Pairs and predicates in expansions of o-minimal structures

Block Gorman, Alexi Taylor

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https://hdl.handle.net/2142/112983 Copy

Description

Title

Pairs and predicates in expansions of o-minimal structures

Author(s)

Block Gorman, Alexi Taylor

Issue Date

2021-07-08

Director of Research (if dissertation) or Advisor (if thesis)

Hieronymi, Philipp

Doctoral Committee Chair(s)

van den Dries, Lou

Committee Member(s)

• Henson, C. Ward
• Günaydin, Ayhan

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• model theory
• o-minimality
• model companions
• Büchi automata

Abstract

This thesis establishes new results concerning the interactions of some model-theoretic notions of tameness with other more geometric notions of tameness, among other properties. In particular, the results range from characterizing the existence of a model companion for a certain kind of extension of o-minimal theories by a unary predicate, to establishing a more robust framework for identifying pairs (of models of geometric theories) with near model completeness, to answering questions about what combinations of model-theoretic tameness and o-minimal open core imply a stronger tameness notion, to characterizing the tameness properties of the expansion of $(\mathbb{R},<,+,0,1)$ by a predicate for an $r$-regular subset of $[0,1]$. The focus of Chapter 3 of this thesis is the extension $T_{\mathcal{G}}$ of a complete o-minimal $\mathcal{L}$-theory $T$ to the language $\mathcal{L}_{\mathcal{G}}:=\mathcal{L} \cup \{ \mathcal{G} \}$, where $\mathcal{G}$ is a unary predicate that picks out a divisible, dense and codense subgroup of either the additive group or the multiplicative group of positive elements. The major result concerning $T_{\mathcal{G}}$ is the full characterization, in terms of $T$ alone, for when the theory $T_{\mathcal{G}}$ has a model companion. By restricting to the setting in which $T$ is o-minimal, a sufficient condition for a more general setting can here be expanded into a necessary and sufficient criterion. Examples are included both in which the predicate is an additive subgroup, and some in which it is a multiplicative subgroup. Chapter 4 expands its setting to geometric theories to examine tameness notions in pairs of models of related theories. The framework introduced applies to many pairs of structures that have been studied previously, as well as new kinds of pairs, such as pairs consisting of a real closed field and a pseudo real closed subfield and pairs of vector spaces with different fields of scalars. Before demonstrating that the set-up introduced applies to some new, interesting examples (as well as the ones mentioned that were previously studied) it is established that pairs of geometric structures that satisfy the criterion have near model completeness. Chapter 5 answers several open questions about o-minimal open cores and its interactions with other model-theoretic properties. This is achieved through the construction of a few new expansions of o-minimal structures, then establishing some results about their open cores. The first construction used is an expansion of an o-minimal structure $\mathcal{R}$ by a unary predicate whose open core is a proper o-minimal expansion of $\mathcal{R}$. Another construction used is a structure that defines a function whose graph is dense, but also has o-minimal open core and the exchange property. The final construction is a structure with o-minimal open core and definable Skolem functions that is not o-minimal. Finally, Chapter 6 considers the expansion of the real ordered additive group by a predicate for a subset of $[0,1]$ whose base-$r$ representations are recognized by a Büchi automaton. In the case that this predicate is closed, a dichotomy is established for when this expansion is interdefinable with the structure $(\mathbb{R},<,+,0,r^{-\mathbb{N}})$ for some $r \in \mathbb{N}_{>1}$. In the case that the closure of the predicate has Hausdorff dimension less than $1$, the dichotomy further characterizes these expansions of $(\mathbb{R},<,+,0,1)$ by when they have NIP and NTP$_2$, which is precisely when the closure of the predicate has Hausdorff dimension $0$.

Graduation Semester

2021-08

Type of Resource

Thesis

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http://hdl.handle.net/2142/112983

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Copyright 2021 Alexi Block Gorman

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