## Notre Dame Journal of Formal Logic

### Decomposable Ultrafilters and Possible Cofinalities

Paolo Lipparini

#### Abstract

We use Shelah's theory of possible cofinalities in order to solve some problems about ultrafilters. Theorem: Suppose that $\lambda$ is a singular cardinal, $\lambda ' \lessthan \lambda$, and the ultrafilter $D$ is $\kappa$ -decomposable for all regular cardinals $\kappa$ with $\lambda '\lessthan \kappa \lessthan \lambda$. Then $D$ is either $\lambda$-decomposable or $\lambda ^+$-decomposable. Corollary: If $\lambda$ is a singular cardinal, then an ultrafilter is ($\lambda$,$\lambda$)-regular if and only if it is either $\operator{cf} \lambda$-decomposable or $\lambda^+$-decomposable. We also give applications to topological spaces and to abstract logics.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 49, Number 3 (2008), 307-312.

Dates
First available in Project Euclid: 15 July 2008

https://projecteuclid.org/euclid.ndjfl/1216152553

Digital Object Identifier
doi:10.1215/00294527-2008-014

Mathematical Reviews number (MathSciNet)
MR2428557

Zentralblatt MATH identifier
1152.03041

#### Citation

Lipparini, Paolo. Decomposable Ultrafilters and Possible Cofinalities. Notre Dame J. Formal Logic 49 (2008), no. 3, 307--312. doi:10.1215/00294527-2008-014. https://projecteuclid.org/euclid.ndjfl/1216152553

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