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Limitations on representing $ \mathcal{P}(X)$ as a union of proper subalgebras


Author: L. Š. Grinblat
Journal: Proc. Amer. Math. Soc. 143 (2015), 859-868
MSC (2010): Primary 03E05; Secondary 54D35
Published electronically: October 27, 2014
MathSciNet review: 3283672
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Abstract: For every integer $ \mu \geqslant 3$, there exists a function $ f_\mu :\mathbb{N}^+\rightarrow \mathbb{N}^+$ such that the following holds: (1) $ f_\mu (k)=2k-\mu $ for $ k$ large enough; (2) if $ \mathfrak{A}$ is a finite nonempty collection of subalgebras of $ \mathcal P(X)$ such that $ \bigcap \mathfrak{B}$ is not $ f_\mu \left (\char93 (\mathfrak{B})\right )$-saturated, for all nonempty $ \mathfrak{B}\subseteq \mathfrak{A}$, then $ \bigcup \mathfrak{A}\neq \mathcal P(X)$.


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Additional Information

L. Š. Grinblat
Affiliation: Department of Mathematics, Ariel University of Samaria, P.O. Box 3, Ariel 40700, Israel
Email: grinblat@ariel.ac.il

DOI: https://doi.org/10.1090/S0002-9939-2014-12220-9
Keywords: Algebras of sets, $\sigma$-algebra, ultrafilter
Received by editor(s): July 17, 2011
Received by editor(s) in revised form: July 16, 2012, and April 4, 2013
Published electronically: October 27, 2014
Communicated by: Julia Knight
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.