On embedding of partially ordered sets in $(\beta\omega,\le_{RK})$

TL;DR

This paper investigates the embeddability of finite partial orders into the Rudin--Keisler order on ultrafilters, demonstrating in ZFC that certain large lattices can be embedded with relations between the Rudin--Keisler and Comfort orders.

Contribution

It proves that in ZFC, large lattices of finite and countable subsets can be embedded into ultrafilters with relations between the Rudin--Keisler and Comfort orders, extending prior results.

Findings

Embeddability of large lattices in ultrafilters in ZFC.

Re-proving Blass' result without CH.

Embedding of ordered lattices with intermediate relations.

Abstract

A natural question, which appeared as Problem 61 in Hart and van Mill's list of open problems on $\beta\omega$ (2024), asks whether every finite partial order is embeddable in the Rudin--Keisler order on (types of) ultrafilters over a countable set. Although the positive answer, even for all countable partial orders, was proved under CH in Blass' thesis (1970), the situation in ZFC alone remains widely open. We show that, in ZFC, not only this result by Blass can be re-proved, but, moreover, the ordered by inclusion lattices of finite subsets of a set of cardinality $2^{\mathfrak c}$, and of countable subsets of a set of cardinality $\aleph_1$, both are embeddable in ultrafilters with any relation lying between the Rudin--Keisler and Comfort orders.