E$ -unitary. Examples are given, such as an HNN extension of a polycylic inverse monoid.