Every $(\lambda ^+,\varkappa ^+)$-regular ultrafilter is $(\lambda ,\varkappa )$-regular

Abstract:

We prove the following:

Theorem. If $D$ is a $(\lambda ^+,\varkappa )$-regular ultrafilter, then either

1. [(a)] $D$ is $(\lambda ,\varkappa )$-regular, or

2. [(b)] the cofinality of the linear order $\prod _D\langle \lambda ,<\rangle$ is $\operatorname {cf}\varkappa$, and $D$ is $(\lambda ,\varkappa ’)$-regular for all $\varkappa ’<\varkappa$.

Corollary. Suppose that $\varkappa$ is singular, $\varkappa >\lambda$ and either $\lambda$ is regular, or $\operatorname {cf}\varkappa <\operatorname {cf}\lambda$. Then every $(\lambda ^{+n},\varkappa )$-regular ultrafilter is $(\lambda ,\varkappa )$-regular.

We also discuss some consequences and variations.