Adequate ultrafilters of special Boolean algebras

Negrepontis, S.

doi:10.1090/S0002-9947-1972-0313052-1

Adequate ultrafilters of special Boolean algebras
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Trans. Amer. Math. Soc. 174 (1972), 345-367 Request permission

Abstract:

In his paper Good ideals in fields of sets Keisler proved, with the aid of the generalized continuum hypothesis, the existence of countably incomplete, ${\beta ^ + }$-good ultrafilters on the field of all subsets of a set of (infinite) cardinality $\beta$. Subsequently, Kunen has proved the existence of such ultrafilters, without any special set theoretic assumptions, by making use of the existence of certain families of large oscillation. In the present paper we succeed in carrying over the original arguments of Keisler to certain fields of sets associated with the homogeneous-universal (and more generally with the special) Boolean algebras. More specifically, we prove the existence of countably incomplete, a-good ultrafilters on certain powers of the a-homogeneous-universal Boolean algebras of cardinality a and on the a-completions of the a-homogeneous-universal Boolean algebras of cardinality a, where $a = a^{[unk]} > \omega$. We then develop a method that allows us to deal with the special Boolean algebras of cardinality $a = 2^{[unk]}$. Thus, we prove the existence of an ultrafilter p (which will be called adequate) on certain powers $\mathcal {S}_\alpha ^\delta$ of the special Boolean algebra ${\mathcal {S}_\alpha }$ of cardinality a, and the existence of a specializing chain $\{ {\mathcal {C}_\beta }:\beta < \alpha \}$ for ${\mathcal {S}_\alpha }$, such that $\mathcal {C}_\beta ^\delta \cap p$ is ${\beta ^ + }$-good and countably incomplete for $\beta < \alpha$. The corresponding result on the existence of adequate ultrafilters on certain completions of the special Boolean algebras is more technical. These results do not use any part of the generalized continuum hypothesis.

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