We study the relationship between κ-Souslin tree T and its reduced powers Tθ/u. Previous works addressed this problem from the viewpoint of a single power , whereas here, tools are developed for controlling different powers simultaneously. As a sample corollary, we obtain the consistency of an -Souslin tree and a sequence of uniform ultrafilters (μn | n < 6) such that Tℵn is ℵ6-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.
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