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Transactions of the American Mathematical Society

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

 
 

 

Applications of a theorem of Lévy to Boolean terms and algebras


Author: Jonathan Stavi
Journal: Trans. Amer. Math. Soc. 205 (1975), 1-36
MSC: Primary 02B25; Secondary 02J05, 02K30
DOI: https://doi.org/10.1090/S0002-9947-1975-0469695-0
MathSciNet review: 0469695
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Abstract: The paper begins with a short proof of the Gaifman-Hales theorem and the solution of a problem of Gaifman about the depth and length of Boolean terms. The main results are refinements of the following theorem: Let $ \kappa $ be regular, $ {\aleph _1} \leq \kappa \leq \infty $. A $ < \kappa $-complete Boolean algebra on $ {\aleph _0}$ generators, which are restricted by just one countably long equation, is either atomic with $ \leq {\aleph _0}$ atoms or isomorphic to the free $ < \kappa $-complete Boolean algebra on $ {\aleph _0}$ generators. The main tools are a Skolem-Löwenheim type theorem of Azriel Lévy and a coding of Borel sets and Borel-measurable functions by Boolean terms.


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DOI: https://doi.org/10.1090/S0002-9947-1975-0469695-0
Article copyright: © Copyright 1975 American Mathematical Society

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