Initial Tukey structure below a stable ordered-union ultrafilter

TL;DR

This paper classifies the Tukey structure below stable ordered-union ultrafilters, revealing exactly four nonprincipal Tukey classes and connecting this classification to Rudin-Keisler classes, using advanced canonization techniques.

Contribution

It provides the first complete classification of Tukey classes below stable ordered-union ultrafilters, extending the understanding of ultrafilter hierarchies.

Findings

Exactly four nonprincipal Tukey classes below any stable ordered-union ultrafilter.

Established a simplified canonization theorem for fronts on FIN^{[ abla]}.

Classified the Rudin-Keisler classes of all ultrafilters Tukey below a stable ordered-union ultrafilter.

Abstract

Answering a question of Dobrinen and Todorcevic, we prove that below any stable ordered-union ultrafilter $\mathcal{U}$, there are exactly four nonprincipal Tukey classes: $[\mathcal{U}], [\mathcal{U}_{\operatorname{min}}], [\mathcal{U}_{\operatorname{max}}]$, and $[\mathcal{U}_{\operatorname{minmax}}]$. This parallels the classification of ultrafilters Rudin-Keisler below $\mathcal{U}$ by Blass. A key step in the proof involves modifying the proof of a canonization theorem of Klein and Spinas for Borel functions on $\mathrm{FIN}^{[\infty]}$ to obtain a simplified canonization theorem for fronts on $\mathrm{FIN}^{[\infty]}$, recovering Lefmann's canonization for fronts of finite uniformity rank as a special case. We use this to classify the Rudin-Keisler classes of all ultrafilters Tukey below $\mathcal{U}$, which is then applied to achieve the main result.