REDUCED POWERS OF SOUSLIN TREES

Abstract

We study the relationship between a $\unicode[STIX]{x1D705}$-Souslin tree $T$ and its reduced powers $T^{\unicode[STIX]{x1D703}}/{\mathcal{U}}$.Previous works addressed this problem from the viewpoint of a single power $\unicode[STIX]{x1D703}$, 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 if and only if $n<6$ is not a prime number.This paper is the first application of the microscopic approach to Souslin-tree construction, recently introduced by the authors. A major component here is devising a method for constructing trees with a prescribed combination of freeness degree and ascent-path characteristics.