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An amalgam of inverse semigroups is embedded into an amalgam with a lower bounded core

Paul Bennett

Department of Mathematics, University of York, York, YO10 5DD, United Kingdom

E-mail Address: paul_bennett89@hotmail.com

https://doi.org/10.1142/S0218196721400129Cited by:0 (Source: Crossref)

This article is part of the issue:

Special Issue on Papers From the Conference “Semigroups and Groups, Automata, Logics” SandGAL 2019
Guest Editors: Achille Frigeri, Victoria Gould, Patrizia Longobardi and Emanuele Rodaro

Abstract

Given any amalgam $[S_{1}, S_{2}; U]$ of inverse semigroups, we show how to construct an amalgam $[T_{1}, T_{2}; Z]$ such that $S_{1} *_{U} S_{2}$ is embedded into $T_{1} *_{Z} T_{2}$ , where $S_{1} \subseteq T_{1}$ , $S_{2} \subseteq T_{2}$ , $S_{1} \cap Z = S_{2} \cap Z = U$ and, for any $z \in Z$ and $h \in E (T_{i})$ with $z \geq h$ in $T_{i}$ , where $i \in {1, 2}$ , there exists $f \in E (Z)$ with $z \geq f \geq h$ in $T_{i}$ ; that is, $Z$ is a lower bounded subsemigroup of $T_{1}$ and $T_{2}$ . A recent paper by the author describes the Schützenberger automata of $T_{1} *_{Z} T_{2}$ , for an amalgam $[T_{1}, T_{2}; Z]$ where $Z$ is lower bounded in $T_{1}$ and $T_{2}$ , giving conditions for $T_{1} *_{Z} T_{2}$ to have decidable word problem. Thus we can study $S_{1} *_{U} S_{2}$ by considering $T_{1} *_{Z} T_{2}$ . As an example, we generalize results by Cherubini, Jajcayová, Meakin, Piochi and Rodaro on amalgams of finite inverse semigroups.

Communicated by all the special editors

Keywords:

AMSC: 20M18

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