Two Results on Cardinal Invariants at Uncountable Cardinals

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.