Permutation monoids and MB-homogeneity for graphs and relational structures

Thomas D. H. Coleman, Robert Gray, David Evans

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7 Citations (Scopus)
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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.
Original languageEnglish
Pages (from-to)163-189
JournalEuropean Journal of Combinatorics
Early online date5 Mar 2019
Publication statusPublished - May 2019


  • Biomorphisms
  • MB-homogeneous
  • Cancellative monoids
  • Permutation monoids
  • Oligomorphic transformation monoids
  • Homomorphism-homogeneous structures
  • Infinite graph theory


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