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Permutation monoids and MB-homogeneity for graphs and relational structures

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ColemanEvansGrayEJCCorrectedFinalRevisedFebruary2019.pdf (410.9Kb)
Date
05/2019
Author
Coleman, Thomas D. H.
Gray, Robert
Evans, David
Keywords
Biomorphisms
MB-homogeneous
Cancellative monoids
Permutation monoids
Oligomorphic transformation monoids
Homomorphism-homogeneous structures
Infinite graph theory
QA Mathematics
T-NDAS
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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 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.
Citation
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
Publication
European Journal of Combinatorics
Status
Peer reviewed
DOI
https://doi.org/10.1016/j.ejc.2019.02.005
ISSN
0195-6698
Type
Journal article
Rights
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
Description
This 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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  • University of St Andrews Research
URI
http://hdl.handle.net/10023/19599

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