Permutation monoids and MB-homogeneity for graphs and relational structures
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In this paper we investigate the connection between infinite permutation monoids and bimorphism monoids of first-order structures. Taking our lead from the study of automorphism groups of structures as infinite permutation groups and the more recent developments in the field of homomorphism-homogeneous structures, we establish a series of results that underline this connection. Of particular interest is the idea of MB-homogeneity; a relational structure M is MB-homogeneous if every monomorphism between finite substructures of M extends to a bimorphism of M. The results in question include a characterisation of closed permutation monoids, a Fraïssé-like theorem for MB-homogeneous structures, and the construction of 2N0 pairwise non-isomorphic countable MB-homogeneous graphs. We prove that any finite group arises as the automorphism group of some MB-homogeneous graph and use this to construct oligomorphic permutation monoids with any given finite group of units. We also consider MB-homogeneity for various well-known examples of homogeneous structures and in particular give a complete classification of countable homogeneous undirected graphs that are also MB-homogeneous.
Coleman , T D H , Gray , R & Evans , D 2019 , ' Permutation monoids and MB-homogeneity for graphs and relational structures ' , European Journal of Combinatorics , vol. 78 , pp. 163-189 . https://doi.org/10.1016/j.ejc.2019.02.005
European Journal of Combinatorics
Copyright © 2019 Elsevier Ltd. All rights reserved. This work has been made available online in accordance with the publisher’s policies. This is the author created, accepted version manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at https://doi.org/10.1016/j.ejc.2019.02.005
DescriptionThis work was supported by the EPSRC (United Kingdom) grant EP/N033353/1 ‘Special inverse monoids: subgroups, structure, geometry, rewriting systems and the word problem’.
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