Hostname: page-component-54dcc4c588-smtgx Total loading time: 0 Render date: 2025-09-12T11:45:30.644Z Has data issue: false hasContentIssue false
  • English
  • Français

The Saturation of a Product of Ideals

Published online by Cambridge University Press:  20 November 2018

Stanley Wagon
Stanley Wagon*
Affiliation:
Smith College, Northampton, Massachussets
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.

In this note we discuss how the saturation I X J, where I, J are k-complete ideals on a regular uncountable cardinal K, depends on the saturation of I and J. We show that if 2k = k+ then the saturation of I X J is completely determined by the saturation of I and J . A consequence of a negative saturation result is that NSk X NSk is not k+-saturated, where NSk is the nonstationary ideal on k (even though it is still open whether NSk can be K+- saturated). We also discuss the preservation of precipitousness under certain products, obtaining a simple example of an ideal on kthat is precipitous but not k+-saturated.

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 1 , 01 February 1980 , pp. 70 - 75
Copyright
Copyright © Canadian Mathematical Society 1980

References

1. Jech, T., Magidor, M., Mitchell, W. and Prikry, K., Precipitous ideals, J. Sym. Logic (to appear).Google Scholar
2. Jech, T. and Prikry, K., Ideals over uncountable sets: application of almost disjoint functions and generic ultrapowers, Memoirs A.M.S. 214 (1979).Google Scholar
3. Kakuda, Y., Saturated ideals in Boolean extensions, Nagoya Math. J. 48 (1972), 159–168.Google Scholar
4. Prikry, K. L., Changing measurable into accessible cardinals, Dissertationes Math. 68 (1970).Google Scholar
5. Solovay, R. M., Real-valued measurable cardinals, in Axiomatic Set Theory, Proc. Symp. Pure Math. 13 (1) (1971), 397–428.Google Scholar
6. Tarski, A., Idéale in vollstdndigen Mengenkorpern II, Fund. Math. 13 (1945), 51–65.Google Scholar
7. Taylor, A., Regularity properties of ideals and ultrafilters, Ann. Math. Logi. 16 (1979), 33–55.Google Scholar
8. Ulam, S., Zur Masstheorie in der Allgemeinen Mengenlehre, Fund. Math. 16 (1930), 140–150.Google Scholar
9. Wagon, S., Decompositions of saturated ideals, Ph.D. Thesis, Dartmouth College (1975).Google Scholar