The domination monoid in o-minimal theories

Randomizing O-minimal Theories

The randomization of a complete first order theory T is the complete continuous theory T R with two sorts, a sort for random elements of models of T , and a sort for events in an underlying atomless probability space. We show that if T has the exchange property and acl = dcl, then T R has a strict independence relation in the home sort, and hence is real rosy. In particular, if T is o-minimal, ...

Types in o - minimal theories

Types in o-minimal theories by Janak Daniel Ramakrishnan Doctor of Philosophy in Mathematics University of California, Berkeley Professor Thomas Scanlon, Chair We extend previous work on classifying o-minimal types, and develop several applications. Marker developed a dichotomy of o-minimal types into “cuts” and “noncuts,” with a further dichotomy of cuts being either “uniquely” or “non-uniquel...

Compact domination for groups definable in linear o-minimal structures

Wild theories with o-minimal open core

Let T be a consistent o-minimal theory extending the theory of densely ordered groups and let T ′ be a consistent theory. Then there is a complete theory T ∗ extending T such that T is an open core of T ∗, but every model of T ∗ interprets a model of T ′. If T ′ is NIP, T ∗ can be chosen to be NIP as well. From this we deduce the existence of an NIP expansion of the real field that has no dista...

The Borel Complexity of Isomorphism for O-Minimal Theories

Given a countable o-minimal theory T , we characterize the Borel complexity of isomorphism for countable models of T up to two model-theoretic invariants. If T admits a nonsimple type, then it is shown to be Borel complete by embedding the isomorphism problem for linear orders into the isomorphism problem for models of T . This is done by constructing models with specific linear orders in the t...