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

Working papers by Cornell University. Series math "arxiv.org". 2025. P. 1–13.

Polyakov N. L., Saveliev D.

A natural question, which appeared as Problem 61 in Hart and van Mill's list of open problems on  (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 , and of countable subsets of a set of cardinality , both are embeddable in ultrafilters with any relation lying between the Rudin--Keisler and Comfort orders.

Keywords: частично упорядоченное множество, ultrafilter, poset, ультрафильтр, Rudin-Keisler preorder, Comfort preorder, предпорядок Рудин-Кейслера, предпорядок Комфорта