Regularity of ultrafilters, Boolean ultrapowers, and Keisler’s order

Parente, Francesco (2019) Regularity of ultrafilters, Boolean ultrapowers, and Keisler’s order. Doctoral thesis, University of East Anglia.

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This thesis investigates combinatorial properties of ultrafilters and their model-theoretic significance. Motivated by recent results on Keisler’s order, we develop new tools for the study of Boolean ultrapowers, deepening our understanding of the interplay between set theory and model theory.
The main contributions can be summarized as follows. In Chapter 2, we undertake a systematic study of regular ultrafilters on Boolean algebras. In particular, we analyse two different notions of regularity which have appeared in the literature and compare their modeltheoretic properties. We then apply our analysis to the study of cofinal types of ultrafilters; as an application, we answer a question of Brown and Dobrinen by giving two examples of complete Boolean algebras on which all ultrafilters have maximum cofinal type. In conclusion, we discuss the existence of non-regular ultrafilters and prove that, consistently, every decomposable ultrafilter on a complete Boolean algebra is regular.
Chapter 3 centres around the study of Keisler’s order. We prove that good ultrafilters on Boolean algebras are precisely the ones which capture the maximum class in Keisler’s order, solving a problem posed by Benda in 1974. We also show that, given a regular ultrafilter on a complete Boolean algebra satisfying a distributivity condition, the saturation of the Boolean ultrapower of a model of a complete theory does not depend on the choice of the particular model, but only on the theory itself. Motivated by this fact, we apply and expand the framework of ‘separation of variables’, recently developed by Malliaris and Shelah, to obtain a new characterization of Keisler’s order via Boolean ultrapowers.

Item Type: Thesis (Doctoral)
Faculty \ School: Faculty of Science > School of Mathematics
Depositing User: Users 11011 not found.
Date Deposited: 16 Oct 2019 12:59
Last Modified: 16 Oct 2019 12:59

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