Combinatorics of ultrafilters on Cohen and random algebras

TL;DR

This paper explores the structure of ultrafilters on Boolean algebras, especially Cohen and random algebras, using Tukey reducibility, and investigates their relation to the ultrafilter number with new consistency results.

Contribution

It introduces techniques for constructing non-Tukey maximal ultrafilters and links these to the ultrafilter number, providing new insights into their possible values.

Findings

Ultrafilters not Tukey maximal can be constructed using new techniques.

Connections established between ultrafilter structure and the ultrafilter number.

Consistency results regarding ultrafilter number on Cohen and random algebras.

Abstract

We investigate the structure of ultrafilters on Boolean algebras in the framework of Tukey reducibility. In particular, this paper provides several techniques to construct ultrafilters which are not Tukey maximal. Furthermore, we connect this analysis with a cardinal invariant of Boolean algebras, the ultrafilter number, and prove consistency results concerning its possible values on Cohen and random algebras.