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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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References

REFERENCES

[1]Bonato, A. and Delić, D.The monoid of the random graph. Semigroup Forum 61 (2000), 138–148.Google Scholar
[2]Cameron, P. J.The random graph. The Mathematics of Paul Erdős, II. Algorithms Combin. 14 (Springer-Verlag, Berlin, 1997), 333–351.Google Scholar
[3]Cameron, P. J. and Nešetřil, J.Homomorphism-homogeneous relational structures. Combin. Probab. Comput. 15 (2006), 91–103.CrossRefGoogle Scholar
[4]Clifford, A. H. and Preston, G. B.The Algebraic Theory of Semigroups, Vol. I. Math. Surveys No. 7 (Amer. Math. Soc., Providence, RI, 1961).Google Scholar
[5]Dolinka, I.A characterization of retracts in certain Fraïssé limits. Math. Log. Quart. 58 (2012), 46–54.Google Scholar
[6]Dolinka, I.The Bergman property for endomorphism monoids of some Fraïssé limits. Forum Math. 26 (2014), 357–376.Google Scholar
[7]de Groot, J.Groups represented by homeomorphism groups. Math. Annalen 138 (1959), 80–120.Google Scholar
[8]Evans, D. M.Examples of ℵ0-categorical structures. Automorphisms of first-order structures. Oxford Sci. Publ. (Oxford Univ. Press, New York, 1994), 33–72.Google Scholar
[9]Fraïssé, R.Sur certaines relations qui généralisent l'ordre des nombres rationnels. C. R. Acad. Sci. Paris 237 (1953), 540–542.Google Scholar
[10]Frucht, R.Herstellung von Graphen mit vorgegebener abstrakter Gruppe. Compositio Math. 6 (1939), 239–250.Google Scholar
[11]Goldstern, M., Grossberg, R. and Kojman, M.Infinite homogeneous bipartite graphs with unequal sides. Discrete Math. 149 (1996), 69–82.Google Scholar
[12]Hodges, W.A Shorter Model Theory. (Cambridge University Press, Cambridge, 1997).Google Scholar
[13]Howie, J. M.Fundamentals of Semigroup Theory. London Math. Soc. Monographs New Ser. 12 (Oxford University Press, Oxford, 1995).Google Scholar
[14]Lockett, D. C. and Truss, J. K.Generic endomorphisms of homogeneous structures. Groups and model theory. Contemp. Math. 576 (Amer. Math. Soc., Providence, RI, 2012), 217–237.Google Scholar
[15]Macpherson, H. D. and Tent, K.Simplicity of some automorphism groups. J. Algebra 342 (2011), 40–52.Google Scholar
[16]Magill, K. D. Jr., and Subbiah, S.Green's relations for regular elements of semigroups of endomorphisms. Canad. J. Math. 26 (1974), 1484–1497.Google Scholar
[17]McPhee, J. D.Endomorphisms of Fraïssé Limits and Automorphism Groups of Algebraically Closed Relational Structures. PhD thesis, University of St Andrews, 2012.Google Scholar
[18]Rhodes, J. and Steinberg, B.The q-Theory of Finite Semigroups. Springer Monographs Math. (Springer, New York, 2009).Google Scholar
[19]Sabidussi, G.Graphs with given infinite group. Monat. Math. 64 (1960), 64–67.Google Scholar
[20]Truss, J. K.The group of the countable universal graph. Math. Proc. Cambridge Philos. Soc. 98 (1985), 213–245.Google Scholar