Boolean Types in Dependent Theories

TL;DR

This paper introduces Boolean types in dependent theories, exploring their properties, especially in NIP theories, and connects them to Keisler measures, providing new insights and tools for model theory research.

Contribution

It generalizes the notion of complete types to Boolean algebra-valued types, analyzes their behavior in NIP theories, and links them to Keisler measures, offering new methods and results.

Findings

Boolean types are well-behaved in NIP theories

Every measure in a dependent theory can be extended to a smooth one

Connections established between Boolean types and Keisler measures

Abstract

The notion of a complete type can be generalized in a natural manner to allow assigning a value in an arbitrary Boolean algebra B to each formula. We show some basic results regarding the effect of the properties of B on the behavior of such types, and show they are particularity well behaved in the case of NIP theories. In particular, we generalize the third author's result about counting types, as well as the notion of a smooth type and extending a type to a smooth one. We then show that Keisler measures are tied to certain Boolean types and show that some of the results can thus be transferred to measures - in particular, giving an alternative proof of the fact that every measure in a dependent theory can be extended to a smooth one. We also study the stable case. We consider this paper as an invitation for more research into the topic of Boolean types.