Reducibility, Degree Spectra, and Lowness in Algebraic Structures

This dissertation addresses questions in computable structure theory, which is a branch of mathematical logic hybridizing computability theory and the study of familiar mathematical structures. We focus on algebraic structures, which are standard topics of discussion among model theorists. The structures examined here are fields, graphs, trees under a predecessor function, and Boolean algebras. For a computable field F, the splitting set SF of F is the set of polynomials in F[X] which factor over F, and the root set RF of F is the set of polynomials in F[X] which have a root in F. Results of Fröhlich and Shepherdson from 1956 imply that for a computable field F, the splitting set SF and the root set RF are Turing-equivalent. Much more recently, in 2010, R. Miller showed that for algebraic fields, if we use a finer measure, the root set actually has slightly higher complexity: for algebraic fields F, it is always the case that SF ≤1 RF, but there are algebraic fields F where we have RF \nleq1 SF . In the first chapter, we compare the splitting set and the root set of a computable algebraic field under a different reduction: the bounded Turing (bT) reduction. We construct a computable algebraic field for which RN \lneq1bT SF. We also define a Rabin embedding g of a field into its algebraic closure, and for a computable algebraic field F, we compare the relative complexities of RF, SF, and g(F) under m-reducibility and under bT-reducibility. Work by R. Miller in 2009 proved several theorems about algebraic fields and computable categoricity. Also in 2009, A. Frolov, I. Kalimullin, and R. Miller proved some results about the degree spectrum of an algebraic field when viewed as a subfield of its algebraic closure. In the second chapter, we show that the same computable categoricity results also hold for finite-branching trees under the predecessor function and for connected, finitevalence, pointed graphs, and we show that the degree spectrum results do not hold for these trees and graphs. We also offer an explanation for why the degree spectrum results distinguish these classes of structures: although all three structures are algebraic structures, the fields are what we call effectively algebraic. Every lown Boolean algebra, for 1 ≤ n ≤ 4, is isomorphic to a computable Boolean algebra. It is not yet known whether the same is true for n \u3e 4. However, it is known that there exists a low5 subalgebra of the computable atomless Boolean algebra which, when viewed as a relation on the computable atomless Boolean algebra, does not have a computable copy. In the third chapter, we adapt the proof of this recent result to show that there exists a low4 subalgebra of the computable atomless Boolean algebra B which, when viewed as a relation on B, has no computable copy. This result provides a sharp contrast with the one which shows that every low4 Boolean algebra has a computable copy. That is, the spectrum of the subalgebra as a unary relation can contain a low4 degree without containing the degree 0, even though no spectrum of a Boolean algebra (viewed as a structure) can do the same. We also point out that unlike Boolean algebras as structures, which cannot have nth-jump degree above 0(n), subalgebras of B considered as relations on B can have nth-jump degree strictly bigger than 0(n)

A guide to the Rado graph : exploring structural and logical properties of the Rado graph

Dissertation (MSc (Mathematics))--University of Pretoria, 2023.The Rado graph, denoted R, is the unique (up to isomorphism) countably infinite random graph. It satisfies the extension property, that is, for two finite sets U and V of vertices of R there is a vertex outside of both U and V connected to every vertex in U and none in V . This property of the Rado graph allows us to prove quite a number of interesting results, such as a 0-1-law for graphs. Amongst other things, the Rado graph is partition regular, non-fractal, ultrahomogeneous, saturated, resplendent, the Fra´ıss´e-limit of the class of finite graphs, a non-standard model of the first-order theory of finite graphs, and has a complete decidable theory. We classify the definable subgraphs of the Rado graph and prove results for finite graphs that satisfy a restricted version of the extension property. We also mention some parallels between the rationals viewed as a linear order and the Rado graph.Mathematics and Applied MathematicsMSc (Mathematics)Unrestricte

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

The Universality Problem

The theme of this thesis is to explore the universality problem in set theory in connection to model theory, to present some methods for finding universality results, to analyse how these methods were applied, to mention some results and to emphasise some philosophical interrogations that these aspects entail. A fundamental aspect of the universality problem is to find what determines the existence of universal objects. That means that we have to take into consideration and examine the methods that we use in proving their existence or nonexistence, the role of cardinal arithmetic, combinatorics etc. The proof methods used in the mathematical part will be mostly set-theoretic, but some methods from model theory and category theory will also be present. A graph might be the simplest, but it is also one of the most useful notions in mathematics. We show that there is a faithful functor F from the category L of linear orders to the category G of graphs that preserves model theoretic-related universality results (classes of objects having universal models in exactly the same cardinals, and also having the same universality spectrum). Trees constitute combinatorial objects and have a central role in set theory. The universality of trees is connected to the universality of linear orders, but it also seems to present more challenges, which we survey and present some results. We show that there is no embedding between an ℵ2-Souslin tree and a non-special wide ℵ2 tree T with no cofinal branches. Furthermore, using the notion of ascent path, we prove that the class of non-special ℵ2-Souslin tree with an ω-ascent path a has maximal complexity number, 2ℵ2 = ℵ3. Within the general framework of the universality problem in set theory and model theory, while emphasising their approaches and their connections with regard to this topic, we examine the possibility of drawing some philosophical conclusions connected to, among others, the notions of mathematical knowledge, mathematical object and proof