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Weakly Normal Filters and Irregular Ultrafilters

A. Kanamori
Transactions of the American Mathematical Society
Vol. 220 (Jun., 1976), pp. 393-399
DOI: 10.2307/1997652
Stable URL: http://www.jstor.org/stable/1997652
Page Count: 7
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Weakly Normal Filters and Irregular Ultrafilters
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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 κ. (a) If κ = λ+, then U is not (λ, λ+)-regular $\operatorname{iff} U$ has a least function f such that $\{\xi < \lambda^+\mid \operatorname{cf}(f(\xi)) = \lambda\} \in U$. (b) If $\omega \leqslant \mu < \kappa$ and U is not (ω, μ)-regular, then U has a least function.

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