Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Refinement properties and extensions of filters in Boolean algebras

Authors: Bohuslav Balcar, Petr Simon and Peter Vojtáš
Journal: Trans. Amer. Math. Soc. 267 (1981), 265-283
MSC: Primary 06E05; Secondary 03E05, 03E35
MathSciNet review: 621987
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Abstract: We consider the question, under what conditions a given family $ A$ in a Boolean algebra $ \mathcal{B}$ has a disjoint refinement. Of course, $ A$ cannot have a disjoint refinement if $ A$ is a dense subset of an atomless $ \mathcal{B}$, or if $ \mathcal{B}$ is complete and $ A$ generates an ultrafilter on $ \mathcal{B}$. We show in the first two sections that these two counterexamples can be the only possible ones. The third section is concerned with the question, how many sets must necessarily be added to a given filter in order to obtain an ultrafilter base.

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Keywords: Boolean algebra, disjoint refinement, ultrafilter, distributivity, Stone space, Gleason space
Article copyright: © Copyright 1981 American Mathematical Society