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

We prove the following:

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

(a)

$D$ is $(\lambda,\varkappa)$-regular, or

(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 B. 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.