In the first part of our talk, we will review some of the combinatorial methods used in inverse semigroup theory.

Inverse semigroups can be viewed as natural generalizations of groups: While every group can be represented via permutations (one-to-one transformations), inverse semigroups can be represented via partial one-to-one transformations. We will introduce graphs (automata), called Schützenberger automata, that are related to presentations of inverse semigroups in a way similar to the way Cayley graphs are related to groups. As shown by the combinatorial approach introduced by Munn and extended by Stephen, these automata are instrumental in the study of structural and algorithmic questions concerning inverse semigroup presentations.

In the second part of the talk, we will turn our attention to HNN extensions of inverse semigroups. HNN extensions, a classical construction originally introduced in group theory, also proved useful in the study of decidability questions in the class of inverse semigroups. We study these extensions via the structure of their Schützenberger automata. These automata are in general infinite, but in specific cases we can detect so called finite core - in the sense that all important information about the automaton is encoded in a “finite” subgraph. This nice property yields in some cases an effective construction of the Schützenberger automata and thus allows us to answer algorithmic questions (such as the Word Problem) about presentations of certain classes of inverse semigroups.

If time permits, we will also consider actions of special groups on Schützenberger graphs, which will allow us to employ the powerful Bass-Serre theory, and to consider structural questions concerning HNN extensions.