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Automorphism groups of countable algebraically closed graphs and endomorphisms of the random graph

Published online by Cambridge University Press:  21 January 2016

IGOR DOLINKA ,
ROBERT D. GRAY ,
JILLIAN D. McPHEE ,
JAMES D. MITCHELL  and
MARTYN QUICK
IGOR DOLINKA
Affiliation:
Department of Mathematics and Informatics, University of Novi Sad, Trg Dositeja Obradovića 4, 21101 Novi Sad, Serbia. e-mail: dockie@dmi.uns.ac.vs
ROBERT D. GRAY
Affiliation:
School of Mathematics, University of East Anglia, Norwich NR4 7TJ. e-mail: robert.d.gray@uea.ac.uk
JILLIAN D. McPHEE
Affiliation:
School of Mathematics & Statistics, University of St Andrews, St Andrews, Fife KY16 9SS, Scotland. e-mail: jdtmcphee@gmail.com; jdm3@st-andrews.ac.uk; mq3@st-andrews.ac.uk
JAMES D. MITCHELL
Affiliation:
School of Mathematics & Statistics, University of St Andrews, St Andrews, Fife KY16 9SS, Scotland. e-mail: jdtmcphee@gmail.com; jdm3@st-andrews.ac.uk; mq3@st-andrews.ac.uk
MARTYN QUICK
Affiliation:
School of Mathematics & Statistics, University of St Andrews, St Andrews, Fife KY16 9SS, Scotland. e-mail: jdtmcphee@gmail.com; jdm3@st-andrews.ac.uk; mq3@st-andrews.ac.uk
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Abstract

We establish links between countable algebraically closed graphs and the endomorphisms of the countable universal graph R. As a consequence we show that, for any countable graph Γ, there are uncountably many maximal subgroups of the endomorphism monoid of R isomorphic to the automorphism group of Γ. Further structural information about End R is established including that Aut Γ arises in uncountably many ways as a Schützenberger group. Similar results are proved for the countable universal directed graph and the countable universal bipartite graph.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 160 , Issue 3 , May 2016 , pp. 437 - 462
Copyright
Copyright © Cambridge Philosophical Society 2016 

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