The directedness of the Rudin-Keisler order at measurable cardinals

Abstract

The manuscript is concerned with the Rudin-Keisler order of ultrafilters on measurable cardinals. The main theorem proved read as follows: Given regular cardinals $\lambda \le \kappa$, the following theories are equiconsistent modulo ZFC: (1) $\kappa$ is a measurable cardinal with $o(\kappa )={\lambda }^{+}$ (resp. $o(\kappa )=\kappa$). (2) The Rudin-Keisler order restricted to the set of $\kappa$-complete (non-principal) ultrafilters on $\kappa$ is ${\lambda }^{+}$-directed (resp. ${\kappa }^{+}$-directed). The theorem reported here is proved after bridging the directedness of the RK-order with the $\lambda$-Gluing Property introduced by the authors in \cite{HP}. Our result provides what seems to be the first example of a compactness-type property at the level of measurable cardinals whose consistency strength is much lower than the existence of a strong cardinal. As part of our analysis we also answer a question of Gitik by showing that the ${\aleph }_0$-Gluing Property fails in his classical model from ''Changing cofinalities and the nonstationary ideal". As a consequence of this, in Gitik's model the Rudin-Keisler order fails to be ${\aleph }_1$-directed.