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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Weakly normal filters and irregular ultrafilters

Author: A. Kanamori
Journal: Trans. Amer. Math. Soc. 220 (1976), 393-399
MSC: Primary 04A20; Secondary 02K35
MathSciNet review: 0480041
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Abstract: For a filter over a regular cardinal, least functions and the consequent notion of weak normality are described. The following two results, which make a basic connection between the existence of least functions and irregularity of ultrafilters, are then proved: Let U be a uniform ultrafilter over a regular cardinal $ \kappa $. (a) If $ \kappa = {\lambda ^ + }$, then U is not $ (\lambda ,{\lambda ^ + })$-regular iff U has a least function f such that $ \{ \xi < {\lambda ^ + }\vert{\text{cf}}(f(\xi )) = \lambda \} \in U$. (b) If $ \omega \leqslant \mu < \kappa $ and U is not $ (\omega ,\mu )$-regular, then U has a least function.

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Keywords: Least functions, weakly normal filters, regularity of uniform ultrafilters
Article copyright: © Copyright 1976 American Mathematical Society

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