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Automorphisms and Endomorphisms of First-Order Structures Automorphisms and endomorphisms of first-order structures A thesis submitted to the School of Mathematics at the University of East Anglia in partial fulfilment of the requirements for the degree of Doctor of Philosophy Thomas David Hawkins Coleman May 2017 © This copy of the thesis has been supplied on condition that anyone who con- sults it is understood to recognise that its copyright rests with the author and that use of any information derived there from must be in accordance with cur- rent UK Copyright Law. In addition, any quotation or extract must include full attribution. Abstract In this thesis, we consider questions relating to automorphisms and endo- morphisms 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 on M, along with generation results when M is the random graph R or the discrete linear order (N; ≤). In addition, we introduce three types of partial map monoid of M, 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 homomorphism- homogeneity 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¨ısse’s´ theorem covering twelve of the eighteen notions of homomorphism-homogeneity; this contains both Fra¨ısse’s´ theorem, and a version of this for MM-homogeneous structures by Cameron and Nesetˇ rilˇ 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 clas- sification, 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. Contents Abstract 1 List of Figures 5 Acknowledgements 7 1 Introduction 10 2 Preliminaries 17 2.1 Relations and functions . 17 2.1.1 Multifunctions . 19 2.2 Group and semigroup theory . 22 2.2.1 Initial definitions . 22 2.2.2 Monoids of transformations . 25 2.2.3 Monoid actions . 28 2.2.4 Topology on Sym(N), End(N) ................. 31 2.3 Model theory . 32 2.3.1 Initial definitions . 33 2.3.2 Maps between structures . 34 2.3.3 Automorphism groups, @0-categoricity and homogeneity 37 2.4 Graph and digraph theory . 42 2.4.1 The random graph . 44 2.4.2 Digraphs and oriented graphs . 45 2.4.3 Homogeneous graphs and digraphs . 49 3 3 Cofinality, strong cofinality and the Bergman property 51 3.1 Definitions and general results . 53 3.2 Initial example: Mon(N) ........................ 58 3.2.1 Semigroup-theoretic properties . 58 3.2.2 Generation results for Mon(N) ................ 64 4 Intermediate monoids of first-order structures 66 4.1 Bimorphisms of σ-structures . 68 4.1.1 Initial semigroup theory of Bi(M) .............. 68 4.1.2 Green’s relations of Bi(M) ................... 72 4.1.3 Representing group-embeddable monoids . 77 4.1.4 Bimorphisms of graphs . 80 4.2 Embeddings of σ-structures . 86 4.2.1 Semigroup theory of Emb(M) ................ 86 4.2.2 Embeddings of (N; ≤) ..................... 91 4.2.3 Generation results for Emb(R) ................ 94 4.3 Monomorphisms of σ-structures . 96 4.3.1 Semigroup theory of Mon(M) ................ 97 4.3.2 Generation properties of Mon(R) ............... 102 5 Partial map monoids of first-order structures 105 5.1 Semigroup-theoretic properties . 107 5.2 Generation results . 114 5.3 The semilattice of idempotents of Inv(M) .............. 120 6 Oligomorphic transformation monoids 126 6.1 Infinite permutation monoids . 128 6.2 Oligomorphic transformation monoids . 131 7 Homomorphism-homogeneous first-order structures 139 7.1 Antihomomorphisms . 142 7.2 Two Fra¨ısse-style´ theorems . 145 7.2.1 Forth alone . 146 4 7.2.2 Back and forth . 152 7.3 Maximal homomorphism-homogeneity classes . 162 7.3.1 Examples . 165 8 MB-homogeneous graphs and digraphs 172 8.1 Properties of MB-homogeneous graphs . 175 8.2 Examples of MB-homogeneous graphs . 182 8.2.1 From countably many to uncountably many . 190 8.3 MB-homogeneous oriented graphs . 198 8.3.1 Properties of MB-homogeneous oriented graphs . 198 8.3.2 Uncountably many MB-homogeneous oriented graphs . 204 8.4 MB-homogeneous digraphs . 209 Bibliography 215 List of Figures 1.1 Transformation monoids on a countable relational structure M . 12 2.1 An example of a partial multifunction f ∗ .............. 