REDUCED POWERS OF SOUSLIN TREES. (16th January 2017)

• Record Type: Journal Article
• Authors: BRODSKY, ARI MEIR; RINOT, ASSAF
• Is Part Of: Forum of mathematics. Volume 5(2017)
• Publication Date: 2017-01-16
• Subjects: 03E05 (primary), -- 03E65, -- 03E35, -- 05C05 (secondary); Mathematics -- Periodicals; 510
• DOI: 10.1017/fms.2016.34
• Languages: English
• ISSNs: 2050-5094

Abstract

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.