Cofinal types below $\aleph_\omega$

TL;DR

This paper explores the structure of directed sets below _, revealing their diversity, connections to Catalan numbers, and embedding of Dyck path posets, advancing understanding of their cofinal types.

Contribution

It establishes a lower bound on the number of non-Tukey-equivalent directed sets and embeds Dyck path posets into the Tukey class _.

Findings

Number of non-Tukey-equivalent directed sets _n is at least c_{n+2} (Catalan number).

The Tukey class _n contains an isomorphic copy of Dyck (n+2)-path poset.

Complete description of the order structure between successive elements in the Dyck path embedding.

Abstract

It is proved that for every positive integer $n$, the number of non-Tukey-equivalent directed sets of cardinality $\leq \aleph_n$ is at least $c_{n+2}$, the $(n+2)$-Catalan number. Moreover, the Tukey class $\mathcal D_{\aleph_n} $ of directed sets of cardinality $\leq \aleph_n$ contains an isomorphic copy of the poset of Dyck $(n+2)$-paths. Furthermore, we give a complete description whether two successive elements in the copy contain another directed set in between or not.