Hostname: page-component-848d4c4894-m9kch Total loading time: 0 Render date: 2024-05-18T14:23:54.734Z Has data issue: false hasContentIssue false
  • English
  • Français

Distributive and related ideals in generic extensions

Published online by Cambridge University Press:  22 January 2016

C. A. Johnson
C. A. Johnson*
Affiliation:
Mathematics Department, Keele University, Keele, Staffs. ST5 5BG, England
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let κ: be a regular uncountable cardinal and I a κ-complete ideal on te. In [11] Kanai proved that the μ-distributivity of the quotient algebra P(κ)I is preserved under κ-C.C. μ-closed forcing. In this paper we extend Kanai’s result and also prove similar preservation results for other naturally occurring forms of distributivity. We also consider the preservation of two game theoretic properties of I and in particular, using a game theoretic equivalent of precipitousness we give a new proof of Kakuda’s theorem ([10]) that the precipitousness of I is preserved under κ-C.C. forcing.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 106 , June 1987 , pp. 91 - 100
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1987

References

[ 1 ] Baumgartner, J. and Taylor, A., Saturation properties of ideals in generic extensions II, Trans. Amer. Math. Soc, 271 (1982), 587–609.Google Scholar
[ 2 ] Baumgartner, J., Taylor, A. and Wagon, S., Structural properties of ideals, Dissertations Mathematicae, vol. 197 (1982).Google Scholar
[ 3 ] Foreman, M., Games played on Boolean algebras, J. Symbolic Logic, 48 (1983), 714–723.CrossRefGoogle Scholar
[ 4 ] Galvin, F., Jech, T. and Magidor, M., An ideal game, J. Symbolic Logic, 43 (1978), 284–291.CrossRefGoogle Scholar
[ 5 ] Jech, T., Some properties of scomplete ideals defined in terms of infinite games, Ann. Pure Appl. Logic, 26 (1984), 31–46.CrossRefGoogle Scholar
[ 6 ] Jech, T. and Prikry, K., Ideals over uncountable sets, Mem. Amer. Math. Soc, 18 (2) (1979).Google Scholar
[ 7 ] Johnson, C. A., Distributive ideals and partition relations, J. Symbolic Logic, to appear.Google Scholar
[ 8 ] Johnson, C. A., Some properties of precipitous and related ideals, Ph.D. Thesis, Leeds University (1984).Google Scholar
[ 9 ] Kakuda, Y., Saturated ideals in Boolean extensions, Nagoya Math. J., 48 (1972), 159–168.CrossRefGoogle Scholar
[10] Kakuda, Y., On a condition for Cohen extensions which preserve precipitous ideals, J. Symbolic Logic, 46 (1981), 296–300.CrossRefGoogle Scholar
[11] Kanai, Y., On quotient algebras in generic extensions, Comm. Math. Univ. Sancti Pauli, 33 (1984), 71–77.Google Scholar
[12] Magidor, M., Precipitous ideals and sets, Israel J. Math., 35 (1980), 109–134.CrossRefGoogle Scholar