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

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.