Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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
DOI: https://doi.org/10.1090/S0002-9939-2014-12220-9
Published electronically: October 27, 2014
MathSciNet review: 3283672
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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)$.


References [Enhancements On Off] (What's this?)

  • [1] P. Erdös, Some remarks on set theory, Proc. Amer. Math. Soc. 1 (1950), 127-141. MR 0035809 (12,14c)
  • [2] Moti Gitik and Saharon Shelah, Forcings with ideals and simple forcing notions, Israel J. Math. 68 (1989), no. 2, 129-160. MR 1035887 (91g:03104), https://doi.org/10.1007/BF02772658
  • [3] Andrew M. Gleason, Projective topological spaces, Illinois J. Math. 2 (1958), 482-489. MR 0121775 (22 #12509)
  • [4] L. Š. Grinblat, On sets not belonging to algebras of subsets, Mem. Amer. Math. Soc. 100 (1992), no. 480, vi+111. MR 1124108 (93d:04001)
  • [5] L. Š. Grinblat, Algebras of sets and combinatorics, Translations of Mathematical Monographs, vol. 214, American Mathematical Society, Providence, RI, 2002. Translated from the Russian manuscript by A. Stoyanovskiĭ; with an appendix by Saharon Shelah. MR 1923171 (2003m:03073)
  • [6] L. Š. Grinblat, Theorems on sets not belonging to algebras, Electron. Res. Announc. Amer. Math. Soc. 10 (2004), 51-57 (electronic). MR 2075896 (2005c:03086), https://doi.org/10.1090/S1079-6762-04-00129-5
  • [7] L. Š. Grinblat, On sets not belonging to algebras, J. Symbolic Logic 72 (2007), no. 2, 483-500. MR 2320287 (2008a:03078), https://doi.org/10.2178/jsl/1185803620
  • [8] L. Š. Grinblat, Theorems with uniform conditions on sets not belonging to algebras, Topology Proc. 33 (2009), 361-380. MR 2471582 (2011a:03050)
  • [9] L. Š. Grinblat, Finite and countable families of algebras of sets, Math. Res. Lett. 17 (2010), no. 4, 613-624. MR 2661167 (2012a:03118)
  • [10] E. Grzegorek, On saturated sets of Boolean rings and Ulam's problem on sets of measures, Fund. Math. 110 (1980), no. 3, 153-161. MR 602883 (82k:04006)
  • [11] Ph. Hall, On representatives of subsets, Journal of London Mathematical Society, vol. 10 (1935), 26-30.
  • [12] Saharon Shelah, Iterated forcing and normal ideals on $ \omega _1$, Israel J. Math. 60 (1987), no. 3, 345-380. MR 937796 (90g:03050), https://doi.org/10.1007/BF02780398
  • [13] S. Ulam, Zur Masstheorie in der allgemeinen Mengenlehre, Fundamenta Mathematicae, vol. 16 (1930), 140-150.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 03E05, 54D35

Retrieve articles in all journals with MSC (2010): 03E05, 54D35


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.

American Mathematical Society