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Israel Journal of Mathematics

, Volume 154, Issue 1, pp 141–155 | Cite as

Chains of well-generated Boolean algebras whose union is not well-generated

  • Robert Bonnet
  • Matatyahu Rubin
Article
  • 47 Downloads

Abstract

A Boolean algebraB that has a well-founded sublattice which generatesB is called awell-generated Boolean algebra. Every well-generated Boolean algebra is superatomic. However, there are superatomic algebras which are not well-generated. We consider two types of increasing chains of Boolean algebras, canonical chains and rank preserving chains, and show that the class of well-generated Boolean algebras is not closed under union of such chains, even when these chains are taken to be countable. A Boolean algebra issuperatomic iff its Stone space is scattered. IfB is superatomic anda∈B, then therank ofa is the Cantor Bendixon rank of the Stone space of{b‖b≤a}. A chain {B α‖α<δ} is acanonical chain if for every α<β<δ,B αis the subagebra ofB βgenerated by all members ofB βwhose rank is <α. For a superatomic algebraB, I(B) denotes the ideal consisting of all members ofB whose rank is less than the rank ofB. A chain {B α‖α<δ} is arank preserving chain if for every α<β<δ anda∈I(Bα), the rank and mutiplicity ofa inB αare equal to the rank and mutiplicity ofa inB β.

Keywords

Intersection System Boolean Algebra Maximal Ideal Filling System Stone Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

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Copyright information

© The Hebrew University 2006

Authors and Affiliations

  • Robert Bonnet
    • 1
  • Matatyahu Rubin
    • 2
  1. 1.Laboratoire de Mathématiques, UMR 5127 (CNRS)Le Chablais Université de SavoieLe Bouget du Lac CEDEXFrance
  2. 2.Department of MathematicsBen Gurion University of the NegevBeer-ShevaIsrael

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