In this thesis, we consider questions relating to automorphisms and endomorphisms of countable, relational first-order structures M, with a particular emphasis on bimorphism monoids. We determine semigroup-theoretic results for three types of endomorphism monoid onM, along with generation results whenMis the random graph R or the discrete linear order (N;_). In addition, we introduce three types of partial map monoid ofM, and prove some semigroup-theoretic and generation results in these cases. We introduce the idea of a permutation monoid, and characterise the closed submonoids of the infinite symmetric group Sym(N). Following this, we turn our attention the idea of oligomorphic transformation monoids, and expand on the existing results by considering a range of notions of homomorphismhomogeneity as introduced by Lockett and Truss in 2012. Furthermore, we show that for any finite group G, there exists an oligomorphic permutation monoid with group of units isomorphic to G. The main result of the thesis is an analogue of Fra¨ıss´e’s theorem covering twelve of the eighteen notions of homomorphism-homogeneity; this contains both Fra¨ıss´e’s theorem, and a version of this for MM-homogeneous structures by Cameron and Neˇsetˇril in 2006, as corollaries. This is then used to determine the extent to which some well-known countable homogeneous structures are also homomorphism-homogeneous. Finally, we turn our attention to MB-homogeneous graphs and digraphs. We begin by classifying those homogeneous graphs that are also MB-homogeneous. We then determine an example of an MB-homogeneous graph not in this classification, and use the idea behind this construction to demonstrate 2@0 many non-isomorphic examples of MB-homogeneous graphs. We also give 2@0 many non-isomorphic examples of MB-homogeneous digraphs.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:716451 |
| Date | January 2017 |
| Creators | Coleman, Thomas |
| Publisher | University of East Anglia |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | https://ueaeprints.uea.ac.uk/64074/ |
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