Applications of a theorem of Lévy to Boolean terms and algebras

Stavi, Jonathan

doi:10.1090/S0002-9947-1975-0469695-0

Applications of a theorem of Lévy to Boolean terms and algebras
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Trans. Amer. Math. Soc. 205 (1975), 1-36 Request permission

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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