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How many Boolean algebras $${\mathcal P}(\mathbb{N})/{\mathcal I}$$ are there? (English) Zbl 1025.03050
The author considers the question in the title for ideals restricted to lie in one of several classes of ideals. Some main results are: (1) under CH, there are just two quotients using dense density ideals; (2) under CH, there are exactly six quotients using ideals associated with certain measures; (3) under CH, there is just one quotient using Louveau-Velickovic ideals; (4) in ZFC, there are at least 21 quotients associated with analytic $$P$$-ideals. In a note added in proof, the author states that M. R. Oliver in his UCLA PhD thesis completely took care of (4) by showing that there are continuum many such quotients. Some of the above results are proved by extending arguments of Just and Krawczyk. There are several other related results, in particular concerning automorphisms of quotients.

##### MSC:
 3e+50 Continuum hypothesis and Martin’s axiom 600000 Structure theory of Boolean algebras 300000 Other combinatorial set theory
##### Keywords:
ideals; quotients; power set algebras
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