86 research outputs found
Infinite Lexicographic Products
We generalize the lexicographic product of first-order structures by
presenting a framework for constructions which, in a sense, mimic iterating the
lexicographic product infinitely and not necessarily countably many times. We
then define dense substructures in infinite products and show that any
countable product of countable transitive homogeneous structures has a unique
countable dense substructure, up to isomorphism. Furthermore, this dense
substructure is transitive, homogeneous and elementarily embeds into the
product. This result is then utilized to construct a rigid elementarily
indivisible structure.Comment: 20 pages, 3 figure
On indivisible structures (Model theoretic aspects of the notion of independence and dimension)
An L-structure M is said to be invisible if for any partition M = X ∨ Y, X or Y contains a copy of M as a substructure. In this note we discuss some examples of indivisible structures and their common properties
Products of Classes of Finite Structures
We study the preservation of certain properties under products of classes of
finite structures. In particular, we examine age indivisibility,
indivisibility, definable self-similarity, the amalgamation property, and the
disjoint n-amalgamation property. We explore how each of these properties
interact with the wreath product, direct product, and free superposition of
classes of structures. Additionally, we consider the classes of theories which
admit configurations indexed by these products.Comment: 33 page
Combinatorial Properties of Finite Models
We study countable embedding-universal and homomorphism-universal structures
and unify results related to both of these notions. We show that many universal
and ultrahomogeneous structures allow a concise description (called here a
finite presentation). Extending classical work of Rado (for the random graph),
we find a finite presentation for each of the following classes: homogeneous
undirected graphs, homogeneous tournaments and homogeneous partially ordered
sets. We also give a finite presentation of the rational Urysohn metric space
and some homogeneous directed graphs.
We survey well known structures that are finitely presented. We focus on
structures endowed with natural partial orders and prove their universality.
These partial orders include partial orders on sets of words, partial orders
formed by geometric objects, grammars, polynomials and homomorphism orders for
various combinatorial objects.
We give a new combinatorial proof of the existence of embedding-universal
objects for homomorphism-defined classes of structures. This relates countable
embedding-universal structures to homomorphism dualities (finite
homomorphism-universal structures) and Urysohn metric spaces. Our explicit
construction also allows us to show several properties of these structures.Comment: PhD thesis, unofficial version (missing apple font
Uniformity, Universality, and Computability Theory
We prove a number of results motivated by global questions of uniformity in
computability theory, and universality of countable Borel equivalence
relations. Our main technical tool is a game for constructing functions on free
products of countable groups.
We begin by investigating the notion of uniform universality, first proposed
by Montalb\'an, Reimann and Slaman. This notion is a strengthened form of a
countable Borel equivalence relation being universal, which we conjecture is
equivalent to the usual notion. With this additional uniformity hypothesis, we
can answer many questions concerning how countable groups, probability
measures, the subset relation, and increasing unions interact with
universality. For many natural classes of countable Borel equivalence
relations, we can also classify exactly which are uniformly universal.
We also show the existence of refinements of Martin's ultrafilter on Turing
invariant Borel sets to the invariant Borel sets of equivalence relations that
are much finer than Turing equivalence. For example, we construct such an
ultrafilter for the orbit equivalence relation of the shift action of the free
group on countably many generators. These ultrafilters imply a number of
structural properties for these equivalence relations.Comment: 61 Page
Multicoloured Random Graphs: Constructions and Symmetry
This is a research monograph on constructions of and group actions on
countable homogeneous graphs, concentrating particularly on the simple random
graph and its edge-coloured variants. We study various aspects of the graphs,
but the emphasis is on understanding those groups that are supported by these
graphs together with links with other structures such as lattices, topologies
and filters, rings and algebras, metric spaces, sets and models, Moufang loops
and monoids. The large amount of background material included serves as an
introduction to the theories that are used to produce the new results. The
large number of references should help in making this a resource for anyone
interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will
appear in physic
- …