Scales in the Point Spectrum

TL;DR

This paper explores the structure of the point spectrum in directed sets using PCF theory, revealing fundamental links between the spectrum's properties and set-theoretic concepts.

Contribution

It establishes a novel connection between the singularity of the supremum of the Tukey spectrum and its cofinality within the spectrum, using PCF techniques.

Findings

If the supremum of the Tukey spectrum is singular, its cofinality is also in the spectrum.

Uncovers fundamental relationships between the point spectrum and PCF theory.

Provides new insights into the structure of directed sets and their spectra.

Abstract

We study the Point/Tukey spectrum of a general directed set using PCF theoretic tools and uncover basic connections between the theories. In particular, we prove that if the supremum of the Tukey spectrum is singular, then its cofinality must also be a member of the Tukey spectrum.