On embedding of partially ordered sets in $({\hat{I}}^2\ddot{I}‰,{\le }_{RK})$

Abstract

A natural question, which appeared as Problem 61 in Hart and van Mill's list of open problems on ${\hat{I}}^2\ddot{I}‰$ (2024), asks whether every finite partial order is embeddable in the Rudin--Keisler order on the set 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 question in ZFC alone remained completely open. We show that, in ZFC, it is possible not only to answer in the positive for all countable orders, but, moreover, to construct embeddings of 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 ${\aleph }_1$, into the set of ultrafilters with any relation lying between the Rudin--Keisler and Comfort orders.