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

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 $\Gamma$, there are uncountably many maximal subgroups of the endomorphism monoid of $R$ isomorphic to the automorphism group of $\Gamma$. Further structural information about End $R$ is established including that Aut $\Gamma$ 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.

Minor revision following referee's comments. 27 pages, 3 figures

References in corpus (1)

• The Bergman property for endomorphism monoids of some Fra\"ıssé limits

Cited by in corpus (1)

• Permutation monoids and MB-homogeneity for graphs and relational structures