The Pseudopower Dichotomy

TL;DR

This paper explores the properties of pseudopowers of singular cardinals, deriving consequences for cardinal arithmetic and establishing specific covariate equalities within ZFC, supported by recent work of Gitik.

Contribution

It introduces new results on pseudopowers of singular cardinals and demonstrates a specific covariate equality in ZFC, emphasizing the necessity of both summands.

Findings

Established a covariate equality involving singular cardinals.

Showed both summands in the covariate equation are necessary.

Connected pseudopower properties to cardinal arithmetic consequences.

Abstract

We investigate pseudopowers of singular cardinals, and show that deduce some consequences for cardinal arithmetic. For example, we show that in {\sf ZFC} that $$\cov(\mu,\mu,\theta,\sigma)=\cov(\mu,\mu,(\cf\mu)^+,\sigma)+\cov(\mu,\mu,\theta,\sigma^+)$$ whenever $\aleph_1\leq\sigma=\cf(\sigma)<\cf(\mu)<\theta<\mu$, and use recent work of Gitik to show that both summands in the equation are required.