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On the compactness of some Boolean algebras

Published online by Cambridge University Press:  12 March 2014

Jacek Cichoń*
Affiliation:
Institute of Mathematics, University of Wrocław, 50-384 Wrocłlaw, Poland

Extract

We say that the Boolean algebra B is λ-compact, where λ is a cardinal number, if for every family ZB∖{0} of power at most λ, if inf Z = 0 then for some finite subfamily Z0Z we have inf Z0 = 0.

On the set of all finite subsets of a cardinal number κ, which is denoted [κ]<ω, the sets of the form for any p Є [κ]<ω generate the filter Tκ.

This filter is a standard example of a κ-regular filter (see [2]). Because of the importance of κ-regular filters in studying the saturatedness of ultraproducts and reduced products by model-theoretic methods, the question of compactness of the algebra Bκ = P([κ]<ω/Tκ was natural. This question in the most optimistical way was formulated by M. Benda [1, Problem 5c]: is the algebra Bκω-compact for every uncountable κ?

In this paper we show that for most of the cardinal numbers which are greater or equal to 2ω the algebra Bκ is not ω-compact. Hence, in view of obtained results, the following question appears: does there exist an uncountable κ such that the algebra Bκ is κ-compact?

We use standard set-theoretical notations. CH denotes the Continuum Hypothesis, GCH denotes the General Continuum Hypothesis and MA denotes Martin's Axiom.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1984

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References

REFERENCES

[1]Benda, M., On saturated reduced products, Pacific Journal of Mathematics, vol. 39 (1971), pp. 557571.CrossRefGoogle Scholar
[2]Chang, C. C. and Keisler, H. J., Model theory, North-Holland, Amsterdam, 1973.Google Scholar