Hostname: page-component-8448b6f56d-wq2xx Total loading time: 0 Render date: 2024-04-25T03:35:27.717Z Has data issue: false hasContentIssue false
  • English
  • Français

The number of openly generated Boolean algebras

Published online by Cambridge University Press:  12 March 2014

Stefan Geschke  and
Saharon Shelah
Stefan Geschke
Affiliation:
Department of Mathematics, Boise State University, 1910 University Drive, Boise, Id 83725-1555, USA Mathematisches Institut, Freie Universität Berlin, Arnimallee 3, 14195 Berlin, Germany, E-mail: geschke@math.fu-berlin.de
Saharon Shelah
Affiliation:
Institute of Mathematics, The Hebrew University of Jerusalem, 91904 Jerusalem, Israel Department of Mathematics, Rutgers University, New Brunswick. NJ 08854., USA, E-mail: shelah@math.huji.ac.il
Get access Rights & Permissions [Opens in a new window]

Abstract

This article is devoted to two different generalizations of projective Boolean algebras: openly generated Boolean algebras and tightly σ-filtered Boolean algebras.

We show that for every uncountable regular cardinal κ there are 2κ pairwise non-isomorphic openly generated Boolean algebras of size κ > ℵ1 provided there is an almost free non-free abelian group of size κ. The openly generated Boolean algebras constructed here are almost free.

Moreover, for every infinite regular cardinal κ we construct 2κ pairwise non-isomorphic Boolean algebras of size κ that are tightly σ-filtered and c.c.c.

These two results contrast nicely with Koppelberg's theorem in [12] that for every uncountable regular cardinal κ there are only 2κ isomorphism types of projective Boolean algebras of size κ.

Keywords

Projective Boolean algebraopenly generatedalmost freetightly σ-filtered
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 73 , Issue 1 , March 2008 , pp. 151 - 164
Copyright
Copyright © Association for Symbolic Logic 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Baumgartner, J. and Hajnal, A., A proof (involving Martin's axiom) of a partition relation, Fundamenta Mathematical, vol. 78 (1973), pp. 193–203.CrossRefGoogle Scholar
[2]Cumminos, J., Foreman, M., and Magidor, M., Squares, scales and stationary reflection, Journal of Mathematical Logic, vol. 1 (2001), no. 1, pp. 35–98.CrossRefGoogle Scholar
[3]Eklof, P. and Mekler, A., Almost free modules: Set-theoretic methods, North Holland Mathematical Library, North Holland, 1990.Google Scholar
[4]Eklof, P. and Mekler, A., Almost free modules: Set-theoretic methods, revised ed., North-Holland Mathematical Library, vol. 65, North Holland, 2002.Google Scholar
[5]Fuchino, S., Set-theoretic aspects of almost free Boolean algebras, Habilitationsschrift, Freie Universität Berlin, 1995.Google Scholar
[6]Fuchino, S., Geschke, S., Shelah, S., and Soukup, L., On the weak Free.se-Nation property of complete Boolean algebras, Annals of Pure and Applied Logic, vol. 110 (2001), no. 1-3, pp. 80–105.CrossRefGoogle Scholar
[7]Fuchino, S. and Shelah, S., More on lκ∞-free Boolean algebras, in preparation.Google Scholar
[8]Fuchs, L., Infinite abelian groups, vol. 1,2, Academic Press, 1970.Google Scholar
[9]Geschke, S., On σ-filtered Boolean algebras, Ph.D. thesis, Freie Universtät Berlin, 2000.Google Scholar
[10]Geschke, S., On tightly kappa-filtered Boolean algebras, Algebra Universalis, vol. 47 (2002), no. 6993.CrossRefGoogle Scholar
[11]Heindorf, L. and Shapiro, L., Nearly projective Boolean algebras. Lecture Notes in Mathematics, vol. 1596, Springer, 1994.CrossRefGoogle Scholar
[12]Koppelberg, S., Projective Boolean algebras, Handbook of boolean algebras (Monk, J.D. and Bonnet, R., editors), vol. 3, North Holland, Amsterdam-New York-Oxford-Tokyo, 1989, pp. 741–773.Google Scholar
[13]Koppelberg, S., Applications of σ-filtered Boolean algebras, Advances in algebra and model theory (Droste, M. and Goebel, R., editors), Gordon and Breach, Science Publishers, 1997, pp. 119–213.Google Scholar
[14]Magidor, M. and Shelah, S., When does almost free imply free? (For groups, transversals, etc.). Journal of the American Mathematical Society, vol. 7 (1994), no. 4, pp. 769–830.CrossRefGoogle Scholar
[15]Shelah, S. and Väisänen, P., Almost free groups and Ehrenfeucht-Fraïssé games for successors of singular cardinals, Annals of Pure and Applied Logic, vol. 118 (2002), no. 1-2, pp. 147–173.CrossRefGoogle Scholar