Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-27T12:47:20.844Z Has data issue: false hasContentIssue false
  • English
  • Français

Cellularity and the structure of pseudo-trees

Published online by Cambridge University Press:  12 March 2014

Jennifer Brown
Jennifer Brown*
Affiliation:
Department of Mathematics, Kenyon College, Ohio, USA. E-mail:[email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Let T be an infinite pseudo-tree. In [2], we showed that the cellularity of the pseudo-tree algebra Treealg(T) was the maximum of four cardinals cT, lT, ϕT, and μT: roughly, cT is the “tallness” of T: lT is the “width” of T, ϕ is the number of “points of finite branching” in T: and μ is the number of “sections of no branching” in T. Here we ask: which inequalities among these four cardinals may be satisfied, in some sense, by a pseudo-tree? We show that the possible inequalities among cT, lT, ϕT, and μT attained by pseudo-trees T are closely related to the existence of generalized Suslin trees.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 72 , Issue 4 , December 2007 , pp. 1093 - 1107
Copyright
Copyright © Association for Symbolic Logic 2007

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., Almost-disjoint sets, the dense set problem and the partition calculus, Annals of Mathematical Logic, vol. 10 (1976), pp. 401–439.CrossRefGoogle Scholar
[2]Brown, J., Cellularity of pseudo-tree algebras, Notre Dame Journal of Formal Logic, vol. 47 (2006), no. 3, pp. 353–359.CrossRefGoogle Scholar
[3]Cummings, J. and Foreman, M., The tree property, Advances in Mathematics, vol. 133 (1998), no. 1, pp. 1–32.CrossRefGoogle Scholar
[4]Devlin, K., Constructibility, Springer-Verlag, 1984.CrossRefGoogle Scholar
[5]Horne, J., Cardinal functions on pseudo-tree algebras and a generalization of homogeneous weak density, Ph.D. dissertation, University of Colorado, 2005.Google Scholar
[6]Jech, T., Set theory, Academic Press, 1978.Google Scholar
[7]Koppelberg, S., General theory of Boolean algebras, Handbook of Boolean algebras (Monk, J. D. and Bonnet, R., editors), vol. 1, North-Holland, 1989.Google Scholar
[8]Koppelberg, S. and Monk, J. D., Pseudo-trees and Boolean algebras, Order, vol. 8 (1992), pp. 359–374.CrossRefGoogle Scholar
[9]Kunen, K., Set theory, Elsevier, 1980.Google Scholar
[10]Laver, R. and Shelah, S., The ℵ2-Suslin hypothesis, Transactions of the American Mathematical Society, vol. 264 (1981), no. 2, pp. 411–417.Google Scholar
[11]Magidor, M. and Shelah, S., The tree property at successors of singular cardinals, Archive for Mathematical Logic, vol. 35 (1996), pp. 385–404.CrossRefGoogle Scholar
[12]Mitchell, W., Aronszajn trees and the independence of the transfer property, Annals of Mathematical Logic, vol. 5 (1973), pp. 21–46.CrossRefGoogle Scholar