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Finiteness conditions and distributive laws for Boolean algebras

✍ Scribed by Marcel Erné

Publisher
John Wiley and Sons
Year
2009
Tongue
English
Weight
141 KB
Volume
55
Category
Article
ISSN
0044-3050

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✦ Synopsis

Abstract

We compare diverse degrees of compactness and finiteness in Boolean algebras with each other and investigate the influence of weak choice principles. Our arguments rely on a discussion of infinitary distributive laws and generalized prime elements in Boolean algebras. In ZF set theory without choice, a Boolean algebra is Dedekind finite if and only if it satisfies the ascending chain condition. The Denumerable Subset Axiom (DS) implies finiteness of Boolean algebras with compact top, whereas the converse fails in ZF. Moreover, we derive from DS the atomicity of continuous Boolean algebras. Some of the results extend to more general structures like pseudocomplemented semilattices (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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