Schutzenberger automata for HNN-extensions of inverse monoids and their use in algorithmic questions

Schutzenberger automata are instrumental in the study of structural and algorithmic questions for inverse semigroup presentations as shown by the combinatorial approach introduced by Munn [1], and extended by Stephen [2]. HNN-extensions, a classical construction originally from group theory, proved useful in the study of decidability questions when introduced to the classes of semigroups and inverse semigroups. In our work [3], we studied HNN-extensions of inverse semigroups via structure of their Schutzenberger automata. The main result of this paper is a characterization of the Schutzenberger automata of a rather rich and interesting class of lower bounded HNN-extensions. The automata have an especially nice lobe structure, and contain a special subgraph - a core - a subgraph with finitely many lobes from which all vital information about the automaton can be restored. These nice properties yield in some cases an effective construction of the Schutzenberger automata and thus have pleasant consequences for algorithmic problems. (C) 2019 Elsevier Inc. All rights reserved.