HNN extensions of inverse semigroups and groupoids. (Q1425087)

In the category of groupoids HNN extensions have a natural definition modeled on the construction of a mapping torus. Properties of groupoid HNN extensions closely follow the properties of HNN extensions of groups, defined by \textit{G. Higman}, \textit{B. H. Neumann}, and \textit{H. Neumann} [J. Lond. Math. Soc. 24, 247-254 (1950; Zbl 0034.30101)]. In the paper under review the author uses 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. The author also describes the maximal subgroups, the maximal group image, and the structure of inverse subsemigroups generated by the stable letters in the HNN extensions. Several examples are discussed at the end of the paper.