Two results on cardinal invariants at uncountable cardinals

TL;DR

This paper establishes two ZFC theorems about cardinal invariants at uncountable regular cardinals, revealing contrasts with their behavior at the continuum and advancing understanding of their relationships.

Contribution

It proves that certain inequalities among cardinal invariants at uncountable regular cardinals hold universally, improving previous results and providing new insights into their structure.

Findings

If ppa is uncountable regular and (ppa)=ppa^{+}, then (ppa)=ppa^{+}

For ppa et;, (ppa)  r(ppa)

Results contrast with known facts about invariants at the continuum

Abstract

We prove two ZFC theorems about cardinal invariants above the continuum which are in sharp contrast to well-known facts about these same invariants at the continuum. It is shown that for an uncountable regular cardinal $\kappa$, $\mathfrak{b}(\kappa) = {\kappa}^{+}$ implies $\mathfrak{a}(\kappa) = {\kappa}^{+}$. This improves an earlier result of Blass, Hyttinen, and Zhang. It is also shown that if $\kappa \geq {\beth}_{\omega}$ is an uncountable regular cardinal, then $\mathfrak{d}(\kappa) \leq \mathfrak{r}(\kappa)$. This result partially dualizes an earlier theorem of the authors.