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HNN extensions of inverse semigroups
Abstract
The concept of HNN extensions of groups was introduced by Higman, Neumann and Neumann in their study of embeddability of groups in 1949. HNN extensions and amalgamated free products have played a crucial role in combinatorial group theory, especially for algorithmic problems. In semigroup theory, Howie introduced the concept of HNN extensions of semigroups and proved the embeddability property in some cases and T. E. Hall showed the embeddability of one type of HNN extensions of inverse semigroups. In this thesis we discuss embeddability, applications, a generalization of Britton's lemma and the structure of maximal subgroups of full HNN extensions to develop the study of inverse semigroup presentation by generators and relations as in group theory. First of all, we give several definitions of HNN extensions of inverse semigroups and investigate embeddability properties. One of our main purposes of the study of HNN extensions is to employ HNN extensions to study some algorithmic problems in inverse semigroups. We prove the undecidability of any Markov property of finitely presented inverse semigroups and the undecidability of several non-Markov properties. We also apply these ideas to give an alternative proof of Reilly's theorem about embeddability of an inverse semigroup into a bisimple inverse monoid. These results indicate that the concept of HNN extensions is a powerful tool to study algorithmic problems in inverse semigroups. For these reasons we are interested in the structure of HNN extensions of inverse semigroups. A normal form for full HNN extensions is provided and used to prove the solvability of the word problem under some conditions. We describe the maximal subgroups of full HNN extensions of an inverse monoid M as the fundamental group of a graph of groups induced from the ${\cal D}$-class structure of M using Bass-Serre theory.
Subject Area
Mathematics
Recommended Citation
Yamamura, Akihiro, "HNN extensions of inverse semigroups" (1996). ETD collection for University of Nebraska-Lincoln. AAI9712534.
https://digitalcommons.unl.edu/dissertations/AAI9712534