HNN- EXTENSIONS OF FINITE INVERSE SEMIGROUPS

Abstract

The concept of HNN-extensions of groups was introduced by Higman, Neumann and Neumann in 1949. HNN-extensions and amalgamated free products have played a crucial role in combinatorial group theory, especially for algorithmic problems. In inverse semigroup theory there are many ways of constructing HNN-extension in order to ensure the embeddability of the original inverse semigroup in the new one. For instance, Howie used unitary subsemigroups , N.D. Gilbert used ordered ideals and Yamamura put some conditions on idempotents. In this thesis we adopt Yamamura’s definition. Let S∗=[S;A,B] be an HNN-extension of inverse semigroups. We show that the word problem of HNN-extensions of inverse semigroups can be undecidable even under some nice conditions on S,A and B. Then we consider HNN-extension S∗ with S finite, because under such hypothesis the word problem is decidable and we prove that the Schützenberger graph of the elements of S∗ is a context-free graph, showing that the language recognized by the Schützenberger automaton is a deterministic context-free language. Moreover, we construct the grammar generating this language. We characterize the HNN-extensions of finite inverse semigroups which are completely semisimple inverse semigroups, using a characterization of HNN-extensions of finite inverse semigroups which have a copy of the bicyclic monoid as subsemigroup. Furthermore, we give some properties of the Schützenberger graph of the elements of HNN-extensions of finite inverse semigroups mainly focusing v Abstract on properties of the hosts, i.e. minimal finite subgraphs that contain all essential information about the automaton. We use the description of such Schützenberger automata and the Bass-Serre theory to study the maximal subgroups of the HNN-extensions of finite inverse semigroups