Reduced powers of Souslin trees

TL;DR

This paper explores the relationship between Souslin trees and their reduced powers, developing tools to control multiple powers simultaneously, and demonstrates a consistency result involving ultrafilters and Aronszajn trees.

Contribution

It introduces new methods for analyzing multiple reduced powers of Souslin trees simultaneously, extending previous single-power approaches.

Findings

Established the consistency of an -Souslin tree with a sequence of ultrafilters.

Showed that certain reduced powers are -Aronszajn iff the index is not prime.

Developed tools for controlling multiple powers of Souslin trees at once.

Abstract

We study the relationship between a $\kappa$-Souslin tree $T$ and its reduced powers $T^\theta/\mathcal U$. Previous works addressed this problem from the viewpoint of a single power $\theta$, whereas here, tools are developed for controlling different powers simultaneously. As a sample corollary, we obtain the consistency of an $\aleph_6$-Souslin tree $T$ and a sequence of uniform ultrafilters $\langle \mathcal U_n\mid n<6\rangle$ such that $ T^{\aleph_n}/\mathcal U_n$ is $\aleph_6$-Aronszajn iff $n<6$ is not a prime number.