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Every -regular

Authors:	Paolo Lipparini

Institution:	Dipartimento di Matematica, Viale della Ricerca Scientifica, II Università di Roma (Tor Vergata), I-00133 Rome, Italy

Abstract:	We prove the following: Theorem A. If is a $(\lambda^+,\varkappa)$ -regular ultrafilter, then either (a) is $(\lambda,\varkappa)$ -regular, or (b) the cofinality of the linear order $\prod _D\langle\lambda,<\rangle$ is $\operatorname{cf}\varkappa$ , and 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.

Keywords:	$(\alpha \varkappa)$-regular ultrafilter cofinality of ultrapowers almost $(\lambda \varkappa)$-regular extensions

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