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Transactions of the American Mathematical Society

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

 

 

Partition algebras for almost-disjoint families


Authors: James E. Baumgartner and Martin Weese
Journal: Trans. Amer. Math. Soc. 274 (1982), 619-630
MSC: Primary 03E05; Secondary 03E50, 06E05
MathSciNet review: 675070
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Abstract: A set $ a \subseteq \omega $ is a partitioner of a maximal almost-disjoint faculty $ F$ of subsets of $ \omega$ if every element of $ F$ is almost contained in or almost-disjoint from $ a$. The partition algebra of $ F$ is the quotient of the Boolean algebra of partitioners modulo the ideal generated by $ F$ and the finite sets. We show that every countable algebra is a partition algebra, and that CH implies every algebra of cardinality $ \leq {2^{{\aleph _0}}}$ is a partition algebra. We also obtain consistency and independence results about the representability of Boolean algebras as partition algebras.


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DOI: https://doi.org/10.1090/S0002-9947-1982-0675070-X
Keywords: Boolean algebra, almost-disjoint, Martin's Axiom, Rothberger property, continuum hypothesis, forcing and generic sets
Article copyright: © Copyright 1982 American Mathematical Society