Tukey types of ultrafilters

TL;DR

This paper explores the structure of Tukey types of ultrafilters on countable sets, providing a canonization of cofinal maps and analyzing the Tukey types of p-points, selective ultrafilters, and block-basic ultrafilters.

Contribution

It introduces a canonization of cofinal maps between ultrafilters and classifies Tukey types for various ultrafilter classes, advancing the understanding of their hierarchical structure.

Findings

Canonization of cofinal maps from p-points to other ultrafilters.

Comparison results for selective ultrafilters with basis elements.

Embedding chains and antichains into Tukey types.

Abstract

We investigate the structure of the Tukey types of ultrafilters on countable sets partially ordered by reverse inclusion. A canonization of cofinal maps from a p-point into another ultrafilter is obtained. This is used in particular to study the Tukey types of p-points and selective ultrafilters. Results fall into three main categories: comparison to a basis element for selective ultrafilters, embeddings of chains and antichains into the Tukey types, and Tukey types generated by block-basic ultrafilters on FIN.