20 2.2 An example of a surjective multifunction g∗ ............. 21 2.3 Containment of Green’s relations . 24 2.4 Some self-map monoids of a first-order structure M ........ 37 2.5 The amalgamation property (AP) . 40 2.6 Alice’s restaurant property in a countable graph Γ ......... 45 2.7 An orientation of a graph . 46 2.8 An orientation of a complete graph . 46 2.9 Oriented Alice’s restaurant property in D .............. 48 2.10 Directed Alice’s restaurant property in D∗ .............. 48 3.1 Strong distortion . 57 4.1 M as in Example 4.1.5 . 70 4.2 A(α) and M(S(α)) for α 2 Bi(M) in Example 4.1.5 . 70 4.3 Construction of Γ ............................ 78 5.1 Γ in R ................................... 118 5.2 The semilattice of idempotents E ................... 123 6.1 Table of XY-homogeneity . 134 6.2 The set H of homomorphism-homogeneity classes . 134 6.3 A rigid graph of [14] . 137 List of Figures 6 7.1 F, B, N, and I in H ........................... 146 7.2 The XY-amalgamation property (XYAP) . 147 7.3 Types of finite partial map . 153 7.4 The XY-amalgamation property (XYAP) . 154 7.5 Construction of Y 0 from Y in Example 7.3.10 . 169 8.1 A diagram of Definition 8.1.5 . 178 8.2 k even in proof of Proposition 8.1.9 . 181 8.3 k odd in proof of Proposition 8.1.9 . 181 8.4 Γ(P ), with P = (0; 1; 0; 1; :::)...................... 183 8.5 Γ(P ), with P = (1; 1; 0; 0; 1; 1; 1; 0; 0; 1; 0; :::) ............. 185 8.6 The cycle graph C4 ........................... 190 8.7 Γ(PA)0 corresponding to the sequence A = (4; 5; 6; :::) ....... 192 8.8 Γ(P )0, with ∆ highlighted in red . 195 8.9 Property () ............................... 199 8.10 k even in proof of Proposition 8.3.4 . 201 8.11 k odd in proof of Proposition 8.3.4 . 202 8.12 H2 in the construction of H(G), where G is a single vertex . 203 8.13 The cycle digraph D4 and an oriented cycle graph G on 4 vertices 205 8.14 H2 in the construction of H(G;S) ................... 207 8.15 A diagram of Definition 8.4.1 . 209 8.16 Viewing a graph as a digraph . 210 8.17 E(PA)0 corresponding to the sequence A = (4; 5; 6; :::) ....... 211 8.18 k even in proof of Proposition 8.4.8 . 213 8.19 k odd in proof of Proposition 8.4.8 . 214 Acknowledgements First, I would like to thank both of my supervisors Dr Robert Gray and Professor David Evans for their excellent supervision, many stimulating discussions, and their endless patience. I would also like to thank the School of Mathematics at the UEA as a whole for the support and opportunities that I have been given throughout my eight years here; it is no small matter to say that without it, this thesis would not have happened. I have been fortunate to meet many immensely talented mathematicians dur- ing my time as a postgraduate student, and have had many insightful and fruit- ful discussions with them. In particular, I would like to thank the NBSAN com- munity and all of the people that I have met there. Thanks also go to my GRIME colleagues Robert and Athina; for all of the workshops and coffees, and gradu- ally turning we into you. On a more personal note, I would like to thank all of my friends for their var- ious influences in my life. From Norwich and UEA, I’d like to thank: Andy for commutes, Friday drinks and a life-saving phone; Peter for being Peter; Jodie, Dan, Bobby T and everyone else in the office for keeping me grounded at ev- ery opportunity; Rob for tolerating frequent coffee breaks; Andi and everyone else at badminton for the fun on and off the court; and anyone who I’ve ever talked to (or at) about my work in a pub. From Stowmarket, thanks go to Josh, Daryl, Matt (and Charlie), Jim, Gav and Adam for all the car football and wel- come pints; and the pool team for many long and enjoyable Tuesday nights over a crib board. Special thanks go to Matt and Dane for their constant encouragement and support, all of the various sports sessions, discussions and life advice about any- thing and everything, and many, many shared drinks.
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