Permutation monoids and MB-homogeneity for graphs and relational structures

Coleman, Thomas, Evans, David and Gray, Robert (2019) Permutation monoids and MB-homogeneity for graphs and relational structures. European Journal of Combinatorics, 78. pp. 163-189. ISSN 0195-6698

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Abstract

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 Fraisse-like theorem for MB-homogeneous structures, and the construction of 2ℵ0 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.

Item Type: Article
Faculty \ School: Faculty of Science > School of Mathematics
Depositing User: LivePure Connector
Date Deposited: 05 Feb 2019 14:30
Last Modified: 21 Mar 2020 01:26
URI: https://ueaeprints.uea.ac.uk/id/eprint/69856
DOI: 10.1016/j.ejc.2019.02.005

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