Hnn - Extensions of F Inite I Nverse S Emigroups
نویسندگان
چکیده
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 HNNextension 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 contextfree 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
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