On embedding of partially ordered sets in $(βω,\le_{RK})$
Nikolai L. Poliakov, Denis I. Saveliev
TL;DR
This work resolves Hart–van Mill Problem 61 by showing in ZFC that finite partial orders embed into the Rudin–Keisler order on ultrafilters over $\omega$ and, more strongly, that lattices $\mathscr P_{\omega}(2^{\mathfrak c})$ and $\mathscr P_{\omega_1}(\omega_1)$ embed into $(\bm\upbeta\omega,\le_{RK})$. The authors develop embeddings that respect not only $\le_{RK}$ but also any intermediate relation $R$ with $\le_{RK}\subseteq R\subseteq \le_C$, using a synthesis of weak $P$-points, ultrafilter tensor products, and careful constructions based on ladder systems. They prove two main theorems: one embedding finite-subset lattices into $\bm\upbeta\omega$ and, separately, an embedding of countable-ordinal lattices $\mathscr P_{\omega_1}(\omega_1)$ via $\omega_1$-indexed sequences and a recursive construction of auxiliary ultrafilters. These results yield universality statements for posets that are locally finite or locally countable up to cardinalities $2^{\mathfrak c}$ and $\aleph_1$, respectively, contributing to the understanding of the RK–C gap and providing new ZFC tools for structuring the Rudin–Keisler order.
Abstract
A natural question, which appeared as Problem 61 in Hart and van Mill's list of open problems on $βω$ (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.