# Maximal Tukey types, P-ideals and the weak Rudin-Keisler order

## TL;DR

This paper explores ideals on natural numbers with maximal Tukey type, examines a refinement called the weak Rudin-Keisler order, and identifies an analytic P-ideal that is maximal in this order.

## Contribution

It introduces new examples of ideals with maximal Tukey type and analyzes the structure of the weak Rudin-Keisler order on these ideals, including the existence of a maximal analytic P-ideal.

## Key findings

- Identified ideals with maximal Tukey type.
- Analyzed the structure of the weak Rudin-Keisler order.
- Proved the existence of an analytic P-ideal above all others in the order.

## Abstract

In this paper, we study some new examples of ideals on $\omega$ with maximal Tukey type (that is, maximal among partial orders of size continuum). This discussion segues into an examination of a refinement of the Tukey order -- known as the "weak Rudin-Keisler order" -- and its structure when restricted to these ideals of maximal Tukey type. Mirroring a result of Fremlin on the Tukey order, we also show that there is an analytic P-ideal above all other analytic P-ideals in the weak Rudin-Keisler. Acknowledgment: this preprint has not undergone peer review or any post-submission improvements or corrections. The Version of Record of this article is published in the Archive for Mathematical Logic, and is available online at https://doi.org/10.1007/s00153-023-00897-z.