Classification of relational structures in terms of embeddability

Relational structures play a central role in my research. In particular, I study structures which have a single binary relation, such as graphs, linear orders and trees and those with extra topological structure such as Boolean algebras and ordered spaces. Relational structures are ubiquitous in all areas of mathematics, computer and information sciences. The methods that I use are set-theoretic, mainly infinite combinatorics and forcing. My work has applications in measure theory, topology, model theory and the theories of orders and graphs. This project aims classify these types of relational structures by their embeddability relations. I concentrate on three sub-projects to this line of research, correpsonding to the top of the embeddability quasi-order (universal models), the internal structure of this ordering and the bottom (bases) of the embeddability quasi-order. The first is to determine and find connections between universality spectra for relational structures. Universal models are not only important for classification, but are important and well-studied structures in their own right. There is a strong programme in universality and much progress has been made on using this indicator to classify structures in a model-theoretic way. However, non-elementary structures and even elementary structures without certain cardinal arithmetic assumptions are not model-theoretically well-behaved and in my project we rely on set- theoretic methods to decide these questions. The second is to examine the internal structure of these embeddability relations. The questions here involve the chains and antichains in these quasi-orders or unbounded and dominating families in the embedding order. For the first type of question, we aim to classify orders (linear and partial) which have generalised notions of dense and scattered. These classifications take the form of a constructive hierarchy, which are useful tools for proving structure and combinatorial theorems about such orders. I plan to extend these results to orders which are scattered in a stronger sense. The third involves finding small bases for different types of structures. It is consistent that there is a five-element basis for linear orders. It could be that a similar basis could be formed for other elementary structures, but it is conjectured that no finite basis exists for most non-elementary structures. In my previous work I have found deep interconnections between different types of relational structures and how they embed into each other. These connections have proved to be useful in finding new approaches to solve the problems above. In the classification projects that I plan to engage in, I will build on this foundation in order to discover and apply new connections.

Relational structures played a central role in my research. In particular, I studied structures which have a single binary relation, such as graphs, linear orders and trees and those with extra topological structure such as Boolean algebras and linearly ordered topological spaces (LOTS). Relational structures are ubiquitous in all areas of mathematics, computer and information sciences. The methods that I used are set-theoretic, mainly infinite combinatorics and forcing. My work has applications in measure theory, topology, model theory and the theories of orders and graphs. This project focussed on the classification of these types of relational structures by their embeddability relations. An embedding between two structures is a structure-preserving map from one to the other. One can therefore define a relation between structures whenever an embedding exists from one into the other. Viewing this relation as a quasi-order (reflexive and transitive) gives a unique abstract perspective from which to compare structures of different types.An important marker in this quasi-order is whether or not a single structure exists at the top; that is, a structure which embeds all other structures in the set. As embeddings are structure-preserving, this so-called a universal model is the quintessential member. We discovered that under order and continuous embeddings, a universal models exists for countable LOTS (the rationals) and for separable spaces (the reals). However, in general, the universal models which exist for linear orders are not universal under this topological embedding, which shows that these embeddings are indeed more complicated. I also examined the internal structure of these embeddability relations. The questions here involved properties which can be viewed as indicators of complexity or richness of structure. We obtained many results in this area with an innovative new technique which applied to many different types of relational structures. A pattern emerged to give us collections of structural types which behave similarly.The bottom of the embeding quasi-order was also studied, that is, the foundational structures, called a basis, which cannot be broken down into simpler ones. It is consistent that there is a five-element basis for linear orders and we extended this result to find that there is an eleven-element basis for all uncountable LOTS. I have found deep interconnections between different types of relational structures and how they embed into each other. This research has also produced fascinating new techniques which can be applied to many other contexts.