On Presburger Arithmetic, Nonstandard Finite Cyclic Groups, and Definable Compactifications

Doctoral Dissertation

Abstract

It is known that any model of the theory of the group of integers can be decomposed into a direct sum of a torsion-free divisible abelian group and an elementary substructure of the profinite group. We give a similar result for models of the theory of Presburger arithmetic and discuss orderings on direct summands. We show that the torsion-free divisible abelian group is densely ordered and we find the number of non-isomorphic expansions of the profinite group to a model of Presburger arithmetic. We also give a description of the f-generic types of saturated models of Presburger arithmetic.

We consider nonstandard analogues of finite cyclic groups as a family of groups defined in an elementary extension of Presburger arithmetic. Since the theory of Presburger arithmetic has NIP, any such group H has a smallest type-definable subgroup of bounded index. Each quotient is a compact group under the logic topology. The main result of this thesis is the classification of these compact groups.

The universal definable compactification of a group G, in a language in which all the subsets of G are definable, coincides with the Bohr compactification bG of G considered as a discrete group. For an abelian group G, in particular the group of integers, we compute the type-connected component. We show that adding predicates for certain subsets of G is enough to get bG as the universal compactification.

Attributes

Attribute NameValues
Author Somayeh Vojdani
Contributor Philipp Hieronymi, Committee Member
Contributor Peter Cholak, Committee Member
Contributor Gabriel Conant, Committee Member
Contributor Anand Pillay, Research Director
Degree Level Doctoral Dissertation
Degree Discipline Mathematics
Degree Name Doctor of Philosophy
Defense Date
  • 2016-04-06

Submission Date 2016-04-15
Subject
  • Model theory

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