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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<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1802.04166v2">arXiv:1802.04166v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/5tj67zxhvre7hlifnrsxgaevty">fatcat:5tj67zxhvre7hlifnrsxgaevty</a> </span>
more »... 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 2^_0 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.
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