Automorphisms and endomorphisms of first-order structures

Coleman, Thomas (2017) Automorphisms and endomorphisms of first-order structures. Doctoral thesis, University of East Anglia.

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Abstract

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.

Item Type: Thesis (Doctoral)
Faculty \ School: Faculty of Science > School of Mathematics
Depositing User: Katie Miller
Date Deposited: 11 Jul 2017 14:05
Last Modified: 11 Jul 2017 14:05
URI: https://ueaeprints.uea.ac.uk/id/eprint/64074
DOI:

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