HNN extensions of inverse semigroups and groupoids

We use the isomorphism between the categories of inverse semigroups and inductive groupoids to construct HNN extensions of inverse semigroups, where the associated inverse subsemigroups are order ideals of the base. Properties of groupoids then ensure that the base inverse semigroup always embeds in an HNN extension constructed in this way. Other properties of HNN extensions are determined, including a description of the maximal subgroups and of the maximal group image, and the structure of the inverse subsemigroup generated by the stable letters. Finally, some examples are given.